Problems

A common mechanical test used to characterize the material properties of engineering and biologic materials is the simple axial compression test. Consider a cylindrical specimen subjected to such a test, and the resulting load-deformation curve sketched below.

1. How is the stiffness (k) of the specimen defined?
2. How do increases in A and L affect K?
3. How is a stress-strain curve obtained from the force-deflection curve?
4. Sketch the stress-strain curve and identify the elastic modulus, yield stress, yield strain, ultimate stress, and ultimate strain. What do these mean (briefly)?

Biological tissues display time-dependent behavior that can be modeled by composite models of springs and dashpots. One such model is shown below:

1. Find the stress-strain relationship for the model
2. Solve the differential equation for sudden stress, applied at time t=0. Use the condition that
3. Sketch the strain versus time or "creep" curve.
4. Would the material represented by the model reach a finite level of strain under a constant stress?

The stress in a beam under bending can be written:

Where M = bending moment

y = distance from the neutral axis

I = area moment of inertia

1. Where are the highest stresses under this bending moment? Please draw the distrabution of stresses for the cross section.
2. Consider a long bone under bending and assume that the bone can be modeled as a hollow cylindrical tube. Calculate the stress at the tensile surface in terms of M if the inner radius r = 1.0 cm and the outer radius R = 1.5 cm.
3. Determine how much the stress at the surface would change if the outer radius increased by 1 mm (such expansions in outer diameter can happen with aging).
4. Assume that the strength of young bone is 120 MPa but decreases to 100 MPa at age 80 while the outer radius increases from 1.5 cm to 1.6 cm. Determine if the loss of strength is offset by the radial expansion when the bone is loaded by bending moment M. Hint: Compare the factor of safety for the two states for a given M.

Soft connective tissues are viscoelastic materials that can be represented by spring-dashpot models, like this Kelvin solid model.

1. Derive the differential equation relating stress to strain for this model.
2. Determine the initial and steady-state strain under a step increase in stress (creep), and sketch strain versus time.
3. What is the coefficient that governs the rate of creep.
4. What property of this composite model governs the equilibrium (steady-state) deformation?

Derive a relationship for the moment of inertia of a rectangular cross-section of width w and depth d about an axis through the center of the cross-section. If d = 2w, what is the ratio of the bending rigidities when the cross-section is flexed against its deep axis compared to against its narrow axis? (Hint: play with a ruler in bending, does the math match the responses of the ruler?)

A 30 year-old women weighing 55 kg, and 1.6 m tall, slips on the ice and falls forward. She lands stiff-armed on one hand, with her arm perpendicular to the ground. Assume that the entire load is carried by the distal radius, which has a cross-sectional area of 0.5 cm² and is predominately trabecular bone with apparent density of 0.5 gm/cm³ and compressive strength of 10 Mpa.

1. Find the impact force using the assumptions that elastic impact occurs, the impact duration is 0.1 seconds, and the fall height is 0.5 times body height. ( v² = 2gh, g = 9.8 m/s², and F=2mv/t)
2. Determine the maximum stress on the radius, and determine the risk factor for radial fracture.
3. If it were her mother who fell, determine the risk factor for radial fracture, assuming that mother and daughter are the same size and that the density of the mother’s bone was 0.1 gm/cm³.

For a healing fracture in stage 2 through stage 4 (final stage);

1. Sketch the stiffness (k) versus time relationship
2. Sketch the force (F) versus time relationship
3. What is the approximate slope of the Strength (ultimate stress) versus Stiffness (k) relationship?

Figures A and B represent patellar tendon grafts which have the same length, but the cross-sectional area of graft A is twice the cross-sectional area of graft B. Each graft undergoes uniaxial tension at a constant strain rate.

1. What is the equation for structural stiffness (k) and for material stiffness or Youngs’s modulus (E)?
2. Sketch the Force versus Displacement curves for grafts A and B, and sketch Stress verse Strain curves for these same grafts.

 

 

 

 

 

 

 

 

 

A B

 

Figures C and D represent the anterior cruciate ligament and the meidal collateral ligament respectively. These ligaments have approximately the same cross-sectional area, but the medial collateral ligament is twice the length of the anterior cruciate ligament. Each ligament undergoes uniaxial tension at a constant strain rate.

3. Sketch the Force verse Displacement curves for ligaments C and D, and sketch Stress versus Strain curves for these same grafts.

C D

1. The area moment of inertia describes the distribution of a cross-sectional area with respect to the neutral axis of bending.
2. When loaded along the longitudinal axis, cortical bone is stronger in tension than in compression.
3. It takes as many days for a unit of bone to be formed as it did for it to be resorbed.
4. Under normal loading conditions, energy absorption in trabecular bone plays an important role in limiting force levels in articular cartilage.
5. Bone regenerates in response to mechanical injury because of the capability of bone cells to divide.
6. The tensile and compressive moduli for cortical bone are equal.
7. The tensile and compressive strengths for cortical bone are equal.
8. Increasing the collagen content increases the static (equilibrium) compressive modulus of cartilage.
9. Cartilage permeability decreases with increasing applied compressive strain.
10. Trabecular bone modulus increases with increasing loading rate, but its strength is constant.
11. The tensile stiffness (modulus) properties of cortical bone tissue are due mainly to its collagen content
12. The modulus of trabecular bone is approximately 500 MPa.
13. All proximal femora have the same strength but their stiffnesses depend on a combination of material and geometric properties.
14. Cortical bone can heal itself by remodeling, but trabecular bone and cartilage can not.
15. Young femora are generally 10 times stronger than old femora.
16. During fracture healing, a bone regains full strength before it regains full stiffness.
17. Patients recovering from hip surgery should be advised to use a cane on the same side as the affected hip in order to reduce forces on that hip.
18. Wolff’s law has not yet been rigorously defined.
19. Age-related remodeling of the diaphysis can actually increase the strength of a long bone despite age-related bone loss.
20. Age-related bone loss is more pronounced in cortical than in trabecular bone.

1. For each fracture pattern identify the type of loading which most likely caused it. Transverse, Oblique, Butterfly, Spiral

 

 

 

*******put in picture*********

 

 

 

2. List four factors (at least two of which are mechanical) associated with increased risk of hip fractures.
3. What type of failure in a bone-ligament-bone test specimen most often results from a slow loading rate mechanical test? Why does this failure type differ from a typical ACL injury suffered during downhill skiing in an adult?
4. What is the "working length" of a fracture fixation device? Illustrate this concept with a simple sketch of a plated long bone.
5. List two ways that a hip pad might reduce the risk of hip fracture.