The Language of Structural Engineering

Constitutive Properties

tructural engineering often concerns itself with “mob action”. That is, it is more concerned with “macro” than with “micro” elements even though it may deal with finite elements of minute order. Hence the oft repeated concept of a continuum. Thus a steel beam is a continuum of iron atoms admixed with a bit of carbon, cobalt and boron atoms for flavor. This continuum will exhibit certain characteristics apart from the fundamental properties of the constituent atoms. So too the cornea. While its fundamental microstructure is important — the individual collagen fibrils and their matrix are lost within the crowd and their individuality lost as well. Consequently the characteristics of the collagen molecules per se need not be considered to obtain useful information about the structure — as any mob psychologist can attest.

The characteristics of the continuum in question will be modified to some extent by the homogeneity of that continuum. Steel and aluminum — even concrete — are homogeneous materials and it is relatively easy to predict their behavior. Nonhomogeneous materials on the other hand are not. For example, when studying the universe, astrophysicists note certain aberrations in their calculations when matter is present in greater density in one spot than elsewhere. It is still possible, nonetheless, to make some generalizations about the universe which can explain some observed phenomena. As the universe is distinctly unhomogeneous so too is most biological tissue.

Biological tissue may be homogeneous in certain areas, however. The scleral coat of the eye is relatively so but the cornea is not. The cornea is really a composite layered material made up of collagen fibrils and ground substance. Each of these distinct layers (Bowman’s, anterior and posterior stroma, and Descemet’s), is homogeneous in its own venue but taken together are not. This of course complicates the study of corneal biomechanics because all of the methods employed to model structures assume homogeneity of that structure in order to produce meaningful results. Thus some average microstructure is assumed.

In general, measurement of gross material properties in substances such as the cornea, with significant internal structure, results in some error. A better approach involves measurement of the more homogeneous subcomponents of the material — in the cornea, the properties of collagen and ground material. Ideally, the material properties of collagen would be measured on a single collagen fibril and likewise the ground substance would be measured in a small, well–defined shape. Measurement of noncorneal collagen material properties (for instance, in rat tail tendon) will give approximate values for collagen, but these may be contaminated by differences in tertiary collagen structure and in ground substance.[36] Nevertheless, it is a place to start.

Displacement

Another concept used in structural engineering is that of displacement. Simply stated, displacement is the movement of a structure in response to an external force; this is also sometimes erroneously called strain. Hence, when someone jumps onto a trampoline, the fabric of the trampoline is displaced downwards. The amount of displacement is dependent upon a number of factors including the extent of the acting force. The example of the trampoline is useful because it is easy to visualize the acting forces and to realize that the device stores some of the energy applied, returning it with great efficiency. This is the concept of elasticity which will be discussed later when we consider a material property known as Young’s Modulus. It should be obvious that the amount of displacement in our example is dependent upon the components which make up the structure of the trampoline. Thus the fabric has a certain stretchiness as do the springs supporting the fabric. The frame holding the whole also has a certain amount of “give” to it. If we were to paint a series of brightly colored dots upon the surface of the fabric, the springs and the frame — each dot in exact relationship to any other — we could learn a great deal about the structure as a whole. That is if we could freeze the motion of the structure at some point when the acrobat jumps onto the trampoline, we could make some assumptions about the entire structure. Thus, were we to employ electronic flash photography to “stop the action” we could measure the displacement of each dot in relation to every other dot. The change in position of the individual dots in response to the acting force is called the displacement field. With homogeneous materials such as the fabric and the steel springs we could model the response to force of the trampoline and predict what would happen if ... Thus, if we knew what the displacement field was for the cornea after a particular force was applied (after relaxing incisions, for example), predicting the outcome of the surgery would be straight forward.

