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Bioviscoelastic SoliBioviscoelastic Solidsds

Prof. Ming-Shaung JuProf. Ming-Shaung JuDept. of Mechanical EngineeriDept. of Mechanical Engineeri

ngngNCKUNCKU

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7.1 Introduction7.1 Introduction Soft tissues: muscles, blood vessel, ligamSoft tissues: muscles, blood vessel, ligam

ent, tendon, cartilage, nerve ent, tendon, cartilage, nerve Elastic materials: abductin, resilin, elastiElastic materials: abductin, resilin, elasti

n, collagenn, collagen Thermodynamics of elastic deformation: Thermodynamics of elastic deformation:

two sources of elasticitytwo sources of elasticity Constitutive equations of soft tissuesConstitutive equations of soft tissues

Uni-axial tension & quasilinear viscoelasticitUni-axial tension & quasilinear viscoelasticityy

Biaxial loadingBiaxial loading

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7.1 Introduction (cont’d)7.1 Introduction (cont’d)

3D stress & strain in large deforma3D stress & strain in large deformation, pseudo-strain energy functiotion, pseudo-strain energy functionn

Example: skinExample: skin General viscoelastic relationGeneral viscoelastic relation Computing strain from known streComputing strain from known stre

ssss

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7-2 Some Elastic 7-2 Some Elastic MaterialsMaterials

Actin Actin 肌動蛋白肌動蛋白 Elastin Elastin 彈性蛋白彈性蛋白 Collagen Collagen 膠原蛋白膠原蛋白 Aldehyde Aldehyde 醛醛 Resilin Resilin 彈性蛋白彈性蛋白 Abductin Abductin 外展蛋白外展蛋白

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7.2.1 Actin7.2.1 Actin In all muscles, leukocytes, red blood cells, endIn all muscles, leukocytes, red blood cells, end

othelial cells & many otherothelial cells & many other Strength of a single actin filament (Kishino & Strength of a single actin filament (Kishino &

Yanagida, 1988)Yanagida, 1988) Single actin filament (~7nm Single actin filament (~7nm ) labelled with ph) labelled with ph

alloidin-tetramethyl-rhodaminealloidin-tetramethyl-rhodamine Two microneedles connected to micromanipulTwo microneedles connected to micromanipul

ators; one very flexible & the other stiff;ators; one very flexible & the other stiff; Needles coated with monometric myosin to inNeedles coated with monometric myosin to in

crease affinity with actincrease affinity with actin

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Actin (cont’d)Actin (cont’d) Stiff needle was pulled until actin broke; forStiff needle was pulled until actin broke; for

ce was calculated from bending of flexible nce was calculated from bending of flexible needleeedle

For actin filaments of length 4 to 32 um, tensFor actin filaments of length 4 to 32 um, tensile force was 108 +/- 5 pN (n=61) without breaile force was 108 +/- 5 pN (n=61) without breaking, comparable to single unit in muscle duking, comparable to single unit in muscle during isometric contractionring isometric contraction

Tensile strength 108pN*4/Tensile strength 108pN*4/7722(nm)(nm)22=2.2x10=2.2x1066N/N/mm22 or 2.2MPa or 2.2MPa

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7.2.2 Elastin (7.2.2 Elastin ( 彈性蛋白彈性蛋白 )) Most linearly elastic biosolid materialMost linearly elastic biosolid material Cylindrical specimen of elastin subjectCylindrical specimen of elastin subject

ed to uni-axial loaded to uni-axial load

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Ligamentum nuchae (Ligamentum nuchae ( 項韌帶項韌帶 ))

denature(denature(變質變質 ))Tensile strain: change of lenTensile strain: change of len

gth / initial lengthgth / initial lengthStress: load / initial cross secStress: load / initial cross sec

tional area (at zero stress)tional area (at zero stress)Almost linear with small hysAlmost linear with small hys

teresisteresisElastic limit up to Elastic limit up to =1.6 =1.6

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Elastin (cont’d)Elastin (cont’d) Protein found in vertebratesProtein found in vertebrates Thin strands in skin & in areolar(Thin strands in skin & in areolar( 蜂窩蜂窩 ) connective ) connective

tissue; walls of arteries & veins, especially near hetissue; walls of arteries & veins, especially near heart; prominent component of lung tissueart; prominent component of lung tissue

Ligamentum nachae is almost pure elastin; Ligamentum nachae is almost pure elastin; small amount of collagen can be denatured by hesmall amount of collagen can be denatured by he

ating to 66ating to 66◦◦C +. C +. Note the process does not change properties of elNote the process does not change properties of el

astinastin Provide elasticity to arteries & lung parenchyma tiProvide elasticity to arteries & lung parenchyma ti

ssue; keep skin tissue smooth; ssue; keep skin tissue smooth;

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7.2.3 Incomplete fixation of ela7.2.3 Incomplete fixation of elastin in Aldehyde (stin in Aldehyde ( 醛醛 ))

Fixation is commonly used in histology studiFixation is commonly used in histology studies. When the tissue has elastin in it artifacts es. When the tissue has elastin in it artifacts may induced by elastic recoverymay induced by elastic recovery

Tissue usually Tissue usually fixedfixed by formalin, formaldehy by formalin, formaldehyde(de( 蟻醛蟻醛 ), or glutaraldehyde then ), or glutaraldehyde then embeddedembedded, , ssectionedectioned & & stained stained..

