effective modulus of elasticity;  Voigt-Reuss;  Hashin-Shtrikman;  Cap;  Lower bound

In the method of calculating the upper and lower limits of the effective modulus of elasticity of composites, the upper and lower upper and lower limit approximations of Voigt and Reuss and the upper and lower limits of Hashin and Shtrikman are more accurate, but the Voigt-Reuss approximation usually ignores the study of anisotropic heterogeneous bodies, which is an incomplete result. Hashin and Shtrikman failed to accurately use variational calculus to study extreme conditions for strain energy. For anisotropic heterogeneous bodies, the variational method and the principle of energy extremum are used to analyze a more accurate and reasonable upper and lower limit calculation formula than the upper and lower limit approximations of Voight-Reuss and Hashin-Shtrikman.