This document defines key concepts in stress and strain analysis: 1) It defines normal stress, shear stress, normal strain, shear strain, and material properties like Young's modulus and Poisson's ratio. 2) It explains that internal forces develop in materials under loading to prevent separation, and that stress is calculated as the limit of force over area as the area approaches zero. 3) It notes that stress is a tensor with directional components, while strain describes changes in size and shape from loading.

1. Week 2: Stress and Strain
…what is essential is invisible to the eye.
~Antoine de Saint-Exupery, The Little Prince
BIOE 3200 - Fall 2015

2. Learning Objectives
 Define internal forces and stresses on
a rigid body
 Calculate internal stresses from
internal forces
 Define strain and Poisson’s ratio
 Relate stress to strain through
constitutive equations
BIOE 3200 - Fall 2015

3. Terminology of Stress and Strain
 σ = Normal Stress
 τ = Shear Stress
 ε = Normal Strain
 γ= Shear Strain
 E = Young’s Modulus (Modulus of
Elasticity)
 G = Shear Modulus (Modulus of Rigidity)
 ν = Poisson’s Ratio
 A = Cross-sectional area
http://www.pbs.org/wgbh/nova/tech/makin
g-stuff.html#making-stuff-stronger
BIOE 3200 - Fall 2015

4. Internal forces: Keeping stuff
together
 Consider a body in equilibrium under the action of forces: why
isn’t it pulled apart?
 Internal forces develop in the body to keep it together
 Recall the mandible in the jaw, eating candy: Where in the bone
is the stress highest?
BIOE 3200 - Fall 2015
From Szűcs, A., Bujtár, P., Sándor, G. K. B., Barabás, J. Finite Element
Analysis of the Human Mandible to Assess the Effect of Removing an
Impacted Third Molar. J Can Dent Assoc 2010;76:a72

5. Internal forces: Keeping stuff
together
 Consider a body in equilibrium under the action of forces: why
isn’t it pulled apart?
 Internal forces develop in the body to keep it together.
 Decrease size of A to a point P
◦ As A  0, F 0, but F/ A ≠0
 Define normal stress: σ = lim A  0 Fn / A
◦ Gives change in size
 Define shear stress τ = lim A  0 Ft/ A
◦ Gives change in shape
BIOE 3200 - Fall 2015

6. Stress: Is it as simple as
force/area?
 If the cut was made in a different direction, the force
and area would be different  stresses would
change
 A vector (like force) is associated with the direction in which it
acts
 A tensor (like stress) is associated with 2 directions:
 Direction in which it acts
 Plane upon which it acts
 Stress is written in terms of its 9 components.
BIOE 3200 - Fall 2015

7. Stress components are
associated with two directions.
BIOE 3200 - Fall 2015
 The stress tensor is symmetric: we can show that τxy =
τyx

8. How many independent components of
stress are there at any point in a body?
BIOE 3200 - Fall 2015

9. Strain: Deformation due to
loading
 Normal strain: change in size (change
in length/original length)
BIOE 3200 - Fall 2015

10. Strain: Deformation due to
loading
 Shear strain: change in shape (change in
angle between lines that were originally
perpendicular)
BIOE 3200 - Fall 2015

11. Constitutive Equations: relating
stress and strain
 Poisson’s ratio
BIOE 3200 - Fall 2015

12. Relating Stress and Strain
Graphically
BIOE 3200 - Fall 2015

13. Constitutive Equations: relating
stress and strain
 For isotropic materials (material properties
the same in all directions), at small strains:
◦ σ = E * ε
◦ Τ = G * ϒ
 For linear elastic materials:
◦ G = E/(2*(1+ν)
BIOE 3200 - Fall 2015