In the case of plane stress, where the in-plane principal strains are given by ε1 and ε2, show that the third principal strain can be obtained from
where v is Poisson’s ratio for the material.
To show that: The third principal strain can be obtained from
Answer to Problem 1RP
The third principal strain can be obtained from
Explanation of Solution
Given information:
The third principal strain
Explanation:
For the case of plane stress
Apply the normal strain in x direction as shown below.
Here, E is the modulus of elasticity,
Substitute 0 for
Multiply both sides of the Equation by
Apply the normal strain in y direction as shown below.
Substitute 0 for
Apply the normal strain in z direction as shown below.
Substitute 0 for
Adding Equation (1) and (2).
Substitute
Substitute
Therefore, the third principal strain can be obtained from
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