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

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1 SEMM Mechanics PhD Preliminary Exam Spring Consider a two-dimensional rigid motion, whose displacement field is given by u(x) = [cos(β)x 1 + sin(β)x 2 X 1 ]e 1 + [ sin(β)x 1 + cos(β)x 2 X 2 ]e 2. Compute the small strain tensor for this motion and comment on its validity relative to the parameter β. 2. Let the mutually orthogonal vectors a, b, and c be expressed in a given Cartesian basis as a 1, b 1, c (a) Consider a state of Cauchy-stress defined as σ = 1aa + 2bb + 3cc. What are the components of σ in the Cartesian basis in which the vectors have been defined? 1 (b) What are the principal stresses associated to this state of stress? What are the corresponding principal directions. 3. Consider an isotropic material that displays a volumetric elastic response p = Ktr[ε] and a deviatoric Maxwell-like response µė = ṡ + 1s, where e = ε 1 tr[ε]1 is the τ 3 deviatoric strain tensor, 1 = e 1 e 1 + e 2 e 2 + e 3 e 3 is the identity tensor, and s is the deviatoric stress tensor. Assume the material is subjected to a time varying strain of the form ε(t) = at1 + H(t)(e 1 e 2 + e 2 e 1 ), where a is a scalar constant and H( ) is the Heaviside step function. Find the stress response. 1 The juxtaposition of vectors indicates a dyadic (or tensor) product in this examination. Thus the other common notation for aa is a a. 1

2 SEMM Mechanics Preliminary Exam Fall (10pts) Consider a two-dimensional body. At a given point X in the reference configuration, one scribes two small vectors V 1 = ae 1 + ae 2 and V 2 = be 1, i.e. V i 1. The body is now deformed, such that V 1 is now v 1 = 3ae 1 + 3ae 2 and V 2 is now v 2 = be 1. Find an expression for the deformation gradient at this point. 2. (10pts) The 1st Piola-Kirchhoff stress at a point in a body (two-dimensional) is given as [ ] 1 2 P. 1 4 The deformation gradient at this point is [ ] 1 1 F. 0 1 What are the principal Cauchy stresses at this point? [Hint: σ = (1/J)P F T.] 3. (10 pts) The following model has been proposed to describe a linear (one-dimensional) viscoelastic solid: A ε + Bε = C σ + Dσ. (a) Find expressions for the storage and loss moduli of the material in terms of the material parameters A, B, C, D. (b) Discuss any restrictions that the material parameters should satisfy. 1

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6 UNIVERSITY OF CALIFORNIA AT BERKELEY SPRING SEMESTER 2012 Department of Civil and Environmental Engineering Structural Engineering, Mechanics and Materials Name Doctoral Preliminary Examination Mechanics Problem 1. (50 points) A two-dimensional anti-displacement field for a screwed dislocation is given as, u x = 0, u y = 0, and u z = b 2π θ where b is the Burger s constant, and θ is the polar coordinate angle from [0, 2π]. Find the shear strains, ɛ xz, ɛ yz, and ɛ θz, ɛ rz? Hints: θ = tan y x, r2 = x 2 + y 2 Figure 1: Problem 1 Problem 2(50 points) Consider a beam with the rectangular cross-section as shown in Figure 2. Assume that the extensional stress inside the cross-section at the position, X, is, σ x = M(x)y I z

7 Figure 2: Problem 2 where M(x) is the bending moment, and I z is moment of inertia of the section with respect to z-axis. Find the shear stress, τ xy, distribution on the cross-section. Hint: Use the equilibrium equation, and appropriate boundary conditions. σ x x + τ xy y = 0, and V (x) = dm dx

8 University of California Department of Civil and Environmental Engineering Spring 2011 Doctoral Preliminary Exam: Mechanics Name: Problem #1 A straight bar of length L and a a square cross section is subjected to an eccentric axial load P as shown in the figure below. Determine the maximum load P max that can be applied without yielding if the material yields when the shear stress reaches the value τ yp. Remarks: The bar can be assumed isotropic linear elastic before yielding, and stress concentrations can be neglected. a/6 a/6 P P P a L a Cross section Problem #2 For the following plane states of strain (in the plane with Cartesian reference system (x, y)) State 1 ε xx =2x + y ε yy =3y 2 ε xy =3xy +2x +1 State 2 ε xx =2x + y ε yy =3y 2 ε xy =3x 2 +2x +1 (all other strain components vanish), determine: 1. Is any of these states of strain compatible? 2. If possible, determine a displacement field associated to each state of strain. 3. In general, how many (if any) displacement fields can be obtained from a strain field? If more than one, what general conditions are required to specify a single one? Illustrate your answers with your results in Item 2 above.

