Abstract
This paper describes the research performed within the scope of H2020 project FIBRESHIP in the development and validation of a thermo-mechanical model to assess the fire performance of composite structures. A one-dimensional thermal model with pyrolysis is used to obtain the temperature profile across the thickness, and later, introduced in the thermo-mechanical model with a quadrilateral shell element approach. The composite constitutive model employed is the so-called Serial/Parallel Rule of Mixtures (SPROM) which has been modified to introduce the effect of the thermal deformation. A set of experimental tests are then used to validate the correctness of the numerical method proposed. The experimental data used to validate the thermal model is the classic Henderson experimental test. The thermo-mechanical coupling is validated against an original vertical furnace test of an FRP ship’s bulkhead, following on the 2010 FTP Code standards. These validations demonstrate the correctness and accuracy of the proposed decoupled thermo-mechanical formulation.
Thermo-mechanical analysis of laminated composites shells exposed to fire Pacheco et al
Fire Safety, Fire Collapse, Thermal, Thermomechanical, Damage Constitutive Model, FIBRESHIP project, Composite Materials, SPROM, Rule of Mixtures, Marine Structures
1 Introduction
Fibre Reinforced Polymers are extensively used today for building the hull structure of crafts with lengths up to about 50 m. In fact, today most of the pleasure crafts and sailing yachts, passenger and car ferries, patrol and rescue crafts, and naval ships below Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 50}
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length are built in composites. These materials are also used in large secondary structures, but only a few complete units above Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 50}
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length – naval vessels – have been built in composite materials.
The main reason for this limitation in the use of composites is the obligation to use steel equivalent structural materials to fulfil the fire-safety requirements of the Convention for the Safety of Life at Sea (SOLAS). However, today there is no question that alternative designs with suitable risk control and the use of fire retardant resins, intumescent coatings, fire insulation and active fire-fighting systems can allow FRP structures to fulfil the strictest fire safety regulations. In addition, despite the fact that many polymer composites are flammable, these materials have other properties that may be beneficial in the event of a fire scenario and that are not present in metals. Composites have a heat conduction rate much slower than metals.
This fact is translated into a slowdown in the speed of fire spread between rooms, and therefore, the composite laminate constitutes a very effective barrier against the spread of fire.
This barrier effect is specially true when considering that composites are well insulated, to avoid auto-ignition, so the fire-spreading cannot be considered as a pure "radiant" heat source, but more like a "convective" heat source, where no flames are produced. Provided that composites present a much lower thermal inertia, the rise of temperature to adjacent rooms is very limited compared to steel.
These advantages were demonstrated in the large fires on two all-composite minehunter ships operated by the Royal Navy — HMS Ledbury and HMS Cattistock. In both ships, the fire started in the engine/machinery room and in the case of HMS Cattistock the fire burned for over four hours before being extinguished. The fires extensively damaged the compartment of both ships, with the composite hull and bulkheads being heavily charred. However, the low thermal conductivity of the composite bulkheads and decks stopped the fire from spreading by heat conduction to surrounding compartments, which is more difficult to stop in steel ships
. Above reasons justify why composite materials are commonly used as thermal insulator of spacecrafts for the re-entry process.
This work relates to the H2020 project FIBRESHIP, which main objective has been to develop the knowledge and technology required to enable the building of the complete hull and superstructure of large-length ships in composites.
This paper describes the research performed within the scope of H2020 project FIBRESHIP in the development and validation of a thermo-mechanical model to assess the fire performance of composite structures.
A thermo-mechanical model to assess the fire performance of composite structures is presented.
Fire in composite materials is a complex phenomenon that involves thermal, chemical, and physical processes that can generate the failure of the laminate structure [3]. The heat flux provided by the fire generates a rise of temperature due to heat conduction. When the temperature of the material reaches a high enough value, chemical reactions (pyrolysis or carbon-silica reactions) begin and the matrix of the composite is decomposed to form gaseous products [4]. These gas products are propagated through the porous structure of the composite by diffusion. This effect generates a reduction of the heat due to conduction. The matrix and fibre components suffer thermal expansion and degradation of its mechanical properties. This process generates a drastic reduction of the stiffness and the strength of the laminate structure, which may provoke its failure.
Several models for the analysis of the distribution of the temperature in laminate composites can be found in the literature. One of the most used models in this field is the one presented by Henderson et al. in [4]. In that paper a 1D transient heat transfer model, which takes into account the pyrolysis and carbon-silica reactions phenomena, is proposed. In [5], in order to take into account the thermochemical expansion and the storage of decomposition gases from the solid material, an extension of the previous model is presented. Other similar 1D models are presented in [6,7,8,9]. A more complex approach is presented in [10] where heat diffusion, polymer pyrolysis with associated gas production and convection through the matrix are coupled together.
In the last twenty years, several thermo-mechanical models have been also presented for laminate composites under fire conditions. Gibson et al. analysed in [1] the post-fire mechanical properties of woven glass laminates with several resins using a two-layer mechanical model proposed by Mouritz and Mathys in [11,12,13]. Feih et al. presented models to calculate the tensile and compressive strength of laminate composites exposed to fire in [14] and [14] respectively. Two models, a thermo-chemical and a thermo-mechanical, were developed by Keller et al. in [15] in order to predict the temperature evolution and strength of slabs subjected to fire conditions. Bai and Keller proposed in [16] a model based on beam theory to analyse time-dependent deflections of cellular FRP slab elements exposed to fire from one side. Zhang et al. presented in [17] a 2D thermo-mechanical model to predict the mechanical response of rectangular GFRP tubes subjected to one-side ISO-834 fire condition. Shi et al. developed in [18] a 3D thermo-mechanical model which took into account the thermo-chemical decomposition and gas diffusion.
One of the most used techniques to evaluate the constitutive behaviour of unidirectional long fibre-reinforced laminates is the classical mixture theory (CMT). This theory was developed in 1960 by Green and Naghdi [19] in a simpler version known as rule of mixtures (ROM). The ROM theory assumes that all component materials of the composite have the same strain in all directions (parallel behaviour). This theory considers the volume fraction of each component material, but not its morphological distribution. To overcome this limitation, some improvements, have been proposed [20,21]. A very interesting solution is the so-called serial/parallel mixing theory (SPROM) presented by Rastellini et al. in [21]. The SPROM theory assumes components behave as a parallel material in the fibre alignment direction and as a serial material in the rest of directions.
This paper presents a thermo-mechanical model for the analysis of laminate composite structures exposed to fire. The model combines a 1D trough-thickness thermal analysis based on the model proposed by Henderson together with a mechanical model based in 3D shells. The constitutive model used for the mechanical behaviour is based in a modification of the SPROM theory to take into account the dilation of thermal strains, i.e., the thermal SPROM (TSPROM). The model is implemented numerically with in-house software. Experimental tests have been also carried out to validate the proposed model. A numerical example for the analysis of time-to-failure of a composite laminated structure exposed to fire is also presented.
2 Thermal model
The thermal model proposed to solve the non-linear one-dimensional heat problem with pyrolysis constitutive model is based on the thermal model proposed by [22] in conjunction to the boundary conditions in [23].
The composite subjected to the action of fire is considered as a saturated porous media [24]. This media is composed by two phases: a solid matrix phase and an interconnected porous space phase. The solid matrix phase is a mixture of resin matrix, reinforcement fibres and residues from the decomposition reaction.
Each phase has a fraction associated to its volume such that
|
(1) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{\phi }}_{i}
is the phase volume fraction , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{\Omega }}
is the total domain , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{\Omega }}_{i}
is the phase domain , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{\Omega }}_{s}
is the solid domain and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{\Omega }}_{g}
is the gas domain .
