7 5 behavior of soft tissues under uniaxial loading
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7.5 Behavior of Soft Tissues under uniaxial Loading. Pure biological materials: actin, elastin, collagen. Tissues: several aforementioned materials & ground substance. Experimental approach to constitutive equation.

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7 5 behavior of soft tissues under uniaxial loading
7.5 Behavior of Soft Tissues under uniaxial Loading

  • Pure biological materials: actin, elastin, collagen.

  • Tissues: several aforementioned materials & ground substance.

  • Experimental approach to constitutive equation.

  • Single axial tension test on cylindrical specimen, load & elongation are recorded, stress-strain relationship.

  • Wertheim (1847): non-Hookean, tissues is under stressed in physiological state, artery shrunk from cut, broken tendon retravts


Cyclic response of dog’s carotid artery

l1: stretch ratio referred to zero-stress length of segment, 37 deg C, 0.21 cycles/min

Hysteresis of rabbit papillary muscle
Hysteresis of rabbit papillary (乳頭) muscle

Increasing strain rate

Long term relaxation
Long-term relaxation


Log 10 t


  • Hysteresis, relaxation, creep at lower stress ranges are common for mesentery of rabbit, cat & dog, ureter of animals, papillary muscles at resting

  • Difference: degree of distensibility

    • Mesentery 100%-200% from relaxed length

    • Ureter 60%

    • Heart muscle 15%

    • Arteries & veins 60%

    • Skin 40%

    • Tendon 2%-5%

  • Notes:

  • For Hookean materials d T/d l = const

  • Piece-wise linear model (practical)

7 5 2 other expressions
7.5.2 Other expressions

  • For finite deformation of elastic body, strain energy (or elastic potential), W, is often used

  • For elastic, isotropic material W is function of strain invariants.

  • Examples: Mooney(1940), Rivlin(1947), Rivlin & Saunders(1951), Green & Adkins (1960)*

7 6 quasi linear viscoelasticity of soft tissues
7.6 Quasi-Linear Viscoelasticity of Soft Tissues

  • Biological materials not elastic, history of strain affects stress, loading unloading difference.

  • Linear theory of viscoelasticity, continuous relaxation spectrum (sec. 2.11), combination of an infinite no. of Voigt & Maxwell elements.

  • Nonlinear theory, a sequence of springs of different natural length with no. of springs increases with increasing strain.

  • Linear viscoelasticity for small oscillation; for finite deformation, nonlinear stress-strain characteristics.


  • Consider a cylindrical specimen subjected to tensile load, a step increase in elongation imposed, stress ~ function of time t & stretch ratio l





Tensile stress =

Instantaneous response +

decrease due to past history

Experimental determination of T(e)[l] and G(t)

7 6 1 elastic response experimental
7.6.1 Elastic Response (experimental)

  • By definition T (e)(l) is instantaneous tensile stress generated by a step stretch; transient stress waves due to sudden loading will be added.

  • Assumptions:

    • G(t) is continuous

    • T (e)(l) can be approximated by T(l) with high loading rate


  • G(0)=1, if l is increased from 0 to l in time interval e, at t= e we have

7 6 2 reduced relaxation function
7.6.2 Reduced Relaxation Function

  • Assume Relaxation function = sum of exponential functions, and identify exponents & coefficients

  • Experiment cut off too early can induce error

  • Non-uniqueness of the fitting

  • Notes

    • A law based on G(t) as t goes to infinite is unreliable

    • Other experiments should be utilized to determine the relaxation function.

7 6 3 special characteristics of hysteresis of living tissues
7.6.3 Special Characteristics of Hysteresis of Living Tissues

  • Hysteresis loop is almost independent of strain rate within several decades of rate variation.

