2. Continuous model

2.1. Behavioral equations: local plastic model

We place ourselves in large logarithmic deformations, noting \(E\) the mechanical deformation (logarithmic) and \(T\) the associated stress (by duality). The internal variables of the model consist of plastic deformation \({E}^{p}\), the work hardening variable \(\kappa\), damage \(d\) (which depends on porosity \(f\)) and cumulative plastic deformation (which governs part of the germination porosity).

First of all, the stress-deformation relationship is based on an additive decomposition into a plastic part and an elastic part:

(2.1)\[ \begin{align}\begin{aligned} T=K\ mathit {tr} (E- {E} ^ {p})\ mathit {Id} +2\ mathrm {\ mu}\ mathit {dev} (E- {E} ^ {p})\\ \ textrm {with}\\ 3K=\ frac {E} {1-2\ mathrm {\nu}}} `and:math: `2\ mathrm {\ mu} =\ frac {E} {1+\ mathrm {\nu}}\end{aligned}\end{align} \]

where :math:`` and :math:`` respectively designate the trace and the deviator of a second-order tensor.

The plasticity threshold function noted \(F\) depends on the level of work hardening \(\kappa\) through the work hardening function :math:``, on an (isotropic) measurement of the intensity of the stresses noted \({T}_{\text{*}}\) which also depends on the damage and the deformation through the Jacobian of the transformation \(J=\mathrm{exp}(\mathrm{tr}E)\). It is written as follows:

(2.2)\[ F (T,\ mathrm {\ kappa}\ text {;} d, E) =\ frac {{T} _ {\ text {*}} (T, d)} {J (E)}} {J (E)}} -R (\ mathrm {\ kappa})\]

The work hardening function is identical to that deployed in law VMIS_ISOT_NL [R5.03.33], i.e.:

\[\]

: label: eq-4

R (mathrm {kappa}) = {R} _ {0} + {R} _ {0} + {R} _ {kappa} + {R} _ {1} (1- {e} ^ {{- {- {mathrm {gamma}} + {mathrm {gamma}}} _ {1}mathrm {kappa}} + {R} _ {1} (1- {e} ^ {2} {- {- {- {mathrm {gamma}}} _ {2} (1- {e} ^ {2} {- {- {mathrm {gamma}}} _ {2} {- {- {mathrm {gamma}}} _ {2} (1- {e} ^ {2} {- {mathrm {gamma}} _ {2}mathrm {kappa}}) + {R} _ {K} {({p} _ {0} +mathrm {kappa})})}} ^ {{mathrm {kappa})}} ^ {{mathrm {kappa})}

It depends on at most nine parameters, some of which may remain null if you do not want to activate all the terms of the expression (choice by default).

Note: we remind you that it is not possible to introduce a Lüders plateau contrary to law VMIS_ISOT_NL.

Gurson’s \({T}_{\text{*}}\) definition of constraint is implicit. Given stress \(T\) and damage \(d\), \({T}_{\text{*}}\) is the solution to the following equation:

\[\]

: label: eq-5

G (T, {T} _ {text {*}}, d) =0text {*}}, d) =0text {*}}, d)equiv {left (frac {{T} _ {frac {{T} _ {mathit {T}}}} {text {*}}}}right)} ^ {2} +2dmathrm {chm}left (frac {3} {2} {2} {q} _ {2}frac {{T}}} {{T} _ {text {*}}}}right) - {(1-d)} {2}} {2}text {;}mathrm {chm} (x)equivmathrm {}}}}right) - {(1-d)}}right) - {(1-d)}}right) - {(1-d)}}}right) - {(1-d)}}}right) - {(1-d)}}} ^ {2}text {2}text {;}mathrm {chm} (x) -1

where \({q}_{1}\) and \({q}_{2}\) are two material parameters and where stress invariants have been introduced:

(2.3)\[ \begin{align}\begin{aligned} {T} _ {H} =\ frac {1} {1} {3}\ mathrm {tr} (T)\\ \ textrm {and}\\ {T} _ {\ mathit {eq}} =\ sqrt {\ frac {3} {2}\ mathrm {dev} (T)\ mathrm {:}\ mathrm {dev} (T)}\end{aligned}\end{align} \]

The evolution of plastic deformation is governed by the flow equation. When the threshold function is differentiable with respect to \(T\), i.e. when \(T\ne 0\), the flow direction is normal to the threshold surface:

(2.4)\[ \ text {if} T\ne 0\ text {} {} {\ dot {E}}} ^ {p} =\ dot {\ mathrm {\ kappa}} N\ text {};\ text {} N=\ frac {\ partial F} {\ partial F}} {\ partial F} {\ partial F} {\ partial F} {\ partial F} {\ partial F} {\ partial F} {\ partial F} {\ partial F} {\ partial F} {\ partial F} {\ partial F} {\ partial F} {\ partial T} {\ partial T} {\ partial T} =-\ partial T} =-\ frac {1} {J} {\ left (\ frac {\ partial G}} {\ partial G} {\ text {}} N=\ frac {\ partial text {*}}}\ right)} ^ {-1}\ left (\ frac {\ partial G} {\ partial T}\ right)\]

In the singular case where \(T=0\), the derivative is generalized via notions of convex analysis (sub-gradient). The flow direction is then only subject to the following condition:

(2.5)\[ \ text {if} T=0\ text {} {} {\ mathrm {\ Pi}}} _ {N} (\ dot {{E} ^ {p}}, d)\ le\ frac {\ dot {\ dot {\ mathrm {\ mathrm {\ kappa}}}} {J}\ text {}}\]

The support function \({\mathrm{\Pi }}_{N}\) has the following expression, with notations similar to ():

(2.6)\[ {\ mathrm {\ Pi}} _ {N} (\ dot {{E}} ^ {p}}, d) =\ frac {2} {q} _ {2}}\ mathrm {arcosh} (1+ {C}} (1+ {C}}) (\ dot {{E}}}\ left|\ dot {{E} _ {H}} {2}}}\ mathrm {arcosh}} (1+ {C}} ^ {h}}}\ arcosh} (1+ {C}} ^ {h}}}\ arcosh} (1+ {C}} ^ {H}} arcosh} (1+ {C}) ^ {C} ^ {p}} (1+ {C} ^ ^ {p}} (1+ ac {2} {3}\ left (1-d\ right)\ sqrt {1-\ mathrm {\ gamma} {C} ^ {\ text {*}}}}\ dot {{E}} _ {\ mathit {eq}}} ^ {p}}\]

(2.7)\[ {C} ^ {\ text {*}}} =-\ mathrm {\ chi}} +\ sqrt {{\ mathrm {\ chi}} ^ {2} +\ frac {2} {\ mathrm {\ gamma}}}} (\ mathrm {\ gamma}}}} (\ mathrm {\ chi}}} =\ frac {\ gamma}}}} {\ mathrm {\ gamma}}}} (\ mathrm {\ gamma}}}} (\ mathrm {\ gamma}}}} (\ mathrm {\ gamma}}}} (\ mathrm {\ gamma}}}} (\ mathrm {\ gamma}}}} (\ mathrm {\ gamma}}}} (\ mathrm {\ chi}}}) 2}\ dot {{E} _ {\ mathit {eq}}} ^ {eq}}}} ^ {p}})} ^ {2} +\ frac {{{E} _ {H} ^ {p}})})} ^ {2}} {2}\ dot {{E} _ {\ mathit {eq}}})} ^ {p}})} ^ {p}})} ^ {2}}\ text {;}\ mathrm {\ gamma} =\ frac {2d} {{\ left (1-d\ right)}} ^ {2}}\]

The evolution of the work-hardening variable is fixed by the consistency condition:

(2.8)\[ \ dot {\ mathrm {\ kappa}}\ ge 0\ text {;} F (T,\ mathrm {\ kappa})\ le 0\ text {;}\ dot {\ mathrm {\ kappa}}\ dot {\ mathrm {\ kappa}}} F (T,\ mathrm {\ kappa}) =0\]

Finally, it remains to define the evolution of the damage. The model is specific to EDF; it coincides with the classic GTN model in charge but it also specifies what happens in landfill. More precisely, damage is defined as follows:

(2.9)\[ d= {q} _ {1} f+ {d} ^ {\ mathit {co}}\ text {;} f= {f} ^ {\ mathit {nu}}} + {f}} ^ {f} ^ {\ mathit {gr}}\]

Where \(f\) is the porosity, the sum of the germination porosity (nucleation in English) \({f}^{\mathit{nu}}\) and the growth porosity (growth) \({f}^{\mathit{gr}}\), and \({d}^{\mathit{co}}\) is the coalescence damage. In the initial (natural) state, we have:

(2.10)\[ {f} ^ {\ mathit {nu}} (0) =0\ text {;} {f} ^ {\ mathit {gr}} (0) = {f} _ {0}\ text {;} {d}} {d} ^ {\ mathit {d}} ^ {\ mathit {co}} ^ {\ mathit {co}}} (0) ^ {\ mathit {co}}} (0) = {f} _ {0}\]

With \({f}_{0}\) the initial porosity.

