P. Gu,     University of California at San Diego, USA

4.1 Introduction

Developments in advanced composite technology offer cost-effective application of polymer matrix composites (FRPs) for large load-bearing structures. In addition to cost effectiveness, these composites are designers’ materials that can be designed to have specific high strength and high stiffness, high fatigue resistance, and high corrosion resistance, for specific needs. One of the major concerns associated with the acceptability of the composite constructions, e.g. ship construction, is fire performance. Fire performance includes both flammability and structural characteristics. Flammability includes the consideration of heat release, flame spread, toxicity, and smoke production. Structural issues include the ability of the composite structures to carry internal and external loads before, during, and after fire.

The objective of the framework for structural fire integrity of polymer matrix composites is to establish a fundamentally sound basis upon which the design criteria of the composites can be developed, and accordingly, the design and operation practice of the composite structures can be performed safely. The goals are: (1) to develop and demonstrate the ability of new composite systems to display high levels of structural integrity when subject to severe fire exposure; (2) to develop and demonstrate the capability to predict and design vis-à-vis combustibility, flame spread, smoke generation, toxicity, and the structural responses to thermal-mechanical loads. As an example of the current status of using these composites, the US ship regulations defined in the Code of Federal Regulations and the international ship regulations defined by the International Maritime Organization (IMO) through the International Convention for the Safety of Life at Sea (SOLAS) have strict restrictions for the use of combustible materials in construction. The International Code of Safety for High-Speed Craft (HSC, 1995) allows the use of combustible constructions, but requires the vessels to meet very strict operation, management and evacuation requirements.

The behavior of polymer matrix composites with combined thermal-mechanical loads is different from that with only mechanical loads. A significant decrease in mechanical properties such as Young’s modulus and failure stress, known as thermal softening, is observed when the temperature rises to 100 °C from room temperature. Hence, the composites significantly lose load-bearing capacity with modest temperature rise. When heated to a higher temperature range (about 300 °C), where chemical decomposition of polymers occurs, the associated phase transition creates char materials which are significantly different in thermal and mechanical properties from virgin materials. In the phase transition, heat generation, the loss of mass, and the variation of thermal properties are influenced by the rate of the decomposition whose effects need to be accounted for in the thermal model. The thermal process introduces various heterogeneous microstructures, such as gas-filled pores, matrix cracking, and matrix-fiber debonding, which can have significant roles in the composites’ macroscopic deformation. Non-uniform temperature field, i.e. thermal gradients, in polymer matrix composites is another aspect that is different from the conventional materials, such as aluminum alloys, which has reasonably assumed isothermal condition and therefore relative simplicity of failure modes via uniform degradation in stiffness and strength. Polymer matrix composites possess much lower thermal conductivities such that they develop thermal gradients and thereby strong gradients in stiffness, strength and other physical properties. The analysis of failure modes and the base for setting allowable limits for thermal loading are, accordingly, more difficult. The deformations and thus mechanical failures are more varied and complex, e.g. even without mechanical loads, thermal expansion induced by the non-uniform temperature field causes non-uniform deformation field. Therefore, investigation of these composites in the thermal-mechanical condition requires the theory of non-homogeneous materials.

The attention of this chapter is confined to structural behavior and load-bearing capacity of the composite panels in fire. Such plate type or shell type structures are common in ship structures, e.g. hull, bulkhead and decks, and in civil infrastructures. Our focus is on analytical modeling projects carried out at the structural engineering department of the University of California at San Diego (UCSD), based on the experimental performance and observations aimed at understanding the structural response in the combined thermal-mechanical environment. Some of the discussion is based on our unpublished work and the extension of our published work.

The chapter builds on the contribution of many researchers, including those at other institutions; some concentrate on structural response while others on material behavior. Broadly speaking, the thermal model discussed in Section 4.2 originates from that developed by Henderson et al., (1985) for considering phase transition in the chemical decomposition of polymers. The power-form degradation model discussed in Section 4.3 was built from that suggested in the book by Hollaway and Head (2001). The buckling load and skin wrinkling load obtained in Sections 4.4 and 4.5 for non-homogeneous sandwich material systems are the development of Allen’s solutions for homogeneous sandwich material systems (Allen, 1969). The micromechanical approach of crystal plasticity (Asaro, 1983) to simulate interlaminar shearing and debonding in Section 4.6 is the latest effort to bridge the gap in investigations between material behavior at the microstructure/specimen level and structural response at the component/system level. In addition, a group of researchers at Royal Melbourne Institute of Technology (RMIT) and the University of Newcastle-upon-Tyne has carried out a set of detailed modeling and experimental work on the material behavior of composite laminates in fire (Mouritz et al., 2009).

