3. L&K model equations#

3.1. Simplifying the model#

In order to best and concisely describe this version of the model, it is necessary to provide an overview of the original model. The difference between the two versions will be better perceived.

3.1.1. A brief overview of the thresholds of the original model#

In the original version of the L&K model as developed under the Flac software in CIH (cf.) or the thesis of A. Kleine (cf. R 4), there are three distinct mechanisms:

  • A pre-peak elastoplastic mechanism, governed by positive work hardening,

  • A viscoplastic mechanism also governed by positive work hardening,

  • A post-peak elastoplastic mechanism governed by negative work hardening describing fracturing.

The particularity of this original model lies in the fact that the coupling of the two pre-peak mechanisms triggers fracturing and therefore the post-peak mechanism. In fact, expansion cracks induce a deterioration in the mechanical properties of materials with the increase in dilatance.

For the elastoplastic mechanism, a load surface evolves through various thresholds. For the viscous mechanism, a viscoplastic surface evolves from an initial threshold to a final threshold.

The various thresholds delimit domains associated with particular physical mechanisms:

  • A damage threshold that is confused with the initial viscosity threshold,

  • An intermediate threshold, referred to as a cleavage limit,

  • A characteristic threshold defined as the envelope of the damage threshold and the cleavage limit, also called the contract/expansion limit (this limit is confused with the maximum viscoplasticity threshold),

  • A macroscopic peak threshold, defined from laboratory tests,

  • A purely conceptual intrinsic threshold defined as extrapolation of the peak threshold, (this threshold is eliminated in the simplified version of the model),

  • A residual resistance threshold.

_images/Object_69.svg

Figure 3.1.1-a *. Original model thresholds shown in the plan* \(({\sigma }_{\text{min}},{\sigma }_{\text{max}})\)

3.1.2. Characteristics of the simplified model#

The simplified version proposed by CIH (see R 2) is based on only two mechanisms: an elastoplastic mechanism and a viscoplastic mechanism.

  • The intrinsic threshold is eliminated in this release.

  • The characteristic threshold delimiting the areas of contraction and expansion in the pre-peak phase is linearized to avoid any numerical problem. It is assumed to be confused with the maximum viscosity threshold.

_images/Object_71.svg

Figure 3.1.2-a *. Simplified model thresholds in the plan* \(({\sigma }_{\text{min}},{\sigma }_{\text{max}})\)

3.2. Description of the mechanisms#

3.2.1. The viscoplastic mechanism#

This mechanism is activated as soon as the load point exceeds the initial viscoelastic threshold (equivalent to the initial elastic limit). Irreversible deformations are generated. These rates of irreversible deformation are proportional to the distance of the load point in relation to the viscosity threshold. The surface associated with the viscoplastic mechanism evolves from the initial elastic limit to the maximum viscoplastic threshold according to the irreversible deformations generated.

3.2.2. The elastoplastic mechanism#

3.2.2.1. Meadow Peak#

The elastoplastic mechanism is activated at the same time as the viscoplastic mechanism. As soon as the load point exceeds the initial elastic limit, the load surface begins to collapse positively.

3.2.2.2. Post Pic#

In the simplified version of the model, this mechanism is governed by:

  • negative work hardening from the peak threshold to the intermediate threshold,

  • negative work hardening from the intermediate threshold to the residual threshold.

3.2.3. Volume behavior#

The volume behavior, during the pre-peak phase, can be contracting or dilating.

Below the initial elastic limit, the behavior is contracting.

Below the contraction/expansion limit, the volume behavior is plastic contracting.

Beyond this limit, the volume behavior is dilating.

N.B: In the simplified version of the model, the contract/dilatance limit is confused with the maximum viscoplastic threshold.

3.3. Deformation tensor decomposition#

The decomposition of the total deformation increment is written as:

\(\underline{\underline{\dot{\varepsilon }}}\mathrm{=}{\underline{\underline{\dot{\varepsilon }}}}^{e}+{\underline{\underline{\dot{\varepsilon }}}}^{p}+{\underline{\underline{\dot{\varepsilon }}}}^{\text{vp}}\)

where \({\underline{\underline{\dot{\varepsilon }}}}^{e}\), \({\underline{\underline{\dot{\varepsilon }}}}^{p}\) and \({\underline{\underline{\dot{\varepsilon }}}}^{\text{vp}}\) are the increments of the elastic, instantaneous irreversible (plastic) and deferred irreversible (viscoplastic) tensors.

3.3.1. Hypo-elasticity#

The elastic law chosen is a hypo-elastic law:

\({\dot{\varepsilon }}_{\text{ij}}^{e}\mathrm{=}\frac{1+\nu }{E}{\dot{\sigma }}_{\text{ij}}\mathrm{-}\frac{3\nu }{E}\dot{p}{\delta }_{\text{ij}}\) or \({\dot{\varepsilon }}_{\text{ij}}^{e}\mathrm{=}\frac{1}{\mathrm{2G}}{\dot{s}}_{\text{ij}}+\frac{1}{\mathrm{3K}}\dot{p}{\delta }_{\text{ij}}\)

which we also note: \({\dot{\sigma }}_{\text{ij}}\mathrm{=}{D}^{e}{\dot{\varepsilon }}_{\text{ij}}^{e}\).

The shear modules \(G\) and the compressibility modulus \(K\) depend on the stress state: \(K={K}_{0}^{e}{\left[\frac{{I}_{1}^{-}}{{\mathrm{3P}}_{a}}\right]}^{{n}_{\text{elas}}}\) and \(G\mathrm{=}{G}_{0}^{e}{\left[\frac{{I}_{1}^{\mathrm{-}}}{{\mathrm{3P}}_{a}}\right]}^{{n}_{\text{elas}}}\) with \({I}_{1}^{\mathrm{-}}\mathrm{=}\text{tr}({\sigma }^{\mathrm{-}})\).

