3. Simo and Miehe theory

3.1. Introduction

Here we recall the specificities of the formulation proposed by SIMO J.C. and MIEHE C. [bib3] to treat large deformations. This formulation has already been used for models of thermo-elasto- (visco) -plastic behavior with isotropic work hardening and Von Mises criterion, [R5.03.21] for a model with no effect of metallurgical transformations and [R4.04.03] for a model with an effect of metallurgical transformations.

Kinematic choices make it possible to treat large displacements and large deformations but also large rotations in an exact manner.

The specific features of these models are as follows:

• just as in small deformations, it is assumed the existence of a relaxed configuration, that is to say locally free of stress, which makes it possible to decompose the total deformation into a thermoelastic part and a plastic part,

• the decomposition of this deformation into thermoelastic and plastic parts is no longer additive as in small deformations (or for large deformation models written in deformation rate with for example a Jaumann derivative) but multiplicative,

• the elastic deformations are measured in the current configuration (deformed) while the plastic deformations are measured in the initial configuration,

• as in small deformations, the stresses depend only on thermo-elastic deformations,

• if the plasticity criterion depends only on the deviatoric stress, then the plastic deformations occur at a constant volume. The volume variation is then only due to thermoelastic deformations,

• During its numerical integration, this model leads to an incrementally objective model (cf. [§3.2.3]), which makes it possible to obtain the exact solution in the presence of large rotations.

Subsequently, we briefly recall some concepts of mechanics in large deformations.

3.2. General information on major deformations

3.2.1. Cinematics

Let us consider a solid subject to great deformations. Let \({\Omega }_{0}\) be the domain occupied by the solid before deformation and \(\Omega (t)\) the domain occupied at the time t by the deformed solid.

Figure 3.2.1-a: Representation of the initial and deformed configuration

In the initial configuration \({\Omega }_{0}\), the position of any particle in the solid is designated by \(X\) (Lagrangian description). After deformation, the position at time \(t\) of the particle that occupied position \(X\) before deformation is given by the variable \(x\) (Eulerian description).

The global motion of the solid is defined, with \(u\) the displacement, by:

\(x=\stackrel{ˆ}{x}(X,t)=X+u\) eq 3.2.1-1

To define the change in metric near a point, we introduce the gradient tensor for transformation \(F\):

\(F=\frac{\partial \stackrel{ˆ}{x}}{\partial X}=\text{Id}+{\nabla }_{X}u\) eq 3.2.1-2

The transformations of the volume element and the density equal:

\(d\Omega =\text{Jd}{\Omega }_{o}\) with \(J=\text{det}F=\frac{{\rho }_{o}}{\rho }\) eq 3.2.1-3

where \({\rho }_{o}\) and \(\rho\) are the density in the initial and current configurations respectively.

Different strain tensors can be obtained by eliminating rotation in the local transformation. For example, by directly calculating the length and angle variations (dot product variation), we get:

\(E=\frac{1}{2}(C-\text{Id})\) with \(C={F}^{T}F\) eq 3.2.1-4

\(A=\frac{1}{2}(\text{Id}-{b}^{-1})\) with \(b={\text{FF}}^{T}\) eq 3.2.1-5

\(E\) and \(A\) are respectively the Green-Lagrange and Euler-Almansi deformation tensors and \(C\) and \(b\) are the right and left Cauchy-Green tensors respectively.

In Lagrangian description, deformation will be described by tensors \(C\) or \(E\) because they are quantities defined on \({\Omega }_{0}\), and in Eulerian description by tensors \(b\) or \(A\) (defined on \(\Omega\)).

3.2.2. Constraints

The stress tensor used in Simo and Miehe’s theory is the Kirchhoff tensor \(\tau\) defined by:

\(J\sigma =\tau\) eq 3.2.2-1

where \(\sigma\) is the Cauchy Eulerian tensor. The \(\tau\) tensor therefore results from a « scaling » by the variation in volume of the Cauchy tensor \(\sigma\).

3.2.3. Objectivity

When we write a law of behavior in large deformations, we must check that this law is objective, that is to say invariant by any change in the spatial frame of reference of the form:

\({x}^{\text{*}}=c(t)+Q(t)x\) eq 3.2.3-1

where \(Q\) is an orthogonal tensor that reflects the rotation of the frame of reference and \(c\) is a vector that translates the translation.

More specifically, if a tensile test is carried out in direction \({e}_{1}\), for example, followed by a rotation of 90° around \({e}_{3}\), which is equivalent to carrying out a tensile test according to \({e}_{2}\), then the danger with a non-objective law of behavior is not to find a uniaxial stress tensor in direction \({e}_{2}\) (which is in particular the case with the kinematics PETIT_REAC).

3.3. Formulation by Simo and Miehe

Subsequently, we will note \(F\) the gradient tensor which makes it go from the initial configuration \({\Omega }_{0}\) to the current configuration \(\Omega (t)\), \({F}^{p}\) the gradient tensor which makes it go from the configuration \({\Omega }_{0}\) to the relaxed configuration \({\Omega }^{r}\), and \({F}^{e}\) from the configuration \({\Omega }^{r}\) to \(\Omega (t)\). Index \(p\) refers to the plastic part, index \(e\) refers to the elastic part.

