Continuum Mechanics with internal variables

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A variational plastic-damage model

This chapter is the core of the present work. Here the coupled plastic-gradient damage model is developed in a three-dimensional setting by means of the energetic formulation introduced in the previous Chapter, in particular Sec. 1.3. The first order stability condition furnishes some (weak) necessary inequalities which the model has to be satisfied and from which it is possible to define in a classical sense the yield functions and disclose their (strong) expression. Nevertheless it is the weak form that reveals the capability of the model to describe a cohesive fracture. From the energy balance it is possible to derive the classical consistency conditions.

Model assumptions

Both plasticity and damage are here considered as rate-independent processes. Hence, the thermodynamic framework with internal variables, Sec. 1.1.6, could ap-ply motivated by the fact that in each instant equilibrium is assured. Moreover, in the context of isothermal processes, the energetic formulation is adopted in order to describe the evolution of the system. A continuous body B is considered, embed-ded in an Euclidean space of dimension d (1, 2, 3) and represented by its geometrical domain W with a sufficiently smooth1 boundary ¶W. For simplicity, deformations gradient is assumed « small » so that the infinitesimal elasticity theory can be applied. The evolution is governed by a « time » parameter t starting from t = 0. The dissi-pation is only due to two phenomena, plasticity and damage while regular energy functionals are also assumed in order to obtain results in closed form. Although not necessary, the material will be always considered initially unstretched, not plas-ticised and undamaged.

State variables and Function Spaces

Throughout the following the definitions of the state variables together with their function spaces are given. A list of the considered state variables with a brief description is given in Tab. 2.1. Moreover also the space of accessible states and the space of the variations will be specified for any state variable. These definitions play a crucial role respectively in the expression of the global stability condition and of the local stability condition, Sec. 1.3.2. For what concerns the global stability con-dition, while for the displacement and plastic strain fields the function spaces are the same as the one for the accessible states this holds not more true for the damage field because of irreversibility, Sec. 1.3.4. The accessible states then has to be explic-itly defined for the damage field. On the other hand for the local stability condition, while the function space of the plastic strain field is the same as its variation space this holds not more true for the displacement and damage fields because respec-tively of the boundary conditions and irreversibility. The variations space then has to be explicitly defined for the displacement and damage field.
The displacement and the total strain are maps defined as u(x, t) : (W, [0, T]) ! F Rd, n o
#(x, t) : (W, [0, T]) ! S := a 2 Rd d : #T = # ,
with #(x, t) = #(u(x, t)) by means of the strain-diplacement relation (1.1). proper function space of the displacement field is u 2 F SBD((W, [0, T]) , F)
where SBD is the space of special bounded deformations, the proper space for the description of fracture or perfect plasticity problems2, Temam and Strang 1980 for details.

Energy functionals and constitutive assumptions

According to the energetic formulation of Sec. 1.3.1 the total potential energy and the dissipation distance, are here defined as well as some constitutive assumptions.

 Stability condition

In this and forth coming sections the theoretical results introduced in Sec. 1.3 are applied to the plastic-damage model. To highlight the capabilities of the model only the differential stability condition Sec. is considered. On the other hand, the global stability condition is in a certain sense preferred for numeric implementations as discussed in Chap. 5.

Second order stability condition

Clearly the first order stability condition is only a necessary condition for stabil-ity when it is satisfied as an equality. In this section it is assumed that both (2.36) and (2.38) are satisfied as an equality. The case when one between the plasticity yield condition and the damage yield condition is still a strictly inequality is not con-sidered since the study becomes simpler then in the presented general case. More specifically, the elastic-perfect plastic problem is always indefinite stable, see Fuchs and Seregin 2000; Temam 1985, while the stability of the elastic-damage model has been already extensively studied, for example in Pham 2010.
In order to achieve a sufficient condition it is essential to investigate the sign of the higher order terms in (1.61) as the second order one. To this aim, refering to Sec., the second variation of the potential energy and the dissipation distance are necessary to express the second-order stability condition (st-2D). The second variation of the potential energy then reads
The proof of the existence of a minimum is assured by compactness and lower semi-continuity properties. A complete proof requires technical mathematical tools of functional analyses and therefore it is left out here.
An efficient procedure for the minimization problem (3.68) in case of an homo-geneous state is suggested in Kohn 1991 and successfully applied for example in Pham 2010 for an elastic-damage model.

