Research Article
Thermodynamically Consistent Models of ElasticPlastic Materials
Claudio Giorgi^{1} and Angelo Morro^{2}*
^{1}DICATAM, Universita` di Brescia, Via Valotti 9, 25133 Brescia, Italy
^{2}DIBRIS, Universita` di Genova, Via all’Opera Pia 13, 16145 Genova, Italy
Angelo Morro, DIBRIS, Universita` di Genova, Via all’Opera Pia 13, 16145 Genova, Italy.
Received Date: July 19, 2021; Published Date: August 05, 2021
Abstract
A threedimensional thermodynamic setting for the modelling of elasticplastic materials is established. The second law of thermodynamics is assumed in the ClausiusDuhem form with the entropy production being given by a constitutive function. An incremental stress strain relation is derived. In essence the free energy is found to describe the elastic behaviour while the hysteretic properties are a joint consequence of the entropy production and the free energy.
Introduction
The purpose of this paper is to establish a thermodynamic threedimensional setting for the modelling of thermoelasticplastic properties of solids. To this end we follow the scheme of rational thermodynamics [1] which is based on the (local) balance equations and the ClausiusDuhem inequality as a way of selecting physically admissible constitutive equations. However, we allow the entropy production too to be given by a constitutive function. This view is quite unusual in the literature and is the key point of the present approach. Mathematically it emerges that new schemes can occur due to common dependences of the free energy, possibly of the stress, and of the entropy production. To describe the elastic plastic behaviour we consider the stress, the strain, and their rates within the set of independent variables. Here it follows that the dependence of the entropy production on the strain rate or on the stress rate combine with the time derivative of the free energy thus providing relations which are qualitatively new; the free energy is dependent on both stress and strain. These models show a joint occurrence of terms arising from the free energy and terms determined by the entropy production. The flexibility of the models and the thermodynamic consistency are the main advantages of the constitutive property of the entropy production.
SecondLaw Inequality and Entropy Production
Consider a continuous body occupying a timedependent region Ω⊂ε 3 . Any point of the body is labelled by the position X occupied in a reference configuration R. The motion is described by the function χ that to each X ∈ R, at time t ∈ ℝ, associates the position x ∈ Ω. Denote by ∇ and ∇_{R} the gradient with respect to x and X. The symbol ∂_{t} denotes the time derivative at a point x in space while a superposed dot denotes the time derivative relative to the observer following the motion of the point under consideration. Hence for any differentiable function f (t, x) we have Instead, if we have g (t, X) then g˙= ∂tg. The body is regarded as a deformable and heatconducting solid. The balances of mass and linear momentum are written in the form
where ρ is the mass density, v is the velocity, T is the symmetric Cauchy stress tensor, b is the body force. The balance of energy is written in the form
where ε is the internal energy density (per unit mass), D is the stretching tensor, q is the heat flux, and r is the heat supply.
Let η be the entropy density and θ the absolute temperature. The balance of entropy is expressed by saying that there is an entropy flux φ and an entropy supply ρs; here we let φ = q/θ and s = r/θ. The balance is stated by assuming that in any process the entropy rate is greater than what is given by φ and s,
The quantity σ so defined is said to be the entropy production. Substitution of ρr − ∇·q from the balance of energy (2) into (3) results in
By means of the Helmholtz free energy ψ = ε−θη we obtain from (4) that
For the present purposes it is convenient to consider (referential) Lagrangian quantities. Let F = ∇_{R} χ be the deformation gradient and J = det F. Hence, we let E = ½ (F^{T} F − 1) and
T_{RR} is the second Piola (or PiolaKirchhoff) stress tensor. It follows that
Observe Jρ =ρ_{R} is the mass density in R. Multiplying inequality (5) by J we obtain
As is standard in continuum mechanics [1,2] we consider any set of constitutive equations as physically admissible if the Clausius Duhem inequality (5), or (6), holds identically for any process subject to the balance equations (1), (2). Equation (3) defines the value of the entropy production; σ > 0 characterizes dissipative phenomena. About (6), and likewise (5), the constitutive functions for ψ, η, T_{RR}, qR are physically admissible only if the lefthand side is nonnegative for any process satisfying eqs. (1), (2). Rather we consider σ too as given by a constitutive function subject to taking nonnegative values. Consequently, the constitutive functions ψ, η, T_{RR}, q_{R}, σ are physically admissible if they satisfy identically both conditions
for any process subject to the balance equations. The assumption that the entropy production σ, or the rate of dissipation θσ, be given by a constitutive equation traces back to [3]. Later on, this assumption was used in [4]. Recently we have shown how the constitutive function for σ is essential in the modelling of ferroelectric hysteresis [5,6]. We now show that the constitutive property of the entropy production allows a thermodynamically consistent class of elasticplastic models.
