Chapter 9, Strain hardening, rate sensitivity, and gradient plasticity Video Solutions, A Course on Plasticity Theory

Problem 1

Verify this claim for $(9.2)_2$. Hint: Use (6.74) and (6.76) together with (6.68) to confirm that $\hat{\mathbf{S}} \cdot \mathbf{G G}^{-1}>0$, to leading order in the small elastic strain, whenever $\dot{\mathbf{G}} \neq 0$.

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01:10

Problem 2

Show that both models are invariant under $\mathbf{G} \rightarrow \mathbf{R}^{\prime} \mathbf{G}$ and $\hat{\mathbf{S}} \rightarrow \mathbf{R}^{\prime} \hat{\mathbf{S}} \mathbf{R}$ for any fixed $\mathbf{R} \in g_{\kappa_i(p)} \subset \mathrm{Orth}^{+}$. Thus, they are meaningful for any type of material symmetry.

Dominador Tan

Numerade Educator

02:35

Problem 3

Prove this.

Manik Pulyani

Numerade Educator

01:03

Problem 4

If $\mathbf{A}$ is uniform and invertible, show that $\operatorname{Curl}\left(\mathbf{A}^{-1} \mathbf{K}^{-1}\right)=$ $\left(\operatorname{Cur} / \mathrm{K}^{-1}\right) \mathbf{A}^{-t}$

Carson Merrill

Numerade Educator

Problem 5

Use (9.34) to determine $\dot{\mathbf{M}}$ in terms of $\dot{\mathbf{G}}$. We have shown that $\mathbf{G}$ and $\mathbf{M}$ are mechanically equivalent in isotropic materials. It should therefore be possible to determine $\dot{\mathbf{G}}$ in terms of $\dot{\mathbf{M}}$. Show that this is indeed the case if we take $\mathbf{G G}^{-1}$ to be symmetric, as stipulated, for isotropic materials, in Chapter 7 . Write the flow rule (7.8) in the form $\dot{\mathbf{M}}=\ldots$.

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Problem 6

The connection $\left\{\begin{array}{l}k \\ i\end{array}\right\}$ also incorporates length-scale effects because it involves the metric and its first-order coordinate derivatives. Would it be reasonable to propose a constitutive function having $\left\{{ }_{i j}^k\right\} \mathbf{m}_k \otimes \mathbf{m}^i \otimes \mathbf{m}^j$ as its only argument?

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01:50

Problem 7

What is the form of the function $\mathcal{F}(\mathbf{M}, \nabla \mathbf{M}, \nabla \nabla \mathbf{M})$ which is such as remain invariant under an arbitrary thrice-differentiable map $\overline{\mathbf{x}}=\lambda(\mathrm{x})$ ?

Anand Jangid

Numerade Educator

04:36

Problem 8

Prove these assertions.

Ameer Said

Numerade Educator

Problem 9

Show that $\mathbb{E}^{\prime} \rightarrow \mathbf{R}^t \mathbb{E}^{\prime} \mathbf{R}$ under material symmetry transformations.

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Problem 10

Show that
$$
\boldsymbol{\mu}^{i j} \cdot \mathbf{r}_{i j j}=\left(\boldsymbol{\mu}^{i j} \cdot \mathbf{m}_i\right)_{j j}-\boldsymbol{\mu}_{j j}^{i j} \cdot \mathbf{r}_i, \quad \text { where } \boldsymbol{\mu}_{l j}^{i j}=\boldsymbol{\mu}_{j j}^{i j}+\boldsymbol{\mu}^{i k} \bar{\Gamma}_{k j}^j+\boldsymbol{\mu}^{k j} \bar{\Gamma}_{k j}^i \cdot
$$

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01:17

Problem 11

Show that $\mathbb{Q} \rightarrow \mathbf{R}^t \mathbb{Q} \mathbf{R}$ under symmetry transformations.

Bryan Lynn

Numerade Educator

Problem 12

Derive (9.87). Hint: Write $\tilde{\mathbf{X}}$ in the form $\xi^j \otimes \mathbf{e}_j$ and show that $\tilde{\mathbf{X}} \cdot \mathbf{R}^t \nabla \dot{\mathbf{G}}=$ $\xi^j \otimes \mathbf{e}_j \cdot \mathbf{R}^i \dot{\mathbf{G}}_{, i} \otimes \mathbf{e}^i=\boldsymbol{\xi}^i \cdot \mathbf{R}^i \dot{\mathbf{G}}_{, i}=\mathbf{R} \xi^i \cdot \dot{\mathbf{G}}_{, i}=\mathbf{R} \xi^j \otimes \mathbf{e}_j \cdot \dot{\mathbf{G}}_{j i} \otimes \mathbf{e}^i=\mathbf{R} \dot{\mathbf{X}} \cdot \nabla \dot{\mathbf{G}}$.

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01:21

Problem 13

Show that $\mathbf{N} \rightarrow \mathbf{N R}$ under symmetry transformations and hence that $\mathbf{N}$. Curl $\dot{\mathbf{G}}$ is invariant. Show that $\eta^{i j} \rightarrow \mathbf{R}^t \eta^{i j}$ and hence that $\mathbf{M} \rightarrow \mathbf{R}^l \mathbf{M}$.

Dominador Tan

Numerade Educator