If elastin specimen is stretched under tensioIf elastin specimen is stretched under tension & soaked in these agents, on release of tensn & soaked in these agents, on release of tension it does not return to unstretched length ion it does not return to unstretched length (40%-70% recovery) but remain elastically(40%-70% recovery) but remain elastically

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Elastic recovery of elastin after fixation in formalin & glutaraldehyde

Note: elastic recovery occurs at all stretch ratios, it is not fixed !

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If a tissue is fixed in a state of tension, e.If a tissue is fixed in a state of tension, e.g., inflated lung, or a disintended artery,g., inflated lung, or a disintended artery, & &

sectioned under no load, residual stress sectioned under no load, residual stress in elastin fibers will be released, in elastin fibers will be released,

length of elastic fibers will be shortened length of elastic fibers will be shortened to length at zero stress state.to length at zero stress state.

Fixed part of tissue will be buckled by sFixed part of tissue will be buckled by shortening of elastinhortening of elastin

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(a)(a) Lung parenchyma of a spidLung parenchyma of a spider monkey, fixed in glutaraler monkey, fixed in glutaraldehyde & embedded in wax.dehyde & embedded in wax.

Note: tissue was allowed to shrNote: tissue was allowed to shrink in a stress-free state.ink in a stress-free state.

(b) Same tissue embedded in c(b) Same tissue embedded in celloidin, a hard plastic. Tisselloidin, a hard plastic. Tissue is not allow to shrink.ue is not allow to shrink.

(a)

(b)

Pulmonary alveoli 肺泡

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Winkle appearance of most photomicWinkle appearance of most photomicrographs of lung tissue is artifact caurographs of lung tissue is artifact caused by unsuspected elastic recovery sed by unsuspected elastic recovery of elastin fibers! of elastin fibers!

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7.2.4 The Elastin Molecule7.2.4 The Elastin Molecule Molecular structure of tropoelastin beMolecular structure of tropoelastin be

en sequenced; tropoelastin formed inten sequenced; tropoelastin formed intracellularly & cross-linked extracellularacellularly & cross-linked extracellulary.ry.

Poly (V PG VG), poly (V PG F GV G AG), Poly (V PG VG), poly (V PG F GV G AG), poly (VPGG) on poly (VPGG) on -irradiation cross-link-irradiation cross-linking are elasticing are elastic

These polypeptides will self-assembly iThese polypeptides will self-assembly into more ordered molecular assembliento more ordered molecular assemblies on raising temperature.s on raising temperature.

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Elastin Molecule (cont’d)Elastin Molecule (cont’d) Sources of elasticitySources of elasticity

Decrease of entropyDecrease of entropy Increase of internal energyIncrease of internal energy

Entropy theory: liberation mechanism or rockEntropy theory: liberation mechanism or rocking of peptides segment contributed to entroping of peptides segment contributed to entropy S =k ln y S =k ln

Self assembly mechanism has a critical tempeSelf assembly mechanism has a critical temperature of 25rature of 25 。。 CC

From lung tissue of rat Debes & Fung suggests From lung tissue of rat Debes & Fung suggests that inverse temperature transition phenomethat inverse temperature transition phenomenon may not a major mechanism for whole elnon may not a major mechanism for whole elastinastin

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7.2.5 Resilin & Abductin7.2.5 Resilin & Abductin Resilin: similar to elastin in mechanical behaResilin: similar to elastin in mechanical beha

vior, different in chemical compositionvior, different in chemical composition Protein in arthropods (Protein in arthropods ( 節足動物節足動物 ), hard when dr), hard when dr

ied, soft & rubbery in natural state 50-60% waied, soft & rubbery in natural state 50-60% water.ter.

Within stretch ratio Within stretch ratio =1-2, E~1.8x10=1-2, E~1.8x1077 dyn/cm dyn/cm22 or 1.8 MPa; G~0.6MPaor 1.8 MPa; G~0.6MPa

Insects uses resilin as elastic joints for wings; Insects uses resilin as elastic joints for wings; fleas & locusts use resilin at base of hind legs fleas & locusts use resilin at base of hind legs as catapults in jumping.as catapults in jumping.

Abductin: in scallops’ hinges to open valvesAbductin: in scallops’ hinges to open valves About same elastic moduli as elastinAbout same elastic moduli as elastin

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Scallop: hinge & Scallop: hinge & abductor muscleabductor muscle

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7.2.6 Elasticity due to entropy 7.2.6 Elasticity due to entropy & Internal Energy Changes& Internal Energy Changes

Elastin, resilin & abductin, long flexiblElastin, resilin & abductin, long flexible molecules jointed together by cross-le molecules jointed together by cross-linking to form 3D networksinking to form 3D networks

Molecules are convolute and thermal eMolecules are convolute and thermal energy keeps them in constant thermal nergy keeps them in constant thermal motion; motion;

Configurations (entropy) change with Configurations (entropy) change with strain; from entropy change elastic strstrain; from entropy change elastic stress appears.ess appears.