  • Incompatible with viscoelastic model with finite no. of springs & dashpots. (discrete relaxation rate constants)

  • A continuous distribution of exponents ni should be considered





7 6 4 g t related to hysteresis
7.6.4 G(t) related to Hysteresis Tissues

ts: time constant for creep at const stress

te: time constant for relaxation at const strain

ER: residual of elastic response

  • Standard linear solid

Frequency response function
Frequency response function Tissues

Lead compensator in control engineering

Note: tand is a measure of internal damping, if it is not frequency dependent,

the peak must spread out – superposing a large no. of Kelvin models

7 6 5 continuous spectrum of relaxation
7.6.5 Continuous Spectrum of Relaxation Tissues

  • Relaxation function

How to find s t
How to find S( Tissuest)

  • Idea: find S that will make G(t), J(t) and M(w) to match with the experimental data. M(w) to be nearly constant for a wide range of frequency.

Constant damping for Tissuest1<1/w<t2

Maximum damping at w=1/√t1t2

Reduced relaxation function
Reduced relaxation function Tissues

Note: continuous relaxation spectrum

Reduced creep function
Reduced Creep Function Tissues

  • Laplace transform is used to solve for creep function from reduced relaxation function

Notes Tissues

  • Relaxation spectrum Eqs.(30)(31) & relaxation function Eq.(34) & complex modulus Eq. (32) & damping (33) work very well for many living tissues.

  • Creep functions Eqs. (45)(46) does not work so well; good for papillary muscle not for blood vessels & lung tissues

  • Creep is a more nonlinear process & does not obey the quasi-linear hypothesis. Microstructure movement in creep process is different from that of relaxation or oscillation, analog to metal at high temperature.

7 6 6 a graphic summary

Maxwell Tissues



Hysteresis vs. frequency

Biological soft tissues

7.6.6 A graphic Summary

  • Generalization

  • Large no. of Kelvin units in series

  • Nonlinear elastic springs, same type

  • Size distribution of springs & dampers

7 6 7 history remarks
7.6.7 History Remarks Tissues

  • Above theory (Fung, 1972)

  • Hysteresis insensitive to frequency (Becker & Foppl 1928)

  • Structure damping (Garrick 1940)

  • Earth’s crust internal friction (Routbart & Sack 1966)

  • Kink in dislocation line & kink energy barrier (Manson 1969)

  • Special plasticity theory (Bodner 1968)

  • Eq (47) by Wagner (1913)

7 6 8 oscillatory stretch
7.6.8 Oscillatory Stretch Tissues

If the amplitude is small then linear viscoelasticity

theory can be used.

arteries: amplitude of strain < 4% on top of l = 1.6

Reproducible up to 16h following removal of tissue

nonlinear elasticity of the tissue

7 6 9 example collagen fibers in uniaxial extension
7.6.9 Example: Collagen Fibers in Uniaxial Extension Tissues

  • Experimental results in 7.3; regimes

    • Small strain toe region, Lagrange stress is nonlinear function of stretch ratio

    • Linear regime

    • Non-physiological, overly extended, & failing regime

  • Stress-strain relationship in toe

  • Both toe and linear region (Mooney-Rivlin material)

T: Lagrange stress, l: stretch ratio, m=finite in toe region, m=0 in linear region

  • Note: Mooney-Rivlin is isotropic, collagen fibers are transverse orthotropic; let x3 axis be axis of fiber then strain energy function

  • If stress T in Eq.(50) is the elastic response T(e) then from quasi-linear theory, stress at time t due to strain history l(t) is:

Limitations extensions
Limitations & Extensions transverse orthotropic; let x

  • QLV model work reasonable for skin, arteries, veins, tendons, ligaments, lung parenchyma, pericardium, muscle & ureter in relaxed state.

  • In reality, specific tissue may have a spectrum with a localized peaks & valleys not considered in QLV

  • No experimental identification of a relaxation function which is dependent on invariants of stress, strain & strain rate.

  • General theory of nonlinear viscoelastic materials (Green & Rivlin 1957), tensorial power series expansion.

7 7 incremental laws
7.7 Incremental Laws transverse orthotropic; let x

  • Mechanical properties of soft tissues such as arteries, muscle, skin, lung, ureter, mesentery are inelastic (hysteresis, anisotropic, nonlinear stress-strain relationship)

  • Incremental laws: linearized relationship between incremental stresses & strains by small perturbation about an equilibrium condition.

