A CONSTITUTIVE EQUATION. FOR PARALLELFIBERED‘ ELASTIC TISSUE Thesis for the Degree of Ph. DI MICHIGAN STATE UNIVERSITY ROLFE B. JENKINS 1973 This is to certify that the thesis entitled A CONSTITUTIVE EQUATION FOR PARALLEL-FIBERED ELASTIC TISSUE. presented by Rolfe B. Jenkins has been accepted towards fulfillment of the requirements for Ph.D. degree in Mechanics @&Efi§g Major professor Datemr IS; I3 "I 5 0-7639 s— smomo av ‘5’ HUAS & SUNS' 800K BINDERY INC. LIBRARY BINDER; 1'" ”R'NBPORT, women! III JIFJ ABSTRACT A CONSTITUTIVE EQUATION FOR PARALLEL-FIBERED ELASTIC TISSUE By Rolfe B. Jenkins The intent of this research is to develop a constitutive equation for the elastic fiber-dominant ligamentum nuchae and to determine the contributions of its two fibrous components to the parameters forming that constitutive equation. Tests specimens for the ligamentum nuchae were obtained from gross bovine specimens. Three types of mechanical tests were conducted on specially prepared test pieces. The first series of tests were stress relaxation tests at various strain levels, strain rates, and temperatures. The second series were loading and unloading cycles conducted at strain levels, strain rates, and temper- atures comparable with those of the relaxation tests. The third test series were cyclic sinusoidally-varying strain tests conducted at various strain amplitudes. frequencies. and at temperatures just be- low and above the physiological range of the specimens considered. Tissue degradation and histologic analysis were employed to deter- mine specifically the contribution of elastic and collagen fibers to the mechanical properties of the ligamentum nuchae as shown in the tests described. -p' :- i'U‘ a )6 ‘5'.;.' F 0- 9-. . r . :- A!» A . (- 1" Rolfe B. Jenkins A “quasi-linear" viscoelasticity law has been formulated based upon a suggestion forwarded by Y. C. Fung. Using this develOpment. certain material constants were determined from relaxation tests so as to establish a constitutive equation. The constitutive equation was then employed to predict the results of constant strain rate loading curves, hysteresis loops, and sinusoidal input tests. P ”5% I‘ t3; A CONSTITUTIVE EQUATION FOR PARALLEL-FIBERED ELASTIC TISSUE BY A Rolfe BS Jenkins A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Metallurgy, Mechanics and Materials Science 1973 ACKNOWLEDGEMENTS I wish to thank Dr. R. w. Little for acting as my thesis director and for patiently awaiting the development of this thesis. Also I express my appreciation to Dr. W. A. Bradley, Dr. G. E. Mase, and Dr. R. Echt for their helpful suggestions. I wish to express my gratitude to Dr. Echt, Dr. A. w. Stinson, Dr. K. Keahey, and Mrs. Alice Middleton for their attention to details with which I was not particularly familiar. I also appreciate the financial support of the Division of Engineering Research and the technical assistance of Tony. ii "'9 no 9' v VI '1 'P’ PP P7 .5- JF I; Ifl‘r an... TABLE OF CONTENTS LIST OF TABLES ....................... LIST OF FIGURES ...................... I. Introduction .................... II. The Elastic Fiber .................. 2.1 Nomenclature .................. 2.2 Physicochemical Properties ........... 2.2.A Physical Properties ........... 2.2.8 Chemical Properties ........... 2.3 Anatomic and Structural Distribution ...... III. Literature Survey .................. 3.l Mechanical Properties ............. 3.2 Mathematical Models .............. IV. Experimental Methods ................ 4.1 Mechanical Testing Procedures ......... 4.2 Histologic Methods ............... V. Mathematical Formulation .............. VI. Presentation of Results ............... 6.1 Experimental and Theoretical Results ...... 6.2 Histologic Interpretation ........... VII. Conclusions ..................... BIBLIOGRAPHY ....................... APPENDICES A. Trigonometric Identities and Integral Evaluations B. Cosine Integrals ................ C. Methods of Quantitative Volume Measurements . . . Page iv 11 36 54 54 93 102 110 116 118 LIST OF TABLES Table Page l. Results of Stress Relaxation Tests for Specimen l . . 62 2. Results of Stress Relaxation Tests for Specimen 2 . . 63 3. Results of Stress Relaxation Tests for Specimen 3 . . 64 4. Results of Stress Relaxation Tests for Specimen 4 at 35.0°C ...................... 65 5. A-Listing of Constants for All Specimens ....... 66 6. Results of Sinusoidal Tests for Specimen 2 ..... 67 7. Results of Sinusoidal Tests for Specimen 3 ..... 68 (2.] Composition of Ligamentum Nuchae .......... 119 iv In L4 4.2 Te 4.} C: 44 Sir 4-5 C07 45 CL.‘ Figure 4.1 4.2 4.3 4.4 4.5 4.6 4.7 5.1 5.2 5.3 5.4 6.1 6.2 (5.:3 €5.41 €5.55 6.6 €5.27 LIST OF FIGURES Complete Recording and Testing Apparatus. . Testing Apparatus ............. Constant Strain Rate Fixture Sinusoidal Strain Input Fixture ...... Complete Specimen Preparation Equipment . . Cutters and Sample Test Strips ....... Specimen Grips and Plexiglas Cubes Experimental Stress Relaxation Test . . . . Constant Strain Rate Test ......... Constant Strain Rate Cyclic Test Sinusoidal Strain Test Stress-Strain Plot for Specimen l ..... Stress-Strain Plot for Specimen 2 ..... Stress-Strain Plot for Specimen 3 ..... Stress-Strain Plot for Specimen 4 . . . . Stress-Strain Plot for all Specimens Breakage Points of Pieces from Specimen 4 Stress-Strain Plot for Specimen l at 43.4% Strain/min. and 30°C Stress-Strain Plot for Specimen 2 at 28.4% Strain/min. and 35.D°C Stress-Strain Plot for Specimen 2 at 73% Strain/min. and 40°C Page 29 30 31 32 33 34 35 39 40 43 47 69 70 71 72 73 74 75 76 77 Figure LIST OF FIGURES (Cont.) 6.10 Stress-Strain Plot for Specimen 3 at 46.0% Strain/min. and 34.7°C ......... 6.11 6.12 6.13 6.14 6.15 6u16 6u17 6.18 Stress-Strain Plot for Specimen 3 at 59.3% Strain/min. and 41.9°C ......... Hysteresis Loop for Specimen 1 at 47.8% Strain/min., 40% Strain, and 30°C . . . . Hysteresis Loop for Specimen l at 22.6% Strain/min., 40% Strain, and 40°C . . . . Hysteresis Loop for Specimen 2 at 55.8% Strain/min., 60% Strain, and 35°C . . . . Hysteresis Loop for Specimen 2 at 15.2% Strain/min., 92.4% Strain, and 40°C . . . Hysteresis Loop for Specimen 2 at 60% Strain/min., 60% Strain, and 40°C . . . . Hysteresis Loop for Specimen 3 at 2.81% Strain/min., 62.9% Strain, and 41.9°C ........ Peak 49.6 Peak 66.2 Peak 50.6 Peak 77.8 Peak 76.2 Peak 76.6 Peak 66.1 Stress-Cycle Number Plot for Specimen Rad./min., 90% Strain, and 34.9°C . Stress-Cycle Number Plot for Specimen Rad./min., 90% Strain, and 34.9°C . Stress-Cycle Number Plot for Specimen Rad./min., 82.4% Strain, and 40.9°C Stress-Cycle Number Plot for Specimen Rad./min., 74% Strain, and 40°C . . Stress-Cycle Number Plot for Specimen Rad./min., 90.1% Strain, and 34.7°C Stress-Cycle Number Plot for Specimen Rad./min., 60% Strain, and 34.7°C . Stress-Cycle Number Plot for Specimen Rad./min., 89.5% Strain, and 40.9°C Formic Acid-Treated Longitudinal Section Untreated Longitudinal Section ..... vi ...... Page 78 79 8O 81 82 83 84 85 86 87 88 89 9O 91 92 98 99 LIST OF FIGURES (Cont.) Figure Page 6.27 Untreated Longitudinal Section with Gomori 's Stain. . 100 6.28 Traverse Section of Treated Specimens ........ 101 vii ,"_e ':- 1' v-- gm: of re 3" ’9‘! ‘- ‘ ArLlEdlc ~55! organs l‘r ' tISSJes ‘ Th€re 10 st I. INTRODUCTION The cell is the basic unit of a living organism. A tissue is a group of cells and cell products specialized in some manner to perform a particular function. Tissues are the binding materials from which tnady organs are formed. Since organ activities are actually those of i‘ts tissues, a knowledge of tissue response to various stimuli is use- ffiJl in characterizing organ activities. There are four basic types of tissues which are functionally and structurally distinct. l) Epithelial Tissue This tissue forms sheet-like coverings with one surface free and the other attached to other tissues. Epithelial tissue lines most body cavities and forms the skin. 2) Muscle Tissue This tissue is a group of elongated cells which respond to stimulation to provide oriented contractile movements. 3) Nervous Tissue This is irritable tissue capable of conducting waves of excitation in the form of nerve impulses. 4) Connective Tissue 'This tissue provides connecting, supporting, and binding functions (1). ?:a::ze 13'] in I reverts 3." :crre: I’J‘S‘CTCQJ 1: Qérfgrf general su Conne: WIS C0731 suzstance Itls gene SJIB'SIIJICe $531011 of 1muefice. The C' 191911363, S SIOrage, i- 1118 I ~I mic“ SUr 2 This particular research is concerned with connective tissue. Connective tissue differs from the other three types in that it con- tains intercellular material as its primary constituent. Epithelial, nancle, and nervous tissues are essentially cellular and thus they are not capable of withstanding mechanical loads or performing mechan- i<:al functions independently. Connective tissue is composed of an organized network of cellular and intercellular substances which may vary in proportion and in structure with anatomic location. The mechanics involved in the formation, maintenance, and degeneration (If connective tissue are not completely understood. However, the TDhysiology of connective tissue is well documented in that it is known to perform the functions of transport, defense, storage, repair, and general support (2). Connective tissue is composed of a sparse array of cells, a fi- brous component lying between these cells, and an amorphous ground. substance which may vary from fluid to semi-sol id in its consistency. I't is generally believed that connective tissue cells secrete ground 30bstance from which connective tissue fibers are formed. The for- mc‘ation of fibers from ground substance is controlled by some cellular ‘1 I’11’luence. The cells of connective tissue perform the functions of transport, Clefense, storage, and repair. They perform these functions by oxygen exChange, phagocytosis and antibody production, fat and serum protein storage, and by cell secretion which aids in the repair of injured ‘11 Ssue. The connective tissue fibers along with the ground substance ”"1 ich surrounds them provide connective tissue with its support :a;a:ility. eksflc, an z'szEcgis a". In va', .as:.?ar w': ‘2) :ocz‘r 5.1331 in 3'8 Sutjev 3“ 117161151 be Ised to fihUms 53518 fib. 3: 5047 ch iii-rt. 0f th ”is the bes 153911 "th ”Crosco;, Crofjbms 1i; para-He She fiber, fibers to m “”11 til FADEar in Ambranes flat bands and d19951 3 capability. Three types of connective tissue fibers; collagenous, elastic, and reticular, have interested anatomists, histologists, physiologists and others for years. These types of fibers are pres- ent in varying proportions in bone, cartilage, tendon, ligament, skin, vascular walls, and in most body organs providing mechanical functions. They combine with smooth and striated muscle to provide motion or support in or around all body organs. Since connective tissue fibers are subject to mechanical loadings, their mechanical properties are of interest. Properties of these fibers determined from testing may be used to define the mechanical characteristics of the organs with which these fibers are associated. Mechanical testing of connective tissue fibers is facilitated by the fact that, by normal standards of body chemical activity, these fibers are almost metabolically inert. 0f the three types of connective tissue fibers, the collagen fiber is the best understood. It is composed primarily of the protein col- lagen which is an arrangement of a large number of amino acid chains. Microscopy reveals the collagen fiber to be composed of very thin mi- crofibrils which combine in parallel to form a fibril. Fibrils combine in parallel to form fibers which are from 1 to 15 microns in diameter. The fiber, barely visible to the naked eye, may combine with other fibers to form a membranous network or they may combine in parallel to form thick bundles. On a macroscopic level, collagen fibers appear in parallel bundles as in ligament and tendon, in sheet-like membranes as in fascia or as in parts of the vascular system, or in flat bands or rings as in the walls of the circulatory, respiratory, and digestive tracts (3). 4 Reticular fibers are, embryologically, the first connective tis- ‘sue fibers. They usually occur as two-dimensional networks rather than parallel fiber bundles and are usually holding collagen and elas- tic fiber bundles together. Reticular microfibrils have the same gen- eral physical structure and chemical composition as collagen micro- fibrils. So, despite some marked differences in staining reactions, many regard reticular fibers as immature collagen fibers. Of primary interest here is the elastic fiber. It has as its primary component the protein albuminoid elastin. An elastic fiber has a much greater extensibility and a much lower modulus than a col- lagen fiber of the same size. The elastic fiber is usually compared with the collagen fiber because the two are very closely associated anatomically, morphologically, chemically, and physiologically. Elastic fibers are always found in close anatomic association with collagen fibers and are usually dominated by collagen fibers. This anatomic association is the reason why elastic fibers are very diffi- cult to isolate and test mechanically. Thus, the mechanical charac- teristics of elastic fibers are not well defined. imposed I IDSed of g a Charact. C'CSCOSx. 