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3G.E and 0.5 material ibleincompressfor

ratioPoisson : modulus, sYoung' :E

(2) 12

re temperatuabsolute :T

next the&link -cross onebetween molecule of weight averaged :M

8.3x10 constant gas :R

excluded. shouldght water wei,g/cm density,

(1)

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3

G)(E

moldeg/erg

:

M

TRG

Treloar (1967)

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Crystalline materials derive elastic stress Crystalline materials derive elastic stress from changes in internal energy, i.e., elastic from changes in internal energy, i.e., elastic modulus related to strain of crystal lattice.modulus related to strain of crystal lattice.

Eq.(1) does Eq.(1) does notnot apply to apply to crystalline crystalline materialsmaterials, fibers whose elasticity comes , fibers whose elasticity comes partly from internal energy changes & partly partly from internal energy changes & partly from entropy changes.from entropy changes.

Most biological materials that can sustain Most biological materials that can sustain finite strain have rubbery elasticity but finite strain have rubbery elasticity but not not allall. .

HairHair can be stretched to 1.7 times & spring can be stretched to 1.7 times & spring back but this is due to keratin has two back but this is due to keratin has two crystalline forms crystalline forms & & helices; when helices; when stretched some stretched some helices change to helices change to helices. helices.

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abductin

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7.2.8 Crystallization due 7.2.8 Crystallization due to strainto strain

Raw rubber can be stretched several times its lengRaw rubber can be stretched several times its length & held extended; stress relaxation is almost coth & held extended; stress relaxation is almost completemplete

Stress relaxation of rubber is due to crystallization.Stress relaxation of rubber is due to crystallization. Stretching extends molecules so they run parallel Stretching extends molecules so they run parallel to each other and crystallize. Heating disrupts the to each other and crystallize. Heating disrupts the crystalline structure.crystalline structure.

Silk is crystallized under a high shear strain rate aSilk is crystallized under a high shear strain rate and emerge as a fiber. Composition: fibroin(nd emerge as a fiber. Composition: fibroin( 繭絲蛋繭絲蛋白白 ) & sericin() & sericin( 絲蛋白絲蛋白 ); ); sericin is gummy & dissolvablsericin is gummy & dissolvable in warm water, e in warm water,

Young’s modulus of silk: 10Young’s modulus of silk: 1044MPa, broke at MPa, broke at =1.2=1.2

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7.3 Collagen7.3 Collagen Basic structural element for soft & hard tiBasic structural element for soft & hard ti

ssues in animals, providing mechanical issues in animals, providing mechanical integrity & strength to our bodies.ntegrity & strength to our bodies.

Important to man as steel to civilization.Important to man as steel to civilization. Main load carrying element in blood vessMain load carrying element in blood vess

els, skin, tendons, cornea, sclera(els, skin, tendons, cornea, sclera( 鞏膜鞏膜 ), b), bone, fascia(one, fascia( 筋膜筋膜 ), dura mater(), dura mater( 硬腦膜硬腦膜 ), ut), uterian cervix(erian cervix( 子宮頸子宮頸 ).).

Collagen molecules, how they wind into fCollagen molecules, how they wind into fibrils and how fibrils organized into fiberibrils and how fibrils organized into fibers, fibers into tissues.s, fibers into tissues.

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In each stage of organization, new In each stage of organization, new features of mechanical properties are features of mechanical properties are acquired.acquired.

In physiology & biomechanics, major In physiology & biomechanics, major attention focused on organ & tissue attention focused on organ & tissue level.level.

Relationship between function & Relationship between function & morphology of collagen in different morphology of collagen in different organs.organs.

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7.3.1 Collagen Molecules7.3.1 Collagen Molecules A protein containing sizable domains of triplet-A protein containing sizable domains of triplet-

helical conformation and functioning as supporhelical conformation and functioning as supporting elements in an extra-cellular matrix.ting elements in an extra-cellular matrix.

Arrangement of amino acids: every 3rd residue Arrangement of amino acids: every 3rd residue glycine(glycine( 甘膠酸甘膠酸 ), proline & OH-proline follow eac), proline & OH-proline follow each other; 3 residues per turn; left hand helicesh other; 3 residues per turn; left hand helices

Chains coiled follow a right-hand twist w. a pitcChains coiled follow a right-hand twist w. a pitch of 8.6 nm; 3 helical chains arranged w. slight dh of 8.6 nm; 3 helical chains arranged w. slight displacement longitudinally; 0.291 nm (-110 deg); isplacement longitudinally; 0.291 nm (-110 deg); distance between each 3rd glycine 0.873 nmdistance between each 3rd glycine 0.873 nm

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Collagen triple helixCollagen triple helix

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Types of collagensTypes of collagens

12 types of collagens has been 12 types of collagens has been identifiedidentified

1(I): 1(I): chain of type I collagen chain of type I collagen

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3 types of collagen:3 types of collagen:Chain composition & glycosylation Chain composition & glycosylation (( 醣苷化醣苷化 ))

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7.3.2 Aggregate Structure7.3.2 Aggregate Structure