\({I}_{1}^{\mathrm{-}}\mathrm{=}\text{tr}({\sigma }^{\mathrm{-}})\) being the trace of the constraints at the time \(\mathrm{-}\).

3.3.2. Plasticity#

As in the simplified version of the model, only the deviatoric mechanism is taken into account, the instantaneous irreversible deformation is written as:

\({\dot{\varepsilon }}_{\text{ij}}^{p}\mathrm{=}\dot{\lambda }{G}_{\text{ij}}\)

\(\lambda\) being the plastic multiplier and \(G\) is the flow function.

Let \({f}^{d}\) be the plasticity criterion:

If \({f}^{d}\le 0\) then \(\lambda \mathrm{=}0\)

If \({f}^{d}=0\) then \(\lambda >0\)

The expression for \(G\) is based on several works cited in note R 2 and has the following form:

\({G}_{\text{ij}}\mathrm{=}\frac{\mathrm{\partial }{f}^{d}}{\mathrm{\partial }{\sigma }_{\text{ij}}}\mathrm{-}(\frac{\mathrm{\partial }{f}^{d}}{\mathrm{\partial }{\sigma }_{\text{kl}}}{n}_{\text{kl}}){n}_{\text{ij}}\), \({n}_{\text{ij}}\mathrm{=}\frac{{\beta }^{\text{'}}\frac{{s}_{\text{ij}}}{{s}_{\text{II}}}\mathrm{-}{\delta }_{\text{ij}}}{\sqrt{{\beta }^{\text{'}2}+3}}\), \({\beta }^{\text{'}}\mathrm{=}\mathrm{-}\frac{2\sqrt{6}\text{sin}(\Psi )}{3\mathrm{-}\text{sin}(\Psi )}\)

The expressions for \(\text{sin}(\Psi )\) are detailed in paragraphs 3.6.1 and 3.6.2.

The calculation of \(\dot{\lambda }\) is the subject of paragraph 4.2.1.2.

The calculation of \(\frac{\mathrm{\partial }f}{\mathrm{\partial }{\sigma }_{\text{ij}}}\) is detailed in paragraph 3.7.1

The evolution of elastoplasticity induces plastic deformation: \({\epsilon }_{p}\) linked through its deviatoric component \({\tilde{\varepsilon }}_{p}\) to the work-hardening parameter \({\gamma }_{p}\) such that: \({\gamma }_{p}\mathrm{=}\mathrm{\int }\sqrt{\frac{2}{3}{\tilde{\dot{\varepsilon }}}_{p}{\tilde{\dot{\varepsilon }}}_{p}}\text{dt}\)

Hence the \({\dot{\gamma }}_{p}\mathrm{=}\dot{\lambda }\sqrt{\frac{2}{3}{\tilde{G}}_{\text{ij}}{\tilde{G}}_{\text{ij}}}\mathrm{=}\dot{\lambda }\sqrt{\frac{2}{3}}{G}_{\text{II}}\) relationship

3.3.3. Viscoplasticity#

The calculation of delayed irreversible deformations \({\underline{\underline{\dot{\varepsilon }}}}^{\text{vp}}\) is based on Perzyna’s theory.

\({\underline{\underline{\dot{\epsilon }}}}^{\text{vp}}=\langle \Phi ({f}^{\text{vp}})\rangle {G}_{\text{ij}}^{\text{visc}}\) where \(\Phi ({f}^{\text{vp}})\) and \({G}^{\text{visc}}\) characterize the magnitude and direction of the speed of irreversible deformations:

\(\Phi ({f}^{\text{vp}})\mathrm{=}{A}_{v}{(\frac{{f}^{\text{vp}}}{\text{Pa}})}^{{n}_{v}}\) and \({G}_{\text{ij}}^{\text{visc}}\mathrm{=}\frac{\mathrm{\partial }{f}^{\text{vp}}}{\mathrm{\partial }{\sigma }_{\text{ij}}}\mathrm{-}(\frac{\mathrm{\partial }{f}^{\text{vp}}}{\mathrm{\partial }{\sigma }_{\text{kl}}}{n}_{\text{kl}}){n}_{\text{ij}}\)

\({f}^{\text{vp}}\) being the viscoplasticity criterion, \({A}_{v}\) and \({n}_{v}\) are parameters of the model. \({P}_{a}\) is air pressure.

The evolution of viscoplasticity induces a viscous deformation: linked through its deviatoric component \({\tilde{\epsilon }}_{\text{vp}}\) to the work-hardening parameter \({\gamma }_{\text{vp}}\) such as: \({\gamma }_{\text{vp}}=\int \sqrt{\frac{2}{3}{\tilde{\dot{\epsilon }}}_{\text{vp}}{\tilde{\dot{\epsilon }}}_{\text{vp}}}\text{dt}\).

Note:

Let \({S}^{\text{vp}}\) be the area defined in the constraint space by: \({S}^{\text{vp}}\mathrm{=}\left\{\sigma ,{f}^{\text{vp}}(\sigma ,{\xi }^{\text{vp}})\mathrm{=}0\right\}\)

The creep speed for a \(\sigma\) stress state is proportional to the distance from \(\sigma\) to \({S}^{\text{vp}}\). Let \({P}_{\sigma }^{\text{vp}}\) be the projection of \(\sigma\) on \({S}^{\text{vp}}\) and \(d=\parallel \sigma -{P}_{\sigma }^{\text{vp}}\parallel\).

You can also write \(d=\parallel \frac{\partial {f}^{\text{vp}}}{\partial \sigma }({P}_{\sigma }^{\text{vp}})\parallel C\). \(C\) being a constant that depends on viscous parameters. As a first approximation, we write: \(d=\parallel \frac{\partial {f}^{\text{vp}}}{\partial \sigma }(\sigma )\parallel C\). But this approximation poses a problem in that the \({f}^{\text{vp}}\) function may not be defined for the value \(\sigma\) while it is for \({P}_{\sigma }^{\text{vp}}\). For the moment, in*Code_Aster* we do not calculate this distance. If the situation occurs, an alarm message alerts the user.