Stress


Figure 9. Stress variable convention for an infinitesimal cube.
Stress is another term used in engineering and while it has some similarities to the stress encountered in medicine — its exact definition requires mathematics. The system of notation in stress–strain expressions are relatively straight forward, however:

, , and                                                                   

where the variable t (tau) is used for stress. Because it relates to certain areas or surfaces, it is customary for this variable to be subscripted with the first subscript associated with a plane perpendicular to a given axis (x in this case) and the second designating the direction of the stress (x, y, z). Figure 9 illustrates the convention. For stresses normal to a surface, the variable is changed from t to s (sigma), with a single subscript sufficient to define the quantity without ambiguity. Shear stress is always designated by t.


Figure 10. The given stress t1 is equivalent in its effect to the two orthogonal stresses t12 and t13 determined graphically. Shear stresses t12 and t13 can be similarly decomposed.
Stress is a measure of the intensity of force acting over an area and is expressed as  or in . Since they can act either perpendicular or parallel to a surface they are also vectors. Stress is often pictured as Cartesian coordinates with arrows showing the direction and the length of the line the magnitude of the force (Figure 10). Most of the readers will be familiar with the diagrams used to show vectors in geometry.


Figure 11. Forces F1 and F2 act parallel and perpendicular, respectively, to area A.
Figure 11 illustrates two forces acting on a surface. The force acting parallel to a given surface is called the shear stress (t) and is derived from  where A is the surface area and F the force applied. A good way to understand shear is to imagine a stack of typewriter paper through which we’ve drilled a perpendicular hole. If we insert a pencil into the hole and then press down upon the stack while moving our hands laterally, the entire pile would take on a different shape with the top layers sliding over the lower. The deviation of the pencil from perpendicular would represent the amount of displacement and hence the amount of shear stress applied to the stack . Shear stress can also be derived by simple summation :

  [1]

By transposing and integrating Equation [1], one obtains the basic relation for the shear V:

  [2]

Except for a positive constant of integration , the shear at a section is, therefore, the negative of the integral of the vertical forces acting on the section or point.

If we look at the intact cornea (in situ) we could deduce that the inner aspect of the cornea is subject only to compressive stress (from the IOP) whereas the anterior, outer, surface is subject to both normal and shear stress from the pressure of the lid. To be sure, there is some shear stress acting on the inner surface as well — but that amount is negligible. This can be shown by conducting an experiment similar to the one carried out with the stack of paper and is true only if the cornea is not severed from its moorings at the scleral ring. We will have more to say about this when we discuss the practical applications of corneal biomechanics to certain refractive surgical procedures. It would be well to keep in mind the paper stack analogy because in some ways it represents the nature of the cornea quite aptly in that it is orthotropic (see below).

The force acting perpendicular to the surface is called the normal stress and is derived from . Such a simplified derivation of the stress can be made more accurate if we were to make A a very small number. Hence it is that an exact definition of stress implies the notion of a limit. [1] Normal stress can be further defined depending upon the direction of the force. Thus the stress created by the reader pushing down on a table top — acting toward the surface — is called compressive or negative stress. If one were to pull up on the table cloth covering the table — away from the surface — one would be exerting tensile or positive stress.

The term tensile strength and compressive strength are often applied to structural materials to indicate the material’s resistance to stretching (tensile) or crushing (compressive) stress. Typically a structural material has different abilities to withstand compressive and tensile loads. It is the never ending attempts to balance these forces that produces the premature alopecia and graying so characteristic of structural engineers.


Figure 12. In three dimensions,the stresses act on areas that may be thought of as faces of a cube. Different normal and shear stresses act on each face, and the subscript 1 and 3 are used to distinguish them.
Stress can, of course, act in any direction and so when we begin to consider stress in a three–dimensional context — the whole business begins to get rather complicated. For example, let us consider a cube of corneal tissue. A cube has six sides each of which taken by itself is a square. For the sake of our discussion we will consider the cube to be made up of three squares each with a mirror image of itself to make up the other three sides (Figure 12). We will assign the subscripts 1, 2, and 3 (x, y, z) to identify the planes upon which the stresses act. The symbol t will represent shear and the symbol s will represent normal or perpendicular stress as described previously.

