Distortions in Media: Compressive andTensile Strain

Let us start with a medium that is uniform and unstrainedand look at how it is distorted by a number of types of strain.  For each point in the medium we have aframed vector that specifies its location, a set of extension vectors, and anorientation. The extension vectorswill be infinitesimals aligned with the frame of reference.  Consequently, we will be concerned witha frame and a location, but will derive two different types of information fromthe strain frame.

The basic procedure will be to describe the anatomy of asituation in terms of a strain and define a convenient framed vector for thecalculations (F). First we will compute the new locationafter the strain (). That basiccalculation is then repeated for a set of locations displaced an infinitesimaldistance from the location along each of the frame axes, before and after thestrain occurs. The extensionvectors are computed by subtracting the location from each of those locations,giving a strained infinitesimal frame, . It is then asimple process to compute the strain quaternion for that distorted extensionframe. The strain quaternion willgive a substantial amount of information about the strain.  We can compute the axes of rotation forthe strain and from those axes compute the orthogonal frame associated withthat strained frame (). Finally, wecan compute the rotation quaternion () for the orthogonalized strained framed vector (). Orthogonalization involves computing a set of orthogonal vectors thatexpress the orientation of the strained frame. A description of the procedure is given elsewhere (Langer 2006b). Forpresent purposes, we will not need a sophisticated understanding.  The appropriate orientation of thestrained frame will be apparent.

Constrained Compression/Expansion

Let us start with a simple anatomy so that we can developour tools before the mathematics becomes too complex. The medium is uniform and isotropic and there is no lateralmovement. It may be visualized asa plunger compressing the medium in a rigid well or pulling it in onedirection.

Let the initial distance between the two plates be and the displacement will be . If there iscompression, then is negative andif there is expansion that ispositive. Since the materialcannot move laterally, it is restricted to the movement in the direction of thecompression or expansion, that is, in the same direction as the displacement ofthe moving plate.

Location is measured relative to a point on the surface ofthe unmoving plate, A₀. The location before the compression is and, after thecompression, it is . The differencebetween the locations is . This isexpressed symbolically as .

The problem comes down to writing an expression for thevector as a function oflocation, . In thissituation the direction of the displacement is perpendicular to the twoplates. Therefore, we construct aunit vector that is perpendicular to the bottom plate, . The part of that is relevantto the compression, , is the projection of upon , , that is, the component of the location vector that isperpendicular to the lower plate. Overall, the material between the plates must compress or expand adistance , but that change is evenly distributed over the distance of . The perpendicular component of , , is multiplied by the interval between the plates, divided by theoriginal distance between the plates, , to give the amount of displacement of the location and thatdisplacement is in the direction from the lower plate to the upper plate.

Normally, the compression is small, on the order of 5%, orless. Most biological materialsare relatively incompressible and inextensible largely because they are mostlywater in colloidal gels or they are mineralized, as in bone.  When it is important that biologicalmaterials do not compress, they are usually provided with a ‘skeleton’ of somesort or they are confined in an inextensible cavity.

We now consider the extension vectors.  Let the first and second vectors of theframe be parallel with the lower plate and the third vector be perpendicular.

If we look at the extension vectors, the vertical axischanges an amount proportional to the compression or expansion, while thehorizontal axes remain the same, so that there is a slight stretching orflattening of the test cube. The test cube is a unit cube so we may replace theinfinitesimals with unity. If the vertical direction is the k axis, then the strain quaternion is a scalar.

The flattening is the same everywhere in the medium,therefore the extension strain is uniform.

The orientation frame for the test box is equal to theorientation frame for the framed vector so the ratio of the two frames isunity.

Unconstrained Expansion/Contraction

The constrained case is in many ways trivial, but itillustrates the basic approach. Wenow move on to a more complex situation, similar to the first, but without aconstraint upon lateral movement. As the plates are brought together the medium may bulge laterally and asthey are separated it may be drawn centrally. Consequently, points in the medium may move vertically andlaterally and they usually do so.

