1. Reference problem

1.1. Theoretical framework

The uniqueness of the solution of a discretized damage problem is defined by the positivity of the following quotient, written from the tangent operator \(K\):

\(\underset{x\mathrm{=}(u,a)}{\mathit{Min}}(\frac{{x}^{T}\mathrm{.}\mathit{Kx}}{{x}^{T}\mathrm{.}x})>0\) Equation 1.1

where \(u\) refers to the degrees of freedom to move and a the degrees of freedom to move. When this criterion is no longer verified, there may be several solutions to the problem discretized by the finite element method, which consists in verifying the initial equilibrium conditions. It is then necessary to discuss the stability of the solution, by verifying the positivity of the second derivative of energy, in the direction of increasing damage (condition of irreversibility on the damage):

\(\underset{x=(u,a\ge 0)}{\mathrm{Min}}(\frac{{x}^{T}\mathrm{.}\mathrm{Kx}}{{x}^{T}\mathrm{.}x})\ge 0\) Equation 1.2

The uniqueness and stability of the homogeneous solution of a bar under tension were studied analytically (Pham, Amor, Amor, Marigo and Maurini*, « Gradient damage models and their use to approximate brittle fracture »,* 2009) and The criteria were written as relationships between the damage*a* of the bar, its length \(L\) and the internal length \(l\). Here we are more particularly interested in the following energy formulation, which corresponds to the law of behavior ENDO_CARRE for modeling GVNO:

\(\phi \mathrm{=}\frac{1}{2}{(1\mathrm{-}a)}^{2}{E}_{0}\varepsilon {(u)}^{2}+\frac{{{\sigma }_{M}}^{2}}{{E}_{0}}a+\frac{{E}_{0}{l}^{2}}{2}\mathrm{\nabla }a\mathrm{.}\mathrm{\nabla }a\) Equation 1.2

where \({E}_{0}\) and \({\sigma }_{M}\) are respectively the healthy stiffness of the material and the limit stress and where \(\varepsilon (u)\) is the deformation of the member associated with the displacement \(u\) .

The criterion of uniqueness is then defined by inequality:

\({L}^{2}<\frac{2{\pi }^{2}{E}_{0}^{2}(1-a)}{6{\sigma }_{M}^{2}}{l}^{2}\) Equation 1.4

and the stability criterion, through inequality:

\({L}^{2}\le \frac{128{\pi }^{2}{E}_{0}^{2}(1-a)}{216{{\sigma }_{M}}^{2}}{l}^{2}\) Equation 1.5

By observing the two inequalities presented (equations 1.4 and 1.5), we see that the loss of uniqueness occurs before the loss of stability. The objective of the test case is then to estimate the stability criterion, firstly between the load for loss of uniqueness and that for loss of stability (the value of the minimum of the Rayleigh quotient (1.2) must then be positive), and in a second time after the load for loss of stability (the calculated minimum must then be negative).

1.2. Geometry

We consider a bar \(\mathrm{2D}\) of length \(L=100\text{m}\) to be of height \(h=1\text{m}\).

Figure 1: Problem Representation

1.3. Material properties

1.3.1. Material parameters

Elastic characteristics:

\(E=1\text{Pa}\)

\(\nu =0\)

Characteristic of the law of damage:

\({\sigma }_{M}\mathrm{=}0.01\text{Pa}\)

Non-linear characteristic:

\(l\mathrm{=}1\text{m}\)

1.4. Boundary conditions and loads

Embedding: Zero imposed displacements \(\mathit{DY}\mathrm{=}0m\) on the nodes at the bottom of the bar (\(y=0.\)), as well as on the nodes at the top (\(y=1.\)). Imposed move draw \(\mathit{DX}\mathrm{=}0m\) on the left side (\(x=0.\)). See figure 1.

Loading 1: Imposed linear displacement \(U\mathrm{=}2\mathrm{\times }t\text{m}\) on the right side (\(x\mathrm{=}100.\)).