The homogenised density, obtained by the rule of mixtures, is
|
(2) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{\rho }}
is the homogenised density, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{\rho }}_{s}
is the solid density and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{\rho }}_{g}
is the gas density.
2.1 Governing equation
The governing equation for the thermal model proposed in [4] can be rewritten in the following manner,
|
(3) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{C_p}}
is the specific heat capacity, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{T}}
is the temperature, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{k}}
is the through-thickness thermal conductivity, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{h}}_{g}
is the gas specific enthalpy, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol{\mathrm{{{w}}_{g}}}}
is the gas mass flux, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{C_p}}_{g}
is the gas specific heat capacity, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{Q}}_{p}
is the polymer degradation energy source and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{h}}_{s}
is the solid specific enthalpy.
The mass balance in 3 for each one of the phases is
|
(4) |
|
(5) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{\dot{m}_{s\to g}}}
is the mass flux rate 4 implies no mass flux (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol{\mathrm{{w}_s}}} =0}
) of the solid phase.
2.1.1 pyrolysis model
In [4] the decomposition of the solid phase to gas phase is obtained by a linear interpolation relationship between the virgin and degraded states
|
(6) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{F}}
is the degradation fraction Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{\rho }}_{0}
is the virgin density and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{\rho }}_{f}
is the char density . The evolution law for the degradation parameter (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {F})
is defined by a nth order Arrhenius equation
|
(7) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{A_{T}}}
is the pre-exponential factor for decomposition reaction of polymer matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{n_{r}}}
is the order of the decomposition reaction of the polymer matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{E_a}}
is the activation energy for decomposition reaction of polymer matrix and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{R}}
is the universal gas constant
2.1.2 Gas transfer model
Assuming negligible accumulation of gas (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \partial ({\phi }_g{\rho }_g)/\partial t=0} ) [4], it can then be concluded that from 4 and 5
|
(8) |
The gas is assumed to not escape through the cold face of the composite (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol{\mathrm{{w}_g}}} ({l_t},t)=0} ), the mass flux field in the gas phase (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol{\mathrm{{w}_g}}} (x,t)} ) is calculated in the following manner
|
(9) |
Note that in 9, the equivalence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \nabla \cdot{ {\boldsymbol{\mathrm{{w}_g}}} } \equiv {\partial ({{w}_g})/\partial x} }
was introduced where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {w}_g}
is the first component of the vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol{\mathrm{{w}_g}}} }
. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{l_t}}
is the thickness of the composite. 9 imposes the direction on which the gas can flow, which is from the cold face to the hot face. This assumption presents instability when the temperature in the cold and hot faces are similar.
2.2 Conditions
2.2.1 Boundary conditions
The boundary conditions are defined similarly to [23], the concept of adiabatic surface temperature – assuming a perfectly insulated surface exposed to radiation and convection – is employed for the hot face
|
(10) |
and assuming opposite heat flux direction in the cold face as
|
(11) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{q}}
is the normal heat flux component, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{ {\boldsymbol{\mathrm{n}}} }}
is the normal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{{\sigma }_{\beta }}}
is the Stefan-Boltzmann constant Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{\epsilon }}
is the emissivity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{T}}_{ad,k}
is the adiabatic hot face temperature in Kelvin, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{T}}_{ad}
is the adiabatic hot face temperature in Celsius, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{h_{conv}}}
is the convection coefficient Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{T}}_{\infty ,k}
is the ambient temperature in Kelvin and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{T}}
is the ambient temperature in Celsius.
The boundary conditions expressed in 10 and 11 from [23] can be generalised in this manner
|
(12) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\bar{{T}}}
is the prescribed temperature which is either, the adiabatic temperature of the hot (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {T})
or cold (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {T})
faces. The subscript Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle k}
denotes the temperature in Kelvin units rather than Celsius.
2.2.2 Initial conditions
Applying ordinary initial conditions to 3
|
(13) |
|
(14) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{T}}_{0}
is the virgin temperature . Note that in the present research, the ambient temperature and initial/virgin temperature are assumed to be the same.
2.3 Weak form
Using the relationship from 9, the mass flux (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol{\mathrm{{w}_g}}} } ) can be found. Thus, the weak form of the problem can be obtained by applying the method of mean weighted residuals in 3.
|
(15) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{\xi }}
is the test function and (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \partial{{\Omega }})
refers to the domain boundary, which arises from applying the divergence theorem to the divergence term that appears from the substitution of the integration by parts of the Laplacian term.
2.4 Finite element discretisation
The temperature field (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol{\mathrm{{T}}}} } ) can be discretised using the interpolation shape functions (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol{\mathrm{{N}}}} } ).
By using the standard Galerkin method, the 15 can be written as
|
(16) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol{\mathrm{{N}}}}
is the nodal shape function matrix, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol{\mathrm{{T}}}}
is the nodal temperature vector and (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n_\hbox{nodes}}
) is the total number of nodes. By using the standard Galerkin method, the 15 can be written as
|
(17) |
where
|
(18) |
|
(19) |
|
(20) |
|
(21) |
|
(22) |
being (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n_\hbox{elm}} ) the total number of elements, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol{\mathrm{{C_{{T}}}{}}}}
is the specific heat matrix , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol{\mathrm{{K_{{T}}}{}}}}
is the conductivity matrix , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol{\mathrm{{B_{{T}}}{}}}}
is the gradient matrix , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol{\mathrm{{{q_{}}}{d}}}}
is the degraded heat flux vector , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol{\mathrm{{{q_{}}}{c}}}}
is the convection heat flux vector , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol{\mathrm{{{q_{}}}{r}}}}
is the radiation heat flux vector .
2.5 Time integration and non-linear solution strategy
The time is discretised in non-overlapping subintervals Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \sum _n (t_{n+1}-t_n)}
and by using a backward-Euler scheme, 17 can be written as
|
(23) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol{\mathrm{{r_{{T}}}{}}}}
is the residual heat flux vector. 23 represents a non-linear problem which depends on the state variable (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {T}}
) and the internal variable (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {F}} ). To solve this problem the Newton-Raphson method can be applied, the solution can be summarised in the 3.4.
3 Mechanical model
The mechanical behaviour of composite materials structure is analysed using the serial/parallel mixing theory [21]. In the present article, the capabilities of the so-called serial/parallel rule of mixtures – in short SPROM – have been extended for composites exposed to high temperatures. The constitutive mechanical model proposed in this work is developed based on Reissner-Mindlin flat shell theory [25,26], where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol{\mathrm{{\varepsilon }}}} := [ {{\varepsilon }}_{x}, {{\varepsilon }}_{y}, {{\gamma }}_{xy}, {{\gamma }}_{yz}, {{\gamma }}_{zx} ]^{T} }
is the mechanical strain vector and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol{\mathrm{{\sigma }}}} := [ {{\sigma }}_{x}, {{\sigma }}_{y}, {{\tau }}_{xy}, {{\tau }}_{yz}, {{\tau }}_{zx} ]^{T} }
the internal stress, both expressed in Voigt nomenclature.