4.2 Temperature distribution

When polymers are heated to sufficient high temperature, say about 300 °C, phase transition occurs during chemical decomposition (pyrolysis) to create char materials and gases. The original thermal model to account for such phase transition was developed by Henderson et al. (1985). The model assumes that the produced gases flow in the space, and no solids or solid particles flow in the space. For uniform heat flux in Fig. 4.1, the heat transfer model is one-dimensional along the y direction. The energy conservation leads to the following equation:

[4.1]

where ρ is the mass density of the composite; k is the thermal conductivity of the composite; is the mass flux of the gas; h_g is the enthalpy of the gas; Q is the heat of decomposition; C_ρ = ∂h/∂T is the specific heat of the composite, where h is the enthalpy of the composite; C_ρg = ∂h_g/∂T is the specific heat of gas; and T = T(y, t) is temperature, where t is heating time. If the last two terms on the right side are removed, it is the conventional heat conduction equation. The last two terms on the right side represent the energy change rate due to the flow of the gas and the heat generation. The thermal conductivity varies along the thickness due to its degradation at elevated temperature.

Since there is no accumulation of decomposition gas in the solid material, the conservation of mass is written as:

[4.2]

To describe the rate of mass density change, the Arrhenius equation is introduced:

[4.3]

where A is a pre-exponential factor; E_a is activation energy; R is the universal gas constant; n is the order of reaction; ρ_v is the mass density of virgin material; and ρ_c is the mass density of char material. The material constants in [4.3], A, n and E_a, are obtained by fitting experimental data, usually the TGA (thermal gravimetric analysis) data which measure the mass loss as the function of temperature for controlled heating rates.

The evolution of the thermal conductivity and specific heat of the composite during the reaction is governed by the mixture rule:

[4.4]

where k_v and k_c are the thermal conductivities of virgin material and char material, respectively; and C_ρv and C_ρc are the specific heats of virgin material and char material, respectively. In general, these thermal conductivities and specific heats of virgin and char materials, as well as the specific heat of gas, are temperature dependent (Henderson et al., 1885; Lattimer and Ouellette, 2006; Lua et al., 2006). The mixture factor F is the ratio of the current mass density and the original mass density, namely F = (ρ – ρ_c)/(ρ_v – ρ_c).

The governing equation [4.1] can be solved along with the supporting equations [4.2–4.4] and thermal boundary conditions, given the material constants, A, n, E_a, k_v, k_c, C_ρv, C_ρc, C_ρg, Q, ρ_v, and ρ_c. The boundary condition at the exposed face is generally convective and radiative exchange with surroundings and can be stated as:

[4.5]

where q₀ is the radiative heat flux; h_conv is the heat transfer coefficient; σ is the Stefan-Boltzmann constant; T_r is ambient temperature; and ε_r and ε_m are the emissivity of the surroundings and the composite surface, respectively. If the boundary condition is temperature versus time curve at the exposed face we have:

[4.6]

The thermal boundary condition at the unexposed face is similar. If the unexposed face is insulated, ∂T(H, t)/∂y = 0. The mass flux of gas satisfies . The initial condition is T(y, 0) = T_r; ρ(y, 0) = ρ_v;.

The solution to the thermal model can be written into dimensionless form. From the expression [4.3], the mass density takes the following non-dimensional form:

[4.7]

The function f_ρ is non-dimensional. Using [4.4] and [4.7], we get the non-dimensional forms of the thermal conductivity and specific heat:

[4.8]

The non-dimensional form for the temperature field is obtained by substituting [4.7] and [4.8] into [4.1] and regrouping the physical parameters. This leads to:

[4.9]

where β = κ_v/ρ_v/C_ρv, and Ψ represents the thermal boundary conditions. For the specified curves of temperature versus time at the exposed and unexposed faces, Ψ = (T_f/T_r, T_b/T_r), where T_b is the temperature of the unexposed face. For insulated unexposed face, the last argument T_b/T_r is removed. If the exposed face is specified by a heat flux q, the argument T_f/T_r is replaced by qH/(T_rk_v). The non-dimensional form [4.9] helps us to characterize the contribution of each physical parameter to the temperature profile.

