Problem 1: Part 6: Continuum elastic-plastic tangent modulus
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The continuum elastic-plastic tangent modulus is defined by the
following relation
![{\displaystyle {\dot {\boldsymbol {\sigma }}}={\boldsymbol {\mathsf {C}}}^{\text{ep}}:{\dot {\boldsymbol {\varepsilon }}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7500b579d219b530018f909aebb20e7363103efc)
Derive an expression for the elastic plastic tangent modulus using the results you have
derived in the previous parts.
The stress rate is given by
![{\displaystyle {\dot {\boldsymbol {\sigma }}}={\boldsymbol {\mathsf {C}}}:\left({\dot {\boldsymbol {\varepsilon }}}-{\dot {\gamma }}f_{\boldsymbol {\sigma }}\right)~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38074c9ad7338a7ce7628333d403e99decdb6b15)
From the previous part
![{\displaystyle {\dot {\gamma }}={\cfrac {f_{\boldsymbol {\sigma }}:{\boldsymbol {\mathsf {C}}}:{\dot {\boldsymbol {\varepsilon }}}}{f_{\boldsymbol {\sigma }}:{\boldsymbol {\mathsf {C}}}:f_{\boldsymbol {\sigma }}-{\sqrt {\cfrac {2}{3}}}~f_{\alpha }~{\cfrac {{\boldsymbol {\varepsilon }}^{p}:f_{\boldsymbol {\sigma }}}{\lVert {\boldsymbol {\varepsilon }}^{p}\rVert _{}}}-{\cfrac {\chi }{\rho ~C_{p}}}~f_{T}~{\boldsymbol {\sigma }}:f_{\boldsymbol {\sigma }}}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad49282469f934e9a55dc619a14f646b374b7042)
Plug in expression for stress rate to get
![{\displaystyle {\begin{aligned}{\dot {\boldsymbol {\sigma }}}&={\boldsymbol {\mathsf {C}}}:\left({\dot {\boldsymbol {\varepsilon }}}-{\cfrac {f_{\boldsymbol {\sigma }}:{\boldsymbol {\mathsf {C}}}:{\dot {\boldsymbol {\varepsilon }}}}{f_{\boldsymbol {\sigma }}:{\boldsymbol {\mathsf {C}}}:f_{\boldsymbol {\sigma }}-{\sqrt {\cfrac {2}{3}}}~f_{\alpha }~{\cfrac {{\boldsymbol {\varepsilon }}^{p}:f_{\boldsymbol {\sigma }}}{\lVert {\boldsymbol {\varepsilon }}^{p}\rVert _{}}}-{\cfrac {\chi }{\rho ~C_{p}}}~f_{T}~{\boldsymbol {\sigma }}:f_{\boldsymbol {\sigma }}}}~f_{\boldsymbol {\sigma }}\right)\\&={\boldsymbol {\mathsf {C}}}:{\dot {\boldsymbol {\varepsilon }}}-{\cfrac {f_{\boldsymbol {\sigma }}:{\boldsymbol {\mathsf {C}}}:{\dot {\boldsymbol {\varepsilon }}}}{f_{\boldsymbol {\sigma }}:{\boldsymbol {\mathsf {C}}}:f_{\boldsymbol {\sigma }}-{\sqrt {\cfrac {2}{3}}}~f_{\alpha }~{\cfrac {{\boldsymbol {\varepsilon }}^{p}:f_{\boldsymbol {\sigma }}}{\lVert {\boldsymbol {\varepsilon }}^{p}\rVert _{}}}-{\cfrac {\chi }{\rho ~C_{p}}}~f_{T}~{\boldsymbol {\sigma }}:f_{\boldsymbol {\sigma }}}}~{\boldsymbol {\mathsf {C}}}:f_{\boldsymbol {\sigma }}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e5be777f981cfc28cf4c03d0e09348b8b6af6ec)
We have to express the above in the form
![{\displaystyle {\dot {\boldsymbol {\sigma }}}={\boldsymbol {\mathsf {C}}}^{\text{ep}}:{\dot {\boldsymbol {\varepsilon }}}\qquad \implies \qquad {\dot {\sigma }}_{ij}=C_{ijkl}^{\text{ep}}~{\dot {\varepsilon }}_{kl}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/654775d7711880dd929953c33cb507bd9a55e719)
Since the denominator is a scalar, we don't have to worry about it
for this calculation. In that case we can write
![{\displaystyle {\dot {\boldsymbol {\sigma }}}={\boldsymbol {\mathsf {C}}}:{\dot {\boldsymbol {\varepsilon }}}-\left({\cfrac {f_{\boldsymbol {\sigma }}:{\boldsymbol {\mathsf {C}}}:{\dot {\boldsymbol {\varepsilon }}}}{\text{denom.}}}\right){\boldsymbol {\mathsf {C}}}:f_{\boldsymbol {\sigma }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7851deddb3012d996637b7e48819925afcb3d647)
In index notation, we can write
![{\displaystyle {\dot {\sigma }}_{ij}=C_{ijkl}~{\dot {\varepsilon }}_{kl}-{\cfrac {f_{pq}^{\boldsymbol {\sigma }}~C_{pqrs}~{\dot {\varepsilon }}_{rs}}{\text{denom.}}}~C_{ijkl}~f_{kl}^{\boldsymbol {\sigma }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7bae1443ce9064ec26378a2d20a741815680848f)