3.1 Serial-parallel rules of mixing
The SPROM acts as a constitutive equation manager, and is capable of successfully predict the structural performance of the composite, taking into account the specific behaviour of the composite in its parallel and serial direction, as well as the non-linearities of the composite components [21]. To do so, the strain vector is split into its parallel and serial components (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol{\mathrm{{\varepsilon }_p}}} } ) and (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol{\mathrm{{\varepsilon }_s}}} } ), respectively, using a projector matrix, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol{\mathrm{P_p}}} }
. This matrix is obtained using the fibre orientation in the composite. The procedure to perform this decomposition is described as follows
|
(24) |
|
(25) |
where
|
(26) |
|
(27) |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol{\mathrm{{\varepsilon }}}}
is the strain , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol{\mathrm{{{\varepsilon }}_{p}}}}
is the parallel strain , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol{\mathrm{{{\varepsilon }}_{s}}}}
is the serial strain , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol{\mathrm{{{\sigma }}_{p}}}}
is the parallel stress , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol{\mathrm{{{\sigma }}_{s}}}}
is the serial stress and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol{\mathrm{P_{p,{\varepsilon }}}}} }
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol{\mathrm{P_{s,{\varepsilon }}}}} } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol{\mathrm{P_{p,{\sigma }}}}} }
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol{\mathrm{P_{s,{\sigma }}}}} }
are their respective parallel and serial projectors, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{\theta }}
is the fibre orientation in the layer with respect to the element local axis Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {{x}'} }
described in 1, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol{\mathrm{I}}} }
is the identity matrix. Note that the projectors have a different relationship between their Voigt and tensor notations, and thus each one has its set of parallel and serial projectors.
Once knowing the parallel and serial components of the composite, it is possible to apply to each one a compatibility conditions that defines how the composite constituents, fibre and matrix, interact between each other:
|
(28) |
|
(29) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol{\mathrm{{{\sigma }}_{p,m}}}}
is the parallel matrix stress , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol{\mathrm{{{\sigma }}_{s,m}}}}
is the serial matrix stress , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol{\mathrm{{{\sigma }}_{p,f}}}}
is the parallel fibre stress , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol{\mathrm{{{\sigma }}_{s,f}}}}
is the serial fibre stress , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol{\mathrm{{{\sigma }}_{s}}}}
is the serial composite stress , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol{\mathrm{{{\sigma }}_{p}}}}
is the parallel composite stress , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol{\mathrm{{{\varepsilon }}_{p,m}}}}
is the parallel matrix strain , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol{\mathrm{{{\varepsilon }}_{s,m}}}}
is the serial matrix strain , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol{\mathrm{{{\varepsilon }}_{p,f}}}}
is the parallel fibre strain , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol{\mathrm{{{\varepsilon }}_{s,f}}}}
is the serial fibre strain , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol{\mathrm{{{\phi }}_{m}}}}
is the matrix volume fraction and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol{\mathrm{{{\phi }}_{f}}}}
is the fibre volume fraction .
Combining 24, 25, 28 and 29, the composite strain and stress yield
|
(30) |
3.2 Constitutive equations of the constituent materials exposed to high temperatures
The constitutive behaviour of matrix and fibre materials are both modelled with isotropic damage models [27]
|
(31) |
|
(32) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{\sigma }}_{m}
is the matrix stress , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{\sigma }}_{f}
is the fibre stress , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{\varepsilon }}_{m}
is the matrix strain , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{\varepsilon }}_{f}
is the fibre strain , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{d}}_{m}
is the matrix isotropic damage index , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{d}}_{f}
is the fibre isotropic damage index , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\mathbb{C}}_{m}
is the matrix elastic constitutive tensor , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\mathbb{C}}_{f}
is the fibre elastic constitutive tensor , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{\varepsilon }}_{{T},m}
is the matrix thermal strain and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{\varepsilon }}_{{T},f}
is the fibre thermal strain .
The thermal strain of each of the constituent materials is anisotropic, which is achieved by
|
(33) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{\alpha }}_{i}
is the constituent material thermal expansion coefficient and is defined as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol{\mathrm{{\alpha }}}} :=[{\alpha }_{x},{\alpha }_{y},0,0,0]^T}
. Note that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i}
refers to properties associated to the constituent material.
Thus, the elastic constitutive matrix, for Reissner-Mindlin flat shell theory, becomes
|
(34) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{E}}_{i}
is the Young modulus , which is considered dependent from the temperature and the degradation fraction and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{\nu }}_{i}
is the Poisson ratio that is constant.
In order to define concepts such as loading and unloading for general 3D stress states, it is necessary to define a scalar positive quantity in terms of normalised equivalent stress. This will permit the comparison of different 3D stress states, even for different degrees of thermal degradation.
|
(35) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{\bar{{\sigma }}}}_{i}
is the effective stress , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{\delta }}_{i}
is the damage threshold , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{\varsigma }}
is the stress weight factor , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{\beta }}_{,i}
is the compress-traction coefficient , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{E}}_{,i}
is the initial Young modulus , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{\sigma }}_{y,i}
is the yield stress, which is considered to be dependent of the temperature and the degradation factor and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\mathbb{C}}_{0,i}
is the initial elastic constitutive tensor .
The effective stress for each constituent material is defined as
|
(36) |
The stress weight factor is equivalent to
|
(37) |
And the compression-traction coefficient is the ratio between the compression and traction yield stress
|
(38) |
The evolution of the damage index is controlled by the internal variable, which is defined by Faria et al. in [28]. Thus, the definition of the damage index with respect to the internal variable is
|
(39) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{r}}_{i}
is the normalised internal variable , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{A}}_{i}
is the pre-exponential factor of the isotropic damage model, which is a mesh independent parameter by means of the characteristic length as shown in [29] and depends on the size (volume, area, or length) of the discretised spatial mesh. The pre-exponential factor can be calculated using [28,29]
|
(40) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{l_c}}_{,i}
is the characteristic length and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{G_{f}}}_{i}
is the fracture energy . The fracture energy of each constituent material depends on temperature and degradation factor. Due to the lack of experimental information to characterise the evolution of the fracture energy with respect to the temperature and the degradation factor, this evolution is in terms of the Young modulus and the yield stress which depend also on both temperature and degradation factor
|
(41) |
The Young's modulus (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {E}} ) and the yield stress (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\sigma }_y} ) follow the evolution law proposed in [2]
|
(42) |
where (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P_{u}} ) is the unrelaxed and (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P_{r}} ) is the relaxed value of a generic property (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P} ), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{T}}_{g,i}
is the glass transition temperature , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{\chi }}_{1,i}
is the first Mourtiz and Gibson fitting parameter , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{\chi }}_{2,i}
is the second Mourtiz and Gibson fitting parameter .
In this case the generic property Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P}
can be substituted by either the Young's modulus (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {E}}
) or the yielding stress (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\sigma }_y} ). Each property will be defined by a pair of unrelaxed (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {E}_{u}, {{\sigma }_{y}}_{u}} ) and relaxed (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {E}_{r}, {{\sigma }_{y}}_{r}} ) values. Also, the glass temperature Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {T}_{g}}
and both calibration parameters (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\chi }_{1},{\chi }_{2}}
) are needed to calibrate the curves.
The parameters found in 42 shall be calibrated with experimental tests at high temperatures for each of the constituent materials. However, calibration of composites at high temperatures is regularly done on the composite laminated material as a whole, which poses an important problem. In the absence of experimental data for each constituent material, the following relationship can be taken into account
|
(43) |
|
(44) |
|
(45) |
which establishes a correlation between the evolution of the property (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P} ) for each constituent material and the composite as a whole.
