8. Appendix: Implicit tangent operator. Option FULL_MECA

In this appendix, we present elements for calculating the coherent tangent operator.

8.1. General case

8.1.1. Treatment of the deviatoric part

We consider here that the load variation is purely deviatory \((\mathrm{\delta P}=0)\).

The deviatoric stress increment is written in the form:

\({\mathrm{\Delta s}}_{\text{ij}}=\mathrm{2\mu }(\Delta {\tilde{\epsilon }}_{\text{ij}}-\Delta {\tilde{\epsilon }}_{\text{ij}}^{p})\) eq 8.1.1-1

Around the equilibrium point \(({\sigma }^{-}+\mathrm{\Delta \sigma })\), we consider a \(\mathrm{\delta s}\) variation in the deviatory part of the stress:

\({\mathrm{\delta s}}_{\text{kl}}=\mathrm{2\mu }(\delta {\tilde{\epsilon }}_{\text{kl}}-\delta {\tilde{\epsilon }}_{\text{kl}}^{p})\) eq 8.1.1-2

Calculation of \(\delta {\tilde{\epsilon }}_{\text{kl}}^{p}\):

We know that:

\(\Delta {\tilde{\epsilon }}_{\text{kl}}^{p}=\mathrm{3\Lambda }{s}_{\text{kl}}\) eq 8.1.1-3

By deriving this equation with respect to the deviatoric stress, we obtain:

\(\delta {\tilde{\epsilon }}_{\text{kl}}^{p}=3\delta \Lambda {s}_{\text{kl}}+3\Lambda \delta {s}_{\text{kl}}\) eq 8.1.1-4

Calculation of \(\delta \Lambda\):

We have:

\(\begin{array}{c}\Lambda \mathrm{=}\frac{1}{{H}_{p}}\left[{(\frac{\mathrm{\partial }f}{\mathrm{\partial }\sigma })}_{\text{mn}}{\mathit{\Delta \sigma }}_{\text{mn}}\right]\mathrm{=}\frac{1}{{H}_{p}}\left[{(\frac{\mathrm{\partial }f}{\mathrm{\partial }s})}_{\text{mn}}{\mathit{\Delta s}}_{\text{mn}}+\frac{\mathrm{\partial }f}{\mathrm{\partial }P}\mathit{\Delta P}\right]\\ \mathrm{=}\frac{1}{{H}_{p}}\left[{\mathrm{3s}}_{\text{mn}}{\mathit{\Delta s}}_{\text{mn}}+{\mathrm{2M}}^{2}(P\mathrm{-}{P}_{\text{cr}})\mathit{\Delta P}\right]\end{array}\) eq 8.1.1-5

If we only consider the evolution of the deviatoric part of \(\sigma\) \((\mathrm{\delta P}=0)\), then:

\(\delta (\Lambda {H}_{p})=\delta {\mathrm{\Lambda H}}_{p}+{\mathrm{\Lambda \delta H}}_{p}=\left[{\mathrm{3\delta s}}_{\text{mn}}{\mathrm{\Delta s}}_{\text{mn}}+{\mathrm{3s}}_{\text{mn}}{\mathrm{\delta s}}_{\text{mn}}\right]-{\mathrm{2M}}^{2}{\mathrm{\Delta P\delta P}}_{\text{cr}}\) eq 8.1.1-6

Gold: \(\delta {P}_{\text{cr}}={\text{kP}}_{\text{cr}}\delta {\epsilon }_{v}^{P}\).

\(\text{Comme}{\mathrm{\Delta \epsilon }}_{v}^{p}={\mathrm{2\Lambda M}}^{2}(P-{P}_{\text{cr}}),\text{on a}{\text{δε}}_{V}^{p}={\mathrm{2\delta \Lambda M}}^{2}(P-{P}_{\text{cr}})-{\mathrm{2M}}^{2}{\mathrm{\Lambda \delta P}}_{\text{cr}}\), eq 8.1.1-7

From where:

\(\mathrm{2\delta }{\mathrm{\Lambda M}}^{2}(P-{P}_{\text{cr}})=\left[\frac{1}{{\text{kP}}_{\text{cr}}}+{\mathrm{2\Lambda M}}^{2}\right]{\mathrm{\delta P}}_{\text{cr}}\text{.}\) eq 8.1.1-8

In addition,

\({H}_{p}=4{\text{kM}}^{4}{\text{PP}}_{\text{cr}}(P-{P}_{\text{cr}})\text{et}\delta {H}_{p}=4{\text{kM}}^{4}P(P-{\mathrm{2P}}_{\text{cr}})\delta {P}_{\text{cr}}\text{.}\) eq 8.1.1-9

