GENG2004 - Lecture 4

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GENG2004 - Lecture 3
Stress in 3D - Slides cont.

Einstein Notation

• Einstein notation is shorthand that simplifies expressions by removing summation $\sum$ symbol when an index is repeated within a single term
  • Used very widely, but here we will use it in the context of stress

For the traction vector $T_i$ acting on surface with unit normal vector $n_j$:
$T_i={\sigma }_{ij}{n}_j$

• The index $j$ is repeated → implicitly summed over 1, 2, 3

For first component $i=1$, this equation expands to:
$T_1={\sigma }_{11}{n}_1+{\sigma }_{12}{n}_2+{\sigma }_{13}{n}_3$

• We can repeat this pattern for $T_2$ and $T_3$

As we can see, using: $T_i={\sigma }_{ij}{n}_j$ simplifies what would otherwise be three longer equations into one short equation → very compact notation

What this notation means verbally → take stress contributions from all directions $j$, project onto direction $i$

You can also think of the equation as matrix multiplication:
$T_i={\sigma }_{ij}{n}_j\to \mathbf{T}=\sigma \cdot \mathbf{n}$

• This is really helpful if you are given the stress tensor $(\sigma )$ and the normal vector of the surface $(\mathbf{n})$ to calculate traction vector on

Stress on Arbitrary Surfaces

Given a stress tensor and a surface, how to find stress acting upon it?

• If surface and stress aligned → no shear stress
  • Normal stress equals full stress, no calculations needed!
• If they aren’t aligned → use stress tensor to find traction vector

Full Method:

1. Identify stress tensor and surface orientation (normal vector)
2. To find traction vector → use Einstein Notation
3. Calculate normal stress:
   ${\sigma }_n=\mathbf{T}\cdot \mathbf{n}$
4. Calculate shear stress:
   $\tau =\sqrt{|\mathbf{T}{|}^2-{\sigma }_n^2}$

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Strain in 3D - Slides

Small Deformation Theory

• When you push/pull an object, every point in the object moves a bit
• Displacement gradient indicates how quickly the movement changes from one point to the next
  $\left|\frac{\partial u_i}{\partial x_j}\right|\ll 1$
• This equation means displacement gradient is much smaller than $1$
  • $u_i$ is displacement in direction $i$
  • $x_j$ is the coordinate

Small deformation theory assumes that any displacement of a materials particles are infinitesimally smaller than any relevant dimension of the body

• Its geometry and properties (such as density and stiffness) are unchanged by the deformation
• Also known as infinitesimal strain theory

This theory helps simplify equations:

• Can ignore complicated terms, such as squaring small numbers
  • We can use our linear equations, (e.g. Hooke’s law)
• Can treat shape of object as the same before and after loading
• Valid for most problems

If an object, such as a ruler, bends a lot, our simple formulas for stress would be wrong → theory doesn’t apply; cannot ignore deformation

• We can apply this theory when objects deform only a tiny fraction of its original size (e.g. tensile/compressive tests, slight beam bend, etc.)

Small Deformation (Strain) Tensor

• Strain is change in geometry per unit length → dimensionless

  • Like stress, we use a tensor to represent it in 3D
  • It comprises of elements ${ϵ}_{ij}$
    ${ϵ}_{ij}=\frac{1}{2}(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i})$
    
    Interpretation of small deformation:

• Diagonal terms: (${ϵ}_{11},{ϵ}_{22},{ϵ}_{33}$) $-$ normal strains

  

  • Represents change in length per unit original length

• Off-diagonal terms $-$ shear strains

  

  • Represents one-half the angle change between two line elements originally at right angles to one another

Volumetric Strain

• Volumetric strain is the sum of normal strains:
  ${ϵ}_v={ϵ}_{11}+{ϵ}_{22}+{ϵ}_{33}$
  ${ϵ}_v$ is also known as dilation
• Represents change in volume of a small element
• Higher order terms are ignored as strains are small ($ϵ\ll 1$)
  • i.e. product of strains → $\cancel{{ϵ}_{11}{ϵ}_{22}}$

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GENG2004 - Lecture 5