Figure 3.3-a: Breakdown of the gradient tensor \(F\) into an elastic part \({F}^{e}\) and plastic \({F}^{p}\)

By composition of the movements, the following multiplicative decomposition is obtained:

\(F={F}^{e}{F}^{p}\) eq 3.3-1

The elastic deformations are measured in the current configuration with the left Cauchy-Green Eulerian tensor \({b}^{e}\) and the plastic deformations in the initial configuration by the \({G}^{p}\) tensor (Lagrangian description). These two tensors are defined by:

\({b}^{e}={F}^{e}{F}^{\text{eT}}\), \({G}^{p}=({F}^{\text{pT}}{F}^{p}{)}^{-1}\) where \({b}^{e}={\text{FG}}^{p}{F}^{T}\) eq 3.3-2

However, one will alternatively use another measure of elastic deformations \(e\) which coincides with the opposite of linearized deformations when the elastic deformations are small:

\(e=\frac{1}{2}(\text{Id}-{b}^{e})\) eq 3.3-3

In the case of an isotropic material, we can show that the free energy potential only depends on the left Cauchy-Green tensor \({b}^{e}\) (or in our case on the \(e\) tensor) and in plasticity on the variable \(p\) linked to isotropic work hardening. In addition, it is assumed that the free energy volume breaks down, just like in small deformations, into a hyperelastic part that depends only on elastic deformation and another linked to the work hardening mechanism:

\(\Phi (e,p)={\Phi }^{\text{el}}(e)+{\Phi }^{\text{bl}}(p)\) eq 3.3-4

If instead of using the Cauchy \(\sigma\) constraint, we use the Kirchhoff constraint \(\tau\), the Clausius-Duhem inequality is written (we forget the thermal part):

\(\tau :D-\dot{\Phi }\ge 0\) eq 3.3-5

expression in which \(D\) represents the Eulerian deformation rate.

Under the previous hypotheses, the dissipation is again written as:

\((\tau +\frac{\partial \Phi }{\partial e}{b}^{e}):D+\frac{1}{2}\frac{\partial \Phi }{\partial e}:(F{\dot{G}}^{p}{F}^{T})-\frac{\partial \Phi }{\partial p}\dot{p}\ge 0\) eq 3.3-6

The second principle of thermodynamics then requires the following expression for the stress-strain relationship:

\(\tau =-\frac{\partial \Phi }{\partial e}{b}^{e}\) eq 3.3-7

Finally, the thermodynamic forces associated with elastic deformation and cumulative plastic deformation are defined in accordance with the framework of generalized standard materials:

\(s=-\frac{\partial \Phi }{\partial e}\text{}\text{soit}\text{}\tau =s{b}^{e}\) eq 3.3-8

\(A=-\frac{\partial \Phi }{\partial p}\) eq 3.3-9

where the thermodynamic force A is the opposite of the isotropic work hardening variable \(R\).

There is then left for the dissipation:

\(\tau :(-\frac{1}{2}F{\dot{G}}^{p}{F}^{T}{b}^{e-1})+A\dot{p}=s:(-\frac{1}{2}F{\dot{G}}^{p}{F}^{T})+A\dot{p}\ge 0\) eq 3.3-10

3.3.1. Original formulation

The principle of maximum dissipation applied from the elasticity threshold \(F\), a function of the Kirchhoff stress \(\tau\) and the thermodynamic force \(A\), makes it possible to deduce the laws of evolution of plastic deformation and cumulative plastic deformation, i.e.:

\(-\frac{1}{2}F{\dot{G}}^{p}{F}^{T}{b}^{e\text{-1}}=\dot{\lambda }\frac{\partial F}{\partial \tau }\) eq 3.3.1-1

\(\dot{p}=\dot{\lambda }\frac{\partial F}{\partial A}\) eq 3.3.1-2

\(\dot{\lambda }\ge 0\text{}F\le 0\text{}F\dot{\lambda }=0\) eq 3.3.1-3

3.3.2. Modified formulation

The approximation introduced here on the original formulation by Simo and Miehe concerns the expression of the flow law, an approximation that is all the more reduced the smaller the elastic deformations, since \(\tau =s{b}^{e}\). In fact, the elasticity threshold is now expressed as a function of thermodynamic forces and no longer of constraints \(F(s,A)\le 0\), and it is in relation to these variables that the principle of maximum dissipation is applied, which leads to the following flow laws:

\(-\frac{1}{2}F{\dot{G}}^{p}{F}^{T}=\dot{\lambda }\frac{\partial F}{\partial s}\) eq 3.3.2-1

\(\dot{p}=\dot{\lambda }\frac{\partial F}{\partial A}\) eq 3.3.2-2

\(\dot{\lambda }\ge 0\text{}F\le 0\text{}F\dot{\lambda }=0\) eq 3.3.2-3

3.3.3. Consequences of approximation

By replacing the stress \(\tau\) by the thermodynamic force \(s\) associated with the elastic deformation in the expression of the plasticity criterion, we are in fact introducing a disturbance of the border of the reversibility domain of the order of magnitude of \(2\parallel e\parallel\). Compared to the initial formulation, this obviously results in an influence on the observed elastic limit but also on the direction of flow: in particular, the derivative with respect to time of the variation in plastic volume is then written as:

\({\dot{J}}^{p}=\dot{\lambda }{J}^{p}{b}^{e\text{-1}}:\frac{\partial F}{\partial s}\) eq 3.3.3-1

so that in the case where criterion \(F\) only depends on the deflector of the stress tensor \(s\), we do not find \({J}^{p}=1\): the isochoric nature of the plastic deformation is no longer perfectly preserved. We will then have to introduce a volume correction afterwards.

Insofar as the elastic deformations remain small, the results obtained with this modified model do not differ significantly from those obtained with the old formulation (cf. [bib4]), while numerical integration will be simplified. In fact, we will see later that this model follows the same integration diagram as that of models written in small deformations.

Note:

This new formulation of large deformations makes it possible to place Simo and Miehe’s theory within the framework of generalized standard materials. From a numerical point of view, this has the consequence of expressing the resolution of the law of behavior as a minimization problem in relation to the increments of internal variables.

Indeed, we recall that in the context of generalized standard materials, the data of the two potentials the free energy \(\Phi (\varepsilon ,a)\) and the dissipation potential \(D(\dot{a})\) , a function of the deformation tensor \(\varepsilon\) and of a certain number of internal variables \(a\) , makes it possible to completely define the law of behavior (this is the case of time-independent materials) .

\(\sigma =\frac{\partial \Phi }{\partial \varepsilon }\) , \(A=-\frac{\partial \Phi }{\partial a}\in \partial D(\dot{a})\) eq 3.3.3-2

where \(\partial D(\dot{a})\) is the subdifferential of the dissipation potential \(D\) .

Generalized standard behavior laws that do not depend on time are characterized by a positively homogeneous dissipation potential of degree 1, which results in the following property:

\(\forall \dot{a}\text{}\forall \lambda >0\text{}D(\lambda \dot{a})=\lambda D(\dot{a})\text{}\Rightarrow \text{}\partial D(\lambda \dot{a})=\partial D(\dot{a})\) eq 3.3.3-3

Now if we write the problem [éq 3.3.3-2] in time discretized form and if we use the property of subdifferentials [éq 3.3.3-3], we get the following discretized problem:

\(\sigma =\frac{\partial \Phi }{\partial \varepsilon }\) , \(A=-\frac{\partial \Phi }{\partial a}\in \partial D(\Delta a)\) eq 3.3.3-4

We can show that the equation [éq 3.3.3-4] is equivalent (cf. [bib5]) to solving the minimization problem with respect to the increments of internal variables \(\Delta a\) following:

:math:`-frac{partial Phi }{partial a}in partial D(Delta a)iff Delta a=text{Arg}underset{Delta {a}^{text{*}}}{text{Min}}left[Phi ({a}^{-}+{mathrm{Delta a}}^{text{*}})+D(Delta {a}^{text{*}})right]`*eq 3.3.3-5*

The application of equation [éq 3.3.3-5] to the Rousselier model in large modified deformations is written as:

\(\underset{\text{énergie continue}}{\Phi (e,p)\text{et}D({D}^{p},\dot{p})}\text{}\underset{\text{discrétisation}}{\text{=>}}\text{}\underset{\text{énergie discrétisée}}{\Phi ({e}^{\text{Tr}}+\Delta e,{p}^{-}+\Delta p)\text{et}D(\Delta e,\Delta p)}\) eq 3.3.3-6

\(\begin{array}{}A=-\frac{\partial \Phi }{\partial a}=\{\begin{array}{}s=-\frac{\partial \Phi }{\partial e}\\ -R=-\frac{\partial \Phi }{\partial p}\end{array}\text{}\in \partial D(\Delta e,\Delta p)\\ \iff \underset{\Delta e,\Delta p}{\text{Min}}\left[\Phi ({e}^{\text{Tr}}+\Delta e,{p}^{-}+\Delta p)+D(\Delta e,\Delta p)\right]\text{}\end{array}\) eq. 3.3.3-7

We will find in paragraph [§4], the relationship that links the plastic deformation rate \({D}^{p}\) once discretized and the elastic deformation increment \(\Delta e\) , as well as the definition of \({e}^{\text{Tr}}\) .

We can clearly see here that if we take the initial formulation of Simo and Miehe, we can no longer write the minimization problem [éq 3.3.3-7] with the Kirchhoff constraint \(\tau\) because of the term en \({b}^{e}\) in the expression:

\(\tau =-\frac{\partial \Phi }{\partial e}{b}^{e}\) eq 3.3.3-8