Constitutive Relation Of Heterogeneous Materials With Effects Of Microstructures Under Kelvin Notation

Metal sheets are mostly polycrystals with orthogonal aggregates of cubic crydtallites.The microstructures(such as grain properties,crystalline orientation distribution,orientation relationship between grains,and grain sizes)in polycrystals determine the macroscopic properties of polycrystals.Fiber-reinforced composite materials are composed of material matrix and fibers.The direction distribution of fibers will affect the macroscopic constitutive relations of composite materials.Both polycrystalline materials and fiber composite materials belong to heterogeneous materials.The study of the constitutive relation between mechanical properties and material microstructure has theoretical and practical value.The Voigt model and the Reuss model are often used in the study of macroscopic constitutive relations of heterogeneous materials.In the Voigt model,it is assumed that any point in the polycrystal has the same strain,and the continuity of the contact force between grains can not be guaranteed.In the Reuss model,it is assumed that all grains in the polycrystal have the same stress,but the deformation continuity among crystallites cannot be satisfied.In this paper,the Kelvin convention is used to give the macroscopic constitutive expressions of polycrystalline materials under the Voigt model and the Reuss model,respectively,with the effect of the material microstructure.Using the Kelvin convention,the elastic stiffness tensor expression of cubic polycrystalline materials based on the self-consistent eigen-strain method is also given in this paper.Compared with the Voigt model and the Reuss model,the self-consistent method has a obvious advantage that the stress continuity and deformation compatibility between grains are satisfied in an average sense.Fiber-reinforced composite materials are widely used in aerospace industry and engineering due to their advantages of light weight and strong impact resistance.The macroscopic elastic constitutive tensor of fiber-reinforced composites is not only related to the material properties of the matrix and fibers but also to the volume fraction and fiber orientation distribution of fibers.Under on the Kelvin convention,using the eigenstrain self-consistent method,we investigate the relationship between the macroscopic constitutive and fiber orientation distribution of fiber-reinforced composites.We give an explicit expression for the effective elastic tensor of fiber-reinforced composites with the effect of fiber orientation distribution.We verify the explicit expression by finite element numerical results.By the physical method of face-centered cubic FCC grain plastic slip,assuming that all grains in the polycrystal have the same(instantaneous)plastic slip shear stress,the Hill criterion yield surface of the FCC grain is derived by the Schmid’s law and the nonlinear optimization theory.The plastic anisotropy tensor of the FCC grain with orientation R is given.Integrating over SO(3),we obtain the volume average of the yield surfaces of all grains in the face-centered cubic polycrystal and present the macroscopic yield function of the face-centered cubic FCC polycrystal in the mean sense.The relationship between the plastic parameters of the macroscopic yield function and the grain slip shear stress is given.The 4th-order constitutive tensors are often represented as the form of 6×6 matrices under the Voigt convention.In this article,the Kelvin subscripting convention is used to replace the Voigt subscripting convention.The matrix form of Kelvin’s convention has the rotation properties on tensors,while the matrix form of Voigt’s convention does not exist such a rotational relationship.The Kelvin’s matrix representation of the constitutive relation has obvious advantages.