StressStrain Response in ThreeDimensional Plasticity
In plasticity, and likewise in any hysteretic model, the dependence of T_{RR} on E would be given by a multivalued function. To avoid this type of dependence it seems convenient to let the pertinent constitutive equations depend on both stress and strain. Moreover, since we look for ratetype models of the stressstrain relation we let E˙ and T˙_{RR} be among the independent variables. Accordingly, we assume ψ, η, qR, σ are continuous functions of the set of variables
The function ψ is assumed to be continuously differentiable. Upon evaluation of ψ˙ substitution in (6) we find
where the derivatives of ψ, η and σ are continuous functions of Ξ. The linearity and arbitrariness of ∇_{R} θ˙, E¨, T¨_{RR} and θ˙ imply
Hence it follows the reduced dissipation inequality
Since ψ is independent of ∇_{R} θ then letting ∇_{R} θ = 0 we have
where σ_{TE} = σ at ∇_{R} θ = 0 and σ_{TE} ≥ 0. Consequently
Given a constitutive equation for q_{R}, eq. (9) yields the constitutive function for σ_{q}, that is
Equation (8) yields different results depending on the function σTE. Anyway it is worth considering the following
Proposition. Let N be a secondorder tensor with N = 1. Any secondorder tensor Z can be represented in the form
In words, Z is split into the longitudinal part (Z · N) N and the transverse part Z⊥.
We first look at (8) in the case σTE = 0 whence
in geometrical words, the thermodynamic restriction (8) determines the longitudinal part of T_{RR} whereas the transverse part is left undetermined. This result yields infinitely many representations of.T_{RR} compatible with thermodynamics. If, e.g., we let then we find
We now let 0 TE σ ≠ . By (8) we have
Assume again for simplicity Hence, we find that
Consequently, the nonzero entropy production σTE results in the additional contribution
For definiteness, in light of the yield criteria, we denote by F (TRR) the hardening function such that F = 0 is the chosen yield criterion ([7], ch. 76). We then define
where H is the Heaviside step function and λ a nonnegative parameter, possibly dependent on temperature. Accordingly, σ_{TE} ≥ 0 as required by (7). Upon substitution we obtain
Some comments are in order about (10). The first term in the righthand side can be viewed as the contribution of the elastic behaviour, the second one as the term providing hysteretic effects. If ψ is quadratic in T_{RR} then N is collinear to T_{RR}. This is consistent with flow rules where the plastic rate E˙p is in the direction of the stress. Sometimes the literature assumes the plastic strain rate in the form
where ζ is an appropriate measure of deformation. If, e.g., as in the von Mises criterion F is quadratic in the deviator of the stress then T_{RR} ∂ F is proportional to T_{RR} and we find again the flow rule in the direction of T_{RR}. Furthermore, the yield criteria and the flow rules are usually stated in terms of the Cauchy stress T rather than of the Piola stress T_{RR}; in linear approximations T_{RR} T .
Acknowledgement
The research leading to this work has been developed under the auspices of INDAMGNFM.
Conflict of Interest
No conflict of interest.
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Claudio Giorgi, Angelo Morro. Thermodynamically Consistent Models of ElasticPlastic Materials. Cur Trends Civil & Struct Eng. 7(5): 2021. CTCSE.MS.ID.000672.

Elasticplastic materials, Thermodynamics, Energy density, Hysteretic effectst

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