GG 5210/6211

Problem Set #6

Due October 23, 2006

 

Strain Problems

 

Note: where applicable use your Mat Lab scripts for the principal stress determinations:

 

1.  A linear (small strain) deformation field is specified by:

• u₁ = 4x₁ - x₂ + 3x₃,
• u₂ = x₁ + 7x₂, and
• u₃ = -3x₁ + 4x₂ + 4x₃

For this deformation:

1. Determine the principal extensional strains, ε_n, and
2. The principal strain deviator for this strain system.

2. A homogeneous deformation field results in the infinitesimal strain tensor of:

 

 

a. Determine the principal strains and their directions,

b. Decompose this tensor into it isotropic and deviatoric parts.

 

3. GPS measurements measures longitudinal strain along the orthogonal and cross axes as shown below.  Note the 45° component.

• ε₁₁ = 5.0x10^-4,
• ε'₁₁ = 4.0x10^-4, and
• ε₂₂ = 7.0x10^-4

1. Determine the shear strain, ε₁₂, at the point P.
2. Strain measurements require an additional component, such as at 45^o to the orthogonal base lines, to resolve the total strain field. Explain why.

 

4. a. Construct a Mohr's circle for the case of plane strain:

 

        

 

b. Determine the maximum shear strain for this case.

c. Verify the results analytically (using your MatLab scripts).

 

 

 and

 

5. A line segment dx lies in the (x₁,x₂ ) plane. Show how the elongation is given by:

 

 

where θ is the orientation of the baseline with respect to the x₁ axis.  Given the strain

 

graph Δl/l between 0 and π. What is the meaning of the maximum and minimum?