By injecting this last equation into equation [éq 5.3.1-6], we get:

\(\delta {\mathrm{\Lambda H}}_{p}+\left[\mathrm{4\Lambda }{\text{kM}}^{4}P(P-{\mathrm{2P}}_{\text{cr}})+{\mathrm{2M}}^{2}\mathrm{\Delta P}\right]\delta {P}_{\text{cr}}=\left[{\mathrm{3\delta s}}_{\text{mn}}{\mathrm{\Delta s}}_{\text{mn}}+{\mathrm{3s}}_{\text{mn}}{\mathrm{\delta s}}_{\text{mn}}\right]\) eq 8.1.1-10

Using the [éq 5.3.1-8] relationship, it then comes:

\(\delta \Lambda =\frac{\left[{\mathrm{3\delta s}}_{\text{mn}}{\mathrm{\Delta s}}_{\text{mn}}+{\mathrm{3s}}_{\text{mn}}{\mathrm{\delta s}}_{\text{mn}}\right]}{({H}_{p}+A)}\) eq 8.1.1-11

with \(A=\left[\mathrm{4k}{\mathrm{\Lambda M}}^{4}P(P-{\mathrm{2P}}_{\text{cr}})+{\mathrm{2M}}^{2}\mathrm{\Delta P}\right]\left[\frac{{M}^{2}(P-{P}_{\text{cr}})}{\frac{1}{2{\text{kP}}_{\text{cr}}}+{\mathrm{\Lambda M}}^{2}}\right]\)

The variation in the deviatoric part of the plastic deformation is then immediately obtained:

Eq 8.1.1-12

\({\mathrm{\delta s}}_{\text{ij}}\) is then written:

\({{\mathrm{\delta s}}_{\text{ij}}=2\text{μδ {}\tilde{\epsilon }}_{\text{ij}}-\frac{\text{18}\mu }{({H}_{p}+A)}\left[({\mathrm{\Delta s}}_{\text{kl}}{s}_{\text{ij}}{\mathrm{\delta s}}_{\text{kl}}+{s}_{\text{kl}}{s}_{\text{ij}}{\mathrm{\delta s}}_{\text{kl}})\right]-\frac{\text{18}\mu }{{H}_{p}}{s}_{\text{kl}}{\mathrm{\Delta s}}_{\text{kl}}{\mathrm{\delta s}}_{\text{ij}}-\frac{\text{12}\mu }{{H}_{p}}{M}^{2}(P-{P}_{\text{cr}}){\mathrm{\Delta P\delta s}}_{\text{ij}}\) eq 8.1.1-13

which becomes, by separating the terms in variation in stress and the term in variation in total deformation:

Eq 8.1.1-14

or in tensor writing:

\(\left\{{I}_{4}^{d}(1+\frac{\text{12}\mu }{{H}_{p}}{M}^{2}(P-{P}_{\text{cr}})\mathrm{\Delta P}+\frac{\text{18}\mu }{{H}_{p}}\mathrm{\Delta s}:s)+\frac{\text{18}\mu }{({H}_{p}+A)}(s+\mathrm{\Delta s})\otimes s\right\}\mathrm{\delta s}=2\text{μδ {}\tilde{\epsilon }\text{}}\) q8.1.1-15

which can still be written by symmetrizing tensor \((s+\mathrm{\Delta s})\otimes s\):

\(\left\{{I}_{4}^{d}(1+\frac{\text{12}\mu }{{H}_{p}}{M}^{2}(P-{P}_{\text{cr}})\mathrm{\Delta P}+\frac{\text{18}\mu }{{H}_{p}}\mathrm{\Delta s}:s)+\frac{\text{18}\mu }{({H}_{p}+A)}\aleph \right\}\mathrm{\delta s}=2\text{μδ {}\tilde{\epsilon }\text{}}\) eq 8.1.1-16

with: \(\aleph =\frac{1}{2}\left[((s+\mathrm{\Delta s})\otimes s)+(s\otimes (s+\mathrm{\Delta s}){)}^{T}\right]\)

Calculate \(\aleph\), by asking: \({T}_{\text{ij}}={s}_{\text{ij}}+{\mathrm{\Delta s}}_{\text{ij}}\)

