Analytical Solution for Flexural Response of Epoxy Resin Materials

Stress-strain curves of polymeric materials are still a challenge for researchers. The difficulty of creating a constitutive stress-strain curve in polymeric material is mainly a result of the characterization of its mechanical behavior under different kinds of loading conditions. The hydrostatic component of stress has a significant effect on the load deformation response of resins, even at low levels of stress (Ward and Sweeny 2004). Hydrostatic stresses are known to affect the yield stress and nonlinear response of epoxy resin materials. To develop a general model for polymeric materials, the behavior of the polymeric materials under different types of loading conditions has to be understood.

Several constitutive models have been proposed for polymeric materials in the past three decades. The most successful models are proposed by groups at Oxford (Buckley and Jones 1995; Buckley and Dooling 2004), Massachusetts Institute of Technology (MIT) (Boyce et al. 1989; Boyce et al. 1994; Hasan and Boyce 1995; Mulliken and Boyce 2006), and Eindhoven (Tervoort et al. 1996; Tervoort et al. 1998; Govaert et al. 2000). Although these models differ in details, they all combine three-dimensional (3D) non-Newtonian viscoelastic flow and elastic strain softening. These models have been proposed in both large deformation and small deformation forms. Although these models are successful in fitting quasi-static in-plane test results, none of them are adequate for describing material response for out-of-plane loading. Wineman and Rajagopal (2000) used a viscoplasticity model to capture the behavior of polymers. Zhang and Moore (1997) and Gilat et al. (2007) modified the Bodner–Partom model, which was originally developed for metals to obtain the nonlinear uniaxial response of polymeric materials. By modifying the definitions of the effective stress and effective inelastic strain rate in the Drucker–Prager yield criteria, Li and Pan (1990), Chang and Pan (1997), and Hsu et al. (1999) developed an approach for the constitutive law of polymeric materials. Jordan et al. (2008) modified the original 3D model of Mulliken and Boyce (2006) for one dimension to capture the compressive mechanical properties of polymeric composites. Lu et al. (2001) used the constitutive model developed by Hasan and Boyce (1995) to simulate the experimental results on the uniaxial compressive stress-strain behavior of Epon E 828/T-403. Chen et al. (1998) modeled the uniaxial compressive response of Epon E 828/T-403 by using the Johnson–Cook model (Johnson 1983). They simulated the experimental compression response up to 10% of true strain but reported experimental stress-strain curves showing elastic deformation, a yieldlike peak, and a strain softening region up to approximately 35%. The majority of the parameters were determined by fitting the model to experimental tensile and compressive data. Naaman and Reinhardt (2006) used piecewise linear stress-strain and stress-crack opening approaches to characterize the mechanical behavior of high-performance fiber-reinforced cement composites. Soranakom et al. (2008) used piecewise linear stress-strain curves in tension and an elastic perfectly plastic model in compression to study the flexural behavior of cement-based composite materials. Hobbiebrunken et al. (2007) and Goodier (1993) studied the correlation between the presence of defects (voids and microcracks) and the volume under stress in epoxy resin glassy polymers. The crack initiation by void nucleation or a preexisting flaw in epoxy resins was observed, and the dependency of the failure behavior and strength on the size effect, stress state, and the volume of the body subjected to stress was studied (Hobbiebrunken et al. 2007, Bazant and Chen 1997; Odom and Adam 1992). Flexural strength distributions and the ratio of flexural strength to tension strength of epoxy resin and polymethyl methacrylate (PMMA) materials were studied using the Weibull model (Giannotti et al. 2003; Vallo 2002). Giannotti et al. (2003) used a modified two-parameter Weibull model to compare the effect of loading systems on the mean stress in polymeric materials and observed that it predicts a mean flexural strength up to 40% higher than the mean tensile strength for a Weibull modulus greater than 14.

This study is motivated by the need to better characterize the flexural behavior of epoxy resin materials. Closed-form solutions for nonlinear moment curvature responses were derived on the basis of nonlinear tension and compression stress-strain curves. The results were expressed in normalized form to eliminate the effects of size and strength of the specimen. A technique based on the uniaxial tension and compression stress-strain relations, strain compatibility in bending, static equilibrium, and softening localization was used to simulate flexural load-deflection response in a statically determinate structure. Because the solution is derived explicitly, iterative procedures required for solving the governing equations of material behavior are not required; hence, this method is extremely powerful for forward solution and inverse analysis. The effects of different segments of tension and compression stress-strain curves for improving the flexural performance of epoxy resin materials were studied. The purpose of this study is two-fold: (1) To correlate the uniaxial tension and compression material response with flexural behavior in epoxy resins; and (2) to evaluate the effects of different segments of tension and compression stress-strain curve on flexural response.

