1. Generalities

The classical approach is the historical approach to the mechanics of rupture. It was initially developed in the context of linear elasticity and then extended to non-linear elasticity.

1.1. Linear elastic mechanics of fracture in quasistatics

This paragraph recalls the characteristic parameters in linear elastic mechanics of rupture.

1.1.1. Stress intensity factors

Stress intensity factors characterize the singularity of stress at the crack point. Their general expression is of the form: \(K\mathrm{=}\underset{r\to 0}{\mathrm{lim}}\sigma (r)\sqrt{r}\). Three stress intensity factors are defined, associated with the three modes of opening the crack.

In linear elasticity, stress intensity factors make it possible to decompose the displacement field \(\mathrm{u}\) into a singular part and a regular part [1] [6]:

\(u\mathrm{=}{u}_{R}+{K}_{I}{u}_{S}^{I}+{K}_{\mathit{II}}{u}_{S}^{\mathit{II}}+{K}_{\mathit{III}}{u}_{S}^{\mathit{III}}\).

1.1.2. Energy return rate

We consider a cracked elastic solid occupying domain \(\Omega\). Be:

\(\mathrm{u}\) the field of movement,

\(T\) the temperature field,

\(\mathrm{f}\) the field of volume forces applied to \(\Omega\),

\(\mathrm{g}\) the field of surface forces applied to a part \(S\) of \(\partial \Omega\),

\(U\) the field of movements imposed on a \({S}_{d}\) part of \(\mathrm{\partial }\Omega\).

\(\sigma\) the stress tensor,

\(\varepsilon\) the strain tensor,

\({\varepsilon }^{\mathrm{th}}\) the tensor of thermal deformations,

\(\psi (\varepsilon ,T)\) free energy density.

The \(G\) energy return rate corresponds to Griffith [5]”s energetic break approach. It is defined by the opposite of the derivative of the potential energy at equilibrium \(W(u)\) with respect to the domain \(\Omega\):

\(G=\frac{-\partial W(u)}{\partial \Omega }\)

with: \(W(\mathrm{u})\mathrm{=}\underset{\Omega }{\mathrm{\int }}\psi (\varepsilon (\mathrm{u}),T)d\Omega \mathrm{-}\underset{\Omega }{\mathrm{\int }}\mathrm{f}\mathrm{u}d\Omega \mathrm{-}\underset{S}{\mathrm{\int }}\mathrm{g}\mathrm{u}d\Gamma\)

We recall that the energy restoration rate is equivalent to the Rice integral in linear elasticity [4].

In plane linear elasticity, the stress intensity coefficients are linked to the energy restoration rate by Irwin’s formula:

\(G\mathrm{=}\frac{1\mathrm{-}{\nu }^{2}}{E}({K}_{I}^{2}+{K}_{\mathit{II}}^{2})\) in plane deformations

\(G\mathrm{=}\frac{1}{E}({K}_{I}^{2}+{K}_{\mathit{II}}^{2})\) in plane constraints

\(G=\frac{1-{\nu }^{2}}{E}({K}_{I}^{2}+{K}_{\mathrm{II}}^{2})+\frac{{K}_{\mathrm{III}}^{2}}{2\mu }\) with \(\mu =\frac{E}{2(1+\nu )}\), in 3D

1.2. Extension to nonlinear elasticity

The previous definitions are only rigorous in linear thermoelasticity but extensions are possible to nonlinear problems. In particular, it is possible to define and calculate the rate of energy restoration in non-linear elasticity, provided that the load remains radial and monotonic.