1. Viscoelasticity – 1
Lumped Parameter Models for time-dependent behavior
DEQ’s as Constitutive Equations

2. Viscoelasticity (Combined viscous and elastic behavior)

3. Deborah Number
= Maxwell relaxation time (time for material to “reach” equilibrium after perturbation)
= Observation time
High De: material is more like elastic solid
Low De: material is more like a viscous, Newtonian fluid
"The mountains flowed before the Lord" in a song by prophetess Deborah (Judges 5:5).

4. Examples of viscoelastic materials

5. Stress Relaxation
When a body is deformed (or strained) and that deformation (or strain) is held constant, stresses in the body reduce with time.
Figure from Fung “Biomechanics” 2nd ed.

6. Creep
When a body is loaded (or stressed) and the stress is held constant, the body continues to deform (or strain) with time.
Figure from Fung “Biomechanics” 2nd ed.

7. Hysteresis
When a body subjected to cyclic loading, load-displacement (or stress-strain) behavior for increasing loads is different than behavior for decreasing loads. The area between the curves represents energy loss (dissipation).
Figure from Fung “Biomechanics” 2nd ed.

8. Viscoelasticity
Behavior exhibited by a material (or tissue) that has both viscous and elastic elements in its response to a deformation (or strain) or load (or stress)
Represented by:
Dashpot/damper for viscous element.
Follows Newtonian fluid constitutive law
Spring for elastic element
Assumed to linearly elastic h m

9. Lumped parameter models
Maxwell
Kelvin-Voigt
Standard-linear
Burger

10. Lumped parameter models (Fung terminology)
Alternate terminology
Maxwell
Kelvin-Voigt
Standard-linear
(alternate form)
The Maxwell model, Voigt model (Kelvin-Voigt model) and the Kelvin model (standard linear model) are all composed of combinations of linear springs and dashpots.
A linear spring with spring constant  theoretically produces a deformation proportional to load.
A dashpot with coefficient of viscosity  produces a velocity proportional to load.
Figure from Fung “Biomechanics” 2nd ed.

11. Maxwell Model
Represented by a purely viscous damper and a purely elastic spring connected in series
The model can be represented by the following DEQ:
Predicts/models a stress that decays exponentially with time to zero with permanent deformation
Stress relaxation experiment
Model doesn’t accurately predict creep (constant stress). Predicts that strain will increase linearly with time. Actually strain rate decreases with time

12. Kelvin-Voigt Model
Represented by a Newtonian damper and Hookean elastic spring in parallel.
The model can be expressed as a linear first order DEQ
Represents a solid undergoing reversible, viscoelastic strain.
Models a solid that is very stiff but will creep (e.g. crystals, glass, apparent behavior of cartilage). At constant stress (creep), predicts strain to tend to σ/E as time continues to infinity
The model is not accurate for relaxation in a material/tissue
Creep and recovery response

13. Standard Linear Model Creep and recovery response
Modeled as Newtonian damper and 2 Hookean springs, one in parallel and one in series.
The model can be expressed as a linear first order DEQ:
Figure from Fung “Biomechanics” 2nd ed.
Represents a solid undergoing an elastic and a reversible, viscoelastic strain.
At constant stress (creep), predicts an initial strain from the spring which will creep over time until the parallel spring carries all the applied load
The model is linear approximation to viscoelastic materials/tissues that are time dependent but completely recover from applied loads
Creep and recovery response

14. Creep and Relaxation Behaviors (from 3 linear models)
Note: steady state and initial values!
Figures from Fung “Biomechanics” 2nd ed.

15. Burgers Model
4 linear elements used to capture “minimum” amount of behavior for many polymers/tissues which are:
“Instantaneous” elasticity or elastic recovery (G1)
Molecular “slip” (η1)
Rubbery elasticity (G2)
“Retarded” elasticity (η2)
Modeled as a Maxwell model in series with a Kelvin-Voigt
Creep and recovery response

16. Generalized Models
Generalized Maxwell model is Maxwell elements in parallel
Generalized Kelvin–Voigt model is Kelvin–Voigt elements in series

17. Linear Model Comparison
Maxwell
Good for predicting stress relaxation
Poor at predicting creep
Used for soft solids with non-recoverable deformations
Kelvin-Voigt
Good for predicting creep with small elastic deformation
Not accurate with predicting stress relaxation
Used for polymers, rubber, cartilage when the load is not too high
Standard Linear Model
Predicts both creep and relaxation
Fully recoverable & initial elastic displacement
Burgers
Predicts essentials of polymer viscoelastic behavior
Used for polymers with non-recoverable behavior
Generalized
Used for fitting of experimental data to an arbitrary level of accuracy

18. Maxwell Model - creep h m
F, u

19. Creep Functions
Solving the DEQs for the Maxwell, Voigt, and Kelvin models for displacement ε = c(t), when the stress (t) is a unit step function 1(t).
We obtain a set of results known as the creep functions.
These functions represent the elongation (strain) in the viscoelastic material which is produced by a sudden application of unit stress at time = 0.
c(t) = (1/ + t/)1(t) Maxwell fluid
c(t) = 1/ (1-e-(/)t)1(t) Voigt solid
c(t) = 1/ER [ 1 – (1-/)e-t/]1(t) Kelvin solid
where  = 1/1 ,  = (1/0)(1 + 0/1), and ER = 0 (Fung’s model parameters)

20. Maxwell Model - relaxationh m
F, u

21. Relaxation Functions k(t) = e-(/)t1(t) Maxwell fluid
Solving the DEQs for stress [(t) = k(t)] when the applied strain is a unit step function ε(t) =1(t) yields relaxation functions.
These represent the resisting stress as a functions of time
k(t) = e-(/)t1(t) Maxwell fluid
k(t) = (t) + 1(t) Voigt solid
k(t) = ER [ 1 – (1 - /)e-t/]1(t) Kelvin solid
where  = 1/1 ,  = (1/0)(1 + 0/1), and ER = 0 (Fung’s model parameters)

22. Summary
Many lumped parameter models can be formulated to represent 1D linear viscoelastic materials. The best one should be chosen based on the material and properties that are being observed.
Models are linear approximations to observed, time-dependent behaviors (elastic and viscous) and are formulated as DEQs
Models can be painful to use because they require a new solution of DEQs for any forcing function.
NEED A MORE GENERAL APPROACH!!!!

23. Creep and failure at higher strains