Mechanical Properties of the Body

Abstract

The composition and structure of the structural components of the body, including bones, ligaments, tendons, and cartilage, are first presented. The mechanical properties of these body components begin by investigating their stress-strain relationships in the harmonic regime. Analysis is improved by modeling their nonlinear, time-dependent properties and then their time-dependent, viscoelastic properties. These models are used to understand how bones can bend under unusual conditions, and the occurrence of fractures and sports injuries, along with ways to avoid these unwanted events.

Problems

Stress and Strain

4.1

Determine the spring constant, k, in SI units for a solid cylinder of cortical bone of length 0.5 m, diameter 2 cm, and $Y = 17.4$ GPa.

4.2

A cylindrical spring of length 2 cm and diameter 3 mm has spring constant $k=1.7\times 10^{5}$ N/m.

(a) How much does it extend when a force of 100 N is applied to it?

(b) What is the strain?

(c) The spring is composed of a uniform material. Find its Young’s modulus (in MPa).

4.3

Equation (4.8) shows how Young’s modulus, the shear modulus, and Poisson’s ratio are interrelated for an elastic isotropic material. The bulk modulus B is the negative of the pressure divided by the fractional change in volume caused by that pressure, and it can be related to any two of these three above parameters. Show that for isotropic harmonic materials $Y=3B(1-2\upsilon )$. (When a stress $\sigma$ is applied to compress a cylinder, the pressure is $\sigma /3$.)

4.4

Calculate the strain and change of length of the femur during a single step while running, using the data near (4.21).

4.5

As we will see in Chap. 11, we are able to see objects that are both near and far from us because of accommodation in the eye. This occurs because the shape of the crystalline lens in the eye changes when the force on it from the suspensory ligaments is changed. This is a fairly complex three-dimensional problem [19–21], which we will simplify. (It is actually a two-dimensional problem because of rotational symmetry, but it is still complex.)

(a) Use the simple one-dimensional model in this chapter to estimate how much stress is on the crystalline lens if it has a Young’s modulus of $1 \times 10^{3}$ Pa (which is that for a 20 year old) and has a strain is $3\%$.

(b) If the total force on the lens is 0.002 N, determine the effective contact area (in mm$^{2}$).

(c) Determine the strain in the lens of someone who is 60 years old, if the stress is the same as in (a), but the Young’s modulus of the person has increased to $3 \times 10^{3}$ Pa.

4.6

Use the information in Table 4.2 to find the stress (in MPa) needed to stretch femoral compact bone, nails, nerves, skin, and coronary arteries to a strain of 0.01. (Assume harmonic behavior, with the low-strain value of Y.)

4.7

Use the information in Table 4.2 to find the strain resulting when femoral compact bone, nails, nerves, skin, and coronary arteries are subjected to a stress of 0.5 MPa. (Assume harmonic behavior, with the low-strain value of Y.)

4.8

Find the energy density for each case in Problem 4.6.

4.9

Find the energy density for each case in Problem 4.7.

4.10

Use the information for the fibula in Fig. 4.18 to:

(a) Calculate the maximum tension a bone with a cross-sectional area of 4 cm$^{2}$ could withstand just prior to fracture.

(b) Determine the elongation of a bone whose initial length is 0.35 m under the maximum tension from part (a).

(c) Calculate the stress on this bone if a tension force of $10^{4}$ N were applied to it [10]. How much would this bone elongate?

Fig. 4.78

Model for the proximal half of the human tibia. (From [45].) For Problem 4.12

4.11

Calculate the energy stored in parts (a) and (c) of Problem 4.10 for a bone that is 0.5 m long, always assuming that $\sigma =Y\epsilon$ (even until fracture).

4.12

Determine the relative amounts of strain energy absorption in the cartilage and bone of the proximal half of the tibia (Fig. 4.78) when it is loaded uniformly over the articular surface by a compressive force. Ignore the fibula and assume the tibia consists of three parts: a hollow cylindrical diaphyseal segment of cortical bone, a solid metaphyseal segment of cancellous (trabecular) bone, and a solid disk-shaped cartilage layer. The dimensions are $a = 10$ mm, $b = 30$ mm, $c = 50$ mm, $e = 4$ mm, $f = 70$ mm, and $g =130$ mm. Assume each material is elastic, with a Young’s modulus of 20,000 MPa for cortical bone, 200 MPa for cancellous bone, and either 20 MPa or 200 MPa for cartilage. Calculate the total strain energy in each segment of the model and the fraction of the total energy in each segment. Do this for both measured parameters for cartilage [45].

