Abstract
The thermodynamic constitutive theory described in the preceding chapter is applied to inelastic materials with isotropic damage. In Section 4.1, one-dimensional elastic-plastic and elastic-viscoplastic constitutive equations of damaged materials will be described as the basis for the succeeding sections. The application of the constitutive theory of Chapter 3 to the three-dimensional case will be discussed in Section 4.2. The strain energy release rate due to damage development and the stress criterion for elastic-plastic damage growth will be considered in Section 4.3, while Section 4.4 is concerned with the inelastic damage theories based on the hypothesis of mechanical equivalence.
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Notes
- 1.
Time-dependence (or rate-dependence) of plastic deformation is observed markedly not only in the quasi-static case due to elevated temperature or due to high level of stress discussed here, but also in the case of high rate of strain due to dynamic loading. Viscoplastic deformation at high rate of strain and its constitutive equations are referred to the excellent reference of Perzyna (1966).
- 2.
Besides these internal variables, another scalar internal variable β, called damage-strengthening variable may be introduced to represent the effect of damage history on its further evolution. As regards the detail of the procedure, see Chapter 5.
- 3.
- 4.
As regards the specific expressions of the Helmholtz free energy function ψ of Eq. (4.30) and the dissipation potential F of Eq. (4.44) later, especially those of the coupled effect between strain-hardening and damage, refer to Section 3.3, and Besson et al. (2010).
- 5.
As will be described in Section 4.3.1 later, the generalized force Y associated with the damage variable D represents the release rate of elastic strain energy per unit volume caused by the damage development. Hereafter the generalized force Y will be called the damage-associated variable, or the strain energy density release rate.
- 6.
In the plastic potential of this expression, the center of the elastic region in stress space deviates from the point A as damage develops, except the case of the isotropic damage. In order to avoid this aspect, another plastic potential function
$$F^P = {\left(\tilde \sigma - \tilde A\right)}_{EQ} - R - \sigma _Y $$((4.48))derived by replacing the kinematic variable A of Eq. (4.47) with its effective variable \(\tilde{\textbf{\textit{A}}} = \textbf{\textit{M}}^{ - 1} :\textbf{\textit{A}}\) is often employed (Lemaitre and Chaboche 1985). In the most standard form of the plastic potential, i.e., Eq. (4.48), the point A remains at the center of the elastic region throughout the damage process. The difference between the results of calculation by Eqs. (4.47) and (4.48), however, is ascertained to be small (Besson et al. 2010).
- 7.
This condition is known as Prager’s consistency condition, and given by \(f = 0\) and \(\dot f = 0\).
- 8.
- 9.
When the viscoplastic strain rate \(\dot{\boldsymbol{\varepsilon}} ^{vp}\) is calculated by differentiating the dissipation potential function F with respect to the stress σ, the damage part F D of F should not affect \(\dot{\boldsymbol{\varepsilon}} ^{vp}\). Thus σ should be excluded from a set of the independent variables of F D. Then the term σ in Eq. (4.158) is interpreted as a parameter (or a function of a parameter ɛ e) of F D.
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Murakami, S. (2012). Inelastic Constitutive Equation and Damage Evolution Equation of Material with Isotropic Damage. In: Continuum Damage Mechanics. Solid Mechanics and Its Applications, vol 185. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2666-6_4
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