1 STRESS-STRAIN RELATIONSHIP Applied Load shape change (strain) stress There must be a relation...

EML 4500 FINITE ELEMENT ANALYSIS AND DESIGN

STRESS-STRAIN RELATIONSHIPApplied Load shape change (strain) stressThere must be a relation between stress and strainLinear Elasticity: Simplest and most commonly usedUni-axial Stress:Axial force F will generate stress In the elastic range, the relation between stress and strain is

Reduction of cross-section

E: Youngs modulus, : Poissons ratio

#UNI-AXIAL TENSION TESTTermsExplanationsProportional limitThe greatest stress for which stress is still proportional to strainElastic limitThe greatest stress without resulting in any permanent strainYield stressThe stress required to produce 0.2% plastic strainStrain hardeningA region where more stress is required to deform the materialUltimate stressThe maximum stress the material can resistNeckingCross section of the specimen reduces during deformationFractureMaterial failure

#LINEAR ELASTICITY (HOOKES LAW)When the material is in the Proportional Limit (or Elastic Limit)In General 3-D Relationship

Stress-Strain Matrix

For homogeneous, isotropic material 36 constants can be reduced to 2 independent constants.

#LINEAR ELASTICITY (HOOKES LAW) cont.Isotropic Material:Stress in terms of strain:

Strain in terms of stressShear Modulus

What are the limits on Poissons ratio and what happens at these limits?#DEGENERATION TO 2DPlane Strain Problem3D engineering problems are often simplified to 2D problemsDeformation in z-dir is constrainedStrains in z-dir are zero

xyxyzNormal stress in z-dir

#DEGENERATION TO 2D cont.Plane Stress Problem:Plate-like structure under in-plane loadsNo constraints in thickness dirStresses in z-dir are zero

Stress-strain relation

#EQUILIBRIUM EQUATIONSEquilibrium Relation (2D)

Equilibrium Relation (3D)

#BOUNDARY-VALUE PROBLEMWhen boundary conditions are given, how can we calculate the displacement, stress, and strain of the structure?Solve for displacement

Equilibrium equationConstitutive equation (Stress-strain relation)Strain definitionGet differential equation for displacementsLoad and boundary conditions

#

F

F

Necking

Strain hardening

Ultimate stress

Yield stress

Proportional limit

Fracture

Plane strain model

x

y

fx

fy

EMBED Equation.DSMT4


x

y







_1071770833.unknown

_1071770995.unknown

_1071771151.unknown

_1071771217.unknown

_1071770859.unknown

_1071770632.unknown

_1071770777.unknown

_1071770612.unknown

{} = [C] {}

Applied loads

Boundary condition

1 STRESS-STRAIN RELATIONSHIP Applied Load shape change (strain) stress There must be a relation...

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