3.4. Criteria expressions#

The expressions of the two criteria: viscoplastic \({f}^{\text{vp}}\) and elastoplastic \({f}^{d}\), depend on the constraints and on the work-hardening functions. In these expressions, we find \({I}_{1}\) the first invariant of the constraints and \({s}_{\text{II}}\) the second invariant of the tensor of the deviatory constraints. In both criteria, the same definitions are adopted for:

\(H(\theta )=\frac{{H}_{0}^{c}+{H}_{0}^{e}}{2}+(\frac{{H}_{0}^{c}-{H}_{0}^{e}}{2})(\frac{\mathrm{2h}(\theta )-({h}_{0}^{c}+{h}_{0}^{e})}{{h}_{0}^{c}-{h}_{0}^{e}})\)

\(h(\theta )={(1-\gamma \text{cos}\mathrm{3\theta })}^{\frac{1}{6}}\), \({h}_{0}^{c}={H}_{0}^{c}=h({0}^{°})={(1-\gamma )}^{\frac{1}{6}}\), \({h}_{0}^{e}=h({\text{60}}^{°})={(1+\gamma )}^{\frac{1}{6}}\),

\({H}_{0}^{e}\) is a model parameter. \(\theta\) is the Lode angle.

3.4.1. The viscoplastic criterion \({f}^{\text{vp}}\)#

\({f}^{\text{vp}}(\sigma )={s}_{\text{II}}H(\theta )-{\sigma }_{c}{H}_{0}^{c}{\left[{A}^{\text{vp}}({\xi }_{\text{vp}}){s}_{\text{II}}H(\theta )+{B}^{\text{vp}}({\xi }_{\text{vp}}){I}_{1}+{D}^{\text{vp}}({\xi }_{\text{vp}})\right]}^{{a}^{\text{vp}}({\xi }_{\text{vp}})}\)

with \({A}^{\text{vp}}({\xi }_{\text{vp}})=-\frac{{m}^{\text{vp}}({\xi }_{\text{vp}}){k}^{\text{vp}}({\xi }_{\text{vp}})}{\sqrt{6}{\sigma }_{c}{h}_{c}^{0}}\), \({B}^{\text{vp}}({\xi }_{\text{vp}})=\frac{{m}^{\text{vp}}({\xi }_{\text{vp}}){k}^{\text{vp}}({\xi }_{\text{vp}})}{{\mathrm{3\sigma }}_{c}}\), \({D}^{\text{vp}}({\xi }_{\text{vp}})={s}^{\text{vp}}({\xi }_{\text{vp}}){k}^{\text{vp}}({\xi }_{\text{vp}})\), \({k}^{\text{vp}}({\xi }_{\text{vp}})={(\frac{2}{3})}^{\frac{1}{{\mathrm{2a}}^{\text{vp}}({\xi }_{\text{vp}})}}\)

The work hardening functions \({A}^{\text{vp}}({\xi }_{\text{vp}})\), \({B}^{\text{vp}}({\xi }_{\text{vp}})\) and \({D}^{\text{vp}}({\xi }_{\text{vp}})\) depend on the work hardening parameters \({a}^{\text{vp}}({\xi }_{\text{vp}})\), \({m}^{\text{vp}}({\xi }_{\text{vp}})\) and \({s}^{\text{vp}}({\xi }_{\text{vp}})\) whose expressions change with the work hardening variables \({\xi }_{\text{vp}}\) (see § 3.5). When \({\xi }_{\text{vp}}\) reaches certain specific values, the surface \({f}^{\text{vp}}\) reaches the corresponding thresholds.

Since the viscoplastic threshold only collapses due to viscosity, we always have \({\dot{\xi }}_{\text{vp}}=\text{Min}\left[{\dot{\gamma }}_{\text{vp}},{\xi }_{v-\text{max}}-{\xi }_{\text{vp}}\right]\). \({\xi }_{\text{vp}-\text{max}}\) corresponds to the maximum viscoplastic criterion and is a parameter of the model

3.4.2. The elastoplastic criterion \({f}^{d}\)#

\({f}^{d}(\sigma )={s}_{\text{II}}H(\theta )-{\sigma }_{c}{H}_{0}^{c}{\left[{A}^{d}({\xi }_{p}){s}_{\text{II}}H(\theta )+{B}^{d}({\xi }_{p}){I}_{1}+{D}^{d}({\xi }_{p})\right]}^{{a}^{d}({\xi }_{p})}\)

with \({A}^{d}({\xi }_{p})=-\frac{{m}^{d}({\xi }_{p}){k}^{d}({\xi }_{p})}{\sqrt{6}{\sigma }_{c}{h}_{c}^{0}}\), \({B}^{d}({\xi }_{p})=\frac{{m}^{d}({\xi }_{p}){k}^{d}({\xi }_{p})}{{\mathrm{3\sigma }}_{c}}\), \({D}^{d}({\xi }_{p})={s}^{d}({\xi }_{p}){k}^{d}({\xi }_{p})\), \({k}^{d}({\xi }_{p})={(\frac{2}{3})}^{\frac{1}{{\mathrm{2a}}^{d}({\xi }_{p})}}\)

The work hardening functions \({A}^{d}({\xi }_{p})\), \({B}^{d}({\xi }_{p})\) and \({D}^{d}({\xi }_{p})\) depend on the work hardening parameters \({a}^{d}({\xi }_{p})\), \({m}^{d}({\xi }_{p})\) and \({s}^{d}({\xi }_{p})\) whose expressions change with the work hardening variables \({\xi }_{p}\) (see § 3.5). When \({\xi }_{p}\) reaches certain specific values, the surface \({f}^{d}\) reaches the corresponding thresholds.

