4. Let N be a unit vector in reference configuration. Its deformed configuration at time t>...

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4. Let N be a unit vector in reference configuration. Its deformed configuration at time t> 0 is n(t) = X(t)f(t) where A is the stretching of the deformed vector and n is the unit vector representing the orien- tation of the deformed vector. Suppose its deformation gradient is Fij(t) = F(t) and the velocity gradient is vij(t) = H (t). We use the symbol to express the material time derivative. (a) Calculate the material time derivative of velocity gradient tensor. (Hint: F = HF and F-¹F = I) (b) Please proof that ₁+= + A = HA or £₁; + +A; = V₁.j^j 4. Let N be a unit vector in reference configuration. Its deformed configuration at time t> 0 is n(t) = X(t)f(t) where A is the stretching of the deformed vector and n is the unit vector representing the orien- tation of the deformed vector. Suppose its deformation gradient is Fij(t) = F(t) and the velocity gradient is vij(t) = H (t). We use the symbol to express the material time derivative. (a) Calculate the material time derivative of velocity gradient tensor. (Hint: F = HF and F-¹F = I) (b) Please proof that ₁+= + A = HA or £₁; + +A; = V₁.j^j