Transformation methods - Examples http://en.wikipedia.org/wiki/Stress_%28mechanics%29#mediaviewer/File:Components_stress_tensor_cartesian.svg

Overview Transformations Mohr’s Circle Eigenvalues and Eigenvectors Direct Approach Transformation Matrix Mohr’s Circle Construction Principal Stresses Eigenvalues and Eigenvectors Principal Scalar Invariants Example Problems

Transformations Transformation of axes:

Transformations Transformation of axes:

Transformations 1. Direct Approach Balance forces in y’ direction: Simple Example: uniaxial stretch Balance forces in y’ direction: Balance forces in τx’y’ direction:

Transformations 1. Direct Approach Generalized Example:

Transformations 2. Transformation Matrix

Transformations 2. Transformation Matrix

Mohr’s Circle Graphical representation of the transformation equations General conventions τxy σx σy Shear stress (τxy) is positive if resulting distortion stretches in 1st and 3rd quadrant Shear stress (τxy) is negative if resulting distortion stretches in 2nd and 4th quadrant Tensile stress (σx or σy) is positive Compressive stress is negative Mohr’s conventions on σ-τ plot Shear stress (τxy) is positive if surface stress rotates element clockwise Shear stress (τxy) is negative ifsurface stress rotates element counter-clockwise Tensile stress (σx or σy) is positive Compressive stress is negative

Mohr’s Circle σy = -3 τxy = 4 σx = 5 σ τ Graphical representation of the transformation equations τxy = 4 σx = 5 σy = -3 σ τ

Mohr’s Circle σy = -3 τxy = 4 σx = 5 σ τ Graphical representation of the transformation equations τxy = 4 σx = 5 σy = -3 σ Points on the circle represent this state of stress at any angle τ

Mohr’s Circle Rotation on Mohr’s circle = 2x angle of transformation Graphical representation of the transformation equations Rotation on Mohr’s circle = 2x angle of transformation

Mohr’s Circle Principal Stress Calculation

Mohr’s Circle Principal Stress Calculation Principal Stresses Maximum Shear Stress Angle to Principal Planes:

Eigenvalues and Eigenvectors Definition: T is a tensor a is a normalized vector which transforms under T into a vector parallel to itself λ is a scalar a is an eigenvector λ is an eigenvalue

Eigenvalues and Eigenvectors Solution: (index notation) (long form)

Eigenvalues and Eigenvectors Characteristic Equation: The eigenvalues of tensor T are the solution to the characteristic equation

Eigenvalues and Eigenvectors Vector Solution: The eigenvectors of tensor T are the solution to the following (for each λ):

Eigenvalues and Eigenvectors Example:

Eigenvalues and Eigenvectors Example:

Eigenvalues and Eigenvectors Principal Values and Principal Directions For symmetric tensors (i.e. stress tensor, strain tensor, rate of deformation tensor, etc.): There exist at least three real eigenvectors, which are called the principal directions, which are mutually perpendicular The corresponding eigenvalues are called the principal values Stresses normal to these principal directions are principal stresses, in which the normal stress vector is maximized

Principal Scalar Invariants Characteristic Equation: * Cubic equation in λ Note: invariants are associated with tensor independent of coordinate system

Principal Scalar Invariants

Principal Scalar Invariants Example: Characteristic Equation: