ENSC1004 - Lecture 3

|← Back to Home

ENSC1004 - Lecture 2
Elastic Moduli 1 - Slides

Introduction

• All materials deform when a force is applied to them
• How does a material extend (elongate, deform) in response to an applied force?

Stress, Strain and Young’s Modulus

In engineering, the effect of a force acting on a material is described by (engineering) stress, which is defined as: $\sigma =\frac{F}{A_0}$

• $F$ is the applied $[N]$
• $A_0$ is the cross-sectional area of the material $[m^2]$
• $\sigma$ has the unit of $\frac{N}{m^2}$ which is known as Pascal $[Pa]$

The deformation of a material in response to an applied force is described by (engineering) strain, which is defined as:
$ϵ=\frac{\Delta l}{l_0}=\frac{l-l_0}{l_0}$

• $\Delta l$ is the length change caused by the applied force
• $l_0$ is the original length
• $l$ is the length under the applied force
• $ϵ$ is dimensionless, and is often expressed in “%”

Why do we use Strain, not Elongation?

A longer rod will stretch ($\Delta l$) more under the same stress than does a shorter one; but the proportion of the stretch, i.e., the strain ($\Delta l$/$l_0$) , is the same for both

Stress-strain relationships

🌟 At low strain levels, metals often exhibit a linear stress-strain relationship, but the relationship deviates from linearity at high strains

🌟 Ceramics generally exhibit a linear stress-strain relationship until fracture. However, they also shatter with very low strain

In applications, we always want to make sure that the material stays within the elastic stage of the deformation

• This is where the strain is small and is self-recoverable force is removed
  • This is the elastic deformation

Hooke's Law

In this region, the linear relationship between stress and strain is expressed as:
$\sigma =Eϵ$

This is the Hooke’s Law and $E$ is the Young’s modulus of the material

• $E$ is a material’s property
• $E$ has the unit of stress, thus $[Pa]$

The Young’s modulus measures the resistance of a material to elastic deformation:
⭐ High $E$, deflect less
⭐ Low $E$, deflect more

Young’s modulus = stress/strain: $E=\frac{\sigma }{ϵ}$

Three Stress-Strain States

• Three types of load:
  1. Tension and Compression
  2. Shear
  3. Hydrostatic pressure

1. Tension and Compression
   • Tension → pulling force
     e.g. poles holding up a bridge
   • Compression → crushing force
     e.g. columns holding up a building
2. Shear is a force that causes an object to slide or deform at an angle
   • It is denoted with $\tau$
     i.e. $\tau =\frac{F}{A_0}$
3. Hydrostatic pressure $P=\rho gh$

---

ENSC1004 - Lecture 4