SaM - MEMORIZE (Stress and Strain • Strain and Stress)

List of things to memorize:

SaM - Stress and Strain • Strain and Stress

• SaM - Definition of Stress and Strain • Definition of Strain and Stress • Stress Vector • Strain Vector
• SaM - Young and Poisson Modulus
• SaM - Hooke’s Law • Collapsed Yieldness Tensor for Isotropic Materials
• SaM - Definition of Isotropic and Anisotropic Materials
• SaM - Collapsed Stiffness Tensor for Isotropic Materials

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SaM - Definition of Stress and Strain • Definition of Strain and Stress • Stress Vector • Strain Vector

• “Stress” ⇒ “Stress”
• “Strain” ⇒ “Deformazione”
• Stress:$\begin{array}{lc}\text{Monodimensional :}& \sigma =\frac{F}{A}\\ \text{In higher dimensions :}& T_{ij}={\lim }_{\Delta A(j)\to 0}\frac{F_i}{\Delta A(j)}\end{array}$
• Strain:$\begin{array}{lc}\text{Monodimensional :}& \epsilon =\frac{\Delta x}{x_o}\\ \text{In higher dimensions :}& {\epsilon }_{ij}=\frac{1}{2}\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)\end{array}$
• Terminology:
  • $\sigma$, ${\sigma }_{ij}$, $T_{ij}$ : stress.
  • $\epsilon$, ${\epsilon }_{ij}$, $s_{ij}$ : strain.
  • $F$ : force.
  • $A$ : area.
  • $\Delta A(j)\to 0$ : ==an infinitesimal small surface area of the solid, perpendicular to the “normal vector” ${\vec{n}}_j$==.
  • $\Delta x$ : displacement.
  • $\vec{u}$ : strain direction vector, so in which direction the strain is applied.
    While $x_k$ : $k$-th axis.

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SaM - Young and Poisson Modulus

• Young Modulus:$E=\frac{\frac{F}{A}}{\frac{\Delta l}{l}}=\frac{\sigma }{{\epsilon }_{\parallel }}$
• Poisson Modulus:$\nu =-\frac{\frac{\Delta r}{r}}{\frac{\Delta l}{l}}=\frac{{\epsilon }_{\perp }}{{\epsilon }_{\parallel }}$
• Terminology:
  • $F$ : force.
  • $A$ : area.
  • $r$ : radius.
  • $l$ : lenght.
  • $\sigma$ : stress
  • $\epsilon$ : strain.
    • ${\epsilon }_{\parallel }$ : parallel strain.
    • ${\epsilon }_{\perp }$ : perpendicular strain.

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

${\sigma }_{ij}={\sum }_{k=1}^3{\sum }_{l=1}^3{c}_{ijkl}\cdot {\epsilon }_{kl}$Or:${\epsilon }_{ij}={\sum }_{k=1}^3{\sum }_{l=1}^3{s}_{ijkl}\cdot {\sigma }_{kl}$

• Hooke’s Law for Isotropic Materials:${\sigma }_i=c_{ij}\cdot {\epsilon }_j$Or:${\epsilon }_i=s_{ij}\cdot {\sigma }_j$
• Terminlogy:
  • ${\sigma }_{ij}$ : stress.
  • ${\epsilon }_{ij}$ : strain.
  • $c_{ijkl}$ 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”)
  • $c_{ij}$ refers specifically for the collapsed stiffness tensor:
    If we expand the terms:
    
    • $c_{11}=c_{22}=c_{33}=\lambda +2\mu$
    • $c_{12}=c_{13}=c_{23}=\lambda$ (lame modulus)
    • $c_{44}=c_{55}=c_{66}=\mu$ (shear modulus)
  • While $s_{ij}$ to the collapsed yieldness tensor:
    

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SaM - Definition of Isotropic and Anisotropic Materials

• Isotropic materials exhibit the same mechanical properties in all directions.
• Anisotropic materials, on the other hand, have different mechanical properties in different directions.

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SaM - Collapsed Stiffness Tensor for Isotropic Materials

Collapsed stiffness tensor:

• $c_{11}=c_{12}+2c_{44}$
• Or we can write:
  • $c_{11}=\lambda +2\mu$
  • $c_{12}=\lambda$ (lame modulus)
  • $c_{44}=\mu$ (shear modulus)

While the collapsed yieldness tensor:

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