The thermal model can be solved numerically using finite difference finite element methods. Past research for the thermal model includes verification with experiments, investigation of composites’ thermal properties, and efficient numerical schemes. Henderson et al. (1985), Henderson and Wiecek (1987) and Wu et al. (1993) employed the finite difference schemes for the thermal model; Looyeh et al. (1997), Looyeh and Bettess (1998) and Krysl et al. (2004) described finite element schemes. Gibson et al. (1995) employed the thermal model to study failure mechanisms for specimens in one-sided fire exposure. In Dodds et al. (2000), various calculated results were verified by measured data from fire testing. Instead of the Arrhenius equation [4.3], Trelles and Lattimer (2007) and Asaro et al. (2009a) used the variation of mass density with respect to temperature at a single heating rate to represent the rate of decomposition in the thermal model. In Feih et al. (2007a, 2007b, 2010), various incident heat fluxes were tested, and it was found that in general the calculated results were in good agreement with experimental results (see Fig. 4.2). The exception was the case of high heat flux intensity where the composite ignited after some heating time. The ignition of the composites, which is not considered in the thermal model, results in higher temperature and increases the rate of decomposition. In Asaro et al. (2009a), the thermal model was used to evaluate and characterize the thermal-physical performance of fire protective coatings for naval ship structures. However, in certain cases, the solution of the thermal model was shown to be sensitive to the parameters characterizing phase transition, e.g., the small change in the mass density of the char can result in significant change in the temperature field (see Fig. 5 in Krysl et al., 2004). In addition, there are limited sources for these material property data describing the phase transition beyond the mentioned publications.

At temperatures below chemical decomposition, the terms and parameters for the decomposition can be removed from [4.1] so that the heat transfer equation becomes:

[4.10]

The thermal material properties in the virgin state may be taken as temperature independent (Asaro et al., 2009a). The non-dimensional form for the temperature field [4.9] becomes:

[4.11]

When the temperature field is obtained from [4.1] or [4.10], the structural analysis can be performed to determine mechanical load-bearing capacity.

In Dao and Asaro (1999) and Asaro et al. (1999), numerical solutions of the temperature field were obtained for representative panels made of E-glassl vinylester composites by solving the thermal model using the measured temperatures at the exposed and unexposed faces. Based on these solutions, simplified temperature distributions were discussed. The simplified temperature distributions capture the major features of the actual temperature variations using the temperature values measured at the key locations in thermal tests. For single skin panels, a two parameter model was proposed, which used two of the key temperature values, i.e. those at the exposed face, the center, and the unexposed face, to construct a linear temperature distribution. For the simplified temperature distribution of sandwich panels, the temperature at the unexposed skin can be taken as ambient temperature because the low conductivities give rise to small temperature change of the unexposed skin for a considerable period of heating time. Figure 4.3 systematically shows the piecewise step temperature distribution, i.e. simplified temperature distribution, derived from the actual temperature field. The three key temperature values that characterize the simplified distribution are the temperature at the exposed face, the temperature at the interface between the exposed skin and the core, and ambient temperature at the unexposed skin. The simplified temperature distribution is also called the upper bound temperature field because the above key temperature values are the maximum temperatures in the skins and the core.

4.3 Material behavior at elevated temperature

The degradation of the Young’s modulus with respect to temperature, T, may be given by the power form:

[4.12]

In [4.12], which is systemically shown in Fig. 4.4, E₀ is the Young’s modulus at ambient temperature T_r, or initial stiffness. The power index m varies from 0 to 1, with m = 0 being no degradation before the temperature reaches the reference temperature T_ref. As shown in Fig. 4.4, the reference temperature characterizes the end of the major stiffness degradation which happens in the temperature range T ≤ T_ref. The parameter , which is with a unit of temperature, and the non-dimensional parameter m are obtained from fitting the experimental data in the temperature range T ≤ T_ref. In Gu and Asaro (2005), it was suggested that m = 0.2 for E-glass fiber/lvinylester matrix composites. The power law [4.12] is a phenomenological form similar to that suggested in Hollaway and Head (2001, page 97). The ambient temperature T_r is taken as the room temperature 20 °C. Experiments showed that the reference temperature T_ref for E-glass fiber/vinylester matrix composites is around 120 °C (Dao and Asaro, 1999).

In [4.12], E₁(T) is stiffness above the reference temperature, which is much smaller than the initial stiffness E₀ and reduces to zero at the starting temperature of chemical decomposition T_cd. The stiffness degrades gradually between the reference temperature and the starting temperature of chemical decomposition. For continuity of the stiffness at the reference temperature, we have the following relation from [4.12]:

[4.13]

The degradation law [4.12] is an extension of that discussed in Gu and Asaro (2005) to include the stiffness above the reference temperature. For the case where E₁(T) is taken to be zero, we have from [4.13] such that [4.12] becomes:

[4.14]

Figure 4.5 shows the degradation law [4.14] for several power indexes and some experimental data. The degradation law was used in the analytical simulations of composite panels’ failure modes in Gu and Asaro (2005, 2008a, 2008b, 2008c, 2009) and Gu et al. (2009), where good agreement with experiments was observed.

