Stress, Strain, and Constitutive Relations PDF

2 Stress, Strain, and Constitutive Relations 2.1 Introduction Consider the two structural members in Figure 2.1, each acted upon by an applied weight W that is much larger than the individual weights mg, which we therefore neglect. From statics, we know that if these two members are in eqUilibrium,...

2 Stress, Strain, and Constitutive Relations 2.1 Introduction Consider the two structural members in Figure 2.1, each acted upon by an applied weight W that is much larger than the individual weights mg, which we therefore neglect. From statics, we know that if these two members are in eqUilibrium, then 'iF = 0 and }:'M = O. Free-body diagrams of the whole structure and the individual parts reveal that the reaction and internal forces are the same: Ry =h = W; that is, from the perspective of statics alone, these two problems are equivalent. Nevertheless, intuition tells us that the behavior of member A need not be the same as that of member B. One may fail before the other. An important question to be answered by mechanics, therefore, may be the following: Which member will likely fail first given increasing weights W? At first glance, we may be inclined to say that A will fail before B, for A is "thinner," and indeed this may well be. Yet, our information is incomplete: We have not specified what A and B are made of; A could be made of a much stronger material than B. Thinking back to statics, we realize that we never specified the prop- erties of the materials or structures that we studied, we simply assumed that they were always rigid (Le., infinitely stiff). In this book, however, we will see that the individual properties of materials are central in biomechan- ics. For example, we often seek to match the properties of man-made or tissue-engineered replacements to those of the native tissue or organ. Indeed, one of the continuing challenges in biomechanics is accurate char- acterization, or quantification, of the material behavior of both living tissues and biomaterials. Returning to Figure 2.1 and the question of whether structure A or B will fail first, we first need to define what is meant by failure. In mechanics, failure simply implies an inability to perform the intended mechanical func- tion. Structures A and B could thus fail by the following: Material failure, including fracture, tearing or rupture, as, for example, in the tearing of an anterior cruciate ligament 45 J. D. Humphrey et al., An Introduction to Biomechanics © Springer Science+Business Media New York 2004 46 2. Stress, Strain, and Constitutive Relations W W Ry=W Lx y fy=W A A Free Body Diagrams W W FIGURE 2.1 Contrast the potentially different responses of two simple structural members, A and B, which have the same type of fixed support at the top, the same initial length, and the same axial loading W at the otherwise free-end. Free body diagrams of the whole and a part reveal the reactions at the fixed support and the internal force. Deforming excessively, which mayor may not include a permanent defor- mation such as a severely bent (e.g., plastically deformed) surgical instru- ment, which does not return to its functional shape Determination of failure criteria for materials is thus an important re- sponsibility of the biomedical engineer. Recalling our intuition earlier that structure A may fail before B because A is thinner (Fig. 2.1) suggests that failure criteria cannot be written in terms of the applied loads alone; one must also consider the geometry. This brings us to the concept of stress. 2.2 Concept of Stress In 1678, Robert Hooke published the anagram (in Latin) ceiiinosssttuv, which can be deciphered as, ut tensio sic vis, and translated, as the force, so the extension; that is, by studying the response of linear metallic springs to  2.2. Concept of Stress 47 force w extension FIGURE 2.2 Force-extension behavior of three different metallic springs, which exhibit linear behaviors and thereby can be quantified by individual spring constants k (or stiffnesses). Although many springs exhibit a linear behavior, nonlinear springs exist as well. the application of various weights, Hooke realized that there is a one-to- one relationship for many materials between the applied load (force) and the motion (extension). Figure 2.2 shows force-extension curves for three similar but different linear springs, each described by the general formula f = k(x - xo) where x is the current length, Xo is the original length, and k is the so-called spring constant or stiffness. The results in Figure 2.2 for three different springs suggest that each is characterized by an individual spring constant kb k2' or k3 (or material property). If we apply the same idea of plotting force versus extension for cylindrical specimens of various materi- als (e.g., aluminum or stainless steel), we quickly discover that such tests do not characterize the material. If the same loads are applied to the same material in two different labs, which use two different diameter specimens, we find different slopes in the force-extension data. Indeed, the thicker sample, albeit composed of the same material, will appear "stiffer" because it will extend less in response to the same force.! Hence, in contrast to Hooke's original idea, there is more to it than just "as the force, so the extension." In 1757, Leonard Euler realized that a better measure for analysis is a "force intensity" or stress. Simply put, Euler defined this intensity as a force acting normal to an area divided by the value of that area (i.e., a pressure- like quantity that we now call a normal stress). During the period 1823-1827, Augustin-Louis Cauchy formalized the concept of stress. Defined as a force acting over an oriented area at any point in a body, it is 1 Differences between structural stiffness, which depends in part on geometry, and true material stiffness are important in clinical measurements, as discussed later. 