Elastoplastic work hardening depends on the position of the load point in relation to the contraction/expansion limit:

  • if the load point is below this limit, \({\dot{\xi }}_{p}={\dot{\gamma }}_{p}\),

  • if the load point is above this limit, \({\dot{\xi }}_{p}={\dot{\gamma }}_{p}+{\dot{\gamma }}_{\text{vp}}\).

3.5. Work hardening functions#

3.5.1. Work hardening functions of the viscous criterion#

The viscoplastic criterion is governed by the following work hardening functions:

\(a({\xi }_{\text{vp}})={a}_{0}+({a}_{v-\text{max}}-{a}_{0})\frac{{\xi }_{\text{vp}}}{{\xi }_{v-\text{max}}}\) with \({a}_{v-\text{max}}=1\text{.}\)

\(m({\xi }_{\text{vp}})={m}_{0}+({m}_{v-\text{max}}-{m}_{0})\frac{{\xi }_{\text{vp}}}{{\xi }_{v-\text{max}}}\)

\(s({\xi }_{\text{vp}})={s}_{0}+({s}_{v-\text{max}}-{s}_{0})\frac{{\xi }_{\text{vp}}}{{\xi }_{v-\text{max}}}\) with \({s}_{v-\text{max}}={s}_{0}\)

3.5.2. Work hardening functions of the elastoplastic criterion and their derivatives#

The expressions of the work hardening functions that govern the elastoplastic criterion vary according to the value of parameter \({\xi }_{p}\):

Evolution between the damage threshold and the peak threshold: If \(0\le {\xi }_{p}<{\xi }_{\text{pic}}\)

\(a({\xi }_{p})={a}_{0}+\text{ln}(1+\frac{{\xi }_{p}}{{x}_{\text{ams}}{\xi }_{\text{pic}}})(\frac{{a}_{\text{pic}}-{a}_{0}}{\text{ln}(1+1/{x}_{\text{ams}})})\) \(\frac{\partial a}{\partial {\xi }_{p}}=(\frac{{a}_{\text{pic}}-{a}_{0}}{\text{ln}(1+1/{x}_{\text{ams}})})(\frac{1}{{\xi }_{p}+{x}_{\text{ams}}{\xi }_{\text{pic}}})\)

\(m({\xi }_{p})={m}_{0}+\text{ln}(1+\frac{{\xi }_{p}}{{x}_{\text{ams}}{\xi }_{\text{pic}}})(\frac{{m}_{\text{pic}}-{m}_{0}}{\text{ln}(1+1/{x}_{\text{ams}})})\) \(\frac{\partial m}{\partial {\xi }_{p}}=(\frac{{m}_{\text{pic}}-{m}_{0}}{\text{ln}(1+1/{x}_{\text{ams}})})(\frac{1}{{\xi }_{p}+{x}_{\text{ams}}{\xi }_{\text{pic}}})\)

\(s({\xi }_{p})={s}_{0}+\text{ln}(1+\frac{{\xi }_{p}}{{x}_{\text{ams}}{\xi }_{\text{pic}}})(\frac{{s}_{\text{pic}}-{s}_{0}}{\text{ln}(1+1/{x}_{\text{ams}})})\) \(\frac{\partial s}{\partial {\xi }_{p}}=(\frac{{s}_{\text{pic}}-{s}_{0}}{\text{ln}(1+1/{x}_{\text{ams}})})(\frac{1}{{\xi }_{p}+{x}_{\text{ams}}{\xi }_{\text{pic}}})\)

with \({s}_{\text{pic}}=1\text{.}\)

Evolution between the peak threshold and the intermediate threshold or the cleavage limit: If \({\xi }_{\text{pic}}\le {\xi }_{p}<{\xi }_{e}\)

\(a({\xi }_{p})={a}_{\text{pic}}+({a}_{e}-{a}_{\text{pic}})(\frac{{\xi }_{p}-{\xi }_{\text{pic}}}{{\xi }_{e}-{\xi }_{\text{pic}}})\) \(\frac{\partial a}{\partial {\xi }_{p}}=\frac{{a}_{e}-{a}_{\text{pic}}}{{\xi }_{e}-{\xi }_{\text{pic}}}\)

\(s({\xi }_{p})=1-(\frac{{\xi }_{p}-{\xi }_{\text{pic}}}{{\xi }_{e}-{\xi }_{\text{pic}}})\) \(\frac{\partial s}{\partial {\xi }_{p}}=\frac{-1}{{\xi }_{e}-{\xi }_{\text{pic}}}\)

\(\frac{\partial m}{\partial {\xi }_{p}}=\frac{\partial m}{\partial a}\frac{\partial a}{\partial {\xi }_{p}}+\frac{\partial m}{\partial s}\frac{\partial s}{\partial {\xi }_{p}}\)

\(\frac{\partial m}{\partial {\xi }_{p}}=\frac{{\sigma }_{c}}{{\sigma }_{\text{po}\text{int}1}}\left[(-\frac{{a}_{\text{pic}}}{a{({\xi }_{p})}^{2}}){({m}_{\text{pic}}\frac{{\sigma }_{\text{po}\text{int}1}}{{\sigma }_{c}}+{s}_{\text{pic}})}^{\frac{{a}_{\text{pic}}}{a({\xi }_{p})}}\text{ln}({m}_{\text{pic}}\frac{{\sigma }_{\text{po}\text{int}1}}{{\sigma }_{c}}+{s}_{\text{pic}})\frac{\partial a}{\partial {\xi }_{p}}-\frac{\partial s}{\partial {\xi }_{p}}\right]\)

Evolution between the intermediate threshold and the residual threshold: If \({\xi }_{e}\le {\xi }_{p}<{\xi }_{\text{ult}}\)

\(a({\xi }_{p})={a}_{e}+\text{ln}(1+\frac{1}{\eta }\frac{{\xi }_{p}-{\xi }_{e}}{{\xi }_{\text{ult}}-{\xi }_{e}})(\frac{{a}_{\text{ult}}-{a}_{e}}{\text{ln}(1+1/\eta )})\) \(\frac{\partial a}{\partial {\xi }_{p}}=(\frac{{a}_{\text{ult}}-{a}_{e}}{\text{ln}(1+1/\eta )})(\frac{1}{{\xi }_{p}+{\text{ηξ}}_{\text{ult}}-(1+\eta ){\xi }_{e}})\)

