Introduction to concepts and equations related to stress and strain

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