The state of strain at a point \(P\) in a structure is

a. Compute the extensional strain in the direction \(\mathbf{n}=\left\{\begin{array}{lll}1 & 1 & 1\end{array}ight\}^{\mathrm{T}}\).

b. If \(\hat{\mathbf{n}}_{1}=\left\{\begin{array}{lll}1 & 1 & 0\end{array}ight\}^{\mathrm{T}} / \sqrt{2}\) and \(\hat{\mathbf{n}}_{2}=\left\{\begin{array}{lll}-1 & 1 & 0\end{array}ight\}^{\mathrm{T}} / \sqrt{2}\) are principal directions, compute the three principal strains at this point.

c. Compute the principal stresses at this point if Young's Modulus E \(=2 \times 10^{11} \mathrm{~Pa}\) and Poisson's ratio \(v=0.3\). (Use principal strains from part (b)).

d. Can the state of stress at this point be described as plane stress? Explain.

e. Compute the strain energy density at the point \(P\).

f. If the yield stress of the material of the plate is \(\sigma_{y}=2 \times 10^{8} \mathrm{~Pa}\), has this plate undergone plastic deformation due to the applied strain? Use maximum shear stress criterion (Tresca's law) to answer this question.

Transcribed Image Text:

[0.125 .25 0 [] = .25 0.1250 x 10-2 0 0 .1