3010 miCl {Eln 9165' with the I elaStjc f Such as t‘ M elasti. Elas fibers as COHWEDQ referEHCe 2'2 phys 2.2. Orm II. THE ELASTIC FIBER 2.1 Nomenclature Elastic fibers are thin thread-like strands which are composed of smaller sub-units. These sub-units are themselves com- posed of smaller sub-units until this progression of sub-units reaches a characteristic filament which can only be observed by electron mi- croscopy. The elastic fiber will be defined as a strand of about 1 to 10 microns in diameter which has as its primary component the pro- tein elastin. The elastic fiber may have collagen fibrils interwoven with the elastic fibrils which are the principal components of the elastic fiber. By this definition of fiber size, elastic membranes such as those in the vascular walls and in the lung are also composed of elastic fibers. Elastic tissue may refer to a connective tissue which has elastic fibers as its principal component. Elastic tissue totally devoid of collagenous tissue and other tissues is called elastica. Thus, elastic tissue or elastic fiber has a structural connotation while elastin references the chemical character of elastic fibers. 2.2 Physicochemical Properties 2.2.A Physical Properties Elastic fibers arranged in parallel bundles, flat bands. or membranes always form a continuous network with no free ends. 6 In the state of relaxation, elastic fibers are highly refractile, almost isotropic, and somewhat kinked in appearance. The drying or stretching of these fibers is a birefringence inducing, vol- ume reducing process. Elastic fibers are paramagnetic and show electrical and magnetic anisotropy. A fine structure description of an elastic fiber on a mi- croscopic level is one offered by Gross (6,7). 8y electron mi- croscopy, he was able to define two distinct chemical and mor- phological components; threads and an amorphous ground substance. The threads are arranged in long, generally parallel bundles which combine in great numbers to form a fiber. The fibers are infil- trated by and imbedded in a trypsin-sensitive ground substance. Collagen filaments are present within the elastic fiber as an incidental component probably incorporated into the fiber extra- cellularly during its formation. Gross was able to fragment the elastic fiber by trypsin digestion and observe a filament of about 70 angstroms in dia- meter. He regarded the filament as the characteristic unit of the elastic fiber because it appeared in elastic fibers from the fish swim bladder, rabbit, rat, and human aortas, and in the ligamentum nuchae of the ox. Filaments wind around one another in pairs to form threads which have a width of about 120 ang- stroms. The exact nature of the ground substance surrounding the threads is not completely understood. However, the ground sub- stance is apparently responsible for the heat resistance of elas- tic fibers because when trypsin disperses the matrix of ground substance, the elastic fiber becomes heat labile (7). bei In SDr 7 Later work by Stack does not confirm all the observations of Gross. Stack (8) defined a different characteristic unit and denies the existence of a structural periodicity of elastic fiber sub-units which Gross suggests. Other authors suggest that tryp- sin does not disperse the matrix of ground substance but the enzyme pepsin does (9). These contradicitons imply that the elas- tic fiber microscopic morphology warrants further investigation. 2.2.8 Chemical Properties The protein elastin, the principal protein of the elastic fiber, is the most resistant of all body proteins to chemical change (10). It resists boiling water and dilute acids and al- kalis. Elastin consists of thirty amino acids only one of which is polar. This nonpolar nature of elastin probably contributes to the great extensibility of the elastic fiber. Meyer and Ferri (11) described elastin as made up of long polymeric chains which are highly contorted and bound together in the lateral direction by only a few cross-links. Stretching a fiber orients these chains in an axial direction reducing entropy and increasing the internal energy of the fiber. There is enough polarity in the peptide linkages of the elastin polymeric chains to cross-link and thus maintain an extended state when the fiber is dried (12). In this last property, the elastic fiber differs from lightly vulcanized rubber with which it is so often compared. Rubber, being completely apolar, will not lock in an extended state (7). In the dry, extended State, water imbibition is necessary to spread polymeric chains of the elastic fiber apart and permit attraz and CI hanCe< thermal contraction. The molecular nature of elastin does not vary significantly from animal to animal (13). There are two molecular chains noted a and 8 wound around one another. The a chain has a molecular weight of between 60,000 to 84,000 while the 8 chain has a weight of approximately 6,000. Both the a and B chains have similar amino-acid compositions. Their hydroxyproline content is low and their proline and valine content is high compared with collagen. The principal acids of elastin are amino-acetic, leucine, alanine, and phenyalanine. These acids are less neutralized by basic amino acids than those of collagen. Elastin contains few of the dia- mino acids which are basic. Thus elastin is not strongly acid- ophilic (4). Staining did not become an important means of differentiating elastic from collagen fibers until it was learned that elastin was more acid than collagen, that collagen swelled in dilute acids while elastin did not, and that elastin possessed certain reducing powers that collagen did not (4). Four common stains developed for elastic fibers are orcein, resorcinol-iron-basic fuchsin, iron- hematoxylin, and aldehyde-fuchsin(l4). The most consistent of these stains, orcein and resorcin-fuchsin, utilize the effect of the reducing power of elastin. By this reducing power, elastin attracts and reduces highly oxidized metallic acids such as osmic and chromic acids. This means that the acidity of elastin is en- hanced, thus increasing its affinity for basic dyes (4). .P"'.IG v «I». In mile that. Alrcgt EFL“! 5:95. 1535“ Ilbm‘ 9 2.3 Anatomic and Structural Distribution The elastic fiber-collagen fiber ratio varies greatly over the human body. In general, the looser the texture of the connective tissue the less abundant the elastic fibers. Thus cartilage, which exhibits great flexibility, contains a great many elastic fibers while adipose tissue has few and bone almost none at all. Elastic fibers are intimately associated with the voluntary muscle in the eye and tongue. In the circulatory system, they are so associated with smooth muscle that the combination has been referred to as myoelastic tissue (4). Almost all vessels in the cardiovascular and lymphatic systems have elastic fibers present in their walls in some arrangement. Gen- erally speaking, the larger the vessel the larger the elastic to col- lagen fiber ratio. In the skin and beneath it, elastic fibers exist in thick networks. If skin folds over easily, elastic fibers are scanty. If the skin is bound lightly to the underlying structure (fac- ial skin), the fibers are abundant (l5). The respiratory tract has many elastic fibers while the alimentary tract has few. The visceral pleura contains many fiber networks of various orientations while the eye has very delicate networks in the sclera and cornea. From an anatomic point of view, there appears to be five func- tions for elastic fiber networks. First, fiber networks disseminate stresses aimed at isolated points. Second, networks are designed for the coordination of the rhythmic motions of the body parts. Third, fiber networks conserve energy by the maintenance of tone during the relaxation of muscle elements. Fourth, the continuous nature of fi- brous networks provides a defense against excessive forces. Fifth, “her Exte' . _,‘ ”A 4 :1: r Urn-e 10 fiber extensibility and retractility assist organs in returning to their undeformed configuration once all forces have been removed. An example of some of the above anatomic—physiological re- lationships might be the arterial walls. The more abundant the elas- tic fiber bands and membranes in the arterial wall, the lower the pulse pressure and the more uniform the blood flow all the way to the capillaries. It has also been observed that the parts of the lung sustaining the greatest amount of fluctuation in normal respiration have the richest supply of elastic fibers (4). III. LITERATURE SURVEY 3.1 Mechanical Properties The mechanical properties of elastic fibers and the mech- anisms which effect changes in them have long been of interest from the point of view of pathology. Workers have tried to determine whether or not there exists a cause and effect relationship between the injury or degeneration of elastic fibers and the susceptibility of the body to arteriosclerosis, emphysema, or aneurytic conditions. Some have suggested that an aneurysm may be the result of the rup- ture of elastic fibers. Others feel that ruptured elastic fibers may be related to arteriosclerosis and emphysema because ruptured elastic fibers have been shown to bind iron and calcium thus inhibiting the motion of associated tissues. Changes in the chemical or structural integrity of an elastic fiber does not always imply a change in func- tional capacity. Neither does a change in functional capacity always accompany a change in staining reaction or structural integrity. Thus the only means of determining the mechanical properties of the elastic fiber and subsequent changes in those properties is by direct measure- ment. Such determinations can also assist in the design of prosthetic devices for replacing or repairing damaged connective tissue. The human aorta, the bovine aorta, and the bovine ligamentum nuchae are the most common specimens used in elastic fiber testing. The aortas are about 35% to 40 % elastin by dry weight and the bovine ll 12 ligamentum nuchae is 78% to 83% elastin by the same measure. Dry weight measurements are used in specifying percentages because they seem to be the most consistent (4). In 1880, C. S. Roy (16) tested human aortic strips and provided what may be the first accurate force-extension curves for an elastic fiber-dominant tissue. For about fifty years, his work provided the data with which others compared their results. In 1937, J. Krafka (l7) calculated the average Young's moduli for aortic and ligamentum nuchae 6 6 strips. He calculated moduli of .238-10 dynes/cm2 and 3.06-10 dynes/cm2 for the aorta and ligamentum nuchae respectively. Shortly afterward, Hass (18,19) tested human aortic rings and introduced the variables of age and degree of purification of the tissue. He attempted to remove non-elastic fiber components from the specimens by prolonged exposure to warm formic acid. He defined the rings as purified fibers (elastica) at seventy-two hours exposure. Elastica showed 32% greater strain and 170% greater retractility than untreated rings at a given stress near the maximum stress the material would sustain. In 1954, G. C. Wood (20) conducted several tests on the ligamen- tum nuchae in small strain. He considered only strains up to 20% in order to avoid any time or hysteresis effects in the material. He used data obtained from the native ligament to serve as a control to determine the relative weakening or strengthening effects upon the load-elongation curve of various enzymes, acids, and testing environ- ments. The chemical treatments employed by Wood were designed to separate the effects upon the load-elongation curve of elastic fibers, collagen fibers, and ground substances. He concluded that the collagen “:e's of t? . \ ‘ a. Q Q s ;.'3 Earl, n: .5 on; .P;K-e‘1: 0" “Joe ":‘at'éve s: l'i'ge in w 13'13 to de‘ In 133 no. ~==-d-'e of t 1:9“- THE} .2“ , . “"335 rat: 51321; efceC C119 fr . M By avg FIRES/[312 OrE‘J‘urton ( 1:3 Oily at Str 13 fibers of the ligamentum nuchae were of a highly extensible (70% strain) nature and did not participate to any significant extent in the load-elongation curve up to a 20% strain level. The presence or absence of ground substance, on the other hand, was very much evidenced by the character of the load-elongation curve. Wood interpreted the relative strengthening or weakening of a test strip in terms of the change in work required to perform a 20% extension. It is not poss- ible to define a stress-strain law for loading from his data. In 1959, Hoeve and Flory (21) analyzed the thermal-mechanical nature of the ligamentum nuchae from a statistical mechanics point of view. They measured load at various elongations after all transient effects had died out but they made no note of how great these tran- sient effects were. From their linear load-elongation graph at 50.5°C one may approximate a lower limit of Young's modulus as 2.65-106 dynes/cmz. This value is somewhat lower than that obtained by Krafka or Burton (22) from Hass' work. Hoeve and Flory concluded that col- lagen fibers had a significant effect upon the load-elongation curve only at strains greater than 70%. In l962, Carton, Dainauskas, and Clark (23) proposed the first curve fit for single elastic fibers in uniaxial tension. These fibers were teased from fresh ligamentum nuchae and had a negligible quantity of collagen fibers surrounding them. Over a 60% strain range the 6 dynes/cmz. No time fibers had an average Young's modulus of 8.8-10 effects were observed in the loadings up to 130% strain. Larger strips of the ligamentum nuchae containing some collagen fibers, ground sub- stance, perhaps some reticular fibers, as well as many elastic fibers represented a lower modulus material of a much different load-strain 14 character. The authors agreed with Wood's conclusion that collagen fibers included in the predominantly elastic fiber ligamentum nuchae did not seem to retard the extensibility of that ligament. The mechanical properties of the lung have recently been reviewed by Mead (24). He emphasized the need for the implementation of the Law of Laplace in evaluating the mechanical properties of elastic fi- bers and of the entire lung from pressure-volume curves. The few works from which quantitative stress-strain data may be obtained depict a curve of two distinct regions. There appears, at small strain, a reversible straight line region which is attributed to elastic fibers. At higher strains there appears a region convex to the stress axis which, according to Krafka (25) and Hoeve and Flory, may be attributed to collagen fibers. The resemblance of this curve to rubber led to a thermodynamic analysis of elastin-rich tissue in terms of the well-documented thermodynmaics of rubber. The work of Jordan and Garrod (12), King and Lawton (26), and Hoeve and Flory are very significant with regard to this approach. The thermal character of the elastic fiber or of elastin-rich composites has been investigated with more interest in thermal con- tractility than in thermal elasticity. McCartney (27), Lennox (28), and Ayer et al (29) indicate that in the case of aortic tissue, hydro- thermal shrinkage is a two-phase reaction. The first phase is a rather continuous contraction up to about 65°C which is reversible and probable attributable to elastic fiber properties. The second phase, occuring at or near 65°C, is mostly irreversible and can be related to the presence of collagen fibers in the aorta. If the thermal properties of elastic fibers are anything like those of 15 collagen fibers, this hydrothermal shrinkage is probably time depen- dent (30). It does appear that elastic fibers offer greater resis- tance to straining at higher temperatures (21). In general, the literature documenting the mechanical properties of elastic fibers is rather sparse. One reason for this is that the elastic fiber is very difficult to isolate in any kind of a pure form suitable for testing. Another reason for this incomplete documentation is the lack of continuity or consistency in the literature that does exist. The work of one author is often difficult to interpret in terms of another. Variation in experimental methods, data representation, and even nomenclature render critical interpretation of existing data very difficult. 3.2 Mathematical Models The need for a mathematical description of the stress-strain law for body tissues has long been recognized. A properly formulated stress-strain law with a certain number of suitable parameters can provide for the unification of all experiments and the normalization of all data obtained from them. The mathematical description of the mechanical properties of tissues and their essential components is necessary before a theoretical approach can be taken with respect to the biomechanics of tendons, ligaments, lung, heart, the circulatory system, or of other organs. A constitutive equation, as the stress- strain law may be called, is not completely known for any biological material. One reason for this is that the experimental verification of any model is very difficult to perform in two or three-dimensional stress fields (31). 16 Most mathematical models attempt to describe biological materials only in uniaxial tension. Some curve fitting and mathematical modeling has been done with the elastic fiber as the material of primary inter- est. Other constitutive equations have been formulated to provide a certain amount of universality among body tissues so that most, if not all, tissues may be classified as special cases of the general consti- tutive equation. The principal work in mathematically modeling the loading curve for the elastic fiber is that of Clark et a1. They proposed the re- lationship c = 1.3(1 - e'°m) where e is strain and T is load in dynes. This relationship can be inverted to yield 1.3 1.3'6) o = 5.89 1n( A where A is area in square centimeters and stress is in dynes per square centimeters. King and Lawton modeled the mechanical behavior of elas- tin-rich tissues in terms of a general elastomer theory derived from statistical mechanics. They derived the pressure-volume relationship for an elastomeric sphere and closed cylinder in terms of the Langevin function which is well documented due to its role in the theory of ferromagnetism. Another method of modeling tissues with elastic fibers not being of primary concern was proposed by Frisen, Magi, Sunnerup, and Viidik (32). They proposed combinations of linear springs. dashpots, and dry friction elements to model the behavior of parallel fibered collagen- ' ‘I .- "r~11uu. . “ ’r13 '7‘." p O éOi:ted .1 (5115,19, to 1518x data Eye were Onl diggjpat The 1. C. Ft the fat: AR ana1C IDIrOdUC S 1‘j‘éI‘ed ”Itch mor l7 rich tissue. The necessity of accounting for the nonlinear response of collagen fibers to strain resulted in the introduction of linear viscoelastic dashpots into the mechanical system at different times. The mathematical model of the resulting mechanical system became en- cumbered with many step functions which made mathematical analysis and parameter quantification very difficult. There are two other methods of mathematically modeling a consti- tutive law for biological materials. One of these is the macroscropic continuum theory based upon the existence of a strain energy function. This method usually employs certain postulates which constrain the form of the strain energy function so as to lead to necessary and sufficient conditions for the existence of a specific number of finite elasticity parameters (33). These postulates usually imply an iso- tropic, incompressible, and elastic material. This approach was adapted by Blatz, Chu, and Nayland (34) and applied to frog striated muscle, human papillary muscle, cat and rabbit mesentery, as well as to latex rubber. Their theory was in good agreement with experimental data even with respect to the two-dimensional problem. Their results were only applicable to the loading problem as theirs' was a non- dissipative or elastic theory. The other method of modeling biological material was proposed by Y. C. Fung (35) in 1967. This method was introduced to account for the fact that biological materials exhibit time and history dependence. An analog of the conventional linear theory of viscoelasticity was introduced. The history dependence of stress upon strain has been con- sidered in the theories of plasticity but biological materials are 1 much more complex. The stress-strain relation in both loading and ap'.;r An- 't- -f‘ - H heredlt 18 unloading in almost all biological materials is highly nonlinear. Also, biological materials exhibit stress relaxation and creep. The time dependence of biological materials, indicated by stress relaxation and creep, has not been incorporated into any statistical mechanics or continuum mechanics formulation of a constitutive equation until just recently (31). The implication that some viscoelastic theory might be applied to body tissues prompted the use of discrete linear viscoelastic models (32,36). However, these models were not entirely successful in de- scribing the nonlinear stress-strain behavior of these tissues. Fung attempted to resolve this problem by postulating the existence of a relaxation function K(A,t) assumed to be of the form K(A.t) = G(t)Te(x) . 6(0) = l where A is the extension ratio and Te(A) is called the "elastic“ re- sponse (31). Assuming the separable nature of the relaxation function along with the principle of superposition of infinitisimal responses, Fung described the stress at any time as t G(t) =6 G(t-;)doe|c(;)| de( C1 d; de(c) dc G(t) is called the "reduced" relaxation function the form of which is not specified. Because 09(2) is not linear in strain, the above theory is called "quasi-linear". The equation is linear in Ge. Although the form of the "reduced" relaxation function was not specified in Fung's integral, the works of others (37,38,39) indicate that it may be taken as logarithmic in form for some tissues. The hereditary integral has been employed to determine the stress response 19 of collagen fibers to constant strain rate loading and unloading and to sinusoidal strain input (38). An adaptation of Fung's hereditary integral will be employed to determine the response of the elastin- rich ligamentum nuchae to different strain inputs. 1m __1 o - IV. EXPERIMENTAL METHODS 4.1 Mechanical Testing Procedures In order to determine a uniaxial stress-strain law for the ligamentum nuchae, three different forms of loading were employed. Stress relaxation tests were conducted at different strain levels, strain rates, and temperatures. 8y loading and unloading specimen strips at constant strain rates, hysteresis 100ps were obtained. Sinusoidal tests at different strain levels, frequencies. and tempera- tures were also performed. The results of these tests were designed to determine and verify a constitutive equation for the ligamentum nuchae which will closely resemble that of the elastic fiber itself. Specimens for testing were obtained from available cattle at a diagnostic laboratory. Holstein cattle of different ages'were consid- ered. Ligamentum nuchae specimens were obtained within about 8 hours after death. Large sections of the specimen of 12 to 15 inches in length and varying cross-sections were easily obtained. Such speci- mens were immediately washed in cold tap water and cut with scissors into smaller sections which were more suitable for further slicing into thin strips to be used in the testing apparatus. The smaller sections cut with scissors were generally about 2 inches long (along the long axis of fiber bundles), 2 inches in height, and of varying thickness. These sections possessed a cross-section similar in contour to that of an airplane wing. The scissor-cut sections were 20 azair. CJI. l I"‘..'; the as we arcs: ahr‘m fr ‘3'”:3' fro: screws of . aTcng the l “5’9 Place: i'iilig NBS it: mid m 7520!“ Made 29‘. Strips "Id grating “11.05 fan I ISir-tentdn. m5 of SD C+- . MW} in 21 again cut in two perpendicular planes. These latter cuts were achieved with the assistance of two piano wire gratings formed by winding piano wire around small regularly-spaced screws which were embedded in an aluminum frame (Figures 4.5, 4.6). The first grating was formed from a 3 x 2 inch frame with wire of .01" diameter wound around screws of .0625" in diameter spaced about .0625" apart. The second grating was formed from a 3 x 1 inch frame with wire of .005" diameter wound around screws of .0312" in diameter spaced about .0312" apart. The wire ran along the long axis of each frame. The scissor-cut thicker sections were placed on paraffin blocks in Lock's saline solution. The larger grating was placed on the tissue section and some pressure was exerted to hold the tissue snugly between the wires and paraffin. Commercial razor blades were then used, with the grating wires as guide wires, to cut strips of a fairly uniform thickness but variable width. The sec- ond grating was then employed in the same manner to make the tissue strips fairly uniform in width as well as in thickness. The resulting ligamentum strip was very similar in size to a office size rubber band 2 in cross-section and which is about .75 mm by 1.5 mm or about 1.15 mm thus of suitable dimensions for testing. Strips were placed soon after cutting in a closed container of Lock's saline solution and stored at 4-6°C in a refrigerator. Although tissue cutting was done at room temperature, tissue strips were refrigerated in saline within 20 min- utes after cutting. Strips were stored in different containers and the first strips cut were generally the first tested. The process of tissue cutting usually took from 10 to 15 hours in order to achieve 60 or more testing strips which were uniform in cross-section along their lengths and with respect to one another. 22 All tests were conducted with tissue strips immersed in a tem« perature controlled saline bath. Strips were tested in the same type of solution in which they were refrigerated. The Lock's saline solu- tion contained 3.78 grams KCl, 1.35 grams NaHCO3, 2.16 grams CaClz, and 82.75 grams of NaCl per 9 liters of distilled water. The tempera- ture of the saline bath in the 5 cubic inch plexiglas bath (Fig. 4.3) was controlled by circulating water from a constant temperature source through a copper heat exchanger coil placed in the bottom of the sa- line bath. The source temperature was monitored by a Bronwill Model 20 constant temperature circulator. The relationship between the tem- perature of the source and that of the saline bath was linear over a large temperature range. The temperature of the saline bath was measured with a iron-constantan thermocouple. One terminal of the thermocouple was submerged in a Rosemount Engineering Co. Model 911 constant termperature ice bath which was full of crushed ice and some distilled water. The temperature difference between the circulating ice bath and saline bath was monitored by voltage reading on a Keithley 160 Digital Multimeter which was capable of resolving a microvolt (Fig. 4.1). The grips used to hold specimens were of the clamping type (4.7). Specimens were thin enough that severe squashing of the specimens was avoided. The gauge length of test specimens was measured at the points of insertion of the specimens in the grips. The gauge length varied somewhat but was usually from 15 to 18 millimeters. In order to check for possible specimen slippage in the grips, a red-colored paste, Microstop, was used to stain some non-control specimens near the grip insertion. Very high strains did not reveal any of the yellow unstained r .I" 14:55- 'E‘FI S 5 SE" US .' ‘- ,Jr i-EI‘. o f 111a Cer 23 portion of the specimen beyond the clamps. Thus, it was determined that slippage was minimized by using specimens of the size indicated. All specimen dimensions were measured with a traveling microscope. For vertical measurements (gauge lengths) an Ames and a Scherr Tumico dial gauge were employed. Each dial gauge was graduated in .001 inch units. The cross-sections of the specimens were measured by the hori- zontal motion of the traveling microscope lens. Horizontal motion was measured by a micrometer graduated in .002 millimeter units. Because the test specimens were approximately rectangular in cross-section, the entire grip assembly had to be rotated 90 degrees to make complete cross—sectional measurements. Five measurements of cross-section were taken along the gauge length of the specimen. The specimen cross- section was taken to be the average of these five readings. No speci- men of cross-section less than .75 mm2 or greater than 1.5 mm2 was used for data taking in any experiment. Two separate testing fixtures were used for conducting the tests. The first fixture (Fig. 4.3), used for generating constant strain rates, is similar to many screw driven tensile testing machines. This machine operates by driving a crosshead by the turning of two finely threaded screws mounted in stainless steel ball bearings. Inserted into the stationary upper plate of the tensile testing fixture is a Statham Model UC3 Universal Transducing Cell along with a Statham UL4-O.5 load cell accessory. This cell is powered by a Statham SClOOl Univer- sal Transducer Readout unit. This load cell, accessory, and readout unit can accommodate slightly over 100 grams load in tension. Dis- placements of the crosshead were measured with a steel clip gage in- serted between the crosshead and load cell plate. The clip gage was - .1- . \‘r-‘ 1' .