A material point in the panel is said to be completely degraded if the temperature at this point is above the reference temperature T_ref, since in this situation the material element around this point can no longer take large external loads according to [4.12] or [4.14]. Recent analytical and experimental works (e.g. Asaro and Dao, 1997; Asaro et al., 1999, 2005, 2009b; Birman et al., 2006; Dao and Asaro 1999; Feih et al., 2007a, 2007b, 2010; Gu and Asaro, 2005; Gu et al., 2009; Ramroth et al., 2006) treated thermal softening as well as the phase transition in chemical decomposition as the major influences on the mechanical properties of polymer matrix composites under fire damage. In the analytical simulation using the rate-dependent micromechanical constitutive model (Ramroth et al., 2006) and the experimental evaluation (Asaro et al., 1999; Dao and Asaro 1999), it was found that rate dependence and plastic deformation were unlikely to be the mechanisms for failure of structure-size panels under considered fire protection time. In Asaro et al. (2009b), by analyzing the thermal gravimetric analysis (TGA) data, it was suggested that the rate dependence may be neglected below the glass transition temperature for the case of modest mechanical load in the considered fire-protection period. It further argued that, in certain cases, the creep influence may even be neglected for elevated temperature up to the point where the chemical decomposition starts. In this article, the constitutive theory describing degradation given in [4.12] or [4.14] is used to discuss structural failure modes such as buckling and skin wrinkling.

Other forms of degradation with respect to temperature were proposed by other researchers. Gibson et al. (2006), Feih et al. (2007a, 2007b, 2007c, 2010) and Mouritz et al. (2009) used a hyperbolic tangent function to describe the temperature dependence of the material strength. Dutta and Hui (2000) related the current elastic modulus to the current temperature as well as the current mass density of the composite. The early model by Springer (1984) related current material properties to the loss of mass density. In Bai et al. (2008), who considered different material states in different temperature ranges, a kinetic theory was employed to model the degradations of stiffness, viscosity, and the coefficient of thermal expansion.

Material variation with position y along the thickness of the panel can be obtained by substituting the temperature field T(y, t) into [4.12] or [4.14]. For simplicity, one approximation approach is to write the material variation as the second order polynomial of y:

[4.15]

Consider the single skin panel of thickness H and denote the moduli at the exposed face (y = 0), the center (y = H/2), and the unexposed face (y = H) as E_f, E_c and E_b, respectively. The coefficients in [4.15] are given by interpolation as

[4.16]

Note that the three moduli in [4.16] are temperature dependent according to [4.12] or [4.14]. The second order polynomial form was used in Asaro and Dao (1997) and Asaro et al. (1999) for analytical modeling. Other forms for material variation with position may also be introduced, e.g. the exponential form (Gu and Asaro, 2008c). These simplified forms allow analytical solutions to be presented in compact forms, as will be seen in later sections.

4.4 Global buckling

For composite plates, experiments showed buckling occurred when heated under the operational loads that were considered to be safe under the design criteria without evaluating fire damage (e.g. Gibson et al., 2004; Dao and Asaro, 1999). The latter reference suggested that global buckling was a fairly common failure mode for panels of structural size. In Gu and Asaro (2005), the buckling load was derived from the non-homogeneous theory of elasticity to account for the material variation along the thickness caused by temperature gradient. For sandwich panels, Allen’s formulation for the additional shear deformation of the core (Allen, 1969) was included. The buckling solution is discussed here.

A beam model, as shown in Fig. 4.1, is considered, where the width of panel in the normal direction of the sheet is taken to be a unit value in the following derivation. Due to the mechanical property gradient induced by the thermal gradient along the thickness, the neutral axis is not located at the center of the cross-section, as shown in Fig. 4.1. We apply the Bernoulli–Euler assumption such that the cross-section remains plane and rotates about the neutral axis during bending deformation. The position of the neutral axis is obtained from equilibrium as:

[4.17]

For simplicity, E[T(y, t)] is denoted by E(y). The bending stiffness for the non-homogeneous material or graded material is:

[4.18]

From the bending formulation, the equilibrium equation is of the same form as that for homogeneous materials except the expression of the bending stiffness. This leads to the buckling load for the non-homogeneous material being of the same form as that for homogeneous materials. For pin support (rollers which allow rotation) at both ends, the buckling load of a single panel is:

[4.19]

where l is the length of the panel. The temperature gradient, via material gradient, reduces the critical load through the reduction of the bending stiffness D. For clamping support (which does not allow translation and rotation) at one end and roller support at another end, the coefficient π² in the above expression is replaced by 4.49²; for clamping support at both ends, the coefficient is replaced by 4π².

