3. Expression of the tangent matrix in the main base

3.1. Cases where only one mechanism is active

We have:

(3.1)\[\begin{split} d {\ mathrm {\ sigma}}} ^ {\ text {+}} =C\ mathrm {.}} \ mathrm {\ delta}\ mathrm {\ epsilon} -\ frac {\ mathrm {\ delta} {\ mathrm {\ sigma}} _ {1}} {K+\ frac {4} {4} {4} {4} {4}} {3} {K+\ frac {4}} {3} {3} {K+\ frac {4}} {3} {3} {3} {K+\ frac {4} {4}} {3} {3} {3} {3} {3} {3} {3} {3} {3} {K+\ frac {4} {4}} {3} {3} {3}} G\\ K-\ frac {2} {3} G\ end {array}\ right)\end{split}\]

Asking \(\{\begin{array}{c}A=K+\frac{4}{3}G\\ B=K-\frac{2}{3}G\end{array}\), we have:

(3.2)\[\begin{split} \ frac {\ mathrm {\ delta} {\ mathrm {\ sigma}} {\ mathrm {\ sigma}}} _ {1}} {A}\ left (\ begin {array} {c} {c} A}\ left (\ begin {array} {c} {c} A\\ B\ B\ end {array}\ right)\ left (\ begin {array} {ccc} 1& 0& 0\ end {array}\ right)\ mathrm {.} C\ mathrm {.} C\ mathrm {.} \ mathrm {\ delta}\ mathrm {\ epsilon} =\ frac {1} {A}\ left (\ begin {array} {c} A\\ B\ B\ end {array}\ right)\ right)\ left (\ begin {array}\ right)\ left (\ begin {array}\ right)\ left (\ begin {array}\ right)\ left (\ begin {array}\ right)\ left (\ begin {array}\ right)\ left (\ begin {array}\ right)\ left (\ begin {array}\ right)\ left (\ begin {array}\ right)\ left (\ begin {array}\ right)\ left (\ begin {array}\ right)\ left (\ begin {array \ mathrm {\ delta}\ mathrm {\ epsilon} =\ left [\ begin {array} {ccc} A& B& B\\ B&\ frac {{B} ^ {2}} {A}} {A}} {A} &\ frac {{B}}} {A}\ frac {{B}} ^ {2}} {A}} {A}} {A} &\ frac {{B}}} {A}} {A} &\ frac {{B}}} {A}} {A} &\ frac {{B}}} {A}} {A} &\ frac {{B}}} {A}} {A} &\ frac {{B}}} {A}} {A} &\ frac {{{B} ^ {2}} {A}\ end {array}\ right]\ mathrm {.} \ mathrm {\ delta}\ mathrm {\ epsilon}\end{split}\]

Consider the following expression for the tangent matrix \(T\) (see for the elasticity matrix \(C\)):

(3.3)\[\begin{split} T=\ frac {\ mathrm {\ delta} {\ delta} {\ mathrm {\ sigma}} {\ mathrm {\ delta}\ mathrm {\ epsilon}}} =C-\ left [\ left [\ left [\ begin {array} {\ sigma}} {ccc}} ^ {2}} =C-\ left [\ left [\ begin {array} {\ sigma}} {ccc}} ^ {2}} =C-\ left [\ begin {array} {\ sigma}} {ccc}} ^ {2}} =C-\ left [\ left [\ begin {array} {\ sigma}} {ccc}} A& B& B\ B&\ frac {{B} ^ {2}} {A}} {A} &\ frac ac {{B} ^ {2}} {A} {A}\\ B&\ frac {{B}} ^ {2}} {A} &\ frac {{B} ^ {2}} {A}\ end {array}\\ right] =\ left [\ left [\ begin {array}\\ right] =\ left [\ begin {array} {\ right]] =\ left [\ begin {array} {\ right]] =\ left [\ begin {array} {\ right] =\ left [\ begin {array} {ccc}} 0& 0\\ 0&\ frac {{A} ^ {2}} {2}\ end {array}\\ right] =\ left [\ begin {array}\ right] =\ left [\ begin {array} {right]] =\ left 2}} {A} &\ frac {B\ left (A-B\ right)} {A}\\ 0&\ frac {B\ left (A-B\ right)} {A} &\ frac {{A} ^ {2} ^ {2}} - {B} ^ {2}} {A}\ end {array}\ right]\end{split}\]

3.2. Cases where two mechanisms are active

We have:

(3.4)\[\begin{split} d {\ mathrm {\ sigma}}} ^ {\ text {+}} =C\ mathrm {.}} \ mathrm {\ delta}\ mathrm {\ epsilon} -\ underset {\ mathrm {\ epsilon} -\ mathrm {\ delta} {\ sigma}} _ {C}}} {\ underset {\ epsilon}} -\ underset {\ epsilon} -\ underset {\ epsilon} -\ underset {\ epsilon} -\ underset {\ epsilon} -\ underset {\ epsilon} -\ underset {\ epsilon} -\ underset {\ epsilon} -\ underset {\ epsilon} -\ underset {\ epsilon} -\ -\ -\ - {\ epsilon} -\ -\ -\ -\ - {\ epsilon} -\ -\ -\ -\ - mathrm {\ delta} {\ mathrm {\ sigma}} _ {\ sigma}} _ {\ sigma}}\ left (\ begin {array} {c} A\\ B\\ B\ end {array}\\ right) +\ frac {\ array}\ right) +\ frac {\ array}\ right) +\ frac {A\ mathrm {\ delta}} {\ mathrm {\ sigma}} _ {1}}} {\ mathit {det}}}\ left (\ begin {array} {c} B\\ A\\ B\ end {array}\ B\ end {array}\ right)\ right)}}\end{split}\]

We get:

\[\]

: label: eq-27

mathrm {delta}mathrm {Delta} {mathrm {delta}} {mathrm {delta} {mathrm {sigma}} _ {1}\ left (begin {array} {delta}} {left (begin {array} {Delta} {Delta} {}}mathrm {delta}}} _ {1}\\\\\\\left (begin {array} {Delta}}}left (begin {array} {Delta}} {left) (begin {array} {Delta}}left (begin {array} {Delta}}left (begin {array} {Delta}}left (begin {array} {Delta}} delta} {mathrm {sigma}} _ {2}left (begin {array} {c} 0\frac {B} {A+B}end {array}right) =left [left (left (left (begin {array} {c} {c} {c} 1\ 0\ frac {B} {A+B}end {array}right)left [left (left (left (begin {array} {c} {c}} 1\ 0\ frac {B} {A+B}end {array}right)left (begin {array}right)left (begin {array} {right) {array} {ccc} A& B& Bend {array}right) +left (begin {array} {c} 0\ 1\ frac {B} {A+B}end {array}right)end {array}right)left)left (begin {array}right)left (begin {array}right)mathrm {.} mathrm {delta}mathrm {epsilon}

Either:

\[\]

: label: eq-28

mathrm {delta}mathrm {Delta} {mathrm {delta}} {mathrm {sigma}}} _ {C} =left [begin {array} {ccc} A& B& B\ 0& 0& 0& 0\ delta} {delta} {delta} {delta}} {delta} {delta}} {delta} {delta} {delta} {delta} {delta} {delta} {delta} {delta} {delta} {delta} {delta} {delta} {delta} {delta} {delta} {delta} {delta} {delta} {delta} {delta} {delta} {delta} {delta} {delta} {ac {{B} ^ {2}} {A+B}end {array}right] +left [begin {array} {ccc} 0& 0& 0\ B& A& B\ frac {{B} {B} {B}} ^ {2}} {2}} {A+B}} {A+B}} {A+B}end {array}right]right]right]mathrm {.} mathrm {delta}mathrm {epsilon} mathrm {delta}mathrm {Delta} {mathrm {delta}} {mathrm {sigma}} _ {C} =left [begin {array} {ccc} A& B\ B& B\ B & B\ BBbdelta} {delta} {delta} {delta} {delta} {delta} {delta} {delta} {mathrm} {delta} {delta} {delta} {mathrm} {delta} {delta} {delta} {mathrm} {delta} {delta} {delta} {mathrm} {delta} {delta} {delta} {mathrm} {delta} { mathrm {delta}mathrm {epsilon}

Consider the following expression from the tangent matrix \(T\):

(3.5)\[\begin{split} T=\ frac {\ mathrm {\ delta} {\ delta} {\ mathrm {\ sigma}}} ^ {\ text {+}}} {\ mathrm {\ delta}\ mathrm {\ epsilon}} =C-\ left [\ left [\ left [\ left [\ begin {array} {\ sigma}}} ^ {\ frac}\\ left [\ begin {array} {\ sigma}} ^ {\ text {+}}} {\ mathrm {\ delta}\ mathrm {\ epsilon}}} =C-\ left [\ left [\ begin {array} {\ sigma}}} ^ {\ mathrm {\ epsilon}} =C-\ left [\ left [\ begin {array} {\ sigma}}} ^ {\ text {2}} {A+B}\ end {array}\ right] =\ left [\ begin {array} {ccc} 0& 0& 0\\ 0& 0&\ 0&\ 0&\ 0&\ frac {A\ left (A+B\ right) -2 {B} (A+B\ right) -2 {B} ^ {2}} {ccc}} {A+B}\ end {array}\ right] =\ left [\ begin {array} {0& 0&\ 0&\ 0&\ frac {A\ left (A+B\ right) -2 {B} ^\ right) -2 {B} ^ {2}}} {A+B}\ end {array}\ right] =\ left [\ begin {array} {0& 0&\ 0&\ frac {A\ left & 0& 0\\ 0& 0& 0\\ 0& 0&\ 0&\ frac {3\ mathit {KG}} {K+\ frac {G} {3}}}\ end {array}\ right]\end{split}\]

3.3. Case of the projection at the top of the cone

We show that we trivially have \(T\mathrm{=}0\).