Constitutive Laws t should be evident that there exists for any given material a unique relationship between stress (intensity of force) and strain (measurement of deformation). This relationship is called the stress–strain or constitutive law for the material under study. It is in the measurement of this relationship that the greatest challenge lies in biomechanics. While such laws have been generated for nonbiologic materials and for some soft tissues — those for the eye are sparse indeed and subject to scrutiny. Figure 21. The stress-strain curve is a straight line representative of linear elastic material behavior. Originally, these definitions were applied to metals and plotted against one another. In such isotropic or homogeneous materials the curve representing the stress–strain relationship would be a straight line (Figure 21). Such materials are said to be linearly elastic. A material is said to be elastic if it returns to its original unloaded dimensions when a load is removed. Hooke’s law (see below) — or the “spring” law, since this also applies to steel springs — states that within certain limits of stress and strain:  [9] Figure 22. Stress-strain relationship of steel showing linear elastic zone, nonlinear plastic zone, and fracture. This holds true in the first or linear portion of the strain/strain curve and in this area the material behaves elastically (i.e., stress is proportional to strain). Young’s modulus is essentially the slope of the stress/strain curve in its linear elastic first portion (Fig 22). When the relationship changes and the curve is no longer straight, as in the second part of the stress/strain curve, the material becomes plastic and Hooke’s law does not apply (i.e.: the material does not return to its unloaded dimensions and may fracture). Figure 23. The stress-strain curve is not a straight-line, but loading and unloading follow the same path — a result representative of nonlinear elastic material behavior. In nonhomogeneous or anisotropic materials the plot would appear as in (Figure 23). Such a plot is obviously nonlinear, hence any material exhibiting such behavior would be called nonlinear as well. Biologic materials, such as the cornea, sclera, and extraocular muscles, exhibit a much more complex stress/strain relationship. We can gain some insight into the behavior of soft tissue by taking another look at our tensile specimen. We have said that we can expect the only significant stress will be normal  and that the only strain will be extensional ( ). If we were to gradually increase the length L from  by gradually increasing F and then at a certain value of F, reverse the process by gradually reducing F to zero, the plot would appear as in (Figure 24). Figure 24. The stress-strain curve is not a straight line, and loading and unloading follow different paths — a result representative of nonlinear viscoelastic material behavior. This is typical of soft tissue. Note that not only is the plot curved but that the extension and relaxation curves do not correspond. Since the only variable in the test is time, it is evident that the stress–strain curve for soft tissue is time dependent. Material that shows such a time dependent stress–strain behavior is said to be viscoelastic. These materials are often termed history–dependent because the strain at any given time is dependent upon the history of the stress to that point. Wound healing may be considered a form of viscoelastic behavior and may be modeled with time–dependent operators within the finite element framework. Nevertheless, in biologic materials, Young’s modulus is less useful because the relationship between stress and strain is nonlinear — hence a series of approximations of Young’s modulus must be made for different stress levels (Fig 25). This linear approximation may be made as the instantaneous slope (tangent modulus) or as a chord between two points on the curve (secant modulus). Thus we can arrive at a reasonable set of values which approximates the nonlinear nature of the material. Figure 25. Linear approximations to nonlinear stress-strain curve at stepped loads (stresses). Tangent modulus (instantaneous slope) and secant modulus (chord of curve) will become equal as approximation distance decreases. There are two further tests that can be performed on viscoelastic materials to determine the importance of the viscous property. These tests are the creep and relaxation tests. The viscoelastic properties of a material may be measured in terms of creep by subjecting the material to constant force (stress) and measuring the extension (strain) over time (Fig 26). Alternatively, the length (strain) may be held constant and the force (stress) required to maintain this length varied. These measurements have traditionally been performed in metals and, although they may be measured in biological materials, the results should be interpreted with caution due to the intervention of active transport mechanisms. Figure 26. a,b.Creep(a)and relaxation(b)tests used to determine the magnitude of the viscous response for a viscoelastic material. A number of earlier studies have been undertaken in an attempt to characterize the nonlinear elastic modulus and creep response of the cornea and sclera.