5. Tangent operator

If the option is: RIGI_MECA_TANG, option used during the prediction, the tangent operator calculated at each Gauss point is said to be in speed:

\({\dot{\sigma }}_{\text{ij}}={D}_{\text{ijkl}}^{\text{elp}}{\dot{\epsilon }}_{\text{kl}}\)

In this case, \({D}_{\text{ijkl}}^{\text{elp}}\) is calculated from the non-discretized equations.

If the option is: FULL_MECA, option used when updating the tangent matrix at each iteration by updating the constraints and the internal variables:

\({\mathrm{d\sigma }}_{\text{ij}}={A}_{\text{ijkl}}{\mathrm{d\epsilon }}_{\text{kl}}\)

In this case, \({A}_{\text{ijkl}}\) is calculated from the implicitly discretized equations.

5.1. Nonlinear elastic tangent operator

The elastic relationship in speed is written as:

\({\dot{\sigma }}_{\text{ij}}=-\dot{P}{\delta }_{\text{ij}}+{\dot{s}}_{\text{ij}}=({k}_{0}P+{K}_{\text{cam}})\text{tr {}\dot{\epsilon }{\delta }_{\text{ij}}+\mathrm{2\mu }\tilde{\dot{\epsilon }}\text{}}\) eq 5.1-1

\({\dot{\sigma }}_{\text{ij}}=({k}_{0}P+{K}_{\text{cam}}-\frac{2}{3}\mu )\text{tr {}\dot{\epsilon }{\delta }_{\text{ij}}+\mathrm{2\mu }{\dot{\epsilon }}_{\text{ij}}\text{}}\) eq 5.1-2

The elastic tangent operator of the Cam_Clay law, noted \({D}^{e}\), is therefore deduced from the following matrix writing:

\(\left\{\begin{array}{}{\dot{\sigma }}_{\text{11}}\\ {\dot{\sigma }}_{\text{22}}\\ {\dot{\sigma }}_{\text{33}}\\ \sqrt{2}{\dot{\sigma }}_{\text{12}}\\ \sqrt{2}{\dot{\sigma }}_{\text{23}}\\ \sqrt{2}{\dot{\sigma }}_{\text{31}}\end{array}\right\}=\underset{{D}^{e}}{\underset{\underbrace{}}{\left[\begin{array}{cccccc}{k}_{0}P+{K}_{\text{cam}}+\frac{4}{3}\mu & {k}_{0}P+{K}_{\text{cam}}-\frac{2}{3}\mu & {k}_{0}P+{K}_{\text{cam}}-\frac{2}{3}\mu & 0& 0& 0\\ {k}_{0}P+{K}_{\text{cam}}-\frac{2}{3}\mu & {k}_{0}P+{K}_{\text{cam}}+\frac{4}{3}\mu & {k}_{0}P+{K}_{\text{cam}}-\frac{2}{3}\mu & 0& 0& 0\\ {k}_{0}P+{K}_{\text{cam}}-\frac{2}{3}\mu & {k}_{0}P+{K}_{\text{cam}}-\frac{2}{3}\mu & {k}_{0}P+{K}_{\text{cam}}+\frac{4}{3}\mu & 0& 0& 0\\ 0& 0& 0& \mathrm{2\mu }& 0& 0\\ 0& 0& 0& 0& \mathrm{2\mu }& 0\\ 0& 0& 0& 0& 0& \mathrm{2\mu }\end{array}\right]}}\left\{\begin{array}{}{\dot{\epsilon }}_{\text{11}}\\ {\dot{\epsilon }}_{\text{22}}\\ {\dot{\epsilon }}_{\text{33}}\\ \sqrt{2}{\dot{\epsilon }}_{\text{12}}\\ \sqrt{2}{\dot{\epsilon }}_{\text{23}}\\ \sqrt{2}{\dot{\epsilon }}_{\text{31}}\end{array}\right\}\) eq 5.1-3

5.2. Plastic tangent operator in speed. Option RIGI_MECA_TANG

In this case, the global tangent operator is \({K}_{i-1}\) (the option RIGI_MECA_TANG called at the first iteration of a new load increment) based on the results known at time \({t}_{i-1}\) [bib3].