Epoxy resin materials share some similarities. Although the compressive and tensile moduli are approximately equal, the first point showing deviation from linearity in the stress-strain curve in tension is weaker than the one in the compression stress-strain curve (Ward and Sweeney 2004). It is critically important to observe that the general shapes of the stress-strain curves in tension and compression in epoxy resin materials are similar because they represent initial linear behavior, followed by an ascending curve with reduced stiffness in the prepeak region and strain softening response in the postpeak region (G’Sell and Souahi 1997; Boyce and Arruda 1990; Buckley and Harding 2001; Shah Khan et al. 2001; Jordan et al. 2008; Littell et al. 2008; Chen et al. 2001). In general, epoxy resin materials exhibit the following distinct behavior in the tension and compression stress-strain behavior: linearly elastic, nonlinearly ascending, yieldlike (peak) behavior, strain softening, and nearly perfect plastic flow.

Fig. 1 shows the complete tension and compression stress-strain curves. The two parameters characterizing the tensile response in the prepeak region are the proportionality elastic limit (PEL) and ultimate tensile strength (UTS). The postpeak region in the tension model is expressed with slope of softening ($E_{\mathrm{soft},t}$), plastic flow (${\sigma }_f$), and the ultimate strain (${\epsilon }_{Ut}$). Yield stress is often assumed to be equal to the first peak stress in the stress-strain curve. The prepeak region in compression is characterized by the PEL in compression and the compressive yield stress (CYS). The postpeak response in compression is determined by the slope of softening ($E_{\mathrm{soft},c}$), compression plastic flow (${\sigma }_{f,c}$), and the compressive ultimate strain (${\epsilon }_{Uc}$). The tension and compression strain stress models are defined in Tables 1 and 2, respectively.

The complete tension and compression stress-strain curves are defined uniquely by two material parameters and 12 normalized parameters: the modulus of elasticity in tension ($E$) and strain at the tensile proportionality elastic limit (${\epsilon }_{\mathrm{PEL}}$) and ${\mu }_{t1}$, ${\mu }_{t2}$, ${\mu }_{Ut}$, ${\mu }_{co}$, ${\mu }_{c1}$, ${\mu }_{c2}$, ${\mu }_{Uc}$, $\alpha$, $\eta$, $\gamma$, $\beta$, and $\xi$. The tensile and compressive stresses at the PEL point are related empirically to the stresses at the UTS and CYS points. Elastic moduli in tension and compression are practically identical (Foreman et al. 2010). However, bimodulus material constants ($\gamma \ne 1$) are considered in tension and compression. Eqs. (1), (2), and (3) show the definitions of the normalized parameters

Strain compatibility in bending is considered to derive moment curvature relationship for a rectangular cross section with the width of $b$ and the depth of $h$. Using the stress-strain relationships in Fig. 1 and the known applied compressive strain at the top fiber ($\lambda {\epsilon }_{\mathrm{PEL}}$), 16 different cases of strain and stress distributions are shown in Fig. 2. The development of the stress-strain relationship across a cross section and the possibilities of tension or compression failures are presented in Fig. 3. In this approach, moving through different stages depends on the transition points ($t{p}_{ij}$), which are functions of material parameters, as shown in Eq. (4). Indexes $i$ and $j$ refer to the origin and destination stages, respectively. Stress strain develops at least to Stage 4, in which compressive and tensile failure is possible if ${\lambda }_{\max }={\mu }_{Uc}$ in Case 10 or ${\lambda }_{\max }=J$ in Case 9

Characteristic points $A–P$ are calculated as functions of material parameters to satisfy the following relationship at each load step:where ${\epsilon }_t$ = tensile strain at the bottom fiber; $\Omega$ = either: 1, ${\mu }_{t1}$, ${\mu }_{t2}$, or ${\mu }_{Ut}$ depending on the case of stress distribution; and ${\epsilon }_t$ is expressed as a linear function of the applied compressive strain at the top fiber (${\epsilon }_c$) as follows:where ${\epsilon }_c$ = $\lambda$ ${\epsilon }_{\mathrm{PEL}}$; and $\kappa$ = depth of the neutral axis, which is a function of material parameters. Characteristic points $A$ and $B$ are presented in Eq. (7) as an example. As the applied strain parameter $\lambda$ is incrementally imposed, the strain and stress distribution is determined, and the internal tension and compression forces are computed. For instance, the internal forces for the tension and compression subzones for Case 16 [Fig. 2(q)], normalized to the tension force at the PEL point ($bhE{\epsilon }_{\mathrm{PEL}}$), are as shown in Eqs. (8), (9), (10), (11), (12), (13), (14), and (15)