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Energy balance

Once the total potential energy (2.25) and the dissipated work (2.27) have been introduced, the energy balance is immediately defined by (EB). The energy balance express the fact that the total energy must remain constant along the evolution.

Table of contents :

1 Background and State of Art 
1.1 Continuum Mechanics with internal variables
1.1.1 Introduction
1.1.2 Fundamental laws in Continuum Mechanics
1.1.3 Thermostatics
1.1.4 The theory of irreversible processes
1.1.5 The « Rational » Thermodynamics of Colemann and Noll
1.1.6 The theory of internal variables
1.2 Generalized Standard Models
1.2.1 Sub-differential formulation
1.2.2 Standard dissipative system: Biot’s equation
1.2.3 Rate-independence
1.2.4 Equivalent formulations KKT – yield surface Normality rule Principle of maximum dissipation
1.2.5 Drucker-Ilyushin’s postulate
1.3 Energetic Formulation (Variational Formulation)
1.3.1 State variables and energy functionals
1.3.2 Stability condition Global stability condition Local stability condition Differential or n-order stability condition
1.3.3 Energy balance Weak energy balance Regular energy balance
1.3.4 Irreversible phenomena
1.4 Fracture, Plasticity and Damage: local and non-local models
1.4.1 Fracture models Classical formulations Variational formulations
1.4.2 Plasticity models
1.4.3 Damage models
1.4.4 Uniqueness, stability and bifurcation
1.4.5 Non-local models overview
1.4.6 Coupled plasticity-damage models
2 A variational plastic-damage model 
2.1 Model assumptions
2.1.1 State variables and Function Spaces
2.1.2 Energy functionals and constitutive assumptions
2.2 Stability condition
2.2.1 First order stability condition
2.2.2 Second order stability condition
2.3 Energy balance
2.4 A particular model
2.5 Conclusions and perspectives
3 Homogeneous evolution 
3.1 Introduction to the 1D model
3.2 The abstract homogeneous evolutions
3.2.1 Elastic phase (t < tI )
3.2.2 First dissipation phase (tI t < tI I )
3.2.3 Second dissipation phase (tI I t)
3.3 Analytic examples
3.3.1 Example: E-D sequence
3.3.2 Example: E-P-D sequence
3.3.3 Example: E-P-DP sequence
3.3.4 Example: E-D-PD sequence
3.4 The stability of homogeneous states
3.4.1 Elastic phase (t < tI )
3.4.2 First dissipation phase (tI t < tI I )
3.4.3 Second dissipation phase (tI I t)
3.5 Conclusions and perspectives
4 Non-homogeneous evolutions 
4.1 Introduction to non-homogeneous evolutions
4.1.1 The setting of the problem
4.1.2 The governing equations Irreversibility Stability condition Energy balance
4.1.3 The general assumptions
4.2 The construction of localizations
4.2.1 E-D-* case E-D-PD case
4.2.2 E-P-* case E-P-D case E-P-DP case
4.2.3 Incipient damage phase
4.2.4 The case of non-uniform plastic strains
4.3 The global response
4.3.1 Stress-displacement response
4.3.2 Cohesive fracture and energy contributions
4.4 Non-homogeneous evolution examples
4.4.1 E-D response
4.4.2 E-D-PD response
4.4.3 E-P-D response
4.4.4 E-P-DP response
4.5 Conclusions and perspectives
5 Numeric implementation and simulations 
5.1 The implementation
5.1.1 Time and space discretisation
5.2 Analyses
5.2.1 Analyses setting
5.2.2 Analyses results
5.3 Conclusions and perspectives
A Partial results
A.1 Gateaux derivative of the dissipation distance
A.2 Rayleigh ratio minimization: 1D case
B Related Pubblications


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