A nonlinear electro-elastic model with residual stresses and a preferred direction

Abstract

In this communication, a spectral model is developed for a residually stressed electro-elastic body with a preferred direction. The model uses a total energy function that depends on the right stretch tensor, the residual stress tensor, a preferred direction structural tensor and one of the electric variables. The proposed spectral invariants have a clear physical meaning; using these invariants, we prove that only [Formula: see text] of the [Formula: see text] classical invariants in the corresponding minimal integrity basis are independent. A method for exclusion or partial exclusion of compressed fibres is proposed. Some boundary value problems with cylindrical symmetry are studied. Results for the inflation of a hollow sphere, where the residual stress is assumed to depend only on the radial position, are also given. The spectral constitutive formulation is useful in a rigorous construction of a specific form of the total energy function via appropriate experiments.

Residually Stressed Fiber Reinforced Solids: A Spectral Approach

“…In this communication, our objective is to develop a novel strain energy function using spectral invariants that contains only single-variable functions. These strain energy types are experimentally attractive [19] and have been fortunate in modelling non-elastic and elastic solids [8][9][10][11][12][13][14][15][16]20,21]. It is important to note that knowing the number of independent invariants facilitates a stringent development of a strain energy via an experiment [22], and in our spectral analysis, we derive that only 11 independent spectral invariants are required in the constitutive equation.…”

“…A review on the presence of residual stress, for example, in fiber reinforced composite materials can be found in [7]. In this paper, 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]. Applications can be found, for example, in the mathematical modelling of the mechanical behaviour of carbon fibre reinforced solids and other types of RSPD, like soft biological tissue.…”

Anisotropic stress softening of residually stressed solids

“…to model the anisotropic mechanical Mullins behaviour. In view of this, following the work of Shariff and co-workers [4][5][6][7][8], we only use some of the 48 invariants, i.e., λ i , s…”

“…The presence of residual stresses in elastic solids has been the essence of numerous publications [1][2][3][4][5][6][7][8][9]. There is 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 (e.g.…”

“…The constitutive equation proposed here is characterized using a set of spectral invariants recently developed in the literature by Shariff and co-workers (e.g. [4][5][6][7][8][24][25][26]). The advantage of using spectral invariants to model the mechanical behaviour of materials is that they have a clear physical meaning, and hence they have an experimental advantage [24] over the standard classical invariants [27] commonly used in dealing with anisotropic problems.…”