Net force is calculated as the difference between the tension and compression forces for each case. By applying internal equilibrium, the value of $\kappa$ is obtained. The expressions of net force in some stages result in more than one solution for $\kappa$. For an isotropic material, the first $\kappa$ value is 0.5 because the neutral axis coincides with the centroid of the rectangular section. Because the neutral axis changes incrementally, the next value of $\kappa$ is the closest to the previous neutral axis. Using a large amount of numerical tests for possible ranges of material parameters, the correct expression for $\kappa$, which yields a valid value $0<\kappa <1$, is determined. For instance, the $\kappa$ for Case 16 [Fig. 2(q)] is as follows:

Moment expressions are obtained by taking the first moment of the compression and tension forces about the neutral axis. Curvature is calculated by dividing the top compressive strain by the depth of the neutral axis $\kappa h$. The general equations for normalized moment and curvature arewhere $M_{\mathrm{PEL}}$ and ${\phi }_{\mathrm{PEL}}$ = moment and curvature (for $\gamma =1$) at the tensile PEL and are defined in Eq. (20). The normalized moment for Case 16 [Fig. 2(q)] is defined as

The closed form solutions for the location of neutral axis ${\kappa }_i$ and normalized moment $M_i^{\prime }$ for all the cases are presented in Appendix I and Appendix II. The normalized ultimate moment for a resin-like material at very large $\lambda$ values ($M^{\prime }\infty$) is computed by substituting $\lambda =\infty$ in the expression for depth of neutral axis in Case 16 in Eq. (16) and by substitution of $\lambda =\infty$ and $\kappa \infty$ in the normalized moment expression in Eq. (21). Eq. (22) presents the value of $\kappa$ for very large $\lambda$ values. As is logically expected, the numerator is a function of material parameters in tension, whereas the denominator is a function of both tension and compression parameters. The normalized ultimate moment is obtained as a function of tension and compression material parameters as follows:

Eq. (19) clearly shows that normalized curvature would be a very large number for very large $\lambda$ values. For elastic perfectly plastic materials with equal tensile and compressive elastic moduli and equal yield stress and strain ($\eta =\xi =0$, $\alpha =\gamma =\beta =1$, ${\mu }_{t1}={\mu }_{c1}=1$), Eqs. (22) and (23) yield 0.5 and 1.5, respectively. This means that the plastic moment capacity is 1.5 times its elastic yield strength for a rectangular cross section (Gere 2001).

A set of analytical parametric studies based on developed closed form solutions for moment curvature response is presented. Although polymeric materials show strain-softening behavior with a percentage of the UTS, a complete set of parametric studies is conducted to examine the effect of postpeak behavior on flexural response. The base set of parameters was defined through curve fitting to represent the material behavior of Epon E 862 studied by (Littell et al. 2008): $E=2,069\mathrm{MPa}$, $Ec=2,457\mathrm{MPa}$, ${\epsilon }_{\mathrm{PEL}}=0.0205$, ${\epsilon }_{\mathrm{Uts}}=0.076$, ${\epsilon }_{St}=0.16$, ${\epsilon }_{Ut}=0.24$, ${\epsilon }_{\mathrm{PEL},c}=0.019$, ${\epsilon }_{\mathrm{CYS}}=0.092$, ${\epsilon }_{Sc}=0.15$, ${\epsilon }_{Uc}=0.35$, ${\sigma }_{\mathrm{Uts}}=70\mathrm{MPa}$, ${\sigma }_f=60.5\mathrm{MPa}$, ${\sigma }_{\mathrm{CYS}}=93\mathrm{MPa}$, and ${\sigma }_{f,c}=87\mathrm{MPa}$.