4.13

As a biomedical engineer you have been assigned to design a replacement for a femur (which consists of compact and trabecular bone). Using the materials (other than bone) listed in Table 4.1 (and others if you like), what materials would you use? (Would you want to use materials that match the properties of bones? Why?)

4.14

Figure 4.79 shows a diagram of parts of a hip replacement, with acetabular and femoral implants and PMMA (poly(methyl methacrylate)). Use the discussion in Chap. 2 to show that the lines of action of the loads on the femoral head and the long axis of the femur do not line up. Also, show that this means that there will bending and twisting moments on the implants, in addition to axial compression.

Fig. 4.79

Schematic of the components of a hip replacement. (From [34].) For Problem 4.14

4.15

In the medieval torture device, the rack, the head is pulled apart from the feet.

(a) Is this compression or tension?

(b) Which body part is likely to break first?

4.16

Someone tells you that UPE $\times$ Y approximately equals the UTS for only a few organs and materials in Table 4.2, such as for femoral compact bone, but not for most of them. Explain why this is either true or false by using several specific examples. Why is this so?

4.17

Use the data in Fig. 4.19a to estimate a power-law relation between the Young’s modulus for bone and its porosity.

4.18

Refer to Fig. 4.21.

(a) Why is it reasonable that the small intestine can stretch more and more easily in the transverse direction than the longitudinal direction?

(b) Is it reasonable that the small intestine can stretch significantly at low stress levels?

(c) How much should the radius of the small intestine be able to change under reasonable stresses and is the UPE large enough for this to occur?

4.19

(advanced problem) Two very simple models of composite material are shown in Fig. 4.80 [45]. In both cases there are two elastic materials, with respective Young’s moduli $Y_{1}$ and $Y_{2}$ and volume fractions $\rho _{1}$ and $\rho _{2}$. In the Voigt composite model in (a) the materials are modeled as slabs in parallel so each material undergoes the same strain, while in the Reuss composite model in (b) the materials are modeled as slabs in series so each material bears the same stress. Show that the effective Young’s modulus for the Voigt composite material is

$$\begin{aligned} Y_{\mathrm {c,Voigt}}=\rho _{1}Y_{1}+\rho _{2}Y_{2}, \end{aligned}$$

(4.98)

while for the Reuss composite model it is

$$\begin{aligned} Y_{\mathrm {c,Reuss}}=\frac{Y_{1}Y_{2}}{(1-\rho _{1})Y_{1}+(1-\rho _{2})Y_{2}}. \end{aligned}$$

(4.99)

These models give the upper and lower limits to the Young’s modulus of the actual composite material.

Fig. 4.80

a Voigt and b Reuss models of composite materials, with components of elastic modulus $Y_{1}$ and $Y_{2}$, with volume fractions $\rho _{1}$ and $\rho _{2}$. In the Voigt model both materials have the same strain, while in the Reuss model both materials are subjected to the same stress. Note that this Voigt model of composite materials is qualitatively different from the Voigt model used to model viscoelastic materials, which is simply called the Voigt model in this chapter. (Based on [45].) For Problem 4.19

4.20

In the text, strain, $\epsilon$, was defined as the fractional deformation $(L-L_{0})/L_{0}$, which equals $\lambda -1$, where $\lambda = L/L_{0}$ is the Lagrangian strain (or the stretch ratio). This is quite common for small fractional deformations, so let us call it now $\epsilon _{\mathrm {small}} = \lambda -1$. More generally, for arbitrary fractional deformation, strain is defined as $\epsilon _{\mathrm {general}} = \frac{1}{2}(\lambda ^{2} -1)$. This is often called finite strain (and sometimes Green’s strain) and is sometimes labeled as E.

(a) Calculate $\lambda$ and compare $\epsilon _{\mathrm {small}}$ and $\epsilon _{\mathrm {general}}$ for $L = 2$ cm and $L_{0}=1$ cm.

(b) Express $\epsilon _{\mathrm {general}}$ in terms of $\epsilon _{\mathrm {small}}$, and vice versa.

(c) Show that for very small $\lambda -1$, the finite strain, $\epsilon _{\mathrm {general}}$, approaches the small-strain approximation, $\epsilon _{\mathrm {small}}$. Specifically compare them for $L = 1.01$ cm and $L_{0}=1.00$ cm.

(d) What is the largest value of $\lambda -1$ for which $\epsilon _{\mathrm {small}}$ is within $10\%$ of $\epsilon _{\mathrm {general}}$?

