2. Formulation of the model#

2.1. Theoretical framework#

In this sub-chapter, we emphasize the specificity of law VENDOCHAB (i.e. damage) compared to the usual viscoplastic models. For more details, refer to [bib2].

The theory of damage describes the evolution of phenomena between the virgin state and the initiation of the macroscopic crack in a material by means of a continuous variable (scalar or tensor) describing the gradual deterioration of this material. This idea, due to Kachanov who was the first to use it to model the creep fracture of metals under uniaxial stress, was taken up in France in the 70s by Lemaitre and Chaboche. The evolution of the material from its virgin state to its damaged state is not always easy to distinguish from the deformation phenomenon that most often accompanies it and is due to several different mechanisms including creep. Visco-plastic creep damage corresponds to the inter-granular de-cohesions accompanying visco-plastic deformations for metals at medium and high temperatures.

To define what this damage variable is, consider the area \(S\) of one of the faces of a volume element \(\Omega\) identified by its normal oriented to the outside \(\text{n}\). On this section, the microcracks and cavities that make up the damage leave traces of various shapes. Let \(\tilde{S}\) be the effective resistant area and \({S}_{D}\) the total area of all traces.

We have:

\({S}_{D}\mathrm{=}S\mathrm{-}\tilde{S}\)

and the damage variable is defined by:

\({D}_{n}=\frac{{S}_{D}}{S}\)

\({D}_{n}\) is the measure of local damage in relation to the \(n\) direction. From a physical point of view, the damage variable \({D}_{n}\) is therefore the relative area of the cracks and cavities cut by the plane normal to the direction \(\overrightarrow{n}\). From a mathematical point of view, by making \(S\) tend towards 0, the variable \({D}_{\overrightarrow{n}}\) is the surface density of the discontinuities of matter in the plane normal to \(n\). \({D}_{n}=0\) corresponds to the undamaged pristine condition. \({D}_{n}=1\) corresponds to the volume element broken into two parts according to a plane normal to \(n\).

The isotropy hypothesis implies that cracks and cavities are uniformly distributed in orientation at one point in the material. In this case, the damage variable becomes a scalar that no longer depends on orientation and is noted \(D\). We have:

\(D={D}_{n}\forall n\)

Here we will only consider the isotropic damage variable.

Global mechanical measurements (modification of elasticity, plasticity or viscoplasticity characteristics) are easier to interpret in terms of damage variables thanks to the concept of effective stress introduced by Rabotnov. The effective stress represents the stress related to the section that effectively resists the forces. In the case of isotropic damage, it is written as:

\(\tilde{\sigma }=\frac{\sigma }{(1-D)}\)

And we have:

  1. \(\tilde{\sigma }=\sigma\) for virgin material

  2. \(\tilde{\sigma }\to +\infty\) at the time of breakup

The principle of deformation equivalence implies that any one-dimensional or three-dimensional deformation behavior of a damaged material is translated by the laws of behavior of the virgin material in which the usual stress is replaced by the effective stress.

There are 2 types of variables to characterize the environment:

Observable (measurable) variables:

  • Temperature \(T\)

  • total deformation \(\underline{\underline{\varepsilon }}\) which breaks down as shown below:

\(\underline{\underline{\varepsilon }}={\underline{\underline{\varepsilon }}}^{e}+{\underline{\underline{\varepsilon }}}^{\text{vp}}+{\underline{\underline{\varepsilon }}}^{\text{th}}\)

The internal variables:

  • viscoplastic deformation \({\underline{\underline{\varepsilon }}}^{\text{vp}}\)

  • the isotropic work hardening variable \(r\)

  • the isotropic damage variable \(D\)

Let’s say \(\Psi =\Psi (\underline{\underline{\varepsilon }},{\underline{\underline{\varepsilon }}}^{\text{vp}},T,r,D)\), the state potential, the state laws describing this potential are:

\(\{\begin{array}{}\underline{\underline{\sigma }}=\rho \frac{\partial \Psi }{\partial \underline{\underline{\varepsilon }}}\\ R=-\rho \frac{\partial \Psi }{\partial r}\\ s=-\rho \frac{\partial \Psi }{\partial T}\\ Y=-\rho \frac{\partial \Psi }{\partial D}\end{array}\)

According to the law of normality, with \(\Phi\), we have the potential for dissipation:

\(\{\begin{array}{}{\underline{\underline{\dot{\varepsilon }}}}^{\text{vp}}=\frac{\partial \Phi }{\partial \underline{\underline{\sigma }}}\\ \dot{r}=\frac{\partial \Phi }{\partial R}\\ \dot{D}=\frac{\partial \Phi }{\partial Y}\end{array}\)