3.3 Constitutive equations of composite materials exposed to high temperatures
Similarly to the approach in [21], 31 and 32 can be written in differential form as
|
(46) |
|
(47) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol{\mathrm{{\mathbb{C}}_{tan,m}}}}
is the matrix tangent elastic constitutive tensor and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol{\mathrm{{\mathbb{C}}_{tan,f}}}}
is the fibre tangent elastic constitutive tensor . The serial and parallel decomposition of the internal stress of each constituent material in differential form can be achieved by
|
(48) |
|
(49) |
where
|
(50) |
In the SPROM theory, the matrix serial strain is selected as an independent internal variable to satisfy 28 and 29. The fibre serial strain can then be expressed as
|
(51) |
The objective function to minimise is defined from 29 and solved by applying a Newton-Rapshon scheme,
|
(52) |
where the objective function (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol{\mathrm{\Delta {\sigma }_{s}}}} } ) is the serial internal stress residue. The Jacobian of the residue can be formulated by using 50 and 51
|
(53) |
Therefore the updated scheme of the design variable is
|
(54) |
The initial prediction (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle k=0} ) is based on the converged serial matrix strain of the previous step. Being
|
(55) |
This initial prediction can be obtained by assuming that in 49 the serial differential internal stress is the same for both matrix and fibre.
|
(56) |
Using 48 and 51 in 56, the initial prediction yields
|
(57) |
|
(58) |
|
(59) |
|
(60) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Delta }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{\sigma }}_{{T},f}
is the fibre incremental thermal stress and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Delta }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {{\sigma }}_{{T},m}
is the matrix incremental thermal stress . Both terms are the result of extending the regular SPROM in [21] in order to take into account the effect of deformation under elevated temperatures.
The definition in 50 is valid for each of the constituent materials, however the homogenised tangent elastic constitutive tensor can be obtained from the derivation of 48 and 49 ([21])
|
(61) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol{\mathrm{\mathbb{C}_{pp}}}} } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol{\mathrm{\mathbb{C}_{ps}}}} } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol{\mathrm{\mathbb{C}_{sp}}}} }
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol{\mathrm{\mathbb{C}_{ss}}}} }
are the parallel-parallel, parallel-serial, serial-parallel and serial-serial tangent elastic constitutive matrices. The expressions of these matrices are described in [21]. The tangent elastic constitutive matrix can be also expressed in in-plane and out-of-plane components using the following relationship
|
(62) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol{\mathrm{{\sigma }_\hbox{in}}}} :=[{\sigma }_{x},{\sigma }_{y},{\tau }_{xy}]^T}
is the in-plane stress vector, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol{\mathrm{{\sigma }_\hbox{out}}}} :=[{\tau }_{yz},{\tau }_{zx}]^T}
is the out-of-plane stress vector, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol{\mathrm{{\varepsilon }_\hbox{in}}}} :=[{\varepsilon }_{x},{\varepsilon }_{y},{\gamma }_{xy}]^T}
is the in-plane stress vector, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol{\mathrm{{\varepsilon }_\hbox{out}}}} :=[{\gamma }_{yz},{\gamma }_{zx}]^T}
is the out-of-plane stress vector and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Big( {\boldsymbol{\mathrm{{\varepsilon }_{{T}}}}} :=[ {\phi }_f{\alpha }_{x,f}+ {\phi }_m{\alpha }_{x,m}, {\phi }_f{\alpha }_{y,f}+ {\phi }_m{\alpha }_{y,m}, 0 ]^T \Delta {T}\Big)}
is the thermal strain.
3.4 Shell finite element
In the thermo-mechanical coupling, the mechanical model is based on the Reissner-Mindlin flat shell theory ([25], [26]). The four-noded QLLL flat shell quadrilateral element proposed in [30] has been chosen as the finite element solution to model the Reisnner-Mindlin flat shell theory. This shell element combines the classical 4-noded plane stress quadrilateral matrix ([31]) and the QLLL plate element ([32]). The constitutive model for this shell element is the one detailed in section 3.3.
The middle plane of the structure is discretised by 4-node quadrilateral shell elements. The local axis of each quadrilateral element is defined as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle ( {{x}'} , {{y}'} , {{z}'} )}
where the vertical local axis Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle ( {{z}'} )}
is taken normal to the middle plane Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle ( {{x}'} {{y}'} )}
and belong to the union of the edges Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \big(\overline{12}\cup \overline{14}\big)}
(1). The thickness of each element is divided in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n_\hbox{layers}}
layers. Each layer Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i}
lies between the plane (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {{z_l}'} }
) and (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {{z_{l+1}}'} } ). It is assumed that each layer satisfy the plane stress hypothesis (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\sigma }_{ {{z}'} }=0}
and the displacements in the directions of the axes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle ( {{x}'} , {{y}'} , {{z}'} )}
are defined as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle ( {{u}'} , {{v}'} , {{w}'} )}
respectively.
Taking into account that the strain (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\varepsilon }_{ {{z}'} }} ) does not generate work, because of the plane stress assumption, the pertinent strains of the Reissner-Mindlin shell theory are written in local axes as
| Local and global description of kinematics for the 4-node quadrilateral shell element. |
| Figure 1: Local and global description of kinematics for the 4-node quadrilateral shell element. |
|
(63) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {{ {\boldsymbol{\mathrm{\hat {\varepsilon }_{m}}}} }'} } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {{ {\boldsymbol{\mathrm{\hat {\varepsilon }_{b}}}} }'} }
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {{ {\boldsymbol{\mathrm{\hat {\varepsilon }_{s}}}} }'} }
are the generalised local strain vectors due to membrane, bending, and shear effects respectively (see [30]), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {{ {\boldsymbol{\mathrm{\hat {\varepsilon }}}} }'} }
is the generalised strain field vector and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol{\mathrm{S}}} }
is the linear transformation matrix for strain decoupling.
The discretisation of the generalised strain field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {{ {\boldsymbol{\mathrm{\hat {\varepsilon }}}} }'} }
presented in this article can be written as
|
(64) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {{ {\boldsymbol{\mathrm{{B}{i}}}} }'} := \big[ {{ {\boldsymbol{\mathrm{{B}{m,i}}}} }'} , {{ {\boldsymbol{\mathrm{{B}{b,i}}}} }'} , {{ {\boldsymbol{\mathrm{{B}{s,i}}}} }'} \big]^T}
is the displacement-strain matrix for the node Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {{ {\boldsymbol{\mathrm{{B}{m,i}}}} }'} } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {{ {\boldsymbol{\mathrm{{B}{b,i}}}} }'} }
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {{ {\boldsymbol{\mathrm{{B}{s,i}}}} }'} }
are the membrane, bending and shear displacement-strain matrices respectively for the node Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i}
. And the local displacements vector for node Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i}
is defined as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {{ {\boldsymbol{\mathrm{{a}_i}}} }'} := \big[ {{u_{0,i}}'} , {{v_{0,i}}'} , {{w_{0,i}}'} ,\theta _{ {{x}'} ,i},\theta _{ {{y}'} ,i}\big]^{T}}
. See [30] for further detail on the procedure to obtain the expressions for the membrane, bending, and shear nodal displacement-strain matrices.
The discretisation of the local generalised strain field (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {{ {\boldsymbol{\mathrm{\hat {\varepsilon }}}} }'} } ) is defined by the local displacement-strain matrix (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {{ {\boldsymbol{\mathrm{{B}}}} }'} } ) and the local displacement vector (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {{ {\boldsymbol{\mathrm{{a}_i}}} }'} := \big[ {{u_{0,i}}'} , {{v_{0,i}}'} , {{w_{0,i}}'} ,\theta _{ {{x}'} ,i},\theta _{ {{y}'} ,i}\big]^{T}} ). See [30] for further detail on the procedure to obtain the expressions for the membrane, bending, and shear nodal displacement-strain matrices.