Relation of function and structure of Relation of function and structure of collagen aggregates (Miller, 1988)collagen aggregates (Miller, 1988)

Fiber-forming collagensFiber-forming collagens

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A: I, II, III, V, KB: IV basement membraneC: VI: placental villi(胎盤絨毛 )D: VII: placental membrane

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Distribution of collagens in human Distribution of collagens in human bodybody

Type I, ubiquitous, can be isolated from any tissue or Type I, ubiquitous, can be isolated from any tissue or organ, e.g., bone, dermis, placental membrane & tenorgan, e.g., bone, dermis, placental membrane & tendondon

Type II, mainly in hyaline cartilage and cartilage-like Type II, mainly in hyaline cartilage and cartilage-like tissues such as nucleus pulposus of vertebral body & tissues such as nucleus pulposus of vertebral body & body of eyes.body of eyes.

Type III, constitutants of dermis, blood vessel walls & Type III, constitutants of dermis, blood vessel walls & other more distensible connective tissues; ubiquitous other more distensible connective tissues; ubiquitous too.too.

Type V, relative minor but has distribution like type IType V, relative minor but has distribution like type I Type K (XI), like Type II, mainly, in cartilageType K (XI), like Type II, mainly, in cartilage Type IX & X, minor constituents of hyaline cartilages; Type IX & X, minor constituents of hyaline cartilages;

short-chain collagens.short-chain collagens.

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7.3.3 Collagen fibrils & 7.3.3 Collagen fibrils & fibersfibers

Fiber-forming collagen molecules: collectioFiber-forming collagen molecules: collection of tropocollagen molecules forms a collagen of tropocollagen molecules forms a collagen fibril;n fibril;

Fibril appears cross-striated (EM graphs), exFibril appears cross-striated (EM graphs), examples: tendon & skinamples: tendon & skin

Period of striation D = 64 nm (native) or 68 nPeriod of striation D = 64 nm (native) or 68 nm (moisture)m (moisture)

A gap of 0.6D is left between ends of successA gap of 0.6D is left between ends of successive molecules. The gap appears as the lighteive molecules. The gap appears as the lighter part of the striation.r part of the striation.

Current view, not perfectly parallel, bent, vaCurrent view, not perfectly parallel, bent, varying spacingrying spacing

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EM graphs of tendon & EM graphs of tendon & skinskin

tendon skin

X 24,000

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Model of organization of Model of organization of moleculesmolecules

Quarter-staggerLength of each molecule/ D = 4.4

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Diameter of fibril 20-40 nm, depends on aniDiameter of fibril 20-40 nm, depends on animal species & tissue.mal species & tissue.

Bundles of fibrils form fibers, diameter 0.2 tBundles of fibrils form fibers, diameter 0.2 to 12 um. Fibers are colorless under light mio 12 um. Fibers are colorless under light microscope and birefringent in polarized light; croscope and birefringent in polarized light;

In tendon as long as tendon itself, in connecIn tendon as long as tendon itself, in connective tissues length varies considerably;tive tissues length varies considerably;

Packaging of collagen fibers has many hieraPackaging of collagen fibers has many hierarchies: parallel-fiber for tendon, (Fig.7.3:6)rchies: parallel-fiber for tendon, (Fig.7.3:6)

Fibers assembled into fascicles & enclosed iFibers assembled into fascicles & enclosed in a sheath of reticular membrane to form a tn a sheath of reticular membrane to form a tendon.endon.

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Hierarchy of structureHierarchy of structure

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7.3.4 Wavy Course of 7.3.4 Wavy Course of FibersFibers

Rat’s tail tendon in polarized light Rat’s tail tendon in polarized light microscope observation light & dark microscope observation light & dark pattern period 100 um: waviness of pattern period 100 um: waviness of collagen fiber in fasciclecollagen fiber in fascicle

When tendon is stretched waviness of When tendon is stretched waviness of crimped fibers decreases and wave shape is crimped fibers decreases and wave shape is planar.planar.

Wave parameters (Table 7.3:2)Wave parameters (Table 7.3:2) When stretched bending angle When stretched bending angle 00 decreases decreases Fiber diameter is age dependent. For rat, it Fiber diameter is age dependent. For rat, it

increases from 100 to 500 nm as rat ages!increases from 100 to 500 nm as rat ages!

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Basic mechanical unit of tendon is bent collagen fiBasic mechanical unit of tendon is bent collagen fibers.bers.

Q: Are the fibers intrinsically bent because of some Q: Are the fibers intrinsically bent because of some fine structural features of the fibrils ?fine structural features of the fibrils ?

A: curvature of fibers might be caused by shrinking A: curvature of fibers might be caused by shrinking of non-collagen components or “ground substancof non-collagen components or “ground substance” of tendon, i.e., curvature is due to buckling of fe” of tendon, i.e., curvature is due to buckling of fibersibers..

Integrity of ground substance is important for mecIntegrity of ground substance is important for mechanical integrity of tendon. hanical integrity of tendon.

Enzymatic digestion directed at non-collagen compEnzymatic digestion directed at non-collagen components changes mechanical properties of tendon.onents changes mechanical properties of tendon.