The constrained case of compression is simple and a bitboring, but it gives us a start on the unconstrained case, which is neithersimple nor boring. In theunconstrained situation, we also assume an incompressible, or minimallycompressible, amorphous matrix between two flat plates, but with room to movelaterally, parallel with the plates. The gel will be assumed to be a circular mass, to simplify thecalculations and because it is the lowest energy state for an unconstrained gelcapable of flow, because it minimizes surface tension.  The shape of the slab of gel isirrelevant to the basic physics.

First, the movement of the plates by distance will cause avolume, , of material to flow laterally from a circular region ofradius about the centerof the mass. The outer radius ofthe circular ring that contains the material that was inside the circular slabof radius . is given by a simple expression.

The volume of displaced material is equal to the volume ofthe ring that it occupies, so, we can set these two expressions equal and solvefor the outer margin of the ring in terms of the vertical shift and the radiusof the original circular slab.

For an initial slab thickness of = 1.0 and radius of = 1.0 the radius of the slab, , is plotted versusthe amount of compression or expansion, .

Consequently, the central circular slab of radius now occupies acircular slab of radius , where is given by theformula that was just derived. Forsmall amounts of compression, the percentage increase in radius is about halfthe percentage decrease in height between the plates. For large compressions the new radius increases rapidly forsmall increments of compression.

However, this is not a correct analysis, because the mediumdoes not move as a vertically flat surface. Rather it bulges. The middle parts move out more that the parts that impinge on thebounding plates. Let us considerhow that might occur.

First we need to determine how much material there is thatis moving. The volume of materialprior to the compression or expansion () is the volume of a circular disc with a radius of and a thicknessof . If the platesare moved, then the volume inside the initial radius is decreased (compression)or increased (expansion). Thatcapacity, , is easily computed and thus the amount of material that hasto shift, , is readily computed.

If the excess material were to flow uniformly peripherally,then the material a distance from the centerof the disc would end up lying at a distance of from the centerof the disc. However, flow is notuniform. It is most likely tooccur as described by the Navier-Stokes equation of hydrodynamics, that is,laminar flow. If that is areasonable approximation to the flow, then the rate of flow is proportional tothe square of the distance from the fixed walls. Close to the wall there is very little flow and, in ahorizontal plane midway between the two plates, the flow would be maximal andproportional to the square of the distance to the nearest wall.

Even if the flow is not laminar, it will often have asimilar profile. For instance, amuscle is often attached to a rigid substrate like bone at its ends and thatattachment does not shrink or expand as the muscle contracts or lengthens.

The lateral flow would be given by an expression like thefollowing, where is taken to bethe middle plane, half way between the two endplates.

The variable ‘d(x)’ is the displacement of the materialparallel to the plates and, ‘x’, is the distance from the middle plane.  The constant is thedisplacement of the material in the middle horizontal plane, at . We do not know or k, thereforewe must solve for them in terms of the variables that we do know.

At the upper and lower plates the horizontal displacement iszero, which allows us to express the constant in terms of the maximaldisplacement.

Let and , to simplify the notation a bit for calculation.  Envision the flow as a stack of thincircular sheets that are expanding or contracting peripherally from the radius to the radius , where d(x) is the diameter given by the expression for thedisplacement, and each sheet has a thickness of . The volume ofthe expansion is the sum of those sheets, where each sheet has a volume that isproportional to the thickness of the sheet times its width.

Passing to the limit for section thickness, dx, theexpression can be written as an integral.

The term within the integral can be expanded into thefollowing expression.

The integral of this expression turns out to be muchsimpler than one might expect.

This expression can be set equal to the volume of materialbetween the plates and, with suitable rearrangement of the terms, it can besolved for the maximal displacement.

The maximal displacement is a function of the radius of thecircular slab, the amount of compression or expansion, and the distance betweenthe plates. With values for themaximal displacement, , and the coefficient of the squared distance from the centerhorizontal plane, k, it is possible to calculate the profiles of the materialfor a series of compressions and expansions. That has been done for an initial disc one unit thick withradius of one unit, and the results are illustrated in the following figure.