Rabbit mesentery

l transverse orthotropic; let xbetween 1 & 1.24

Rabbit Mesentery


Small loops are not parallel to each other, neither are tangent to loading unloading curves

Incremental moduli should be determined by incremental experiments

Pseudo-elasticity laws is simpler for full range of deformation

7 8 pseudo elasticity
7.8 Pseudo-Elasticity transverse orthotropic; let x

  • Simplification of QLV to pseudo-elastic equation (for preconditioned tissue)

  • For loading & unloading branches, the stress-strain relationship is unique

  • Treat the material as one elastic material in loading and another in unloading

  • Hysteresis independent of strain rate

  • 1000-fold change of strain rate vs 1 to 2 fold change of stress of a given strain. Ultrasound experiments suggest lower limit of relaxation time 10-8,

  • To describe stress-strain relationship in loading & unloading by a law of elasticity and it can be further simplified if assuming a strain energy function exists.

7 9 biaxial loading experiments on soft tissues
7.9 Biaxial Loading Experiments on Soft Tissues transverse orthotropic; let x

Rectangular specimen of uniform thickness in biaxial loading

Typical display of specimen on vda monitor
Typical display of specimen on VDA monitor transverse orthotropic; let x


Testing of skin

Lung tissue, with thickness measurement

Digital computer control of stretching in two directions

  • Key Issues:

  • Need to control boundary conditions, edges must be allowed to expand freely

  • In target region stress and strain should be uniform, away from outer edges

  • Strain is measured optically to avoid mechanical disturbance

7 9 1 whole organ experiments
7.9.1 Whole organ experiments transverse orthotropic; let x

  • Alternative: test whole organ

  • For lung, whole lobe in vivo or in vitro

  • For artery, deformation when internal/external pressure are changed or longitudinal tension is imposed.

  • In whole organ test, tissues not subjected to traumatic excision, close to in vivo condition

  • Difficulty in analyzing whole organ data

  • Complement to excised experiments

7 10 three dimensional stress and strain states
7.10 Three-dimensional Stress and Strain States transverse orthotropic; let x

  • Consider a rectangular plate of uniform thickness, orthotropic material

  • Two pairs of forces F11, F22 act on the edges; no shear stress and x, y are principal axes

    • s– Cauchy stress (~equilibrium Eq. )

    • T- Lagrange stress (~lab)

    • S- Kirchhoff stress (~strain energy)

    • r0– density at zero stress

For large deformation transverse orthotropic; let x

7 11 strain energy function
7.11 Strain-energy Function transverse orthotropic; let x

  • Strain potential or Strain-energy function

  • W: strain energy per unit mass of tissue (J/kg)

  • r0 density in zero-stress state (kg/m3)

  • r0W: strain energy per unit volume (J/m3)

  • Let W be expressed in terms of strain components E11, E22, E33, E12, E21, E23, E32, E31, E13, and using symmetric properties.

General relationship between cauchy lagrange kirchhoff stresses
General relationship between Cauchy, Lagrange & Kirchhoff Stresses

  • s– Cauchy stress (~equilibrium Eq. )

  • T- Lagrange stress (~lab)

  • S- Kirchhoff stress (~strain energy)

Pseudoelasticity pseudo strain energy function
Pseudoelasticity & pseudo-strain energy function Stresses

  • If a material is perfectly elastic then a strain-energy function exists (thermodynamics)

  • Living tissues are not perfectly elastic a strain energy function not exist.

  • Fact: after preconditioning, cyclic loading & unloading stress-strain relationship are strain-rate independent.

