Chapter 4, Deformation and stress in convected coordinates Video Solutions, A Course on Plasticity Theory

02:17

Problem 1

The contravariant components of $\mathbf{C}$ are not equal to $g^{i j}$. What are they?

Meghan Mulcahy

Numerade Educator

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Problem 2

Show that $e^{i j}$ are the contravariant components of the left Cauchy-Green deformation tensor $\mathbf{B}=\mathbf{F F}^t$.

Victor Salazar

Numerade Educator

02:37

Problem 3

Show that $\mathbf{F}^{-t}=\mathbf{g}^k \otimes \mathbf{e}_k$ and hence that (4.21) is consistent with (1.36).

Erika Bustos

Numerade Educator

01:20

Problem 4

Show that (4.33) implies (4.34). Hint: Make repeated use of (3.124).

Manik Pulyani

Numerade Educator

01:22

Problem 5

Invert this relation to obtain $R^{i j}=\Pi^{i j}-\Pi^{i j}$, where $\Pi=\Pi_{\cdot n}^n$ and $\Pi_{\cdot j}^i=$ $g_{j k} \Pi^{i k}$.

Carson Merrill

Numerade Educator

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Show that the Einstein tensor is divergence-free, i.e., $\Pi_{; j}^{i j}=0$. This is a consequence of the Bianchi identities (see, for example, Section 86 of Lichnerowicz' book, cited in Chapter 3). Incidentally, this result plays a decisive role in Einstein's theory-hence the name.

Victor Salazar

Numerade Educator

Problem 7

Prove this.

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Problem 8

Demonstrate that the symmetry of the Ricci tensor follows directly from $(4.23)-(4.25)$.

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01:07

Problem 9

It is not immediately obvious that the right-hand side of (4.46) possesses the required symmetry with respect to interchange of $i$ and $j$. The symmetry of the second and fourth terms is obvious, while that of the third follows from $S_{\cdot m i}^k S_{\cdot k j}^m=$ $S_{\cdot k i}^m S_{\cdot m j}^k=S_{\cdot m j}^k S_{\cdot k i}^m$. As for the first term, establish that
$$
S_{\cdot j k}^k=\left(\mathscr{F}_F\right)_{, j} / \mathscr{J}_F
$$
and use this to show that $S_{-j k \mid i}^k=S_{-i k \mid j^*}^k$

Adriano Chikande

Numerade Educator

Problem 10

Obtain the strain compatibility conditions by substituting (4.47) into (4.46). Along the way you will encounter terms like $g_{\mid i}^{j m}$. Show that $g_{\mid i}^{j m}=$ $-g^{j k} g^{m l} g_{k|i|}=-2 g^{j k} g^{m l} E_{k l \mid i}$ and use this in $R_{i j}=0$ to derive the long-awaited strain compatibility equations. These indicate that the strain field cannot be arbitrary, but instead that it satisfies certain differential constraints.

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02:18

Problem 11

Prove that $S^{i j}=\mathfrak{J}_F T^{i j}$, as claimed.

Amany Waheeb

Numerade Educator

08:58

Problem 12

Derive the Doyle-Ericksen formula
$$
\mathfrak{f}_F T^{i j}=2 \frac{\partial W}{\partial g_{i j}}
$$
for hyperelastic materials, where $W$ is the strain-energy function.

Andrew Eddins

Emory University

00:28

Problem 13

Show that $\mathrm{P}^i=\mathrm{Pe}^i$ and hence that $\mathrm{P}^i$ is proportional to the traction exerted on a surface where $\xi^i$, with $i \in\{1,2,3\}$ fixed, is constant.