Chapter 9 Biological elastomers Introduction 9 1 Constitutive

Chapter 9 Biological elastomers Introduction 9. 1 Constitutive equations for soft biopolymers 9. 1. 1 Worm-like chain model 355 9. 1. 2 Power equation 358 9. 1. 3 Flory–Treloar equations 359 9. 1. 4 Mooney–Rivlin equation 359 9. 1. 5 Ogden Equation 360 9. 1. 6 Fung equation 361 9. 1. 7 Molecular dynamics calculations 362 9. 2 Skin 362

Chapter 9 Biological elastomers 9. 3 Muscle 375 9. 4 Blood vessels 378 9. 4. 1 Nonlinear elasticity 381 9. 4. 2 Residual stresses 383 9. 5 Mussel byssus 384 9. 6 Whelk eggs 384 9. 7 Extreme keratin: hagfish slime and wool 390 Summary

Worm-like Chain (WLC) Model n where k. B is the Boltzmann constant, T is the absolute temperature, Lp is the persistence length, L is the contour (total) length of the fibers, and z is the extension (displacement). This equation is especially useful in the prediction of DNA unfolding. Such a curve is shown in Figure 9. 2 and compared with experimental results. It can be seen that f, the force, is equal to zero for z=0; as z approaches L, f→∞.

(a) Domain deformation and unfolding of a multidomain protein under stretching with AFM. (From T. E. Fisher et al. [1999]); each unfolding event starts a new tooth; (b) WLC modeling of a DNA chain having a length L=3. 958μm when completely stretched (contour length of molecule); Lp is the characteristic length of folds

Two types of tensile response exhibited by biological materials: (a) J curve (d/dε > 0); (b) curve with inflection point d 2/d 2ε = 0.

n Force (f ) and extension (z ) for a protein at 300 K and 500 K using the WLC models.

Skin

Human Skin • Soft tissue covering the body (16% of total body weight) Skin anatomy • Large surface area (1 -2 m 2) contacting external environment • Multi-functionality: – Protection Epidermis – Sensation – Heat regulation Dermis – Prevent excessive water loss – Water resistant barrier Hypodermis – Storage and synthesis • Epidermis, Dermis, Hypodermis • Heterogeneous, anisotropic, and non-linear viscoelastic material http: //cancer. stanford. edu/information/cancer. Diagnosis/images/ei_0390. gif

Epidermis • The outmost layer of the skin • Protects body from the environment • Contains no blood vessels • Major types of cells: – Keratinocyte: keratin formation – Merkel cell: sensing – Melanocyte: melanin production – Langerhens cell: immune • Sublayers: – Stratum corneum: protection; prevents water loss – Stratum lucidum: layers of dead keratinocytes – Stratum granulosm: contains keratinocytes – Stratum spinosum: secretes bipolar lipids – Stratum basale: source of epidermal stem cells http: //missinglink. ucsf. edu/lm/Dermatology. Glossary/epidermis. html

Dermis • The middle layer of the skin; load bearing • Consists of cells, connective tissues, and ground substance • Contains blood and lymphatic vessels, nerves, sweat glands, and hair follicles • Two layers: Papillary & Reticular dermis • Papillary dermis: collagen fibers are thinner and loosely packed • Reticular dermis: collagen fibers are thicker and densely arranged • Elastin fibers are found in both layers; more numerous in reticular dermis http: //missinglink. ucsf. edu/lm/Dermatology. Glossary/dermis. html

Mechanical Behavior of Skin • Collagen fibers are straightened and realign parallel to each other under tensile loading Load • Non-linear viscoelastic behavior • Hysteresis loop corresponds to energy loss during loading-unloading process Elongation • Creep is a skin mechanical failure – the result of water molecules displacement from collagen fiber network Load hysteresis g g in d oa un loa din l Elongation C. H. Daly and G. F. Odland, J. Invest. Derm. , 73, 84 -87 (1979). 12

Stress-strain curves of elastin n (From Y. C. Fung, Biomechanics: Mechanical properties of living tissues, 2 nd edition. 1993, p. 244, Fig 7. 2: 1)

n Young’s modulus of rat skin plotted as a function of strain rate. (Adapted from Vogel (1972), with permission from Elsevier. )

Rhinoceros Skin • The rhinoceros dorsolateral skin forms a unique protective armor • Three times thicker than the belly skin • Dense and organized 3 -D array of relatively straight and highly cross-linked collagen fibers • High elastic modulus, tensile strength, toughness, and work of fracture compare to normal mammalian skin http: //mirceaeliade. wikispaces. com/file/view/rhino. jpg/32484363/rhino. jpg • Ideal impact resistant material Rhino dorsalateral skin Rhino belly skin 18 R. E. Shadwick, A. P. Russell, R. F. Lauff, Philos Trans Roy Soc London B 337, 419 -428 (1992).

