Prediction of Stress-Strain Curves for the Three Types of Laminates

Figures 5 and 6 display the predicted global stress-strain curves of the three types of laminates and the comparison with the test results. All the calculations use the same set of material constants and the same set of constants used in the post-damage constitutive model. The test specimens were made of “Scotchply 1003” prepregs of 3M Company. The test data of cross-ply laminates were taken from the technical data of the 3M Company and that of the [±45°]ns angle-ply laminates and unidirectional laminates (UDC) under 45° off-axis loading were from Ellyin and Kujawski (1995). It is seen that the predictions of the drastically different responses of the three types of laminates are in good agreement with the experimental data.

In the case of the unidirectional laminate, it is noted that the predicted trend is in good agreement with the test results, Fig. 5. The predicted initial stiffness is 10.1 GPa, and the maximum load is 68 MPa, while the corresponding test results are 10.1 GPa and 82 MPa.

The effect of viscoelastic behavior of the matrix is manifested by the nonlinearity of the stress-strain curve, which is noticeable once the stress exceeds 40 MPa (about 0.5% strain). Since damage has not yet occurred at this load level (for the unidirectional laminate, damage initiates at the peak of the stress-strain curve), therefore this nonlinearity is mainly caused by the viscoelasticity of the epoxy matrix.

For the cross-ply and angle-ply laminates the test and predicted results are shown in Figure 6. For the cross-ply laminate, a bilinear stress-strain curve is predicted in which the two ‘moduli’ are approximately 25.5 GPa and 17.4 GPa. The corresponding test values are 25.5 GPa and 15.6 GPa, respectively. The knee between the two straight lines corresponds to the load level at which the transverse cracking of matrix occurs in the laminate. And the final failure of the specimen is due to the fracture of fibers in the 0° plies. From the technical data of the ‘Scotchply’, the test value of tensile strength of cross-ply laminate is 480 MPa, and the present prediction of 470 MPa is very close to that of the test. Note, however, that for the other two laminates, no fiber fracture occurs within the strain range of the present calculations.

The unique nonlinear stress-strain curve of the [±45°]ns laminate is also well predicted. For example, the predicted initial stiffness of 10.1 GPa agreed very well with the test value of 10.1 GPa. In contrast to the unidirectional laminate under 45° off-axis loading which failed at a relatively low global strain of 0.9%, at the same strain level, the [±45°]ns laminate is capable of carrying the applied load albeit at a reduced stiffness. However, prior to the ‘yield’ point, the stress-strain curve also manifested a nonlinear response. Since the damage has not yet occurred at this load level (50 MPa), this nonlinearity is mainly caused by the viscoelasticity of the epoxy matrix. Note that the nonlinearity of the stress-strain curve has different causes at different strain levels: at lower strain levels, it is mainly caused by the viscoelasticity of the matrix, while at higher strain values, the onset of damage and its evolution is the main contributor to the nonlinear response. Finally, it should be noted that the simulation is carried out up to about 2.3% applied global strain. Thereafter, it is difficult to continue the simulation, since the local deformation is very large and the present FEM model is based on the small deformation formulation.


To perform an effective micro/meso-mechanical analysis of composite materials and to obtain reliable predictions, the following three prerequisites are essential:

(1) Correct periodic boundary conditions must be applied to the repeated unit cells (RUCs).

• A unified form of periodic boundary conditions for RUCs of composites has been presented.

• Application of the proposed periodic boundary conditions can guarantee both the displacement and traction continuity. The solution is also independent on the choice of RUCs.

(2) The constitutive models must accurately represent the constituents’ behaviours.

• A nonlinear viscoelastic constitutive model has been developed for epoxy polymers.

• The model is capable of predicting complicated time – and loading-history-dependent response of the polymer matrix materials.

(3) Proper damage criteria and post-damage constitutive model must be included in the analyses to simulate different damage modes in composite materials.

• A post-damage constitutive model based on smeared crack concept has been developed.

• Initiation and propagation of matrix cracking in composites are well simulated by using this post-damage constitutive model.


The Work presented here is supported, in part, by grants to F. E and Z. X. from the Natural Sciences and Engineering Research Council (NSERC) of Canada.


Aboudi, J., (1991). Mechanics of Composite Materials, A Unified Micromechanical Approach. Elsevier Science Publishers, Amsterdam.

Ellyin, F., Kujawski, D. (1995). Tensile and fatigue behaviour of glass fibre/epoxy laminates. Construction and Building Materials, 9, 425-430.

Hashin, Z. (1966). Viscoelastic fiber reinforced materials. AIAA Journal, 4, 1141-1147.

Hu, Y., Xia, Z, Ellyin, F. (2003). Deformation behavior of an epoxy resin subjected to multiaxial loadings, part I: experimental investigations. Polymer Engineering & Science, 43, 721-733. Nemat-Nasser S., Hori M. (1993). Micromechanics: Overall Properties of Heterogeneous Materials. Elsevier Science Publishers, Amsterdam.

Shen, X., Xia, Z., Ellyin, F. (2004). Cyclic deformation behavior of an epoxy polymer, part I: experimental investigation. Polymer Engineering & Science, 44, 2240-2246.

Sun, C. T., Vaidya, R. S. (1996). Prediction of composite properties from a representative volume element. Composites Science and Technology, 56, 171-179.

Suquet, P. (1987). Elements of homogenization theory for inelastic solid mechanics. In: Sanchez- Palencia, E., Zaoui, A., (eds.). Homogenization Techniques for Composite Media. Springer – Verlag, Berlin, 194-275.

Xia, Z., Zhang, Y., Ellyin, F. (2003a). A unified periodical boundary condition for representative volume elements of composites and applications. International Journal of Solids and Structures, 40, 1907-1921.

Xia, Z, Hu, Y., Ellyin, F. (2003b). Deformation behavior of an epoxy resin subjected to multiaxial loadings, part II: constitutive modeling and predictions. Polymer Engineering & Science, 43, 734­748.

Xia, Z., Shen, X., Ellyin, F. (2005). Cyclic deformation behavior of an epoxy polymer, part II: prediction of constitutive model. Polymer Engineering & Science, 45, 103-113.

Xia, Z., Zhou, C., Yong. Q., Wang, X. (2006). On selection of repeated unit cell model and application of unified periodic boundary conditions in micro-mechanical analysis of Composites. International Journal of Solids and Structures, 43, 266-278.

Yuan, F. G., Pagano, N. J., Cai, X. (1997). Elastic moduli of brittle matrix composites with interfacial debonding. International Journal of Solids and Structures, 34, 177-201.

Zhang, Y., Xia, Z., Ellyin, F. (2005). Viscoelastic and damage analyses of fiber reinforced polymer laminates by micro/meso-mechanical modeling. Journal of Composite Materials, 39, 2001-2022.

3M Minnesota Mining & Manufacturing CO., Technical Data of ‘Scotchply’ Reinforced Plastic Type 1002 & 1003, St. Paul, Minnesota, USA.