1. Reference problem

1.1. Theoretical framework

In the case of a conservative problem, the stability of the equilibrium state is defined by the strict positivity of all the eigenvalues of the tangent operator. What is written, in the case of a symmetric tangent \(K\) operator:

\(\underset{x}{\mathit{Min}}(\frac{{x}^{T}\mathrm{.}\mathit{Kx}}{{x}^{T}\mathrm{.}x})>0\)

In the case of non-conservative problems, unilateral irreversibility conditions are imposed on certain components of the vector \(x\). The previous inequality then becomes sufficient but not necessary to deduce the stability of a state of equilibrium.

One of the functionalities developed in the constrained optimization algorithm makes it possible to limit oneself to the calculation of the smallest eigenvalue (quantity that can be quickly accessed), when it has a positive sign. In this specific case, the value referenced for the stability criterion is exactly that of the lowest eigenvalue, recalculated using the power method. Additional projection steps are not performed to determine the exact value of the minimum under inequality constraints. This saves calculation time.

In the test case presented here, we are interested in a bar under uniform tension, whose behavior is purely elastic. The elastic character is conservative and ensures the strict positivity of the smallest eigenvalue of the tangent operator. We are therefore in the specific case where the functionality described above is triggered.

1.2. Geometry

We consider a bar \(\mathrm{2D}\) of length \(L\mathrm{=}4m\) to be of height \(h\mathrm{=}0.5m\)

Figure 1: Representation of the problem

1.3. Material properties

1.3.1. Elastic law: material ELAS

Features: \(E=1\text{Pa}\), \(\nu \mathrm{=}0.\)

1.4. Boundary conditions and loads

Embed: Null imposed displacements \(\mathit{DY}\mathrm{=}0m\). on all nodes and \(\mathit{DX}\mathrm{=}0m\). on the left face (\(x=0.\)). See figure 1.

Loading 1: Imposed linear displacement \({U}_{1}\) on the right side (\(x=4.\)): \({U}_{1}\mathrm{=}t\mathrm{\cdot }{10}^{\mathrm{-}6}\text{m}\)