\(s({\xi }_{p})=0\) \(\frac{\partial s}{\partial {\xi }^{p}}=0\)

\(m({\xi }_{p})=\frac{{\sigma }_{c}}{{\sigma }_{\text{po}\text{int}2}}{({m}_{e}\frac{{\sigma }_{\text{po}\text{int}2}}{{\sigma }_{c}})}^{\frac{{a}_{e}}{a({\xi }_{p})}}\) \(\frac{\partial m}{\partial {\xi }_{p}}=\frac{{\sigma }_{c}}{{\sigma }_{\text{po}\text{int}2}}\left[(-\frac{{a}_{e}}{a{({\xi }_{p})}^{2}})\text{ln}({m}_{e}\frac{{\sigma }_{\text{po}\text{int}2}}{{\sigma }_{c}}){({m}_{e}\frac{{\sigma }_{\text{po}\text{int}2}}{{\sigma }_{c}})}^{\frac{{a}_{e}}{a({\xi }_{p})}}\right]\frac{\partial a}{\partial {\xi }_{p}}\)

On the residual criterion: If \({\xi }_{p}\ge {\xi }_{\text{ult}}\)

\(a({\xi }_{p})\mathrm{=}{a}_{\text{ult}}\mathrm{=}1\text{.}\) \(\frac{\partial a}{\partial {\xi }_{p}}=0\)

\(s({\xi }_{p})=0\) \(\frac{\partial s}{\partial {\xi }_{p}}=0\)

\(m({\xi }_{p})\mathrm{=}{m}_{\text{ult}}\) \(\frac{\partial m}{\partial {\xi }_{p}}=0\)

3.6. Laws of dilatance#

The elastoplastic and viscoplastic mechanisms are not associated. The laws of evolution of \({\dot{\varepsilon }}_{\text{ij}}^{p}\) and \({\dot{\epsilon }}_{\text{ij}}^{\text{vp}}\) are governed respectively by a function \(G\) and a function \({G}^{\text{visc}}\), such as:

\({G}_{\text{ij}}\mathrm{=}\frac{\mathrm{\partial }{f}^{d}}{\mathrm{\partial }{\sigma }_{\text{ij}}}\mathrm{-}(\frac{\mathrm{\partial }{f}^{d}}{\mathrm{\partial }{\sigma }_{\text{kl}}}{n}_{\text{kl}}){n}_{\text{ij}}\) and \({G}_{\text{ij}}^{\text{visc}}\mathrm{=}\frac{\mathrm{\partial }{f}^{\text{vp}}}{\mathrm{\partial }{\sigma }_{\text{ij}}}\mathrm{-}(\frac{\mathrm{\partial }{f}^{\text{vp}}}{\mathrm{\partial }{\sigma }_{\text{kl}}}{n}_{\text{kl}}){n}_{\text{ij}}\) with

\({n}_{\text{ij}}\mathrm{=}\frac{{\beta }^{\text{'}}\frac{{s}_{\text{ij}}}{{s}_{\text{II}}}\mathrm{-}{\delta }_{\text{ij}}}{\sqrt{{\beta }^{{\text{'}}^{2}}+3}}\) and \({\beta }^{\text{'}}\mathrm{=}\mathrm{-}\frac{2\sqrt{6}\text{sin}(\Psi )}{3\mathrm{-}\text{sin}(\Psi )}\)

The calculation of the expansion angle \(\Psi\) differs according to the viscous or elastoplastic pre-peak and elastoplastic post-peak mechanisms.

3.6.1. Angle of expansion of elastoplastic pre-peak and viscoplastic mechanisms#

\(\text{sin}(\Psi )\mathrm{=}{\mu }_{\mathrm{0,}v}(\frac{{\sigma }_{\text{max}}\mathrm{-}{\sigma }_{\text{lim}}}{{\xi }_{\mathrm{0,}v}{\sigma }_{\text{max}}+{\sigma }_{\text{lim}}})\) with \({\mu }_{\mathrm{0,}v}\) and \({\xi }_{\mathrm{0,}v}\) model parameters.

where

\({\sigma }_{\text{lim}}\mathrm{=}{\sigma }_{\text{min}}+{\sigma }_{c}{({m}_{v\mathrm{-}\text{max}}\frac{{\sigma }_{\text{min}}}{{\sigma }_{c}}+{s}_{v\mathrm{-}\text{max}})}^{{a}_{v\mathrm{-}\text{max}}}\) with \({s}_{v-\text{max}}={s}_{0}\) and \({a}_{v\mathrm{-}\text{max}}\mathrm{=}1\). \({\sigma }_{c}\) and \({m}_{v\mathrm{-}\text{max}}\) are model parameters.

There are conditions on parameters \({\mu }_{\mathrm{0,}v}\) and \({\xi }_{\mathrm{0,}v}\) which are:

  • \({\mu }_{\mathrm{0,}v}<{\xi }_{\mathrm{0,}v}\) or

  • \(\mathrm{\{}\begin{array}{c}{\mu }_{\mathrm{0,}v}>{\xi }_{\mathrm{0,}v}\\ \frac{{({s}_{\text{pic}})}^{{a}_{\text{pic}}}}{{({s}_{0})}^{{a}_{0}}}\mathrm{\le }\frac{1+{\mu }_{\mathrm{0,}v}}{{\mu }_{\mathrm{0,}v}\mathrm{-}{\xi }_{\mathrm{0,}v}}\end{array}\)