Growth porosity results from the conservation of plastic volume:

(2.11)\[ \ dot {{f} ^ {\ mathit {gr}}}} = (1-f)\ mathrm {tr}\ dot {{E} ^ {p}}\ text {;} {f}} ^ {\ mathit {gr}} ^ {\ mathit {gr}}} {\ mathit {gr}}}}\ ge {f} _ {0}\]

It cannot go below \({f}_{0}\) but it is reversible (you can close the cavities).

With regard to germination porosity, two families of models are implemented:

(2.12)\[ {f} ^ {\ mathit {nu}} = {f} _ {(1)}} ^ {\ mathit {nu}} + {f} _ {(2)}} ^ {\ mathit {nu}} ^ {\ mathit {nu}}\]

The first family adopts a change in porosity linked to that of the work-hardening variable:

(2.13)\[ {\ dot {f}} _ {(1)}} ^ {\ mathit {nu}}} = {B} _ {n} (\ mathrm {\ kappa})\ dot {\ mathrm {\ kappa}}\ text {}\ kappa}}}\ text {}\ Rightarrow\ text {}\ Rightarrow\ text {}\ Rightarrow\ text {}\ Rightarrow\ text {}\ Rightarrow\ text {}\ Rightarrow\ text {} Rightarrow\ text {} {Rightarrow\ text {} {f}} _ {n}} (\ mathrm {\ kappa}}) =\ underset {0} {\ overset {\ mathrm {\ kappa}} {\ int}} {\ int}} {B} _ {n} (s)\ mathit {ds}\]

To cover two models used in the literature, we choose the following function \({B}_{n}\):

(2.14)\[ {B} _ {n} (\ mathrm {\ kappa}) =\ frac {{f}} _ {N}} {{s} _ {N}\ sqrt {2\ mathrm {\ pi}}}}\ mathrm {exp}}\ left [-\ frac {1} {2}} {\ left (\ frac {\ mathrm {\ pi}}}}}}\ mathrm {\ pi}}}}}\ mathrm {exp}}\ left [-\ frac {1} {2} {\ left (\ frac {\ mathrm {\ pi}}}}}}\ mathrm {exp}}\ left [-\ frac {1} {2}} {\ left (\ frac {\ mathrm {mathrm {\ kappa}} _ {N}} {{s}} {{s}} _ {N}}\ right)} ^ {2}\ right] +\ frac {{c} _ {0}} {{\ mathrm {\ kappa}} {\ kappa}} _ {\ kappa}} _ {\ mathrm {\ chi}}} {{\ mathrm {\ kappa}}} _ {\ kappa}} _ {\ kappa}} _ {\ kappa}} _ {\ kappa}} _ {\ kappa}} _ {\ kappa}} _ {\ kappa}} _ {\ kappa}} _ {{\ kappa}} _ {\ kappa}} _ {\ kappa}} _ {\ kappa}} _ {\ mathrm {\ kappa}} _ {i}\ text {;} {\ mathrm {\ kappa}} _ {f}]]}\]

where \({\chi }_{I}\) is 1 in the \(I\) range and 0 outside. Taking into account the primitives of these two terms, we deduce:

