Homework 5

AE 321 Homework 5 Due in class on October 4 1.

The components of a displacement field are (in meters):

u x

 x2  20 104 ,



3

u y  2 yz  10 , u z   z 

2

 xy 103,

(a) Consider two points (2, 5, 7) and (3, 8, 9) in the undeformed configuration. Find the change in distance between these points. (b) Compute the components of the Lagrangian and the infinitesimal strain tensors. (c) Compute the components of the rotation tensor. (d) Compute and compare the Lagrangian strains and infinitesimal strains at location (2, -1, 3). (e) Show that this displacement field satisfies:  

ij ,kl 

  kl ,ij   ik , jl     jl ,ik   0

The equation above is called Compatibility Equation.  This equation must be satisfied by any valid strain field.

2. Assume

a “candidate” strain field is given by

 Ax22   sym.  sym.

  

ij

 Bx1 x2 2

 Ax1

0 0 0 

Find the relationship between A and B such that this is a valid strain field.

3. 

Compute the infinitesimal strain tensor and sketch the deformed shape of an initially

rectangular volume element aligned with the axes under the following displacement fields (A, B, are constants):

 

(a) Simple extension: u1

  Bx2,

(b) Simple shear: u1 4. The

x1, u2   u3  0 u2   u3   0

components of a strain tensor referred to the coordinate frame in the figure are 1

0.02 0.003  0.01  sym.

  

ij

 0.02 0.01 0

and are constant in the region shown. The direction cosines of AC are (1/√2, 0, -1/√2) and of BD are (-1/√6, √2/√3, -1/√6). x2 B

 A O

C

x1 D

x3 Find: (a) The extension ratios of lines AC and DB. (b) The change of the initially right angle ADB. (c) The three invariants of this strain. (d) The principal strains and principal directions.

2