The above expression for the pin-supported panel may be scaled by the buckling load of temperature independent materials so that:

[4.20]

D* is non-dimensional buckling load P*_cr. One obtains the buckling load by evaluating the bending stiffness using [4.17] and [4.18]. Dimensional analysis considering the temperature field [4.11] leads to the non-dimensional form of the buckling load:

[4.21]

where Π is the group of non-dimensional parameters characterizing the material degradation discussed in Section 4.3, e.g. m for the power law [4.14].

For a sandwich panel, the low shear stiffness of the core introduces additional shear deformation. The approach to include this effect can be found in Plantema (1966) and Allen (1969). For pin support at both ends, the buckling load of sandwich panels with material property variation induced by the thermal gradient was derived in Gu and Asaro (2005) as:

[4.22]

[4.23]

The expressions for the bending stiffness of the sandwich panel D, bending stiffness of the skins D_f, and shear stiffness of the core G were given in Gu and Asaro (2009). For other end conditions, clamping/clamping, roller/clamping, the π² in [4.23] is replaced by appropriate coefficients such as those in the single skin case. The non-dimensional form of the buckling load for the sandwich panel is:

[4.24]

The subscript c is for the variables of the core, and the subscript s is for the variables of the skin; E_0s, E_0c and G_0c are the Young’s modulus of the skin, the Young’s modulus of the core, and the shear modulus of the core, at ambient temperature, respectively. With [4.21] and [4.24], the buckling solutions are presented in terms of non-dimensional parameters. The non-dimensional representations help us to characterize buckling loads, for both understanding structure behavior and designing components in engineering practice.

The staggered approach can be employed for the thermal-mechanical problem. The temperature field is first obtained from the thermal model in Section 4.2. The obtained temperatures are then substituted into the degradation law in Section 4.3 to find the degraded material properties, which, with geometrical properties, determine the buckling load from [4.19] and [4.22]. It is noted that since in such approach the mechanics solutions are independent of the thermal solutions, these buckling solutions, as well as the skin wrinkling solutions discussed in the next section, are applied to the temperature range with or without chemical decomposition. In Gu and Asaro (2005) and Gu et al. (2009), the predicted buckling loads using [4.19] and [4.22] were compared with the measured data of Asaro et al. (1999) and Dao and Asaro. (1999). The tests were carried out using the UCSD multi-axial fire test apparatus (Asaro and Dao, 1997; Asaro et al., 2009b; Dao and Asaro 1999) designed for the panels to expose one-sided fire while maintaining the flexibility of in-plane and out-of-plane mechanical loads. The material properties and dimensional lengths of the specimens were given in the above references. Figure 4.6 shows buckling loads of a single skin panel at some heating times obtained from actual temperature profiles and the estimated buckling load curve obtained by connecting the discrete points with straight lines. The intersection of the buckling load curve and applied load curve is the failure time, which defines the fire protection time period. From the figure, the estimated failure time with applied in-plane compressive load at 6670 N is around 72 minutes, whereas the actual failure time as measured in tests is 65 minutes. A small out-of-plane load was applied in the test to simulate initial imperfection of the panel. The measured deflection is also plotted in Fig. 4.6, which shows that near the failure the deflection increases dramatically due to instability. For sandwich panels, the estimated buckling load curve is plotted for applied compressive load 17 800 N in Fig. 4.7. Also plotted is the measured deflection for the same panel, which shows that near the failure time, around 33 minutes, the deflection again increases dramatically. For recent tests performed at Hughes Associates, Inc. with the same fire test apparatus, the predicted results were again in agreement with the measured results (Asaro et al., 2009b).