\(\begin{array}{}\\ T\otimes \text{s=}\left[\begin{array}{cccccc}{T}_{\text{11}}{s}_{\text{11}}& {T}_{\text{11}}{s}_{\text{22}}& {T}_{\text{11}}{s}_{\text{33}}& \sqrt{2}{T}_{\text{11}}{s}_{\text{12}}& \sqrt{2}{T}_{\text{11}}{s}_{\text{23}}& \sqrt{2}{T}_{\text{11}}{s}_{\text{31}}\\ {T}_{\text{22}}{s}_{\text{11}}& {T}_{\text{22}}{s}_{\text{22}}& {T}_{\text{22}}{s}_{\text{33}}& \sqrt{2}{T}_{\text{22}}{s}_{\text{12}}& \sqrt{2}{T}_{\text{22}}{s}_{\text{23}}& \sqrt{2}{T}_{\text{22}}{s}_{\text{31}}\\ {T}_{\text{33}}{s}_{\text{11}}& {T}_{\text{33}}{s}_{\text{22}}& {T}_{\text{33}}{s}_{\text{33}}& \sqrt{2}{T}_{\text{33}}{s}_{\text{12}}& \sqrt{2}{T}_{\text{33}}{s}_{\text{23}}& \sqrt{2}{T}_{\text{33}}{s}_{\text{31}}\\ \sqrt{2}{T}_{\text{12}}{s}_{\text{11}}& \sqrt{2}{T}_{\text{12}}{s}_{\text{22}}& \sqrt{2}{T}_{\text{12}}{s}_{\text{33}}& {\mathrm{2T}}_{\text{12}}{s}_{\text{12}}& {\mathrm{2T}}_{\text{12}}{s}_{\text{23}}& {\mathrm{2T}}_{\text{12}}{s}_{\text{31}}\\ \sqrt{2}{T}_{\text{23}}{s}_{\text{11}}& \sqrt{2}{T}_{\text{23}}{s}_{\text{22}}& \sqrt{2}{T}_{\text{23}}{s}_{\text{33}}& {\mathrm{2T}}_{\text{23}}{s}_{\text{12}}& {\mathrm{2T}}_{\text{23}}{s}_{\text{23}}& {\mathrm{2T}}_{\text{23}}{s}_{\text{31}}\\ \sqrt{2}{T}_{\text{31}}{s}_{\text{11}}& \sqrt{2}{T}_{\text{31}}{s}_{\text{22}}& \sqrt{2}{T}_{\text{31}}{s}_{\text{33}}& {T}_{\text{31}}{s}_{\text{12}}& {\mathrm{2T}}_{\text{31}}{s}_{\text{23}}& {\mathrm{2T}}_{\text{31}}{s}_{\text{31}}\end{array}\right]\end{array}\) eq 8.1.1-17

\(\aleph =\frac{1}{2}\left[(T\otimes s)+(T\otimes s{)}^{T}\right]\) eq 8.1.1-18

Either:

\(C=\left\{{I}_{4}^{d}(\frac{1}{\mathrm{2\mu }}+\frac{6}{{H}_{p}}{M}^{2}(P-{P}_{\text{cr}})\mathrm{\Delta P}+\frac{9}{{H}_{p}}\mathrm{\Delta s}:s)+\frac{9}{({H}_{p}+A)}\aleph \right\}\) eq 8.1.1-19

We ask:

\(c=\frac{9}{{H}_{p}}(\mathrm{\Delta s}:s)\) eq 8.1.1-20

and

\(d=\frac{6}{{H}_{p}}{M}^{2}(P-{P}_{\text{cr}})\mathrm{\Delta P}\) eq 8.1.1-21

The symmetric matrix \(C\) of dimensions (6,6) is too big to be presented as a whole, it is broken down into 4 parts \({C}_{1}\), \({C}_{2}\), \({C}_{3}\) and \({C}_{4}\):

\(C=\left[\begin{array}{cc}{C}_{1}& {C}_{2}\\ {C}_{3}& {C}_{4}\end{array}\right]\)

with

\({C}_{1}=\left[\begin{array}{ccc}\frac{1}{\mathrm{2\mu }}+c+d+\frac{9}{({H}_{p}+A)}{s}_{\text{11}}{T}_{\text{11}}& \frac{9}{2({H}_{p}+A)}({T}_{\text{11}}{s}_{\text{22}}+{T}_{\text{22}}{s}_{\text{11}})& \frac{9}{2({H}_{p}+A)}({T}_{\text{11}}{s}_{\text{33}}+{T}_{\text{33}}{s}_{\text{11}})\\ \frac{9}{2({H}_{p}+A)}({T}_{\text{22}}{s}_{\text{11}}+{T}_{\text{11}}{s}_{\text{22}})& \frac{1}{\mathrm{2\mu }}+c+d+\frac{9}{({H}_{p}+A)}{T}_{\text{22}}{s}_{\text{22}}& \frac{9}{2({H}_{p}+A)}({T}_{\text{22}}{s}_{\text{33}}+{T}_{\text{33}}{s}_{\text{22}})\\ \frac{9}{2({H}_{p}+A)}({T}_{\text{33}}{s}_{\text{11}}+{T}_{\text{11}}{s}_{\text{33}})& \frac{9}{2({H}_{p}+A)}({T}_{\text{22}}{s}_{\text{33}}+{T}_{\text{33}}{s}_{\text{22}})& \frac{1}{\mathrm{2\mu }}+c+d+\frac{9}{({H}_{p}+A)}{T}_{\text{33}}{s}_{\text{33}}\end{array}\right]\) eq 8.1.1-22

