Midterm Examination. Please initial the statement below to show that you have read it

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1 EN75: Advanced Mechanics of Solids Midterm Examination School of Engineering Brown University NAME: General Instructions No collaboration of any kind is permitted on this examination. You may use two pages of reference notes Write all your solutions in the space provided. No sheets should be added to the exam. Make diagrams and sketches as clear as possible, and show all your derivations clearly. Incomplete solutions will receive only partial credit, even if the answer is correct. If you find you are unable to complete part of a question, proceed to the next part. Please initial the statement below to show that you have read it `By affixing my name to this paper, I affirm that I have executed the examination in accordance with the Academic Honor Code of Brown University. PLEASE WRITE YOUR NAME ABOVE ALSO!. (3 points). (5 points) 3. (5 points) 4. (5 points) 5. ( points) TOTAL (35 points)

2 . A thin circular membrane with radius R and mass per unit area m is stretched by a uniform force per unit length of perimeter T.. Re-write the equation ω = f( T, mr, ) for the natural frequency of vibration of the membrane ω in dimensionless form - The units are ω = / s m= kg/m T = kg.m.s /m R= m e e 3 x x e T We need to make the LHS and the function arguments dimensionless. There is no way to combne the function arguments into a dimensionless group, but they can be combined to make something with dimensions of seconds. Hence the dimensionless formula is mr ω = f (.) T [ POINTS]. A membrane with mass per unit area grams and radius cm is stretched by a force per unit length of 4N. It is observed to have a vibration mode with natural frequency 7.7 Hz. If the radius is doubled, but other parameters are unchanged, what is the new natural frequency (in Hz)? We have mr ω = C T If the radius is doubled, the frequency will halve, i.e. ω = 3.85Hz. Express the following equations in index notation (a) η = ab (b) c= a b (a) η = ab i i c = (b) i ijk ab j k Hence, use index notation to show that a b = a b ( ab ). Recall that = δ δ δ δ. Please show your work clearly ijk ipq jp kq jq kp From (a), (b) a b ( ijk ab j k )( ipq ab p q ) Recall the identities ijk ipq = δ jpδkq δ jqδkp and aj = δ jiai. Hence ( )( ) ( δ δ δ δ ) a b ab ab = abab ijk j k ipq p q jp kq jq kp j k p q = abab abab = ( aa )( bb ) ( ab ) j k j k j k k j a [ POINTS] = aa and [3 POINTS]

3 3. The figure shows a small element of material in a deformable solid before and after deformation. Determine the components of the D Lagrange strain tensor (eg as a x matrix). (c) e (c) (b) (a) e e (c) L (a) (b) e l l We apply the identity = E ijmm i j, taking m to be the two sides of the square and the l diagonal. This gives E E [ ] E E E = = E E [ ] E E E = = E E 4 [ ] ( E E E) E E = + + = 4 E = E = E = / (Strictly speaking this is not a physically admissible deformation since the volume of the deformed solid goes to zero, but the numbers were chosen to make the arithmetic easy!) [5 POINTS] 3

4 4. For the stress tensor please calculate 4. The hydrostatic stress 4 3 σ = σ h = trace( σ ) / 3 = 6 / 3 = 4. The principal stresses and their directions det( σ λi) = ( λ)( λ λ ) 6 (4 )(4 + ) 3 = λ 5 = λ =± 5 So [ σ, σ, σ 3] = [6,5, 5] The principal stress directions are the null vectors of 4 λ λ 6 λ 3 m = [,,] m = [3,9, ] / 9 m = [ 9,3, ] / 9 [3 POINTS] 4.3 The tractions acting on a plane with normal n= ( e+ e + e 3)/ n σ = [,,] 3 4 [,7,6] / 3 3 = 6 4

5 5. The figure shows a cross-section U through a joint connecting two hollow e r D eθ cylindrical shafts. The joint is a hollow C cylinder with external radius b and Joint b internal radius a. It is bonded to the e z two rigid shafts AB and CD. Shaft AB B is fixed (no translation or rotation), and an axial displacement u= Ue z is a A Rigid applied to the hollow cylinder CD. Rigid L The goal of this problem is to estimate the axial force necessary to produce displace the shaft CD, and hence determine the stiffness of the joint. 5. Assume that the displacement field in the joint can be approximated by u= ur () e z, where ur () is a function to be determined. Calculate the infinitesimal strain tensor in the joint in terms of ur () and its derivatives (you can assume that the gradient of a vector in cylindrical-polar coordinates is vr vr vθ vr r r θ r z v v v ( ) r v vr r v v θ θ e + θ θeθ + ze z + r r θ r z vz vz vz r r θ z T The strain is ε= { u+ ( u ) } Substituting the given displacement gives du dr ε du dr 5. Assume that the joint can be idealized as an isotropic, linear elastic material with Youngs modulus E and Poisson s ratio ν. Find a formula for the stress in the joint a<r<b, in terms of derivatives of ur () and the material properties. The only nonzero stress component is σ zr Eε zr E du = = ( + ν) ( + ν) dr 5

6 [ POINTS] 5.3 Use the equation of static equilibrium σ = to show that ur () must satisfy d u du d du + r = = dr r dr r dr dr You can use the formula for the divergence of a tensor S in cylindrical-polar coordinates Srr Srr Sθ r SzR Sθθ r r r θ z r Sθθ Srθ Srθ Sθ r S S zθ r θ r r r z Szz Srz Srz S θ z z r r r θ Substituting the stress from the previous problem into the third equation (the first two are zero) gives σzr σzr E d u du + = + = r r ( + ν ) dr r dr [ POINTS] 5.4 Write down the boundary conditions for displacements and/or stresses on the four external surfaces of the joint (i.e. give any known values for displacement components ur, uθ, uz, or stress σ, σ, σ, σ, σ, σ ) rr θθ zz rθ rz zθ On () and (4) σ = σ = σ = zz zθ rz () (3) C eθ e r B () (4) On () u = u = u = r θ z On (3) ur = uθ = uz = U [3 POINTS] 6

7 5.5 Find a solution for ur () that satisfies 5.3 and boundary conditions on r=a and r=b. Does the solution satisfy all the boundary conditions in 5.4? If not, which boundary conditions are not satisfied? d du du du C r = r = C = u = A+ Clog( r) r dr dr dr dr r The boundary condition at r=a gives A= Clog( a) u = Clog( r/ a) The boundary condition at r=b gives U = Clog( b/ a) C = U / log( b/ a) Hence U u = log( r/ a) log( b/ a) All the boundary conditions are satisfied except that on () and (4) σ rz [ POINTS] 5.6 Find a formula for the axial force F z that must be applied to shaft CD to cause the necessary axial displacement U. Hence, find a formula for the stiffness of the joint. The force can be found by integrating the traction exerted by the joint on the rigid outer cylinder. The force magnitude is E du E U π LEU Fz = σzrda = πbl = πbl = ( + ν) dr ( + ν) log( b/ a) b ( + ν)log( b/ a) A r= b The stiffness is therefore π LE k = ( + ν )log( b/ a) [ POINTS] 7