STRESS-STRAIN RELATIONSHIP

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Presentation transcript:

STRESS-STRAIN RELATIONSHIP Applied Load shape change (strain) stress There must be a relation between stress and strain Linear Elasticity: Simplest and most commonly used Uni-axial Stress: Axial force F will generate stress In the elastic range, the relation between stress and strain is Reduction of cross-section E: Young’s modulus, : Poisson’s ratio

UNI-AXIAL TENSION TEST Terms Explanations Proportional limit The greatest stress for which stress is still proportional to strain Elastic limit The greatest stress without resulting in any permanent strain Yield stress The stress required to produce 0.2% plastic strain Strain hardening A region where more stress is required to deform the material Ultimate stress The maximum stress the material can resist Necking Cross section of the specimen reduces during deformation Fracture Material failure

LINEAR ELASTICITY (HOOKE’S 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 (HOOKE’S LAW) cont. Isotropic Material: Stress in terms of strain: Strain in terms of stress What are the limits on Poisson’s ratio and what happens at these limits? Shear Modulus

DEGENERATION TO 2D Plane Strain Problem 3D engineering problems are often simplified to 2D problems Deformation in z-dir is constrained Strains in z-dir are zero y y Normal stress in z-dir x x z

DEGENERATION TO 2D cont. Plane Stress Problem: Plate-like structure under in-plane loads No constraints in thickness dir Stresses in z-dir are zero Stress-strain relation

EQUILIBRIUM EQUATIONS Equilibrium Relation (2D) Equilibrium Relation (3D)

BOUNDARY-VALUE PROBLEM When boundary conditions are given, how can we calculate the displacement, stress, and strain of the structure? Solve for displacement Equilibrium equation Constitutive equation (Stress-strain relation) Strain definition Get differential equation for displacements Load and boundary conditions