Chapter 8, Small-deformation theory Video Solutions, A Course on Plasticity Theory

Problem 1

Using the convected-coordinate formalism we have $\nabla \mathbf{u}=\mathbf{u}_{, i} \otimes \mathbf{e}^i$ in which $\mathbf{u} \in T_{\mathcal{K}_i}$ in accordance with (8.3). We may thus write $\mathbf{u}=u^j \mathbf{e}_j^*$. Then, $D \mathbf{u}=\mathbf{1}^t \mathbf{u}_{, i} \otimes \mathbf{e}^i=\left(\mathbf{1}^t \mathbf{u}\right)_{, i} \otimes \mathbf{e}^i$, where $1^t \mathbf{u}=u^j \mathbf{e}_j$. Show that the components $u_{\mid j}^i$, where $(\cdot)_{\mid i}$ is the covariant derivative with respect to the referential connection $\bar{\Gamma}_{i j}^k$, are invariant under a change of frame. Derive the representation (4.12) for the strain, in which
$$
E_{i j}=\epsilon_{i j}+\frac{1}{2} u_{\mid i}^k u_{k \mid j}, \quad \text { with } \quad \epsilon_{i j}=\frac{1}{2}\left(u_{i \mid j}+u_{j \mid i}\right)
$$
where $u_i=e_{i j} u^j$ in which $e_{i j}=\mathbf{e}_i \cdot \mathbf{e}_j$ is the referential metric. Show that $\bar{\Gamma}_{i j}^{* k}=\bar{\Gamma}_{i j}^k$.

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Problem 2

Use (5.2) and (5.3) with (1.20) to obtain
$$
\mathbf{E}=\mathbf{G}^t \hat{\mathbf{E}} \mathbf{G}+\hat{\mathbf{P}}
$$
where $\hat{\mathbf{E}}$ is the elastic strain, given by (5.46), and $\hat{\mathbf{P}}$, the plastic strain (not to be confused with the Piola stress), is defined by
$$
\hat{\mathbf{P}}=\frac{1}{2}\left(\mathbf{G}^t \mathbf{G}-\mathbf{I}\right)
$$
Show that $\hat{\mathbf{E}}=\hat{E}_{i j} \mathbf{m}^i \otimes \mathbf{m}^j$ and $\hat{\mathbf{P}}=\hat{P}_{i j} \mathbf{e}^i \otimes \mathbf{e}^j$, where $\hat{E}_{i j}=\frac{1}{2}\left(g_{i j}-m_{i j}\right)$ and $\hat{P}_{i j}=\frac{1}{2}\left(m_{i j}-e_{i j}\right)$, and thus establish the relation $E_{i j}=\hat{E}_{i j}+\hat{P}_{i j}$, valid for covariant components only.

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Problem 3

Use (8.13) and $(8.14)_2$ together with the polar factorization $\mathbf{F}=\mathbf{R U}$, in which $\mathbf{R}$ is a (two-point) rotation tensor and $\mathbf{U}$ is the symmetric, positive definite right-stretch tensor, to show that
$$
\mathbf{U}=\mathbf{I}+\varepsilon+\mathbf{O}\left(\varepsilon^2\right), \quad \mathbf{U}^{-1}=\mathbf{I}-\boldsymbol{\epsilon}+\mathbf{O}\left(\varepsilon^2\right) \quad \text { and } \quad \mathbf{R}=\mathbf{1}\left\{\mathbf{I}+\omega+\mathbf{O}\left(\varepsilon^2\right)\right\}
$$
where $\varepsilon=|D \mathbf{u}|,\left|\mathbf{O}\left(\varepsilon^2\right)\right|=O\left(\varepsilon^2\right)$ and
$$
\omega=\frac{1}{2}\left\{D \mathbf{u}-(D \mathbf{u})^t\right\}
$$

