3. Principle of minimizing energy

3.1. Energy formulation in the elastic framework

An energetic formulation characterizes the response of a structure as a minimum of one energy in relation to the set of variables that have been chosen to describe the mechanical state of the structure [LOR 08]. In the elastic case, it is a question of minimizing the potential energy in relation to the field of movement. In order to take into account dissipative phenomena, the \({G}_{p}\) approach is based on the work of Frankfurt and Marigo [FRA 93, FRA 98] dedicated to a model of brutal partial damage. Unlike the elastic case, the energy depends on two fields, the displacement field \(u\) and a damage field \(\mathrm{\chi }\) with a value in \(\left\{\mathrm{0,1}\right\}\), \(0\) corresponding to the healthy material with free energy volume \({\mathrm{\varphi }}_{s}\) and \(1\) to the damaged material with free energy volume \({\mathrm{\varphi }}_{d}\):

(3.1)\[ {E} _ {\ mathit {tot}}\ left (u,\ mathrm {\ chi}\ right) =\ underset {\ mathrm {\ Omega}} {\ int}\ mathrm {\ varphi} (\ mathrm {\ varphi}} (\ mathrm {\ varphi}} (\ mathrm {\ varphi}} (u) (u),\ mathrm {\ epsilon} (u),\ mathrm {\ epsilon} (u),\ mathrm {\ chi}) + {w} _ {c}\ mathrm {\ varphi} (\ varphi}) (\ mathrm {\ varphi}} (\ mathrm {\ varphi}) (u),\ mathrm {\ chi})\ text {;}\ mathrm {\ varphi}} (\ mathrm {\ epsilon} (u),\ mathrm {\ chi}) =\ mathrm {\ chi} {\ mathrm {\ varphi}}} _ {d}} _ {d} (\ mathrm {\ epsilon}) + (1-\ mathrm {\ chi}) {\ mathrm {\ chi}) {\ mathrm {\ varphi}} _ {d}} _ {d} (\ mathrm {\ epsilon}) + (1-\ mathrm {\ chi}}) {\ mathrm {\ varphi}}} _ {d}} (\ mathrm {\ epsilon}) {\ mathrm {\ varphi}}} phi}} _ {s} (\ mathrm {\ epsilon})\]

where we restrict ourselves to imposed trips (the potential for external efforts \({W}_{\mathit{ext}}\) is zero). The additional term \({w}_{c}\mathrm{\chi }\) measures the energy required to go from healthy to damaged. After discretization in time, Frankfurt and Marigo postulate that the displacement fields \(u\) and damage fields \(\mathrm{\chi }\) realize a minimum of incremental potential energy:

(3.2)\[ (u,\ mathrm {\ chi}) =\ underset {u,\ mathrm {\ chi}} {\ mathit {argmin}} {E} _ {\ mathit {tot}}}\ left (u,\ mathrm {\ chi}\ right)\]

We then consider the sudden damage model presented by Frankfurt and Marigo, except that the residual stiffness is zero for a damaged condition (abrupt total damage model). So, we learn from the theory of Frankfurt and Marigo that at a given displacement, the evolution of damage is governed by the minimization of the total energy compared to \(\mathrm{\chi }\). In this case, total damage or external forces are no longer a problem. Thus, in the elastic framework, the evolution of the damage is obtained by minimizing the following total incremental potential energy:

(3.3)\[ {E} _ {\ mathit {tot}}\ left (u,\ mathrm {\ chi}\ right) = {\ int} _ {\ mathrm {\ Omega}}\ left [\ left (1-\ mathrm {\ chi}\ right) {\ mathrm {\ chi}}\ right) {\ mathrm {\ chi}}\ right) {\ mathrm {\ chi}\ right) {\ mathrm {\ chi}\ right) {\ mathrm {\ chi}\ right) {\ mathrm {\ chi}\ right) {\ mathrm {\ chi}\ right) {\ mathrm {\ chi}\ right) {\ mathrm {\ chi}\ right) {\ mathrm {\ chi}\ right) {\ mathrm {{C}\ right] d\ mathrm {\ omega} - {W} _ {\ mathit {ext}} (u)\]

where \({\mathrm{\Phi }}_{\mathit{el}}\) is the free energy density, \({\Phi }_{\mathit{el}}\mathrm{=}{\mathrm{\int }}_{\Omega }\frac{1}{2}\mathrm{[}\sigma \mathrm{:}{A}^{\text{-}1}\mathrm{:}\sigma \mathrm{]}d\Omega\) with \(A\) the tangent behavior matrix, and \({w}_{c}\) the energy volume dissipated in the damage process at each material point. In this new approach, we can take imposed force loads: the potential for external \({W}_{\mathit{ext}}\) efforts is not necessarily zero.

