Microstructure damage related to stress–strain curve for grain composites

Mathematical model of micro-heterogeneous medium with random deformation and strength properties of microstructure is developed assuming that the tensor of macroscopic deformations is known for the structure. Green–Somigliana tensor is used to obtain the formulas for random stress distribution in microstructure elements. The probability of the stress exceeding the ultimate strength in an element determines the probability of fracture in this element and the relative micro-damage. The correlation functions of stochastic microstructure ultimate strength condition are calculated for various types of stress. Normal distribution is used to calculate the damage. The distribution density can be adjusted through the stress moments to the fourth order.Micro-fractions change the composite’s macro modules of elasticity. Therefore, changes the relationship between stress and strain. Setting an increment step on the macro-strain axis, the stress–strain curve is plotted taking into account changes in composite properties. Stress–strain curves are obtained for different types of load.The increase of the factor of safety corresponds with the reduction of microstructure damage permitted in the design. Critical microstructure damage also depends on the dispersion of the microstructure properties. It is shown that the microstructure properties of composite significantly influence the behavior of materials under load and the shape of stress–strain curve. Findings are compared with experiment data.