- J45, ' I..., g: 0': ,- d v '5 o 1' fir“ )l'13' ‘ ‘p T: W UV la a 15‘.‘ ‘ (lg ‘: ‘I :- 1 .5: DOWN 1Dtr0c or the 24 constructed from steel shim stock .01 inches in thickness, .20 inches in width, and about 6.0 inches in length. Attached to the convex side of the clip gauge was a Budd Metalfilm Strain Gauge Type Cl5-624. The strain gauge was balanced in the bridge circuit by a Heath Decade Box Resistance. The clip gauge and variable resistor were used in con- junction with a Ellis Associates Model 8AM-1 Bridge Amplifier Meter. Another testing device was a variable amplitude scotch yoke used for sinusoidal tests. This fixture was used to induce displacements up to 15 millimeters (Fig. 4.4). Both testing fixtures were powered by a 1/70 horsepower Bodine Model NSR-12R shunt wound motor. The speed of the motor was controlled by a Minarik Model N-l4 speed control unit. In order to obtain both high and low rates of crossbar diSplace- ment without radically changing motor speeds, a gearbox was used to achieve 5 to 1, 3 to 1, and l to 1 speed reduction ratios. The loads and displacements of all specimens were recorded on a Sanborn 322 Dual Channel DC Amplifier Recorder. Occasionally, con- stant strain rate loading and unloading tests were monitored on a Varian F-BO X-Y Recorder. At times, both recorders were engaged in parallel. Because the testing of all specimens was done at high strain, the fixed upper grip and part of the universal joint attaching the grip to the load cell were submerged in the saline bath in order to insure that the entire test specimen remained in solution during straining. This immersion of grip and universal joint required that a slight adjustment be made in the load zeroing to account for the bouyant effect of the saline solution. Once a very small strain was introduced to fix the specimen in the grips, the initial gage length of the specimen was judged to be that length at which the specimen 25 just began to record load. In the stress relaxation tests, specimens were loaded to a given level of strain and all motion of the crossbar ceased by manually turning the Minarik power unit dial to the zero speed setting. Stress relaxation was usually measured for six minutes. In constant strain rate loading and unloading tests, the strain rate reversal was achieved by manually flipping a switch on the power unit. In the case of sinu- soidal testing, specimens were subjected to a sinusoidal input for about 25 to 30 minutes. All tests run on specimens from a given animal were run during a 4 day or 96 hour period. Literature concerning the effects of time and manner of post-mortal storage on connective tissue mechanical pro- perties (16,17,40,4l) is contradictory. However, preliminary testing done on the ligamentum nuchae specimens stored in the refrigerator in preparation for this work implied that the mechanical character of specimens was not significantly altered within 96 hours after the gross ligament was taken from the cow. From 4 to 7 days storage, specimens lost their ability to dissipate load while retaining their general "elastic" or stiffness character. After 7 days storage, specimens began to putrify very noticeably. In order to preserve the specimens as best as one could, the saline solution in which they were stored in the refrigerator was changed and the specimens rinsed twice daily. 26 4.2 Histologic Methods The mechanical character and the mathematical description of that character of any tissue depends on three factors. The defor- mational characteristics of the tissue fibers is certainly important. The proportion of the tissue fibers within the composite tissue is also significant. Finally, the geometrical relationship of the tissue fibers with each other and with other components must effect the defor- mation nature of the gross tissue (33). Since the elastic fibers very strongly dominates all other components of the ligamentum nuchae, it is felt that the mechanical nature of the ligamentum nuchae is very closely related to that of the mechanical properties of the elastic fibers composing it. However, the effect of collagen fiber, reticular fibers, and ground sustance upon the mechanical properties of the lig- amentum nuchae cannot be ignored. Some work has been directed toward determining the effects which various components of the ligamentum nuchae have on its mechanical properties (20,21,42). There is gen- erally little disagreement among these works as to what these effects are in a qualitative sense. An effort will be made to determine how the components of the ligamentum nuchae affect the quantitative mathe- matical model of the composite tissue. The most certain way of determining the properties of the com- ponents of any tissue is to isolate the various components by some physical or chemical means. In the case of the ligamentum nuchae, physical isolation of components for mechanical testing has proved infeasible. Chemical methods have proved more successful but the results of chemical treatments must be studied histologically to de- termine their effectiveness. Elastic fibers have been removed from 27 the ligament by various methods which have different effects upon the ground substance. No treatments are available for removing all but reticular fibers or all but ground substance. In view of the difficulties of removing elastic fibers, only methods designed to re- move nonelastic fiber components were considered here. The effec- tiveness of these methods was studied by prepared slides which also provided information concerning the structural arrangement of the ligamentum nuchae in different stages of purification. In order to determine the relative proportion of elastic and collagen fibers, some of the very basic methods of stereology were applied (Appendix C). Perhaps the best analysis of the various methods for removing nonelastic fibers from a predominantly elastic-fiber tissue is that given by Ayer (14). Boiling water removes young collagen fibers and ground substance from the ligamentum nuchae very effectively. Prolonged exposure to concentrated (88%) formic acid at 45°C to 50°C is probably the most effective means of removing collagen and ground substance. The boiling water and formic acid methods were employed in this study. The ligamentum nuchae of a certain animal was cut into four sepa- rate sections. All four sections were cut and prepared for testing as prescribed. The pieces from one section were heated in distilled water for 90 minutes at 58°C and then stored in a refrigerator. The pieces~ from a second section were boiled in distilled water for 7.5 hours and refrigerated. The pieces from a third section were heated in 88% for- mic acid for 24 hours in a temperature controlled oven at 48°C and refrigerated. The pieces from the fourth seetion were not treated at all. The refrigeration times for all sections were similar. A number of pieces from all four sections were tested in the same testing 28 environment. Only stress relaxation tests were conducted. Pieces from the untreated, boiled, and formic acid treated speci- mens were placed in a 10% formalin solution at room temperature as soon as they were available. These pieces were fixed in formalin for 36 to 48 hours (43). A number of slides were made of each of these three types of tissues. These slides were made on the Technicon Tissue Processor according to the prescribed methods for that machine (44). All tissue was rinsed in mixes of alcohol, cleared with xylene, em- bedded in paraffin, cut in 6 micron sections, and fixed to the slides with albumin. Three stains were employed to stain each of the three types of tissue pieces three different ways. Hematoxylin and Basin served to give the slides contrast and to display the connective tissue cells. Gomori's One Step Trichrome stain was used to stain collagen specifically. Verhoeff's stain was used to stain elastic fibers spec- ifically. An analysis of these nine slides was intended to show any changes in the structural integrity or chemical composition of the ligamentum nuchae as affected by means of chemical purification. Thus, an attempt was made to see how such changes affected the mechanical characteristics of the ligamentum nuchae and how the mathematical model established reflected an alteration in these mechanical char- acteristics. Fig. 4 29 A; I :1," K5 - I III I'IlllllllllllllllllllllI'lll! ‘1 I I} M I II no” ‘4 H to... 11L. I V I I ' ratus. Fig 4 1 Complete Recording and Testing Appa 30 Fig. 4.2 Testing Apparatus. 31 Fig. 4.3 Constant Strain Rate Fixture. U9. 4.: Fig. 4.4 Sinusoidal Strain Input Fixture. 33 Fig. 4.5 Complete Specimen Preparation Equipment. 34 Fig. 4.6 Cutters and Sample Test Strips. 35 Fig. 4.7 Specimen Grips and Plexiglas Cube. 5.1 Detemi Be IESLOnse to I‘Ton must be Ilse may be Here. This $011 to 611 1 V. MATHEMATICAL FORMULATION 5.1 Determination of a Relaxation Function Before the hereditary integral can be used to determine the response to any kind of strain input, the form of the relaxation func-. tion must be known. The assumption that the variables of strain and time may be separated in the relaxation function will not be employed here. This is to say, there is no “reduced" relaxation function com- mon to all strain levels. Thus the relaxation function may be written K(e,t) = F(e,t) 06(5) . 5.1 ce(g) may be referred to as the “elastic" stress or "elastic" re- sponse and F(c,t) may be called a "normalized" or "modified" relaxation function. F(€.t) will replace the "reduced" relaxation function as defined previously (31,38). Fung (45) has described the "elastic" response as exponential in character and modified by a nonlinear function of the stretch ratio which is a zero factor resulting from finite elasticity considerations. He described the "elastic" response as gem = An - 1_ 1e“ 5.2 A2 36 were A and tires of tre °I Expandin be written 1n“Escrib h the StHass Hurf. I. g a 0‘, ti) TDEre are E r th‘ln tr 37 where A and a are constants. If one writes the Lagrangian strain in terms of the extension ratio A as Equation 5.2 may be written as 2 + 4:3 + §£f_+...]e°° . 5.3 09(6) = B[e - e ____ 3 8y expanding the exponential term in a power series, Equation 5.3 may be written 0°(e) = B[e+ (b-l):2 + (g? — b + fl)c3 + ...1 . 5.4 2 3 ‘ It is reasonable to assume that in a power series expansion such as Equation 5.4, certain terms will be much more important than others in describing the "elastic" response of different tissues. It has been shown (37,46) that the second term of Equation 5.4 approximates the stress-strain relation of collagen fibers. For the ligamentum nuchae, the first and second terms will be used to say that oe(e) = C6 + 052 . 5.5 There are many reasons for using the polynomial in Equation 5.5 rath- er than the expressions Ol’ stare C, 8, iii three e azurately, for curve f :c'istants 1‘ tion means ”1th the or II‘. 1111811“ pa It makes me The NC hmOglcal 3738,39). IIOrk, this the tend‘Enc 38 01‘ o°(e) = G(eBE - 1) where C, D, F, G, a, and B are constants. One reason is that, although all three expressions may be made to fit the "elastic" curve fairly accurately, the polynomial is linear in the constant parameters used for curve fitting. This linearity implies no correlation between the constants in a statistical manner of speaking. This lack of correla- tion means that a more accurate data curve fit is possible compared with the other two "elastic" response expressions which are nonlinear in their parameters. Another reason for picking a polynomial is that it makes mathematical analysis simpler. The work of others has shown that in stress relaxation tests on biological tissues the stress relaxes with the logarithm of time (36, 37,38,39). In preliminary tests run on the ligamentum nuchae for this work, this was true for tests run at a given level of strain. However, the tendency of a given specimen to dissipate stress was determined to be strongly strain dependent. This strain dependency may be accounted for if one writes F(e,t) = l + “62(t) ln t . 