The investigations of the UCSD group focused on panels of approximately 1 m² in in-plane size. Mouritz and coworkers (Feih et al., 2007a, 2007b, 2010) studied failures of specimens of approximately 0.1 m² in in-plane size (effective area of fire exposure). They first determined the variation of the compressive strength with temperature from coupon tests. For composite laminates, with the known temperature field from thermal analysis, the variation of compression strength obtained from coupon tests was integrated through the thickness to obtain the strength of the panel. It appears that no insulation layers were placed at the exposed face in Mouritz and coworkers’ tests. Even though the higher heat flux can result in ignition without insulation at the exposed face, the predicted compressive strength was in good agreement with experimental data. The buckling load was also evaluated using a two-layer model (Gibson et al., 2004), where the fire protection time was shown to be significantly increased by adding constraint at the non-mechanical-loading sides. Laminate theory was employed to treat each ply’s load-bearing capacity individually (Gibson et al., 2006). The work carried out by the groups at UCSD, RMIT and the University of Newcastle showed that the degradation law of mechanical properties with respect to elevated temperature, i.e. those discussed in Section 4.3, can be used to model the deformation behavior of the composites in compression, when overlooking micromechanical behaviors under the fire protection time period. If the panel is subjected to tensile in-plane load during fire, the situation can be different since for the tensile load the deformation mechanism of fiber is different from that of the matrix. In addition to thermal softening, the time-dependence (creep) of the fiber may need to be considered. In Feih et al. (2007c), a degradation model with both time and temperature dependences was proposed for the glass fiber in tension. In addition to in-situ fire behavior, residual fire behavior was investigated, for example by Gibson et al. (2003, 2004), Mouritz and Mathys (1999, 2001), Mouritz (2002, 2003) and Mouritz et al., (2004).

Since the staggered approach treats the thermal analysis and mechanical analysis separately, the non-homogeneous mechanics solutions described in this section and in the following sections, as well as those given in the references, can be used to evaluate the structural behavior in fire and after-fire when the appropriate material degradation law, with respect to temperature and time, is determined.

4.5 Skin wrinkling of sandwich panels

Skin wrinkling is a failure mode of sandwich polymer matrix composite panels under in-plane compression. The low transverse stiffness of the core with relative small skin thickness can make skin wrinkling to be the favored failure mode against buckling. Two analytical approaches to evaluate the wrinkling stress, one based on energy balance and another equilibrium equation, were described in Plantema (1966), Allen (1969), and Zenkert (1995). Recent modeling and experimental wrinkling analyses for various loading conditions and material behaviors include Vonach and Rammerstorfer (2000), Gdoutos et al. (2003), Birman and Bert (2004), Fagerberg and Zenkert (2005), and Kardomateas (2005). However, in the above references, the influence of mechanical property gradients, e.g. those caused by one-sided fire heating or those in functionally graded materials, was not investigated. In Gu and Asaro (2008c), using a constructed Airy stress function for the graded material, the analytical wrinkling load for sandwich system with material gradients was derived for the case in which the core’s thickness was infinitely large. Later, Gu and Asaro (2011) extended the wrinkling solution to the general case with finite core thickness, and also discussed the wrinkling solution using the Winkler model (see Allen, 1969). This section summarizes the discussion in Gu and Asaro (2008c, 2011).

The material variation for the Young’s modulus and Poisson’s ratio of the core is taken as the exponential form:

[4.25]

The shear modulus relates to the above by:

[4.26]

The E_1c and v_1c in [4.25] are the Young’s modulus and Poisson’s ratio at y = 0, the upper face of the core as shown in Fig. 4.8. The positive y axis points to the unexposed skin; the x axis is on the interface between the exposed skin and the core. The β, having the dimension 1/[length], represents the material gradient in the core and is given by the Young’s modulus at the upper face of the core and that at the lower face of the core as:

[4.27]

Here E_2c is the Young’s modulus at the lower face of the core. The two Young’s moduli are related to the temperature field through the material degradation law in Section 4.3.

Assuming that during wrinkling the exposed skin displaces in the y direction following sinusoidal wave with a half wavelength L_w:

[4.28]

the membrane stress in the exposed skin is derived from the non-homogeneous theory of elasticity as:

[4.29]

In [4.29], the normalized wavelength is θ = πH_c/L_w; the non-dimensional parameters, β* = β, H_c, η = H_s/H_c(D*E_1s/E_1c)^1/3, and , where E_1s is the Young’s modulus at the upper face of the exposed skin, and D_e is the bending stiffness of the exposed skin. The wrinkling stress σ_cr is achieved by minimizing the membrane stress with respect to the wavelength:

[4.30]

The numerical values of f_cr for considerable ranges of the two non-dimensional parameters, β* and η, are tabulated in Gu and Asaro (2011).