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00:52

Problem 4

Use (8.13) and (8.14) with $\mathfrak{F}_F^2=\operatorname{det}\left(\mathbf{F}^t \mathbf{F}\right)$ to obtain
$$
\mathfrak{f}_F^2=\operatorname{det}(\mathbf{A}-\lambda \mathbf{I})_{\mid \lambda=-1}, \quad \text { where } \quad \mathbf{A}=2 \varepsilon+(D \mathbf{u})^t D \mathbf{u} .
$$
Use the cubic characteristic equation for the eigenvalues of A to show that
$$
\mathfrak{f}_F=1+\operatorname{tr} \varepsilon+O\left(\varepsilon^2\right)
$$
and hence that $\mathfrak{f}_F \simeq 1$ if $\varepsilon \ll 1$.

Victor Salazar

Numerade Educator

01:01

Problem 5

Show that
$$
\mathrm{F}^*-1 \simeq 1\left\{(\text { tre }) \mathbf{I}-(D \mathbf{u})^t\right\}
$$
if $\varepsilon \ll 1$, where $\mathbf{F}^*$, given by (1.36), is the cofactor of the deformation gradient. Thus, $\mathrm{F}^* \simeq 1$.

Raj Bala

Numerade Educator

Problem 6

Use (5.3) to conclude, again if $\varepsilon \ll 1$, that $\mathbf{H} \simeq 1 \mathrm{~K}$. Note that this approximation preserves the transformation formulas (6.88) for changes of frame.

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Problem 7

Prove this claim.

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03:42

Problem 8

A fixed frame of reference used to describe the motion of a body may be regarded as vector space spanned, at a particular point, by $\left\{\mathbf{e}_i^*\right\}$ in which the $\mathbf{e}_i^*$ are independent of time. This implies that 1 is also independent of time. Use this with $\mathbf{v}=\dot{\chi}$ to show that $1^t \dot{\mathbf{v}}=\ddot{u}^i \mathbf{e}_i$. In effect, we then regard the vector space $T_{\mathcal{K}_t}$ as being fixed while $k_t$ itself evolves in the associated point space.

Urvashi Arora

Numerade Educator

04:08

Problem 9

It follows easily, from (1.22), (1.32), (1.38), and (1.40), that $\mathcal{F}_F$ div $\mathbf{T}=$ $\operatorname{Div}\left(\mathbf{T F}^*\right)$. Use the divergence and localization theorems with the Piola-Nanson formula (1.35), or direct calculation, to establish the identity
$$
\mathfrak{f}_F \operatorname{div} \mathbf{w}=\operatorname{Div}\left\{\left(\mathbf{F}^*\right)^t \mathbf{w}\right\},
$$
where $\mathrm{w}$ is a spatial vector field. Show, under our assumptions, that
$$
\operatorname{div} \mathbf{T} \simeq 1 \operatorname{Div}\left(1^t \mathrm{~T} 1\right) \quad \text { and } \quad \operatorname{div} \mathbf{\mathrm { w }} \simeq \operatorname{Div}\left(1^t \mathbf{w}\right) .
$$

Jacob Fry

Numerade Educator

Problem 10

With reference to Problem 6.3 and the Appendix to Section 6.1 , show that, in the case of isotropy,
$$
\mathcal{L}[\hat{\mathbf{S}}]=\frac{1}{9 \mathcal{K}}(t r \hat{\mathbf{S}}) \mathbf{I}+\frac{1}{2 \mu} D e v \hat{\mathbf{S}}
$$
where $\mu$ and $\kappa\left(=\lambda+\frac{2}{3} \mu\right)$ are the (positive) shear and bulk moduli, respectively.

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Problem 11

Use (6.9) and (6.11) to show, granted our assumptions, that $\hat{\mathbf{S}} \simeq \mathbf{S}$.

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01:03

Problem 12

Use convected coordinates, as in Problem 8.1, to show that $(D \mathbf{u})^{\prime}=\dot{u}_{\mid j}^i \mathbf{e}_i \otimes \mathbf{e}^j$.

Khoobchandra Agrawal

Numerade Educator