Consider a cubic workpiece of rigid perfect plastic material with side length lo. The cube is deformed plastically to the shape of a rectangular parallelepiped of dimensions l1,l2, and l3. Show that the volume constancy requires that the following expression be satisfied: e1+e2+e3 = 0

 ε₁+ε₂+ε₃ = 0

Explanation: Considering the initial and final volume to be constant which gives rise to the relation:-

                         l₀l₀l₀=l₁l₂l₃

                      taking natural log on both sides

Considering the logarithmic Laws of division and multiplication :

                                ln(AB) = ln(A)+ln(B)

                                ln(A/B) = ln(A)-ln(B)

Use the image attached to see the definition of true strain defined as

                         ln(l1/1o)= ε₁

which then proves that ε₁+ε₂+ε₃ = 0

answer:

put more,