2. Benchmark solution#

In this part we show an analytical solution with a non-constant jump along \({\Gamma }_{0}\), then we give a condition of uniqueness of the solution.

2.1. Analytical solution#

The Airy function \(\Phi (x,y)\) governed by the equation \(\Delta \Delta \Phi =0\) su \(\Omega\) r, in the case where the external forces are zero, leads to constraints satisfying the equations of equilibrium and elasticity compatibility (see Fung [bib1]). The components of the \({\sigma }_{\mathrm{xx}}\), \({\sigma }_{\mathrm{yy}}\), and \({\sigma }_{\mathrm{xy}}\) constraint derive from the \(\Phi (x,y)\) as follows:

\({\sigma }_{\mathrm{xx}}=\frac{{\partial }^{2}\Phi }{\partial {y}^{2}}\), \({\sigma }_{\mathrm{yy}}=\frac{{\partial }^{2}\Phi }{\partial {x}^{2}}\) and \({\sigma }_{\mathrm{xy}}=\frac{{\partial }^{2}\Phi }{\partial x\partial y}\) eq 2.1-1

Let’s choose a bi-harmonic function \(\Phi (x,y)\) defined by:

\(\Phi (x,y)=\beta \frac{{y}^{3}}{6}+(\alpha x+\gamma )\frac{{y}^{2}}{2}+\eta xy\)

with \(\alpha ,\beta ,\gamma\) and \(\eta\) arbitrary real constants. The constraint field is deduced from [éq 2.1-1]:

\(\{\begin{array}{ccc}{\sigma }_{\mathrm{xx}}& \text{=}& \alpha x+\beta y+\gamma \\ {\sigma }_{\mathrm{yy}}& \text{=}& 0\\ {\sigma }_{\mathrm{xy}}& \text{=}& -\alpha y-\eta \end{array}\) eq 2.1-2

By integrating the elastic law, if we note \(E\) the Young’s modulus and \(\nu\) the Poisson’s ratio (which we take zero), we deduce the field of displacement in \(\Gamma\) verifying the balance:

\(u=\left\{\begin{array}{}u(x,y)\\ v(x,y)\end{array}\right\}=\left\{\begin{array}{}\frac{1}{E}(\alpha (\frac{{x}^{2}}{2}-{y}^{2})+x(\beta y+\gamma ))\\ -\frac{1}{E}(\beta \frac{{x}^{2}}{2}+2\eta x)\end{array}\right\}\) eq 2.1-3

Note \({U}_{0}\) and \(U\) respectively the movements on \({\Gamma }_{0}\) and \(\partial \Omega \setminus {\Gamma }_{0}\) given by [éq 2.1-3]. The latter correspond to the boundary conditions leading to the stress field [éq 2.1-2]. From this data, it is easy to build a displacement field with a discontinuity on edge \({\Gamma }_{0}\). In fact, knowing the normal constraint \(\sigma n\) on \({\Gamma }_{0}\), which is noted as \(F(y)\), we obtain the displacement jump \(\delta (y)\) by inverting the Barenblatt law of exponential behavior: CZM_EXP (see documentation on elements with internal discontinuity and their behavior: [R7.02.12]):

\(\delta (y)=-\frac{{G}_{c}F(y)}{{\sigma }_{c}\parallel F(y)\parallel }\mathrm{ln}(\frac{\parallel F(y)\parallel }{{\sigma }_{c}})\)

for all \(y\) in \(\left[\mathrm{0,}H\right]\). Thus, the new displacement imposed on \({\Gamma }_{0}\) generating such a jump is equal to \({U}_{0}-\delta\). We have therefore constructed an analytical solution of the flat plate verifying the equations of balance and compatibility with a discontinuity in \({\Gamma }_{0}\) along which the displacement jump \(\delta\) is not constant. Let us recall the boundary conditions of the problem:

\(\{\begin{array}{ccc}u=U(x,y)& \mathrm{sur}& \partial \Omega \setminus {\Gamma }_{0}\\ u={U}_{0}(y)-\delta (y)& \mathrm{sur}& {\Gamma }_{0}\end{array}\) eq 2.1-4

2.2. Uniqueness of the solution#

After building an analytical solution, it is important to ensure that it is unique in order to be able to compare it with the digital solution. We show, see [bib2], that uniqueness is guaranteed as soon as the following condition, on the geometry of the domain as well as on the material parameters, is verified:

\(L<2\mu \frac{{G}_{c}}{{\sigma }_{c}^{2}}\). Eq 2.2-1

The dimensions of the plate and the material parameters given above verify this condition.

2.3. Bibliographical references#

  1. FUNG Y.C.: Foundation of Solid Mechanics, Prentice-Hall, (1979).

  2. laverne J.: Energetic formulation of rupture by cohesive force models: theoretical considerations and numerical implementations, Doctoral thesis of the University of Paris13, November 2004.