\({C}_{2}=\left[\begin{array}{ccc}\frac{9\sqrt{2}}{2({H}_{p}+A)}({T}_{\text{11}}{s}_{\text{12}}+{s}_{\text{11}}{T}_{\text{12}})& \frac{9\sqrt{2}}{2({H}_{p}+A)}({T}_{\text{11}}{s}_{\text{23}}+{s}_{\text{11}}{T}_{\text{23}})& \frac{9\sqrt{2}}{2({H}_{p}+A)}({T}_{\text{11}}{s}_{\text{13}}+{s}_{\text{11}}{T}_{\text{13}})\\ \frac{9\sqrt{2}}{2({H}_{p}+A)}({T}_{\text{22}}{s}_{\text{12}}+{s}_{\text{22}}{T}_{\text{12}})& \frac{9\sqrt{2}}{2({H}_{p}+A)}({T}_{\text{22}}{s}_{\text{23}}+{s}_{\text{22}}{T}_{\text{23}})& \frac{9\sqrt{2}}{2({H}_{p}+A)}({T}_{\text{22}}{s}_{\text{13}}+{s}_{\text{22}}{T}_{\text{13}})\\ \frac{9\sqrt{2}}{2({H}_{p}+A)}({T}_{\text{33}}{s}_{\text{12}}+{s}_{\text{33}}{T}_{\text{12}})& \frac{9\sqrt{2}}{2({H}_{p}+A)}({T}_{\text{33}}{s}_{\text{23}}+{s}_{\text{33}}{T}_{\text{23}})& \frac{9\sqrt{2}}{2({H}_{p}+A)}({T}_{\text{33}}{s}_{\text{13}}+{s}_{\text{33}}{T}_{\text{13}})\end{array}\right]\) eq 8.1.1-23

\({C}_{3}={C}_{2}\) eq 8.1.1-24

\({C}_{4}=\left[\begin{array}{ccc}\frac{1}{\mathrm{2\mu }}+c+d+\frac{\text{18}}{({H}_{p}+A)}{s}_{\text{12}}{T}_{\text{12}}& \frac{9}{({H}_{p}+A)}({T}_{\text{12}}{s}_{\text{23}}+{T}_{\text{23}}{s}_{\text{12}})& \frac{9}{({H}_{p}+A)}({T}_{\text{12}}{s}_{\text{23}}+{T}_{\text{23}}{s}_{\text{12}})\\ \frac{9}{({H}_{p}+A)}({T}_{\text{23}}{s}_{\text{12}}+{T}_{\text{12}}{s}_{\text{23}})& \frac{1}{\mathrm{2\mu }}+c+d+\frac{\text{18}}{({H}_{p}+A)}{T}_{\text{23}}{s}_{\text{23}}& \frac{9}{({H}_{p}+A)}({T}_{\text{23}}{s}_{\text{13}}+{T}_{\text{13}}{s}_{\text{23}})\\ \frac{9}{({H}_{p}+A)}({T}_{\text{13}}{s}_{\text{12}}+{T}_{\text{12}}{s}_{\text{13}})& \frac{9}{({H}_{p}+A)}({T}_{\text{13}}{s}_{\text{23}}+{T}_{\text{23}}{s}_{\text{13}})& \frac{1}{\mathrm{2\mu }}+c+d+\frac{\text{18}}{({H}_{p}+A)}{T}_{\text{13}}{s}_{\text{13}}\end{array}\right]\) eq 8.1.1-25

Calculation of the rate of change in volume:

\({\mathrm{\Delta \epsilon }}_{v}^{p}={\mathrm{2M}}^{2}\Lambda (P-{P}_{\text{cr}}),{\text{δε}}_{v}^{p}={\mathrm{2M}}^{2}\mathrm{\delta \Lambda }(P-{P}_{\text{cr}})-{\mathrm{2M}}^{2}\mathrm{\Lambda \delta }{P}_{\text{cr}}=\mathrm{B\delta }\Lambda =\frac{\mathrm{3B}}{({H}_{p}+A)}(s+\mathrm{\Delta s})\text{.}\delta s\) eq 8.1.1-26

with: \(B={\mathrm{2M}}^{2}(P-{P}_{\text{cr}})-{\mathrm{2M}}^{2}\Lambda \frac{{M}^{2}(P-{P}_{\text{cr}})}{\frac{1}{2{\text{kP}}_{\text{cr}}}+{M}^{2}\Lambda }\text{.}\) eq 8.1.1-27

or using [éq 5.3.1-11]

\({\text{δε}}_{v}^{p}=\frac{\mathrm{3B}}{({H}_{p}+A)}(s+\mathrm{\Delta s})\text{.}\delta s\) eq 8.1.1-28

So we have:

\(\delta {\epsilon }_{\text{ij}}=({C}_{\text{ijkl}}-\frac{B}{({H}_{p}+A)}(s+\mathrm{\Delta s}{)}_{\text{kl}}{\delta }_{\text{ij}})\delta {s}_{\text{kl}}\) eq 8.1.1-29

8.1.2. Treatment of the hydrostatic part

It is now considered that the load variation is purely spherical (\(\delta s=0\)).

The \(P\) increment is written as:

\(\mathrm{\Delta P}={P}^{-}\text{exp}({k}_{0}{\mathrm{\Delta \epsilon }}_{v}^{e})-{P}^{-}\) eq 8.1.2-1

Derivating this equation gives:

\(\mathrm{\delta P}={k}_{0}P({\text{δε}}_{v}-{\text{δε}}_{v}^{p})\) eq 8.1.2-2

Calculation of \({\text{δε}}_{v}^{p}\):

We know that:

\({\mathrm{\Delta \epsilon }}_{v}^{p}={\mathrm{\Lambda 2M}}^{2}(P-{P}_{\text{cr}})\) eq 8.1.2-3

Differentiating this equation, we obtain:

\({\text{δε}}_{v}^{p}={\mathrm{2M}}^{2}(\delta \Lambda (P-{P}_{\text{cr}})+\Lambda (\mathrm{\delta P}-{\mathrm{\delta P}}_{\text{cr}}))\) eq 8.1.2-4

We know the expression for \(\Lambda\):

\(\Lambda =\frac{{\mathrm{2M}}^{2}(P-{P}_{\text{cr}})\mathrm{\Delta P}+\mathrm{3s\Delta s}}{{H}_{p}}=\frac{b}{{H}_{p}}\) eq 8.1.2-5

By posing

\(b={\mathrm{2M}}^{2}(P-{P}_{\text{cr}})\mathrm{\Delta P}+\mathrm{3s\Delta s}\) eq 8.1.2-6

By differentiating \(\text{ΔΛ}\), it comes from:

\(\mathrm{\delta \Lambda }=\frac{{\mathrm{2M}}^{2}}{{H}_{p}}\left[(P-{P}_{\text{cr}})\mathrm{\delta P}+(\mathrm{\delta P}-{\mathrm{\delta P}}_{\text{cr}})\mathrm{\Delta P}\right]-\frac{4{\text{kM}}^{4}b}{{H}_{p}^{2}}\left[\delta {\text{PP}}_{\text{cr}}(\mathrm{2P}-{P}_{\text{cr}})+{\mathrm{\delta P}}_{\text{cr}}P(P-{\mathrm{2P}}_{\text{cr}})\right]\) eq 8.1.2-7

We’re looking for the expression for \({\mathrm{\delta P}}_{\text{cr}}\) according to \(\mathrm{\delta \Lambda }\):

We have

\({\mathrm{\delta P}}_{\text{cr}}={\text{kP}}_{\text{cr}}{\text{δε}}_{v}^{p}\) eq 8.1.2-8

We can write:

\(\frac{{\mathrm{\delta P}}_{\text{cr}}}{{\text{kP}}_{\text{cr}}}={\mathrm{\delta \Lambda 2M}}^{2}(P-{P}_{\text{cr}})+{\mathrm{\Lambda 2M}}^{2}(\mathrm{\delta P}-{\mathrm{\delta P}}_{\text{cr}})\) eq 8.1.2-9

