SaM - Hooke's Law • Collapsed Yieldness Tensor for Isotropic Materials

Simarly to the Young and Poisson Modulus in one dimension, we can define a relationship between stress and strain in higher dimensions:

So if we deform a material, an object, in the elastic regime (so linear regime), and for small deformation we can say:IMPORTANTE ${\sigma }_{ij}={\sum }_{k=1}^3{\sum }_{l=1}^3{c}_{ijkl}\cdot {\epsilon }_{kl}$Or:${\epsilon }_{ij}={\sum }_{k=1}^3{\sum }_{l=1}^3{s}_{ijkl}\cdot {\sigma }_{kl}$Where:

• $c_{ijkl}$ is the elastic constant (depends on the material), and comes from the Stifness Tensor (In Italian “Tensore di Rigidezza”)
• $s_{ijkl}$ is the yieldness constant (depends on the material), and comes from the Yieldness Tensor (In Italian “Tensore di Cedevolezza”)

For isotropic materials, we will see why, we can reduce the Hooke’s law to just:${\sigma }_i=c_{ij}\cdot {\epsilon }_j$Where: this particular vectors ($\vec{\sigma }$ and $\vec{\epsilon }$) have dimensions $\left[1\times 6\right]$ and the collapsed stifness matrix is reduced to just a $\left[6\times 6\right]$.
The vectors can be reduced from a $\left[3\times 3\right]$ matrix to a $\left[1\times 6\right]$ vectors without loss of information, because the strain and stress matrix are both symmetrical.

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==As it stands the Stifness tensor $\overline{\overline{C}}$ (which elements are $c_{ijkl}$) has $81$ constants, however for physcial reason only $36$ of them are independent. However for isotropic materials the indipendent constants are only $2$==. Here you can find the collapsed stiffness tensor for isotropic materials.

We can write the ==Stifness Tensor $(\overline{\overline{C}})$ for isotropic materials== is defined, in the collapsed Einstein notation, as:${\sigma }_i=c_{ij}\cdot {\epsilon }_j$If we expand the terms:

• Source: Internet
• $c_{11}=c_{22}=c_{33}=\lambda +2\mu$
• $c_{12}=c_{13}=c_{23}=\lambda$ (lame modulus)
• $c_{44}=c_{55}=c_{66}=\mu$ (shear modulus)

We have also seen other “moduluses” used to describe the stifness tensor: ==The lame modulus $(\lambda )$, the shear modulus $(\mu )$ and the bulk modulus $(K)$==, while their definition are NOT IMPORTANT, you need to know that they can be defined using the Young and Poisson modulus:$\begin{array}{l}\lambda =\frac{\nu E}{(1+\nu )(1-2\nu )}\\ \mu =\frac{E}{2(1+\nu )}\\ K=\frac{E}{3(1-2\nu )}\end{array}$

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While the ==Yieldness Tensor $(\overline{\overline{S}})$ for isotropic materials== is defined, in the collapsed Einstein notation, as:${\epsilon }_i=s_{ij}\cdot {\sigma }_j$If we expand the terms:

• Source: Internet
• We write the yieldness tensor, instead of the stifness one, since it is more simple.
• $E$ and $\nu$ are the Young Modulus and Poisson Modulus respectevely.
• In this course when talking about this matrix we will ignore the effect of temperature variation.
• NOTE: how the strain vector is defined: $⟨{\epsilon }_{11},{\epsilon }_{22},{\epsilon }_{33},2\cdot {\epsilon }_{23},2\cdot {\epsilon }_{13},2\cdot {\epsilon }_{12}⟩$And how the stress vector is defined:$⟨{\sigma }_{11},{\sigma }_{22},{\sigma }_{33},{\sigma }_{23},{\sigma }_{13},{\sigma }_{12}⟩$

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