GENG2004 - Lecture 5

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GENG2004 - Lecture 4
Generalised Hooke’s Law - Slides

Generalised Hooke’s Law

• Normal stress causes shear strain on certain planes:
  • There must be a relationship between Young’s modulus $(E)$, Poisson’s ratio $(\text{upnu})$ and shear modulus $(G)$

• This is established through generalised Hooke’s law $-$ relationship between stress and strain in 3D (multiaxial loading)
  • Multiaxial loading $-$ application of stress in multiple directions

Assumptions:

• Linear elasticity (no permanent deformation)
• Small strain → see Small Deformation Theory
• Isotropic material $-$ same properties in all directions

Given an object undergoing multiaxial loading:

• Stress in $x$ direction → direct strain
  ${ϵ}_x^{\prime }=\frac{{\sigma }_x}{E}$
• Stress in $y$ and $z$ direction → still creates strain in $x$ direction
  • Think Poisson’s Ratio → lateral contraction when stretched
    ${ϵ}_x^{\prime \prime }=-\text{upnu}\frac{{\sigma }_y}{E}{ϵ}_x^{\prime \prime \prime }=-\text{upnu}\frac{{\sigma }_z}{E}$
• Total normal strain in $x$ direction:
  ${ϵ}_x={ϵ}_{11}={ϵ}_x^{\prime }+{ϵ}_x^{\prime \prime }+{ϵ}_x^{\prime \prime \prime }$
  ${ϵ}_{11}=\frac{1}{E}[{\sigma }_x-\text{upnu}({\sigma }_y+{\sigma }_z)]$
• This can be repeated for normal strain in $y$ and $z$ direction

Now for shear strain:
${ϵ}_{12}=\frac{1+\text{upnu}}{E}{\sigma }_{12}=\frac{1}{2G}{\tau }_{12}$

• @ See: Elastic Properties of Materials

  

It is important to note:
> Normal stresses → normal strains and lateral strains (Poisson)
> Shear stresses → shear strains only

Additionally:
> Volume change (bulk behaviour)
> Shape change (shear behaviour)

All of these equations can be expressed using Einstein Notation as:
${ϵ}_{ij}=\frac{1+\text{upnu}}{E}{\sigma }_{ij}-\frac{\text{upnu}}{E}{\delta }_{ij}{\sigma }_{kk}$

Kronecker Delta Property

The Kronecker delta (${\delta }_{ij}$) is a tool used to compare two variables. It acts as a ‘switch’ in an equation.

In the case above, $i=j$ for normal components and $i\ne j$ for shear components. When $i\ne j$, delta becomes 0 (${\delta }_{ij}=0$), removing the term.

Lamé Constants

Lamé constants are another way to describe the elastic behaviour of an isotropic, linear elastic material (just like our conditions above)

• $\lambda$ $-$ describes how normal strains in one direction influence stresses in other directions
  $\lambda =\frac{E\text{upnu}}{(1+\text{upnu})(1-2\text{upnu})}$
• $\mu =G$ (shear modulus)
  $\mu =G=\frac{E}{2(1+\text{upnu})}$

These constants are useful because they make generalised Hooke’s Law more compact as seen below:

Use in Hooke’s Law:
${\sigma }_{ij}=\lambda {\delta }_{ij}{ϵ}_{kk}+2\mu {ϵ}_{ij}$
Also used for normal stress:
${\sigma }_{11}=(\lambda +2\mu ){ϵ}_{11}+\lambda {ϵ}_{22}+\lambda {ϵ}_{33}$

Principal Axes

• Principal directions are special directions where shear stress disappears
  • Stress becomes purely normal
• Principal axes theorem states that for any state of stress at a point there is a set of three perpendicular axes where all shear stresses vanish
  • Calculations become much easier when shear vanishes
• 3D questions are much harder and unlikely to be covered
  • Thus, we will focus on the method for 2D
  • This method can also be used to find maximum normal stress

Finding principal axes:

1. Write stress matrix
2. Use formula to solve for $\theta$
   $\tan (2\theta )=\frac{2{\tau }_{xy}}{{\sigma }_x-{\sigma }_y}$
3. Rotate axes (mentally)
4. New axes = principle axes; shear becomes zero

Principal stress formula:
${\sigma }_{1,2}=\frac{{\sigma }_x+{\sigma }_y}{2}\pm \sqrt{(\frac{{\sigma }_x-{\sigma }_y}{2}{)}^2+{\tau }_{xy}^2}$

• ? Use this method when question asks for:
  • Principal stress
  • Maximum normal stress
  • Orientation/angle where shear $=0$

Material Stiffness vs Strength

• Young’s modulus measures stiffness
  • Slope of elastic region on stress-strain curve
• Yield stress/UCS measures strength/failure
  • UCS → Unconfined Compressive Strength

These are different material properties

• One cannot be inferred from another
• Stiffness $\ne$ strength

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GENG2004 - Lecture 6