5. Consider a viscoelastic tissue for which the stress-strain relation is -Be +

Question

5. Consider a viscoelastic tissue for which the stress-strain relation is -Be + the tissue's modulus and viscosity, respectively. You apply a constant stress only for a short time, de _ . dt Here, B and n are between 0 ando. As a result, the tissue will gradually deform while the stress is applied, and then relax back to zero strain after the stress is removed: A) Find the constants Ao, Al, and , and plot (t) and (t). B) Plot the deformation on a stress-strain diagram for t20 C) Starting with Jod: J _dt (or using a more creative approach), find the overall work dt performed on the tissue. The resting volume of the tissue is Vo

Explanation / Answer

5. consider

sigma = B*epsilon + n*d(epsilon)/dt

sigmao is applied for 0 < t < to

epsilon = Ao(1 - exp(-t/tau))

hence

sigma = B*Ao(1 - exp(-t/tau)) + n*Ap(exp(-t/tau)/tau)

A. now at t = 0, strain = 0

hence

Ao(1 - 1) = 0

at t= to, strain = Ao(1 - exp(-to/tau))

also

for t = to ( from continuity of strain)

strain = A1*exp(-to/tau)

hence

comparing

A(1 - exp(-to/tau)) = A1*exp(-to/tau)

also

at for 0 < t < to

sigmao = B*Ao(1 - exp(-t/tau)) + n*Ao(exp(-t/tau)/tau)

d(sigmao)/dt for this time = 0

B*Ao(0 + exp(-t/tau)/tau) + n*Ao(-exp(-t/tau))/tau^2 = 0

hence

BAo = nAo/tau

B*tau = n

tau = n/B

sigmao = B*Ao(1 - exp(-tB/n)) + B*Ao(exp(-tB/n))

Ao = sigmao/B

A1 = sigmao(exp(toB/n) - 1)

hence

sigma(t) = sigmao(1 - exp(-t/tau)) + sigmao(exp(-t/tau))

sigma(t) = sigmao, for 0 < t < to

sigma(t) = 0 for t > to

and

epsilon(t) = sigmao(1 - exp(-Bt/n))/B ( for 0 < t < to)

epsilon(t) = sigmao(exp(Bto/n) - 1)exp(-tB/n) = sigmao(exp(-B(t - to)/n) - exp(-tB/n))

the plots are as under

B) deformation = dV

dV = V*epsiloin(t) = Vo*sigma(1 - exp(-Bt/n))/B for 0 < t < to

and

for t > to

dV = V*epsilon = Vo*sigmao(exp(-B(t - to)/n) - exp(-Bt/n))

the plot is asme as strian curve freom the last diagram

C) now

work performed = inte3gral(sigmad(epsilon))

W = B*epsilon^2/2 - n*sigmao*(exp(-to/tau) - 1)/B

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