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Multiplying decomposition of stress/strain, constitutive/compliance relations, and strain energy

To account for phenomenological theories and a set of invariants, stress and strain are usually decomposed into a pair of pressure and deviatoric stress and a pair of volumetric strain and deviatoric strain. However, the conventional decomposition method only focuses on individual stress and strain, so that cannot be directly applied to either formulation in Finite Element Method (FEM) or Boundary Element Method (BEM). In this paper, a simpler, more general, and widely applicable decomposition is suggested. A new decomposition method adopts multiplying decomposition tensors or matrices to not only stress and strain but also constitutive and compliance relation. With this, we also show its practical usage on FEM and BEM in terms of tensors and matrices.