3.2. Changing the form of energy

One of the limitations of the proposed damage model is that it does not distinguish tension from compression, due to the shape of the free energy density, so that the restoration of compression energy contributes just as much to the propagation of the defect as tensile energy [HAB 16]. We choose to artificially eliminate the areas in compression from the integral (hypothesis H6). For this we consider a traction/compression asymmetry in the formulation of the energy considered, based on the method developed by Badel in [BAD 01]. Thus, by placing ourselves in the proper coordinate system of the deformations, the following elastic energy will be adopted:

(3.4)\[ {\ mathrm {\ Phi}} _ {t} ^ {\ mathit {el}} ({\ mathrm {\ epsilon}}} ^ {\ mathit {el}}) =\ frac {\ mathrm {\ lambda}} ^ {\ mathrm {\ lambda}} ^ {\ mathrm {\ lambda}}} {2}\ mathit {\ lambda}}} {2}\ mathit {tr}} {2}\ mathit {tr}} {2}\ mathit {tr} {2}\ mathit {tr}} {({\ mathrm {\ epsilon}})} ^ {\ mathit {el}})} ^ {2} H (\ mathit {tr} ({\ mathrm {\ epsilon}}} ^ {\ mathit {el}}))) +\ mathrm {\ mu}\ sum _ {i} {({\ mathrm {\ epsilon} {({\ mathrm {\ epsilon}}}} _ {\ epsilon}}} _ {i}} ^ {el}}))} ^ {2} H ({\ mathrm {\ epsilon}}} _ {\ epsilon}} _ {i}} ^ {el}})} ^ {2} H ({\ mathrm {\ epsilon}}} on}} _ {i} ^ {\ mathit {el}})\]

where \(\mathrm{\lambda }\) and \(\mathrm{\mu }\) designate the Lamé coefficients that characterize the stiffness tensor. The eigenvalues of the elastic deformation tensor are noted \({\mathrm{\epsilon }}_{i}^{\mathit{el}}\). \(H\) is the Heaviside function such as:

(3.5)\[\begin{split} H (x) =\ {\ begin {array} {c} 1\ text {si} x≥0\\ 0\ text {si} x<0\ end {array}\end{split}\]

3.3. Extension to the elastoplastic frame

In the case of elastoplastic behavior, the definition of total incremental energy \({E}_{\mathrm{tot}}\) is extended by Lorentz et al. [LOR 00], by defining new global potentials (free energy and dissipation). Here we are restricted to the framework of isotropic materials. The state of a material point is described by its deformation \(\mathrm{\epsilon }\), its damage \(\mathrm{\chi }\), but also by its plastic deformation \({\mathrm{\epsilon }}^{p}\) and internal variables \(\mathrm{\alpha }\) characterizing cold working. Assume:

that**plastic dissipation is decoupled from that related to damage**(almost fragile materials), (hypothesis* H7). The plastic dissipation potential \({D}_{\mathrm{pl}}\) therefore depends on plastic internal variables alone.

that the energy blocked by work hardening in dislocations :math:`{E}_{mathrm{bl}}` cannot be restored by the cracking mechanism (phenomenological hypothesis* H8). Thus, consider the following expression for incremental potential energy:

(3.6)\[ {E} _ {\ mathit {tot}}\ left (u, {\ mathrm {\ epsilon}}} ^ {p},\ mathrm {\ alpha},\ mathrm {\ chi}\ right) = {\ int}\ right) = {\ int}\ right) {\ int} _ {\ int} _ {\ mathrm {\ omega}}}\ left [\ left (1-\ mathrm {\ chi}\ right) {\ mathrm {\ chi}\ right) {\ int} _ {\ mathrm {\ omega}}}\ left [\ left (1-\ mathrm {\ chi}\ right) {\ mathrm {\ chi}\ right) {\ int}\ right) {\ int} _ {\ mathrm {\ omega}\ Phi}} _ {t} ^ {\ mathit {el}}} +\ mathrm {\ chi} {w} _ {C}\ right] d\ mathrm {\ Omega} + {E} _ {\ mathit {bl}}}\ left}\ left (\ mathrm {\ alpha}\ right) + {D} _ {\ mathit {pl}}\ left (\ mathrm {bl}}}\ left (\ mathrm {bl}}}\ left (\ mathrm {bl}}}\ left (\ mathrm {bl}}}\ left (\ mathrm {bl}}}\ left (\ mathrm {bl}}}\ left (\ mathrm {bl}}}\ left (\ mathrm {bl}}}\ left (\} {\ mathrm {\ epsilon}}} ^ {p},\ mathrm {\ Delta}\ mathrm {\ alpha}\ right) - {W} _ {\ mathit {ext}} (u)\]

where \(\Delta\) designates the variation of a quantity during the increment in question.