Buckling model (Dale & Baer, 1974): hyaluronic aciBuckling model (Dale & Baer, 1974): hyaluronic acid(d( 玻尿酸玻尿酸 ) may be responsible for the buckling of co) may be responsible for the buckling of collagen fibers.llagen fibers.

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In some tissues, elastin & collagen together form a In some tissues, elastin & collagen together form a composite material. Straight elastic fibers are attaccomposite material. Straight elastic fibers are attached to bent collagen fibers. Not for pulmonary alvehed to bent collagen fibers. Not for pulmonary alveolar.olar.

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7.3.5 Ground Substance (strom7.3.5 Ground Substance (stroma)a)

Collagen integrated with cells & intercellular subCollagen integrated with cells & intercellular subatanceatance

Dense connective tissues: fibrocytes, fibers of collDense connective tissues: fibrocytes, fibers of collagen, elastin, reticulin, hydrophilic gel (ground sagen, elastin, reticulin, hydrophilic gel (ground substance)ubstance)

Loose connective tissues have more amount of grLoose connective tissues have more amount of ground substance than dense connective tissuesound substance than dense connective tissues

Composition: mucopolysaccharides(Composition: mucopolysaccharides( 黏多醣黏多醣 ) or glyc) or glycosaminoglycans(osaminoglycans( 葡萄糖胺葡萄糖胺 ), tissue fluid; ), tissue fluid;

Hydration of collagen is an important problem in Hydration of collagen is an important problem in biomechanics!biomechanics!

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7.3.6 Structure of Collagenous Tiss7.3.6 Structure of Collagenous Tissuesues

Parallel fibers: tendons & ligaments; 2D & 3D nParallel fibers: tendons & ligaments; 2D & 3D networks: skins & blood vessels, intestinal mucoetworks: skins & blood vessels, intestinal mucosa & female genital tracts.sa & female genital tracts.

Tendon functions transmit tension & parallel-fTendon functions transmit tension & parallel-fiber;iber;

Ligament similar but less regular, curved & oblLigament similar but less regular, curved & oblique to motion direction; most collagenous, ligique to motion direction; most collagenous, ligamenta flava of spine & ligamentum nuchae mamenta flava of spine & ligamentum nuchae mostly elastin.ostly elastin.

Insertion of ligaments to bones, gradual transitInsertion of ligaments to bones, gradual transition from ligament to bones; rows of fibrocytes, ion from ligament to bones; rows of fibrocytes, chondrocytes, groups of osteocytes;chondrocytes, groups of osteocytes;

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Tendon inserts one end to main fibrous layer of pTendon inserts one end to main fibrous layer of periosteum. The other end to muscle, collagenous feriosteum. The other end to muscle, collagenous fibrils bound to plasma membranes & collagen fibibrils bound to plasma membranes & collagen fibers.ers.

In fasciaeIn fasciae(( 筋膜筋膜 )) & aponeuroses( & aponeuroses( 腱膜腱膜 ), parallel fiber), parallel fibers spread into sheets.s spread into sheets.

Others include: diaphragm, periosteum, perichonOthers include: diaphragm, periosteum, perichondrium, membrana fibrosa of capsules, dura mattedrium, membrana fibrosa of capsules, dura matter, sclera, fasciae, organ capsules.r, sclera, fasciae, organ capsules.

Collagen fibers in skin – 3D networks of fibrils, wCollagen fibers in skin – 3D networks of fibrils, woven into rhombic parallelgram pattern. In dry deoven into rhombic parallelgram pattern. In dry dermis, 75% collagen & 4% elastin.rmis, 75% collagen & 4% elastin.

Collagen structure in blood vessel (Sec 8.2)Collagen structure in blood vessel (Sec 8.2) Female genital tract muscle organ, smooth musclFemale genital tract muscle organ, smooth muscl

e cells in circular & spiral pattern; in uterus 30-40e cells in circular & spiral pattern; in uterus 30-40%v muscle, in cervix 10%v muscle, other connecti%v muscle, in cervix 10%v muscle, other connective tissues, ve tissues,

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7.3.7 The Stress-Strain 7.3.7 The Stress-Strain RelationshipRelationship

Nonlinear load-elongationNonlinear load-elongation Hysteresis & stress relaxationHysteresis & stress relaxation Cyclic loading variationCyclic loading variation

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Typical load-elongation for rabbit limb tendon

• O-A toe: exponential increase physiological region, normal• A-B linear, strength of tendon• B-C nonlinear• C 50-100MPa, maximum load• Rupture elongation 10-15 %• Slope tan : elastic stiffness

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韌帶機械特性韌帶機械特性

Toe

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韌帶之應力應變曲線與其組成成韌帶之應力應變曲線與其組成成分布很大關係分布很大關係

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前十字韌帶 (ACL) 與黃韌帶特性比較

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Load-elongation & stress relaxation Load-elongation & stress relaxation (ACL)(ACL)

loaded to 1/3 of failure load & unloaded at loaded to 1/3 of failure load & unloaded at same ratesame rate

Stretched until load reach FStretched until load reach F00 then stop then stop stretching, the load then relaxedstretching, the load then relaxed

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Preconditioning of ACLPreconditioning of ACL

• Cycle of loading & unloading at constant rate, unstressed for 10 min until it relaxed length recovered• Right shifting of load-elongation curves• Increased toe region• Preconditioned: repeated until successive cycle difference gone

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Reasons for preconditionReasons for precondition Internal structureInternal structure of tissue changes of tissue changes

with the cyclingwith the cycling At steady-state, no further change will At steady-state, no further change will

occur unless cycling routine changed.occur unless cycling routine changed. Upper & lower limits Upper & lower limits of loadof load should should

notnot be changed. be changed.