For small compressions, there is a small bulging of thematrix, approximately 8/10^ths the percentage of the compression, sothat a 10% compression causes the maximal lateral displacement to be about 108%of the radius of the slab. Whenthe compression is 50%, the maximal displacement is about 160% of the slabradius. Finally, when thecompression is large, like 90%, then the maximal displacement is large, 400% ofthe slab radius. For largecompressions it is likely that the matrix spreads to adhere to the endplates,so this is clearly an idealization.

When stretching the medium, the center of the slab dips in,making the middle thinner, much as a rubber band becomes thinner in the middlewhen it is stretched. The minimalthickness when the medium is twice its original thickness is 0.54 of theoriginal radius. The minimalthickness becomes zero when the distance between the endplates is five timesthe original. Clearly, that is anidealization as well.

Thedisplacement profiles for different amounts of compression and expansion. The slab is compressed or expanded from0.1 to 0.9 of its height (d). This causes thematerial in the slab to be displaced laterally. The volume of the displaced material is the displacementprofile rotated in a circle about the center of the slab. The denser green area indicates theoriginal profile.

Distortions of framed vectors in an unconstrained compressed/stretchedmedium

We now turn to the calculation of the components of framedvectors in the gel. To start, the framed vector will have a location relativeto the center of the disc of strained material, in the horizontal plane midwaybetween the two moving plates, on the central axis. The extension and orientation frames will be aligned withthe vertical and horizontal axes of the anatomy, with the first two axesparallel with the horizon and the third vector parallel with the vertical axis.

Thedisplacement vectors for compression vary in obliquity as a function of thedepth of the location and the distance from the center of the slab.  If we take the middle horizontal planeas our reference plane, then the locations in that plane are carried directlylaterally a distance equal to the maximal displacement.  At the bounding planes, thedisplacement is entirely vertical. Between those two extreme locations, the displacement is the sum of avertical displacement () similar to that computed in the first example and ahorizontal lateral displacement ().

The lateral displacement in the horizontal plane is computedas follows. First, compute thehorizontal expansion or contraction at the new horizontal level of thelocation. To do that, it isnecessary to compute the horizontal component of the location, , and the new horizontal level, x. The contraction or expansion at that level, , is the parameter that we just finished computing in theprevious section.

Once we know the amount of change at the level of thelocation, then it is simply a matter of scaling the horizontal component of thelocation by multiplying it by the ratio of the dilated radius to the originalradius. The change in thehorizontal component is the new horizontal location minus the originalhorizontal location. The total changein locations is the change in vertical position plus the change in horizontalposition and the new location is the original location plus the change inlocation.

Compressionwithout lateral constraint causes the matrix to move vertically and laterally. The vertical movement is greatest nearthe moving plates and the lateral movement is greatest near the middlehorizontal plane.

Many of these relationships are illustrated in the abovefigure. This is a situation wherethe order of the vectors in the vector sum is important.  The magnitude of the horizontal changein location is a function of depth in the slab. Therefore, it is necessary to determine the depth bysubtracting the vertical change in location from the initial location beforecomputing the horizontal change in location.

If the origin of the coordinate system is in the middlehorizontal plane at the center of the slab and the initial location is , then, assuming that the plates are parallel to the i,j-plane and equal distances above and below, that thecompression brings them symmetrically towards the plane, and that the center ofthe slab is at the origin of the coordinate system, then the displacementvector is given by the following expression.

where

If we plug the value for ‘x’ into the expression for thechange in location, the result is complex, but almost all of the symbols standfor constant parameters of the anatomy.

The anatomy of this system is symmetrical about the centralaxis, so it is possible to examine most of the behavior of the system in aradial slice through the slab of material. If we examine the positive j,k-plane, then the above expression reduces to the following equation.

We can use this formula to calculate the amount of movementat any point in the slab of material for any amount of compression orextension. The next couple figuresare examples of the results of such calculations in a slab of unit radius andthickness when it is compressed or stretched by a factor of 0.30.

The greater the distance to the middle horizontal plane themore vertical the flowlines. Forpoints on the upper or lower surface of the slab, the flowlines are verticaland for points in the middle plane they are horizontal.  The greater the distance from thecentral axis, the longer the flowlines, particularly for points in the middleof the slab.

