Lecture 9:Generalized Hooke’s Law

σ=Eε, z=Gγ

 Poisson’s Effect: As the bar elongates/shortens the cross section contracts/expands.

ν=(-εtransversal)/(εaxial)=(-εyy)/(εxx)

νranges from 0.2 (concrete) to 0.3 (steel)

ν=0.5 for rubber, incompressible (no change of volume)

 

Generalized Hooke’s Law

εx=(σx/E)- ν(σy/E)- ν(σz/E);

εy=(σy/E)- ν(σx/E)- ν(σz/E);

εz=(σz/E)- ν(σx/E)- ν(σyE);

γxy=(τxy)/G; γyz=(τyz)/G; γzx=(τzx)/G

Assumptions: isotropic linear elastic material—same response in all directions model

Εx=(σx)/E

Remark (1): G= E/(2(1+ν))

ε=α(T-T0)

ΔT=α(δT)L

 

**STATICALLY INDETERMINANT SYSTEMS

Axial Deformation: solution involves Three Steps

  1. EQUILIBRIUM to get the reactions, the resultants, and the stress σ=P/A
  2. GEOMETRY to measure deformation ε=Δ/L
  3. MATERIAL PROPERTIES and in particular elastic modulus: σ=Eε

Strain Energy Density: U=dU/dV=σε/2=( Eε)ε/2=(σ2)/(2E)

Modulus of Resilience: (σ2)/(2E)

 

TOUGHNESS= Area under the stress-strain curve (to failure)

An energy method for obtaining deflections: U=We

Lecture 8: General Concepts of Strain

Shear strains are caused by shear stresses (γ).

Shear modulus of elasticity, G

τ=Gγ, this refers to the stress versus "angular strain" graph.

Strain: Measures Relative Displacement

εx=δu/δx, εy=δv/δy, εz=δw/δz

γxy=γyx=(δv/δx)+(δu/δy)