(e) Say that the stress–strain relation can be written as $\sigma = Y \epsilon _{\mathrm {general}}$. Find the relationship between $\sigma$ and $\epsilon _{\mathrm {small}}$ and sketch it. Compare it to the linear and exponential relations between $\sigma$ and $\epsilon _{\mathrm {small}}$.

4.21

Another way used to define strain in general is $e = \frac{1}{2}(1-1/\lambda ^{2})$ (Almansi’s strain). Repeat Problem 4.20, replacing $\epsilon _{\mathrm {general}}$ by e.

4.22

Yet another way used to define strain in general is as $\ln \lambda$ (“true” strain). Repeat Problem 4.20, replacing $\epsilon _{\mathrm {general}}$ by $\ln \lambda$.

Viscoelasticity

4.23

(a) The faster you try to open or close a screen door, the more resisting force you encounter. Does that mean you should use include a dashpot in the mechanical model of the door closer unit? Why?

(b) When the door is open and released, it returns to its initial position. Does that mean you should include a spring in the mechanical model of the door closer? Why?

(c) When the door is open and released, the length of the closer decreases linearly with time until it returns to its initial length. Do any of the three viscoelastic models (Maxwell, Voigt, Kelvin) model this behavior? If so, which? If not, which one comes closest?

4.24

A dashpot of length 3 cm is characterized by the constant $c =2\times 10^{4}$ N-s/m. A constant force of 10 N is applied to it. Find its length and $\mathrm{d}x/\mathrm{d}t$ after the force has been applied for 2 s.

4.25

Say that the constant c of a dashpot varies linearly with its cross-sectional area. Repeat Problem 4.24 for a dashpot that is smaller in all dimensions by a factor of 2.

4.26

Consider a function f(t) that is 0 before time $-T/2$, then increases linearly in time until it becomes 1 at time T / 2, and remains at 1 thereafter.

(a) Sketch it.

(b) What function does it become as T approaches 0?

(c) For arbitrary T, find and plot $\mathrm{d}f/\mathrm{d}t$.

(d) What does this derivative become as T approaches 0?

4.27

The text says that the time derivative of the step function $\mathrm{d}\theta (t)/\mathrm{d}t$ is the Dirac delta function $\delta (t)$. Show that the integral over the Dirac delta function is the step function.

4.28

Use the step function to compose the following functions:

(a) A function that is 0 for $t<-1$ s, 2 from $t= -1$ to 3 s, and 0 thereafter, which is a type of square pulse (Fig. 4.45b).

(b) A function that is 0 for $t<0$ s, 1 from $t= 0$ to 3 s, $-2$ from $t= 3$ to 4 s, and 0 thereafter.

(c) A function that is 0 for $t<0$ s, 1 from $t= 0$ to 1 s, 0 from $t= 1$ to 2 s, 1 from $t= 2$ to 3 s, and so on, which is a type of square wave (Fig. 4.45c).

4.29

Verify the solutions of the Maxwell model for applied force (4.55) and then for applied deformation (4.56), by inserting them into (4.52)—and verifying that you get an equality—(for $t>0$ where the step function $\theta (t)$ can be replaced by 1 (unity)) and by checking initial conditions (i.e., solutions for $t=0$).

4.30

Show that you obtain the solutions of the Maxwell model ((4.55) and (4.56)) by integrating (4.52) from early times to time t for the sudden application of force $F_{0}$ as described in the text. Hint: This involves the integral

$$\begin{aligned} x(t)= \int _{-\infty }^{t} \left( \frac{F(t')}{c }+\frac{\mathrm{d}F(t')/\mathrm{d}t'}{k}\right) \mathrm{d}t'. \end{aligned}$$

(4.100)

4.31

Verify the solutions of the Voigt model for applied force (4.58) and then for applied deformation (4.59), by inserting them into (4.57)—and verifying that you get an equality—(for $t>0$ where the step function $\theta (t)$ can be replaced by 1 (unity)) and by checking initial conditions (i.e., solutions for $t=0$).

4.32

Verify the solutions of the Kelvin/standard linear solid model for applied force (4.69) and then for applied deformation (4.70), by inserting them into (4.68)—and verifying that you get an equality—(for $t>0$ where the step function $\theta (t)$ can be replaced by 1 (unity)) and by checking initial conditions (i.e., solutions for $t=0$).

4.33

What happens to the Kelvin model when either $k_{1}$ goes to 0 or $k_{2}$ goes to $\infty$, or when both occur? Why?

