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Material Damage and Continuum Damage Mechanics Read chapter Read first chapter

2012 | OriginalPaper | Chapter

3. Thermodynamics of Damaged Material

Author : Sumio Murakami

Published in: Continuum Damage Mechanics

Publisher: Springer Netherlands

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Abstract

In the preceding chapter, we discussed the mechanical modeling of the damaged state of materials by means of internal variables, and called them damage variables. The present chapter is concerned with the thermodynamic constitutive theory with internal variables, which furnishes a consistent basis to formulate the mechanical behavior of damaged materials. In Section 3.1, the fundamental principles and the basic laws of the non-equilibrium continuum thermodynamics are presented as the foundations for the succeeding discussions. In Section 3.2, the notion and the procedure of the thermodynamic constitutive theory with internal variables will be described in detail. It is shown that the inelastic constitutive equations and the evolution equations for internal variables are formulated as a set of generalized normality rule defined by a dissipation potential function and a common multiplier.

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previous chapter Mechanical Representation of Damage and Damage Variables
next chapter Inelastic Constitutive Equation and Damage Evolution Equation of Material with Isotropic Damage
Footnotes
1
Excellent review of the development of the non-equilibrium thermodynamics and its major results are found in Germain et al. (1983) and Chaboche (1997).
 
2
This equation together with the analogous equations, e.g., \(s = s{\left(e,\;\chi _i \right)}, e = e{\left(T,\;{\textbf{\textit{F}}}\right)}, s = s{\left(T,\;{\textbf{\textit{F}}}\right)}\), describes the thermodynamic property of the material, and thus they are called caloric equations of state (Truesdell and Toupin 1960; Coleman and Mizel 1964), where F is the deformation gradient.
 
3
Called also Gibbs function, or Gibbs free energy.
 
4
Thermal and mechanical behavior of a material is governed not only by various conservation laws of general validity but also by material relations specifying the intrinsic response of the material. In a broad sense, equations characterizing the individual material are called constitutive equations. Thus the state functions of Eqs. (3.24) through (3.27) are called state equations, or constitutive equations.
 
5
In the definitions (3.47) and (3.48) of A k and g (e.g., Maugin 1992), we have arbitrariness in the selection of the sign on their right-hand sides. In Chapter 4 and thereafter, the definition of Eq. (3.47) without the minus sign “−” will be used for a part of the associated variables A k .
 
6
The equation which governs the change of internal state is a material function, and is called a constitutive equation as already described in Section 3.1.4. However the equation which describes the progress of an irreversible process, in particular, is usually termed an evolution equation or a complementary equation.
 
7
A k defined here is identical to that of Eq. (3.47), and has the same value.
 
8
 Note that the associated variables A j , Y and B are defined by Eq. (3.47) in this chapter. However, in Chapter 4 and thereafter, the different sign is used in the right hand side of the definition of these variables for the expedience of the argument.
 
9
As regards the consistency condition, see the footnote of Section 4.​2.​3 afterward.
 
Literature
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Rice JR (1971) Inelastic constitutive relations for solids: an internal-variable theory and its application to metal plasticity. J Mech Phys Solids 19:433–455MATHCrossRef
Rice JR (1975) Continuum mechanics and thermodynamics of plasticity in relation to microscale deformation mechanics. In: Argon AS (ed) Constitutive equations in plasticity. MIT Press, Cambridge, pp 23–79
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Chaboche JL (1997) Thermodynamic formulation of constitutive equations and application to the viscoplasticity and viscoelasticity of metals and polymers. Int J Solids Struct 34:2239–2254
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Metadata
Title
Thermodynamics of Damaged Material
Author
Sumio Murakami
Copyright Year
2012
Publisher
Springer Netherlands
Book
Continuum Damage Mechanics

Print ISBN: 978-94-007-2665-9

Electronic ISBN: 978-94-007-2666-6

Copyright Year: 2012

DOI
https://doi.org/10.1007/978-94-007-2666-6_3

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