An important way to illustrate transformation of stress and strain in 2-0 3002 MT-1620 Fall via Mohr's circle(recall from Unified ) This was actually used B.C. (before calculators). It is a geometrical representation of the transformation (See handout (you will get to work with this in a problem set) Also recal‖l (Three) Important Aspects Associated with Stress/Strain Transformations 1. Principal Stresses/Strains(AXes): there is a set of axes into which any state of stress /strain can be resolved such that there are no shear stresses/strains >O depend on applied loads Ei depend on applied loads and material response Thus note For general materials axes for principal strain axes for principal stress Generally Chave nothing to do with) material principal axes* principal axes of stress/strain Paul A Lagace @2001 Unit 7-p 6
MIT - 16.20 Fall, 2002 An important way to illustrate transformation of stress and strain in 2-D is via Mohr’s circle (recall from Unified). This was actually used B.C. (before calculators). It is a geometrical representation of the transformation. (See handout). (you will get to work with this in a problem set). Also recall… (Three) Important Aspects Associated with Stress/Strain Transformations 1. Principal Stresses / Strains (Axes): there is a set of axes into which any state of stress / strain can be resolved such that there are no shear stresses / strains --> σij depend on applied loads --> εij depend on applied loads and material response Thus, note: For general materials… axes for principal strain ≠ axes for principal stress Generally: (have nothing to do with) material principal axes ≠ principal axes of stress / strain Paul A. Lagace © 2001 Unit 7 - p. 6
MT-1620 al.2002 Find via roots of equation 12 13 12 0 eigenvalues:o1,σ1 I9 II (same for strain) 2. Invariants: certain combinations of stresses strains are invariant with respect to the axis system Most important: E(extensional stresses /strains)=Invariant very useful in back-of-envelope /quick check" calculations 3. EXtreme shear stresses/strains:(in 3-D)there are three planes along which the shear stresses /strains are maximized These values are often used in failure analysis(recall Tresca condition from unified These planes are oriented at 45 to the planes defined by the principal axes of stress strain(use rotation to find these) Paul A Lagace @2001 Unit 7-p. 7
MIT - 16.20 Fall, 2002 Find via roots of equation: σ11 − τ σ12 σ13 σ12 σ22 − τ σ23 = 0 σ13 σ23 σ33 − τ eigenvalues: σI, σII, σIII (same for strain) 2. Invariants: certain combinations of stresses / strains are invariant with respect to the axis system. Most important: Σ (extensional stresses / strains) = Invariant very useful in back-of-envelope / “quick check” calculations 3. Extreme shear stresses / strains: (in 3-D) there are three planes along which the shear stresses / strains are maximized. These values are often used in failure analysis (recall Tresca condition from Unified). These planes are oriented at 45° to the planes defined by the principal axes of stress / strain (use rotation to find these) Paul A. Lagace © 2001 Unit 7 - p. 7