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

In Italian:IMPORTANTE

• “Stress” ⇒ “Stress”
• “Strain” ⇒ “Deformazione”

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Stress (In 1 Dimension)

While for stress ($\sigma$) using this formla:$\sigma =\frac{F}{A}$Where: - $F$ : force applied, unit of measaure $\left[N\right]$ (newton), $\left[N=\frac{\text{Kg}\cdot \text{m}}{{\text{s}}^2}\right]$. - $A$ : surface area over which the force ($F$) is applied, unit of measure $\left[{\text{m}}^2\right]$.

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Strain (In 1 Dimension)

• We can calculate the strain ($\epsilon$) as:$\epsilon =\frac{\Delta x}{x_o}$Where:
  • $x_0$ is the size of a body at rest, unit of measure $\left[\text{m}\right]$.
  • $\Delta x$ is the size of the deformation, unit of measure $\left[\text{m}\right]$.

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Stress (In Higher Dimensions)

We define the strain ($T_{ij}$ or ${\sigma }_{ij}$) as:$T_{ij}={\sigma }_{ij}\triangleq {\lim }_{\Delta A(j)\to 0}\frac{F_i}{\Delta A(j)}$Where:

• $F_i$ : $i$-th component of the force vector $\vec{F}$
• $\Delta A(j)\to 0$ : ==an infinitesimal small surface area of the solid, perpendicular to the “normal vector” ${\vec{n}}_j$==.

We can distinguish between two stresses:

• compressive or extensive preassure: ${\sigma }_{kk}$ (the force is perpendicular to the face)
• shear stress: ${\sigma }_{ij}:i\ne j$.
  Here’s an example:
  

Good explenation, especially for visualizing stress in higer dimensions, can be found here (Youtube) (Don’t listen, skip through the video)

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Strain (In Higher Dimensions)

==In higher dimensions both stress and strain are symmetrical tensors==.

The complete formula for strain ($s_{ij}$ or ${\epsilon }_{ij}$) is given by: $s_{ij}={\epsilon }_{ij}\triangleq \frac{1}{2}\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)$Where:

• $\vec{u}$ is the strain direction vector, so in which direction the strain is applied, and $u_k$ is one of the dimension of the strain direction vector.
• $x_k$ is the $k$-th axis.

We can distinguish between two stains:

• normal strain: ${\epsilon }_{kk}$ (change in lenght along the direction $k$).
• angular deformation: ${\epsilon }_{ij}:i\ne j$.

Simple way to understand:

• normal strain: ${\epsilon }_{kk}$ provoked by a compressive or extensive preassure: ${\sigma }_{kk}$:
  
• angular deformation: ${\epsilon }_{ij}:i\ne j$, provoked by a shear stress: ${\sigma }_{ij}$:
  
• Source: Youtube

Let’s make two examples, to better understand the strain formula:

• Source: Youtube
• Extensive or compressive strain:
  
• Shear strain:
  

Good explenation, especially for visualizing strain in higer dimensions, can be found here (Youtube) (Don’t listen, skip through the video)

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• From this definitions we have (specifically when talking about collapsed yieldness tensor for isotropic materials) seen also:
  • Defintion of the strain vector:IMPORTANTE $⟨{\epsilon }_{11},{\epsilon }_{22},{\epsilon }_{33},2\cdot {\epsilon }_{23},2\cdot {\epsilon }_{13},2\cdot {\epsilon }_{12}⟩$And the stress vector is defined:$⟨{\sigma }_{11},{\sigma }_{22},{\sigma }_{33},{\sigma }_{23},{\sigma }_{13},{\sigma }_{12}⟩$
  • Young and Poisson Modulus.
  • Hooke’s Law.

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Memory Card

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