1. Topic 3: Constitutive Properties of Tissues

2. The Continuum Mechanics Problem
1. Geometry and Structure
2. Boundary Conditions
3. Governing Equations:
Conservation Laws
Mass
Eulerian
Lagrangian
Momentum
linear
angular
Energy
Constitutive Equations

3. Conservation Laws
Mass
Momentum
Energy

4. The Constitutive Law
describes the mechanical properties of a material, which depend on its constituents
is a mathematical relation for stress as a function of kinematic quantities, such as strain or strain-rate
is an idealization and an approximation
the validity of the idealization depends not only on the material but on the mechanical conditions
must typically be determined by experiment
is constrained by thermodynamic and other physical conditions, e.g. conservation of mass and energy
should be derived from considerations of material microstructure

5. Solids and Fluids Solids Fluids
Can support shear stress indefinitely without flowing
Assume an unloaded natural shape
Deform with minimal or substantial energy dissipation
Usually composites
Fluids
Liquids and gases
Gases have lower density and higher compressibility than liquids; dependent on temperature
Phase transition as function of temperature and pressure
Support stress as fluid hydrostatic pressure at rest
Can not resist a shear stress indefinitely without flowing
No unique unloaded natural state; conform to the shape of their container
Dissipate energy as heat when they flow
Usually mixtures

6. Elastic solids Stress Strain Stress Strain
Stress depends only on strain, Tij = Tij(ekl)
Example: Isotropic Hookean elastic solid, Tij = lekkdij + 2meij
Return to a unique natural state when loads removed
Work done during loading is stored as potential energy without dissipation (a reversible process)
Hookean solids have a constant elastic modulus, E
In nonlinear solids, Etangent is dependent on strain
Stress
Strain E
Linear (Hookean)
Stress
Strain
Etangent
Nonlinear

7. Exponential Stress-Strain Relation of Soft Tissues
Etangent
Stress vs. Strain
Stiffness vs. Stress
Etangent
Stress

8. Plasticity Stress Stress Strain Strain Ductile failure Brittle failure
elastic limit (yield point)
elastic limit
failure
UTS
Stress
Strain
Stress
Strain
strain
hardening
failure
strain
hardening

9. Viscous Fluids Viscometer Couette Flow Blood Viscosity
Shear stress depends on the rate of shear strain, Tij = Tij(Dkl)
Example: Newtonian viscous fluid, Tij = -pdij + 2mDij
Linear viscous (Newtonian) fluids have constant viscosity m
Viscosity measures resistance to shear,
Work done on flowing viscous fluids is dissipated as heat
In non-Newtonian fluids, apparent viscosity depends on the shear rate , e.g. whole blood is shear-thinning
Fig 2.14:2
Viscometer
Fig 2.7:1
Couette Flow
Fig 3.1:1
Blood Viscosity

10. Viscoelastic Solids and Fluids
Stress depends on strain and strain-rate, Tij = Tij(ekl,D kl)
Creep at constant stress
Stress relaxation at constant strain
Hysteresis, energy dissipation during loading and unloading
Maxwell Fluid m T E e1 e2
Voigt Solid T m E T1 T2
Kelvin Solid
m" T
E" E'
Elastic stress depends on strain (spring)
Viscous stress depends on strain-rate (dashpot)
Strains add in series, stresses are the same
Stresses add in parallel, strains are the same

11. Linear Viscoelastic Models: Creep Functions
Maxwell Fluid m F E u1 u2
Voigt Solid F m E F1 F2
Kelvin Solid
m" F
E" E'

12. Linear Viscoelastic Models: Relaxation Functions
Maxwell Fluid m F E u1 u2 F
Voigt Solid m E F1 F2
Kelvin Solid
m" F
E" E'

13. Viscoelasticity in Soft Tissues Bovine Coronary Artery
STRESS (kPa)
Hysteresis and Preconditioning
STRETCH
data from Humphrey JD, Salunke N, Tippett B, 1996
STRESS (kPa)
TIME (seconds)
Stress Relaxation

14. Other Material Properties
Viscoplastic
behaves like a viscous fluid after shear stress exceeds a finite yield stress (e.g. whole blood)
Thixotropic
sol-gel transformation from solid (gel) to fluid (sol) properties
induced by shear stress such as agitation or stirring (e.g. actin)
Strain softening
Also known as the Mullin’s effect
progressive, irreversible reduction in elastic stiffness induced by increased maximum previous strain
e.g. elastomers and small intestine

15. Considerations in Biomechanics

16. Topic 3: Summary of Key Points
The constitutive law describes the mechanical properties of a material, which depend on its constituents
Unlike fluids, solids can support a shear stress indefinitely without flowing
In an elastic solid, the stress depends only the strain; it returns to its undeformed natural state when unloaded.
In a viscous fluid, the shear stress depends only on the shear strain rate.
Stress depends on strain and strain rate in viscoelastic materials; they exhibit creep, relaxation, hysteresis.
Viscoelastic properties can be modeled by combinations of springs and dashpots.