\({\mathrm{\delta P}}_{\text{cr}}(\frac{1+{\mathrm{\Lambda 2M}}^{2}{\text{kP}}_{\text{cr}}}{{\text{kP}}_{\text{cr}}})={\mathrm{\delta \Lambda 2M}}^{2}(P-{P}_{\text{cr}})+{\mathrm{\Lambda 2M}}^{2}\mathrm{\delta P}\) eq 8.1.2-10

\({\mathrm{\delta P}}_{\text{cr}}=(\frac{{\mathrm{2M}}^{2}(P-{P}_{\text{cr}}){\text{kP}}_{\text{cr}}}{1+2{\text{kP}}_{\text{cr}}{\mathrm{\Lambda M}}^{2}})\mathrm{\delta \Lambda }+(\frac{{\mathrm{2\Lambda M}}^{2}{\text{kP}}_{\text{cr}}}{1+2{\text{kP}}_{\text{cr}}{\mathrm{\Lambda M}}^{2}})\mathrm{\delta P}\) eq 8.1.2-11

We pose

\(c=\frac{{\mathrm{2M}}^{2}{\text{kP}}_{\text{cr}}(P-{P}_{\text{cr}})}{\left[1+{\mathrm{2M}}^{2}{\text{kP}}_{\text{cr}}\Lambda \right]}\), eq 8.1.2-12

\(a=\frac{{\mathrm{2M}}^{2}{\text{kP}}_{\text{cr}}\Lambda }{\left[1+{\mathrm{2M}}^{2}{\text{kP}}_{\text{cr}}\Lambda \right]}\) eq 8.1.2-13

We then have:

\({\mathrm{\delta P}}_{\text{cr}}=\mathrm{a\delta P}+\mathrm{c\delta \Lambda }\) eq 8.1.2-14

Replacing the expression for \({\mathrm{\delta P}}_{\text{cr}}\) in \(\mathrm{\delta \Lambda }\) [éq 5.3.2-7], we find:

\(\begin{array}{}\mathrm{\delta \Lambda }=\left[{\mathrm{2M}}^{2}(P-{P}_{\text{cr}})\mathrm{\delta P}+{\mathrm{2M}}^{2}(\mathrm{\delta P}-\mathrm{c\delta \Lambda }-\mathrm{a\delta P})\mathrm{\Delta P}\right]\text{.}\frac{1}{{H}_{p}}\\ -\frac{4{\text{kM}}^{4}b}{{H}_{p}^{2}}\left[\delta {\text{PP}}_{\text{cr}}(\mathrm{2P}-{P}_{\text{cr}})+(\mathrm{c\delta \Lambda }+\mathrm{a\delta P})P(P-{\mathrm{2P}}_{\text{cr}})\right]\end{array}\) eq 8.1.2-15

Combining the terms in \(\mathrm{\delta \Lambda }\) and those in \(\mathrm{\delta P}\), we find:

\(\mathrm{\delta \Lambda }=\frac{f}{e}\mathrm{\delta P}\) eq 8.1.2-16

with,

\(\begin{array}{}f=\frac{1}{{H}_{p}}\left[{\mathrm{2M}}^{2}(P-{P}_{\text{cr}})+{\mathrm{2M}}^{2}\mathrm{\Delta P}-2{\text{aM}}^{2}\mathrm{\Delta P}\right]\\ -\frac{4{\text{kM}}^{4}b}{{H}_{p}^{2}}\left[(\mathrm{2P}-{P}_{\text{cr}}){P}_{\text{cr}}+\text{aP}(P-{\mathrm{2P}}_{\text{cr}})\right]\end{array}\) eq 8.1.2-17

\(e=1+\frac{2{\text{cM}}^{2}\mathrm{\Delta P}}{{H}_{p}}+\frac{4{\text{bckM}}^{4}}{{H}_{p}^{2}}P(P-{\mathrm{2P}}_{\text{cr}})\) eq 8.1.2-18

So the expression for \({\text{δε}}_{v}^{p}\) becomes:

\({\text{δε}}_{v}^{p}={\mathrm{2M}}^{2}\left[\Lambda -\mathrm{a\Lambda }-\mathrm{\Lambda c}\frac{f}{e}+\frac{f}{e}(P-{P}_{\text{cr}})\right]\mathrm{\delta P}\) eq 8.1.2-19

Hence the expression for \({\text{δε}}_{v}\) in terms of \(\mathrm{\delta P}\):

\(\mathrm{\delta P}=\frac{{k}_{0}P}{G}{\text{δε}}_{v}\) eq 8.1.2-20

\(G=1+{\mathrm{2M}}^{2}{k}_{0}P(\Lambda -\mathrm{a\Lambda }-\Lambda \frac{f}{e}c+\frac{f}{e}(P-{P}_{\text{cr}}))\) eq 8.1.2-21