  • Loading & unloading curves can be treated as two uniquely defined stress-strain relationships, each associated with a strain-energy function

  • Pesudo-elasticity curve, pseudo-strain energy function

  • Must be justified by experiments

Modification for incompressible materials
Modification for incompressible materials Stresses

  • Eqs.(6)(8) should be modified for incompressible materials as

P: pressure, indeterminante quantity solved from

equation of motion or equilibrium

7 12 constitutive equation of skin
7.12 Constitutive Equation of Skin Stresses

  • For membranous material, biaxial test sufficient to yield 2D constitutive equation

  • 3 components of stress & 3 components of strains

  • Biaxial experiments on rabbit abdomen skin (Lanir & Fung 1974, Tong & Fung 1976 pesudo-elasticity)

  • x1 direction along rabbit’s head to tail, x2 direction of width ; Green strains E1=E11, E2=E22

Pseudo strain energy function

Biphasic Stresses

Low stress

High stress

Pseudo-strain energy function

c, ai, ai, gi are constants to be determined

E12, shear strain is kept zero in the experiment

  • From Eq.(2) we compute the derivatives ∂S Stresses11/ ∂E1, ∂S22/ ∂E2, ∂S11/ ∂E2=∂S22/ ∂E1,

  • By requiring that the equations fit the data at some selected points, we can obtain the coefficients of the strain-energy function

  • Four points A, B, C, D were used in Fig. 7.12.1

l Stressesx=1


□ experimental data Stresses

●: a1=a2=1020, a4=254, g’s=0, a1=3.79, a2=12.7, a4=.587, c=.779

x: a1=a2=1020, a4=254, a1=3.79, a2=18.4, a4=.587, c=.779, g1= g2=0, g4=g5=15.6

Notes: Stresses

the fitting is remarkably good!

g’s can be omitted

Summary Stresses

  • Pseudo-strain energy function is suitable for rabbit skin for stress and strain in the physiological range.

  • The third order terms are unimportant and can be neglected

  • Further simplification

For higher stress & strains range

7 13 generalized viscoelastic relationship
7.13 Generalized Viscoelastic Relationship Stresses

  • Constitutive equations for biosolids, viscoelastic bodies;

  • Generalization of QLV model of tissues from 1D to 2D & 3D

Sij: Kirchhoff stress tensor

Eij: Green’s strain tensor

Gijkl: reduced relaxation function tensor


S(e): elastic stress tensor corresponding to strain tensor E

  • Notes:

  • Assume elastic response can be approximated by pseudoelastic stresses

  • Gijkl is tensor of rank 4, for isotropic it has two indept. components

  • and more for anisotropic

  • Gijkl has a continuous relaxation spectrum

7 14 complementary energy function inversion of stress strain relationship
7.14 Complementary Energy Function: Inversion of stress-strain relationship

  • Given S = f(E) can we find E = g(S) ?

  • Linear elasticity, Hooke’s law can be expressed in both way, but not for nonlinear elasticity

  • Strain energy function in Sec. 7.12 Eq.(7) can be easily inverted; useful for calculating strains from known stresses or using complementary energy for numerical analysis.

  • Stress function in finite deformation analysis

Complementary energy function
Complementary energy function stress-strain relationship

If a complementary energy function can be found then Eq.(3) will give the inversion of the stress-strain relationship

  • Solve Eq. (10) stress-strain relationship

  • Sub. Into Eq.(12) will yield complementary function which is

  • function of stresses

example stress-strain relationship

  • Consider a 2D problem

By this Eq. (12) can be expressed as function of P

Universal for all materials with exponential strain-energy function

7 5 constitutive equation derived according to microstructure
7.5 Constitutive Equation Derived according to Microstructure

  • Two ways to build constitutive equations of a continua

    • Ground up approach: from elementary particle to atoms, from atoms to molecules, from molecules to macromolecules, to proteins, cells, tissues, and organs

    • Top down analysis:

  • Biomechanics

    • Explain constitutive equation in terms of its microstructure or ultrastructure, or to derive it from microstructure

    • Examples in literature

Summary Microstructure

  • Elastic materials in biosolids

  • Collagen

  • Thermodynamics of Elastic deformation

  • Soft tissues under uniaxial loading

  • QLV model of soft tissues

  • Incremental laws,

  • Pseudo-elasticity, biaxial loading experiment, strain energy function, skin example

  • Generalized viscoelastic relationship

  • Complementary energy function