The generalised stress vector (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {{ {\boldsymbol{\mathrm{\hat {\sigma }}}} }'} } ) at the middle plane of an element can be written as
|
(65) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {{ {\boldsymbol{\mathrm{\hat {\sigma }_{m}}}} }'} =[N_{ {{x}'} },N_{ {{y}'} },N_{ {{x}'} {{y}'} }]^T} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {{ {\boldsymbol{\mathrm{\hat {\sigma }_{b}}}} }'} =[M_{ {{x}'} },M_{ {{y}'} },M_{ {{x}'} {{y}'} }]^T}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle { {{ {\boldsymbol{\mathrm{\hat {\sigma }_{s}}}} }'} }=[Q_{ {{x}'} },Q_{ {{y}'} }]^T}
are the resultant stress vector corresponding to membrane, bending, and shear effects respectively. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {{ {\boldsymbol{\mathrm{{\sigma }_{\hbox{out},l}}}} }'} }
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {{ {\boldsymbol{\mathrm{{\sigma }_{\hbox{in},l}}}} }'} }
are the in-plane and transverse stress vectors of the layer Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle l}
that are defined in 62, however in this case it is expressed in the element local axis.
The residual vector of forces expressed in the local axes can obtained starting from the principle of virtual work and applying the classical procedure in finite element methodology.
|
(66) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {{ {\boldsymbol{\mathrm{{f_{\hbox{int}}}}}} }'} _{{\Omega }_{e}}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {{ {\boldsymbol{\mathrm{{f_{\hbox{ext}}}}}} }'} _{{\Omega }_{e}}}
are the elemental internal and external force vector in local description, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol{\mathrm{{N}}}}
is the classic mechanical shape function [31] and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {{ {\boldsymbol{\mathrm{\hat {\tau }}}} }'} :=[q_{ {{x}'} },q_{ {{y}'} },q_{ {{z}'} },m_{ {{x}'} },m_{ {{y}'} },m_{ {{z}'} }]}
is the external traction force vector.
In order to transform these magnitudes from local to global axes, it typically is carried out by multiplying by a linear transformation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle T_{{\Omega }_e}}
[30]. Then the residual vector of forces in global axes is expressed
|
(67) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol{\mathrm{{f_{\hbox{int}}}}}} _{{\Omega }_{e}}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol{\mathrm{{f_{\hbox{ext}}}}}} _{{\Omega }_{e}}}
are the element internal and external force vector in global description. The residual vector of forces of the whole structure is obtained by the assembling of the elements and the equilibrium equations yield
|
(68) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol{\mathrm{{a}}}} }
is the global displacements vector. 68 represents a non-linear problem due to geometrical and material non-linearities. The solution can be found by the Newton-Rapshon method together with the application of an incremental-iterative scheme ([33]). Linearising 68 yields
In order to transform these magnitudes from local to global axes, it typically is carried out by multiplying by a linear transformation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle T_{{\Omega }_e}} . The solution of the global system can be found by applying the Newton-Rapshon method to the definition of the residual ([33]).
|
(69) |
|
(70) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left. {\boldsymbol{\mathrm{{K}}}} ^{-1}\right|_{n+1,k}}
is the global tangent stiffness matrix and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left.{ \dfrac{\partial {\boldsymbol{\mathrm{{{r_{{a}}}}}}} _{{\Omega }_{e}}}{\partial {\boldsymbol{\mathrm{{a}}}} } }\right|_{n+1,k}}
is calculated using the expression
|
(71) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left. {{ {\boldsymbol{\mathrm{{D}}}} }'} \right|_{n+1,k}}
is the generalised constitutive matrix. This matrix can be calculated, in terms of the tangent constitutive matrices of each layer defined in 61, as follows
|
(72) |
69 represents a non-linear problem that depends on the state variable (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {a} ) and its non-linear constitutive model in 72 depends on the thermal model (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {T}, {F} ). The thermo-mechanical model solution is shown in alg:thermo-mechanical.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): n=0
Initialise
while Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): n\le n_{max}
do
- THERMAL PROBLEM
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): k=0
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. {\boldsymbol{\mathrm{{T}}}} \right|_{n+1,k}=\left. {\boldsymbol{\mathrm{{T}}}} \right|_{n}
- while Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): k\le k_{max}
do
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. {\boldsymbol{\mathrm{{T}}}} \right|_{n+1,k+1}=\left. {\boldsymbol{\mathrm{{T}}}} \right|_{n+1,k}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. {\boldsymbol{\mathrm{{L_{{T}}}}}} \right|_{n+1,k} \left. {\boldsymbol{\mathrm{\Delta {T}}}} \right|_{n+1,k+1}=-\left. {\boldsymbol{\mathrm{{r_{{T}}}}}} \right|_{n+1,k}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. {\boldsymbol{\mathrm{{L_{{T}}}}}} \right|_{n+1,k}=\left.{\dfrac{\partial {\boldsymbol{\mathrm{{r_{{T}}}}}} }{\partial {\boldsymbol{\mathrm{{T}}}} }}\right|_{n+1,k}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. {\boldsymbol{\mathrm{{T}}}} \right|_{n+1,k+1}=\left. {\boldsymbol{\mathrm{{T}}}} \right|_{n+1,k}+\left. {\boldsymbol{\mathrm{\Delta {T}}}} \right|_{n+1,k+1}
Update Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left.{F}\right|_{n+1,k+1}
- if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \dfrac{\left|\left|\left. {\boldsymbol{\mathrm{{r_{{T}}}}}} \right|_{n+1,k}\right|\right|}{\left|\left|\left. {\boldsymbol{\mathrm{{{q_{\hbox{ext}}}}}}} \right|_{n+1,k}\right|\right|}\leq \hbox{tolerance}
- then
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. {\boldsymbol{\mathrm{{T}}}} \right|_{n+1}=\left. {\boldsymbol{\mathrm{{T}}}} \right|_{n+1,k+1}
- then
- break;
- else
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): k=k+1
- end
- end
- THERMO-MECHANICAL PROBLEM
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): k=0
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. {\boldsymbol{\mathrm{{a}}}} \right|_{n+1,k}=\left. {\boldsymbol{\mathrm{{a}}}} \right|_{n}
- while Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): k\le k_{max}
do
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. {\boldsymbol{\mathrm{{a}}}} \right|_{n+1,k+1}=\left. {\boldsymbol{\mathrm{{a}}}} \right|_{n+1,k}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. {\boldsymbol{\mathrm{{K}}}} \right|_{n+1,k} \left. {\boldsymbol{\mathrm{\Delta {a}}}} \right|_{n+1,k+1}=-\left. {\boldsymbol{\mathrm{{r_{{a}}}({a},{T})}}} \right|_{n+1,k}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. {\boldsymbol{\mathrm{{K}({a},{T})}}} \right|_{n+1,k}=\left.{\dfrac{\partial {\boldsymbol{\mathrm{{r_{{a}}}}}} ( {\boldsymbol{\mathrm{{a}}}} , {\boldsymbol{\mathrm{{T}}}} )}{\partial {\boldsymbol{\mathrm{{a}}}} }}\right|_{n+1,k}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. {\boldsymbol{\mathrm{{a}}}} \right|_{n+1,k+1}=\left. {\boldsymbol{\mathrm{{a}}}} \right|_{n+1,k}+\left. {\boldsymbol{\mathrm{\Delta {a}}}} \right|_{n+1,k+1}
- if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \dfrac{\left|\left|\left. {\boldsymbol{\mathrm{{r_{{a}}}({a},{T})}}} \right|_{n+1,k}\right|\right|}{\left|\left|\left. {\boldsymbol{\mathrm{{f_{\hbox{ext}}}}}} \right|_{n+1,k}\right|\right|}\leq \hbox{tolerance}
- then
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. {\boldsymbol{\mathrm{{a}}}} \right|_{n+1}=\left. {\boldsymbol{\mathrm{{a}}}} \right|_{n+1,k+1}
- then
- break;
- else
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): k=k+1
- end
- end
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): n=n+1
end Non-linear thermo-mechanical coupling
4 Validation of the numerical model
4.1 Henderson experimental tests
In order to validate the thermal part of the thermo-mechanical model developed in this work, the experimental data presented in Henderson et al. in [4] will be used. In the experimental tests, a composite material consisting of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\phi }_m=39.5%}
of phenolic resin and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\phi }_f=60.5%}
of glass and talc filler was studied. The test samples were of cylindrical shape of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 1}
cm diameter by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 3} cm height. The tests consisted of exposing one side of these samples to a radiant heat flux of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 279.7} kW/mFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle ^2} . The temperature evolution of the samples was monitored using four thermocouples at depths of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0.1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0.5} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 1.0}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 2.9}
cm from the heated side. The instrumentation, sensors and experimental procedure is described in [34].
