A particular biological tissue can be modeled with the incompressible neo-Hookea

ID: 181403 • Letter: A

Question

A particular biological tissue can be modeled with the incompressible neo-Hookean model, so that the strain-energy function can be written as W = mu/2 (lambda_1^2 + lambda_2^2 + lambda_3^2 - 3), lambda_1 lambda_2 lambda_3 = 1 where mu is a material parameter and lambda_1, lambda_2, lambda_3 are the principal stretches. Under uniaxial loading the deformation of the body is given by x = lambda_1 X, y = lambda_2 Y, z = lambda_3 Z, where the coordinates are coincident with the principal axes, and lambda_2 = lambda_3. a) Calculate the deformation gradient [F] in terms of lambda_1 only. b) Given the principal, second Piola-Kirchhoff stresses S_a = 1/lambda_a partial differential W/partial differential lambda_a, a = 1, 2, 3, find the maximum shear stress tau_max = (simga_1 - simga_3) in terms of lambda_1 and the material parameter mu .

Explanation / Answer

We have given Incompressible neo-Hookean model for principal stretches

W = u/2 (12 + 22 + 32 -3) where 1 x 2 x 3 = 1

Because 1 x 2 x 3 = 1 so

1 > 2 = 3 but not equals to 1

The linear consistency (u) can be converted to material constant C1 which is u/2.

W = C1 (12 + 22 + 32 -3) equation 1

To calculate strain energy density function W we use the formula

W= C1 (I1- 3)+ D1(J-1)2 , now from equation 1

I1 = 12 + 22 + 32

I1 is the first invariant which calculated by J-2/3

J = det(F) = 1 x 2 x 3 where F is deformation gradient, and J is Jacobian determinant

Here J is 1 because 1 x 2 x 3 = 1

For this Incompressible neo-Hookean model, we use the equation to calculate Cauchy stress differences ():

Cauchy stress differences = 2 x Piola- Kirchoff stresses (S)

1 – 3 = 1 x (W/1) – 3 x(W/3)

W/1 = [ u/2 (12 + 22 + 32 -3)] /1 = 2C1 1

So by putting all values we get

1 – 3 = 1 x 2C1 1 – 3 x 2C1 3

1 – 3 = 2C1 12 -2C1 (1/ 11/2) [1 > 3 and 132 =1, so 3 = 1/ 11/2]

1 – 3 = 2C1 [12 -(1/ 11/2)

1 – 3 = u [12 -(1/ 11/2) [ C1 = u/2]