The parameter β* is the natural logarithm of the ratio of the core’s Young’s modulus at interface with the unexposed skin to that at the interface with the exposed skin. If the highest possibility of this ratio is 10 000, β* ranges from 0 to 10. The typical material properties and dimensional lengths are used to estimate the values for η. Consider that H_s/H_c = 0.1, the non-dimensional bending stiffness D* = 2.9 (this is the case that E_2s/E_1s = 1000 for the exponential variation form), and E_1s/E_1c = 1000. Using the above numbers, η = 1.43. An estimate is that η ranges between 0 and 10 for most material systems. For the homogeneous core, β* = 0 and η is the only parameter that the wrinkling stress is dependent on. When the core thickness H_c approaches infinity, η and β approach zero, and f_cr = 1.288.

The Winkler model, in which the core is approximated by a set of continuously distributed springs in the transverse direction, is a simplified model that overlooks the contribution of the shear deformation of the core. Considering the material gradients and following the similar procedure as that in Allen (1969), the wrinkling stress is written as:

[4.31]

where

[4.32]

Given the stiffness variation along the thickness, the wrinkling stress can be expressed by using the material degradation law [4.14]:

[4.33]

where E_0s and E_0c are the Young’s moduli of the skin and core at ambient temperature, respectively. The three temperatures, T₁, T₂ and T₃, are denoted in Fig. 4.3. This expression separates the contributions of temperature field, materials and dimensional lengths. For linear stiffness variations in the skin and the core, we have:

[4.34]

where R(T) = [(T_ref – T)/(T_ref – T_r)]^m for T ≤ T_ref; R(T) = 0 for T > T_ref. For constant temperature T₁ in the skin and constant temperature T₂ in the core, i.e. the upper bound temperature distribution, we have:

[4.35]

which is in a fairly simple form.

A thermal-mechanical simulation was performed to evaluate the wrinkling load in Gu and Asaro (2011). The exposed face is uniformly heated, where the temperature was linearly ramped up to the reference temperature 120 °C in 3600 seconds, and the unexposed face was kept at the room temperature. When the skin is above the reference temperature, it loses a significant portion of load bearing capability as shown by the degradation law in Section 4.3. So, the wrinkling load for a sandwich structure with thin skins in that temperature range may not be of interest. Figure 4.9 shows the wrinkling stresses normalized by the Young’s moduli at ambient temperature versus fire-heating time. The wrinkling stresses decrease as the fire-heating time increases. Examining the exact solution, at 3600 seconds the panel loses 33% of wrinkling load bearing capacity from that before heating. This shows again that the panel’s thermal performance considerably influences the mechanical load bearing capacity. The wrinkling stress obtained from the exact solution is larger than that from the Winkler model since the Winkler model does not consider the contribution of the shear stiffness of the core. When the heating time approaches zero, the two curves for the Winkler model approach each other since there are no thermal gradients before heating. The largest difference between the exact solution and the solution from the Winkler model with linear variation of material properties is about 18%.

4.6 Plastic micro-buckling

Micro-buckling which can lead to catastrophic failure is another failure mechanism of polymer matrix composite panels in compressive loading. A constitutive model, based on crystal plasticity theory (Asaro, 1983), was proposed for polymer matrix composites that experience both temperature rise resulting from fire exposure and strain rate dependence via interlaminar slip (Asaro et al., 2005; Ramroth et al. 2006). The constitutive model was implemented in finite element simulations to capture the plastic micro-buckling phenomena.

The kinematics of micromechanical deformation is illustrated in Fig. 4.10. The principal directions of the fibers are described by a set of mutually orthogonal unit base vectors, a₁ a₂ and a₃. The bold faced variables denote vectors or tensors. The elastic response of the composite laminates is formulated on these base vectors. The material also deforms via slipping in the plane of the laminates, i.e. interlaminar shear. We introduce two slip systems aligned with the slipping directions s₁ and s₂, where the normal of the interlaminar plane is m. These three vectors in the deformed state are called , and m*, respectively. The total deformation from the reference configuration B₀ consists of two portions: a viscoplastic deformation followed by an elastic deformation. The viscoplastic deformation occurs by the flow of the material through the lattice of the laminate via interlaminar shear. The spatial velocity gradient of the plastic flow is:

[4.36]

where F^p is the plastic deformation gradient, and is the rate of shearing on the α slip system. The value of F^p is obtained from the integration of [4.36] based on the incremental values of the associated variables. The viscoplastic deformation produces an intermediate configuration B_p. The deformed configuration B_c is reached by elastically distorting and rigidly rotating the laminate from B_p. The second portion of deformation is represented by the elastic deformation gradient F*, such that the total deformation gradient is:

[4.37]

The driving force for interlaminar shear is the resolved shear stress on the slip system, namely

[4.38]

where

[4.39]

In the above two expressions, τ = Jσ is the Kirchhoff stress, where σ is the Cauchy stress and J is the Jacobian of the deformation gradient. Here, and are along the α slip direction and slip plane normal, respectively, in the deformed configuration.

