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

The imposed loading allows us to obtain a homogeneous solution, equivalent to a bar of length :math:`L` subjected to uniaxial tensile loading. We can then express elastic energy in the following way:

:math:`{W}_{\mathrm{el}}=\frac{E}{2}{(\frac{\mathrm{dx}}{L})}^{2}`

The damage values associated with the moments :math:`{t}_{1}`, :math:`{t}_{2}` and :math:`{t}_{3}` are deduced analytically from the formula:

:math:`d=1-{(\frac{{\sigma }_{y}L}{E\mathrm{dx}})}^{2}`

That is: :math:`{d}_{1}=0.`, :math:`{d}_{2}=0.36` and :math:`{d}_{3}=0.75`. We then consider that the test is verified if Newton returns the same damage values to us, at a precision of :math:`{10}^{-6}`.