1. Viscoelasticity Soft tissues and cells exhibit several anelastic properties: –hysteresis during loading and unloading –stress relaxation at constant strain –creep at constant stress –strain-rate dependence i.e., in general, stress in soft tissues depends on strain and the history of strain These properties can be modeled by the theory of viscoelasticity

2. Simple Linear Viscoelastic Models Stress depends on strain and strain-rate: Kelvin Solid Voigt Solid Maxwell Fluid T  T E T1T1 T2T2  T T E 11 22 "" T T E" E' Elastic stress depends on strain (spring) Viscous stress depends on strain-rate (dashpot) Strains add in series, stresses are equal Stresses add in parallel, strains are equal

3. Maxwell Fluid Model Total strain rate = spring strain rate + dashpot strain rate: A linear first-order ODE  T T E 11 22

4. Maxwell Model: Relaxation Solution A monoexponential decay Exercise: check that the original ODE (Eq. 1) is satisfied by this solution Integrating Eq. 1 for constant applied strain,  0 → t=0

5. Voigt Solid Model Total stress = spring stress + dashpot stress: T  T E T1T1 T2T2 A linear first-order ordinary differential equation (ODE) Integrating, for constant applied stress, T 0 →

6. Voigt Model: Creep Solution Initial condition, ε (0) = 0 Exercise: check that the original ODE (Eq. 1) is satisfied by this solution t=0  T

7. Asymptotic and Instantaneous Elastic Moduli Voigt Solid Maxwell Fluid E0E0 ∞E E∞E∞ E0

8. Three-Parameter Models "" T T E" E' "" T T E" E' "" '' There are various possible configurations The Kelvin Standard Linear Solid Model: And the equivalent 3-parameter model constructed from a Voigt model and an elastic element in series:

9. Three-Parameter Models "" T T E" E' "" ''

10. Creep Solution

11. Relaxation

12. 3-Parameter Model "" T T E" E' "" ''

13. Stress Relaxation

14. Harmonic Strain Input

15. Linear Viscoelastic Models: Creep Functions Kelvin Solid Voigt Solid Maxwell Fluid T  T E T1T1 T2T2  T T E 11 22 "" T T E" E'

16. Linear Viscoelastic Models: Relaxation Functions Kelvin Solid Voigt Solid Maxwell Fluid T  T E T1T1 T2T2  T T E 11 22 "" T T E" E'

17. Generalized Voigt Model Total strain = ∑  strains in each Voigt model) T T 11 E1E1 22 E2E2 nn EnEn

18. Generalized Maxwell Model Total stress = ∑  stress in each Maxwell model) T T 11 E1E1 22 E2E2 nn EnEn

19. Quasilinear Viscoelasticity Soft tissues exhibit several viscoelastic properties: –hysteresis –stress relaxation –creep –strain-rate dependence Linear viscoelastic models also display many of these properties However, soft tissue elasticity is nonlinear Quasilinear viscoelasticity combines the time- history dependence of linear viscoelasticity with nonlinear elasticity (static nonlinearity)

20. Linear Viscoelasticity: General Property In all linear viscoelastic models: the relaxation function k(t) is proportional to  0 and therefore to T 0 the creep function c(t) is proportional to T 0 and therefore to  0

21. Quasilinear Viscoelasticity K(t,ε) = T (e) (ε).G(t) and c(t,T) = ε (e) (T).J(t) i.e. in quasilinear viscoelasticity, the instantaneous stress may be a nonlinear function of strain but the strain-dependence and time-dependence are separable. The reduced relaxation and creep functions are independent of strain.

22. Reduced Relaxation Function in Tissue Fig 7.5:4 page 272 from text book showing mean normalized relaxation curve for 16 loads from 2-16 g (solid line)

23. Reduced Creep Function Figure 8.3:5 in textbook. A typical creep curve, plotted as a reduced creep function J(t). Dog carotid artery.

24. Linear Viscoelasticity: Summary of Key Points In viscoelastic materials stress depends on strain and strain-rate They exhibit creep, relaxation and hysteresis Viscoelastic models can be derived by combining springs with syringes 3-parameter linear models (e.g. Kelvin Solid) have exponentially decaying creep and relaxation functions; time constants are the ratio of elasticity to damping The instantaneous elastic modulus is the stress:strain ratio at t=0 The asymptotic elastic modulus is the stress:strain ratio as t→∞

25. Quasilinear Viscoelasticity: Summary of Key Points The stress-strain relation is not unique, it depends on the load history. The elastic modulus depends on the load history, e.g. the instantaneous elastic modulus E 0 at t=0 is not, in general, equal to the asymptotic elastic modulus E  at t= . The instantaneous elastic response T (e) (t) = E 0  (t). Creep, relaxation and recovery are all properties of linear viscoelastic models. Creep solution can be normalized by the initial strain to give the reduced creep function J(t). J(0)=1. Relaxation solution can be normalized by the initial stress to give the reduced relaxation function G(t). G(0)=1.