Chapter 5, Elastic and plastic deformations Video Solutions, A Course on Plasticity Theory

Problem 1

Show that if the tractions acting on the disjoint sub-bodies $\pi_t^{(n)}$ vanish, then they become stress free in the limit $d\left(\pi_t^{(n)}\right) \rightarrow 0$ whether or not they are in equilibrium, provided that the norm of $\rho(\mathbf{b}-\dot{\mathbf{v}})$, where $\mathbf{b}$ is the body force, is bounded.

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Problem 2

Show that the Burgers vectors $\mathbf{b}_\gamma$ and $\mathbf{B}_{\Gamma}$ are one and the same.

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03:22

Problem 3

Use the relevant version of (3.124) to invert (5.45), and thus show that
$$
\hat{T}_{\cdot n m}^l=\frac{1}{2} \hat{\epsilon}_{m n k} \alpha^{k l}
$$

George Dekermenjian

Numerade Educator

05:52

Problem 4

Prove the identity $\operatorname{Div}\left[(\text { CurlA })^t\right]=0$, and establish the equivalence of the following statements:
(a) $\operatorname{Div}\left[\left(\operatorname{Curl} \mathbf{G}^t\right]=\mathbf{0}\right.$, where $\mathbf{G}$ is the plastic part of the deformation gradient.
(b) $\hat{\epsilon}^{i j k}\left(\hat{T}_{. j i k}^n+\hat{T}_{. j i}^l \hat{\Gamma}_{l k}^n\right)=0$, where $\hat{\epsilon}, \hat{T}$, and $\hat{\Gamma}$, respectively, are the permutation tensor, the torsion tensor, and the connection in the intermediate state.
(c) $\hat{\epsilon}^{m l j} \hat{R}_{k m l j}=0$, where $\hat{R}$ is the curvature tensor based on $\hat{\Gamma}$.
(d) $\hat{\Pi}^{[i j]}=0$.

Sriram Soundarrajan

Numerade Educator

Problem 5

Establish the formula
$$
\hat{S}_{-i j}^k=m^{k m}\left(\hat{E}_{m i j j}+\hat{E}_{m j ; i}-\hat{E}_{i j ; m}\right)
$$

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