4. Calculation of tangent stiffness

In order to allow the global problem (equilibrium equations) to be solved by a Newton method, it is necessary to determine the coherent tangent matrix of the incremental problem. To do this, we once again adopt the convention for writing symmetric tensors of order 2 in the form of vectors with 6 components. So, for a \(a\) tensor:

\(a\text{=}{}^{t}\text{}\left[\begin{array}{ccc}{a}_{\text{xx}}& {a}_{\text{yy}}& {a}_{\text{zz}}\end{array}\begin{array}{ccc}\sqrt{2}{a}_{\text{xy}}& \sqrt{2}{a}_{\text{xz}}& \sqrt{2}{a}_{\text{yz}}\end{array}\right]\) [éq4-1]

If we also introduce the hydrostatic vector \(1\) and the deviatoric projection matrix \(P\):

\(1\text{=}{}^{t}\text{}\left[\begin{array}{cccccc}1& 1& 1& 0& 0& 0\end{array}\right]\) [éq4-2]

\(P\text{=}\text{Id}\text{-}\frac{1}{3}1\otimes 1\) [éq4-3]

So the coherent tangent stiffness matrix is written for elastic behavior:

\(\frac{\partial s}{\partial \Delta \varepsilon }\text{=}K1\otimes 1\text{+}2\mup\) [éq4-4]

and for plastic behavior:

\(\frac{\partial s}{\partial \Delta \varepsilon }\text{=}K1\otimes 1\text{+}2\mu (1\text{-}\frac{3\mu \Deltap }{{s}_{\text{eq}}^{e}})P\text{+}9{\mu }^{2}(\frac{\Deltap }{{s}_{\text{eq}}^{e}}\text{-}\frac{1}{{R}^{\text{'}}(p)+\frac{3}{2}(2\mu \text{+}C)})(\frac{{\tilde{s}}^{e}}{{s}_{\text{eq}}^{e}}\otimes \frac{{\tilde{s}}^{e}}{{s}_{\text{eq}}^{e}})\) [éq4-5]

The initial tangent matrix, used by option RIGI_MECA_TANG, is obtained by adopting the behavior of the previous step (elastic or plastic, signified by an internal variable \(x\) equal to 0 or 1) and by taking \(\Deltap \text{=}0\) into the equation [eq].

Note:

RIGI_MECA_TANG is the operator linearized with respect to **time (cf.* [R5.03.01] , [R5.03.05] ) and corresponds to what is called the speed problem; in this case, linearization with respect to \(\Deltau\) , in \(\Deltau \text{=}0\) , provides the same expression.

It is now proposed to demonstrate the expression [eq]. By differentiating between [eq] and [eq] at a fixed temperature, we immediately obtain:

\(\delta \sigma \text{=}\left[K1\otimes 1\text{+}2\mup \right]\delta \varepsilon \text{-}2\mu \delta {\varepsilon }^{p}\) [éq4-6]

If the behavior regime is plastic, the incremental flow law [eq] then provides:

\(\delta {\varepsilon }^{p}=\frac{3}{2}\deltap \frac{{\tilde{s}}^{e}}{{s}_{\text{eq}}^{e}}\text{+}\frac{3}{2}\Deltap \delta (\frac{{\tilde{s}}^{e}}{{s}_{\text{eq}}^{e}})\) [éq4-7]

As for \(\mathrm{dp}\), it is obtained by differentiating the implicit equation [eq]:

\(\left[\frac{3}{2}(2\mu \text{+}C)\text{+}{R}^{\text{'}}(p)\right]\deltap \text{=}\delta {s}_{\text{eq}}^{e}\) [éq4-8]

Finally, all that’s left to do is provide the \({\tilde{s}}^{e}\) variations:

\(\delta {\tilde{s}}^{e}\text{=}2\mu \delta \tilde{\varepsilon }{\mathrm{ds}}_{\text{eq}}^{e}\text{=}3\mu \frac{{\tilde{s}}^{e}}{{s}_{\text{eq}}^{e}}\cdot \delta \tilde{\varepsilon }\delta (\frac{{\tilde{s}}^{e}}{{s}_{\text{eq}}^{e}})\text{=}\frac{1}{{s}_{\text{eq}}^{e}}(2\mu \text{-}3\mu \frac{{\tilde{s}}^{e}}{{s}_{\text{eq}}^{e}}\otimes \frac{{\tilde{s}}^{e}}{{s}_{\text{eq}}^{e}})\cdot \delta \tilde{\varepsilon }\) [éq4-9]

By then replacing [eq], [eq] and [eq] in [eq], we get the expression [eq].

This expression is formally identical to that defined in R5.03.02: [eq] and is written as:

\(\frac{\partial \sigma }{\partial \Delta \varepsilon }\text{=}K1\otimes 1\text{+}2\mu (1\text{-}\frac{3\mu \xi \Deltap }{{s}_{\text{eq}}^{e}})(\text{Id}-\frac{1}{3}1\otimes 1)\text{+}9{\mu }^{2}\xi (\frac{\Deltap }{{s}_{\text{eq}}^{e}}\text{-}\frac{1}{{R}^{\text{'}}\text{+}\frac{3}{2}(2\mu \text{+}C)})(\frac{{\tilde{s}}^{e}}{{s}_{\text{eq}}^{e}}\otimes \frac{{\tilde{s}}^{e}}{{s}_{\text{eq}}^{e}})\)

with \(\xi \text{=}1\) if \(\Delta \varepsilon\) leads to lamination, and \(\xi \text{=}0\) otherwise.

Using [eq], we find:

\(\frac{\partial s}{\partial \Delta \varepsilon }\text{=}{\lambda }^{\text{*}}\overrightarrow{1}\otimes \overrightarrow{1}\text{+}2{\mu }^{\text{*}}\text{Id}\text{-}\xi \frac{9{\mu }^{2}}{H(p)}(1\text{-}\frac{{R}^{\text{'}}\text{.}\Delta p}{R(p)})\frac{1}{{R}^{\text{'}}+\frac{3}{2}(2\mu \text{+}C)}(\frac{{\sigma }^{\text{dev}}}{R(p)}\otimes \frac{{\sigma }^{\text{dev}}}{R(p)})\)

with \({\lambda }^{\text{*}}\text{=}K\text{-}\frac{2\mu }{3}\frac{G(\Delta p)}{H(\Delta p)}2{\mu }^{\text{*}}\text{=}2\mu \frac{G(\Delta p)}{H(\Delta p)}\)

for option FULL_MECA: \({\sigma }^{\text{dev}}\text{=}\tilde{\sigma }-X\)

for option RIGI_MECA_TANG: \({\sigma }^{\text{dev}}\text{=}{\tilde{\sigma }}^{-}\text{-}{X}^{-}\)

with \(H(\Delta p)\text{=}1+\frac{\frac{3}{2}(2\mu \text{+}C)\xi \Delta p}{R(p)}\)

and \(G(\Delta p)=1+\frac{3}{2}C\xi \frac{\Delta p}{R(p)}\)