Fig. 4 shows the effect of tensile flow stress on the moment curvature and the location of the neutral axis; $\eta =0.3$ and $\eta =0.001$ correspond to tensile plastic flow equal to 25% and almost 100% of the UTS, respectively. Fig. 4 shows that the moment curvature response is extremely sensitive to the variations in constant tensile flow as the location of maximum flexure and the postpeak regime completely changes with changing tensile plastic flow stress. For the parameters given, Eq. (23) yields to $\eta =0.306$ for $M^{\prime }=1$; values of $\eta >0.306$ leads to moment capacity at failure less than the elastic moment capacity at the PEL. To obtain the bending moment of large top compressive strains equal to or greater than the elastic bending capacity, the required tensile plastic flow should be equal to or greater than 25% of the UTS. The material behavior of Epon E 862, for which Eq. (23) indicates $M^{\prime }=2.55$ at the ultimate point, is exactly characterized by $\eta =0.05$. Fig. 4 also shows that decreasing the level of tensile flow decreases the neutral axis depth, especially for $\eta$ values greater than 0.2. A value of $\eta =0.2$ corresponds to a tensile plastic flow stress equal to 50% of ${\sigma }_{\mathrm{UTS}}$. It is observed that the strain-softening region of tensile response contributes to the flexural load-carrying capacity and nonlinear energy dissipation when subjected to the flexural stress.

Fig. 5 shows the effect of different values of ${\sigma }_{\mathrm{UTS}}$ at constant ${\epsilon }_{\mathrm{UTS}}$ on the moment curvature and neutral axis location. Because the flow stress in tension is constant, the post PEL and the softening slopes are calculated for different ${\sigma }_{\mathrm{UTS}}$ values. The strength gain is almost proportional to ${\sigma }_{\mathrm{UTS}}$; there is almost no change in stiffness, whereas ductility slightly increases. However, the amount of $M_{\infty }^{\prime }$ is not affected as much as the flexural strength because for cases $\alpha =0.4$, $\eta =0.16$, and $\alpha =0.5$, $\eta =0.226$ the moment at infinity is less than the flexural strengths. Fig. 5 illustrates that by increasing the UTS, the neutral axis moves downward and exceeds $\kappa =0.5$ for the case of $\alpha =0.5$, $\eta =0.226$. Fig. 6 shows the effects of different post-PEL slopes, strain at the UTS point, and softening slopes with constant ${\sigma }_{\mathrm{UTS}}$ on flexural response. Results show that changes in the location of the UTS point with a constant value slightly change the moment curvature response. It is observed that the location of the UTS point, for a wide range of normalized top compressive strains between 1 and 4, changes the location of the neutral axis and stress distributions. Fig. 7 illustrates the effect of compressive plastic flow on the moment curvature and location of the neutral axis. Because the epoxy resin Epon E 862 is stronger in compression than tension, changes in compressive plastic flow do not change the moment capacity but affect the moment at failure considerably. It illustrates that a decrease of compression plastic flow increases the neutral axis depth for top compressive strains greater than 0.103. Fig. 8 shows the effects of ${\sigma }_{\mathrm{CYS}}$ values at constant strain. Like tension, an increase of peak strength in compression at constant strain increases the flexural capacity of the epoxy resin. It is observed that a change in ${\sigma }_{\mathrm{CYS}}$ values at constant strain affects moment at failure less than flexural capacity. Results show that an increase of compression peak stress decreases the neutral axis depth considerably.

Researchers have observed different compression behavior in postpeak responses for epoxy resins with different specimen shapes and dimensions. Strain softening at yield, followed by strain stiffening at higher strains in compression for different low and high strain rates has been reported (Littell et al. 2008; Jordan et al. 2008; Fiedler et al. 2001; Behzadi and Jones 2005; G’Sell and Souahi 1997; Boyce and Arruda 1990; Buckley and Harding 2001). However, Shah Khan et al. (2001) and Chen et al. (2001) did not observe any strain stiffening at high strains. Fig. 9 illustrates the effect of tension and compression behavior at high strains at stress development at a point of material for Epon E 862 under flexural loading. Tensile failure is the governing mechanism for all cases. Materials with $\eta \ge 0.2$ do not experience compression plastic flow, and their stress-strain relationship in the compression side always is in the ascending region and/or first part of the softening regime. This is the reason that their neutral axis depth and moment capacity drops sharply by increasing the top compressive strain. Results show that the shape of the stress-strain curve for high strain values in compression do not influence the flexural response of materials in which compression is stronger than tension.

