ENSC1004 - Lecture 4

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ENSC1004 - Lecture 3
Elastic Moduli 2 - Slides

Complex Stress-Strain States

• In real life situations, there are many other stress-strain states
• They can all be described by the three basic states, including:
  • Biaxial tension or compression
  • Bending, which is a combination of tension and compression
  • Torsion, which is a shear
  • Any combinations of above

1. Biaxial stress
   • Occurs when two perpendicular normal stresses act on a material’s surface within the same plane
     
     * Note: this diagram only features one instance of biaxial stress
2. Bending
   • A beam in bending experiences tension on one side and compression on the other with the middle line being strain neutral
     
   • Stress and strain are non-uniform across cross-section of a beam
3. Torsion

   • Torsion is a form of pure shear
     

     • Stress and strain are non-uniform

   • Shear strain: $\gamma =\frac{\varphi }{l_0}r$
     ${\gamma }_{max}=\frac{\varphi }{l_0}{r}_0$

   • Shear stress: $\tau =\frac{G\varphi }{l_0}r$
     ${\tau }_{max}=\frac{G\varphi }{l_0}{r}_0$

Poisson’s Ratio

• When a bar is stretched longer by tensile stress, it usually becomes thinner at the same time

• The ratio of the lateral strain to the axial strain is found to be a material’s constant, known as Poisson’s ratio $v=-\frac{{ϵ}_{lateral}}{{ϵ}_{axial}}$
  Note: $-$ve sign is included for convenience of making $v$ a $+$ve value

  

  • For a cylindrical rod shown here: ${ϵ}_{axial}=\Delta l/l_0{ϵ}_{lateral}=\Delta d/d_0$
    Therefore: $v=-\frac{\Delta d/d_0}{\Delta l/l_0}$

• More examples are on the slides

What does the value of Poisson's Ratio tell us?

• When $v=0.5$, the volume of the material does not change
• If $v<0.5$, the volume increases
  • Most real life materials fall in this range
• If $v>0.5$ the volume decreases

Elastic Properties of Materials

• Young’s modulus, 𝑬
• Shear modulus, 𝑮
• Bulk modulus, 𝑲
• Poisson’s ratio, 𝒗

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ENSC1004 - Lecture 5