Consider a square region R in the x – y plane, aligned parallel to the x and y axes….

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Consider a square region R in the x – y plane, aligned parallel to the x and y axes. Let V be a two–dimensional vector representing a point in R. (a) Consider the following linear transformations acting on R. Tell whether each of the following transformations is a dilation, contraction, rotation, shear, or a combination of these. Also give a rough sketch of the effect of each transformation on R.

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Continuum Mechanics Open-book Examination, 2017 1. Consider a square region R in the x y plane, aligned parallel to the x and y axes. Let V be a two{dimensional vector representing a point in R. (a) Consider the following linear transformations acting on R. Tell whether each of the following transformations is a dilation, contraction, rotation, shear, or a combination of these. Also give a rough sketch of the e ect of each transformation on R. i. ” # p 1=2 3=2 p V! V: 3=2 1=2 ii. ” # 1 1 V! V: 0 1 iii. ” # 2 4 V! V: 4 2 (b) Give a linear transformation which squashes R into a rectangle of the same area, but where the width is double the height. (c) Give a linear transformation which projects R into the line x =y. (d) Show that the result of any linear transformation acting on R is a parallelogram. (n) T 2. The traction vector on an internal surface with outward unit normal n is t = n = n. The component of this traction vector in the normal direction, called the normal stress and denoted by , is then given by = n n. If we maximize this normal n n stress with respect to n subject to the constraint n n = 1, we end with the eigenvalue problem n = !n, where ! is a Lagrange multiplier associated with the constraint. The eigenvalues of are called principal stresses and the associated directions principal directions of stress. The vector n (n n)n is the component of n tangential to the surface segment and is called the shear stress. Relative to an orthonormal basisfe ; e ; eg the stress tensor at a certain point has 1 2 3 components 0 1 1 c b B C c 1 a : @ A b a 1 Find the values of a;b and c for which the traction vector vanishes on a surface with p normal (e e e )= 3. Deduce that, for these particular values of a;b and c, one of 1 2 3 the principal stresses is zero and nd the other principal stresses. 13. We consider equilibrium elasticity in two dimensions. In the absence of body forces the Navier equation reads 2 0 = ( + )rr u r u where u…

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