Bounds on the distribution of extreme values for the stress in composite materials

Suitable macroscopic quantities beyond effective elastic properties are used to assess the distribution of stress within a composite. The composite is composed of N anisotropic linearly elastic materials and the length scale of the microstructure relative to the loading is denoted by ε. The stress distribution function inside the composite λε(t) gives the volume of the set where the norm of the stress exceeds the value t. The analysis focuses on the case when 0 < ε << 1. A rigorous upper bound on limε→0λ(t) is found. The bound is given in terms of a macroscopic quantity called the macro stress modulation function. It is used to provide a rigorous assessment of the volume of over stressed regions near stress concentrators generated by reentrant corners or by an abrupt change of boundary loading.