Calculation of the deviatoric deformation variation:

\(\delta {\tilde{\epsilon }}_{\text{ij}}=\delta {\tilde{\epsilon }}^{p}=\mathrm{3\delta }\Lambda s=3\frac{f}{e}\delta {\text{Ps}}_{\text{ij}}\) eq 8.1.2-22

So we finally have:

\({\text{δε}}_{\text{ij}}={F}_{\text{ij}}\mathrm{\delta P}\) eq 8.1.2-23

with

\(F=\frac{\mathrm{3f}}{e}s-\frac{G}{{\mathrm{3k}}_{0}P}{1}^{d}\) eq 8.1.2-24

8.1.3. Tangent operator

The tangent operator relates the total stress change to the total strain change. Since the total deformation increment under deviatory loading is written as:

\(\delta {\epsilon }_{\text{ij}}=({C}_{\text{ijkl}}-\frac{B}{({H}_{p}+A)}(s+\mathrm{\Delta s}{)}_{\text{kl}}{\delta }_{\text{ij}}){D}_{\text{klmn}}^{1}\delta {\sigma }_{\text{mn}}\), eq 8.1.3-1

with:

\({D}^{1}=\left[\begin{array}{cccccc}2/3& -1/3& -1/3& 0& 0& 0\\ -1/3& 2/3& -1/3& 0& 0& 0\\ -1/3& -1/3& 2/3& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]\) eq 8.1.3-2

the projection into deviatoric space,

and that under spherical loading we have:

\(\delta {\epsilon }_{\text{ij}}={F}_{\text{ij}}{D}_{\text{kl}}^{2}\delta {\sigma }_{\text{kl}}\) eq 8.1.3-3

with:

\({D}^{2}=\left[\begin{array}{c}-1/3\\ -1/3\\ -1/3\\ 0\\ 0\\ 0\end{array}\right]\) eq 8.1.3-4

hydrostatic projection, we then have:

\({\text{δσ}}_{\text{ij}}={A}_{\text{ijkl}}{\text{δε}}_{\text{kl}}\) eq 8.1.3-5

with:

\({A}_{\text{ijkl}}={\left[({C}_{\text{ijmn}}-\frac{B}{({H}_{p}+A)}(s+\mathrm{\Delta s}{)}_{\text{mn}}{\delta }_{\text{ij}}){D}_{{\text{mnkl}}^{1}}+{F}_{\text{ij}}{D}_{\text{kl}}^{2}\right]}^{-1}\) eq 8.1.3-6

the discretized tangent operator.

8.2. Tangent operator at the critical point

If the load point is at critical point \((P={P}_{\text{cr}})\), the general expression for the tangent operator is no longer valid. This is shown in particular by divisions by 0 (see the equations in [§ 5.3.1]). In what follows, the coherent tangent operator at the critical point is detailed by using, as in the general case, the decomposition that is partly deviatoric and partly hydrostatic.

8.2.1. Treatment of the deviatoric part

Recall that at the critical point, the expressions for the plastic multiplier \(\Lambda\) and its derivation \(\mathrm{\delta \Lambda }\) are written as follows:

\(\Lambda =(\frac{{Q}^{e}}{Q}-1)/\mathrm{6\mu }\) and \(\mathrm{\delta \Lambda }=\frac{{\mathrm{\delta Q}}^{e}}{\mathrm{6\mu Q}}-\frac{{Q}^{e}\mathrm{\delta Q}}{{\mathrm{6\mu Q}}^{2}}\) eq 8.2.1-1

with,

\({\mathrm{\delta Q}}^{e}=\frac{3}{2}\frac{{s}^{e}{\mathrm{\delta s}}^{e}}{{Q}^{e}}\) and \(\mathrm{\delta Q}=\frac{3}{2}\frac{\mathrm{s\delta s}}{Q}\) eq 8.2.1-2

Hence the expression for \(\mathrm{\delta \Lambda }\):

\(\mathrm{\delta \Lambda }=\frac{1}{\mathrm{6\mu }}\frac{3}{2}\left[\frac{{s}^{e}{\mathrm{\delta s}}^{e}}{{Q}^{e}Q}-\frac{{Q}^{e}\mathrm{s\delta s}}{{Q}^{3}}\right]\) eq 8.2.1-3

Likewise, let’s recall the expression for \(\mathrm{\delta s}\):