48 2. Stress, Strain, and Constitutive Relations clear that there can be different "stresses" at the same point depending on the orientation of the applied force and the orientation of the area of inter- est, which implicitly says depending on the choice of a coordinate system (i.e., an origin and basis)-that is, stress is a mathematical construct; its "value" is not unique. For example, consider a force having only an x component, say f1fx, which acts over an area M in the current (deformed) configuration of the body (Fig. 2.3). Intuitively, the effect of the same force f1/x on the same area M will have different effects depending on the orientation of M, which is denoted by the outward unit normal vector n (i.e., Inl = 1). For example, if n is in the direction of f1fx, we call the force a normal force and its inten- sity (per unit area) a normal stress; if n is perpendicular to the direction of f1/x, we call the force a shearing force and its intensity a shear stress. Note, therefore, that although a force could act on an area at any angle, it is gen- erally convenient to resolve the force vector into components that are normal and parallel to the surface. Specifically, then, if we let Mx denote that M has an outward normal n = ex and take the limit as M tends to zero, then we obtain the normal stress: (2.1) n normal ---+-~-Afx AA force L~ shear force FIGURE 2.3 Schema of the x component of a differential force Mx (actually, the mean value of a uniformly applied force) that acts on an area !lA. Clearly, the effect of this single component of force on the underlying material will depend on the ori- entation of the area over which it acts: If the area is oriented in the same direction as the force, we expect a tension or compression, whereas if the area is oriented orthogonal to the force, we expect a shearing action. The directions of both the force f and the area (given by its outward unit normal vector n) are equally important.  2.2. Concept of Stress 49 that is, we denote stress (i.e., a force acting over an oriented area in the current deformed configuration) with the Greek lowercase sigma, with the first and second subscripts (or indices) associated with the oriented area (i.e., face) on which the force acts and the direction of the applied force, respectively. Hence, with O"(face)(direction) , (2.2) then lim Nx = dfx =0" (2.3) oMy dAy M.... yx for a shear stress in the x-y plane. Although stresses act in the direction of that component of the force that acts at the point of interest, they are not vectors. Rather, because stress is a force acting over an oriented area, it is associated with two directions, one each for the direction of the force and the outward unit normal n. Mathe- matically, such quantities are called tensors, but we will not exploit this character. It is useful nonetheless to represent the components of stress by arrows that act on the appropriate faces of a body in the appropriate direc- tions. See, for example, Figure 2.4, which shows the so-called positive sign convention for a 2-D state of stress relative to a Cartesian ·coordinate system. In particular, we shall assume that normal stresses are positive (Tyy (Tyx ~ (Txy + SIGN CONVENTION (Txy 0 --- CJYx FIGURE 2.4 Positive sign convention for a 2-D state of stress, relative to Cartesian coordinates, that exists at point p but is shown over a square domain for illustrative purposes. The direction of each component is positive on a positive face (i.e., a face with an outward unit normal in the positive coordinate direction) but negative on a negative face. This convention is consistent with normal stresses being positive when tensile and it is consistent with equilibrium at a point (i.e., the balancing of equal and opposite pushes and pulls). 50 2. Stress, Strain, and Constitutive Relations when tensile; this requires that O"xx be directed in a positive direction on a positive face (i.e., one having an outward unit normal in a positive coordi- nate direction) and conversely that O"xx be directed in a negative direction on a negative face (i.e., one having an outward unit normal in a negative coordinate direction). Indeed, for consistency, we assume the same for the shear stresses, as seen for O"yx and O"xy in Fig. 2.4. (Note: An easy way to remember this positive sign convention is that a positive times a positive is positive and a negative times a negative is a positive; hence, the positive sign convention requires a negative direction stress on a negative face.) As in statics, it is best to use the directions associated with the sign convention; if the computed value turns out to be negative, it simply tells us to switch the assumed direction of that