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 Thomas Young, 1773-1829, England, polymath and physician with contribution to Rosetta stone, astigmatism, electric theory, ‘The last person who knew everything’/ Simeon Denis Poisson 1781-1840, France, student of Lagrange and Laplace, most known for the Poisson equation for potential fields. Robert Hooke 1635-1703 English natural philosopher, architect and polymath. Came close to experimental proof of the inverse distance square in the law of gravity, and an early proponent of evolution.

UNI-AXIAL TENSION TEST Definitions in table are material for quiz 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 Energy considerations dictate that C is symmetric (21 constants) For homogeneous, isotropic material 21 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? See notes page. Shear Modulus From the equations it is obvious that we get singularities when Poisson’s ratio is either 0.5 or -1. From the equation for G we see that at -1 we will get infinite shear rigidity, and at 0.5 we will need infinite stresses to generate any normal strains. When Poisson’s ratio approaches 0,5 the material becomes incompressible, and this is true of many plastics. Negative Poisson’s ratios are more exotic and do not normally happen in isotropic materials, but one can generate foams with negative Poisson’s ratios.

Quiz-like questions What is the proportional limit? What are the limits on Poisson’s ratio? What happens when Poisson’s ratio is 0.5? How many independent elastic constants can a general material have? For an isotropic material with E=3GPa, and G=1.2GPa, if the only non-zero strain component is exx=0.005, what are the non-zero stress components? Answers in the notes page. Proportional limit: The greatest stress for which stress is still proportional to strain -1<n<0.5 n=0.5 corresponds to incompressible material where you need infinite stresses to change volume, so we must have exx+eyy+ezz=0 General elastic material can have up to 21 independent elastic constants To calculate the stresses we ues Hooke’s law. Using Matlab we get exx=0.005; E=3;G=1.2;nu=E/(2*G)-1 nu = 0.2500 >> sig=E/((1+nu)*(1-2*nu))*[(1-nu), nu, nu]'*exx sig = 0.0180 0.0060 So

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

Quiz-like Questions Airplane wings and fuselages are analyzed using plane stress, but almost all their loading is pressure, which is in the z direction and generates szz. How come? A point in the skin of the wing has exx=eyy=0.001, gxy=0. What is ezz if n=0.25? Answers in the notes page. The szz is a fraction of atmospheric pressure, that is a few psi. On the other hand the in-plane stresses due to bending of the wing or fuselage are thousands or ten of thousands psi, so the z stress can be neglected. From Hooke’s law for plane stress

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