Given the strain energy potential psi E lambda2 tr E2 mu t

Given the strain energy potential psi (E) = lambda/2 (tr E)^2 + mu tr (E E), determine the second Piola-Kirchhoff stress tensor S in terms of the Green strain tensor E.

Solution

solution:

1) here we are provided with strain energy density potential function or strain energy per unit volume function in term of strain tensor as

here psi=f

strain tensor=E

stress tensor=S

where constant=

lamda=m

mu=n

so strain potential function as

f=m/2*E^2+nE^2

where relation between stress and strain tensor as

s=df/dE

hence on differentiating above eqution woth respect to strain tensor, we get

df/dE=(d/dE)(m/2*Etranslational^2)+(d/dE)(nElateral^2)

m and n are constant so we get

S=mEtrans+2*n*Elateral

in this way above equation gives value of stress tensor and obeys hooks law which is

f=1/2*S*E