Mechanical Properties of Rhino Skin 80 tendon Stress (MPa) Rhino dorsalateral skin (armor) Rhino belly skin Donkey flank skin 60 40 20 0. 0 cat skin rhino skin 0. 2 0. 4 0. 6 0. 8 1. 0 Strain R. E. Shadwick, A. P. Russell, R. F. Lauff, Philos Trans Roy Soc London B 337, 419 -428 (1992). 19

n Skin of the rhinoceros: (a) flank; (b) belly. (Used with kind permission of Professor R. E. Shadwick. )

n Langer lines indicating the direction of alignment of the collagen fibrils

n Collagen fibers and fibrils in chicken skin: (a) curvy structure of fibers; (b) fibril assemblages in fibers; (c) characteristic 67 nm bands in fibrils. (Figure courtesy W. Yang. )

n Stress–strain curves for pig belly skin parallel and perpendicular to the Langer lines and for human and rat skin. (Reprinted from Shergold et al. (2006), with permission from Elsevier. )

n Force–stretch ratio test in tension for rabbit skin along two directions: along width and along length of body.

Stress vs. strain rate for slow-twitch and fast-twitch muscles

n (a) Network of arteries and veins in rabbit dermis. (b) Cross-section of an artery and a vein composed of the endothelium, tunica intima, tunica media, and tunica adventitia.

n Stress–strain response of human vena cava: circles, loading; squares, unloading. (Adapted from Fung (1993, p. 329), with kind permission from Springer Science+Business Media B. V. )

n Residual stresses in arteries; the artery is sliced longitudinally and the angle is measured. (From Fung (1990, p. 389), with kind permission from Springer. Science+Business Media B. V. )

Functional Biomaterials: Mussel Byssus [1] • 5 times tougher • 16 times more extensible than a human tendon[2]. First known protein with both collagenous and elastin-like domains: • "a stiff tether" • "shock absorber" • distal cells secrete primarily the silk- and polyglycinecollagen diblocks, • proximal cells secrete the elastin- and polyglycinecollagen diblocks. [1] J. H. Waite et al. , Biochemistry, Vol. 43, No. 24, 2004 [2] Smeathers, J. E. , Vincent, J. F. V. , Mechanical properties of mussel byssus threads , J. Molluscan Stud. 49, 219 -230 (1979).

Mussel Byssal Threads Under Tension [1] Stress vs. Strain in Tension Anisotropically oriented bundles of collagen fibrils Extension rate: 10 mm/min [2] [1] Qin X. , Coyne, K. , Waite, J. H. , Tough Tendons, Journal of Biochemistry 272, 51, 32623 -32627 (1997). [2] Bell, E. Y. C. , Gosline, J. M. , Mechanical design of mussel byssus: material yield enhances attachment strength, The Journal of Experimental Biology 199, 1005– 1017 (1996)

Modulus Mismatch Mussel Two joined materials with VERY high modulus mismatch • Expect failure at interface due to accumulation of residual stresses [1] 30 MPa DOESN’T HAPPEN! (“modulus management”)[2]. Near interface: Proximal portion stiffens from 30 to ~60 MPa 500 MPa Distal portion shows stress softening from 500 to ~60 MPa Rock [1] Rabin, B. H. , Williamson, R. L. , Suresh, S. , Fundamentals of residual stresses in joints between dissimilar materials. Materials Res. Soc. Bull. 20, 37 -39 (1995). [2] Vaccaro, E. , Waite, J. H. , Yield and post-yield behavior of mussel byssal thread: a self-healing bimolecular material. Biomacromolecules 2, 906 -911 (2001).

Whelk eggs n (a) “Mermaid necklace” of interconnected capsules forming a helical pattern around strand. (b) Cross-plywood structure of fibers from whelk egg capsules, each with 0. 2– 0. 5 μm diameter. (Reprinted by permission from Macmillan Publishers Ltd. : Nature Materials (Miserez et al. , 2009 b), copyright 2009. )

Whelk eggs (a) Engineering tensile stress–strain curves of whelk egg capsules in loading and unloading, up to a strain of ~1; note the superelastic effect and hysteresis; (b) schematic representation of presumed reversible structural transformation. (Reprinted by permission from Macmillan Publishers Ltd. : Nature Materials (Miserez et al. , 2009 b), copyright 2009. )

hagfish slime

hagfish slime