Use (7.52) to write the stress in the polar form 𝐓=Tr r 𝐞r ⊗𝐞r+Tφφ 𝐞φ ⊗𝐞φ+Tr φ(𝐞r ⊗𝐞φ+𝐞φ ⊗𝐞r)+Tz

Name: Problem 13, Chapter 6: A Course on Plasticity Theory
Uploaded: 2021-02-23T14:47:53-08:00
Duration: 3 min 20 s
Channel: Himanshu Kushwaha
Description: Use (7.52) to write the stress in the polar form 𝐓=Tr r 𝐞r ⊗𝐞r+Tφφ 𝐞φ ⊗𝐞φ+Tr φ(𝐞r ⊗𝐞φ+𝐞φ ⊗𝐞r)+Tz z 𝐤 ⊗𝐤. Show that Tπr=Tφφ=Tz z=-p and Tr φ=k in region B.

step 1 In this problem of polar coordinates, we have to convert the polar equation into the rectangular

Question

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 $T_{\pi r}=T_{\varphi \varphi}=T_{z z}=-p$ and $T_r \varphi=k$ in region $B$.

   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 $T_{\pi r}=T_{\varphi \varphi}=T_{z z}=-p$ and $T_r \varphi=k$ in region $B$.

A Course on Plasticity Theory

David J. Steigmann 1st Edition

Chapter 6, Problem 13

Instant Answer

Solved by Expert Himanshu Kushwaha

02/23/2021

Step 1

52): $\mathbf{T}=T_{rr}\mathbf{e}_r\otimes\mathbf{e}_r+T_{\varphi\varphi}\mathbf{e}_{\varphi}\otimes\mathbf{e}_{\varphi}+T_{r\varphi}(\mathbf{e}_r\otimes\mathbf{e}_{\varphi}+\mathbf{e}_{\varphi}\otimes\mathbf{e}_r)+T_{zz}\mathbf{k}\otimes\mathbf{k}$ Now, we need Show more…

Show all steps

Thanks for your feedback!

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 $T_{\pi r}=T_{\varphi \varphi}=T_{z z}=-p$ and $T_r \varphi=k$ in region $B$.