9 UNIVERSITY OF CALIFORNIA AT BERKELEY SPRING SEMESTER 2010 Department of Civil and Environmental Engineering Structural Engineering, Mechanics and Materials Name Doctoral Preliminary Examination Mechanics Problem 1. (40 points) A two-dimensional displacement field and shear strain field in a continuum domain is given as u x = a 0 y 4 + a 1 x 2 y 2 + a 2 xy 2 + a 3 x 2 y, u y = b 0 x 2 + b 1 x 2 y + b 2 xy 2, γ xy = c 0 x 2 y + c 1 xy + c 2 x 2 + c 3 y 2 + c 4 x. What relationships connecting the constants (a s, b s, and c s) make the foregoing expression possible? Problem 2(60 points) Consider a large thin plate with a small hole (plane stress state) under an in-plane uniform bi-axial remote loading, σ 0 and 3σ 0 (see Fig. 1 ). Given the stress solution (see below) for a plate with a hole subjected to uniaxial tension boundary condition in X-axis direction, i.e. σ xx x = σ 0. Find: (1) the total stress solution, (2) the maximum stresses of σ xx and σ yy, and (3) the location at where σ r and σ θ reach maxima. (Hint: use superposition) Figure 1: Problem 2 Hint: σ r = 1 2 σ 0 [(1 a2 ) r 2 + (1 + 3a4 r 4 4a2 ) )] r 2 cos 2θ

10 σ θ = 1 2 σ 0 [(1 a2 ) r 2 τ rθ = 1 2 σ 0 (1 3a4 r 4 (1 + 3a4 ) )] r 4 cos 2θ + 2a2 r 2 ) sin 2θ

11 UNIVERSITY OF CALIFORNIA AT BERKELEY FALL SEMESTER 2009 Department of Civil and Environmental Engineering Structural Engineering, Mechanics and Materials Name Doctoral Preliminary Examination Mechanics Problem 1. A solid steel shaft of radius a = 0.1m is pressed onto a steel cylinder (the value of its outer radius b is not explicitly given) inducing a contact (interfacial) pressure p 1 and a maximum tangential stress value = 2p 1 inside the cylinder. If an axial tensile load P L = 45kN is applied to the shaft, what will be the change in the contact pressure p 1? Assume that the axial load is superposed to the initial state. Hints: (a) The relation between the shrinking allowance, δ (misfit margin between the outer radius of the shaft and the inner radius of the cylinder), and the contact pressure p 1 is given as: σθθ max p 1 = Eδ(b2 a 2 ) 2ab 2 where E is Young s modulus; the shaft and the cylinder are made by the same material. Note that both the value of δ, p 1, and E are not explicitly given. (b) Poisson s ratio ν = 1/3; (c) The circumference stress distribution for a thick-cylinder is: σ θθ = a2 p i b 2 p o b 2 a 2 + (p i p o )a 2 b 2 (b 2 a 2 )r 2 (1) (50 points) Figure 1: Problem 1

12 Problem 2. A simply supported beam has a span L and a uniform cross section dimension b h (see Figure 2 (a) and (b)). There is a concentrated load P acting at the middle span of the beam (Figure 2 (a)), and the beam is made of linear elastic-perfectly plastic material (see Fig 2b). (1) Draw moment diagram; (5 points) (2) Assume that in a given cross section the elastic core size is 2y 0, draw axial stress σ xx distribution along the depth of the beam; (10 points) (3) Find the relation between the elastic-plastic bending moment, yield stress, and y 0 for a given cross section; (15 points) (4) Assume that the beam is under monotonic loading (no unloading). Find the expressions for the applied load P at which the middle section of the beam is completely yielded. (10 points) (5) Find the envelop of the elastic region and plastic region, i.e. find y 0 (x)? (10 points) Figure 2: Problem 2 (a) Figure 3: Problem 2 (b)

13 University of California Department of Civil and Environmental Engineering Fall 2008 SEMM Doctoral Preliminary Exam: Mechanics Name: Problem #1 Write and prove the expression giving the bulk modulus (relating the volumetric strain and the hydrostatic pressure) in terms of the Young modulus and Poisson ratio in a linear elastic isotropic material. Can you identify a value of these parameters (the Poisson ratio in particular) that captures a somehow special response of the material? Problem #2 1. If a material is known to follow Tresca s yield criterion for an uniaxial yield limit of σ y, under what value of σ the material will yield in the plane stress state depicted in the figure for a rectangular block? 2. What would be the value in plane strain if the material can be considered to be isotropic linear elastic before yielding? (discuss your answer for different values of the elastic material parameters, considering in particular typical ranges of these parameters, as needed) 5¾ in compression 4¾ shear ¾ in tension

14 University of California Department of Civil and Environmental Engineering Spring 2008 SEMM Doctoral Preliminary Exam: Mechanics Name: Problem #1 A linear isotropic solid is subjected to a state of plane strain in the plane xy, withthe in-plane components of the stress given by σ xx = Ax 2 σ yy =0 σ xy = Bx(x + y) for two constants A and B. LetE and ν be the Young modulus and Poisson ratio of the material, respectively, both being constant. 1. Determine the body force per unit volume in static equilibrium with these stresses. 2. Determine any conditions between the constants A and B so the above stresses define a valid stress field besides the equilibrium already considered in the previous item. 3. In that case, determine the displacement field u(x, y) and v(x, y) for its x and y components, respectively, so u(0, 0) = v(0, 0) = 0 and v(l,0) = L for some length value L. Problem #2 Determine the yield limit in simple shear τ y predicted by the (1) Tresca and (2) von Mises yield criteria if both criteria have been calibrated to match the yield limit σ y in uniaxial tension. Based on these values, which criterion is more conservative?

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