For small amounts of compression, the lateral component issmall. However, if the matrix isloosely constrained, like the nucleus pulposus of an intervertebral disc withinthe ligamentous sheath of the annulus fibrosus, then the lateral displacementmay be the limiting parameter upon compression.

Theflow vectors for compression between parallel plates are greater and morehorizontal for middle levels and greater radii. The illustration is for a compression that is 0.3 times thedistance between the plates.

Theflow vectors for stretching between parallel plates are greater and morehorizontal for middle levels and greater radii. The illustration is for an expansion that is 0.3 times thedistance between the plates. Flowlines are drawn for vertical lines at 0.2, 0.4, 0.6, 0.8, and 1.0times the horizontal distance from the central axis to the outer surface.

In general, biological structures are not as neat as thecomputed examples considered here, but, these examples do give reasonablemodels for the consideration of biological situations.  While exact solutions of the biologicalgeometry may not be possible, it is possible to generate order of magnitudesolutions that support or deny particular interpretations.

The extension frame

Now, we turn to a consideration of the extension frame.  As before, the unstrained frame is aset of infinitesimals. Unlikebefore, the strained frame is a function of the location of the sampled point.

We can simplify the problem by noting the radial symmetryand looking at the extension frame in the j,k-plane (a=0.0).

The fractional change in the extension vectors is obtainedby dividing by the unstrained frame.

The rate of stretching in the horizontal plane becomesgreater as one moves to more peripheral locations. That is because the area of a ring of a given widthincreases in proportion to the square of the radius of the ring.  In the present situation the matrix hasmore volume to occupy, for a given increase in radius, as it spreadsperipherally. The proportionalchange in the vertical component of the ratio of the stretched or compressedmatrix to the unstressed matrix is not a function of the radial offset, but thefurther from the middle horizontal plane one moves the greater the tilt in thedirection of j.

The following figures show how the medium is strained bycompression () and stretching (). Two types ofstrain are examined, volume strain (V) and strain in a plane parallel with theend plates (A). The verticalstrain is uniform throughout the depth of the slab, because it was assumed tobe so as one of the initial conditions of the anatomy.  However, the resulting distortion ofthe medium causes the strain in the perpendicular plane to vary as a functionof depth in the slab ().

In fact, the direction of the volume strain reverses so thatstretching the medium causes the material near the endplates to experience anet increase in local volume and the material midway between the plates todecrease in volume. That is alsoimplicit in the illustrations of the flow lines. With stretching, material flows from the middle towards theupper and lower surfaces and during compression the flow is towards the middleand laterally.

Thelocal volume strain is computed for different amounts of compression andstretching of a slab between parallel endplates. The amount of strain upon the slab is given by the relativechange in the distance between the plates (). The depth in the slab is . There is areversal in the direction of the local volumetric strain so that material tendsto flow from the center of the slab to towards the endplates (stretching) orvice versa (compression).

Since the amount of vertical strain is constant throughoutthe depth of the slab, the flow must be largely due to spreading or contractionin the perpendicular plane. As thematerial is forced centrally and laterally during compression, the onlyavailable direction for compensation is to cause the lateral walls of thestrain box to move laterally. Consequently, there is an area strain (A). Conversely, as the slab is stretched and its center is drawnmedially, the walls of local strain boxes are drawn in and the areadiminishes. Unlike the volumestrain, the area strain is in a consistent direction for all depths in theslab, but is becomes greatest or least in the middle plane.  As noted above, there is no dependenceupon the radial distance in the slab, so the area strain is uniform in a planeparallel to the endplates.

Thelocal area strain in a plane parallel with the endplates depends upon depth inthe slab or material. Thedirection of the strain is consistent for all depths, but the amount of strainis maximal in the middle plane parallel to the endplates. The magnitudes of area strain aregreater than the amount of volume strain.

The maximal amount of areal strain is greater than theamount of volume strain. Forinstance, a compression of 0.5, that is rendering the slab half as thick,causes the area in the middle plane to expand to more than 2.5 times itsinitial value. Consequently, allother things being equal in this model, we would expect damage to the materialin the slab to occur most often near the middle of the slab.

