Solid Mechanics

Abstract

This chapter provides an overview of solid mechanics with applications in biomechanics. It begins wih a coverage of tensors, including the tensor transformation law and tensor invariants, before proceeding to the basic mechanical concepts of stress and strain. Stress and strain are then related via material constitutive laws, the most basic being that of linear elasticity. The concepts of hydrostatic stresses and strains are also introduced, along with the notion of von Mises stress as a representative scalar stress indicator. Other material laws relevant to biological tissues are presented, including linear viscoelasticity and hyperelasticity, as well as anisotropic hyperelastic material laws. Detailed examples of models solved in COMSOL include a strap tension testing device for a respirator mask, as well simulations of simple shear experiments in myocardial tissue. The chapter ends with a set of theoretical and computational COMSOL problems, with fully-worked answers provided in the solution section of the text.

Problems

8.1

(a) Show that the triple scalar product $(\mathbf {a}\times \mathbf {b})\cdot \mathbf {c}$ is given by

$$\begin{aligned} (\mathbf {a}\times \mathbf {b})\cdot \mathbf {c} = \left| \begin{array}{ccc} a_1 &{} a_2 &{} a_3 \\ b_1 &{} b_2 &{} b_3 \\ c_1 &{} c_2 &{} c_3 \\ \end{array} \right| = \varepsilon _{ijk}a_ib_jc_k \nonumber \end{aligned}$$

where $\varepsilon _{ijk}$ is the permutation symbol (a third-order tensor) defined by

$$\begin{aligned} \varepsilon _{123}= & {} \varepsilon _{231} = \varepsilon _{312} = 1 \nonumber \\ \varepsilon _{321}= & {} \varepsilon _{132} = \varepsilon _{213} = -1 \nonumber \\ \varepsilon _{ijk}= & {} 0 \quad (\mathrm {if~any~two~of~}i, j, k \mathrm {~are~equal}) \nonumber \end{aligned}$$

(b) Verify that $\sigma _{ij}^{\mathrm {H}}\varepsilon _{ij}^{\mathrm {D}} = \sigma _{ij}^{\mathrm {D}}\varepsilon _{ij}^{\mathrm {H}} = 0$.

8.2

Determine the Cauchy and Green strain tensors for the following deformations:

(a) Simple shear:

(b) Uniform inflation:

(c) Rotation:

8.3

Consider the left ventricle to be approximated by a spherical shell with inner and outer radii A and B. A uniform pressure is applied to the endocardial (i.e. inner) surface such that the ventricle passively expands to new inner and outer radii of a and b respectively.

(a) Assuming that the myocardium is incompressible, find b as a function of a, A, and B.

(b) Determine the resulting Cauchy strain tensor in the ventricular wall.

8.4

A $100\times 100\times 1$ mm square piece of rabbit skin tissue is subjected to a tensile test in the laboratory as shown schematically below:

Assume that the constitutive law of the skin tissue obeys a Mooney–Rivlin relationship with material parameters $C_1$ and $C_2$ equal to 20 kPa, density $\rho$ = 1100 kg m$^{-3}$ and bulk modulus $K = 1\times 10^6\,\mathrm {kPa}$ . Assume that the leftmost edge of the tissue is clamped (fixed) in place, and that the rightmost edge is subjected to a prescribed displacement only in the x-direction. All other edges are free.

(a) Use COMSOL’s stationary solver to simulate this tensile test and plot the applied force magnitude F (in mN) against skin displacement from 0 to 50 mm.

(b) Plot the spatial distribution of strain energy in the deformed skin sample at its maximum displacement of 50 mm.

8.5

The left ventricle of the heart can be approximated by a circular half-ellipsoidal shell, with endocardial semi-axes of $a_{\mathrm {endo}}$ and $b_{\mathrm {endo}}$, and epicardial semi-axes of $a_{\mathrm {epi}}$ and $b_{\mathrm {epi}}$, as shown in the figure below:

Assuming the myocardium follows a two-parameter Mooney–Rivlin strain energy formulation , use COMSOL to simulate passive inflation of the venticle using a 2D axisymmetric implementation , and determine endocardial volume in ml as a function of filling pressures from 0 to 50 mmHg. Use the model parameters in the table below, and employ a stationary solver for each pressure. Assume the upper boundary (base) of the ventricle is held fixed, and take the external pressure on the epicardial surface to be zero .

Parameter	Value	Description
$a_{\mathrm {endo}}$	2 cm	Endocardial radius
$a_{\mathrm {epi}}$	3 cm	Epicardial radius
$b_{\mathrm {endo}}$	6 cm	Endocardial long semi-axis
$b_{\mathrm {epi}}$	7 cm	Epicardial long semi-axis
$\rho$	1200 kg m$^{-3}$	Myocardial density
K	$1\times 10^6$ kPa	Bulk modulus
$C_1$	3 kPa	Mooney–Rivlin material parameter
$C_2$	5 kPa	Mooney–Rivlin material parameter