5.6 Equation 5.6 implies that relaxation is linearly dependent upon the logarithm of time only at a specified strain. The strain dependence of Equation 5.6 precludes the separation of variables in the total re- laxation function. ‘2 iith i .nc‘erstandi ray write 1 5.2 Relaxe WDEre H1t) functhn. fnte‘gr‘al I 39 with the form of the stress relaxation determined and with the understanding that F(e,t) now replaces the G(t) of Equation 3.1, one may write the generalized nonlinear hereditary integral as t G(t) = 3 [l + p.2(t - g) ln (t - .)] [c + 20.(.)] agggz a. . 5.7 C 5.2 Relaxation Test ‘é Fig. 5.1 Experimental Stress Relaxation Test. In the stress relaxation test, s = 80 H(t) and dc = 50 G(t) t where H(t) is the unit step function and 6(t) is the Dirac Delta function. The substitution of these expressions into the hereditary integral in Equation 5.7 yields EC‘JSLI 0n 5 which is U 5'3 Conste 40 t G(t) = 3 [l + neozH2(t - c) 1n (t - c)] [c + ZDeoH(C)]eo 5(a) dc . 5.8 By definition of the Dirac Delta function o(t) is simply the integrand evaluated at c = o. In performing this evaluation it might be noted that 0 1f a>O H(a) = 1 1f a;_(1nx - 13)]0 o 42 Each of the expressions in Equation 5.11 is zero at the lower limit of zero by a single application of L'Hospital's rule. Thus, Equation 5.11 becomes 1] = (C + 20812)}: (ln t - _1_)- 2083: (1n t - _l_) 3 3 4 4 = t3 [(_c_ + _2_ DBt - l_DBt) lnt - (g + _2_ Oct - _l_ D8t)] 3 3 2 9 9 8 I1 =t_3_[(C+D§_t)lnt-1_(C+1D3t)] . 5.12 3 2 3 8 Substituting Equation 5.12 for the evaluation of the integral in Equation 5.10 yields o(t) = Cat + Dazt2 + 831:3 [(C + D 1:) Int - l_ (C + _7_ Det)] 3 2 3 8 which in terms of strain becomes 2 3 G(t)=Cc+De+Ee [(C+D€)1"_§_‘_1_(C+_7_D)J 3 2" a 3 5'13 on which is the response to a constant strain loading. 43 5.4 Constant Strain Rate Loading and Unloading Test ot.----- I I I B l a ‘t 'tt Fig. 5.3 Constant Strain Rate Cyclic Test. For a constant strain rate loading and unloading, the strain at any time may be described as at for Ogtgt1 e(t) = 28t1 - St for t15t52t1 - Substitution of this description of strain into the hereditary integral yields t1 2 2 G(t) = g [1 + u8 (t - c) 1n(t - 4)][C + 208:] 8d; t + {I [I + “32(2t] - t + ;)2 1n(t - 4)][0 + 208(2t1 - £1] (- 8) dc . 5.14 Equation 5.14 can be rewritten as 44 t t G(t) = 3 1SEC + 208:] dc - I] [C + 203(2t] - 4)] dc 3 t 2 t 2 + DB {3 (t - 4) 1n(t - c)[c + 2DBC] dc - I (t - c) 1n(t - c) 1 t [C + 2084] dc - I t (2t1 - t + C)2 1n(t - CILC + 208(2t1 - c)] dz}. 1 This equation is valid for time greater than t1 since for time less than t], Equation 5.13 applies. This equation can be rewritten as t t G(t) = 3 IBIC + 203;] dc - BI] [C + 208(2t] - :1] dc 3 t 2 3 t 2 + us {3 (t - c) 1H(t - CIIC + 2D8c] dcl- vs {]{(C + 208;)(t - t) + (2t1 - t + ;)2[c + 203(21:1 - c)]} 1n(t - c)dc 5.15 The first two integrals in Equation 5.15 represent the strictly strain dependent portion of the stress response. They will be written in terms of strain. The rest of the response is left in separate "elastic" and "time dependent" parts of the stress response. With this idea in mind one may write 45 t t 1 2 a B[C + 208;] dg - £1 B[C + 208(2t] ' C)] dC H II 1 + 0821:]2 - CBt + CBt1 - 4Dth1t + 4082t12 - 082t]2 + 082t2 Cet ce(2t1 - t) - 032(4tt1 - 4t]2 - t2) )2 C3(2t1 - t) + 032(21;1 — t C5 + De 5.16 The first integral in the braces in Equation 5.15 is simply I1 of Equation 5.11 which is evaluated in Equation 5.12. With this know- ledge and with the result Equation 5.16, Equation 5.15 becomes 3(t) = Cc + 022 + [s3t3 [(c + Qgt)ln t - 1_(c + Z_DBt)] 3 2 3 8 3 t 2 2 -u8 { {(c + 208c)(t - t) + (2t1 - t + c) [C + 208(21;1 - c)]} 1 ln (t - c) dc . 5.17 Now let t I3 = It]{(c + 203;)(t - c)2 + I2t1 - t + c)2[C + 203(21;1 - C)]}ln(t - ;)d¢ . 46 Using the change of variables x = t - c and defining t = t - t1 , t I3 = 6 {[C + 203(t - x)] x2 + (2t1 - x)2[C + 208(2t1 - t + x)]}ln x dx which upon expanding the integrand becomes t 13 = 3 {[4t12C + 8D8t12(2t] - t)] + (- 4t1C + 308t1t) x + (zc - 4Det]) x2} ln x dx or integrating I = [4t 2c + 8DBt 2(2t - t)] x (ln x - I) + [- 4t c + 8D8t E] x2(ln x - l) t + [2C - 4D8t1] x3 (1n x - 1)] . 5.18 3" 3 o In evaluating this integral, it should be noted that, by L'Hospital's rule, the whole expression in Equation 5.18 is zero at the lower limit of zero. Thus, the integration becomes, after sorting terms, I = 2t[C(2t 2 - t E + E2 ) + 403t (2t 2 - t t + if)] In t 3 I I -§— I I I 3 1t + .73in 9 36 Nl Since I3 is the evaluation of the integral in braces in Equation 5.17, the substitution of I3 from the above into Equation 5.17 yields 47 o(t)=c.+o.2+_n____s33t [(C+_I§_i_;_)lnt-_1_(C+_7_D8t)] 3 2 3 8 - 2u83t {[C(2t]2 - tlt + if_) + 403t](2t]2 - t1t + §3_)J 1" i 3 3 2 -2 - [c (2t - t t+ t2 ) + 4D8t (2t2 t + 7 t )]l 5.19 I 1" T 1 1 —36 'no+ which is the response to unloading in the single cycle constant strain rate test. 5.5 Sinusoidal Test Fig. 5.4 Sinusoidal Strain Test. For the sinusoidal input, the strain is defined as e = 60 (1 - COS wt) 2' and the derivative of strain with respect to time is written as 48 _geo Sln wt GIG. g-I- m N The substitution of these expressions into the hereditary integral of Equation 5.7 yields t G(t) = 3 {l + nEOZII - cos (t - 2)]2 ln (t - 2)][c + n.0(I - cos mc)]I 4 mco Sln w; d; 2 This expression can be separated into the time-dependent and time-in- dependent parts to give t 950 [C + 080(1 - COS w§)] sin w; dc G(t) = g 2 3 + 1150 8 t w Io [l - COSm(t - 2)]2 1n (t - CILC + 050(] ' COSPCIJ sin w; d g . , 5.20 The first integral is integrated to give t 3:3 Io[c + Deo(1 - cosw;)] sinwr, a; fig.“ - COSwt) + :30 - com)2 2 2 4 CE+DI 49 Substituting this back into Equation 5.20 yields G(t) = C6 + 062 + ueo3w I4 5.21 8 t where I4 = 5 [1 - COSw(t - 4)]2 1n(t-c)[C + Deo(l - COSwC)]Slan d2 The integral I4 may be separated to give 1’. I4 = (C + DEo)O [l - COSw(t ‘C)]251nw§ 1n(t -2) d2 t - Dec 3 [1 - c05m(t ~C)]2sin 2w: 1n(t - a) dc 5.22 2 where the trigonometric identity 2 sinmc c05wg = sin 2w; has been employed. The change of variables x = t - C may be employed to write Equation 5.22 as 1: I4 = (C + 0:0)6 [l - COwa]251nw(t - x) 1n x dx t 2 - Dec 3 [l - COwa] sin 2w(t - x) 1n x dx . 5.23 "27' Using the trigonometric identity sin 01- 3) = sina cosB - COSu sinB . t I4 = (C + Deo)sinwt’o[l - COSwXJZCOSwX 1nx dx 50 t - (C + DCO)COSwt 3 [l - c05wx]2sinwx 1nx dx t - Duosin 2M: 3 [I - c05mx]2cos aux 1nx dx 2 t + Dwocos amt 3 [1 - COSNXJZSTH me lnx dx . 5.24 2 In order to write Equation 5.24 in terms of simple integrals, the square term in the integrand must be expanded to yield t t t 4 (C + Dco)sinmt [3 COwalnxdx - 2 3 coszmxlnxdx + 3 costxlnxdx] r—o II t t (C + Deo)c05wt [é sinwxlnxdx - é sianxlnxdx + 3 coszmxsinwxlnxdx] t t - DeosinZwt [3 COSZwXIHXdX - 2 3 c05wxc052wxlnxdx + 3 coszwxc052wxlnxdg] '2" t t t 2 + DCOCOSZwt [3 sin2wx1nxdx - 2 3 COSwXSthwXIflXdX + 3 cos mxsin2wxlnxdx]. '1?’ 5.25 Equation 5.25 appears to contain twelve different integrals. However, trigonometric identities reduce these twelve integrals to only two distinct ones. The resulting integrals involve sine and cosine inte- grals. The derivation of these integrals and the trigonometric iden- tities used to obtain them are considered in Appendix A. Only the results of these integrations are listed in this derivation. Using the results of Appendix A, Equation 5.25 becomes 51 I = (C + Deo)sinmt {sinwt 1nt - Si(wt) - l[2wt(lnt - l) + sin2wt 1nt 2 OJ 4 - sin(Zwt)] + l_[3$inwt 1nt - 3Si(wt) + siant Int 4 3 " 31‘3mt)]} 3 - (C + DEO)COSwt {Ci(wt) - COSwt lnt - v - l[ Ci(26t) - cosZwt 1nt — y] 2 (I) - cos3nt 1nt + eight) + Ci(3wt) - 11 "3' 4 12 3 ‘DEOSTHZwt {lIsinZwtlnt - Si(th)] - sinwtlnt + sinZwtlnt - Si(wt) 2 3 ’27»— - Si(3wt) + [[wt(lnt - l) + sinZwtlnt + sin4wtlnt 3 4 ______.___4 - 51(4wt) - $1(2wt)]} 4 + DEOCOSZwt{ lICi(Zwt) - cosZwtlnt - y] - 4_[ - cos3wtlnt - y 2 3 227' + 614(3nt) + 3mm] + y - cosawtlnt + E33_1nI; - 5} + Ci(2wt) + Ci(4wt)]} . 5.26 8 2 8 52 where Ci(a) and Si(a) are the cosine and sine integrals respectively of the argument a as defined in Abramowitz and Stegun (47). The sub- stitution of Equation 5.26 into equation yields G(t) = Ce + 082 + 3 us (C + De )[Slanth[1 - 1nt] + [7Slnwt - Sln2wt + sin3wt31nt o 0 Z’ 2 12 8 - L%§i(wt) - Si(2wt) + Si(3wt)]} 2 12 + cos6t{§1_+ [chSwt - cosZwt + cos3wt| Int 6 4 2 12 - [§Ci(wt) - Ci(2wt) + C1(3wt)]}] 4 2 12 - neo4D [sin26t{wt[lnt - l] + [-sinwt + 3 sin26t - sinwt + sin4mt]lnt ‘TTF' 1“ 4' 3 16 -[-51(wt) + 351(2wt) - 31(3wt) + $1 4wt ]I 4 3 6 - cosZwt { + g§1_+ [COSwt - 3_cos26t + cos3wt - cos4mt] Int 48 4 3 16 -[Ci(wt) - 3_Ci(2wt) + Ci(3wt) - 61(46I1JI] 4 3 16 5.27 which is the total response to a sinusoidal input as prescribed. 53 In order to render Equation 5.27 suitable for computation it is reasonable to read the sinusoidal response only at peak strain. That is, when 6 = :2-(1 - COSwt) = $0 2 or when COSwt = -1 which implies wt = (2n - 1)n thus costh = 1 cos3wt = l cos4wt = 1 sin(nwt) = 0 With these simplifications, Equation 5.27 simplifies to G(t) = Ce + De 2+ De 3(C + Dc )[- §1_+ 11 1nt + 5C1(wt) - CIIZwt) P ° °'—§2 °66” 4 2 i-Cfl4gfl] +n.04o[_3_2_1-211_g_3_1nt - C1(wt) + 236mm) - c13g3nt) + “1(2th 5.28 which is the response to peak strain in cyclic sinusoidal strain input. VI. PRESENTATION OF RESULTS 6.1 Experimental and Theoretical Results Ligamentum nuchae specimens from four different animals were tested. The general constitutive equation formulation developed in Chapter 5 was used to describe the mechanical characteristics of the test pieces from all four animals. The three parameter constants specified in Chapter 5 were somewhat different from one specimen to another to describe the difference in mechanical character of those specimens. The results of a number of relaxation tests at various strains, strain rates, and temperatures are listed in Tables 1 - 5. It was observed that stress decayed as a linear function of thelogarithm of time for any given relaxation test. However, the magnitude of decay was strongly strain dependent. As shown in Tables 1 - 4, there is a strain level in each specimen at which test pieces display no relaxation or at least an undetectable amount. To account for the strain dependence of the coefficient of the logarithm of time in the "modified" relaxation function, and to account for the negligible relaxation of test pieces at lower strains, the power term in strain has been incorporated into the "modified" relaxation function. The slight curvature noted in the loading curves for relaxation tests suggested that the "elastic" stress be described as in Equation 5.5. Thus the response to stress relaxation from Equation 5.9 is written as 54 55 2 G(t) = (cg + 0.02) (l + uco In t) o where u. C, and D are constants for a given ligamentum nuchae speci- men and dependent only upon temperature. As may be observed from Tables 1 - 4, the constant u varies considerably among the various pieces tested at a given temperature. The tests in which no relax- ation was observed (p = D) were not included in the averaging of u. The time-dependent characteristics of test pieces at low strains were approximated by the determination of u from higher strains and from the incorporation of strain into the "modified" relaxation function. It may be observed from Tables 1 - 3 that an increase in the temper- ature of the testing environment reduces u in each case. Table 4 implies that heating pieces to 58°C and testing them in the same testing environment as untreated pieces seems to reduce u also. The boiled test pieces from Specimen 4 showed a very slight stress re- laxation in two cases but the relaxation was not logarithmic and u could not be determined. The formic acid-treated pieces showed no relaxation even at strains at which untreated pieces from the same specimen (Specimen 4) showed a definite amount of stress relaxation and u was measurable. Thus, increased temperature, in the testing environment or by previous treatment, does seem to reduce the time dependent character of the average test piece in all four cases tested. The determination of the constant C and D appearing in the re- laxation response expression was made from an analysis of the average loading curve for all test pieces at all strain rates for a given specimen at a given temperature. These constants were determined by 56 curve fitting the expression O=Ce+De to the average stress-strain curve at low strains where time-depen- dent effects were assumed negligible as evidenced by negligible stress relaxation at the maximum strain in this curve fitting strain range. The experimental data from the average stress relaxation loading curve was fitted by a finite difference approximation as a first estimate of the constants C and D. Finer adjustments in the values of C and D were made to obtain the best possible curve fit as was determined by minimizing the standard deviation of the fitted curve from the aver- age experimental loading curve. 