1. The second gradient model

1.1. Brief presentation of microstructure media

At the origin of this theory, we find the work of Mindlin ([6], [7]) in the context of linear elasticity. This work was then taken up by Germain ([8], [9]) who gave an expression of it by applying the principle of virtual work, the basis of numerical methods for application by finite elements.

This theory involves the definition of rich kinematics. In addition to the classical field of displacements \({u}_{i}\), we consider the second-order tensor, noted \({f}_{\text{ij}}\) and called microscopic kinematic gradient, which models both the deformations and the rotations at the grain scale of the structure. Attention is drawn here to the fact that, in the context of microstructured media, the microscopic deformation gradient has no reason to be linked to the gradient of any field dependent on macroscopic displacement. It is not necessarily symmetric. The microscopic deformation gradient \({f}_{\text{ij}}\) is a variable in the same way as the macroscopic displacement \({u}_{i}\), in contrast to the classical (macroscopic) deformation field, which is obtained by derivation

\[\]

: label: EQ-None

{epsilon} _ {text {ij}} =frac {1} {2}} (frac {partial {u} _ {i}} {partial {x} _ {j}}} +frac {partial {j}}} +frac {partial {i}}} +frac {partial {u}} _ {j}})

An essential fact at the basis of the writing of this theory concerns the statement of the principle of objectivity or also of material indifference:

The virtual power of the forces within a system is zero in any virtual movement rigidifying the system at the moment in question.

By neglecting the expression of external volume efforts for reasons of simplification of writing, the consequence of the axiom of the virtual powers of internal forces leads to the expression of the variational formulation, for any kinematically admissible field \(({u}_{i}^{\text{*}},{f}_{\text{ij}}^{\text{*}})\)

(1.1)\[ \ underset {\ omega} {\ int} ({\ sigma} _ {\ sigma} _ {\ text {ij}}}\ frac {\ partial {u} _ {i}} ^ {\ text {*}}}} {\ partial {x} _ {j}}} _ {\ tau}} _ {\ text {ij}}} ^ {\ text {*}}}} {\ text {*}}}} {\ partial {x} _ {j}}} + {\ tau}}} + {\ tau}}} + {\ tau}} _ {\ tau} _ {\ text {ij}}} -\ text {}\ frac {\ partial {u} _ {i}} ^ {\ text {*}}} {\ partial {x} _ {j}}) + {\ Sigma} _ {\ text {ijk}}}\ frac {\ partial {f}}}\ frac {\ partial {f}}}\ frac {\ partial {f}}}\ frac {\ partial {f}}}\ frac {\ partial {f}} _ {k}}\ frac {\ partial {f}} _ {k}}\ frac {\ partial {f}} _ {k}}\ frac {\ partial {f}} _ {k}}\ frac {\ partial {f}} _ {k}})\ frac {\ partial {f}} _ {k}}\ frac} =\ underset {\ partial\ Omega} {\ int} {\ int} ({t} _ {i} {u} _ {i} ^ {\ text {*}}} + {T} _ {\ text {ij}}} {\ text {ij}}} {f}} {f}} _ {\ text {ij}}} {f}} _ {f}} _ {\ text {ij}}} {f}} _ {f}} _ {f}} _ {f}} _ {f}} _ {f}} _ {f}} _ {f} _ {f}} _ {f}} _ {f} _ {f}\]

where \({t}_{i}\) and \({T}_{\text{ij}}\) are respectively the tensile forces and the double forces corresponding to the boundary conditions, on the border \(\partial \mathrm{\Omega }\), combined with kinematic variables.

The variational formulation () is another way to express balance relationships that are expressed

(1.2)\[ \ frac {\ partial ({\ sigma} _ {\ text {ij}}} - {\ tau} _ {\ text {ij}})} {\ partial {x}} _ {j}} =0\]

(1.3)\[ \ frac {\ partial {\ Sigma} _ {\ text {ijk}}}} {\ partial {x} _ {k}} - {\ tau} _ {\ tau} _ {\ text {ij}}} =0\]

and we find for the expression boundary conditions

\[\]

: label: eq-4

{t} _ {i} = ({sigma}} _ {text {ij}} - {tau} _ {text {ij}}) {n} _ {j}

\[\]

: label: eq-5

{T} _ {text {ij}}} = {Sigma} _ {text {ijk}} {n} _ {k}

where \({n}_{j}\) refers to the outgoing normal at border \(\partial \mathrm{\Omega }\).

To complete the problem, it is necessary to define the laws of behavior that will link the static variables \({\sigma }_{\text{ij}}\), \({\tau }_{\text{ij}}\), \({\Sigma }_{\text{ijk}}\) respectively to the history of the kinematic variables of \(\frac{\partial {u}_{i}}{\partial {x}_{j}}\), \(\left({f}_{\text{ij}}-\frac{\partial {u}_{i}}{\partial {x}_{j}}\right)\) and \(\frac{\partial {f}_{\text{ij}}}{\partial {x}_{k}}\).