The evolution of the viscoplastic deformation is cast in terms of the relationship between the rate of shearing and the resolved shear stress on the slip system by:

[4.40]

where sgn is the sign function; n_p is strain rate sensitivity exponent; is the reference strain rate; and g_α is the slip system hardness, which varies with the shear strain of the slip system. A simple form for the slip system hardness is given by

[4.41]

where g₀, g_∞, h₀, and h_∞ are material constants.

Finally, the constitutive relation between the rate of Piola–Kirchhoff stress S and the rate of Green–Lagrange strain E is given by:

[4.42]

where L and X_α are related to the deformation gradient, the current stress, the slip system and the elasticity tensor.

An example to simulate micro-buckling progress using the constitutive model was given in Asaro et al. (2005). The specimen geometry was taken to be stubby so that the global buckling (Euler buckling) was precluded. The length of the specimen was 100 mm, the core thickness was 50 mm, and the skin thickness was 1.75 mm. The plane strain condition in the width direction was considered. The skin was initially perfectly bonded to the core through an adhesive thin layer of elements, which had the capability to simulate debonding when its failure criterion was reached. The degradation with respect to temperature rise for elastic material properties was modeled by the degradation law in Section 4.3; the degradation of the slip system hardness was also modeled similarly. The micromechanical constitutive model was used for skins with slip plane parallel to the surfaces of the skins in the reference configuration. The core was taken as isotropic elastic material. The micro-buckling was introduced by a small geometrical imperfection for the slip plane normal at the middle of the panel. The boundary condition was that one end of the specimen moved toward the other end along the axial direction with a velocity of 100 mm per second, whereas the other end was fixed in the axial direction. Figure 4.11 shows the load in the axial direction versus the displacement in the axial direction. The curve is linear up to the point where slip begins, as analyzed in the contour plots of plastic strain evolution at various representative times. After yielding, the micro-buckling or kink band mode develops, and the load drops dramatically. The plastic strain in the examined contour bands is between –5% and 5%. Figure 4.12 shows that after yielding and following the kink band formation, there is a reduction in the magnitude of the resolved shear stress on the slip system, which is similar for each node along the cross-section in the micro-buckled region.

The finite element program implemented through the micromechanically-based constitutive model is capable of simulating a variety of failure modes, including the structural failure modes of global buckling and skin wrinkling, in addition to the material failure mode of plastic micro-buckling. In Ramroth et al. (2006), the constitutive model was used to simulate the failure of the single-skin panel discussed in Section 4.4. The panel was slender, not stubby, so that it was inherently prone to global buckling. With the same thermal and mechanical loads used in the example in Section 4.4, the calculated transverse displacement versus time curve did match that obtained from experiment given in Figure 4.6. The micromechanical constitutive model predicted the structural collapse behavior similar to that observed in experiments and formulated by the global buckling model in Section 4.4. The global buckling failure mode was expected since for the slender geometry one would not expect significantly local plastic shear to occur. This was confirmed by examining the contour plots of plastic strain which showed only very modest amounts of viscoplastic deformation developed up to the point of imminent collapse. Continued deformation from the onset of global buckling may cause local finite viscoplastic deformation which ultimately results in plastic mirco-buckling. But, the precise sequences of such phenomena remain to be analyzed in future studies. By the comparison of the results of the slender panel with the stubby panel, Ramroth et al. (2006) were able to conclude that micro-buckling may not be the primary mechanism that contributes to the onset failure of the structural panel, i.e. the panel with large in-plane dimensions. Experiments in Asaro et al. (1999, 2009b) did not show the local failure mechanism in structural panels.

Viscoelastic material behavior can be included in micromechanically based constitutive models. The viscoelastic behavior of polymer matrix composites under fire was tested and modeled by Boyd et al. (2006). Micro-buckling was investigated by applying the Budiansky–Fleck kink band solution (Budiansky and Fleck, 1993) with time and temperature dependent shear modulus, where time–temperature superposition was employed for the viscoelastic property (Bausano et al., 2006; Boyd et al., 2007).

4.7 Other aspects of structural integrity in fire

In addition to those aspects discussed in the above sections, there are others to explore the failure mechanisms of polymer matrix composites in structural fire integrity.

4.8 References