1. Expectations after this section
What is a constitutive equation?
What is elasticity?
What is Hooke’s Law?
What is generalized Hooke’s Law applied to mechanical behaviors?
What are assumptions and limitations, i.e. when can we use it?
What are compliance tensor and condensed compliance matrix?
What are stiffness tensor and condensed stiffness matrix?
What is orthotropic behavior and how do we describe it under Hookean assumptions?
What is transversely isotropic behavior and how do we describe it under Hookean assumptions?
What is isotropic behavior and how do we describe it under Hookean assumptions?
What are Lame equations?
What are standard mechanical parameters used to describe Hookean behavior?
How can we deal with off axis properties?

2. Elastic Moduli for linearly elastic behaviors

3. Terminology
Tangent Modulus is slope of stress-strain curve at any point
Young's modulus is typically the linear portion of a stress-strain curve
Secant modulus is point to point slope on non-linear curve

4. Young’s Modulus Young's modulus (tensile modulus)
Normalized stiffness of an elastic material
Quantity used to characterize materials (or composites)
Ratio of uniaxial stress over uniaxial strain in range where Hooke's Law holds
Commonly called elastic modulus
Other elastic moduli, e.g. bulk modulus, shear modulus, aggregate modulus, etc.
where
E is the Young's modulus (modulus of elasticity)
F is the applied force on an object under tension
A0 is the original cross-sectional area
ΔL is the length change
L0 is the original length

5. Thomas Young (1773-1829) English polymath
Contributions in vision, light, solid mechanics, energy, physiology, language, music, and Egyptology
He helped decipher the Rosetta Stone

6. Bulk Modulus

7. Modulus of elasticity
Modulus of elasticity (or Young's Modulus) is the slope of the straight-line portion of a stress-strain curve under an axial stretching load.
Common tensile test results include elastic limit, tensile strength, yield point, yield strength, elongation, and Young's Modulus.
Typical graph showing modulus of elasticity/Young's Modulus.
True stress here would be more nonlinear through the linear portion of this curve….Why?

8. Hooke’s Law (which really isn’t a law)
Hooke's law of elasticity is an approximation that states that the extension of a spring is in direct proportion with the load applied to it.
Many materials (or composite materials) exhibit this behavior as long as the load does not exceed the elastic limit.
Materials (or composites) where Hooke's law approximates behavior are known as linearly elastic or "Hookean" materials.
Hooke's law for mechanical behavior simply says that stress is directly proportional to strain.
Mathematically, Hooke’s law states that
where
x is the displacement of the spring's end from its equilibrium position
F is the restoring force exerted by the spring on that end
k is a constant called the rate or spring constant

9. Constitutive Equations
In mechanics, constitutive relations connect applied stresses (or forces) to strains (or deformations). The constitutive relation for linear materials is termed Hooke’s law.
In physics, a constitutive equation is a correlation between two physical quantities (often tensors) that is specific to a material or substance and does not follow directly from physical law. An empiricism!
Constitutive equations must be combined with other equations that do represent physical laws to solve physical problem.
Some constitutive equations are simply phenomenological; others are derived from first principles. A constitutive equation frequently has a parameter taken to be a constant of proportionality in ideal systems.

10. Constitutive Equations - Examples
Friction - Ff = Fpμf
where Ff is frictional force, Fp is normal force and μf is the coefficient of static friction.
Drag equation -  
where D is drag, Cd is the coefficient of drag, ρ is density, A is projected area, and v is velocity.
Linear elasticity -
where C is stiffness tensor, σ is engineering stress and ε is engineering strain
Ohm’s law - V = IR
where V is voltage, I is current, and R is resistance
Diffusion constant, specific heat constant, electrical permittivity constant, etc., etc.

11. Generalized Hooke’s Law
“CEIIINOSSSTTUV”
Robert Hooke’s Latin anagram (1676) (ut tension sic vis) meaning – “as is the deformation, so is the force”.

12. Hooke’s Law
Plot of applied force F vs. elongation X for a helical spring according to Hooke's law (red line) and what the actual plot might look like (dashed line). At bottom, pictures of spring states corresponding to some points of the plot; the middle one is in the relaxed state (no force applied).

13. Robert Hooke
( ) English natural philosopher, architect and mathematician
Relatively obscure for all his accomplishments: “cell”, gravity, light, elasticity, modes of vibration, astronomy, paleontology, microscopy, dome of St. Paul’s cathedral used his method of construction, architectural planning for London, buildings, etc.
Life in three periods
Scientific inquiry lacking wealth
Acquiring wealth
Fighting jealous intellectual disputes with Newton over work on gravity, planets, and light
Thought to be morose and a recluse, cantankerous, vengeful, etc. Someone who claimed far more credit than he deserved for things.