These models have already proven to be effective in terms of regularization. However, they are complex in their use because of the different laws of behavior to be specified. In addition, discretization by the finite element method in 3D induces the addition of 9 additional degrees of freedom per node corresponding to components \({f}_{\text{ij}}\). The calculation times are then relatively long and therefore not compatible with the type of studies we want to carry out.

1.2. The second gradient model

Starting from the previous model, expressed by the relationship (), we can restrict the kinematics by forcing the microscopic gradient to be equal to the macroscopic gradient (see Chambon et al [2] for a detailed analysis)

(1.4)\[ {f} _ {\ text {ij}}\ mathrm {=}\ frac {\ mathrm {\ partial} {u} _ {i}} {\ mathrm {\ partial} {\ partial} {x}} _ {j}}\]

The advantage of this hypothesis is to reduce the number of independent variables and to introduce simpler laws of behavior. The expression of virtual powers reworked after some algebraic manipulations is then written for any kinematically admissible field \({u}_{i}^{\text{*}}\)

(1.5)\[ \ underset {\ omega} {\ int} ({\ sigma} _ {\ sigma} _ {\ text {ij}}}\ frac {\ partial {u} _ {i}} ^ {\ text {*}}}} {\ partial {x} _ {j}} _ {j}}} + {\ sigma}}} + {\ sigma}} _ {j}}} + {\ sigma}} + {\ Sigma}} _ {\ sigma}} _ {i}}} + {\ Sigma}} _ {\ sigma}} _ {i}}} + {\ Sigma}} _ {\ Sigma}} _ {\ Sigma} _ {\ Sigma}} _ {\ Sigma} _ {\ text {ijk}}}\ frac {{\ partial} ^ {u}}} {*}}} {\ partial {x} _ {j}\ partial {x} _ {k}})\ text {dv} =\ underset {\ partial\ Omega} {\ int} ({p} _ {i} {u} {u} _ {u} _ {u} _ {u} _ {i}} ^ {i}} ^ {i}} ^ {i}} ^ {i}} ^ {i}} ^ {i}} ^ {i}} ^ {i}} ^ {i}} ^ {i}} {\ text {\ text}} *}})\ text {ds}\]

where \({p}_{i}\) and \({P}_{i}\) are the boundary conditions defined by

(1.6)\[ {p} _ {i} = {\ sigma} _ {\ text {ij}} _ {\ text {ij}}} {n} _ {k} {n} _ {j} {\ mathit {D\ Sigma}}} _ {\ text {\ Sigma}}} _ {\ text {\ Sigma}}} _ {\ text {\ Sigma}}} _ {\ text {\ Sigma}}} _ {\ text {\ Sigma}}} _ {\ text {\ Sigma}}} _ {\ text {\ Sigma}}} _ {\ text {\ Sigma}}} _ {\ text {\ Sigma}}} _ {\ text {\ Sigma}}} _ {k}} {n} _ {j} -\ frac {{\ mathit {D\ Sigma}}} _ {\ text {ijk}}} {{\ text {Dx}} _ {j}} {n} _ {k} -\ k} +\ frac {{\ mathit {D\ Sigma} _ {k}} +\ frac {{\ text {Dn}}} +\ frac {{\ text {Dn}}} +\ frac {{\ text {Dn}}} {\ text {Dn}}} {\ text {Dx}} _ {l}} {\ Sigma} _ {k}} +\ frac {{\ text {Dn}}} {\ text {Dn}}} {\ text {Dx}}} {\ ijk}} {n} _ {j} {n} {n} _ {k} -\ frac {{\ text {Dn}} _ {j}} {{\ text {Dx}}} _ {k}} _ {k}} {\ Sigma}} {\ Sigma} _ {\ text {ijk}}\]

(1.7)\[ {P} _ {i} = {\ Sigma} _ {\ text {ijk}} {n} _ {n} {n} _ {k}\]

with

\(\mathit{Dq}\) which refers to the normal derivative of the variable \(q\): \(\mathit{Dq}\mathrm{=}\frac{\mathrm{\partial }q}{\mathrm{\partial }{x}_{j}}{n}_{j}\)

\(\frac{Dq}{D{x}_{j}}\) which designates the tangential derivative of the variable \(q\): \(\frac{Dq}{D{x}_{j}}\mathrm{=}\frac{\mathrm{\partial }q}{\mathrm{\partial }{x}_{j}}\mathrm{-}{n}_{j}\mathit{Dq}\)

The hypothesis on the equality between microscopic and macroscopic deformation fields (6) has a direct impact on the expression of boundary conditions because the variables \({u}_{i}^{\text{*}}\) and \({f}_{\text{ij}}^{\text{*}}\) are no longer independent.

It has already been shown that this model corrects the dependence of the thickness of the localization bands on the discretization of the mesh (see Chambon et al [1] or Matsushima et al [3]). For this, the model can be used by taking into account two different laws of behavior, one to describe the first classical gradient part and the other for the second gradient. As far as the latter is concerned, any relationship could be considered, but up to now, linear elasticity has generally been chosen.