The horizontal components of the strain frame are identical,therefore the frame does not rotate in the i,j – plane. The thirdcomponent, the vertical component, does rotate, about the - i axis. In this instance, a slice in the j,k-plane, the i axis is the tangent to the circumference of the slab, so, the verticalaxis generally tilts radially. Theangular excursion of the tilt is the ratio of the new direction to the originaldirection.

The tilt angle is computed for compressions from -0.9 to-0.1 and expansions from 0.1 to 0.9. The curves are symmetrical about the middle horizontal plane.  Tilt is less for locations near themiddle horizontal plane and greater as one approaches the upper and lower facesof the slab.

There is also a dependence upon the radial distance from thecentral axis of the slab (b).  As one moves away from the centralaxis, the amount of tilt at a given vertical position in the slab increases forany given amount of compression or stretching.

Thetilt angle is about the i axis if the slice is in the j,k plane. For stretching (), the vertical axis tilts peripherally.  With compression, the vertical axis ofthe strained box tilts centrally. The magnitudes of tilt are more affected by compression.  The amount of tilt is greater at largerdistances from the center of the slab.

When compression occurs, the vertical axis of the strainedbox tilts towards the central axis of the slab and when stretching occurs thetilt is away from the central axis. The tilt is least near the central axis and greatest at the peripheralboundary. At the central axis thematerial either expands or contracts along the vertical axis.  Peripherally the material further fromthe middle horizontal plane is dragged relative to the material that isslightly closer. If the gel isbulging (compression) then the vertical axis is dragged centrally and if thegel is sucking in (stretching) the vertical axis is dragged peripherally.  This may also be expressed by notingthat the axis of rotation for compression is in the direction of i and for stretching it is in the direction of –i.

Atgreater distances from the center of the slab the magnitude of the tilts of thevertical axis of the strained box increase. Compare this plot, sampled at the outer boundary, with theprevious figure, which is sampled midway between the center of the slab and itsouter boundary. The verticaldistances () in both figures are in the unstrained slab.

The orientation frame

We may now consider the orientation of the frame vector forthis type of strain. Since thefirst two vectors of the strain frame do not rotate, that is, the frame movesperipherally along a radial trajectory, one can orthogonalize the strain frameby computing the vector that is the ratio of the first two vectors, which isobviously a vector in the direction of the k axis. Consequently theorientation frame does not rotate. That agrees with our intuition that the material moves radially in orout, but it does not rotate.

Summary

Compressive or tensile strain is intuitively a simpledistortion. The material simplyexpands or contracts without twisting. That is clearly the case when the material is constrained.  If the constraints are removed and thematerial is allowed to flow, then the situation becomes somewhat morecomplex. Most of this chapter wastaken up in defining the distortion that occurs in a relatively simple anatomy.

In order to obtain a definite solution, it was necessary toassume some basic flow characteristics, namely laminar flow. In unconstrainedmaterials, it is natural to assume constant volume and equal forces throughoutthe medium. Flow under thosecircumstances will tend to be laminar. Those assumptions may or may not be valid in any particular situation,but the computed deformations are consistent with observations of changes inelastic media that are compressed or stretched. When pinched between two surfaces they bulge laterally andwhen stretched they narrow in the middle.

The intent of this analysis was not to describe anyparticular instance of this type of strain, but to demonstrate how one might goabout analyzing the distortions in terms of strains as characterized by strainboxes. This is not the usualapproach of elasticity theory, but the intent was not to describe flows perse, but to express the anatomical movementsimplicit in particular types ofdistortions. If better oralternate descriptions of the flow become available, then the same methodologymay be used to explore the implications of that flow to the anatomicalmovements within the medium.

There are a number of anatomical situations that might beexplored with this approach. Themovements of the annulus fibrosus of the intervertebral disc might beinteresting as a case of compressive strain. Strains in muscles, tendons, and ligaments might be examinedas instances of tensile strain.

Langer, T. P. (2006b). Strained Boxes and Products of Three Vectors.: http://homepage.mac.com/tlanger_sasktel_net/Strain/Strain.htm.