1. Cracking energy model#
The aim of this part is to draw up the general framework of a crack initiation and propagation model based on the principle of energy minimization. This model is inspired by those developed in the literature under the name « Cohesive Force Models » in the sense that one seeks to take into account a residual interaction between the lips of the crack. However, the major difference in our approach is the formulation of the problem in a larger framework by adopting the energy perspective introduced by Frankfurt and Marigo [3].
The idea consists in taking into account the process of energy dissipation during cracking thanks to an energy defined on a discontinuity surface and dependent on the jump in displacement through the latter. Here we will content ourselves with posing the problem in its continuous form in space and semi-discretized in time without trying to explain the form of surface energy.
The next part will be devoted to the implementation of the finite element with internal discontinuity based on this energy formulation.
1.1. General principle#
We consider an elastic structure defined by domain \(\Omega\) and a crack defined as a surface of discontinuity noted \(\Gamma\) through which the displacement \(u\) admits a jump \(\delta\):
We define the total energy \({E}_{T}\) of this structure as the sum of its elastic energy noted \(\Phi\), of a surface energy noted as \(\Psi\) and of \({W}^{\text{ext}}\) potential of external forces:
We denote \(\Phi ={\int }_{\Omega }\phi \mathrm{.}d\Omega\) and \(\Psi ={\int }_{\Gamma }\psi \mathrm{.}d\Gamma\), where \(\phi\) and \(\psi\) designate the elastic energy density and the surface energy density respectively. The total energy is a function of the displacement \(u\) and the displacement jump \(\delta\), the minimization problem is written as:
Recall that we are looking for a minimum local because the total energy is not bounded lower in the presence of external forces. Now, let’s see what hypotheses can be formulated on surface energy in order to take into account the condition of non-inter-penetration of the crack lips and depending on whether we consider cracking to be a reversible process or not.
1.2. Reversible model#
Let us present a first model that is simple, but not very realistic from a physical point of view at the macroscopic level, where it is assumed that the cracking of a material is a reversible process. It is considered that the total closure of the lips of a crack makes it possible to find healthy material. Surface energy can be written as the sum of a function of the Euclidean norm of the displacement jump and an indicator accounting for the condition of non-interpenetration of the lips:
: label: eq-4
Psi (delta) = {Psi} _ {mathrm {rev}} (paralleldeltaparallel) + {I} _ {{390} ^ {text {+}}}}} ({delta} _ {n})
with \({\delta }_{n}=\delta \cdot n\) where \(n\) is the normal to the discontinuity and \({I}_{{ℝ}^{\text{+}}}({\delta }_{n})\) is the indicator function:
: label: eq-5
{I} _ {{328} ^ {text {+}}}} ({delta} _ {n}) ={begin {array} {cc} +infty &text {si} {delta} {delta} _ {delta} _ {delta} _ {n} {delta} _ {n}text {array} _ {n}ge 0end {array} _ {n}end {array} _ {n}
The model developed in Code_Aster makes it possible to take into account the irreversibility of cracking; it is presented in the next section.
1.3. Memory model#
Based on the writing of the previous reversible model, an internal variable noted \(\kappa\) is introduced, making it possible to memorize the state of cracking at a given moment and thus to translate its irreversible nature. Its evolution is defined as follows:
We can equivalently express this law of evolution by means of a threshold function \(f\) defined by:
The law of evolution then takes the form of a classical coherence condition:
We define surface energy \(\Psi\) as a function of the displacement jump and the variable \(\kappa\):
with \(H\) Heaviside function defined by:
Surface energy \(\Psi\) will therefore be equal to \({\Psi }_{\mathrm{dis}}\) if the threshold is positive, representing the dissipation when the crack evolves. And it will be worth \({\Psi }_{\mathrm{lin}}\), translating an evolution without energy dissipation, if the threshold is negative. \({\Psi }_{\mathrm{lin}}\) takes into account the case where the crack closes; we choose a linear discharge, so the corresponding surface energy density \({\Psi }_{\mathrm{lin}}\), will be of the form:
where \(R(\kappa )\) and \({C}_{0}\) are chosen to ensure the continuous derivability of energy. Surface energy density \({\Psi }_{\mathrm{dis}}\) when cracking evolves can take many forms; we will give an example for the finite element model in chapter 5. From the surface energy densities \({\Psi }_{\mathrm{dis}}\) and \({\Psi }_{\mathrm{lin}}\) we can define the stress vector \(\overrightarrow{\sigma }\) through the discontinuity:
This is the interface law that we will adopt.
1.4. Temporal discretization#
Let us now present temporal discretization. First of all, it should be noted that we only consider semi-static evolutions, so the time is set by loading increments. By performing a time semi-discretization, we define the surface energy \(\Psi\) at an instant \(i\) as a function of the displacement jump and of \({\kappa }^{i\mathrm{-}1}\) internal variable at time \(i-1\):
The minimization problem can then be written as of now
:
The internal variable at the moment
is updated once the jump just now
known: