Explicit formulations of tangent stiffness tensors for isotropic materials

A compact explicit expression for the tangent stiffness tensor is presented. Throughout the analysis, the formulation holds for general isotropic elastic materials and does not require solving eigenvector problems. On the theoretical side, a very simple solution of a tensor equation is obtained. Then the expressions for the derivatives of general symmetric isotropic tensor functions of a symmetric tensor are developed. On the computational side, particular attention is given to the consideration of the special case, Green elastic materials, in which the strain energy does not admit a closed-form expression in terms of principal invariants. Finally, a simple formulation of the tangent stiffness tensor for Ogden material model is supplied. Copyright © 2006 John Wiley & Sons, Ltd.

[1]  R. Hill,et al.  On constitutive inequalities for simple materials—I , 1968 .

[2]  Shen Zhu-jiang,et al.  Time Rates of Hill's Strain Tensors , 1999 .

[3]  Twirl tensors and the tensor equation AX−XA=C , 1992 .

[4]  Guansuo Dui,et al.  The Derivative of Isotropic Tensor Functions, Elastic Moduli and Stress Rate: I. Eigenvalue Formulation , 2004 .

[5]  Mikhail Itskov,et al.  On the theory of fourth-order tensors and their applications in computational mechanics , 2000 .

[6]  Christian Miehe,et al.  Comparison of two algorithms for the computation of fourth-order isotropic tensor functions , 1998 .

[7]  A. Saleeb,et al.  On the development of explicit robust schemes for implementation of a class of hyperelastic models in large-strain analysis of rubbers , 1992 .

[8]  R. M. Bowen,et al.  Acceleration waves in inhomogeneous isotropic elastic bodies , 1970 .

[9]  J. C. Simo,et al.  Quasi-incompressible finite elasticity in principal stretches. Continuum basis and numerical algorithms , 1991 .

[10]  R. W. Ogden,et al.  A theorem of tensor calculus and its application to isotropic elasticity , 1971 .

[11]  Donald E. Carlson,et al.  The derivative of a tensor-valued function of a tensor , 1986 .

[12]  E. A. de Souza Neto,et al.  On general isotropic tensor functions of one tensor , 2004 .

[13]  E. A. Repetto,et al.  The computation of the exponential and logarithmic mappings and their first and second linearizations , 2001 .

[14]  Guansuo Dui,et al.  Basis-free representations for the stress rate of isotopic materials , 2004 .

[15]  Christian Miehe,et al.  Computation of isotropic tensor functions , 1993 .

[16]  Yavuz Başar,et al.  Finite element formulation of the Ogden material model with application to rubber-like shells , 1998 .

[17]  A basis-free formula for time rate of Hill's strain tensors , 1993 .