Chapter 7, Isotropy Video Solutions, A Course on Plasticity Theory

Problem 1

For isotropic materials, show that the Cauchy stress $\mathbf{T}$ is invariant under the replacement $\mathbf{H} \rightarrow \overline{\mathbf{H}}=\mathbf{H B}$. Because $\mathbf{F}$ is invariant, this means that the Piola stress $\mathbf{P}$ is likewise invariant.

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01:59

Problem 2

Why not?

Khoobchandra Agrawal

Numerade Educator

01:19

Problem 3

With reference to Problem 6.9, derive an expression for the yield function $F(\hat{\mathbf{S}})$ in the case of cubic symmetry. Assume this to be a quadratic function of Dev $\hat{\mathbf{S}}$, and derive an expression for $F_{\mathrm{S}}$. State the form of the flow rule (6.105) in this case, together with appropriate restrictions on the plastic spin arising from considerations of material symmetry. Note that it is necessary to provide a constitutive prescription for the plastic spin. You need not do so here, but it would be worthwhile to research this issue independently. See, for example, the paper by Edmiston et al.

Dominador Tan

Numerade Educator

13:37

Problem 4

A popular yield function for isotropic soils is the Drucker-Prager function
$$
F(\hat{\mathbf{S}})=\frac{1}{2}|D e v \hat{\mathbf{S}}|^2-[\alpha(t r \hat{\mathbf{S}})+k]^2,
$$
where $\alpha$ and $k$ are material constants. What is the corresponding flow rule for $\mathbf{G G}^{-1}$ ? Reduce this to an equation for the straining tensor $\mathbf{D}$ in the case of rigidplastic response. Is the material compressible or incompressible? (For a thorough discussion of soil plasticity, consult the book by Yu.)

Mahnoor Amin

Numerade Educator

Problem 5

Position in a cylindrical polar-coordinate system $\{r, \varphi, z\}$ is given by
$$
\mathbf{y}=r \mathbf{e}_r(\varphi)+z \mathbf{k}
$$
where $r=\sqrt{x^2+y^2}, \tan \varphi=y / x, \mathbf{e}_r(\varphi)=\cos \varphi \mathbf{i}_1+\sin \varphi \mathbf{i}_2$, and $\mathbf{k}=\mathbf{i}_3$. Using the basis $\left\{\mathbf{e}_r, \mathbf{e} \varphi, \mathbf{k}\right\}$, where $\mathbf{e}_{\varphi}(\varphi)=\mathbf{k} \times \mathbf{e}_r$, we can write the Cauchy stress tensor in the form
$$
\begin{aligned}
\mathbf{T}= & T_{r r} \mathbf{e}_r \otimes \mathbf{e}_r+T \varphi \varphi \mathbf{e}_{\varphi} \otimes \mathbf{e}_{\varphi}+T_{z z} \mathbf{k} \otimes \mathbf{k}+T_{r \varphi}\left(\mathbf{e}_r \otimes \mathbf{e}_{\varphi}+\mathbf{e}_{\varphi} \otimes \mathbf{e}_r\right) \\
& +T_{r z}\left(\mathbf{e}_r \otimes \mathbf{k}+\mathbf{k} \otimes \mathbf{e}_r\right)+T_{\varphi z}\left(\mathbf{e}_{\varphi} \otimes \mathbf{k}+\mathbf{k} \otimes \mathbf{e}_{\varphi}\right)
\end{aligned}
$$
Assuming all components $T_{r r}, T_r \varphi$, etc., to depend only on $r$ and $\varphi$, use the methods of Chapter 3, or otherwise, to show that
$$
\begin{aligned}
\operatorname{div} \mathbf{T}= & {\left[\frac{\partial}{\partial r} T_{r r}+\frac{1}{r}\left(T_{r r}-T_{\varphi \varphi}+\frac{\partial}{\partial \varphi} T_{r \varphi}\right)\right] \mathbf{e}_r } \\
& +\left[\frac{\partial}{\partial r} T_r \varphi+\frac{1}{r}\left(2 T_{r \varphi}+\frac{\partial}{\partial \varphi} T_{\varphi \varphi}\right)\right] \mathbf{e}_{\varphi} \\
& +\left[\frac{\partial}{\partial} T_{r z}+\frac{1}{r}\left(T_{r z}+\frac{\partial}{\partial \varphi} T_{\varphi z}\right)\right] \mathbf{k} .
\end{aligned}
$$

