Answer Happy

4. Let T₁ > T₂ > T3 be the principal components of the stress tensor I at a certain point P within an elastic body and choose an appropriate Cartesian Ox₁x₂x3 such that e₁, 2 and 3 are the corresponding principal axis directions. a) From your knowledge of principal components and principal directions of a tensor, state the form the stress tensor I will take under the Cartesian axes defined above. (1 Mark) b) Calculate the surface traction at P for a surface with arbitrary unit normal n = npep, and hence using normal vector operations find the magnitude of the surface traction acting in the direction of n (i.e. the normal stress). (4 Marks) c) Use the method of Lagrange multipliers, or otherwise, to find the maximum value of the normal stress found in b) in terms of T₁, T₂ and T3. State also the directions in which the maximum stress acts. (20 Marks)