ME 601 – Stress Analysis

Homework 3 – Due Date: 12th August 2015

1. a) Show that, dual vector of an antisymmetric tensor T is

𝑡!! = −12𝜖!"#𝑇!"𝒆!

Show that this expression can be inverted to get Ajk ijk iT tε= −

b) Decompose the following tensor into a symmetric and antisymmetric

parts. Also find the dual vector for the antisymmetric part. Verify 𝑇!𝑎 = 𝒕!×𝒂 for 𝒂 = 𝒆𝟏 + 𝒆𝟑

𝑇 =1 2 34 2 11 1 1

2. For the following tensor

𝑇 =5 4 04 −1 00 0 3

a) Find the scalar invariants, the principal values and corresponding principal directions of the tensor 𝑇.

b) If n!, n!, n! are the principal directions, write [𝑇]!!,!!,!! c) Could following matrix represent the tensor 𝑇 in respect to some

basis?

𝑇 =7 2 02 1 00 0 −1

3. Show that

a) For a real symmetric tensor there always exist three principal directions which are mutually perpendicular.

b) The principal values of a tensor T include the maximum and minimum values that the diagonal elements of any matrix of T can have.

4. What is the gradient of a scalar and vector field, derive it? Show that divergence of a vector field is the trace of gradient of vector field.

5. The angles between the respective axes of the 𝑥′!𝑥′!𝑥′! and the 𝑥!𝑥!𝑥! Cartesian systems are given by the table below.

Determine the transformation matrix between the two sets of axes, and show that it is a proper orthogonal transformation.

6. The Lagrangian description of a deformation is given by: ( )( )

2 21 1 3

22 2 3

23 3

X X

X X 1

X

x e e

x e

x e

−= + −

= − −

=

Determine the components of ,i jF and check if it is symmetric or not. Also show that the Jacobian J does not vanish. Obtain the Eulerian description of the deformation.

7. Show that the six compatibility equations may also be represented by the three independent fourth-order equations,

8. Show that the following strain field gives continuous, single valued displacements in a simply connected region only if the constants are related by A = 2B/3.

9. A three-dimensional elasticity problem of a uniform bar stretched under its

own weight gives the following strain field,

where A and B are constants. Integrate the strain-displacement relations to determine the displacement components and identify all rigid-body motion terms.

10. Show that the isotropic second-rank tensor is given by, ij ijT λδ=

11. Show that an arbitrary tensor A can be expressed as the sum of a spherical

tensor (i.e., a scalar times the identity tensor) and a tensor with zero trace (trace A = Aii). Prove that this decomposition is unique, and that Aʹ′ , the traceless part of A, is given by, Aʹ′ = A – 1/3(tr A)I, where Aʹ′ is the deviator of A.

12. The deformation of a body is given by

x1  =λ1X1,  x2  =-­‐λ3X3,  x3  =λ2X2  

(a) Find the deformed volume of the unit cube shown in Figure. (b) Find the deformed area OABC. (c) Find the rotation tensor and the axial vector of the antisymmetric part of

the rotation tensor

 

13. (a) Determine the principal scalar invariants for the strain tensor given here at left and (b) Show that the matrix given at the right cannot represent the same state of strain.

   

14. Given x1=X1+3X2 , x2 =X2 , x3=X3. Obtain (a) the deformation gradient F and the right Cauchy-Green tensor C, (b) the eigenvalues and eigenvector of C, (c) the matrix of the stretch tensor U and U-1 with respect to the ei-basis and (d) the rotation tensor R with respect to the ei -basis.

15. Given the following large shear deformation:

x1=X1+2X2, x2=X2, x3=X3 (a) What is the stretch for the element that was in the direction e2? (b) Find the stretch for an element that was in the direction of e1 + e2. (c) What is the angle between the deformed elements of dS1e1 and dS2e2? (d) Show that the eigenvalues of any given symmetric tensor are real.

16. Are the following two velocity fields isochoric (i.e., no change of volume)? 1 1 2 2

2

x e x evr+

= , 2 2 21 2r x x= +

2 1 1 22

x e x evr

− += , 2 2 2

1 2r x x= +  

17. Derive the strain vs. displacement relations in cylindrical polar coordinates in 3D.

18. Derive the strain vs. displacement relations in spherical coordinates in 3D.