The direction of shear stress is generally not known — after all it can act in any vector whereas normal stress acts at 90°. Hence, for the purposes of our demonstration we will assign direction of the ambiguous shear stress to 2 unique shears whose direction is parallel to the sides of each surface. We will make the magnitude of the combination of the two known shears equal to that of the unknown one. Remember — we do not know the direction of the original shear (hence its ambiguity) but we can know its magnitude. We therefore make the two equivalent to the one by applying the rule of adding forces.

Thus it is that every small volume (which can be decomposed to a single point — keep in mind the concept of limits) has nine associated stresses: six shears ( ) and three normals: ( ). By contrast, a force vector P has only three components: , , and . This can be written in a column vector, thus:

                                                                                                                                          

Analogously, the stress components can be assembled as follows:

  [3]

 

This is a matrix form of the stress tensor, which is a second–rank tensor requiring two indices to identify its elements. For brevity, a stress tensor is sometimes written in indicial notation such as , where it is understood that i and j can assume designations x, y, and z as in Equation [4]. It should be noted that the stress tensor is symmetrical, i.e.: . This follows directly from the equilibrium requirements for an element.

Figure 13. An element of a body in pure shear.

We can neglect the higher order infinitesimal and show that this process is equivalent to taking the moment around the z–axis (Figure 13). Thus:

 +,

                                                                                   

where the expressions in parentheses correspond respectively to stress, area, and moment arm. Simplifying:

 [4]

It can be shown in the same manner that  and . Hence the subscripts for the shearing stresses are commutative — that is their order may be interchanged and the stress tensor is symmetric. The implication of this symmetry is very important. The fact that the subscripts can be interchanged means that the shearing stresses on mutually perpendicular planes of an infinitesimal element are numerically equal. It also follows from this that it is possible to have an element in equilibrium only when the shearing stresses occur on all four sides of an element simultaneously.

It is a given that all of these forces are in equilibrium. That is, as each volume (point) moves, the object deforms and the stresses change until there is a balance of forces and movement ceases. For each volume to have equilibrium there are six relationships between the stresses. While an exact demonstration requires mathematics, suffice to say that these relationships — the equilibrium equations — are not alone sufficient to determine the nine stresses.

For the equilibrium of a solid body, the equations of statics require the fulfillment of the following conditions:

 [5]

 

This expression states that the sum of all forces acting on a body in any (x, y, z) direction must be zero. The same would hold true for momentum — not discussed in this treatise. In a planar problem where all members and forces lie in a single plane such as the x–y plane, the relation , although still valid, is trivial.


Figure 14. Infinitesimal element with stresses and body forces acting on it.
An infinitesimal element of a body must be in equilibrium. For a two–dimensional case the stresses acting on such an element (dx)(dy)(1) is shown in Figure 14. Hence:

Simplifying and recalling that  holds true, one obtains the equilibrium equation for the x direction. This equation and an analogous one for the y direction, are shown below:

[6]

 

For a cube, a typical equation from a set of three is:

 [7]

 

In deriving the above equation, material properties of the element are not needed. Thus these equations are applicable whether the material is plastic, elastic, or viscoelastic.

In three–dimensional cases, there are not enough equations of equilibrium to solve for all the unknown stresses. In a two–dimensional case, there are only three unknown stresses: . In a three–dimensional case, however, there are six unknown stresses but still only three equations. Consequently, all problems in stress analysis are indeterminate. We either have to introduce additional ideas and equations or else consider problems where intuition tells us that some of the stresses are zero; then equilibrium equations can be used to obtain the remaining stresses. Fortunately for us, in many cases the stresses encountered in biomechanics do equate to zero or at least nearly so. It is unfortunate that refractive surgery is not one of these.

Stress Calculations

Stress is never measured directly. What is measured are loads such as pressures and forces. The stress is calculated by using a mechanical model of varying complexity with the measured loads as input variables. Two of the more simple models are:

  • stresses in a strip of tissue extended along its length
  • stresses in a thin–walled shell

To measure the stress/strain characteristics of a material, the geometrical measurements must be precise (length, width, thickness]. Traditionally. a rectangular shape or strip has been used since the geometry is easy reproduced and the constitutive properties are easily inferred. In a complex Nonhomogeneous layered material, such as the cornea, the act of cutting the cornea into strip can introduce errors in measurement however. Excessive corneal edema may result which can affect the results. Reproducible external conditions of temperature and humidify thus become important components of the measurement.