Let us manipulate the numerator of the second term above so that we get what we need. Thus
![{\displaystyle {\begin{aligned}f_{pq}^{\boldsymbol {\sigma }}~C_{pqrs}~{\dot {\varepsilon }}_{rs}~C_{ijkl}~f_{kl}^{\boldsymbol {\sigma }}&=\left(C_{pqrs}~f_{pq}^{\boldsymbol {\sigma }}\right)~\left(C_{ijkl}~f_{kl}^{\boldsymbol {\sigma }}\right)~{\dot {\varepsilon }}_{rs}\\&=\left(C_{rspq}~f_{pq}^{\boldsymbol {\sigma }}\right)~\left(C_{ijkl}~f_{kl}^{\boldsymbol {\sigma }}\right)~{\dot {\varepsilon }}_{rs}\qquad {\text{Major symmetry of}}~{\boldsymbol {\mathsf {C}}}\implies C_{pqrs}=C_{rspq}\\&=\left(C_{ijkl}~f_{kl}^{\boldsymbol {\sigma }}\right)~\left(C_{rspq}~f_{pq}^{\boldsymbol {\sigma }}\right)~~{\dot {\varepsilon }}_{rs}\\&\equiv A_{ij}~B_{rs}~{\dot {\varepsilon }}_{rs}\equiv M_{ijrs}~{\dot {\varepsilon }}_{rs}\\&={\boldsymbol {\mathsf {M}}}:{\dot {\boldsymbol {\varepsilon }}}~.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f40134109cece51e74230d9f3694c991aac0ea83)
In the above
![{\displaystyle M_{ijrs}=A_{ij}~B_{rs}\qquad \implies \qquad {\boldsymbol {\mathsf {M}}}={\boldsymbol {A}}\otimes {\boldsymbol {B}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f2f57a5d2d13163aa6c6dceb2525689aa94ed9f)
and
![{\displaystyle {\begin{aligned}A_{ij}&=C_{ijkl}~f_{kl}^{\boldsymbol {\sigma }}\qquad \implies \qquad {\boldsymbol {A}}={\boldsymbol {\mathsf {C}}}:f_{\boldsymbol {\sigma }}\\B_{rs}&=C_{rspq}~f_{pq}^{\boldsymbol {\sigma }}\qquad \implies \qquad {\boldsymbol {B}}={\boldsymbol {\mathsf {C}}}:f_{\boldsymbol {\sigma }}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/481a3ccb1b5d5af86bf815f7f0707a146fe03dbf)
Therefore,
![{\displaystyle {\boldsymbol {\mathsf {M}}}=({\boldsymbol {\mathsf {C}}}:f_{\boldsymbol {\sigma }})\otimes ({\boldsymbol {\mathsf {C}}}:f_{{\boldsymbol {\sigma }})}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9338acc008c85233b64c62b8bc5e2a1fdc29bf0)
This gives us
![{\displaystyle {\dot {\boldsymbol {\sigma }}}={\boldsymbol {\mathsf {C}}}:{\dot {\boldsymbol {\varepsilon }}}-\left({\cfrac {({\boldsymbol {\mathsf {C}}}:f_{\boldsymbol {\sigma }})\otimes ({\boldsymbol {\mathsf {C}}}:f_{{\boldsymbol {\sigma }})}}{\text{denom.}}}\right):{\dot {\boldsymbol {\varepsilon }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bae90412ea704fa23a763c740576f5ffb6894b6)
or,
![{\displaystyle {\dot {\boldsymbol {\sigma }}}=\left[{\boldsymbol {\mathsf {C}}}-\left({\cfrac {({\boldsymbol {\mathsf {C}}}:f_{\boldsymbol {\sigma }})\otimes ({\boldsymbol {\mathsf {C}}}:f_{{\boldsymbol {\sigma }})}}{\text{denom.}}}\right)\right]:{\dot {\boldsymbol {\varepsilon }}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec2131b199a233ce2772a8433fb640e838d7c556)
Hence
![{\displaystyle {\boldsymbol {\mathsf {C}}}^{\text{ep}}={\boldsymbol {\mathsf {C}}}-\left({\cfrac {({\boldsymbol {\mathsf {C}}}:f_{\boldsymbol {\sigma }})\otimes ({\boldsymbol {\mathsf {C}}}:f_{{\boldsymbol {\sigma }})}}{\text{denom.}}}\right)~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/448b9dd3ff148c45f71719afd718a90d19993646)
The continuum elastic-plastic tangent modulus is therefore
![{\displaystyle {{\boldsymbol {\mathsf {C}}}^{\text{ep}}={\boldsymbol {\mathsf {C}}}-\left({\cfrac {({\boldsymbol {\mathsf {C}}}:f_{\boldsymbol {\sigma }})\otimes ({\boldsymbol {\mathsf {C}}}:f_{{\boldsymbol {\sigma }})}}{f_{\boldsymbol {\sigma }}:{\boldsymbol {\mathsf {C}}}:f_{\boldsymbol {\sigma }}-{\sqrt {\cfrac {2}{3}}}~f_{\alpha }~{\cfrac {{\boldsymbol {\varepsilon }}^{p}:f_{\boldsymbol {\sigma }}}{\lVert {\boldsymbol {\varepsilon }}^{p}\rVert _{}}}-{\cfrac {\chi }{\rho ~C_{p}}}~f_{T}~{\boldsymbol {\sigma }}:f_{\boldsymbol {\sigma }}}}\right)~.}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64f1eee66a11119ee6ce20609f245efa8b7c694d)