(2.15)\[ {f} _ {(1)}} ^ {\ mathit {nu}}} =\ frac {{f}} _ {N}} {2}\ left [\ mathrm {erf}\ left (\ frac {\ mathrm {\ kappa}} - {\ mathrm {\ kappa}} - {\ mathrm {\ kappa}} - {\ mathrm {\ kappa}} - {\ mathrm {\ kappa}}} _ {N}}} {\ sqrt {2} {s}} _ {N}}\ left (\ frac {\ mathrm {\ kappa}} - {\ mathrm {\ kappa}} - {\ mathrm {\ kappa}} - {\ mathrm {\ kappa}} - {) +\ mathrm {erf}\ left (\ frac {{\ mathrm {\ kappa}} _ {N}} {\ sqrt {2} {s} _ {N}}\ right)\ right)\ right]\ right]\ text {+} {right]]\ text {+} {c}}\ text {+} {c} _ {c} _ {c} _ {0}\ phantom {\ rule {0.5em} {0ex}}\ mathrm {min}\ right)\ right]\ right]\ text {+} {c} _ {c} _ {c} _ {c} _ {0}\ phantom {\ rule {0.5em} {0ex}}\ mathrm {min}\ right)\ right]\ right]\ text {+} {c} _\ mathrm {\ kappa} - {\ mathrm {\ kappa}}} _ {i} ⟩} {{\ mathrm {\ kappa}} _ {f} - {\ mathrm {\ kappa}}} _ {i}} ,1\ right)\]

where McCauley’s brackets \(⟨x⟩\) refer to the positive part of \(x\).

The second family of germination models depends on the cumulative plastic deformation which is defined as follows:

(2.16)\[ {\ dot {E}} _ {\ mathit {eq}}} ^ {eq}} ^ {p}}\ equiv\ sqrt {\ frac {2} {3} {\ dot {E}}} ^ {p}} ^ {p}}\ mathrm {:}\ mathrm {:}:} {\ dot {E}}\]

The established germination function is proportional to \({E}_{\mathit{eq}}^{p}\) starting from a threshold:

(2.17)\[ {f} _ {(2)} ^ {\ mathit {nu}}} ({E}} _ {\ mathit {eq}} ^ {p}) = {b} _ {0} {E} _ {\ mathit {eq}} _ {e} ^ {p}}) = {b} _ {c} ⇒ {e} ^ {p}\]

As the work-hardening variable \(\kappa\) and the cumulative plastic deformation \({E}_{\mathit{eq}}^{p}\) are increasing by definition, the same is true of the germination porosity; its evolution is irreversible.

The evolution of coalescence damage remains to be defined. For this, the definition (usual for GTN) of porosity is introduced in the \({f}^{\text{*}}\) coalescence regime, where \({f}_{c}\) designates the coalescence porosity threshold and \({f}_{F}\) the rupture porosity:

\[\]

: label: eq-21

{f} ^ {text {*}}} ={begin {array} {cc} {cc} f&text {si} f< {f} _ {c}\ {f} _ {c} +mathrm {delta} (f-delta}} (f- {delta}} (f- {f}) {f} _ {c} +mathrm {delta} (f- {delta}} (f- {f} _ {c}) &text {f} _ {c}) &text {f} _ {c}end {array}

Coalescence damage then obeys the following law of evolution:

(2.18)\[ {d} ^ {\ mathit {co}} (t) =\ underset {\ mathrm {\ tau}\ le t} {\ text {max}} [{q} _ {1} {f}} ^ {\ text {*}} ^ {\ text {*}}} (\ mathrm {*}}}} (\ mathrm {*}}}} (\ mathrm {*}}}} (\ mathrm {\ tau}}}) - {q} _ {1} f (\ mathrm {\ tau})]\]

It is also increasing by definition, and therefore irreversible.

For numerical reasons, we also introduce a breaking damage threshold \({d}_{f}\) beyond which we consider that the point is broken, that is to say \(T=0\) (there is therefore a jump in stress that can disturb the resolution of the equilibrium equations but facilitates the integration of the behavior). Once a stitch is broken, it stays that way for good. This mechanism is not activated by default (\({d}_{f}=1\)).

2.2. Taking viscosity into account

In the presence of viscosity, plastic flow is delayed. More exactly, the speed of evolution of work hardening is no longer fixed by the consistency condition () but it is now a function of the intensity of the threshold function according to Norton’s law:

(2.19)\[ \ dot {\ mathrm {\ kappa}} = {\ left (\ frac {⟨F (T,\ mathrm {\ kappa}) ⟩} {\ stackrel {~} {K} {K}} {K}}}\ right)} ^ {n}\]

The other equations in the model remain the same. When \(\stackrel{~}{K}\to 0\), we find the elastoplastic model.

