1. Boundary Value Problems in Elasticity
Theoretical development and
closed form solution (Cliff Note version) for
Pressurized cylinder
Stress concentration around a hole
Contact stress (Hertz, Boussinesq)
Computational methods

2. Boundary Value Problems
Dynamic equilibrium is defined as
Static equations of equilibrium (3 equations)
or, if body force is zero
Equilibrium for the 2-D case (2 equations):
and

3. Boundary Value Problems
Strain-displacement relations (6 equations):
Constitutive relations (6 eq.) or
For 3D,
15 equations (3 equilibrium, 6 strain displacement, 6 constitutive)
15 parameters (6 stresses, 6 strains, 3 displacements)
BVP in mechanics always involve simplifying the above equations as much as possible and then finding solutions.
If a problem becomes too complex and closed form solution cannot be found, computational methods (e.g. finite elements) solve these same equations.

4. Stress concentrations from holes (also solved for inclusions)
Sharp edges, cracks, or holes induce stress concentrations.
A hole increases stress 3x
Stress concentrations in bone resolve themselves over time with remodeling
See pdf on webpage for more detailed solutions

5. Mechanics of a Pressurized Cylinder under Axial Tension
See pdf on webpage for more details

6. Eq. of equilibrium can be derived from above differential element
By ignoring higher order term and making a simplification for small angles
This equation of equilibrium can then be written as or
Carry this eq. forward

7. (1) (2) Carry this eq. forward
Strain in the z-direction is assumed to be constant
(constitutive eq., Hooke’s law) .
is known
Tangential stress can be given as
(2)
where,
.Substituting Eq. (2) into (1), we get
Carry this eq. forward

8. The radial stress will be given as the solution of this differential equation as
(3)
(2)
From Eq. (2) and (3), the tangential stress is P1
C and K are from boundary conditions:

9. P1
From boundary conditions and Eq. (2):
Then, the stresses are:
These are field equations that define the above BVP for stress.
It required the use of equilibrium equations, constitutive equations, and BCs.
To find strain, use the inverse form of Hooke’s law, i.e.
To find displacement, use strain displacement relationships.

10. Typical blood vessel testing (can use cylinder BVP solution)

11. Contact BVPs
Heinrich Hertz (1882) described localized deformation and distribution of pressure between to elastic bodies with certain BCs
S, e satisfy DEQ of equilibrium and s vanish far from contact
Contact is frictionless
Pressure is equal and opposite
Integral of pressure distribution is contact force
Found that an ellipsoidal distribution of pressure would satisfy BCs for two spheres in contact.
Joseph Boussinesq (1885) described a solution for point contact on a surface.
Allows solution for any distribution of pressure with a contact area by the principle of superposition, e.g. cylindrical flat punch

12. Hertzian contact
When an elastic sphere is pressed against an elastic material, the contact area increases.

13. Classic Hertz Solutions
Contact between a sphere and an elastic half-space
The applied force F is related to the displacement d by
where
Contact between two spheres
where p0 is the maximum contact pressure given by
where

14. Boussinesq Solution
A rigid cylinder is pressed into an elastic half-space
where a is the radius of the cylinder and
The relationship between indentation depth and normal force is

15. Indentation methods nano or standard
Indenter pushes down into tissue
Modulus of tissue estimated from the unloading curve

16. Commercial Indentor

17. Bone indentation
Johnston JD, et al. Clin Biomech, 26(10):

18. Indentation Testing
Indentation testing is used in situ to map strength & Young’s modulus E.
Tests are typically done with a blunt end indenter of varying diameter d.
A load-displacement curve is created.
Strength is estimated from the maximum loaded divided by the indenter area.
The Young’s modulus is estimated from slope S of the unloading curve S
Loading
Unloading
Displacement
Force
Note: Poisson’s ratio is squared in this equation. Therefore, errors in estimating the Poisson’s ratio do not strongly affect modulus estimates.

19. Micro or “Nano” Indentation Testing
Micro- and nano-indentation testing can measure biomechanical properties in a region as small as 1 m.
Consequently, it can provide mechanical properties of individual trabecula or lamellae in bone.
It estimates Young’s modulus from the following equation S
Loading
Unloading
Displacement
Force
where subscripts b and i refer to bone and indenter, respectively, and contact area is A

20. Bone “nano-indentation”
Zysset PK, et al. J Biomech, 32(10):

21. Cartilage Indentation
Simplest cartilage models have two phases and require two parameters to describe mechanical behavior, Aggregate Modulus and Permeability
More complex BVPs solutions (because of more complex constitutive equations are used) and will be discussed later in course

22. Cartilage indentation testing

23. FEM model of cancellous bone
Note:
Model is built from micro-CT image where each cubic element is one voxel in image.
This gives a stair step rather than smooth surface in model and adds error

24. FEM model of spinal motion segment with prosthesis