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04:27

Problem 6

Use Bingham's model to solve the problem of anti-plane viscoplastic flow in a circular pipe aligned with the $z$-axis. Assume a velocity field of the form $\mathbf{v}=w(r) \mathbf{k}$, where $r=\sqrt{x^2+y^2}$ is the radius in a system of cylindrical polar coordinates. Suppose the material occupies the annular region $(0<) a \leq r \leq b$ and that $w(b)=0$ and $w(a)=W$. Assume that the axial pressure gradient vanishes and find the velocity field $w(r)$.

Chai Santi

Numerade Educator

03:16

Problem 7

Show that the principal stresses are
$$
T_1=-p+k, \quad T_2=-p-k \quad \text { and } \quad T_3=-p
$$
It follows that
$$
\mathbf{T}=-p \mathbf{i}+k\left(\mathbf{u}_1 \otimes \mathbf{u}_1-\mathbf{u}_2 \otimes \mathbf{u}_2\right)
$$
where $\left\{\mathbf{u}_i\right\}$, with $\mathbf{u}_3=\mathbf{i}_3$, are the orthonormal principal stress axes. Let $\mathbf{t}$ and $\mathbf{s}$ be plane orthonormal vector fields such that
$$
\mathbf{u}_1=\frac{\sqrt{2}}{2}(\mathbf{s}+\mathbf{t}), \quad \mathbf{u}_2=\frac{\sqrt{2}}{2}(\mathbf{s}-\mathbf{t})
$$
Then,
$$
\mathbf{T}=-p \mathbf{i}+\tau, \quad \text { where } \tau=k(\mathbf{t} \otimes \mathbf{s}+\mathbf{s} \otimes \mathbf{t}),
$$
and this implies that $k$ is the shear stress resolved on the $\mathbf{s}, \mathbf{t}$-axes.

Chai Santi

Numerade Educator

03:04

Problem 8

If $\mathbf{a}(\mathbf{y})$ and $\mathbf{b}(\mathrm{y})$ are vector fields, show that $\operatorname{div}(\mathbf{a} \otimes \mathbf{b})=(\operatorname{grada}) \mathbf{b}+$ (divb)a, where grad and div are the gradient and divergence operators based on position $\mathbf{y}$.

James Kiss

Numerade Educator

03:34

Problem 9

Show that Eqs. (7.64) are equivalent to the pair
$$
\mathbf{t} \cdot\left(\operatorname{grad} v_t-v_s \operatorname{grad} \theta\right)=0 \quad \text { and } \quad \mathbf{s} \cdot\left(\operatorname{gradv}_s+v_t \operatorname{grad} \theta\right)=0
$$

Monique Rousselle Maynard

Numerade Educator

02:47

Problem 10

Using (7.54), show that $\mathbf{T}=T_{\alpha \beta} \beta^{\mathbf{i} \alpha} \otimes \mathbf{i}_\beta-p \mathbf{i}_3 \otimes \mathbf{i}_3$, where
$$
T_{11}=-p-k \sin 2 \theta, \quad T_{22}=-p+k \sin 2 \theta, \quad \text { and } \quad T_{12}=T_{21}=k \cos 2 \theta .
$$

Subhadeepta Sahoo

Numerade Educator

03:23

Problem 11

Use (7.52) to verify that the zero-traction condition, $\mathbf{T i}_1=\mathbf{0}$, is satisfied.
To illustrate a simple state, consider the problem of a straight, semi-infinite tractionfree crack and the tentative slip-line field sketched in Figure 7.1. This is partitioned into the three regions labeled $A, B$, and $C$, in which $A$ and $C$ are constant states and $B$ is a simple state. In region $A, T_{22}$ and $T_{12}$ vanish, and, assuming $T_{11}$ to be positive, $T_{11}=2 \mathrm{k}$. We also have $\theta=3 \pi / 4$ and $p=-k$.