If the stress tensor at \({t}_{i-1}\) is on the border of the elasticity domain, we write the condition: \(\dot{f}=0\) which must be satisfied in conjunction with the condition \(f=0\). If the stress tensor at \({t}_{i-1}\) is inside the domain, \(f<0\), then the tangent operator is the elasticity operator.

\(\dot{f}=(\frac{\partial f}{\partial \sigma })\dot{\sigma }+\frac{\partial f}{\partial {P}_{\text{cr}}}{\dot{P}}_{\text{cr}}=0\) eq 5.2-1

like \({\dot{P}}_{\text{cr}}=\frac{\partial {P}_{\text{cr}}}{\partial {\epsilon }_{v}^{p}}{\dot{\epsilon }}_{v}^{p}\), so:

\(\dot{f}=(\frac{\partial f}{\partial \sigma })\dot{\sigma }+\frac{\partial f}{\partial {P}_{\text{cr}}}\frac{\partial {P}_{\text{cr}}}{\partial {\epsilon }_{v}^{p}}{\dot{\epsilon }}_{v}^{p}=0\) eq 5.2-2

On the other hand \({\dot{\epsilon }}^{e}=\dot{\epsilon }-{\dot{\epsilon }}^{p}\)

So:

\({D}^{{e}^{-1}}\dot{\sigma }=\dot{\epsilon }-\dot{\Lambda }\frac{\partial f}{\partial \sigma }\), eq 5.2-3

that is to say:

\({\dot{\sigma }}_{\text{ij}}={D}_{\text{ijkl}}^{e}{\dot{\epsilon }}_{\text{kl}}-\dot{\Lambda }{D}_{\text{ijkl}}^{e}{(\frac{\partial f}{\partial \sigma })}_{\text{kl}}\) eq 5.2-4

The plastic work hardening module is written according to equation [éq 3.5-7] and using the flow rule:

\({H}_{p}=\frac{\partial f}{\partial {P}_{\text{cr}}}\frac{\partial {P}_{\text{cr}}}{\partial {\epsilon }_{v}^{p}}\frac{\partial F}{\partial {P}_{\text{cr}}}=-\frac{1}{\dot{\Lambda }}\frac{\partial f}{\partial {P}_{\text{cr}}}\frac{\partial {P}_{\text{cr}}}{\partial {\epsilon }_{v}^{p}}{\dot{\epsilon }}_{v}^{p}\) eq 5.2-5

The equations [éq 5.2-1] and [éq 5.2-5] give:

\({(\frac{\partial f}{\partial \sigma })}_{\text{ij}}{\dot{\sigma }}_{\text{ij}}-\dot{\Lambda }{H}_{p}=0\) eq 5.2-6

Multiplying equation [éq 5.2-4] by \({(\frac{\partial f}{\partial \sigma })}_{\text{ij}}\) gives:

\({(\frac{\partial f}{\partial \sigma })}_{\text{ij}}{\dot{\sigma }}_{\text{ij}}={(\frac{\partial f}{\partial \sigma })}_{\text{ij}}{D}_{\text{ijkl}}^{e}\dot{\epsilon }-\dot{\Lambda }{(\frac{\partial f}{\partial \sigma })}_{\text{ij}}{D}_{\text{ijkl}}^{e}{(\frac{\partial f}{\partial \sigma })}_{\text{kl}}\) eq 5.2-7

The two previous equations allow us to find:

\({H}_{p}\dot{\Lambda }={(\frac{\partial f}{\partial \sigma })}_{\text{ij}}{D}_{\text{ijkl}}^{e}{\dot{\epsilon }}_{\text{kl}}-\dot{\Lambda }{(\frac{\partial f}{\partial \sigma })}_{\text{ij}}{D}_{\text{ijkl}}^{e}{(\frac{\partial f}{\partial \sigma })}_{\text{kl}}\) eq 5.2-8

Hence the expression for the plastic multiplier:

\(\dot{\Lambda }=\frac{{(\frac{\partial f}{\partial \sigma })}_{\text{ij}}{D}_{\text{ijkl}}^{e}{\dot{\epsilon }}_{\text{kl}}}{{(\frac{\partial f}{\partial \sigma })}_{\text{ij}}{D}_{\text{ijkl}}^{e}{(\frac{\partial f}{\partial \sigma })}_{\text{kl}}+{H}_{p}}\) eq 5.2-9

Let H be the elastoplastic module defined as:

\(H={(\frac{\partial f}{\partial \sigma })}_{\text{ij}}{D}_{\text{ijkl}}^{e}{(\frac{\partial f}{\partial \sigma })}_{\text{kl}}+{H}_{p}\) eq 5.2-10

The plastic multiplier is written as:

\(\dot{\Lambda }=\frac{{(\frac{\partial f}{\partial \sigma })}_{\text{ij}}{D}_{\text{ijkl}}^{e}{\dot{\epsilon }}_{\text{kl}}}{H}\) eq 5.2-11

Replacing \(\dot{\Lambda }\) with its expression in equation [éq 5.2-4], we get:

\({\dot{\sigma }}_{\text{ij}}={D}_{\text{ijkl}}^{e}{\dot{\epsilon }}_{\text{kl}}-\frac{1}{H}\left[{(\frac{\partial f}{\partial \sigma })}_{\text{mn}}{D}_{\text{mnop}}^{e}{\dot{\epsilon }}_{\text{op}}\right]\text{.}{D}_{\text{ijkl}}^{e}{(\frac{\partial f}{\partial \sigma })}_{\text{kl}}\) eq 5.2-12

The elastoplastic operator \({D}^{\text{elp}}={D}^{e}-{D}^{p}\) is therefore deduced from this:

\({\dot{\sigma }}_{\text{ij}}=\underset{{D}^{\text{elp}}}{\underset{\underbrace{}}{\left[{D}_{\text{ijkl}}^{e}-\frac{1}{H}{(\frac{\partial f}{\partial \sigma })}_{\text{op}}{D}_{\text{ijop}}^{e}{D}_{\text{mnkl}}^{e}{(\frac{\partial f}{\partial \sigma })}_{\text{mn}}\right]}}{\dot{\epsilon }}_{\text{kl}}\) eq 5.2-13

with,

\({D}_{\text{ijkl}}^{p}=\frac{1}{H}{(\frac{\partial f}{\partial \sigma })}_{\text{op}}{D}_{\text{ijop}}^{e}{D}_{\text{mnkl}}^{e}{(\frac{\partial f}{\partial \sigma })}_{\text{mn}}\) eq 5.2-14

Calculation of \(H\):

\({(\frac{\partial f}{\partial \sigma })}_{\text{ij}}=-\frac{2}{3}{M}^{2}(P-{P}_{\text{trac}}-{P}_{\text{cr}}){\delta }_{\text{ij}}+{\mathrm{3s}}_{\text{ij}}\), eq 5.2-15

Which is written in vector notation:

\(\left[\begin{array}{c}-\frac{2}{3}{M}^{2}(P-{P}_{\text{trac}}-{P}_{\text{cr}})+{\mathrm{3s}}_{\text{11}}\\ -\frac{2}{3}{M}^{2}(P-{P}_{\text{trac}}-{P}_{\text{cr}})+{\mathrm{3s}}_{\text{22}}\\ -\frac{2}{3}{M}^{2}(P-{P}_{\text{trac}}-{P}_{\text{cr}})+{\mathrm{3s}}_{\text{33}}\\ 3\sqrt{2}{s}_{\text{12}}\\ 3\sqrt{2}{s}_{\text{23}}\\ 3\sqrt{2}{s}_{\text{31}}\end{array}\right]\) eq 5.2-16

Hence the expression of:

\({D}_{\text{ijkl}}^{e}{(\frac{\partial f}{\partial \sigma })}_{\text{kl}}:\left[\begin{array}{c}-{\mathrm{2k}}_{0}{M}^{2}(P-{P}_{\text{trac}})(P-{P}_{\text{trac}}-{P}_{\text{cr}})+{\mathrm{6\mu s}}_{\text{11}}\\ -{\mathrm{2k}}_{0}{M}^{2}(P-{P}_{\text{trac}})(P-{P}_{\text{trac}}-{P}_{\text{cr}})+{\mathrm{6\mu s}}_{\text{22}}\\ -{\mathrm{2k}}_{0}{M}^{2}(P-{P}_{\text{trac}})(P-{P}_{\text{trac}}-{P}_{\text{cr}})+{\mathrm{6\mu s}}_{\text{33}}\\ \mathrm{6\mu }\sqrt{2}{s}_{\text{12}}\\ \mathrm{6\mu }\sqrt{2}{s}_{\text{23}}\\ \mathrm{6\mu }\sqrt{2}{s}_{\text{31}}\end{array}\right]\) eq 5.2-17

and

\({(\frac{\partial f}{\partial \sigma })}_{\text{ij}}{D}_{\text{ijkl}}^{e}{(\frac{\partial f}{\partial \sigma })}_{\text{kl}}={\mathrm{4k}}_{0}{M}^{4}(P-{P}_{\text{trac}})(P-{P}_{\text{trac}}-{P}_{\text{cr}}{)}^{2}+\text{12}{\mathrm{\mu Q}}^{2}\) where