These experimental tests have been simulated with the numerical model developed in this work. The cylindrical samples have been modelled with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 30}
finite elements of 1D. The material properties and boundary conditions that have been used in the simulations are the same as those reported in [4].
The 2 shows the time evolution of the temperatures obtained from the experimental test and the numerical model is compared.
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\begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix}
| Evolution of the temperature (T(x,t)) of the experimental and numerical results at different thickness positions. |
| Figure 2: Evolution of the temperature (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {T}(x,t)
) of the experimental and numerical results at different thickness positions. |
It can be observed that the agreement between the numerical and experimental results is
excellent.
very good. The discrepancies at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x=0.1}
from 50 to 300s, according to [4], are attributed to minor differences of the thermal and kinetic material properties of the test compared to the calibration. The neglection of certain physical processes such as the thermo-chemical expansion, non-steady decomposition gas diffusion or the heat transfer between the char and the gases.
This is further evidenced by the loss of experimental mass provided in [4] and shown in 3. There is an initial discrepancy at time 50 seconds and as time evolves, and pyrolysis becomes more extensive, the experimental and numerical distribution of pyrolysis through-thickness presents greater difference between the calibrated and testing thermo-chemical properties.
| Evolution of the fraction mass remaining through-thickness of the experimental and numerical results. |
| Figure 3: Evolution of the fraction mass remaining through-thickness of the experimental and numerical results. |
Note that the mass loss is intrinsically related to the pyrolysis fraction. An Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 80%}
corresponds to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {F}=0}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 100%}
to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {F}=1}
.
4.2 Fire-resistant test on a load-bearing bulkhead
The second validation exercise will be done against the experimental test of an FRP bulkhead following the standards of the International Code for Application of Fire Test Procedures (2010 FTP Code) [35]. This test has been carried out within the scope of the H2020 FibreShip project in the vertical furnace of Eurofins Expert Services. The test was aimed at demonstrating the performance of the bulkhead as a 60 minutes load bearing fire-resistant division. In order to study the performance of the bulkhead beyond those 60 minutes, the test was extended until 5100s, where it was terminated due to safety reasons. At that instant, the panel deformation created a significant gap with the frame, allowing the flames to escape.
An FRP division was manufactured with the dimensions and characteristics shown in 5. A total of 5 sensors – represented as groups of three squared pattern markers in 5 – are placed at (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 25%} ,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 25%} ), (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 75%} ,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 25%} ), (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 50%} ,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 50%} ), (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 25%} ,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 75%} ) and (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 75%} ,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 75%} ) where each coordinate is relative to the width (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 2.9} m) and height (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 2.98} m) of the panel. Each of these groups of sensors are composed of 3 through-thickness thermocouples placed on the unexposed surface, in the middle of the PVC and behind the monolithic laminate bounded to the insulation (see 5).
The experiment was performed as established by the standards([36], [37]), which determines the heat flux based on a design temperature curve. The calibrated heat flux is correct, as the furnace temperature matches perfectly the ISO 834 curve . The standard emissivity value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0.9}
is used for both unexposed and exposed surfaces, the standard convection coefficient for the unexposed surface is 9 (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {{W}}/{{m}^2{^{\circ}}{K}}}
) and for the exposed should be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 25}
(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {{W}}/{{m}^2{^{\circ}}{K}}}
), however the sensitivity tests have shown that its real value is more of the order of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 15}
(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {{W}}/{{m}^2{^{\circ}}{K}}}
). The temperature, at which the unexposed surface is considered to be the ambient, is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 17} (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {^{\circ}}{C}} ).
| Experimental setup for the bulkhead test. |
| Figure 4: Experimental setup for the bulkhead test. |
The experimental test in 5, according to the requirements in [35], is performed to achieve a class "load bearing fire-resisting divisions 60". The performance criterion determines that the test has failed when is not able to support the test load. The specimen has failed if either the axial contraction is greater than "h/100 (mm)" or the rate of axial contraction is greater than "3h/1000 (mm/min)", where "h" is the initial height in millimetres. Another limiting criterion is the presence of an opening between the specimen and the gap gauges, and this happens in the test after 85 minutes, with the potential risk of fire spreading.
| Schematic of the test panel. |
| Figure 5: Schematic of the test panel. |
The International Maritime Organisation (IMO) in their Fire Test Procedures guidelines[35] establish also the mechanical load conditions used in the test. The load condition is a compression load of 7.0 kFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {N}/{m}}
placed on the top edge.The mechanical deflection in the centre of the panel was measured by an actuated cable sensor and later compared with the numerical results.
The composite laminate is conformed by the following stack shown in 5:
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- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 2\times{17.5}} mm layers of PVC which works as the core of the sandwich (PVC-H80 from Diab Group; this core is part of the Divinycell HFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle ^\hbox{®}} materials) [39].
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 16\times{0.375}} mm layers of unidirectional glass/vinylester (LEO Injection Resin Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 8500} from BÜFA; this resin is part of the Saertex LEOFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle ^\hbox{®}} fire retardant composite system) and the fibre orientation of the stack is [0 90 ,0 ,90 ,-45 ,45 ,0 ,90 ,0 ,90 ,-45 ,45 ,0 ,90 ,0 ,90 ,0 ] degrees.
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The thermal properties of each layer material are presented in 1
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(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {W}/{m}{^{\circ}}{K}} ) |
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(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {J}/{m}^2{^{\circ}}{K}} ) |
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(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {J}/{m}^2{^{\circ}}{K}} ) |
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(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {kg}/{m}^3} ) |
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| glass/vinylester | 0.5135 | 858.55 | 1000 - 1200 | 1780 | 2Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \cdot 10^5} |
| PVC | 0.02-0.06 | 1170 | 1200 | 80 | 0 |
| rockwool | 0.03-0.8 | 1000-750 | 0 | 60 | 0 |
The decomposition energy of the PVC is neglected, since the temperature measured in the core of the bulkhead throughout the test was lower than the degradation temperature threshold.The pyrolysis model of the laminate used to build the bulkhead was calibrated against the experimental thermo-gravimetric test which has also been carried out within the context of the Fibreship project. 6a shows the agreement of the mass fraction evolution of the model in the thermo-gravimetric test of the laminate after the calibration. On the other hand, the pyrolysis of the core material of the laminate is taken from the thermo-gravimetric analysis presented in [41] and 6b compares the experimental data from the literature to selected the pyrolysis model for the PVC.
| Material | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {A_{T}}
(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {s}^{-1}} ) |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {E_a}
(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {J}/{kmol}} ) |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {n_{r}}
(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {J}/{m}^2{^{\circ}}{K}} ) |
| glass/vinylester | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 6\cdot{10}^{20} | 2.8Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \cdot 10^5} | 6 |
| PVC | 1202604.28 | 90000 | 2 |
| Leo System^®. | PVC-H80 [41]. |
| (a) Leo SystemFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): ^\hbox{®}
. |
(b) PVC-H80 [41]. |
| Figure 6: Evolution of the experimental and modelled mass fraction Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \Big(\dfrac{\partial {\rho }_s({T})}{\partial t}\Big)
. | |
The values for the Arrhenius law (7) of the parameters of the calibrated pyrolysis model in 6a and 6b are given in 2. The insulation can be considered pyrolysis free. This means that the degradation factor will remain equal to one and the evolution to zero.