The cornea may also be measured by inflating it within an artificial anterior chamber using internal pressure as the stress (calculated using La Place’s rule or finite element modeling) and measuring the strain by means of marks on the cornea. Conceptually, this is a better method, although it requires special instrumentation and very precise measurement of strain.

Stress Measurement in a Strip


Figure 15. A representative tension specimen subjected to force F and having a cross-sectional area A.
A simple state of stress exists when a strip of tissue having a uniform cross–section (a tension specimen) is extended (stretched) by a force acting lengthwise (Figure 15). Every cross–sectional area A transverse to the length of the specimen must act to support the pressure F. A reasonable expectation is that the stress acts only on A and is a uniform, normal stress wherein magnitude is . However, this relationship is only true far from the ends of the specimen. The concept of far, though, is elastic and depends upon whom you are talking to. Far has been facetiously defined as that quantity which is equal to twice the distance from the center to the end — the end in this case being defined as 10 times the cross–sectional area of the specimen divided by its perimeter.

Thin Shell Stress Measurement Method


Figure 16. An element of a thin shell is pictured. The loads P 1, P 2, and P 3 and the stresses ó1, ó2, and ó3 are shown.
In the literature there are two models related to an estimation of curvature change due to RK surgery in which the cornea is considered as a uniform sphere.[26, 37] Shell elements are a natural choice given the small thickness to radius of curvature ratio for the cornea ( ). For purpose of our discussion, the stresses  may be calculated in terms of the loads  from consideration of equilibrium only (Figure 16). Since the shell is thin, the stresses can be assumed to be constant through the thickness and the shear transverse to the wall negligible. Generally partial differential equations must be solved. However, if we assume a spherical shell subjected to pressure only — the shear stresses will be zero and the result will be . Some of you may recognize this as La Place’s Law . La Place’s law or formula refers to pressure within a hollow sphere:

Internal pressure is transmitted to the wall as a force that produces tangential, and therefore circumferential, stress. Factors modifying this effect are the radius of curvature of the sphere and the wall thickness, thus:  where S = stress in grams per square millimeter, p = intrasphere pressure in grams per square millimeter, r = radius of curvature in millimeters, and t = wall thickness, also in millimeters.

Note that the calculation of the stress in terms of pressure, radius, and thickness is independent of the kind of material making up the wall. For the cornea, which is an idealized thin–walled sphere, we may apply La Place’s rule to obtain circumferential or hoop stress:

circumference/hoop stress = (pressure x radius)/(2 x thickness)

This relationship can be quite helpful in a variety of circumstances to estimate the stress at the surface of the model cornea. In reality, it is an approximation whose application implies a certain degree of error. In general, if the radius/thickness ratio is in excess of 15, the error will be held to approximately 5%. A closed treatment of these problems can be found with the Lamé equations and constants, particularly with layered materials such as those seen in the cornea.[38] In a thick–walled sphere under internal radial pressure, the radial components of force are progressively converted to hoop or circumferential forces. Indirect measurements indicate the distribution of tension across the cornea is uniform in its normal state.[39, 40] Direct measurements have not been performed to confirm these findings, however.

The application of the law of La Place in studying the action of incisions upon the cornea is, in this author’s opinion, misplaced. In engineering, the law of La Place is usually not used with radius/thickness ratios of 10–20. The mean radius of the cornea is @ 7.0 mm and the thickness about 0.5 mm yielding a ratio of 14. Furthermore, La Place’s law would only hold for that portion of the cornea far (1–2 mm) from the limbus. The area of the limbus itself is mechanically different from the rest of the cornea and the stresses much more complex in their nature. This difference cannot be accounted for by the formula of La Place, hence a more complex model for the cornea must be used.

