1. 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

2. 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

3. Worm-like Chain (WLC) Model where k B is the Boltzmann constant, T is the absolute temperature, L p 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→∞.

4. (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

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

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

8. Skin

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

10. Epidermis http://missinglink.ucsf.edu/lm/DermatologyGlossary/epidermis.html 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

11. Dermis http://missinglink.ucsf.edu/lm/DermatologyGlossary/dermis.html 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

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

13. Stress-strain curves of elastin (From Y.C. Fung, Biomechanics: Mechanical properties of living tissues, 2nd edition. 1993, p.244, Fig 7.2:1)

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

18. 18 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 Ideal impact resistant material http://mirceaeliade.wikispaces.com/file/view/rhino.jpg/32484363/rhino.jpg Rhino belly skin Rhino dorsalateral skin R.E. Shadwick, A.P. Russell, R.F. Lauff, Philos Trans Roy Soc London B 337, 419-428 (1992).

19. 19 Mechanical Properties of Rhino Skin Rhino dorsalateral skin (armor) Rhino belly skin Donkey flank skin 0.2 20 40 60 80 0.40.6 0.8 1.0 0.0 Stress (MPa) Strain tendon rhino skin cat skin R.E. Shadwick, A.P. Russell, R.F. Lauff, Philos Trans Roy Soc London B 337, 419-428 (1992).

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

21. Langer lines indicating the direction of alignment of the collagen fibrils

22. 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.)

23. 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.)

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

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

28. (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.

31. 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.)

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

33. Functional Biomaterials: Mussel Byssus 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 " [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). distal cells secrete primarily the silk- and polyglycine- collagen diblocks, proximal cells secrete the elastin- and polyglycine- collagen diblocks. [1]

34. [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) Extension rate: 10mm/min [2] Stress vs. Strain in Tension Anisotropically oriented bundles of collagen fibrils Mussel Byssal Threads Under Tension [1]

35. Modulus Mismatch [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). Rock Mussel 30MPa 500MPa Two joined materials with VERY high modulus mismatch Expect failure at interface due to accumulation of residual stresses [1] DOESN’T HAPPEN! (“modulus management”)[2]. Near interface: Proximal portion stiffens from 30 to ~60 MPa Distal portion shows stress softening from 500 to ~60 MPa

36. Whelk eggs (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., 2009b), copyright 2009.)

37. 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., 2009b), copyright 2009.)

38. hagfish slime