8y adjustments of C and D it was always possible to fit the "elastic" curve to the average experimental loading curve within a standard deviation of less than one percent of the stress of the experimental curve at the maximum strain under con- sideration. The constants C and D for all testing situations are listed in Table 5. The curves which they represent are shown in Figures 6.1 - 5. Table 5 shows that the increased temperature of the testing environment affects both C and D but in different ways. Thus, a single purely temper- ature-dependent parameter is very difficult to define. The average stress-strain ratios indicated in Table 5 serve to show that increasing temperature makes the average test piece more resistant to straining in all three cases. The average moduli were increased by l.O7%/°c, .l%/°C, and .64%/°C in Specimens 1, 2, and 3 respectively. Temper- ature variation tests at low strains were employed on single pieces 57 from each of the three specimens in order to verify the data in Table 5. These tests showed the average moduli were increased by .93%/°C, .6l%/°C, and .69%/°C in Specimens l, 2, and 3 respectively. The single piece temperature variation tests verified the information in Table 5 except for Specimen 2. The theoretical stress-strain plots were determined from Equation 5.13 as dn=co+m2+fiiuc+k)mg-lic+1mn 3 2 B 3 8 Typical comparisons of the theoretical and experimental stress- strain plots are shown graphically in Figures 6.7 - 11. The constants C, D, and u determined from the stress relaxation tests for each speci- men at each temperature were used to make the theoretical loading plots. A further check upon the validity of the herditary integral is de- picted in Figures 6.12 - 17. The equation for the loading curve of the hysteresis loops is the same as for a simple constant strain rate loading plot in Equation 5.13. The unloading cycle is described by Equation 5.19 as O(t) = CC + 082 + 283:3 [(c +.2§£) In t -3[ (c + Z_DBt)] 3 2 3 8 - 28831: {[C(2t2 - t T: + 132) + 4Dst (2t2 - t t + £2” In t 1 1 '3' 1 1 1 3 - [C(2t2 - t E + £2) + 4DBt (2t2 - t t + z__t2)]I 1 _j__ 1 l 1 36 2 tZt 58 where t = t - t1 and t1 is the time of strain rate reverseal. The hysteresis loops depicted graphically were performed at strain levels much higher than those used to determine the parameter constants C and D. It may be noted from Figures 6.12 - 17 that the theoretical hys- teresis loop is somewhat shifted and open relative to the experimen- tal loop. That is to say, if the theoretical loop and experimental loop are of the same height at a strain about half way to the peak strain in the cycle, the theoretical loop will have a smaller height than the experimental loop at strains close to the peak strain and a greater height than the experimental at very low strain. The theo- retical unloading cycle always predicts a residual strain of some low magnitude. The experimental loops usually show residual strain but not to the extent that the theory predicts it. The final attempt to verify the hereditary integral was made through cyclic tests of sinusoidal strain input at different strain levels, frequencies, and at different temperatures. These tests were performed only on Specimens 2 and 3. The results of these tests are listed in Tables 6 and 7. The peak stresses at the peak strains at the end of the first and final cycles are noted there. From these tables it may be observed that the theory is able to predict the peak stress at the end of the first cycle fairly accurately in most cases. This is because the constants C and D which were evaluated by curve fit at low strains still apply at high strains to a test piece which has only weak time dependence in loading. However, the peak stress decay! predicted by the theory is much too weak to even approximate the stress decay realized experimentally. The theoretical response to a sinusoidal input at peak strain is given by Equation 5.28 as 59 Ci(wt) - 21(wt) + 91(3th 2 2 + ”8040 [_2_§_y - 193m t - Ci(wt) + _3_Ci(2urt) - _TE- 48 48 4 £1 (amt) + Cilggmfl] 3 6 where v is Euler's constant and Ci(x) is the cosine integral of x. In Appendix 8, some of the significant cosine integrals are evaluated to show that after only a few cycles of sinusoidal strain input they can all be neglected. The resulting equation for the theoretical peak stress indicates a stress decay in the logarithm of time considerably less than that which would be realized by a stress relaxation test at the same strain level for the same period of time. This theoretical implication is not realized in any of the twenty-nine cyclic sinu- soidal tests run. The stress decay predicted by the hereditary integral is very small and would appear as an almost horizontal line on graphs such as Figures 6.l7 - 24. In order to accurately describe the experi- mental results an auxiliary model was attempted. The form of this model was motivated by two factors. One, is that the hereditary integral was reasonably accurate in predicting peak stresses at the end of the first cycle. Secondly, the stress decay with time over a large number of cycles was similar in form to although larger in 60 rnagnitude than stress relaxation under similar conditions. Thus an auxiliary model designed to fit the experimental data was of the form G(t) = o.I (1 + a(c) ln n) . 6.] In Equation 6.l, the term 01 is the peak stress at the end of the first cycle, n is the cycle number, and a(e) was judged to be a con- stant for a cyclic sinusoidal test at a given strain level, temperature, and on a given specimen. Since the experimental and theoretical peak stresses at the end of the first cycle compared relatively'well in most cases, 01 was taken to be the "elastic" stress associated with a given specimen at a given temperature. Thus, the experimental constants C and D already determined were applicable to this auxiliary model. The function a(e) was observed to have a strong strain dependence so the use of a power term in strain was used in the same way and for the same reason it was used in the stress relaxation tests. With these considerations, the auxiliary model for stress decay at peak strain in the cyclic sinu- soidal tests may be written as o = (cc 2) 2 p ln n) . 6.2 + 06° (1 + GEO O The a in Equation 6.2 is analogous to the u in relaxation tests and the cycle number argument of the logarithm, n, is analagous to time in the relaxation tests. Equation 6.2 describes the stress decay for any given test very accurately up to as many as 350 cycles. However, the constant a varies somewhat among the test pieces of a given Etna; 61 specimen at a given temperature. The auxiliary model of Equation 6.2 introduces a new constant parameter and uses two parameters used before. The a for eact test is recorded in Tables 6 and 7 with an average a computed below each table. Although the two temperatures at which stress relaxation and cyclic tests were conducted were not exactly the same for either speci- men, the constants C and D determined for the higher and lower temper- ature at stress relaxation were used for the higher and lower temper- ature cyclic tests of the same specimen. The analogy between the u and a for a given specimen seems all the more consistent when one {1‘ compares their values at different temperatures. For Specimen 2, the E‘ average p decreases in magnitude 2.1%/°C while the average a decreases 3.3%/°C. For Specimen 3, the average p decreases 5.0%/°C while the average a decreases 5.l%/°C. It may also be observed from Tables 2,3, 6, and 7 that at the low temperature the average a is 2.85 times average u for Specimen 2 and average a is 3.01 times average u for Specimen 3. Thus temperature affects both a and u in about the same way. Figures 6.18 - 24 compare the experimental results with those predicted by the auxiliary model of Equation 6.2. The a used in Equation 6.2 was the average a for a given specimen at a given tem- perature . 62 Table 1. Results of Stress Relaxation Tests for Specimen l. Temp _Strain Rate Strain tea? 5 °c Wmin. z (in 1111-11.)" (in mm" 22.7 20.0 0 o 3.1 21.1 o 0 45.9 22.3 0 0 30 45.9 30.2 -.0055 .0501 45.8 40.8 .0092 .0553 22.1 53.2 .0141 .0497 45.7 55.7 .0109 .0353 2.3 19.1 0 0 42.8 20.0 0 0 24.7 22.2 0 0 43.4 30.0 .0050 .0552 40 25.7 32.1 .0055 .0535 42.9 40.0 .0053 .0332 25.4 44.0 .0094 .0473 2.1 45.0 .0092 .0455 43.5 50.0 .0134 .0535 23.3 58.3 .0110 .0323 5 32 x 105 dynes/cm2 , At 30°C , C = 40 x 10 dynes/cm2 , D and u =-. 501. At 40°C , c = 45 x 105 dynes/cm2 , 0 30 x 105 dynes/cm2 , and u = -.O455 . 63 Table 2. Results of Stress Relaxation Tests for Specimen 2. Temp Strain Rate Strain use? 0 °c %fimm % unmm.r‘ (HimmJJ 32.6 60.0 -.0025 -.0070 29.2 60.0 -.0040 -.0110 3.0 62.5 0 0 51.6 70.0 -.0047 -.0096 35.0 53.2 70.0 -.0052 —.0107 32.5 70.0 -.0056 -.0114 28.2 80.0 -.0056 -.0088 3.1 80.0 -.0058 -.0090 54.7 80.0 -.0071 -.0111 54.0 80.0 -,0033 -.0129 33.0 83.7 -.0077 -.0110 3.1 90.0 -,oo73 -.0096 66.7 60.0 0 0 35.0 60.0 -.0025 -.0068 12.7 64.6 0 0 39.3 70.0 -.0038 -.0077 40.4 13.0 70.0 -.0049 -.0099 67.6 80.0 -.0045 -.0070 36.1 80.0 -.0045 -.0071 15.3 86.7 -.0090 -.0120 66.1 94.7 -.0116 -.0130 13.5 103.8 -.0107 -.0099 At 35.0°c , c = 32 x 105 dynes/cm2 , 0 and p = -.0102 At 40.0°C , C and u = -.0092 30 x 10 dynes/cm2 , D 16.5 X 105 dynes/C1112 , 22 x 10 5 dynes/cm2 , 64 Table 3. Results of Stress Relaxation Tests for Specimen 3. Temp Strain Rate Strain neoz 0 °C %/min. % (ln min.)-] (1n min.)-1 60.0 40.0 0 0 3.1 50.0 0 0 47.2 50.0 0 0 19.2 50.0 -.0031 -.0123 47.1 60.0 -.0063 -.0176 2.7 65.0 -.0066 -.0157 19.4 67.4 -.0121 -.0266 34.7 3.1 69.4 - 0049 -.0102 3.0 70.0 - 0085 -.0174 3.0 70.0 -.0127 -.0260 56.3 71.3 - 0129 -.0254 49.8 73.8 - 0164 -.0312 9.5 76.1 -.0197 -.0341 54.5 76.6 - 0214 -.0365 54.7 79.8 - 0193 -.0303 28.1 81.4 - 0150 -.0226 63.6 39.1 0 0 61.9 41.8 0 0 3.1 50.0 0 0 20.0 50.8 0 0 19.6 53.8 -.0038 -.0133 41.9 3.1 60.0 -.0058 -.0160 62.6 61.8 -.0043 -.0114 2.5 68.8 -.0105 -.0222 64.3 68.9 -.0100 -.0211 20.0 70.2 -.0104 -.0212 65.4 71.4 -.0101 -.0198 19.4 74.3 -.0150 -.0272 At 34 7°C 0 - 44 5 2 - 5 2 . , - x 10 dynes/cm , 0 — 14 x 10 dynes/cm , and u = -.0255. At 41.9°C , c = 47 x 105 dynes/cm2 , 0 = 11 x 105 dynes/cm2 , and u = -.0188. 65 Table 4. Results of Stress Relaxation Tests for Specimen 4 at 35.0°C. Piece Type Strain Rate Strain ueoz 0 %/min. % (1n min.)-] (ln min.)-1 3.1 39.9 -.0024 -.0153 2.9 40.0 -.0042 -.0263 Untreated 3.5 40.0 -.0028 -.0177 11.3 40.0 -.0056 -.0350 Pieces 12.3 43.0 -.0073 -.0396 51.4 43.7 -.0074 -.0386 10.5 79.9 -.0153 -.0240 10.2 84.2 -.0087 -.0123 55.1 87.7 -.0164 -.0213 62.3 105.9 -.0191 -.0173 19.5 29.0 0 0 12.3 40.0 -.0032 -.0198 Heated 19.8 40.0 -.0056 -.0347 12.5 40.0 -.0164 -.1024 Pieces 10.2 73.1 -.0095 -.0177 11.4 87.1 -.0146 -.0193 20.4 102.6 -.0132 -.0125 For Untreated Pieces: C = 23 x 105dynes/cm2, D = 18 x 105dynes/cm2, and For Heated Pieces: C = 32.5 x 105dynes/cm2, D = 20 x 10 For Boiled Pieces: 5 5 u = -.0247 . dynes/cmz, and u = -.0208 . 0 , and u is undetermined . c = 38.85 x 10 dynes/cmz, 0 For Formic Acid-Treated Pieces: C = 27.95 x 105dynes/cm2, D 0 , and u = 0 . The value of u of the heated piece at 40% strain and 12.5% strain/min. has been disregarded. 66 Table 5. A Listing of Constants for All Specimens. * (Specimen) C D u Ave. (ye Type Pieces 105dynes/cm2 105dynes/cm2 (1n min.)'] 106dynes/cm2 1) At 30.0°C 40.0 32.0 -.0501 4.74 1) At 40.0°C 46.0 30.0 -.0455 5.25 2) At 35.0°C 32.0 16.5 -.0102 3.81 2) At 40.0°C 30.0 22.0 -.0092 3.83 3) At 34.7°C 44.0 14.0 -.0247 4.90 3) At 41.9°C 47.0 11.0 -.0188 5.13 4) Untreated 23.0 18.5 -.0247 2.85 4) Heated 32.5 20.0 -.0208 3.81 4) Boiled 38.86 0 UND 3.63 4) Acid Treated 27.95 0 0 2.72 * The stress over strain ratio is computed over 50% strain in all 10 cases above for standardization. Over higher strain ranges, this ratio will invariably be higher. 67 Table 6. Results of Sinusoidal Tests for Specimen 2. Temp Strain Frequency Experimental Theoretical a Peak_§tnessfe Peak Stress °C % Rad./min. 105 dynes/cm2 105 dynes/cm2 (1n n)"1 63.1 ‘3.6 24.55/24.01 26.60/26.51 -.0232 76.3 29.0 28.64/24.70 34.07/33.88 -.0378 76.5 13.0 33.63/32.05 34.20/34.08 -.0267 34.9 77.4 3.3 35.39/33.22 34.67/34.47 -.0240 78.7 49.6 35.75/30.62 35.40/35.15 -.0353 81.8 28.6 40.82/35.18 37.38/37.l8 -.0780 83.5 3.7 36.10/32.87 38.19/37.98 -.0353 90.0 66.2 44.60/38.58 42.39/41.96 -.0285 91.1 3.3 44.79/42.28 42.92/42.64 -.0211 63.3 3.7 27.83/27.26 29.56/29.42 -.0194 67.0 34.6 31.33/30.43 30.07/29.98 -.0209 40.9 81.0 12.0 35.36/34.15 38.82/38.71 -.0242 82.4 50.6 38.22/36.58 39.79/39.62 -.0214 84.5 23.5 38.55/35.10 40.56/40.34 —.0322 89.5 3.0 42.00/38.75 44.67/44.41 -.0232 92.4 66.1 52.03/47.51 46.51/46.25 -.0272 Average 6 For 34.9°C is -.0290. Average 5 For 40.9°C is -.0241. The value of a at 34.9°C and 81.8% strain has been disregarded. 68 'Table 7. Results of Sinusoidal Tests for Specimen 3. Strain Frequency Experimental Theoretical 0 Peak Stress Peak Stress % Rad./min. 105dynes/cm2 105 dynes/cm2 (ln 11)-1 61.0 76.6 30.29/26.73 32.21/31.86 -.0587 71.5 76.2 42.40/35.00 38.89/38.44 -.0806 77.7 75.4 42.39/35.00 42.75/42.06 -.0802 77.9 3.2 42.69/36.39 42.77/42.38 -.0548 80.8 3.3 44.01/39.61 44.71/44.36 -.0734 90.1 76.2 50.50/39.97 51.61/51.04 -.0950 90.0 3.0 50.98/42.30 51.61/51.20 -.0944 64.3 2.9 33.23/28.01 34.74/34.62 -.l728 68.5 3.1 28.27/27.16 37.37/37.22 -.0341 68.8 76.4 30.60/27.67 37.72/37.40 -.0554 73.5 3.7 41.83/32.92 40.49/40.29 -.1054 74.0 77.8 39.41/35.24 41.04/40.74 -.0465 78.1 76.3 43.7 [36.32 43.66/43.24 -.0594 Average 0 For 35°C is -.0767. Average a For 40°C is -.0602. The value of a at 40°C and 64.3% strain has been disregarded. 69 ----- Average of 40°C Pieces -———-Average of 30°C Pieces 20 “it / 1s 4 / Stress, o, 105dynes/cm2 1 —: 4 1 4 10 20 30 40 50 Strain, e, % Fig. 6.1 Stress Strain Plot for Specimen l. 7O Average of 35°C Pieces ----- Average of 40°C Pieces 1 30—~ 251- N E U \ VI 2 >. .59 20-— O 6‘ .3 m m .‘5 15? V) 10-~ 5» / t~ : i i *7 1 1o 20 30 40 50 60 Strain, e, % Fig. 6.2 Stress-Strain Plot for Specimen 2. 71 ----- Average of 41.9°C Pieces Average of 34.7°C Pieces 30 ~- 25'9- Stress, o, 105dynes/cm2 15'*—' /////// 10‘” //// 21 l l I T 10 20 30 Strain, e. % Fig. 6.3 Stress-Strain Plot for Specimen 3. 9—1— 01-)- 3.1L 72 Average of Runs at 35°C For: A Heated Pieces 35—— B - Boiled Pieces C - Untreated Pieces —1 A D - Formic Acid-Treated Pieces 30—4L J- T 25-~ N 8 __ 3- \ m 0 C >. 'O ‘?2 20~F 3 6' § 0 .5 155- (I) 10" 5-— . 4- 4 I. 1 4 1r 10 20 30 50 60 70 Fig. 6.4 Stress-Strain Plot for Specimen 4. Strain, e, % 73 Average of Pieces From: A - 6 year old at 30°C (Specimen 1) 35—F B - 6 month old at 34.7°C (Specimen 3) C - 2 year old at 35°C (Specimen 2) D - 5 year old at 35°C (Specimen 4) 30" 25.41— N E U \ U) Q) C >3 “3.0 20“” O 6 3 Q) .5- 15— U) 10‘” 5.__ l l I 1 10 20 30 Strain, e, % l 1 Fig. 6.5 Stress-Strain Plot of all Specimens . 74 20 o - Formic Acid-Treated Pieces Z5 23 - Boiled Pieces 0 ‘1e 15. 