# Mechanics PhD Preliminary Spring 2017

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1 Mechanics PhD Preliminary Spring (10 points) Consider a body Ω that is assembled by gluing together two separate bodies along a flat interface. The normal vector to the interface is given by n = (1 1 1) T. Assume that Ω is subjected to a homogeneous (Cauchy) stress field σ = β MPa where β is a non-dimensional load factor. Assume that the interface fails when the normal stress on the interface exceeds 100 MPa. Determine the maximal permissible value of β. 2. (10 points) Consider a small-strain hyperelastic body Ω, free of body forces, with strain energy density W (ε) subjected to dead-loads t on Ω t Ω and restrained with zero displacement on Ω u Ω, where Ω t Ω u = Ω. Starting from the principal of minimum potential energy, derive the strong form of the governing equilibrium equations. 3. (10 points) A three dimensional viscoelastic body Ω = {x x x 2 2 R 2 and x 3 L/2} has a measured displacement field u = (αx 1 /R)e 1 + (αx 2 /R)e 2, where R and L are given and α > 0 is much smaller than R and L. Determine the algebraically maximum and minimum normal strains in the body as well as the maximum shear strain in the body. 1

2 Mechanics PhD Preliminary Fall (10 points) Consider a linear elastic, isotropic material, under two-dimensional plane strain. Assume that the material is subjected to a body force, b, that is described by a known potential, V ; i.e. b i = V,i. Assume further that σ 11 = σ 22 = φ,22 + V φ,11 + V where φ is an unknown stress function. σ 12 = φ,12 (a) Show that the equilibrium equations are identically satisfied by this assumption. (b) Noting that compatibility requires ε 11,22 + ε 22,11 2ε 12,12 = 0, find the governing equation for φ in terms of V and the elastic constants. 2. (10 points) The general form of an infinitesimal rigid motion in two-dimensions can be described by three scalars A, B, C where A, B describe an infinitesimal translation and C describes an infinitesimal rotation. In Cartesian form the resulting rigid displacement field is given by: u 1 (x 1, x 2 ) = A + Cx 2 u 2 (x 1, x 2 ) = B Cx 1 Find the form of this rigid displacement field in polar coordinates; i.e. determine the corresponding u r (r, θ) and u θ (r, θ). 3. (10 points) Starting from the global statement of linear momentum balance b(x) dv + t(x, n) da = ρ(x)ü(x) dv B B derive Newton s 3rd Law t(x, n) = t(x, n). Hint: Employ a pillbox construction. B 1

3 UNIVERSITY OF CALIFORNIA AT BERKELEY SPRING SEMESTER 2016 Department of Civil and Environmental Engineering Structural Engineering, Mechanics and Materials Name Doctoral Preliminary Examination Mechanics Problem 1. (50 points) Consider the double cantilever beam shown in Figure 1. The beam length is a, width is B and height is h. Assume that the beam is made by linear elastic material, and the Young s modulus is E, and the moment inertia is I = Bh 3 /12. Figure 1: A double cantilever beam Calculate: (1) The bending moment M(x) and the total strain energy stored inside the beam U E (for the sum of the two beams); (2) The cantilever beam deflection (under the transverse load) at the tip (EIv (x) = M(x))? (3) The total external potential W L (for the two forces)? (4) The total fracture energy release per unit width, i.e. G = 1 B dπ da, Hints: (1) Do not need to calculate numerical values. where Π = U E W L (2) Only consider the classical bending part of the strain energy of the beam.

4 Problem 2.(30 points) Consider a simply connected continuum media Ω under infinitesimal deformation. The following displacement boundary condition is prescribed, where E ij is a constant symmetric tensor. u i = E ij x j, x Ω, Show that 1 ɛ ij dω, = E ij Ω Ω for the infinitesimal deformation (strain) field ɛ ij. Problem 3.(20 points) Derive the weak formulation for the following static linear elastic (small deformation) problem, i.e. σ ij,j + f i = 0, x V, (1) where σ ij = C ijkl ɛ kl is the Cauchy stress; ɛ ij = (1/2)(u i,j + u j,i ) is the strain field, and f i is the body force. The following displacement boundary condition is being prescribed, where ū i is the prescribed boundary displacement. u i = ū i, x V, (2)

5 UNIVERSITY OF CALIFORNIA AT BERKELEY FALL SEMESTER 2015 Department of Civil and Environmental Engineering Structural Engineering, Mechanics and Materials Name Doctoral Preliminary Examination (Solid Mechanics) Problem 1. (40 points) Consider a single-connected domain Ω. The following boundary condition is prescribed, σ ij n j = t i = Σ ij n j, x Ω where Σ ij is a constant stress tensor. Assume that the body force is zero, and the equilibrium equation inside the domain has the following form, σ ij,j = 0. where Show that 1 σ ij ɛ ij dω = Σ ij < ɛ ij > Ω Ω < ɛ ij >:= 1 Ω Ω ɛ ij dω. Problem 2. (30 points) Consider the following displacement field, 1. Find the strain field? 2. Find the rotation field? u x = 1 2 y y4 + xz u y = 1 2 x x4 + yz u z = xy 3. At the point (1, 1, 0), there is a principal strain that has value 2.0. Find the other two principal strains.

6 Problem 3. (30 points) Consider a stress tensor in plane stress state, [σ ij ] = [ Find suitable planes, i.e. n = (n 1, n 2 ) T, such that σ n = 0. ]

7 UNIVERSITY OF CALIFORNIA AT BERKELEY SPRING SEMESTER 2015 Department of Civil and Environmental Engineering Structural Engineering, Mechanics and Materials Name Doctoral Preliminary Examination Mechanics Problem 1(25 points) Consider the following plane strain state, ɛ xx = a 1 y 2 + a 2 y 4 ɛ yy = b 1 x 2 + b 2 x 4 γ xy = cxy(x 2 + y 2 ) 1. Is this strain field a compatible strain field? If it is, are any restrictions required on the constants a s, b s and c? 2. Find the principal strains at the point (1, 1) with the constant a 1 = 1 and c = 1. Problem 2. (50 points) The stress components at a point P is given as [σ ij ] = (a) Find the traction components on a plane passing through point P, whose outward normal is n = (3/5, 0, 4/5) T ; (b) Find the normal and shear stresses at the point P on that given plane, and (c) Find the principal stresses at the point P. Problem 3. (25 points) Consider the isotropic linear elasticity stress-strain relation, σ ij = λɛ pp δ ij + 2µɛ ij. where λ and µ are Lamé constants.

8 Consider that an isotropic elastic sphere of radius r centred at the origin undergoes the following uniform dilatation deformation, u 1 = αx 1, u 2 = αx 2, and u 3 = αx 3, α > 0. (a) Find the strain components; (b) Find the stress components. (c) Derive the bulk modulus of the material. Hint : K := σ m ɛ ii, σ m = 1 3 σ ii.

### SEMM Mechanics PhD Preliminary Exam Spring Consider a two-dimensional rigid motion, whose displacement field is given by

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