The load-deflection response is obtained by using the nonlinear moment curvature response, static equilibrium, and the softening localization concept. In displacement control, the normalized top compressive strain is incrementally imposed to generate a stress distribution profile in a given cross section. For resins, if the compressive strength is greater than the tensile strength, the shape of the moment curvature diagram greatly depends on the value of the postpeak tensile stress, as observed in the parametric study. Fig. 10 shows a typical nonlinear moment curvature diagram for epoxy resins, consisting of a linear elastic part from 0 to $M_{\mathrm{LOP}}$, followed by an ascending curve with reduced stiffness from $M_{\mathrm{LOP}}$ to $M_{\max }$ in the prepeak region and a descending curve from $M_{\max }$ to $M_{\mathrm{failure}}$ in the postpeak region.

The first deviation from linearity in a moment curvature or load-deflection curve is called limit of proportionality (LOP), and the first peak moment or load is called the modulus of rupture (MOR), as determined in Fig. 10. When a beam is loaded beyond MOR in a material with strain-softening behavior, the increase of the deformation decreases load. Polymeric materials are characterized by the existence of a fracture process zone, with distributed cracking damage (Bazant and Chen 1997). Fig. 11 shows a three-point bending (3PB) with softening localization in the cracking region at the vicinity of the load at the groove (Region 2), whereas other zones outside the groove (Region 1) undergo unloading during softening. Static equilibrium is used to obtain a series of load steps in the 3PB setup from the moment curvature diagram. For each load step, the moment diagram along the length of the structure is calculated, and the corresponding curvature is obtained from the moment curvature relationship. The deflection at the midspan is calculated using the moment-area method for curvature points at each load step. This procedure is repeated for the number of load steps until a complete load-deflection response is obtained.

The specimen is loaded from 0 to $P_{\mathrm{LOP}}$ in the ascending portion of the moment curvature diagram from 0 to $M_{\mathrm{LOP}}$. The curvature for this portion is determined directly from the moment-curvature diagram. Beyond the LOP, as the specimen undergoes softening, the curvature distribution depends on the localized or nonlocalized zones and prior strain history. The strain and curvature unloads elastically for an undamaged section. If the section is loaded beyond $M_{\mathrm{LOP}}$, the unloading curvature of the damaged section follows a recovery path, as shown in Fig. 10 and as observed by Littell et al. (2008) in the cyclic loading. However, because analytical simulation and the experiment are done for 3PB under displacement control, there is no recovery in the unloading path. For sections located in the localized zone, the unloading curvature is determined from the postpeak response of the moment-curvature diagram from $M_{\max }$ to $M_{\mathrm{failure}}$. The main steps to calculate load-deflection response are summarized as follows:

Tension, compression, and 3PB tests were conducted on epoxy resin Epon E 863 with a hardener (EPI-CURE 3290) by using a $100/27$ weight ratio at room temperature. A digital image correlation technique, ARAMIS (2006), was used to study the strain fields. Dog bone samples with a 14-mm gauge length and an averaged rectangular cross section of $3.18\times 3.43\mathrm{mm}$ were selected to conduct the monotonic tensile tests. Small cubic samples (side = 4 mm) were tested under monotonic compression. Small beams with 4-mm average width, 10-mm thickness, and 60-mm length (50-mm span), with a groove (radius of approximately 3.5 mm) in the middle of the beam were selected to conduct 3PB tests. Sections 1 and 2 in Fig. 12 show that the length of deformation localization at $493\mu \mathrm{str}/\mathrm{s}$ and $59\mu \mathrm{str}/\mathrm{s}$ in the softening stage, obtained from strain field analysis, are 5 and 4.5 mm, respectively. Figs. 13(a), 13(b), 14(a), and 14(b) illustrate the representative experimental tension and compression true stress-strain curves at $493\mu \mathrm{str}/\mathrm{s}$ and $59\mu \mathrm{str}/\mathrm{s}$. Because the compression stress-strain curve was not available at $59\mu \mathrm{str}/\mathrm{s}$, it was built on the basis of the linear relationship between the mechanical properties and the logarithm of the strain rate. Experimental results show a strain-softening behavior beyond the peak point followed by a constant plateau before failure. A simulation was made to study the load-deflection response of Epon E 863 and to evaluate the effects of out-of-plane loading.