3.6.2. Angle of expansion of the post-peak elastoplastic mechanism#

\(\text{sin}(\Psi )\mathrm{=}{\mu }_{1}(\frac{\alpha \mathrm{-}{\alpha }_{\text{res}}}{{\xi }_{1}\alpha +{\alpha }_{\text{res}}})\) with \({\mu }_{1}\) and \({\xi }_{1}\) model parameters

where

\(\alpha \mathrm{=}\frac{{\sigma }_{\text{max}}+\tilde{\sigma }}{{\sigma }_{\text{min}}+\tilde{\sigma }}\) and \({\alpha }_{\text{res}}\mathrm{=}\frac{{\sigma }_{\text{max}}}{{\sigma }_{\text{min}}}\mathrm{=}1+{m}_{\text{ult}}\)

\(\tilde{\sigma }\mathrm{=}\mathrm{\{}\begin{array}{c}\frac{\tilde{c}({\xi }_{p})}{\text{tan}(\tilde{\varphi }({\xi }_{p}))}\text{si}{\xi }_{p}\mathrm{\le }{\xi }_{e}\\ 0\text{si}{\xi }_{p}>{\xi }_{e}\end{array}\)

with \(\tilde{c}({\xi }_{p})\mathrm{=}\frac{{\sigma }_{c}\text{.}s{({\xi }_{p})}^{a({\xi }_{p})}}{2\sqrt{1+a({\xi }_{p})m({\xi }_{p})s{({\xi }_{p})}^{a({\xi }_{p})\mathrm{-}1}}}\) and \(\tilde{\varphi }({\xi }_{p})\mathrm{=}2\text{.}\text{arctg}(\sqrt{1+a({\xi }_{p})m({\xi }_{p})s{({\xi }_{p})}^{a({\xi }_{p})\mathrm{-}1}})\mathrm{-}\frac{\pi }{2}\)

\({\sigma }_{\text{min}}\) and \({\sigma }_{\text{max}}\) are calculated using the invariants of the constraints:

\({\sigma }_{\text{min}}\mathrm{=}\frac{1}{3}({I}_{1}\mathrm{-}(\frac{3}{2}\mathrm{-}\frac{\mathrm{2H}(\theta )\mathrm{-}({H}_{0}^{c}+{H}_{0}^{e})}{2({H}_{0}^{c}\mathrm{-}{H}_{0}^{e})})\sqrt{\frac{3}{2}}{s}_{\text{II}})\)

\({\sigma }_{\text{max}}\mathrm{=}\frac{1}{3}({I}_{1}+(\frac{3}{2}+\frac{\mathrm{2H}(\theta )\mathrm{-}({H}_{0}^{c}+{H}_{0}^{e})}{2({H}_{0}^{c}\mathrm{-}{H}_{0}^{e})})\sqrt{\frac{3}{2}}{s}_{\text{II}})\)

3.7. Derivatives of the criterion#

3.7.1. Calculation of \(\frac{\mathrm{\partial }f}{\mathrm{\partial }{\sigma }_{\text{ij}}}\)#

\(\frac{\mathrm{\partial }{I}_{1}}{\mathrm{\partial }{\sigma }_{\text{ij}}}\mathrm{=}\frac{\mathrm{\partial }\text{tr}({\sigma }_{\text{ij}})}{\mathrm{\partial }{\sigma }_{\text{ij}}}\mathrm{=}{\delta }_{\text{ij}}\)

\(\frac{\partial ({s}_{\text{II}}H(\theta ))}{\partial {\sigma }_{\text{ij}}}=\frac{\partial ({s}_{\text{II}}H(\theta ))}{\partial {s}_{\text{kl}}}\frac{\partial {s}_{\text{kl}}}{\partial {\sigma }_{\text{ij}}}=(\frac{\partial H(\theta )}{\partial {s}_{\text{kl}}}{s}_{\text{II}}+H(\theta )\frac{\partial {s}_{\text{II}}}{\partial {s}_{\text{kl}}})\frac{\partial {s}_{\text{kl}}}{\partial {\sigma }_{\text{ij}}}\)

\(\frac{\partial {s}_{\text{II}}}{\partial {s}_{\text{kl}}}=\frac{{s}_{\text{kl}}}{{s}_{\text{II}}}\); \({s}_{\text{II}}\mathrm{=}\sqrt{{s}_{\text{kl}}\text{.}{s}_{\text{kl}}}\)

\(\frac{\partial {s}_{\text{kl}}}{\partial {\sigma }_{\text{ij}}}=\frac{\partial ({\sigma }_{\text{kl}}-\frac{1}{3}\text{tr}(\sigma ){\delta }_{\text{kl}})}{\partial {\sigma }_{\text{ij}}}={\delta }_{\text{ik}}\text{.}{\delta }_{\text{jl}}-\frac{1}{3}{\delta }_{\text{ij}}\text{.}{\delta }_{\text{kl}}\)

Note: \({s}_{\text{kl}}\text{.}({\delta }_{\text{ik}}\text{.}{\delta }_{\text{jl}}-\frac{1}{3}{\delta }_{\text{ij}}\text{.}{\delta }_{\text{kl}})={s}_{\text{ij}}\)

\(\frac{\partial H(\theta )}{\partial {s}_{\text{kl}}}=(\frac{{H}_{0}^{c}-{H}_{0}^{e}}{{h}_{0}^{c}-{h}_{0}^{e}})\frac{\partial h(\theta )}{\partial {s}_{\text{kl}}}\)

Hence \(\frac{\mathrm{\partial }({s}_{\text{II}}H(\theta ))}{\mathrm{\partial }{\sigma }_{\text{ij}}}\mathrm{=}((\frac{{H}_{0}^{c}\mathrm{-}{H}_{0}^{e}}{{h}_{0}^{c}\mathrm{-}{h}_{0}^{e}})\frac{\mathrm{\partial }h(\theta )}{\mathrm{\partial }{s}_{\text{kl}}}{s}_{\text{II}}+H(\theta )\frac{{s}_{\text{kl}}}{{s}_{\text{II}}})\text{.}({\delta }_{\text{ik}}\text{.}{\delta }_{\text{jl}}\mathrm{-}\frac{1}{3}{\delta }_{\text{ij}}\text{.}{\delta }_{\text{kl}})\)

