Continuum Mechanics Examination exercises

Transcription

1 Continuum Mechanics Examination exercises Name:... Result: 1:1... 1:... 1:3... :1... :... :3... :4... 3:1... 3:... 3:3... 4:1... 4:... 5:1... 5:... LTH, Box 118, S-1 00 LUND Sweden Tel: +46 (0) Fax: +46 (0)

2 Problem 1:1 A homogenous deformation of a cubic body with side length l=1m is given by x= ax ( + Y) y = by z = cz where a, b and c are positiv contants. Determine, a) the length of the diagonale OC after the deformation.(see the figure) b) the angel between OA and AC after the deformation, c) give a relation for the constants in order to have an isochoric deformation.

3 Problem 1: The displacement field for a deformation is given by u = AYZ v = AZ w = AX where A is a small constant. Determine, a) the Green deformation tensor C b) the infinetisimal stretching tensor E 0 and the corresponding rotation tensor R 0. c) for the position r = ( 1,1,0) in the undeformed configuration, calculate the principle stretches for C using the exact expression and the approximate expression C=1+ E 0.

4 Problem 1:3 A deformation i given by x= 3X + Y y = Y z = Z a) Determine the deformation tensors V and U. b) Show by a direct calculaton that the eigen values of V and U are the same.

5 Problem :1 The motion of a material body is given by, x = X + Yt y = Y + Xt z = Z a) Find the components of the velocity at t = 1,5 (sek) for the particle that was found at (,3,4) at t = 1 (sek). b) Give the components for the acceleration of the same particle at t = (sek).

6 Problem : The temperature field in a fluid flow is given by the relation 3t e Θ=Θ ( xyz,, ) = where r = x + y + z r The velocity field for the flow is given by, v = y+ z, v = z x, v = x+ 3y x y z Determine the material time derivative for temperture field.

7 Problem :3 A velocity field is given by v = txsin z, v = ty cos z, v = 0 x y z Determine at t = 1 (sek) in the position (1,-1,0), a) the rate of deformation tensor D and the vorticity tensor W, b) the rate of stretching for the material line element having the direction n= ( e1+ e + e 3)/ 3 c) the largest rate of stretching and the direction in the case, d) the vorticity vector.

8 Problem :4 Show that the velocity field describes v = 1.5z 3 y, v = 3 x z, v = y 1.5x x y z a) an isochoric motion, and b) a rigid rotation. What will the vorticity vector ω be?

9 Problem 3:1 In a given position in a material body the stress tensor is given by, T = a) Determine the stress vector to the surface element having the normal direction 1 n= ( e1+ 4 e + 8 e 3 ) 9 b) Determine the magnitude of the stress vector. c) Give the normal stress component. d) Determine the shear stress for the directions n and m where m= 1 ( e ) e 3 5

10 Problem 3: The stress tensor in a Cartesian system (O,x,y,z) is given by, 3 1 T = 3 3 (1 x ) y+ y (4 y ) x 0 3 (4 y ) x ( y 1 y) (3 x ) a) Show that the body is in equilibrium if the body forces can be neglected. b) Determine the stress vector in the position (,-1,6) for a surface element given by 3x+6y+z=1. y

11 Problem 3:3 The stress tensor for a material body is given, 4 b b T = b 7 b 4 where the constant b is unknown. If you know that one of the principle stresses is 3 MPa and that we have for two remaining principle stresses the relation σ1 = σ, then determine a) the constant b, b) the principle stresses and the principle directions.

12 Problem 4:1 A linearly viscous, incomprerssible fluid with the density ρ and viscosity coefficient µ is flowing steady between two horisontal plates. The upper plate is moving with a constant speed u 0 in the x-direction (see figure), while the lower plate is in rest. The distance between the plates is z 0 and the velocity field is r = v(z) ex Determine the pressure drop, as a function of u 0, if the mass flow through the cross section A (see figure) is zero. (The body forces can be neglected.) Give the velocity field in a diagram.

13 Problem 4: In a Couette flow sometimes climbing can be observed (see figure in the lecture notes). To be able to describe the phenomenon the stress tensor in the flowing fluid is assumed to be given by, T= p 1+ µ D+ βd where the parameters µ and β are considered as constants. Use cylindrical coordinates (, r θ, z), and write down Navier-Sokes equations in cylindrical cordinats, then try to find a restriction on β so that the normal traction on the inner cylinder will be greater than the normal traction on the outer cylinder in the Coutte-flow considered. Assume that the normal traction is approximately equal to the T rr component of the stress tensor. The radii of the inner and outer cylinders are R 1 and R, angular velocities to be ω and 0, neglect boundary effects and the body forces. Assume constant pressure in the r-dir.

14 Problem 5:1 A non-homogeneous deformation of a material body, which is constituted by the relation Tr () = p() r1+ CBr (), is given by, 1 1 ( ) x= BX + Y Y y = arctan X z = AZ ( AB = 1) (In the common notations: T(r) stress tensor, B(r) the left Cauchy-Green deformation tensor,p(r) the pressure, C 1 material coefficient, r:(x,y,z) coordinates in the deformed configuration, r 0 : (X,Y,Z) coordinates in the undeformed configuration and A,B are positive constants. The deformation can be characterized as, that a section is cylindrical shell is straighten out and stretched. a) Determine the deformation tensor B=B(r) b) Give the stress tensor T=T(r)

15 Problem 5: A cylinder of an elastic material has in the undeformed configuration the radius R and the symmetry axis along the Z-axis. The body is deformed and the deformation is given by: x= µ ( X cos( Y Z) + Ysin( YZ)) y = µ ( X sin( Y Z) + Ycos( YZ)) z = λ Z ( µλ,, Y constants) a) Give a relation between the constants, if the deformation is given for an incompressible elastic body b) Determine the normal components of the stress vector on the end surfaces of the 1 cylinder if the material of the body is constituted by T= p1+ C1B C B, where C 1 and C are material constants. (You can use a software such as MAPLE to invert the matrix)