Strain

We have defined stress as a measure of intensity of force at a given surface point and displacement as the movement of that point in response to an applied force. Strain is tied to the idea of displacement and can be of two types — neither of which has dimension.


Figure 17. The line AB on the surface of the globe deforms into the line A'B' giving rise to the geometric interpretation of the extensional strain å = ( A'B' — AB)/ AB.
The first type of strain is extensional strain and can be readily visualized by drawing a very short line (AB) on an extensible globe (Figure 17). Note the requirement that the line be very short. If we measure the length of the line, , in the unloaded or resting state and again, L, after applying a load, the ratio after the change in length of the line is the extensional strain:

[8]

The accuracy of our equation would increase were we to reduce the line length almost to zero.


Figure 18. Lines AB and BC are perpendicular lines on the surface of the globe that deform into lines A'B' and B'C' giving rise to the geometric interpretation of shear strain . Shear strain is a measure of the amount of change in the angles associated with a deformation.
The other type of strain is shear strain and can be illustrated by using the previous example with the addition of another line (BC) drawn at right angles to the other (Figure 18), thus forming an orthogonal pair. When the surface is again loaded, the angle between AB–BC will change. This change in angle is a measure of the shear strain (g). Hence shear strain is defined as the change in angle from 90° of any orthogonal pair of short lines.


Figure 19. A small square shown deforming into a parallelogram. The sides extend giving rise to two extensional strains, while the right angles distort giving rise to a shear strain. A cube distorts into a parallelepiped.
For any such pair of lines there are, therefore, three associated strains — two extensional (one for each line) and one shear (for the angle between the pair). There can, of course, be any number of lines radiating out from a single point on the surface. Nonetheless, there are only three unique strains associated with that point regardless of the number of lines one chooses to draw — all the rest are derived from the three. This can be shown to be true by examining a small square drawn on our extensible surface (Figure 19). It follows that deformation of this square can only occur through extension of its sides and by one pair of angles becoming smaller and the other pair larger than 90°. If we make the square small enough, the magnitude of the change for all four angles would be the same — hence there is only one shear strain associated with our square. Since we are dealing with a very small square (sides approaching zero in length) an extension of one side opposite to any other side will be almost equal to the side itself. In the limit these extensions will be the same. Thus, there are only two extensional strains, one for each of two adjacent sides.

Strain in three–dimensions

We used a cube in our discussion of three–dimensional stresses. We will use the same cube to demonstrate the derivation of shear in three–dimensions. A cube can be drawn using twelve lines — three radiating out from each of the four vertices. By making the cube small enough we can consider each vertex to be essentially the same point. We therefore have reduced our cube to three lines. We can therefore associate six strains with our cube — three extensional (one for each line) and three shear (one for each unique orthogonal pair of lines). Another way to show this is to recall that a cube is made up of six squares — three the mirror image of the other three. There are three strains associated with each square (see our discussion of stress above) hence 3+3+3=9 strains. However, some of the sides are common to more than one square. When these are accounted for the strains are reduced to three extensional and three shear — hence six strains in all.


Figure 20. The tension specimen shown deforming from its original length L0 to length L under the action of axial force F.
Strains can only be measured indirectly because applying any instrument to the surface would distort the measurements taken. Such data is usually gathered by using optical methods such as particle trackers which measure the displacement of marks or other reference points on the surface (hearken to our trampoline). With the usual small loads encountered in the eye (and the subsequent small displacements) this task can be difficult.

An example of a simple strain state can be seen in our previously described tensile test model (Figure 20). If the tissue is loaded along its length, we would not expect any shear and therefore the most important strain would be along its length. If we assume the strain to be constant then .

Strain is usually assumed to be the same under a given stress in compression and extension. However, this assumption is probably incorrect for biological materials since these are comprised primarily of water which resists compression to an extraordinary degree. On the other hand, its resistance to extension is negligible. Also, most strain measurements are made on the surface of the element. Internal strains are derived through mechanical modeling or by using photoelastic methods.

[1]    The idea of limits is essential to a branch of mathematics known as the calculus.

© Leo D. Bores, MD - 2002