Benchmark solution
=====================

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


Analytical solution
-------------------

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


.. _RefEquation 2.1-1:

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

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

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

with :math:`\alpha ,\beta ,\gamma` and :math:`\eta` arbitrary real constants. The constraint field is deduced from [:ref:`éq 2.1-1 <éq 2.1-1>`]:


.. _RefEquation 2.1-2:

:math:`\{\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 :math:`E` the Young's modulus and :math:`\nu` the Poisson's ratio (which we take zero), we deduce the field of displacement in :math:`\Gamma` verifying the balance:


.. _RefEquation 2.1-3:

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

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

for all :math:`y` in :math:`\left[\mathrm{0,}H\right]`. Thus, the new displacement imposed on :math:`{\Gamma }_{0}` generating such a jump is equal to :math:`{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 :math:`{\Gamma }_{0}` along which the displacement jump :math:`\delta` is not constant. Let us recall the boundary conditions of the problem:


.. _RefEquation 2.1-4:

:math:`\{\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


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 [:ref:`bib2 <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:


.. _RefEquation 2.2-1:

:math:`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.


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.