We have the relationship: \(\text{cos}(\mathrm{3\theta })\mathrm{=}\sqrt{\text{54}}\frac{\text{det}(s)}{{s}_{\text{II}}^{3}}\) (see Documentation R7.01.13-A: Law CJS in mechanics)

\(\begin{array}{c}\frac{\mathrm{\partial }h(\theta )}{\mathrm{\partial }{s}_{\text{kl}}}\mathrm{=}\frac{1}{6}{(1\mathrm{-}\gamma \text{cos}(\mathrm{3\theta }))}^{\mathrm{-}\frac{5}{6}}\frac{\mathrm{\partial }(1\mathrm{-}\gamma \text{cos}(\mathrm{3\theta }))}{\mathrm{\partial }{s}_{\text{kl}}}\mathrm{=}\frac{1}{\mathrm{6h}{(\theta )}^{5}}\frac{\mathrm{\partial }}{\mathrm{\partial }{s}_{\text{kl}}}(\frac{{s}_{\text{II}}^{3}\mathrm{-}\gamma \sqrt{\text{54}}\text{det}(s)}{{s}_{\text{II}}^{3}})\end{array}\)

\(\begin{array}{c}\frac{\mathrm{\partial }h(\theta )}{\mathrm{\partial }{s}_{\text{kl}}}\mathrm{=}\frac{1}{\mathrm{6h}{(\theta )}^{5}}\left\{\left[\frac{\mathrm{\partial }{s}_{\text{II}}^{3}}{\mathrm{\partial }{s}_{\text{kl}}}\mathrm{-}\gamma \sqrt{\text{54}}(\frac{\mathrm{\partial }\text{det}(\underline{\underline{s}})}{\mathrm{\partial }{s}_{\text{kl}}})\right]\frac{{s}_{\text{II}}^{3}}{{s}_{\text{II}}^{6}}\mathrm{-}({s}_{\text{II}}^{3}\mathrm{-}\gamma \sqrt{\text{54}}\text{det}(\underline{\underline{s}}))\frac{{\mathrm{3s}}_{\text{kl}}{s}_{\text{II}}}{{s}_{\text{II}}^{6}}\right\}\\ \mathrm{=}\frac{1}{\mathrm{6h}{(\theta )}^{5}}\left\{\frac{{\mathrm{3s}}_{\text{kl}}}{{s}_{\text{II}}^{2}}\mathrm{-}\gamma \sqrt{\text{54}}(\frac{\mathrm{\partial }\text{det}(\underline{\underline{s}})}{\mathrm{\partial }{s}_{\text{kl}}})\frac{1}{{s}_{\text{II}}^{3}}\mathrm{-}(1\mathrm{-}\gamma \text{cos}(\mathrm{3\theta }))\frac{{\mathrm{3s}}_{\text{kl}}}{{s}_{\text{II}}^{2}}\right\}\\ \mathrm{=}\frac{\gamma \text{cos}(\mathrm{3\theta })}{\mathrm{6h}{(\theta )}^{5}}\frac{{\mathrm{3s}}_{\text{kl}}}{{s}_{\text{II}}^{2}}\mathrm{-}\frac{\gamma \sqrt{\text{54}}}{\mathrm{6h}{(\theta )}^{5}{s}_{\text{II}}^{3}}(\frac{\mathrm{\partial }\text{det}(\underline{\underline{s}})}{\mathrm{\partial }{s}_{\text{kl}}})\end{array}\)

So we find:

\(\begin{array}{c}\frac{\mathrm{\partial }({s}_{\text{II}}H(\theta ))}{\mathrm{\partial }{\sigma }_{\text{ij}}}\mathrm{=}\\ ((\frac{{H}_{0}^{c}\mathrm{-}{H}_{0}^{e}}{{h}_{0}^{c}\mathrm{-}{h}_{0}^{e}})(\frac{\gamma \text{cos}(\mathrm{3\theta })}{\mathrm{6h}{(\theta )}^{5}}\frac{{\mathrm{3s}}_{\text{kl}}}{{s}_{\text{II}}^{2}}\mathrm{-}\frac{\gamma \sqrt{\text{54}}}{\mathrm{6h}{(\theta )}^{5}{s}_{\text{II}}^{3}}(\frac{\mathrm{\partial }\text{det}(\underline{\underline{s}})}{\mathrm{\partial }{s}_{\text{kl}}})){s}_{\text{II}}+H(\theta )\frac{{s}_{\text{kl}}}{{s}_{\text{II}}})\text{.}({\delta }_{\text{ik}}\text{.}{\delta }_{\text{jl}}\mathrm{-}\frac{1}{3}{\delta }_{\text{ij}}\text{.}{\delta }_{\text{kl}})\end{array}\)

Finally:

For the elastoplastic criterion:

\(\begin{array}{c}\frac{\mathrm{\partial }{f}^{d}}{\mathrm{\partial }{\sigma }_{\text{ij}}}\mathrm{=}\\ \frac{\mathrm{\partial }({s}_{\text{II}}H(\theta ))}{\mathrm{\partial }{\sigma }_{\text{ij}}}\mathrm{-}{a}^{d}({\xi }_{p}){\sigma }_{c}{H}_{0}^{c}{\left[{A}^{d}({\xi }_{p}){s}_{\text{II}}H(\theta )+{B}^{d}({\xi }_{p}){I}_{1}+{D}^{d}({\xi }_{p})\right]}^{{a}^{d}({\xi }_{p})\mathrm{-}1}\\ ({A}^{d}({\xi }_{p})\frac{\mathrm{\partial }({s}_{\text{II}}H(\theta ))}{\mathrm{\partial }{\sigma }_{\text{ij}}}+{B}^{d}({\xi }_{p}){I}_{d})\end{array}\)

and for the viscous criterion:

\(\begin{array}{c}\frac{\mathrm{\partial }{f}^{\text{vp}}}{\mathrm{\partial }{\sigma }_{\text{ij}}}\mathrm{=}\\ \frac{\mathrm{\partial }({s}_{\text{II}}H(\theta ))}{\mathrm{\partial }{\sigma }_{\text{ij}}}\mathrm{-}{a}^{\text{vp}}({\xi }_{\text{vp}}){\sigma }_{c}{H}_{0}^{c}{\left[{A}^{\text{vp}}({\xi }_{\text{vp}}){s}_{\text{II}}H(\theta )+{B}^{\text{vp}}({\xi }_{\text{vp}}){I}_{1}+{D}^{\text{vp}}({\xi }_{\text{vp}})\right]}^{{a}^{\text{vp}}({\xi }_{\text{pp}})\mathrm{-}1}\\ ({A}^{\text{vp}}({\xi }_{\text{vp}})\frac{\mathrm{\partial }({s}_{\text{II}}H(\theta ))}{\mathrm{\partial }{\sigma }_{\text{ij}}}+{B}^{\text{vp}}({\xi }_{\text{vp}}){I}_{d})\end{array}\)

with \(\frac{\mathrm{\partial }({s}_{\text{II}}H(\theta ))}{\mathrm{\partial }{\sigma }_{\text{ij}}}\mathrm{=}\)

\(((\frac{{H}_{0}^{c}\mathrm{-}{H}_{0}^{e}}{{h}_{0}^{c}\mathrm{-}{h}_{0}^{e}})(\frac{\gamma \text{cos}(\mathrm{3\theta })}{\mathrm{6h}{(\theta )}^{5}}\frac{{\mathrm{3s}}_{\text{kl}}}{{s}_{\text{II}}^{2}}\mathrm{-}\frac{\gamma \sqrt{\text{54}}}{\mathrm{6h}{(\theta )}^{5}{s}_{\text{II}}^{3}}(\frac{\mathrm{\partial }\text{det}(\underline{\underline{s}})}{\mathrm{\partial }{s}_{\text{kl}}})){s}_{\text{II}}+H(\theta )\frac{{s}_{\text{kl}}}{{s}_{\text{II}}})\text{.}({\delta }_{\text{ik}}\text{.}{\delta }_{\text{jl}}\mathrm{-}\frac{1}{3}{\delta }_{\text{ij}}\text{.}{\delta }_{\text{kl}})\)

3.7.2. Calculation of \(\frac{\partial {f}^{d}}{\partial {\xi }_{p}}\)#

Expression of the threshold in constraints:

\({f}^{d}(\sigma )={s}_{\text{II}}H(\theta )-{\sigma }_{c}{H}_{0}^{c}{\left[{A}^{d}({\xi }_{p}){s}_{\text{II}}H(\theta )+{B}^{d}({\xi }_{p}){I}_{1}+{D}^{d}({\xi }_{p})\right]}^{{a}^{d}({\xi }_{p})}\)

with \({A}^{d}({\xi }^{p})=-\frac{{m}^{d}({\xi }_{p}){k}^{d}({\xi }_{p})}{\sqrt{6}{\sigma }_{c}{h}_{c}^{0}}\), \({B}^{d}({\xi }_{p})=\frac{{m}^{d}({\xi }_{p}){k}^{d}({\xi }_{p})}{{\mathrm{3\sigma }}_{c}}\), \({D}^{d}({\xi }_{p})={s}^{d}({\xi }_{p})k({\xi }_{p})\), \({k}^{d}({\xi }_{p})={(\frac{2}{3})}^{\frac{1}{{\mathrm{2a}}^{d}({\xi }_{p})}}\)

\(\frac{\partial {f}^{d}}{\partial {\xi }_{p}}=\frac{\partial {f}^{d}}{\partial {a}^{d}}\text{.}{\dot{a}}^{d}({\xi }_{p})+\frac{\partial f}{\partial {m}^{d}}\text{.}{\dot{m}}^{d}({\xi }_{p})+\frac{\partial f}{\partial {s}^{d}}\text{.}{\dot{s}}^{d}({\xi }_{p})\)

\(\frac{\partial {f}^{d}}{\partial {s}^{d}}=-{a}^{d}{k}^{d}{\sigma }_{c}{H}_{0}^{c}{\left[{A}^{d}{s}_{\text{II}}H(\theta )+{B}^{d}{I}_{1}+{D}^{d}\right]}^{{a}^{d}-1}\)

\(\frac{\partial {f}^{d}}{\partial {m}^{d}}=-{a}^{d}{\sigma }_{c}{H}_{0}^{c}\left[\frac{{A}^{d}}{{m}^{d}}{s}_{\text{II}}H(\theta )+\frac{{B}^{d}}{{m}^{d}}{I}_{1}\right]{\left[{A}^{d}{s}_{\text{II}}H(\theta )+{B}^{d}{I}_{1}+{D}^{d}\right]}^{{a}^{d}-1}\)

\(\frac{\mathrm{\partial }{f}^{d}}{\mathrm{\partial }{a}^{d}}\mathrm{=}\)

\({\sigma }_{c}{H}_{0}^{c}{\dot{a}}^{d}{\left[{A}^{d}{s}_{\text{II}}H(\theta )+{B}^{d}{I}_{1}+{D}^{d}\right]}^{{a}^{d}}\text{.}\left[\text{ln}\left[{A}^{d}{s}_{\text{II}}H(\theta )+{B}^{d}{I}_{1}+{D}^{d}\right]\mathrm{-}\frac{\frac{{s}^{d}}{{\mathrm{2a}}^{d}}\text{ln}(\frac{2}{3}){(\frac{2}{3})}^{(\frac{1}{\mathrm{2a}})}}{\left[{A}^{d}{s}_{\text{II}}H(\theta )+{B}^{d}{I}_{1}+{D}^{d}\right]}\right]\)