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FeaturesFeatures Nonlinear stress-strain, hysteresis loop in cyclic loNonlinear stress-strain, hysteresis loop in cyclic lo

ading/unloading, stress relaxation, preconditioninading/unloading, stress relaxation, preconditioningg

Common in tendons, ligaments, skin, mesentery, bCommon in tendons, ligaments, skin, mesentery, blood vessels & muscleslood vessels & muscles

Differences:Differences: Hysteresis loop small for elastin & collagen, large for muHysteresis loop small for elastin & collagen, large for mu

sclescle Relaxation: elastin small, collagen large, smooth muscle Relaxation: elastin small, collagen large, smooth muscle

very largevery large Precondition: blood vessel quick if blood flow in, slow if Precondition: blood vessel quick if blood flow in, slow if

blood flow is cut. Vasa vasorum blood flow is cut. Vasa vasorum (( 血管滋養管血管滋養管 ))

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7.3.8 Change of collagen Molecular 7.3.8 Change of collagen Molecular Structure with TensionStructure with Tension

Low-angle XRD of collagen, 67nm periLow-angle XRD of collagen, 67nm periodicity it increased when specimen is sodicity it increased when specimen is stretched.tretched.

Mechanical properties & long period wMechanical properties & long period with age, Riedl et al (1980), Nemetschek ith age, Riedl et al (1980), Nemetschek et al (1980)et al (1980)

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7.3.9 Change of fiber configuration 7.3.9 Change of fiber configuration with strainwith strain

Electro microscopy findings: 10% stretch of Electro microscopy findings: 10% stretch of tendon yield 9% increase of light/dark pattern tendon yield 9% increase of light/dark pattern spacing, 1% stretch due to strengthening of spacing, 1% stretch due to strengthening of fiber.fiber.

In basic alignment of collagen molecules, fifth In basic alignment of collagen molecules, fifth segment contribute most of the stretch.segment contribute most of the stretch.

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7.3.10 Critical 7.3.10 Critical TemperatureTemperature

At 65 deg C mammalian collagen shrinks to At 65 deg C mammalian collagen shrinks to 1/3 of initial length1/3 of initial length

Mechanism: breakdown of crystalline Mechanism: breakdown of crystalline structure.structure.

No X-ray diffraction pattern, rubbery, E~1MPaNo X-ray diffraction pattern, rubbery, E~1MPa

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7.3.11 Change with Life 7.3.11 Change with Life CycleCycle

Mechanical properties of tissues changMechanical properties of tissues change with function of organ in life.e with function of organ in life.

Uterine cervix in childbirth of rat pregnUterine cervix in childbirth of rat pregnancy/after birth (Harkness 1959a)ancy/after birth (Harkness 1959a) Non-pregnant: 5-10% wt of collagenNon-pregnant: 5-10% wt of collagen Transverse section of horns of uterus, flatteTransverse section of horns of uterus, flatte

ned ellipses with fibers concentric arrangened ellipses with fibers concentric arrangementment

Extension of cervix under constant loadExtension of cervix under constant load

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Creep curves of cervixes of rat in Creep curves of cervixes of rat in various stages of pregnancyvarious stages of pregnancy

No pregnant & pregnant 12 No pregnant & pregnant 12 day show little creepday show little creep

Later in pregnancy cervix Later in pregnancy cervix enlarges & more extensible.enlarges & more extensible.

Pregnant 21 days, high Pregnant 21 days, high creep. Stretching resisted creep. Stretching resisted by by viscous forceviscous force, not elastic , not elastic restoring force.restoring force.

Within a day after birth Within a day after birth cervix reverts to its original cervix reverts to its original properties although it is properties although it is still larger than before still larger than before pregnancy.pregnancy.

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Mechanical behavior of uterine Mechanical behavior of uterine cervix of ratcervix of rat

Virginal vs. 1, 5, 18D post partumVirginal vs. 1, 5, 18D post partum((產後產後 )) Rapid change of size, distensibility increases but stRapid change of size, distensibility increases but st

rength decreasesrength decreases ‘‘stress’ exclude ground substance only collagen fibstress’ exclude ground substance only collagen fib

er area; strength decreases during resorptive & rester area; strength decreases during resorptive & restorative phasesorative phases

63

ReasonsReasons

Creep characteristics of near-term uterine cerCreep characteristics of near-term uterine cervix due to change in ground substance.vix due to change in ground substance.

Enzyme trypsin(Enzyme trypsin( 胰蛋白胰蛋白 ) treatment (not attack c) treatment (not attack collagen) increase creep rateollagen) increase creep rate

Uterus of non-pregnant rat undergoes cyclic cUterus of non-pregnant rat undergoes cyclic changes in water content but not in dry weight.hanges in water content but not in dry weight.

Swollen cervix with Swollen cervix with increased water contentincreased water content & changed & changed ground substance propertiesground substance properties may may be the reason for dramatic change in mechanbe the reason for dramatic change in mechanical properties.ical properties.

64

7.4 Thermodynamics of Elastic 7.4 Thermodynamics of Elastic DeformationDeformation

Relationship between elasticity and Relationship between elasticity and internal constitution of material internal constitution of material (crystalline or amorphous)(crystalline or amorphous)

Thermodynamics consideration: Thermodynamics consideration: (molecular structure & material (molecular structure & material constitution)constitution)

Sources of elastic response to Sources of elastic response to deformationdeformation Change of internal energyChange of internal energy Change of entropyChange of entropy

65

)(eee

)(p

)(eSdTEd

)(TdSdQ

)(QdEd

,ijijij

,ijijij

ijij

ijij

53

1

deviationsstrain

4

deviations stress

31

:yield (2)&(1) Combine

2

entropy of change re temperatuabsolute input heat

:law Second

11

done work ferredheat trans energy internal of change

:namics theromodyof lawFirst

66

)()V

ST

V

E(p

)()e

ST

e

E

/ρedV

)()e

ST

e

E(

)(eVdpSdTEd

T

T,ij

,ij

,ij

αα

Tijij

ij

'ij

,ij

9

8(

:becomes Eq.(7) deviation,strain and using

strain r.t.entropy w. of decrease andstrain w.r.t

energy internal specific of increase from arises stress

7

(3) From

volumeof change dV

61

:becomes (3) Eq. ,Eqs.(4)(5) Using

67

body. ain atomsby assumed becan that ionsconfigurat

possible of no. of logarithm toalproportion isentropy

strain r.t.entropy w. of decrease from arises stress

11

energy internalcontsant For

10

process isentropicFor

)(e

ST

)(e

E

T,Eij

ij

T,Sij

ij

Two ways of changing entropy:• Conduction of heat through boundary• Internal irreversible process: viscous friction, thermal current between crystals, polymer chain changes, structural configuration change

68

Note: in laboratory experiment, isentropic condition is not easy to achieve.Isothermal condition is easier to maintain.

)(e

F

)()e

TS

e

F(

)(TdSed

TdSSTdEdFd

STEF

Tij

ij

Sijij

ij

ijij

15

have weprocess isothermalFor

14

have weEq.(7), Similar to

131

energy free specific:F

variabledependent new a Introduce

69

)(T

)(T

e

E

)(eT

F

e

S

T

e

)(T

FS

e

F

)(ede

FTd

T

FFd

ijeij

Tij

ij

ijij

ij

ij

e

Tij

ij

ijTij

e

20

have weEq.(7) back to Sub.

19

toEq.(18) and T w.r.t.Eq.(15) ateDifferenti

18

1

yield Eq.(13) with Compared

16

:by derived be alsocan (15)(11),Eqs.(10),

2

70

Usage of Eq.(20)Usage of Eq.(20)

No need for measuring entropy changeNo need for measuring entropy change Change of stress w.r.t temperature when strChange of stress w.r.t temperature when str

ain is kept constant.ain is kept constant. Procedures:Procedures:

Held material at a const. strain while T is alteredHeld material at a const. strain while T is altered Equilibrium stress is measured Equilibrium stress is measured ijij==ijij(T)(T) Compute Compute Plot Plot ij ij against lnT, slope~ stress due to entropyagainst lnT, slope~ stress due to entropy

TlnTT ijij

Note: the method has been applied for rubber & rubber elasticity is mainly from entropy change

71

Strain EnergyStrain Energy For elastic material and strain energy For elastic material and strain energy

function W a function of strain componentsfunction W a function of strain components

)(e

W

ijij 21

• W= internal energy /unit volume in isentropic process

• W= free energy/unit volume in isothermal process

• General process: (E, S) or (F, T) both change, W depends on thermodynamic process

• Note: E and F are functions of state variables (T, S, e) but history of strain or strain rate or any other factors such as pH, electric charge, chemical reaction. Limited to elastic bodies and elastic stress!

72

7.5 Behavior of Soft Tissues under 7.5 Behavior of Soft Tissues under uniaxial Loadinguniaxial Loading

Pure biological materials: actin, elastin, collagen. Pure biological materials: actin, elastin, collagen. Tissues: several aforementioned materials & groTissues: several aforementioned materials & gro

und substance.und substance. Experimental approach to constitutive equation.Experimental approach to constitutive equation. Single axial tension test on cylindrical specimen, Single axial tension test on cylindrical specimen,

load & elongation are recorded, stress-strain relatload & elongation are recorded, stress-strain relationship.ionship.

