2. Reference solution

2.1. Calculation method

Ratings:

\(u\): displacement;

\(\sigma\): constraint;

\(\epsilon\): deformation;

\(\delta =⟦u⟧\): displacement jump through cohesive discontinuity;

\({\delta }_{c}=\frac{2{G}_{c}}{{\sigma }_{c}}\): displacement jump corresponding to the rupture of the bar (zero stress).

It is an analytical solution. Since the problem is one-dimensional, all quantities are scalar and the constraint is constant in space. The problem is symmetric and can therefore be restricted to interval \([\mathrm{0,}L]\). We then have the following set of equations:

In the bar:

\(\sigma =E\epsilon\) (eq1)

\(\frac{\mathit{du}}{d\widehat{x}}=\epsilon\) (eq2)

At the level of discontinuity (cohesive law refines in a softening regime):

\(\delta ={\delta }_{c}(1-\frac{\sigma }{{\sigma }_{c}})\) (eq3)

By integrating (eq2) over the interval \([\mathrm{0,}L]\) and using (eq1), we obtain the following relationship:

\(u(\widehat{x}=L)-u(\widehat{x}=0)=L\epsilon\), or \(U-\frac{\delta }{2}=L\frac{\sigma }{E}\) (eq4)

The imposed displacements that correspond to stress levels \({\sigma }_{c}\) (threshold for opening the discontinuity) and 0 (broken material) respectively are denoted \({U}_{c}\) and \({U}_{f}\). By applying (eq4), it comes:

\({U}_{c}=L\frac{{\sigma }_{c}}{E}\) and \({U}_{f}=\frac{{\delta }_{c}}{2}\) (eq5)

It is assumed that the values chosen for the material parameters \(E\), \({\sigma }_{c}\) and \({G}_{c}\), and for the length of the bar \(L\), lead to a stable response of the bar (absence of « snap-backs »). We also assume increasing monotonic loading, so we have \({U}_{c}⩽U⩽{U}_{f}\) in a non-linear regime. The values of the material and geometric parameters must therefore verify the following inequality:

\(L\frac{{\sigma }_{c}}{E}<\frac{{\delta }_{c}}{2}\), or \(\frac{L}{E}<\frac{{G}_{c}}{{\sigma }_{c}^{2}}\) (eq6)

When this inequality is true, we can then express the constraint as a function of the imposed displacement:

\(\sigma =\frac{\frac{{\delta }_{c}}{2}-U}{\frac{{\delta }_{c}}{2{\sigma }_{c}}-\frac{L}{E}},\forall U\in [:ref:\)frac{L{sigma }_{c}}{E},frac{{delta }_{c}}{2} <frac{L{sigma }_{c}}{E},frac{{delta }_{c}}{2}>`] `(eq7)

2.2. Reference quantities and results

At the instant corresponding to the imposed displacement \({U}_{\mathit{test}}=0.0199\mathit{mm}\), we test the constraint \(\sigma\) (constant in space) as well as the displacement \(u\) at the point \(\widehat{x}={0}^{\text{+}}\) (i.e. the half-jump in displacement) are tested.

\(\sigma =\frac{\frac{{\delta }_{c}}{2}-\mathrm{2L}\frac{{\sigma }_{c}}{E}}{\frac{{\delta }_{c}}{2{\sigma }_{c}}-\frac{L}{E}}\) digital application: \(\sigma =1.72344975053\mathit{MPa}\)

\(u(\widehat{x}={0}^{\text{+}})=2{\delta }_{c}(1-\frac{\sigma }{{\sigma }_{c}})\) digital application: \(u(\widehat{x}={0}^{\text{+}})=0.0141838916607\mathit{mm}\)

Identification	Reference type	Reference value
Under load \({U}_{\mathit{test}}=0.0199\mathit{mm}\), at any point: \(\sigma\)	“ANALYTIQUE”	\(1.72344975053\mathit{MPa}\)
Under load \({U}_{\mathit{test}}=0.0199\mathit{mm}\), in \(\widehat{x}={0}^{\text{+}}\): \(u\)	“ANALYTIQUE”	\(0.0141838916607\mathit{mm}\)