[41-48] Unfortunately, much of the available data is not taken at low load (stress levels below 30 ) and this is the area in which physiologic intraocular pressures would indicate that stress–strain data should be obtained; this is essential if accurate computational (finite element) models of the cornea are to be developed. Most studies have concentrated on constitutive properties within the plane of the cornea. Figure 27. Relationship of elastic modulus taken at high load values to age in years. (from Arciniegas A, Amaya LE, Ruiz LA: Myopia: a bioengineering approach. Ann Ophthalmol, 805-810, 1980). Battaglioli has measured perpendicular properties of sclera and has performed stress–strain analysis at high loads. Although his data is not conclusive, some indication of a relationship between age and these constitutive properties is apparent (Fig 27).[49] Note the relatively flat area of the graph for ages 20 to 60 years, indicating wound healing may contribute to more variation in refractive procedures than constitutive properties. This was also found at low load by Nash et al.[42] Poisson’s Ratio Poisson’s ratio is the negative ratio of the strain transverse to the length of the tension specimen and the longitudinal strain. It is a measure of the transverse contraction of the specimen in response to the tensile loading. If a bar of homogeneous material is subjected to a tensile load and increases in length by some fraction, it will exhibit a reduction in dimensions laterally. If the bar is maintained within the elastic limits of the material, the ratio of the strains will be a constant as illustrated in (Figure 28): Figure 28. Poisson’s ratio Again, this expression is really only true for homogeneous materials, such as metals. Biologic materials are very unhomogeneous consisting of many separate cells and even active processes opposing change. A more exact expression of angular transmission of strains within such a material involves the use of tensors (see below) a detailed description of which is beyond the scope of this discussion. Since the cornea is composed predominately of water which is an incompressible material, extensions longitudinally will be matched by lateral contractions of approximately half the magnitude of the longitudinal extension. For cornea, Young’s modulus is said to be , thus Poisson’s ratio is assumed to be 0.5. Hooke’s Law In linear elastic materials such as steel and concrete — the experiments described previously are sufficient to establish the six stress–strain relationships which characterize the material. These relationships collectively considered constitute Hooke’s law. This law depends on Young’s modulus, E, given in either  or , and Poisson’s ratio n, which is nondimensional. This dependency upon Young’s modulus makes the application of Hooke’s law to the study of soft tissue inappropriate. Nevertheless, Yamada has calculated results for many soft tissues.[50] Boundary Conditions Everything is connected to everything else — “neck bone connected to head bone, etc.”. The boundaries between something and something else are very important in mechanics. These boundary conditions present another and very difficult problem in biomechanics. The forces exerted by the eyelids, IOP, extraocular muscles and ocular support structures are not well differentiated but their importance cannot be overemphasized. In our tensile specimen, for example, the stresses at the ends of the specimen depend upon the nature of the application of the stretching force while the strain depends upon the material. If the material is clamped at each end, the clamps will induce a distortion of the material which will affect the measurement of stress/strain. These “grip–to–grip” measurements are less accurate than measurements made between lines oriented internal to the specimen (Fig. 6). Thus the stress–strain relationship can only be calculated far from the ends of the specimen. Accurate lines are difficult to create on biologic specimens but cyanoacrylate glue has been found to be provide a reasonable grip without excessive distortion, thus adding more accurate “grip–to–grip” stress/ strain measurements in biological materials. In a layered material, such as the cornea, the glue must connect to all layers (i.e., Bowman’s, stroma, and Descemet’s) or the contribution of an individual layer maybe lost in strip testing due to slippage. Until we can accurately determine the stress–strain relationship at these boundaries we have only a partial solution to the problem. We must therefore find ways to precisely model the individual constitutive properties of each of the structures of the eye and then by combining them one by one determine the different properties of each boundary and its the overall response of the combination to applied forces. Until we can accomplish that, any attempt to apply the results of corneal modeling to refractive surgery will be an exercise in futility and fraught with peril. © Leo D. Bores, MD - 2002