Abstract
We use a spectral approach to model residually stressed elastic solids that can be applied to carbon fiber reinforced solids with a preferred direction; since the spectral formulation is more general than the classical-invariant formulation, it facilitates the search for an adequate constitutive equation for these solids. The constitutive equation is governed by spectral invariants, where each of them has a direct meaning, and are functions of the preferred direction, the residual stress tensor and the right stretch tensor. Invariants that have a transparent interpretation are useful in assisting the construction of a stringent experiment to seek a specific form of strain energy function. A separable nonlinear (finite strain) strain energy function containing single-variable functions is postulated and the associated infinitesimal strain energy function is straightforwardly obtained from its finite strain counterpart. We prove that only 11 invariants are independent. Some illustrative boundary value calculations are given. The proposed strain energy function can be simply transformed to admit the mechanical influence of compressed fibers to be partially or fully excluded.
Highlights
IntroductionThe presence of residual stresses in solids has been the essence of numerous publications [1,2,3]
The presence of residual stresses in solids has been the essence of numerous publications [1,2,3].There is a considerable interest in the mechanical behaviour of residually stressed materials in recent years and attempts to comprehend the mechanical behaviour of residual stresses on solid materials can be found in the literature [2,4,5,6]
To the best of our knowledge, since we are not able to find an appropriate residual stress experiment data of materials with a preferred direction, this paper focuses on the development of a rigourous theoretical spectral constitutive equation based on a systematic and rigorous use of the restrictions imposed by thermodynamics, the derivation and use of the representation formulae, a consequence of the rigorous definitions of the different material behaviors and of the concept of material symmetry, and a priori restrictions that are required by well posed mathematical models
Summary
The presence of residual stresses in solids has been the essence of numerous publications [1,2,3]. We focus on the modelling of the mechanical anisotropic response of a residually stressed fiber material with a preferred direction (RSPD) based on the spectral method (method that used the eigenvalues and eigenvectors of tensors) developed recently in the literature [8,9,10,11,12,13,14,15,16]. A discussion of the advantages of spectral invariants over classical invariants is given in [20] In this communication, our objective is to develop a novel strain energy function using spectral invariants that contains only single-variable functions.
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