Week 2: Stress and Strain…what is essential is invisible to the eye.

~Antoine de Saint-Exupery, The Little Prince

BIOE 3200 - Fall 2015

Learning ObjectivesDefine internal forces and

stresses on a rigid bodyCalculate internal stresses from

internal forcesDefine strain and Poisson’s ratioRelate stress to strain through

constitutive equations

BIOE 3200 - Fall 2015

Terminology of Stress and Strainσ = Normal Stressτ = Shear Stressε = Normal Strainγ= Shear StrainE = Young’s Modulus (Modulus of Elasticity)G = Shear Modulus (Modulus of Rigidity)ν = Poisson’s RatioA = Cross-sectional areahttp://www.pbs.org/wgbh/nova/tech/making-stuff.html#making-stuff-stronger BIOE 3200 - Fall 2015

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

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

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

Stress components are associated with two directions.

BIOE 3200 - Fall 2015

The stress tensor is symmetric: we can show that τxy = τyx

How many independent components of stress are there at any point in a body?

BIOE 3200 - Fall 2015

Strain: Deformation due to loadingNormal strain: change in size

(change in length/original length)

BIOE 3200 - Fall 2015

Strain: Deformation due to loadingShear strain: change in shape (change

in angle between lines that were originally perpendicular)

BIOE 3200 - Fall 2015

Constitutive Equations: relating stress and strainPoisson’s ratio

BIOE 3200 - Fall 2015

Constitutive Equations: relating stress and strainFor 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