4.34

Show that the solution for a force $F(t)=F_{0}(\theta (t)-\theta (t-T))$ applied to a Maxwell model material is $x(t)=F_{0}(1/k \,{+}\, t{\!/\!}c )\theta (t)-F_{0}(1/k \,{+}\, (t-T){\!/\!}c )\theta (t-T)$.

4.35

(a) Verify (4.75) by substitution into the Kelvin constitutive equation (4.68).

(b) Verify (4.77)—evaluated at the end of the force ramp. (Hint: Use the fact that $\exp (-x)$ is approximately $1-x$ when $x\ll 1$. Remember that $\exp (-x)$ approaches 0 as x gets very large.)

(c) Now explain why this Kelvin solid model successfully describes the cited viscoelastic properties of bone (at least qualitatively). Carefully sketch x(T) as a function of time from $T=0$ to $T=$ many times $\tau$.

4.36

Consider the deformation x resulting from a force that linearly increases from 0 to $F_{0}$ from $t=0$ to T, as described by (4.75) and (4.77). The force is then maintained at $F_{0}$ for $t>T$. Use the Kelvin model to find x for $t>T$.

4.37

Repeat Problem 4.36 if instead at $t=T$ the force is very suddenly decreased to 0 and is maintained at that value.

4.38

Apply a deformation $x_{0}$ in a linearly increasing manner over a time T ($x=x_{0}(t/T)$ from $t=0$ to $t=T$) to a material described by the Kelvin standard linear model. Determine the stress relaxation F(t) from $t=0$ to T and obtain the deformation when the total deformation $x_{0}$ has been applied, F(T). Examine how F(T) depends on T, and explain why this qualitatively agrees or disagrees with observations for bone in Fig. 4.46 with increasing strain rates.

4.39

(advanced problem) One way to model the phase during running when a foot is in contact with the ground is with a one dimensional model with the force of a mass on a muscle (which is modeled as a passive material—a Voigt material with elastic and viscous elements in parallel) acting on the track (which is modeled as an elastic element in series with the muscle). (The effective mass is that of the body minus that of the leg in contact with the ground. The force on the mass includes the effects of gravity and that due to downward acceleration.) This is mathematically equivalent to the Kelvin model of a material. Show that there is a range of stiffnesses for which the period of this damped oscillation is shorter than that for a very hard track (very high stiffness, very low elasticity). (Is this consistent with the data in Table 3.6?) Because the running speed is thought to be inversely proportional to the time the foot is in contact with the track and the foot would be in contact for about a half of an oscillation period, this would mean that people could run faster on a track with the right elasticity—and this has been demonstrated.

Breaking

4.40

The goal is to find how much a tibia of length L bends before it breaks when a force F is applied to the center of it, as shown in Fig. 4.35.

(a) Show that the vertical deflection for a beam is $L^{2}/8R$, where R is the radius of curvature. (Define the vertical deflection as the maximum difference in vertical position of the neutral axis, laterally across the beam. Assume that $R\gg L$.)

(b) Show that this maximum vertical deflection is $L^{2}($UBS) / 4Yd, by using (4.81). d is the thickness, which can be taken as the tibia diameter here.

(c) If the tibia is 440 mm long and its diameter is 20 mm, find this deflection. (Use the data in the tables such as Table 4.6.)

(d) Is your answer reasonable to you? Why?

(e) Quantitatively, how does this deflection compare to how much the tibia deforms laterally in length (i) at the top (compression) and bottom (tension) of the bone at the UBS (where the magnitude of the relevant strain is $d/2R_{\mathrm { min}}$) and (ii) before fracture in ordinary compression or tension?

4.41

Based on the discussions of bone breaking, below what porosity level (the fraction of maximum bone density) will bones break fairly easily?

4.42

(a) Estimate the critical load for Euler buckling of an adult femur. Assume the femur is solid.

(b) Do forces on the femur during exercise ever reach the levels needed for buckling?

(c) If your answer to (b) is no, how small would Y have to be for buckling to occur during exercise?

(d) If Y were proportional to porosity, how porous would the bone in the femur need to be for one to be concerned about buckling?

4.43

Find the critical load to buckle a hair (on your head). Use the Euler buckling equation, (4.88), Table 4.2 for Y, $L =$ 3 mm, and a thickness of 0.02 mm.

4.44

Replot (i.e., carefully sketch) Fig. 4.67 on linear–linear axes.

4.45

Use Fig. 4.67 to determine relationships for the number of cycles needed to create stress fractures in cortical bone for a given strain range, for both compressive and tensile loading. Also express these relations as the number of miles to failure as a function of the strain range.