The mechanical properties of the components in the Leo SystemFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle ^\hbox{®}}
material are taken from the experimental data presented in [42]. The PVC is calibrated against the data provided in [39] and any missing information is completed by standard low density PVC values.
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(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {Pa}} ) |
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(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {N}/{m}} ) |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\beta } | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\phi } |
| Matrix | 3.35 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \cdot{10}^{9}} | 0.26 | 20 | 1.2 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \cdot 10^4} | 1 | 0.40 |
| Fibre | 72.4 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \cdot{10}^{9}} | 0.21 | 1800 | 8.0 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \cdot 10^5} | 1 | 0.60 |
| PVC | 49 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \cdot{10}^{6}} | 0.4 | ||||
| rockwool | 2466060.9905 | 1.17647Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \cdot 10^{-6}} |
The characterisation of the evolution of the Young modulus respect to the temperature is obtained from a dynamic mechanical thermal analysis (DMTA) of the Leo SystemFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle ^\hbox{®}}
material and for PVC, the storage modulus, is obtained from the research by Earl and Shenoi in [44].
| File:Test--Posttikz-output-figure.png | Leo System^®. |
| (a) Leo SystemFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): ^\hbox{®}
. | |
| Figure 7: Experimental and numerical evolution of the storage modulus respect to the temperature. | |
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| Matrix | 3350 | 1507.5 | -0.0691 | 6 | 96 | 36Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \cdot 10^{-6}} |
| Fibre | 72400 | 32580 | -0.0691 | 6 | 96 | 36Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \cdot 10^{-6}} |
| PVC | 49 | 1.47 | -0.0475 | 6 | 90 | 40Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \cdot 10^{-6}} |
| rockwool | 60Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \cdot 10^{-6}}
|
Note that in 4, the insulation only takes into account the thermal expansion coefficient which is considered isotropic. The fitting equation in 6 represents the evolution of the Young modulus described in 42 and with the assumptions in 43, 44 and 45. The glass transition temperature or the thermal expansion coefficient are assumed to be identical for matrix and fibre since the experimental data is referred to the composite material, therefore the properties of the composite are ingrained into its constituent materials for simplicity.
4.2.1 Results of the fire-resistant test on the bulkhead
This section presents the results of the fire-resistant test on the bulkhead and compares them with those of the computational model. 9 compares the time evolution of the temperatures obtained from the experimental test and the numerical model.
The thermal numerical model
(1D)
consists of a discretisation of the thickness of the composite, with a total of 112 1D finite elements. The glass/vinyl-ester layers are discretised with 16 uniform spaced elements each, the PVC core is discretised with 20 uniform spaced elements and the insulation with a total of 60 uniform spaced elements.
The temperature of the furnace is prescribed on the exposed surface using the ISO 834 curve and the ambient temperature prescribed in the unexposed surface is considered to be initially at 17 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {^{\circ}}{C}}
and by the end of the experimental test it is near 25 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {^{\circ}}{C}}
. Note that these two temperatures do not refer to the temperature in the exposed/unexposed surfaces themselves, but rather to the nearby temperatures to these surfaces. Hence, the flux can be calculated as established in 21 and 22 and the convection coefficient and emissivity of both surfaces were described in the setup of the experiment in 4.2.
The thermo-mechanical model
(3D layer-wise shell)
consists of a total of 256 equally distributed linear geometric quadrilateral elements with a non-linear constitutive model. From 8 and defining the horizontal axis as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x} , the vertical axis as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y}
and the out-of-plane axis as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle z}
. The lower horizontal edge fixes the translation degrees of freedom in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x,y,z} , the upper horizontal edge fixes the translation degree of freedom Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle z}
and has a load applied of 7 kFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {N}/{m}}
as described in 4.2.
| Boundary conditions applied to the thermo-mechanical model. |
| Figure 8: Boundary conditions applied to the thermo-mechanical model. |
Both left and right vertical edges fix the translation degree of freedom in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle z}
and have a dynamic variable elastic constraint. These two elastic constraints are symmetric and are controlled by the horizontal dilatation of the edges. There are two stages, first the translation degree of freedom in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x}
is given a certain rigidity to simulate the friction of the panel and the frame and later the translation degree of freedom in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x}
is fixed completely. The plausible phenomenon that is addressed in this example is that the panel is able to slide in the test frame, meaning there is a little space reserved, and it is occupied as the bulkhead starts dilating due to the increase of temperature. The first stage blocks gradually the movement in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x}
and in the second stage considers that there is no extra space and the boundary condition has become fully fixed. The specific values of these two elastic boundary conditions can be found in 12.
Both thermal and thermo-mechanical analysis are solved incrementally with a total simulation time of 5100 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {s}}
in intervals of 100 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {s}}
.
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0.0 \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} 28.9 \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} 46.4 \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} 157.8 \begin{matrix}\end{matrix} \begin{matrix}\end{matrix}
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solid, 0.5mm \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix} \begin{matrix}\end{matrix}
| Evolution of the temperature (T(x,t)) at different positions of the thickness. The temperature through thickness represents the sets in red, blue and green and the temperature furnace the orange set. Thermocouples T1, T2, T3, T4 and T5 in red were placed at x=0.0 mm, thermocouples T11, T12, T13, T14 and T15 in blue were placed at x=25.5 mm and thermocouples T6, T7, T8, T9 and T10 in green were placed at x=41.0 mm. All measures are respect to the unexposed surface. |
| Figure 9: Evolution of the temperature (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {T}(x,t)
) at different positions of the thickness. The temperature through thickness represents the sets in red, blue and green and the temperature furnace the orange set. Thermocouples T1, T2, T3, T4 and T5 in red were placed at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): x=0.0 mm, thermocouples T11, T12, T13, T14 and T15 in blue were placed at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): x=25.5
mm and thermocouples T6, T7, T8, T9 and T10 in green were placed at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): x=41.0
mm. All measures are respect to the unexposed surface.
|
Note that both the experimental and computational data closely agree. There is a minor fluctuation of the temperature (green) at the interface between the glass/vinyl-ester layer and the core, closest to the fire exposed surface. This fluctuation is produced near the time the upper right corner of the panel starts to bulge outwards producing a gap between the specimen and the test frame, these the two thermocouples, which have registered these fluctuations, coincide exactly with those that are situated in the upper right and upper left of the panel.
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\begin{matrix}\end{matrix}
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| Sectional degradation fraction (F) at time 5100s. | Sectional temperature (T) at time 5100s. |
| (a) Sectional degradation fraction (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {F}
) at time 5100Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {s} . |
(b) Sectional temperature (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {T}
) at time 5100Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {s} . |
| Figure 10: Computed final profile of the degradation and temperature through thickness of the section in the mid point of the panel. | |
The profile of the degradation fraction at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 5100{s}} , computed by the pyrolysis model at the mid-point of the panel, is presented in 10a. Notice that the degradation can be considered zero, since the minimum degradation fraction is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0.985} . It implies that the fire-resistant performance of the panel is excellent.
The corresponding profile of the temperature can be found in 10b, agreeing that the composite is well insulated. The 4 layers of the composite described previously can be observed by checking the slope variation. Within those 4 regions, only two regions show a temperature over the range of 200 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {^{\circ}}{C}} .