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

73

Preconditioning Preconditioning

Cyclic response of dog’s carotid artery

1: stretch ratio referred to zero-stress length of

segment, 37 deg C, 0.21 cycles/min

74

Hysteresis of rabbit papillary Hysteresis of rabbit papillary (( 乳乳

頭頭 )) muscle muscle

Increasing strain rate

75

Relaxation of rabbit mesenteryRelaxation of rabbit mesentery(( 腸繫膜腸繫膜 ))

76

Long-term relaxationLong-term relaxation

G(t)

Log 10 t

77

Creep of papillary muscle of Creep of papillary muscle of rabbitrabbit

Log 10 t

78

summarysummary

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

Difference: degree of distensibility Difference: degree of distensibility Mesentery 100%-200% from relaxed lengthMesentery 100%-200% from relaxed length Ureter 60%Ureter 60% Heart muscle 15%Heart muscle 15% Arteries & veins 60%Arteries & veins 60% Skin 40%Skin 40% Tendon 2%-5%Tendon 2%-5%

79

7.5.1 Stress Response in 7.5.1 Stress Response in loading and unloadingloading and unloading

(2) stressEulerian

by stretched isit when 1

to

reduce area sectional cross material, ibleincompressFor

stress zeroat length initial & area sectional cross :

:ratiostretch

:stress tensile

Strain & Stress Largange

0

00

0

0

TA

P

A

P

, LA

L

L

A

PT

Lagrange stress

80

(6) 1

1 when 0,T state natural For the

(5)

yield ),(T curve on thepoint a ngsubstitutiBy

(4)

(3)

linestraight afit with

*

)(

)(*

)(*

*

*

*

*

e

eTβ

e)T(T

ecT

)T(d

Td

Notes:1) For Hookean materials d T/d = const2) Piece-wise linear model (practical)

12122

1111 10

,TTTfor),T(d

Td

,TTfor),T(d

Td

81

7.5.2 Other expressions7.5.2 Other expressions For finite deformation of elastic body, strain For finite deformation of elastic body, strain

energy (or elastic potential), W, is often useenergy (or elastic potential), W, is often usedd

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

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

82

1:formanother

rubberlatex for 1.5

state restingin muscle skeletalfor 8

mesenteryrabbit for 18

1

al(1969)et Blatz

determined be ofunction tcertain a :

ratiosstretch principal theare31

(9)

(1967) Landel & Valanis

1

i

3

1

2

)(i

ii

ii

ieC)(lnf

)(C)(lnf

f

,i,

)(lnfW

83

..,.C,.C

,C,C

)I(g)I(CeCW

I

,I,I

eeG

)I(

]/[)(

035019850 003940

skin scat'for

0.g(1) constants, are

(13)31

invariantsstrain material, isotropic

1

1

ibleincompress tension,uniaxial

21

21

3223

1

3213

21

23

23

22

22

212

23

22

211

112

1

1

2

84

Mechanical properties of Mechanical properties of CorneaCornea

bearing) load initiate strain tosmallest and

strain zerobetween e(differencstrain slack :s

s

s

)(

)ln(lnln

Table 7.5:1

85

7.6 Quasi-Linear Viscoelasticity of Soft T7.6 Quasi-Linear Viscoelasticity of Soft Tissuesissues

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

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

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

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

86

hypothesishypothesis Consider a cylindrical specimen Consider a cylindrical specimen

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

(3)

t ),(stretch in

change malinfinitesian to(t) response stress

:

10

d)()]([T

)t(G)t(T

)()]([T

)t(G),t(Td

response: elastic)(λT

functionrelaxationreducedG(t)

)(G),(T)t(G)t,(K

functionrelaxation

t)e(

)e(

(e)

)e(

G(t)

1

t0

G

87

heoryasticity tin viscoelstrain of role theplays T :Note

system of response impulse :G

tat time stress tensile:output

e w.r.t timT of change :input

systeminvariant -elinear tim

(4)

(e)

(e)

t )e( d)(T)t(G)t(T

J of F inverse is stress stepunit todue ratioStretch

6

5

Let

function creep reduced :J(t)

(5)

Eq.(4) of Inverse

1

1

)([J(t)]Fλ(t)

)t(J)(F

)(from),t()t(T,stepunit)t(TIf

)T(Fand)(FT

d)(T)t(J)t(T

)e()e(

t)e(

88

)(d)(G

)]t([T)]t([T)t(T

)(G

)b(d)(G

)t(Tt

)a(d)(G

)t(T)t(T)(G)t(T

:toequivalentis).(Eqthen

],[incontinuousaret/G,t/TIf

)(d)]([T

)t(G)t(G)(T)t(T

e, σi.e., at td system, For relaxe

t )e()e(

t )e(

t )e()e(

)e(

t)e(

)e(

ijij

9

10

8

80

7

0

70

00

0

0

0

0

Tensile stress = Instantaneous response +decrease due to past history

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

89

7.6.1 Elastic Response 7.6.1 Elastic Response (experimental) (experimental)

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

Assumptions:Assumptions: G(t) is continuousG(t) is continuous T T (e)(e)(() can be approximated by T() can be approximated by T(with hiwith hi

gh loading rategh loading rate

90

JustificationJustification

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

cwhere

)]c(G

)[(T

d)(G

)]([T)(T)(T

)e(

)e()e(

0

1

0

91

92

93

94

95

96

97

98

99

100

101

102

103

104