7.5 Behavior of Soft Tissues under uniaxial Loading - PowerPoint PPT Presentation

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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. – PowerPoint PPT presentation

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Title: 7.5 Behavior of Soft Tissues under uniaxial Loading


1
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

2
Preconditioning
Cyclic response of dogs carotid artery l1
stretch ratio referred to zero-stress length of
segment, 37 deg C, 0.21 cycles/min
3
Hysteresis of rabbit papillary (??) muscle
Increasing strain rate
4
Relaxation of rabbit mesentery(???)
5
Long-term relaxation
G(t)
Log 10 t
6
Creep of papillary muscle of rabbit
Log 10 t
7
summary
  • 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

8
7.5.1 Stress Response in loading and unloading
Lagrange stress
9
  • Notes
  • For Hookean materials d T/d l const
  • Piece-wise linear model (practical)

10
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)

11
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12
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13
Mechanical properties of Cornea
14
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.

15
hypothesis
  • Consider a cylindrical specimen subjected to
    tensile load, a step increase in elongation
    imposed, stress function of time t stretch
    ratio l

G(t)
1
0
t
16
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17
Tensile stress
Instantaneous response
decrease due to past history
Experimental determination of T(e)l and G(t)
18
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

19
Justification
  • G(0)1, if l is increased from 0 to l in time
    interval e, at t e we have

20
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.

21
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

Discrete spectrum
Continuous spectrum
22
7.6.4 G(t) related to Hysteresis
ts time constant for creep at const stress te
time constant for relaxation at const strain ER
residual of elastic response
  • Standard linear solid

23
Frequency response function
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
24
7.6.5 Continuous Spectrum of Relaxation
  • Relaxation function

25
How to find S(t)
  • 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.

26
Constant damping for t1lt1/wltt2 Maximum damping at
w1/vt1t2
27
Maximum damping
28
Reduced relaxation function
Note continuous relaxation spectrum
29
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30
Reduced Creep Function
  • Laplace transform is used to solve for creep
    function from reduced relaxation function

31
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32
Notes
  • 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.

33
7.6.6 A graphic Summary
  • Generalization
  • Large no. of Kelvin units in series
  • Nonlinear elastic springs, same type
  • Size distribution of springs dampers

34
7.6.7 History Remarks
  • Above theory (Fung, 1972)
  • Hysteresis insensitive to frequency (Becker
    Foppl 1928)
  • Structure damping (Garrick 1940)
  • Earths 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)

35
7.6.8 Oscillatory Stretch
If the amplitude is small then linear
viscoelasticity theory can be used. arteries
amplitude of strain lt 4 on top of l
1.6 Reproducible up to 16h following removal of
tissue
nonlinear elasticity of the tissue
36
7.6.9 Example Collagen Fibers in Uniaxial
Extension
  • 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, mfinite in
toe region, m0 in linear region
37
  • 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

38
Limitations Extensions
  • 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.

39
7.7 Incremental Laws
  • 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.

40
Rabbit Mesentery
Note 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
41
7.8 Pseudo-Elasticity
  • 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.

42
7.9 Biaxial Loading Experiments on Soft Tissues
Rectangular specimen of uniform thickness in
biaxial loading
43
Typical display of specimen on VDA monitor
Applications 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

44
7.9.1 Whole organ experiments
  • 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

45
7.10 Three-dimensional Stress and Strain States
  • 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

46
For large deformation
47
7.11 Strain-energy Function
  • 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.

48
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49
General relationship between Cauchy, Lagrange
Kirchhoff Stresses
s Cauchy stress (equilibrium Eq. ) T- Lagrange
stress (lab) S- Kirchhoff stress (strain energy)
50
Pseudoelasticity pseudo-strain energy function
  • 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

51
Modification for incompressible materials
  • Eqs.(6)(8) should be modified for incompressible
    materials as

P pressure, indeterminante quantity solved from
equation of motion or equilibrium
52
7.12 Constitutive Equation of Skin
  • 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 rabbits head to tail, x2
    direction of width Green strains E1E11, E2E22

53
Pseudo-strain energy function
c, ai, ai, gi are constants to be determined
E12, shear strain is kept zero in the experiment
54
  • From Eq.(2) we compute the derivatives ?S11/ ?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

55
lx1
ly1
56
  • ? experimental data
  • ? a1a21020, a4254, gs0, a13.79, a212.7,
    a4.587, c.779
  • x a1a21020, a4254, a13.79, a218.4, a4.587,
    c.779, g1 g20, g4 g515.6

57
Notes the fitting is remarkably good! gs can
be omitted
58
Summary
  • 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
59
7.13 Generalized Viscoelastic Relationship
  • Constitutive equations for biosolids,
    viscoelastic bodies
  • Generalization of QLV model of tissues from 1D
    to 2D 3D

Sij Kirchhoff stress tensor Eij Greens strain
tensor Gijkl reduced relaxation function
tensor Gijkl(0)1 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

60
7.14 Complementary Energy Function Inversion of
stress-strain relationship
  • Given S f(E) can we find E g(S) ?
  • Linear elasticity, Hookes 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

61
Complementary energy function
If a complementary energy function can be found
then Eq.(3) will give the inversion of the
stress-strain relationship
62
  • Solve Eq. (10)
  • Sub. Into Eq.(12) will yield complementary
    function which is
  • function of stresses

63
example
  • Consider a 2D problem

By this Eq. (12) can be expressed as function of
P Universal for all materials with exponential
strain-energy function
64
Universal function Q(P) for inverting a nonlinear
stress-strain relationship
65
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

66
Summary
  • 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
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