Show that the principal directions of stress and strain are the same for isotropic linear elastic materials. Note: this means that if you know the direction of largest principal stress at a point, you also know the direction of largest stretch/strain at that point. Also note: this is not the case for anisotropic solids.

His can you find answers for these please and explain how ? Thank you

The state of stress at a point in an engine piston is given in the x-y-z coordinates by: 90 180 [0]=0, MPa = 180 240 yx yy 120 zy Using the matrix transformation law, determine the state of stress at the same point for an element rotated about the x-axis (in the y-z plane) 60° clockwise from its original position. Calculate the stress invariants and write the characteristic equation for the original state of stress, Calculate the deviatoric invariants for the original state of stress, Calculate the principal stresses and the absolute maximum shear stress at the point. What are the stress invariants and the characteristic equation for the transformed state of stress, а. b. С. d. e

The stress tensor for a three-dimensional system for some x, y z axis system is 1. 30 20 -15] 0 30 MPa 20 [-15 30 (a) Verify that one of the principal stresses is 40 MPa, and hence find the other two principal stresses. (b) Find the directional cosines of the plane on which the principal stress of 40 MPa acts.