The modeling of work hardening and material damage is done through internal variables (or hidden). In the case of model VENDOCHAB, the internal variables introduced in*Code_Aster* are:

  • \({\underline{\underline{\varepsilon }}}^{\text{vp}}\): tensor of inelastic deformations

  • \(p\): cumulative plastic deformation

  • \(r\): viscosity work-hardening variable

  • \(D\): isotropic damage scalar variable

2.2. Model equations#

The equations of the model are then written:

\(\{\begin{array}{}\underline{\underline{\varepsilon }}={\underline{\underline{\varepsilon }}}^{e}+{\underline{\underline{\varepsilon }}}^{\text{th}}+{\underline{\underline{\varepsilon }}}^{\text{vp}}\\ \underline{\underline{\sigma }}=(1-D)\underline{\underline{\underline{\underline{\Lambda }}}}{\underline{\underline{\varepsilon }}}^{e}\\ {\underline{\underline{\dot{\varepsilon }}}}^{\text{vp}}=\frac{3}{2}\dot{p}\frac{{\underline{\underline{\sigma }}}^{\text{'}}}{{\sigma }_{\text{eq}}}\\ \dot{p}=\frac{\dot{r}}{(1-D)}\\ \dot{r}={\langle \frac{{\sigma }_{\text{eq}}-{\sigma }_{y}(1-D)}{(1-D)K{r}^{}}\rangle }^{N}\\ \dot{D}={\langle \frac{\chi (\underline{\underline{\sigma }})}{A}\rangle }^{R}(1-D{)}^{-k(\chi (\underline{\underline{\sigma }}))}\end{array}\)

with:

\(\begin{array}{}\chi (\underline{\underline{\sigma }})=\alpha {J}_{0}(\underline{\underline{\sigma }})+\beta {J}_{1}(\underline{\underline{\sigma }})+(1-\alpha -\beta ){J}_{2}(\underline{\underline{\sigma }})\\ \text{où :}{J}_{0}(\underline{\underline{\sigma }})\text{est la contrainte principale maximale}\\ \text{}{J}_{1}(\underline{\underline{\sigma }})=\text{Tr}(\underline{\underline{\sigma }})\\ \text{}{J}_{2}(\underline{\underline{\sigma }})={\sigma }_{\text{eq}}=\sqrt{\frac{3}{2}{\tilde{\sigma }}_{\text{ij}}^{\text{'}}{\tilde{\sigma }}_{\text{ij}}^{\text{'}}}\\ \langle x\rangle \text{partie positive de}x\end{array}\)

where:

\(\underline{\underline{\varepsilon }},{\underline{\underline{\varepsilon }}}^{e},{\underline{\underline{\varepsilon }}}^{\text{th}}\) and \({\underline{\underline{\varepsilon }}}^{\text{vp}}\)

are the total, elastic, thermal, and plastic deformations respectively,

\(\underline{\underline{\underline{\underline{\Lambda }}}}=({\Lambda }_{\text{ijkl}})\)

is the elastic stiffness tensor,

\({\underline{\underline{\sigma }}}^{\text{'}}\mathrm{=}\underline{\underline{\sigma }}\mathrm{-}\frac{1}{3}\text{Tr}(\underline{\underline{\sigma }})\underline{\underline{\text{Id}}}\)

is the deviatoric part of the stress tensor,

\(p\)

is the cumulative plastic deformation,

\(r\)

is the viscoplastic isotropic work hardening variable

\(D\)

is the isotropic damage scalar variable

Note Bene:

All the parameters of the model \(\alpha ,\beta ,N,M,K,A,R\text{et}k\) can be a function of temperature (in \(°C\)). \(k\) can be constant, depending on the temperature or \(\chi (\underline{\underline{\sigma }})\) (in \(\mathit{MPa}\)) and on the temperature.

Moreover, we can see that this model considers that there may be a viscoplastic threshold \({\sigma }_{y}\) that depends on temperature.

We can see that this model is reduced to Lemaitre’s viscoplastic model if we consider \(D=0\) and if we neglect the evolution equation of \(D\). \(M,N,\text{et}K\) are coefficients characteristic of the purely viscoplastic behavior of the material.

The evolution of damage is governed by a law with three parameters: \(A,R,\text{et}k\) . The equivalent stress \(\chi (\underline{\underline{\sigma }})\) makes it possible to take into account a possible effect of the spherical part of the stress tensor on the damage (a bit like in the laws of cavity growth at the base of the Gurson and Rousselier models). The fact that maximum principal stress can play a role in \(\chi (\underline{\underline{\sigma }})\) is hard to imagine for materials like steel but makes the model more general.