In general, parameter \(\stackrel{~}{K}\) coincides with parameter \(K\) in the Norton model. However, when damage \(d\) approaches 1, the gap between \({T}_{\text{*}}\) and \(T\) increases (since \(T\) tends to zero) and viscoplasticity can become too brutal to maintain an effective role, especially in the presence of numerical integration errors. This is why a modification of the usual Norton’s law is proposed by coupling viscoplasticity with damage, beyond a threshold \({d}_{v}\):

(2.20)\[\begin{split} \ begin {array} {cc}\ text {si} d\ le {d} _ {v} &\ stackrel {~} {K} =K\\\ text {si} d\ ge {d} _ {v} &\ text {} _ {v} &\ text {} {v} _ {v} _ {v} _ {v} _ {v} _ {v} _ {v} _ {v}} {v} {v}} {v} {v}} {v} {v}} {v} &\ text {} {v}} {v} {v}} {v} &\ text {} {v}} {v} {v}} {v} &\ text {} {v}} {v} {v}} {v} {v}} {v}\end{split}\]

Finally, viscoplasticity makes it possible to accommodate potential stress jumps by smoothing them over time. It is important to maintain this mechanism, including for broken spots. In this case, the cancellation of the constraints (broken point) is delayed in time, in a manner similar to a relaxation mechanism, provided you have chosen \({d}_{v}<1\). More precisely, in a viscoplastic broken point regime, plasticity and constraints obey the following law of evolution:

(2.21)\[ \ dot {\ mathrm {\ kappa}} =0\ text {;}\ dot {{E} ^ {p}} =0\]

(2.22)\[\begin{split} \ begin {array} {cc}\ text {si}\ Green T\ Vert\ Vert\ ge K (1- {d} _ {v}) &\ text {}\ frac {\ dot {T}} {E} =- {\ left [\ left [\ frac {\ left [\ frac {\ green T\ Vert\ Vert}} {K (1- {d} _ {v})}\ frac {v})}\ right]} ^ {n}\ frac {T}} =- {\ left [\ frac {T}}\ Vert T\ Vert}\\ text {si}\ Vert T\ Vert\ Vert\ le K (1- {d} _ {v}) &\ frac {\ dot {T}} {E} =-\ frac {T} {T} {T} {K} {K (1- {d} _ {v})}\ end {array}\end{split}\]

The transition to an exponent equal to 1 when \(\Vert T\Vert \le K(1-{d}_{v})\) avoids delaying the erasure of the constraint too much.

2.3. Taking into account the work-hardening gradient

A non-local formulation with an internal variable gradient is suitable when the \(\nabla \kappa\) work hardening gradient becomes important. It consists in introducing an additional term \({\mathrm{\Phi }}^{\mathit{grad}}\) into free energy that reflects the interactions between neighboring material points, see [R5.04.01]:

\[\]

: label: eq-27

{mathrm {Phi}}} ^ {mathit {grad}} (nablamathrm {kappa}) =frac {1} {2} c {left (nablamathrm {mathrm {kappa}right)} ^ {2} c {left (nablamathrm {mathrm {kappa}right)} ^ {2}

Using the relaxed augmented formulation presented in [R5.04.01], the impact on the behavioral relationship is reflected in a modification of the thermodynamic force associated with the work hardening variable and therefore, in practice, in a modification of the work hardening function:

(2.23)\[ \ stackrel {~} {R} (\ mathrm {\ kappa}) =R (\ mathrm {\ kappa}) +r\ mathrm {\ kappa} -\ mathrm {\ varphi}\ text {\ varphi}\ text {;}\ mathrm {\ kappa}) =R (\ mathrm {\ kappa}) +r\ mathrm {\ kappa} -\ mathrm {\ varphi} -\ mathrm {\ varphi}\ text {;}\ text {;};}\ mathrm {\ varphi} =\ mathrm {\ varphi} =\ mathrm {\ lambda} +\ mathit {ra}\]

where \(\lambda\) is the Lagrange multiplier associated with relaxation, \(r>0\) the increase coefficient and \(a\) the interpretation of the work-hardening variable at the scale of the structure. These three quantities are data as far as the behavioral relationship is concerned. In the end, the work hardening function is corrected by an affine term, which does not therefore add complexity compared to the numerical processing of the model. On the other hand, the sign and the amplitude of \(\varphi\) not being fixed, this explains why it is necessary to take into account the case of a singular flow for which \({T}_{\text{*}}=0\).