The two main parameters and 12 nondimensional parameters for the models at $493\mu \mathrm{str}/\mathrm{s}$ and $59\mu \mathrm{str}/\mathrm{s}$ are $E=3,049\mathrm{MPa}$, ${\epsilon }_{\mathrm{PEL}}=0.0162$, ${\mu }_{c0}=1.148$, ${\mu }_{c1}=3.52$, ${\mu }_{c2}=6.79$, ${\mu }_{Uc}=15.70$, ${\mu }_{t1}=2.55$, ${\mu }_{t2}=8.64$, ${\mu }_{Ut}=20.98$, $\gamma =1.09$, $\alpha =0.395$, $\beta =0.298$, $\eta =-0.0385$, and $\xi =-0.117$ for $493\mu \mathrm{str}/\mathrm{s}$ and $E=2,877\mathrm{MPa}$, ${\epsilon }_{\mathrm{PEL}}=0.0154$, ${\mu }_{c0}=1.331$, ${\mu }_{c1}=3.896$, ${\mu }_{c2}=6.79$, ${\mu }_{Uc}=19.48$, ${\mu }_{t1}=2.753$, ${\mu }_{t2}=8.05$, ${\mu }_{Ut}=19.87$, $\gamma =0.83$, $\alpha =0.33$, $\beta =0.285$, $\eta =-0.0352$, and $\xi =-0.122$ for $59\mu \mathrm{str}/\mathrm{s}$. Figs. 13(c) and 14(c) show the 3PB load-deflection curve compared with the simulation results. The figures illustrate that the tension and compression stress-strain curves underestimate the load-deflection response as a result of the difference between stress distribution profile in the uniaxial tests and the bending test. In tension and compression tests, the entire volume of the sample is subjected to the same load and has the same probability of failure. However, in a bending test, only a small fraction of the tension and compression regions are subjected to the maximum peak stress. Therefore, the probability of crack nucleation, propagation, and failure development in tension and compression samples is higher than in bending samples. Results of the parametric study show that simulation of the flexural response can be improved by changing the ultimate tensile and compressive level and by further adjustments of the other parameters. To quantify these effects, and on the basis of the results of the parametric study, one scaling factor ($C_1$) is proposed to modify the strength of the material.

Imperfections in the material directly affect $C_1$. However, an inverse analysis of the load-deflection response showed that the $C_1$ values for Epon E 863 for $493\mu \mathrm{str}/\mathrm{s}$ and $59\mu \mathrm{str}/\mathrm{s}$ are around 1.14 and 1.24, respectively. The authors are currently determining the ratio of flexural strength to tension and compression peak strength for PR-520. More studies at different strain rates and on different epoxy resin materials need to be done before an average flexural overstrength factor can be recommended. An inverse analysis approach of flexural results will establish a powerful relationship between the compression, tension stress-strain curves, and the flexural response.

Explicit moment curvature equations using nonlinear tension and compression stress-strain relationships for epoxy resin materials have been developed. A piecewise linear stress-strain relationship for epoxy resin materials, consisting of strain softening and flow stress in tension and compression, has been used. The material model is described by two intrinsic material parameters (the tensile modulus of elasticity and tensile strain at the PEL point) in addition to five nondimensional parameters for tension and seven nondimensional parameters for compression. A parametric study showed that the moment curvature response is primarily controlled by the postpeak tensile and compressive strengths, ${\sigma }_{\mathrm{UTS}}$ and ${\sigma }_{\mathrm{CYS}}$. It was concluded that compression stress-strain parameters have less of an effect on flexural behavior than tension parameters, as long as the compression strength is higher than the tension strength. For materials with small postpeak tensile strength values, the moment at failure is much lower than the moment carrying capacity, and the response terminates at a relatively low compressive strain. Materials with higher normalized postpeak tensile strength have a gradual reduction in the height of the compressive zone; therefore, larger deformations are possible. Epoxy resin materials with a considerable amount of postpeak tensile strength have a moment capacity around 2.5 times the moment at the PEL point. An increase of ${\sigma }_{\mathrm{CYS}}$ by increasing the postcompressive PEL stiffness at high CYS values, marginally affects the moment capacity in polymeric materials. It is observed that the flexural response in polymeric materials that are stronger in compression than in tension is independent of the shape of the compression stress-strain curve at high strain values. Simulation of the load-deflection response of epoxy resins in 3PB test revealed the effect of the stress gradient on the material behavior. Results indicate that the direct use of tension and compression data underestimates the flexural strength. By applying a scaling factor ($C_1$) to uniaxial tension and compression strength in a stress-strain curve, the flexural behavior of epoxy resins was predicted accurately.

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