\(\text{12}{\mathrm{\mu Q}}^{2}=\text{18}\mu \text{tr}(s\text{.}s)\) eq 5.2-18

From the equations [éq 3.5-7] and [éq 5.2-18], we can deduce the expression for \(H\):

\(H={\mathrm{4M}}^{4}(P-{P}_{\text{trac}})(P-{P}_{\text{trac}}-{P}_{\text{cr}})(k{}_{0}\text{}(P-{P}_{\text{trac}}-{P}_{\text{cr}})+{\text{kP}}_{\text{cr}})+\text{12}{\mathrm{\mu Q}}^{2}\) eq 5.2-19

By posing:

\({A}_{\text{ij}}=-{\mathrm{2k}}_{0}{M}^{2}(P-{P}_{\text{trac}})(P-{P}_{\text{trac}}-{P}_{\text{cr}}){\delta }_{\text{ij}}+{\mathrm{6\mu s}}_{\text{ij}}\), eq 5.2-20

we can write the following symmetric plastic matrix:

\({D}^{p}=\frac{1}{H}\left[\begin{array}{cccccc}{A}_{\text{11}}^{2}& {A}_{\text{11}}{A}_{\text{22}}& {A}_{\text{11}}{A}_{\text{33}}& 6\sqrt{2}{\mathrm{\mu A}}_{\text{11}}{s}_{\text{12}}& 6\sqrt{2}{\mathrm{\mu A}}_{\text{11}}{s}_{\text{23}}& 6\sqrt{2}{\mathrm{\mu A}}_{\text{11}}{s}_{\text{31}}\\ \text{.}& {A}_{\text{22}}^{2}& {A}_{\text{22}}{A}_{\text{33}}& 6\sqrt{2}{\mathrm{\mu A}}_{\text{22}}{s}_{\text{12}}& 6\sqrt{2}{\mathrm{\mu A}}_{\text{22}}{s}_{\text{23}}& 6\sqrt{2}{\mathrm{\mu A}}_{\text{22}}{s}_{\text{31}}\\ \text{.}& \text{.}& {A}_{\text{33}}^{2}& 6\sqrt{2}{\mathrm{\mu A}}_{\text{33}}{s}_{\text{12}}& 6\sqrt{2}{\mathrm{\mu A}}_{\text{33}}{s}_{\text{23}}& 6\sqrt{2}{\mathrm{\mu A}}_{\text{33}}{s}_{\text{31}}\\ \text{.}& \text{.}& \text{.}& \text{36}{\mu }^{2}{s}_{\text{12}}^{2}& \text{36}{\mu }^{2}{s}_{\text{12}}{s}_{\text{23}}& \text{36}{\mu }^{2}{s}_{\text{12}}{s}_{\text{31}}\\ \text{.}& \text{.}& \text{.}& \text{.}& \text{36}{\mu }^{2}{s}_{\text{23}}^{2}& \text{36}{\mu }^{2}{s}_{\text{23}}{s}_{\text{31}}\\ \text{.}& \text{.}& \text{.}& \text{.}& \text{.}& \text{36}{\mu }^{2}{s}_{\text{31}}^{2}\end{array}\right]\) eq 5.2-21

5.3. Implicit tangent operator. Option FULL_MECA

The coherent tangent operator in option FULL_MECA is calculated as the speed tangent operator for the current stress state.

However, theoretical elements for calculating it are given in the appendix, in paragraph 8. Note that the equations in the appendix assume that the criterion passes through a zero stress state, \({P}_{\text{trac}}\) and \({K}_{\text{cam}}\) were not yet introduced. It is necessary to remember to take them into account and if necessary to reactivate these equations for the coherent tangent operator.