\({\mathrm{\delta s}}_{\text{ij}}=\mathrm{2\mu }(\delta {\tilde{\epsilon }}_{\text{ij}}-{\mathrm{3\delta \Lambda s}}_{\text{ij}}-{\mathrm{3\Lambda \delta s}}_{\text{ij}})\)

By replacing \(\Lambda\) and \(\mathrm{\delta \Lambda }\) with their expressions, we can write:

\({{\mathrm{\delta s}}_{\text{ij}}=2\text{μδ {}\tilde{\epsilon }}_{\text{ij}}-\frac{3}{2}\frac{{s}_{\text{kl}}^{e}{\mathrm{\delta s}}_{\text{kl}}^{e}}{{Q}^{e}Q}{s}_{\text{ij}}+\frac{3}{2}\frac{{Q}^{e}}{{Q}^{3}}{s}_{\text{kl}}{\mathrm{\delta s}}_{\text{kl}}{s}_{\text{ij}}-(\frac{{Q}^{e}}{Q}-1){\mathrm{\delta s}}_{\text{ij}}\) eq 8.2.1-4

\({\mathrm{\delta s}}_{\text{kl}}\left[{\delta }_{\text{ijkl}}+\frac{{Q}^{e}}{Q}{\delta }_{\text{ijkl}}-{\delta }_{\text{ijkl}}-\frac{3}{2}\frac{{Q}^{e}}{{Q}^{3}}{s}_{\text{kl}}\text{.}{s}_{\text{ij}}\right]=\mathrm{2\mu }\left[{\delta }_{\text{ijkl}}-\frac{3}{2}\frac{{s}_{\text{kl}}^{e}\text{.}{s}_{\text{ij}}}{{Q}^{e}Q}\right]\delta {\tilde{\epsilon }}_{\text{kl}}\) eq 8.2.1-5

or in tensor writing:

\(\mathrm{\delta s}\underset{G}{\underset{\underbrace{}}{\left[\frac{{Q}^{e}}{Q}{I}_{4}^{d}-\frac{3}{2}\frac{{Q}^{e}}{{Q}^{3}}s\otimes s\right]}}=\mathrm{2\mu }\underset{H}{\underset{\underbrace{}}{\left[{I}_{4}^{d}-\frac{3}{2}\frac{{s}^{e}\otimes s}{{Q}^{e}Q}\right]}}\delta \tilde{\epsilon }\) eq 8.2.1-6

Since \(\mathrm{\delta s}\) doesn’t depend on \({\text{δε}}_{v}\), you can mistake \(\delta \tilde{\epsilon }\) for \(\text{δε}\).

Using the projection tensor in deviatory stress space \({D}^{1}\) [éq 5.3.3-2], we can write:

\(\text{δε}=\frac{{D}^{1}\text{.}G\text{.}{H}^{-1}}{2\mu }\text{.}\text{δσ}\) eq 8.2.1-7

8.2.2. Treatment of the hydrostatic part

In tensor writing, we have the following relationship:

\({I}^{d}\mathrm{\delta P}={k}_{0}P{\text{δε}}_{v}\). eq 8.2.2-1

based on equation [éq 5.3.2-2] with \({\text{δε}}_{v}^{p}=0\) at the critical point.

Since \(\mathrm{\delta P}\) does not depend on \(\delta \tilde{\epsilon }\) then you can confuse \(\delta \tilde{\varepsilon }\) with \(\text{δε}\).

\({I}^{d}\mathrm{\delta P}={k}_{0}P\text{δε}\) eq 8.2.2-2

Using the projection tensor in hydrostatic stress space \({D}^{2}\) [éq 5.3.3-4], we can write:

\(\text{δε}=\frac{{I}^{d}}{{k}_{0}P}{D}^{2}\text{δσ}\) eq 8.2.2-3

8.2.3. Tangent operator

By combining the contributions of the two deviatoric and hydrostatic parts, we find the writing of the tangent operator that relates the variation in total stress to the variation in total deformation at the critical point:

\(\text{δε}=\left[\frac{{D}^{1}\text{.}G\text{.}{H}^{-1}}{\mathrm{2\mu }}+\frac{{I}^{d}}{{k}_{0}P}{D}^{2}\right]\text{.}\text{δσ}\)

or

\({\text{δσ}}_{\text{ij}}={A}_{\text{ijkl}}{\text{δε}}_{\text{kl}}\) eq 8.2.2-4

with

\({A}_{\text{ijkl}}={\left[\frac{{D}^{1}\text{.}G\text{.}{H}^{-1}}{\mathrm{2\mu }}+\frac{{I}^{d}}{{k}_{0}P}{D}^{2}\right]}^{-1}\) eq 8.2.2-5