Approximation of non-linear in-plain shear stress-strain diagrams of unidirectional and cross-ply reinforced polymer matrix composites

Approximation of the normalized in-plane shear stress-strain diagrams under the quasi-static loading of a variety of 25 unidirectional and cross-ply reinforced polymer matrix composites was studied. Approximation was carried out with the use of piecewise prescribed functions for two cases of deformation curve splitting into separate sections, i.e., into two and three parts. The first part is linear and corresponds to Hooke’s law. Description of the second and the third nonlinear parts required the development of a number of different functions relatively simple in structure with three or four independent parameters (coefficient) determined from boundary conditions in characteristic points of the in-plane shear stress-strain diagrams of polymer matrix composites. This approach provided using the minimum set of experimental characteristics of the material for approximation of the deformation curve. All the developed functions and their derivatives are continuous on all parts of the stress-strain diagrams. Assessment of the error of approximating functions was carried out by the criteria based on deviations between design and experimental values of the shear stress. The best functions having the smallest approximation error were determined for all considered materials both on average and separately for each material. It is shown that on average the error of approximation over three parts is 2.5 times less than that over two parts, however for some specific materials approximation over two parts appeared more accurate. The examples of approximation of in-plane shear stress-strain diagrams for three of 25 considered materials are presented for the cases of dividing the deformation curve into two or three parts. The approximating functions obtained can be recommended for use in modeling the stress-strain behavior of layered polymer matrix composites with allowance for nonlinearity of in-plane shear strain of the materials by taking into account the minimum required number of experimental characteristics.

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