3-(30%) Consider the following second order tensor F shown in its 2D component form: Later we will see that F is deformation gradient tensor defining a simple shear deformation shown in the Figure 1. Using the polar decomposition theorem, decompose F to positive definite symmetric tensors U and V, and orthogonal rotation tensor R such that F=RU=VR. (Hint: consider the general 2D form of a rotation tensor in form of sine and cosine functions, i.e., R= cos - sin 0 to find the angle of rotation 0 in terms of y. Then find sin e cos e U and V) X2 - Y - XI Figure 1: Simple shear deformation Page 1 of 24- (35%) Consider the following skewed coordinate y, with basis {g } and the Cartesian coordinate x with unit basis {e;). V2 60 a) Find the relation (transformation) between the coordinate of a point P in xi: (x1, x2, x3) and yu: (vi, yz, V3). b) Using these relations find the covariant and contravariant base vectors of yi. c) Find the metric coefficients g; and g", mixed Kronecker delta o', and & and the scale factors. d) Assume the strain tensor of the simple shear problem shown in problem 3 is where ' n = asinh! VAty z and the stress response is 0 = [H(1 - Cosy) H Siny H Siny -H(1 - Cosy)] in x; for linear elastic material with shear modulus p = 0.5 MPa. For y = 1.05 find the covariant, contravariant, and mixed components of stress (S) and strain (E) in y, axis. e) Find the stored strain energy density (S:E). Confirm this result is the same as (0: E)