The degradation shown is in accordance to what was presented in 6a. Notice that the mass fraction presents a substantial change for temperatures higher than Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 300-400}
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, and this is the primary reason why the Leo SystemFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle ^\hbox{®}}
closest to the exposed surface does not present a significant degradation. Similarly, 6b shows that the mass fraction of the PVC starts degrading for temperatures above Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 200} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {^{\circ}}{C}} and the PVC region almost reaches this temperature in the union with the Leo SystemFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle ^\hbox{®}} layer closest to the exposed surface.
The deflection in the middle of the panel is measured with a cable actuated sensor.
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| Deflection evolution in mm. |
| Figure 11: Deflection evolution in mm. |
In 11 the deflection in the mid-point of the unexposed surface is registered experimentally and compared to the numerical simulation. A good global agreement between computed and measured data is found. However, relevant differences are found in some phases of the test, likely due to the complexity of the problem and the uncertainties in some material properties. This is discussed below.
Throughout the experiment, we can identify four phases, defined by the points 1, 2, 3 and 4 in 11. The panel start deflecting negatively due to the compression load that the panel has to endure according to [35]. Note from 5, the section is non-isotropic and hence the load that is theoretically placed in the middle of the thickness which does not coincide with the neutral axis of the section. This generates two moments in the upper and lower edges each bending the structure, in addition, since the elastic modulus of the section is lower in the insulation than in the glass/vinyl-ester layer, the composite initially experiments a higher compression on the exposed surface (insulation). Thus, the exposed surface is relatively compressed and the unexposed is relatively tensed. The numerical simulation does not fit exactly the early deflection, however it captures the phenomenology, starting with a negative deflection and once the temperature is high enough, it bends to the opposite direction (point 1).
Since the panel is inserted into the test frame, which means it is not fixed in any other direction other than the out-of-plane direction, and the increase of temperature makes the panel to proportionally dilate. It happens around the time step of point 2, in 11, that the specimen edges have started to partially contact the frame and to be fixed to it. In order to model this phenomenon, two boundary conditions – namely two dynamic elastic constraints – were introduced on both vertical edges (right and left) which add an extra fixation on the horizontal direction opposite to the dilatation of the panel. As shown from point 2 onward, this dynamic elastic constraints are able to reproduce the friction condition between the bulkhead and the test frame.
Once the simulation arrives to point 3, the panel is fully fixed due to experimenting an elevated temperature and the fact that there is no reserved extra space to dilate. The dynamic boundary conditions from the third point onward fix the in-plane horizontal translation.
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| File:Test--Posttikz-output-figure.png | Left mid-point. |
| (a) Left mid-point. | |
| Right mid-point. | |
| (b) Right mid-point. | |
| Figure 12: Horizontal displacement in terms of time. Note the dynamic boundary condition stages. | |
The numerical model is able to reproduce the behaviour of the third stage to a certain extent, note that the uncertainties and limitations of the experimental test play a fundamental role in this stage. The panel approaching point 4 starts to present a non-symmetrical behaviour, and finally the panel collapses when the upper right corner bulges from the test frame. At the fourth point, the structure collapses due to the augment of effective mechanical load, since the upper right corner is no longer enduring any load.
The collapse could be reproduced, since the non-symmetric behaviour is very complex to be incorporated in this example. It is not even likely to happen in a real bulkhead fire collapse scenario. However, the bulging can be explained through the deflection of a sectional cut passing through the mid-point of the panel (13a) and the maximum damage index (13b).
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| Symmetric vertical section. | Damage index (d). |
| (a) Symmetric vertical section. | (b) Damage index (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {d}
). |
Figure 13: Final snapshot of the panel at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 5100
s before the experimental test collapses. Observe the damage localisation | |
In 13 it can be seen that the deflection of the panel presents two protuberances. This is equivalent to the two blue regions found in 13b, observe that the lower end of the section cut presents a mild stepper rotation than the upper end, this explains why the bulging occurs on the top rather than on the bottom. The bulging is considered to occur due to uncertainties that cannot be controlled, however the methodology used in this thermo-mechanical analysis shows that this phenomenon is feasible to occur since the degradation has a double triangle shape which localises the damage in all four corners, however – numerically speaking – it is still able to conserve the perfect contact between the frame and the specimen.
The experimental test has some certain limitations when it comes to maintaining a perfect contact between the bulkhead and the test frame. Once the upper right corner bulges, fire security protocol, establishes to cease the testing. However, it is quite interesting to extend the simulation beyond the scope of 5100 seconds and see that the bulging phenomenon can be addressed in the model. Not the phenomenon per se, but the prelude to it. Short after the experimental model collapses, the numerical model shows a good agreement with this bulging behaviour since around all four corners a negative deflection starts to form and this very change of the deflection explains the sudden bulge of one of the corners.
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| Deflection close to the upper right corner. |
| Figure 14: Deflection close to the upper right corner. |
14 shows the evolution of the deflection in the closest node to the corner and does not belong to any of the edges. This node is inside the domain and has no boundary condition applied of any kind. Notice how just close to time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 5000{s}}
it bulges outwards of the test frame. The rest of the analysis contemplates a scenario that has not been capable to test in the experiment, however the numerical model assesses the bulging of the node closest to the right upper corner and maintains it for the rest of the extended simulation time. So this clearly shows that the bulging phenomenon – which is the sole reason for the collapse, since the effective section of the bulkhead is reduced considerably – is the product of the inability to perfectly retain the edges fixed to the test frame and that the thermo-mechanical model allows to extend the empirical testing to the hypothetical real scenario.
5 Conclusion
The numerical model detailed in the present work has shown a very good agreement between the predicted results and the thermal experimental tests, as shown firstly in the experimental data provided in [4] (see 2) and also the bulkhead test.
The results from these original experimental tests are almost identical to the numerical results shown in 9, where the numerical model predicts correctly the temperatures registered at different positions of the thickness.
The thermo-mechanical model, despite not agreeing perfectly, is able to capture the overall behaviour. Note the difficulty to carry out a controlled experiment when fire is involved. The specimen contour is one of the most difficult problems to control (variable boundary conditions) or the fact that little imperfections in the material or mechanical load may result in a non-symmetrical collapse of the structure. Even the instrumentation itself may be limiting to properly register the real deflection.
Even though, the thermo-mechanical analysis shown in 11 is able to assess the structural integrity of the specimen and despite not reproducing the exact collapse phenomenon (non-symmetrical upper right bulging), it has shown to be an excellent tool to perform a fire collapse assessment for composite materials which indeed is one of the novel and unique goals framed in the FibreShip project. The combination of the present pyrolysis model with the simplicity of the SPROM formulation, adapted to high temperatures, grants a unique tool to incorporate the thermal deformation in composites and note the simplicity of using isotropic constituent materials instead of the common and general orthotropic approach.
The importance of using isotropic constituent materials does not reside uniquely in the linear constitutive model, but in the non-linear constitutive model, namely the isotropic damage model, which is a common and standard constitutive model for FRP composite materials. This proposed model is able to introduce the effects of thermal degradation in the non-linear constitutive model and poses a framework to analyse the non-linear structural integrity of composite marine structures as shown in 10. Also, the thermo-mechanical coupling is suitable to analyse different hypothetical fire scenarios when experimental data is missing or limited.
Finally, the original case presented in this work has not addressed the buckling phenomenology, which seemed not to be present in the observed experimental data.
6 Acknowledgement
This work was funded thanks to H2020 project FIBRESHIP sponsored by the EUROPEAN COMMISSION under the grant agreement 723360 "Engineering, production and life-cycle management for complete construction of large-length FIBRE-based SHIPs". www.fibreship.eu/about
7 Disclosure statement
No potential conflict of interest was reported by the authors.
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Published on 30/06/01
Accepted on 30/06/01
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