If we trace a $\beta$-line-a curve on which $\alpha$ is constant-from region $A$ into region $C$, where $\theta=\pi / 4$, and make use of $(7.89)_2$, we obtain $-p / 2 k+\pi / 4=k / 2 k+3 \pi / 4$. Thus, $p=-(1+\pi) k$ in region $C$.

Manish Jain

Numerade Educator

01:47

Problem 12

Use (7.52) to derive the Cartesian stress components $T_{11}=\pi k, T_{22}=$ $(\pi+2) k$, and $T_{12}=0$ in region $C$.

The state in region $B$ is similarly obtained by tracing a $\beta$-line from region $A$. Thus, $-p / 2 k+\varphi=k / 2 k+3 \pi / 4$, where $\varphi$ is the azimuthal angle in a system of plane polar

Aamir Mithaiwala

Numerade Educator

03:20

Problem 13

Use (7.52) to write the stress in the polar form $\mathbf{T}=T_{r r} \mathbf{e}_r \otimes \mathbf{e}_r+T \varphi \varphi \mathbf{e} \varphi \otimes$ $\mathbf{e}_{\varphi}+T_{r \varphi}\left(\mathbf{e}_r \otimes \mathbf{e}_{\varphi}+\mathbf{e}_{\varphi} \otimes \mathbf{e}_r\right)+T_{z z} \mathbf{k} \otimes \mathbf{k}$. Show that $T_{r r}=T_{\varphi} \varphi=T_{z z}=-p$ and $T_{r \varphi}=k$ in region $B$.

Himanshu Kushwaha

Numerade Educator

Problem 14

If we reflect this solution through the $y$-axis passing through the rightmost point of the constant state labelled $C$, we obtain a solution for a body containing two cracks lying on the $x$-axis with crack tips equidistant from the $y$-axis and with the slip-line fields meeting at the origin. Assuming the rigidly deforming parts of the body in the region of positive (resp., negative) $y$ to move vertically with velocity $V$ (resp., $-V$ ), use the Geiringer equations (7.90) to show that a tangential velocity discontinuity occurs on the curves separating the rigid and plastic portions of the body.

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00:48

Problem 15

Use the stress field to show that $\pm p / 2 k+\theta$ are constants on the appropriate logarithmic spirals that define the slip lines.

R M

Numerade Educator

03:29

Problem 16

Generalize this problem to accommodate a shear traction $\tau \mathbf{e} \varphi$ applied at the hole boundary $(r=a)$, with $|\tau|<k$. Assuming an axisymmetric stress field, with $T_{r r}, T_{\varphi}$, and $T_r \varphi$ depending only on $r$, obtain the equilibrium stress distribution assuming the material to be completely yielded. Can you find a velocity field that is consistent with this solution?

Chai Santi

Numerade Educator

04:00

Problem 17

Investigate the equilibrium state of stress under the assumptions that the stress $T_r \varphi$ depends only on $\varphi$ and the material is in a state of yield.

Chai Santi

Numerade Educator

01:06

Problem 18

Fill in the steps leading to (7.130).
This yields $p, \alpha=0$ and $p=P\left(y_3\right)$, with $P^{\prime}\left(y_3\right) / k=\operatorname{divt}$. Then, $P^{\prime \prime}\left(y_3\right)$ vanishes and $(7.130)$ reduces to
$$
C=k d i v \mathbf{t}
$$

Amy Jiang

Numerade Educator

03:29

Problem 19

Generalize this problem to accommodate a shear traction $\tau \mathrm{k}$ applied at the hole boundary $(r=a)$, with $|\tau|<k$. Find an axisymmetric equilibrium stress field in the region $r>a$ and show that it approaches the solution for the traction-free hole as $r / a \rightarrow \infty$. What restrictions does the velocity field $w$ satisfy?

Chai Santi

Numerade Educator