48 U R g o >, A 18° [5 o 6 o I; 8 10~~ o .5 m 5-. 1B 20 30 40 50 0 Strain, e, % Fig. 6.6 Breakage Points of Pieces from Specimen 4 . :1 ‘_'\. 75 ----- Experiment ---—'Theory 30*— N m l 1 211" 1 Stress, o, 105dynes/cm2 N O 1 1 10‘— 5—.. / t 1 1 1 4 -~ --+ 10 20 30 40 50 Strain, e, % Fig. 6.7 Stress-Strain Plot for Specimen 1 at 43.4% Strain/min. and 30°C. 76 ----- Experiment Theory 35 —- 30—~ 25—r N E U / \ a: / c 15’ mo 20‘? / 5" 8 8 3,,” ifs—- 10‘“‘ S-F //////’ 1 1 1 1 1 1 1 10 20 30 40 . 50 60 70 Strain, a, % Fig. 6.8 Stress-Strain Plot for Specimen 2 at 28.4% Strain/min. and 35°C. 77 ----- Experiment --——-Theory 351* 30-t 25*» _1 0'1 1 T Stress, o, 105dynes/cm2 N O l 1 105- 1 1 1 1 1 1 1 10 20 30 40 50 60 70 Strain, o, % Fig. 6.9 Stress-Strain Plot for Specimen 2 at 73% Strain/min. and 40°C. 78 ----- Experiment Theory 404 1 301’ Stress, o, 105dynes/cm2 20‘- 10—4 L l 1 1 1 1 10 20 30 Strain, e, % J 1 50 60 70 «b—A. D Fig. 6.10 Stress-Strain Plot for Specimen 3 at 46% Strain/min. and 34.7°C. 79 ----- Experiment 71 -———-Theory 50‘- 40—- N 8 \ In (D 5: >5 LOT) 0 g 3071 .3 In 0) I... a 209* lO-P 1 1 J t L 1 1 1 *1 1 I 1 1 1 10 20 30 40 50 60 70 Strain, e, % Fig. 6.11 Stress-Strain Plot for Specimen 3 at 59.3% Strain/min. and 41.9°C. gt .. 80 ----- Experiment -——-—-Theory \ 20‘ \\ 10+ / Stress, o, 105dynes/cm2 1 1 1 12 10 20 30 40 Strafil, E, % Fig. 6.12 Hysteresis Loop for Specimen 1 at 47.8% Strain/min., 40% Strain, and 30°C. 81 ----- Experiment -————-Theory //// 20— / // NE 15 '1)" / g / $1 / .112 / :2 / g; 10 4 .5- U) 5 1.. / / / / 13 26 3'5 46 Strain, e, % Fig. 6.13 Hysteresis Loop for Specimen 1 at 22.6% Strain/min., 40%.Strain, and 40°C. 82 ----- Experiment Theory 25-- N E U \ v: Q) c 15’ m -1— o 20 D 13 U1 3.’ H 1- m 151 10 “ 1 J l 1 I I T 10 20 30 40 50 60 Strain, e, % Fig. 6.14 Hysteresis Loop for Specimen 2 at 55.8% Strain/min., 60% Strain, and 35°C. 83 ----- Experiment ---Theory 55* r 50‘ \ 45* \\ 15 / 25- // 20*' Stress, o, 105dynes/cm oo ‘7‘ 1 15— 10m 5—11— / / 1 1 1 1 1 L 1 1 I I 10 20 30 40 50 60 70 80 90 Strain, e, % Fig. 6.15 Hysteresis Loop for Specimen 2 at 15.2% Strain/min., 92.4% Strain, and 40°C. cl ‘1— ----- Experiment Theory 25—11— 151 / Peak Stress, o, 105dynes/cm2 10 20 30 40 50 60 Strain, e, % Fig. 6.16 Hysteresis Loop for Specimen 2 at 60% Strain/min., 60% Strain, and 40°C. 85 ----- Experiment --——-Theory 35? \\‘§§~. BO‘F 25~e 20"r 154~ Stress, o, 105dynes/cm2 10+ 1 1 1 1 t 10 20 30 40 50 60 Strain, e, % Fig. 6.17’ Hysteresis Loop for Specimen 3 at 2.81% Strain/min., 62.6% Strain, and 41.9°C. 86 ----- Experiment Auxiliary Model N \ E “77“““ -————____ :1 ‘*-—~—- ————______ g 30‘" >1 m‘U E? 2525-1 O 1; 2’ 8 20-1— .12 1‘6 a“: 15“ 10“ 5—41— 10 30 50 70 90 110 130 Cycle, 11 Fig. 6.18 Peak Stress-Cycle Number Plot for Specimen 2 at 49.6 Rad./min., 90% Strain, and 34.9°C. 87 ----- Experiment Auxiliary Model U1 0 1 1 on O 1 1 Peak Stress, op, 105dynes/cm2 .b O 20.1.. 10-— 1 1 1 1 1 30 60 90 120 150 Cycle, 11 Fig. 6.19 Peak Stress-Cycle Number Plot for Specimen 2 at 66.2 Rad./min., 90% Strain, and 34.9°C. ----- Experiment Auxiliary Model N s U \ 1n 0) 5: >1 mu 0 . 40—4 ‘_‘____~_______¥ on \ ‘ fl 1; 8 L 4.: (I) x 30 «1— 1‘6 0 O. 20‘- 104- 1 4 H 1 1 5 10 15 20 25 Cycle, n Fig. 6.20 Peak Stress-Cycle Number Plot for Specimen 2 at 50.6 Rad./min.. 82.4% Strain, and 40.9°C. 89 ----- Experiment Auxiliary Mode] N E U \ U) Q) C >5 L013 0 . 40— CD. a \ 3 *— 0 .‘3 V) x 30 .1.— f6 8’ 20 ~1- 10—e 1 1 1 e 1 20 4O 60 80 100 Cycle, n Fig. 6.21 Peak Stress-Cycle Number Plot for Specimen 3 at 77.8 Rad./min., 74.0% Strain, and 40°C. 90 ----- Experiment Auxiliary Model 50~1 NE 507’ \\\\\\\\ U E; (D g \ 10.0 \ 2 \ x . 40+ T‘T“ D. D 1; 1n 0) S. .p (I) .12 30-— 8 D. 201* 10" 1 1 1 l 5 10 Cycle, n Fig. 6.22 Peak Stress-Cycle Number Plot for Specimen 3 at 76.2 Rad./min., 90.1% Strain, and 34.7°C. 91 ----- Experiment Auxiliary Model 354' N E ii 30 m —d 0:, \ :; “‘*-.. m o r “ca. 25 " b u? ‘5’} s. 7,; 20 A. .v. to 11’ 15 11- 1011- 5.... 20 60 100 140 180 220 260 Cycle, n Fig. 6.23 Peak Stress-Cycle Number Plot for Specimen 3 at 76.6 Rad./min., 60% Strain, and 34.7°C. 92 ----- Experiment Auxiliary Model b O 1 1 30‘ 1 Peak Stress, op, lOdenes/cm2 1011 L 1 L 1 T T 1 1 i 15 30 45 60 75 Fig. 6.24 Peak Stress-Cycle Number Plot for Specimen 3 at 66.1 Rad./min-. 89.5% Strain, and 40.9°C. 93 6.2 Histologic Interpretation Three specimens were tested in order to determine certain parameter constants and to illustrate the validity of the hereditary integral in using these constants to predict the response of ligamentum nuchae test pieces to other loadings. Experiments with Specimen 4 were designed to achieve a different objective. In Specimen 4, the same parameter constants were determined but no indication of their validity was attempted. Experiments on Specimen 4 were designed to determine the variations in the parameter constants as effected by chemical treatments which alter the elastic-collagen fiber balance in the native ligamentum nuchae test pieces. Only stress relaxation tests were conducted on this final specimen. The parameters C, D, and p for the four types of pieces run on Specimen 4 are listed in Tables 4 and 5. Graphical portrayal of these results is displayed in Figures 6.4 and 6.6. The radical difference in the mechanical properties of the four types of tests pieces sug- gests that a histological analysis might offer some insight into the interrelationship of components in the composite ligamentum nuchae. In order to facilitate such an analysis, three types of specimen sec- tions were stained in three different ways as has been described. Gomori's stain was the best of the three stains in differentiating the components of the ligamentum nuchae. Light microscopy of 100 mag- nification reveals many deep purple cell nuclei dispersed between fiber components. It is assumed that the cells are fibroblasts. Elastic fibers from the ligamentum nuchae were stained a rich red by Gomori's stain. They occur in compact bundles which have a generally parallel arrangement with one another. These long fiber bundles show 94 a slight waviness as they traverse the longitudinal dimension of the tissue. The collagen fibers from the ligamentum nuchae are stained green by Gomori's stain. A loose and rather irregular arrangement of collagen fibers is discernible in untreated sections. In untreated sections, individual collagen fibers may be seen clearly to display a large amount of waviness so as to simulate loops. These loops were so large that if individual collagen fiber networks could be sepa- rated from the ligamentum nuchae and stretched until they were straight, the straightened fiber network would probably be half again as long as the strip from which that fiber network was taken. This latter phenom- enon seems to lend credence to the suggestion made by others (14,21) that collagen fibers contribute to the stress-strain character of ten- sile specimens only at some finite strain level estimated to be from 50% to 70%. However, the experimental and theoretical results of this work seem to imply that the entrance of the tensile characteristics of the collagen fiber into the stress-strain curve at a non-zero strain level does not break the continuity or smoothness of the curve. Neither does the tensile character of collagen preclude the curve fit of "elastic" stress at low strain from applying to high strains as well. This latter fact seems to confirm the assertion made by Wood that collagen from the ligamentum nuchae is of a highly extensible nature. The examination of a cross-section of untreated ligamentum nuchae section showed that many groups of collagen fibers surrounded all elastic fiber bundles. By employing one of the most basic methods of stereology as described by Elias and Pauly (48) and Elias et al (49) it was possible to estimate the volume ratio of elastic to collagen fibers. 95 The grid point method employed resulted in the approximation that collagen fibers occupy 30.4% and elastic fibers 67.8% of the volume of the specimen under consideration (Appendix C). Large specimen sections and test pieces which were boiled for 7.5 hours were also investigated microscopically. The elastic fiber bundles from boiled sections seemed to look the same as those of untreated sections. Cells were still present in boiled sections. However, the collagen fiber networks so evident in untreated sections were dispersed. Traces of collagen fibers were attached to the elastic fiber bundles but no collagen was observed in the gaps be- tween fiber bundles. Cross-sections of boiled strips stained with Verhoeff's stain showed large black areas (elastic fiber bundles) surrounded closely by a reddish tint (collagen fibers) with empty spaces in between the elastic fiber bundles where collagen fibers had been observed in untreated sections. Stained sections of formic acid-treated strips revealed no apparent alteration in the elastic fiber bundles, no cells, and almost no col- lagen. Cross-sections of formic acid-treated sections revealed empty gaps where collagen had been but no reddish tint with Verhoeff's stain as was observed in boiled sections. The gaps*were much larger than those which collagen fibers occupied in the native ligament. This seems to imply that collagen fibers are capable of binding elastic fiber bundles together laterally or that reticular fibers which bind these bundles are dispersed by the same treatment which disperses collagen fibers. Observation of large boiled ligamentum nuchae sections with the naked eye showed noticeable differences from untreated sections. 96 Boiled sections were white in color while native sections were yellow- ish. Boiled sections were very dry and compact and resisted lateral loading much more than native sections. Formic acid-treated sections were a rich gold in color and were very moist and soft. These gross physical properties were unaltered by six months of formalin fixation. The apparent compactness of boiled specimens may have been due to the hydrothermal contraction of collagen fibers or perhaps some hydro- thermal effect on elastic fibers. Specimens heated for 90 minutes at 58°C showed this same conpactness. It is suggested that this compact- ness is only apparent because stereological studies (Appendix C) indi- cate that boiled specimens are no more compact than untreated ones and boiled specimens do have gaps in between elastic fiber bundle5. An examination of Table 4 and Figures 6.4 and 6.6 reveals that the untreated test pieces from Specimen 4 yielded the stress-strain curve of the lowest modulus of the four specimens. This average stress- strain curve showed the greatest curvature or the lowest C/D ratio of the four specimens. Heated pieces yielded a curve of similar form but of much higher modulus than that of the untreated pieces. The large increase in modulus of heated pieces is probably attributable to the thermal contraction of collagen fibers and to some degree to the increase in modulus of elastic fibers with temperature elevation. If this is true, thermal changes must be irreversible to a large extent since all testing was done in the same testing environment. The load- ing curves for formic acid-treated and boiled pieces were linear. Many pieces from both these latter types were broken at various strain levels as indicated in Figure 6.6. No untreated or heated piece was broken in any of the four specimens. Formic acid-treated pieces were 97 shown histologically to contain negligible amounts of collagen fibers and several stress relaxation experiments on such pieces revealed no stress relaxation at all. Boiled pieces showed traces of collagen fibers histologically and stress relaxation experiments showed very slight relaxation in two cases. Untreated and heated pieces showed considerable amounts of collagen and stress relaxation tests showed relaxation in both types of pieces at strains greater than 40%. The above considerations imply that collagen fibers are responsible for the nonlinearity in the ligamentum nuchae stress-strain curve. The nonlinearity may be inherent in the stress-strain character of the col- lagen fibers themselves or may result from the interaction of collagen and elastic fibers structurally during deformation. It may be seen that the stress-strain curve of test pieces with no collagen fiber net- works (boiled and formic acid-treated) were linear over the entire strain range. The curves of test pieces with collagen fiber networks were nonlinear over the entire strain range. This implies a collagen fiber effect in the stress-strain curve of the ligamentum nuchae even at low strain. Table 4 and histologic examinations of the three types of tissue imply that collagen fibers are primarily responsible for the time dependent or dynamic effects of the ligamentum nuchae as evidenced by stress relaxation. The large difference in moduli be- tween the boiled and formic acid-treated pieces indicates that elastic fibers respond strongly to temperature treatment and that such thermal effects are largely irreversible. 98 Fig. 6.25 Formic Acid-Treated Longitudinal Section. The elastic fiber bundles are stained black with Verhoeff's stain. The very few collagen fibers are stained lightly (x 100). 99 F”ig. 6.26 Untreated Longitudinal Section. Elastic fiber bundles are stained black and meshes of collagen fibers appear gray with Verhoeff's stain. Just to the right of center an artery traverses the section. Collagen fibers are very dense around arteries and tissue cells (x 100). 