By J.P. Boehler
Contents: J.P. Boehler: actual motivation.- J.P. Boehler: advent to the invariant formula of anisotropic constitutive equations.- J.P. Boehler: Representations for isotropic and anisotropic non-polynomial tensor functions.- J.P. Boehler: Anisotropic linear elasticity.- J.P. Boehler: Yielding and failure of transversely isotropic solids.- J.P. Boehler: On a rational formula of isotropic and anisotropic hardening.- J.P. Boehler: Anisotropic hardening of rolled sheet-steel.- A.J.M. Spencer: Isotropic polynomial invariants and tensor functions.- A.J.M. Spencer: Anisotropic invariants and extra effects for invariant and tensor representations.- A.J.M. Spencer: Kinematic constraints, constitutive equations and failure principles for anisotropic materials.- J. Betten: Invariants of fourth-order tensors.- J. Betten: formula of anisotropic constitutive equations.- J. Betten: Interpolation equipment for tensor functions.- J. Betten: Tensor functionality conception and classical plastic strength.
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Extra resources for Applications of Tensor Functions in Solid Mechanics
The values of the functions a.. are thus polynomial 1 scalar invariants of the arguments (1) under the group s. The representation (6) is irreducible if the integrity basis is irreducible and if none of the generators G. can be expressedas a linear combination of the others, -1 with coefficients a. 1 Explicit representations for isotropic and anisotropic polynomial scalar-valued and tensor-valued functions are presented in Chapters 8 and 9 by Professor SPENCER. 3 Representations for non-polynomial scalar and tensor functions In this Section, we suppose that the function f and the components of the function F are general functions, not necessarily polynomials, of the components, in a reference frame, of the arguments (1).
Now, for a given arbitrary ortho- -2 tropic distribution in the interval [0, TI/2], the distribution in the interval [0, 2TI] is obtained by the same symmetries. , arbitrary scalar-valued orthotropic 1 functions a(S) and b(8) can be constructed. Thus, from representation (25), the following structural tensor can be constructed: ~ - = a(S)M- + b(S)M -2 b(S)I + [a(S) - b(S)]M (41) which is the polar representation for an arbitrary orthotropic structural tensor. 7. CONCLUSIONS The theory of representations for tensor functions is a powerful and efficient tool for the invariant formulation of non-linear anisotropic constitutive equations.
E. the type and the minimal number of independent scalar variables which must appear in the constitutive law and which constitute the set of independent anisotropic mechanical variables. The theorems of representationsfor anisotropic tensor functions allow canonical forms which fulfil these two requirements to be developed. 2 Theorem of representation The theorem of representation for orthotropic tensor functions () indicates that the relation (22), tagether with the invariance condition (23), admits the following irreducible canonical form: T a.