14. Hooke’s Law
Linear relationship between σ and ε, i.e. a constant of correlation between each component of σ and each component of ε (many = 0)
2nd order tensors for σ and ε related via a 4th order tensor having 81 constants (one constant relating each of 9 ε terms to each of 9 σ terms). That is:
and
where:
is a 4th order mechanical stiffness tensor
is a 4th order mechanical compliance tensor
Since σ and ε are symmetric tensors and since many terms in and = 0, the compliance tensor can be condensed to 36 terms: or
What is definition for each compliance term in the matrix?

15. Hooke’s Law from Compliance Homogeneous, Isotropic, Linearly Elasticsy sx sz sx sy sx y x z
x direction
What if cube is incompressible?
y direction
What if cube is not isotropic?
z direction

16. Poisson’s Ratio Vol Vol Incompressible Vol Vol
What happens to density in each case?

17. Matrix Form for compliance behavior
In matrix form, Hooke's law for isotropic materials is:
What is definition for each compliance term in the matrix? or

18. Compliance tensor and matrix for general Hookean behavior (condensed from compliance tensor)
Note: when equations are condensed,
Shear strains have been redefined to twice the value of uncoupled
Shear stresses have been redefined to twice the value of uncoupled

19. Condensed stiffness matrix inverting the compliance matrix
What is definition for each stiffness term in the matrix?
Definition of aggregate modulus!
Incompressibility? or

20. Stiffness tensor and matrix for generalized Hookean behavior (condensed from stiffness tensor)
Then, generalized Hooke's law can be written as:

21. Condensed stress and strain vectors
Express σ and ε as six-dimensional vectors in orthonormal coordinate system

22. Biomedical
Engineering

23. Orthotropic behavior 1
When a material has different stiffnesses (or compliances) in three orthogonal directions, the material is orthotropic.
Note for a symmetric matrix, the lower terms are not repeated.
And, the stiffness and compliance matrices are:
Note: usually no shear coupling with normal stresses or strains or with other shear stresses or strains, hence many zeros in the above

24. Orthotropic behavior 2
Values of the compliance coefficients for an orthotropic material in terms of engineering constants Eij , νij , and Gij are:
Eii is the Young's modulus along axis i
Gij is the shear modulus in direction j on the plane whose normal is in direction i
νij is the Poisson's ratio that corresponds to a contraction in direction j when an extension is applied in direction i.
Note: There are 9 independent engineering constants required to describe this type of behavior
Note: If all Poisson’s ratios are ~0, there is no coupling between normal stains and stresses (i.e. cork)

25. Orthotropic behavior 3
In general (with non-zero ν), the stiffness coefficients are:
where

26. Ligament Fibers from SEM
Biomedical
Engineering

27. Transversely isotropic behavior 1
When a material is isotropic in one plane (say the 2-3 plane) but has a different stiffness (or compliance) in a direction orthogonal to this plane, the material is transversely isotropic.
In this special case of linear elasticity, the compliance coefficients are:
Note that there are 5 independent engineering constants!

28. Transversely isotropic behavior 2
In this special case of linear elasticity, the stiffness coefficients are:
where

29. Isotropic behavior
When a material is isotropic it has the same stiffness (or compliance) in all directions. In this special case of linear elasticity, the compliance and stiffness coefficients are:
Note that there are 2 independent engineering constants

30. Isotropic Hookean behavior (condensed from stiffness tensor),
Isotropic Hookean behavior (condensed from stiffness tensor)
Lame cast this stiffness behavior into an alternative form

31. Plane Stress (all out of plane stresses = 0)
The inverse relation is usually written in the reduced form

32. Elastic constant pairs
Elastic constant pairs are related for isotropic, linearly elastic (Hookean) behavior

34. Off Axis Elastic Properties
With bone or other anisotropic materials, we sometimes have a specimen to test with testing axes (x,y,z) that are not coincident with the principal material directions (1,2,3).
The following method shows interrelationships between these data and the principal properties of interest.
To simplify equations and fix ideas, the development will be in 2D, but the method can extrapolated to 3D in the same fashion, just more terms.

35. Cut out, test in specimen coordinates, and transform to principal coordinates

36. Transformation of 2nd order tensors
Use transformation matrix equations previously developed
where transformation matrix T is given by

37. Derive expression for principal material properties
Substituting the transformation expressions
Pre-multiplying both sides by
Thus, the interrelation of the stiffness matrices is or

38. Expectations after this section
What is a constitutive equation?
What is elasticity?
What is Hooke’s Law?
What is generalized Hooke’s Law applied to mechanical behaviors?
What are assumptions and limitations, i.e. when can we use it?
What are compliance tensor and condensed compliance matrix?
What are stiffness tensor and condensed stiffness matrix?
What is orthotropic behavior and how do we describe it under Hookean assumptions?
What is transversely isotropic behavior and how do we describe it under Hookean assumptions?
What is isotropic behavior and how do we describe it under Hookean assumptions?
What are Lame equations?
What are standard mechanical parameters used to describe Hookean behavior?
How can we deal with off axis properties?