Mathematica Assignment and Solution

Connect to Mathematica using the following procedure:

1. login on an HP workstation

2. Open a shell (window)

3. Type xhost+

4. type rlogin mxsgl

5. On the newly opened shell, enter your password first, and then type setenv DISPLAY xxx:0.0 where xxx is the workstation name which should appear on a small label on the workstation itself.

6. Type mathematica &

and then solve the following problems:

1. The state of stress through a continuum is given with respect to the cartesian axes 0x1x2x3 by

3xix2

5x2

0

T ■ =

5x2

0

2x3

MPa

0

2x3

0

Determine the stress vector at point P(1,1, a/3) of the plane that is tangent to the cylindrical surface x2 + X3 = 4 at P.

2. For the following stress tensor

"6-30

(a) Determine directly the three invariants Ia, IIa and IIIa of the following stress tensor

(b) Determine the principal stresses and the principal stress directions.

(c) Show that the transformation tensor of direction cosines transforms the original stress tensor into the diagonal principal axes stress tensor.

(d) Recompute the three invariants from the principal stresses.

(e) Split the stress tensor into its spherical and deviator parts.

(f) Show that the first invariant of the deviator is zero.

3. The Lagrangian description of a deformation is given by xi = X1+X3(e2 — 1), x2 = X2 +X3(e2 — e-2, and x3 = e2X3 where e is a constant. SHow that the Jacobian J does not vanish and determine the Eulerian equations describing this motion.

4. A displacement field is given by u = X1X|e1 + XlX2e2 + X|X3e3. Determine independently the material deformation gradient F and the material displacement gradient J and verify that J = F—I.

5. A continuum body undergoes the deformation x\ = X1, x2 = X2 + AX3, x3 = X3 + AX2 where A is a constant. Compute the deformation tensor C and use this to determine the Lagrangian finite strain tensor E.

6. A continuum body undergoes the deformation xi = Xi + AX2, x2 = X2 + AX3, x3 = X3 + AX2 where A is a constant.

(a) Compute the deformation tensor C

(b) Use the computed C to determine the Lagrangian finite strain tensor E.

(c) COmpute the Eulerian strain tensor E* and compare with E for very small values of A.

7. A continuum body undergoes the deformation x1 = X1 + 2X2, x2 = X2, x3 = X3

(a) Determine the Green's deformation tensor C

(b) Determine the principal values of C and the corresponding principal directions.

(c) Determine the right stretch tensor U and with respect to the principal directions.

(d) Determine the right stretch tensor U and U_1 with respect to the ej basis.

(e) Determine the orthogonal rotation tensor R with respect to the ej basis.

8. A continuum body undergoes the deformation x1 = 4X1, x2 = —iX2, x3 = —1X3 and the Cauchy stress tensor for this body is

100 0 0

(a) Determine the corresponding first Piola-Kirchoff stress tensor.

(b) Determine the corresponding second Piola-Kirchoff stress tensor.

(c) Determine the pseudo stress vector associated with the first Piola-Kirchoff stress tensor on the ei plane in the deformed state.

(d) Determine the pseudo stress vector associated with the second Piola-Kirchoff stress tensor on the ei plane in the deformed state.

9. Show that in the case of isotropy, the anisotropic stress-strain relation cAniso cijkm

c1111

c1112

c1133

c1112

c112s

CllSl

c2222

c22ss

c2212

c222s

C2231

cssss

css12

css2s

C3331

c1212

c122s

c12s1

SYM.

c2s2s

C1122 Cllss C2222 C2233 C3333

reduces to cijkm c1111

C1122 Cllss C2222 C2233 C3333

SYM.

0 0 0 0 0 0 0 0 0 a 0 0 b0 c with a = 1 (ciiii - C1122), b = 2(c2222 - c2233), and c = 2(c3333 - cn33). 10. Determine the stress tensor at a point where the Lagrangian strain tensor is given by

30

B0

20

B0

40

0

X 10-6

20

0

30

and the material is steel with A = 119.2 GPa and ¡ = T9.2 GPa.

0 0

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