100 Fig. 6.27 Untreated Longitudinal Section With Gomori's Stain.1 Very little color contrast between elastic fiber bundles and collagen fibers is shown here. Oval cell nuclei are dark. Irregular meshes of collagen fibers around tissue cells break up the otherwise longi— tudinal orientation of the tissue section (x 100). Fig. 6.28 Transverse Section of Treated Specimens. Top: A boiled section with Verhoeff's stain showing the ends of elastic fiber bundles as black. Bottom: Formic acid-treated section. Note that the boiled section is more dense with elastic fiber bundles (x 100). VII. CONCLUSIONS An attempt has been made to develop a uniaxial constitutive equation for elastic fibers. A "quasi-linear" viscoelasticity law based upon a suggesti0n by Y. C. Fung has been developed. An adapta- tion of Fung's relaxation function has been employed to define an average loading curve and a "modified" relaxation function analagous to Fung's "elastic" curve and "reduced“ relaxation function. The average stress-strain curve is depicted as the sum of a linear and a squared term in strain as was suggested by H. R. Elden (46) in his studies on skin. The "modified" relaxation function differs from the ”reduced" relaxation function in that it possesses strain dependence. The "modified" relaxation function is logarithmic in form and intro- duces no irregularities or discontinuities in the process of inte- gration due to the zero argument of the lagarithm. The bovine ligamentum nuchae was the souce of elastic fibers be- cause it is the most elastic fiber-dominant tissue known. Collagen fibers and other components of the ligamentum nuchae were removed to varying degrees by chemical treatment. Such an isolation of elastic fibers was designed to interpret the mechanical properties of the composite tissue in terms of its basic fibrous components and their interaction in the process of deformation. Verification of the structural composition of the untreated and treated ligamentum was achieved by the histologic methods of fixation, sectioning, staining, 102 103 and light microscopy. Three basic tests were performed on ligamentum nuchae test pieces from four different animal specimens. Stress relaxation tests were conducted at various strain levels and strain rates on each specimen to determine three parameter constants believed to be characteristic of the material at a given temperature. Variations of these para- meters with temperature was discussed. Parameter evaluations deter- mined from relaxation tests were used to predict hysteresis loops. The theoretical-experimetnal comparison of hysteresis loops was gen- erally good. Theoretical computation predicted a slightly shifted and more open loop relative to experimental data. The theory pre- dicted a slight compressive stress which implies the existence of an experimentally realizable residual strain. This experimental resid- ual strain was usually observed but not to the degree to which an extrapolated theoretical unloading hysteresis loop curve predicted. Theoretically generated solutions of stress responses to sinusoidal loadings, as predicted from stress relaxation-determined parameters, were less satisfactory. For convenience, only stress response to peak strain was considered. In general, the theoretical and exper- imental comparison of peak stress at the end of the first sinusoidal cycle was good. However, the theory could not predict the large ex- perimental stress decay with cycling. An auxiliary model to describe experimental stress decay with cycling was described. This auxiliary model suggested the existence of a fatigue factor in biological mate- rial. Cyclic stress relaxation tests were conducted upon some test pieces and the results showed that a fatigue factor must be included in the relaxation function in order for the hereditary integral to 104 adequately predict experimental cyclic test stress decay. Neither the experimental results of cyclic relaxation tests nor the attempt to incorporate a cyclic factor into the relaxation function are docu- mented in this work. Stress-strain ratio for all four tissues are listed. They are averaged only for 50% strain for standarization among the tissues. These ratios are listed in order to compare results for an average Young's modulus with the works of others. Although these average moduli vary considerably among the four specimens tested, they gen- erally are higher than those described from the works of Krafka, Haas, and Hoeve and Flory. However, these same moduli are much lower than that determined from the data obtained from Clark et a1. It is be- lieved that such variation in moduli is the result of variation in the cross-sections of test specimens among the various works. In gen- eral, the larger the test specimen cross-section the lower the aver- aged Young's modulus for the test specimen. This cross-section pro- blem was evident in test specimens in this work. Thus the need to define a certain size cross-section for uniformity among future tests and test data is obvious. In any work, specimen cross-sections and grip style should be stated explicity. It is hoped that this work might suggest a basic method of anal- ysis of biological material and especially of composite material. This method of collecting basic experimental results, mathematital modeling, and histologically interpreting test sections should be applicable to all biological material. Once control specimens are documented, the effect of age, fatigue, temperature, and testings environment upon the mechanical properties of elastic fibers need to 105 be clarified and defined in terms of some suitable parameters. From a more mathematical point of view, the mathematics of different forms of "elastic" stress equations and different "reduced" or "normalized" relaxation functions has yet to be worked out. 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M., Ed.), Advances in Biomedical Engineering, Academic Press, London and New York,'V01. l, 1971. APPENDIX APPENDIX A Trigonometric Identities and Integral Evaluations The integral t t t I cosnwxlnxdx = [sin nwxlnxjo -.1_ 1 sin nwx dx A.1 O nw nm 0 X as is determined by a single integration by parts. Since lim sin an x+o 1nx = O by a double application of L'Hospitals' rule, Equation A.1 may be written t 3 cos nwxlnxdx = sin nmtlnt - 1__gtsin nwx dx . A.2 no nm x To integrate the expression on the right hand side of Equation A.2 one may employ the change of variables y = nmx. Thus = Inwt x o It sin nwx d o x sin y dy Y hfliich is by Abramowitz and Stegun (47) simply Si(nwt) where Si(nmt) is tltie sine integral of the argument nwt. Thus t 3 cos nwxlnxdx = 1__[sin nmtlnt - Si(nmt)] A.3 nw 110 111 The integral t t t a sin nmxlnxdx = [- cos nwxlnx]o + nw cos nwx dx A.4 x 1 I EE'O which is obtained by a simple integration by parts. The integral on the right hand side of Equation A.4 can by the change of variables y = nwx be rewritten as It IHwt o cosxnwx dx - o co; y dy Substituting this into Equation A.4 yields t t nwt é sin nwxlnxdx = [- cos nwxlnx]0 + 1__é cos y dy 0w nw y t [- cos nwxlnx]o + 1n(nwt) - lim 1n(nwt) 5+0 hm hm hm nwt + l I (cos y - l) dy ——-o “w y nwt = - cos nwtlnt + [1n(nmt) + l__$ (COS y ' 1) dY]-A-5 hm hm nw lWae quantity in the bracket is defined (47) as 01 Ci(nmt) - y_ since Ci(a) = y + lna + 3 cos 2 dz nw Z Where Ci(nmt) is the cosine integral and y is Euler's constant. 112 Thus by Equation A.5 and the definitive of the cosine integral t 3 sin nwxlnxdx = l__[- cos nwtlnt + Ci(nwt) - v] . A.6 1101 Equations A.3 and A.6 along with the trigonometric identities A.7 through A.13 contain all the information necessary to perform all the integrations for Equation 5.25. sinza = J. (1 - cosza) A.7 2 sin3a = l (35ina - sin3a) A.8 5' 51114.. = _1_ (3 - 4cosZa + c0545) A.9 8 c0525 = l_(l + c0525) A.10 2 C0S3a = 1_(3c055 + cos3a) A.ll 4 c0545 = l_(3 + 4cosZa + c0545) A.12 8 sinZu = 25ina C052a A.13 The integrals from Equation 5.25 are evaluated as follows: C 1 . é COSwX1nXdX = llsinmtlnt - Si(wt)] by A.3 t 2 t t 2.. I cos mxlnxdx = l I lnxdx + I coswalnxdx by A.10 o 2'0 o-——7f—- = §11nt - l) + 1_Isin28tlnt - Si(20t)] by A.3 4w 113 t 3 t t I cos wxlnxdx 3 I cOwalnxdx + l I 0 '4-0 '50 cos3wxlnxdx by A.ll = 1__[3sinwtlnt + sin3wtlnt - 351(wt) - Si(3wt)| by A.3 401 3 3 t é sinwxlnxdx = l_[Ci(mt) - y - COSwt1nt] by A.6 (A) t a sinwalnxdx = 1__[Ci(2wt) - y - costhlnt] by A.6 2w t 2 3 t t 3 é cos wxsinmxlnxdx = [- cos wxlnx]o + 1_.é gh§_wxdx by integration 301 301 X by parts 3 t t = l_j[- cos wxlnx]O + §_é COwa dx 3w 4 x t + l_é cos3wx dx} by A.11 4 x ‘ 3 t = l_j- cos wtlnt + (3.1m c05wxlnx + 3_é COSwX dx) 301 4 x->o 4 x t + (+ l_1im cos3wxlnx - 1_é costx dx)} 4 x+o 4 x = 1_{- cos3wtlnt + 31011.1] - Y) + 110113.11] - y 1 3w 4 4 by definition of cosine integral 114 t 7. 3 coswalnx = 1_lsin2wt1nt - Si(2wt)] by A.3 2w t t t 8. a COSwX coswainxdx = 2 a cos3mx1nxdx - é cOwainxdx by A.10 t t = 113 c05wx1nxdx + 3 costxlnxdx] by A.1] 2 = l_[sinwt1nt + sin3wt1nt - 51(wt) - Si§3 t2] 2w 3 3 by A.3 t 2 t 4 t 2 9. 3 cos wXCOSZwX1nXdX = 2 3 cos mxlnxdx - 3 cos wxlnxdx t t t = 1 I lnxdx + 1 I coswalnxdx + 1 I cos4wx1nxdx 3'0 “2‘0 KO by A.1O & A.12 1_th(1nt - 1) + sinZwtlnt + sin4mt 1nt 4w - Si(2wt) - Si§4wtll by A.3 4 t 10. a sinwalnxdx = 1_ICi(2om) - cosZwtint - y] by A.4 2 t t 2 11. a c05mxsin2wx1nxdx = 2 3 cos wxsinmxlnxdx by A.13 3 t t 3 . = Z_j[cos wx1nx]o - é cos wx dX} by integration 3m x by parts 115 = - g_{cos3mtlnt - lim cos3wxlnx - lim chwalnx 3w X+O 4 X+0 4 t t - 3 costde — 3 3c05wxdx} by A-11 4x 4X -_2__{cos3mtlnt + y + [3Ci(mt) + Ci(3wt)J} by definition of 3m 4 cosine integral t t 12. 3 coszmxsinwalnxdx = 2 a cos3wxsinmx1nxdx by A.l3 4 tt4 = l_j[- cos wxlnx]o + 3 cos mx dx} by integration 200 X by parts [- 31nt - 4cosZwtlnt - cos4wtlnt 1... 16m + lim (3lnx + 4cosmelnx + cos4wxlnx) x+o t t t dx + I 4coswa dx + I cos4mx dx] 0 x 0 x + I o .i X = l__I- 4c052wtlnt - cos4wtlnt + 4Ci(2wt) + Ci(4wt) - 5Y1 16w by definition of the cosine integral APPENDIX B Cosine Integrals Table B.l. Evaluation of Cosine Integrals a Ci(mn) +.07373 .02257 +.01063 -.00507 +.00396 .00277 +.00204 -.00lS7 +.00124 -.00103 .00083 .00070 .00060 .00052 .00045 OQmNO'iUW-wa-J l I .J-fld—lu—ld 01-5de + i + i + For a > l0 , Ci(a) may be approximated by Ci(a) = - COSéa! , a = mu 0 In the cyclic sinusoidal strain input tests, wt = (2n - 1)n where n is the cycle number. For n greater than 3, either the logarithmic 116 117 term or the term involving Euler's constantLdominates all cosine inte- grals in the equation for the stress response to sinusoidal input. This is to say that the stress response as written in Equation 5.28 is dependent only upon strain and the logarithm of time after the first three cycles of sinusoidal strain input. APPENDIX C Methods of Quantitative Volume Measurements In order to obtain some quantitative information concerning the relative proportion of elastic fibers and collagen fibers to the vol- ume of a given test specimen, some very basic methods of stereology as described by Elias and Pauly (48) and Elias et al (49) were em- ployed. Information acquired from these methods provides a means of determining the effect of boiling and formic acid upon the balance of components in test specimens. Nine microscopic slides of ligamentum nuchae sections were em- ployed. From these nine microscopic slides twenty color film slides were made. These film slides included cross-sections and longitudinal sections of the various tissues considered. A six by six square grid was constructed which defined 49 points of intersection between per- pendicular lines. These points may be referred to as grid points. The projection of a film slide over the array of grid points showed elastic fiber bundles superposed on some grid points, collagen fibers on other grid points, and tissue cells and empty spaces on others. The grid points associated with elastic fibers, collagen fibers, and other tissue components were added up. A given film slide was re— oriented somewhat 6 or 7 times and grid point counting was performed each time. By adding up all the grid points associated with the various tissue components, dividing by the total number of grid points, 118 ll9 and converting to a percentage one may determine the percentage of grid points associated with each component of the section of tissue shown on the film slide. It was then assumed that the grid point per- centages closely approximate the percent composition each component has relative to the tissue section. Because the percentage composition of the components of the tissue seem consistent from slide to slide of a similar type tissue (untreated, boiled, or formic acid-treated) and because the results were the same for cross-sections and longitudinal tissue sections, it was determined that percentages determined by grid points approximate the percentages of their respective components in a three-dimensional test specimen. These percentages are listed in Table C.l. Table C.l. Composition of Ligamentum Nuchae. Elastic Collagen Other Fibers Fibers Components Untreated 67.8% 30.4% l.8% Boiled 63.5% 15.8% 20.7% Formic 45.4% 1.6% 53.0% Acid-Treated For untreated specimens, the designation "other components" refers mainly to connective tissue cells. However, in Verhoeff's stain, small blood vessels are also visible. The small reduction in the per- centage of elastic fibers from the untreated specimens to boiled speci- mens may be too small to suggest any qualitative inference. However, there is a large reduction in collagen fiber percentage in boiled l20 pieces. The collagen fibers remaining in boiled specimens are on the periphery of the elastic fibers. Although collagen is present, the continuous collagen network linking elastic fiber bundles has been dis- persed. The regions formerly occupied by collagen show only empty spaces. Some cells were still seen in boiled sections. There was almost no collagen fibers in formic acid-treated pieces. This fact is best shown by Verhoeff's stain. Elastic fiber bundles were very loosely packed.