component of stress. Recall that if a body is in equilibrium, then each of its parts must also be in equilibrium-this holds true for any material point p. Note, therefore, that because of our sign convention (Fig. 2.4), the "two" normal stresses O"xx in the figure balance and so too for the x-direction action of the "two" shear stresses O"yx. If the mathematical point p represents a material particle even of infinitesimal dimension, however, we see that eqUilibrium is not neces- sarily satisfied; that is, the two O"yx'S would tend to create a couple (Le., a pure moment, or force acting at a distance) that would tend to rotate the differential region centered at particle p. Equilibrium could be ensured by the addition of an opposing pair O"xy, as seen in Figure 2.4, wherein we have preserved both the positive sign convention (e.g., positive direction on a positive face) and the notation sigma subscript (face, direction). Conse- quently, O"xy == O"yx numerically at every point, which can be proven rigor- ously via the balance of angular momentum (Le., I.M = 0 in this case) as shown below. Committing the sign convention represented in Figure 2.4 to memory serves one well throughout mechanics. In general, however, we note that each point p could be thought of as an infinitesimal cube that is reduced in size in a limiting process. As such, each point can be thought to have six faces relative to each Cartesian coordinate system. For (x, y, z) coordinates, this implies positive and negative M.., My, and M z faces. Moreover, given that each point can be acted upon by a force 4f, which has a component representation relative to (x, y, z) as 4f =f1fi +i1fy)+ f1f:k ==i1fx ex +i1fiy +i1fzez , (2.4) where i == ex and so forth, there are nine possible measures (Le., compo- nents) of stress at each point p relative to (x, y, z). They are dfx dfr dfz dAx dAx dAx O"(face)(direction) = dfx dfy dfz == [un O"yx 0" xy 0"yy Un]yz , 0" (2.5) dAy dAy dAy o"zx 0" zy 0" zz dfx dfr dfz dA z dA z dA z  2.2. Concept of Stress 51 which we have written in matrix form for convenience (matrices are reviewed in Appendix 6). The stresses 0'"", O'yy, and O'zz are normal stresses; they can cause extension or compression. The stresses O'xy, O'x" O'yx, O'y" O'zx, and O'Zy are called shear stresses; they can cause a body to distort, which is to say to experience changes in internal angles. Consistent with the above, this matrix is symmetric (i.e., O'xy = O'yx, O'XZ = O'zx, and O'yz = O'ZY)' This can be shown formally by letting the dimensions of an infinitesimal element be Llx, Lly, and Llz. Because the components of stress have units of force/area, to sum the moments about an axis such as the z axis in Figure 2.4, we must first mUltiply the respective component of stress by the area over which it acts and then multiply by the associated moment arm about any point, say o (because two of the four stress components have lines of action that go through 0, this point is convenient for computing the moments). Hence, tBLMz)o = 0 ~ -O'yx(LlxLlz)Lly + O'xy(LlyLlz)Llx = O. (2.6) Simplifying, therefore, we have the result: (2.7) Similarly, show that O'xz = O'zx and O'yz = O'zy. Example 2.1 Referring to Figure 2.5, what are the values of 0'"", O'xy, O'yx, and O'yy in this 2-D state of stress. 150kPa B 120 kPa --+-1- 120 kPa A x 150kPa FIGURE 2.5 An illustrative 2-D state of stress acting at a point, components of which act over oriented areas that are expanded for ease of visualization. Although the magnitudes of the components can be considered arbitrary, these values are con- sistent with in-plane values of stress within a large artery. 52 2. Stress, Strain, and Constitutive Relations Solution: Noting that the right face is an x face (with outward unit normal ex) and that the top face is a y face, we have O"xx = 120 kPa, O"xy = 0 kPa, O"yy = 150 kPa, and O"yx = 0 kPa. Being able to identify components of stress O"(face)(direction) is an important step in understanding the mechanics. It cannot be overemphasized that stress is a mathematical concept; it is defined as a measure of a force acting over an oriented area at a point. Mathematically, stress is a tensor, which is defined independent of a coor- dinate system. Yet, to solve practical problems, one must always compute components of stress relative to a particular coordinate system. Because coordinate systems (which are defined by an origin and a set of base vectors) are not unique but can be defined in many different ways, many different sets of components of stress exist at the same point in a body that is subjected to a single set of applied loads. For example, for the three Carte- sian coordinate systems shown in Fig. 2.6-defined by (0; ex, ey, e z ), (0; e;, e;, e;), and (0; e;, e;, e;)-the point p admits three different sets of com- ponents of the same stress O"(face)(direction): [au O"yx 0" xy O"yy au] O"yz , (2.8) O"ZX O"zy O"zz [a~ a~l O"~y O"~x O";y O"~z , (2.9) O"zx O";y O"zz Y y' x' ~~--~~----------~ x FIGURE 2.6 Interrelations, via single angles, between different Cartesian coordinate systems that share a common origin and a z-axis. The angle a is taken to increase in the direction given by the right-hand rule: counterclockwise, with the z-axis coming out of the paper.  2.2. Concept of Stress 53 and (2.10) [Note: The (... )' and (... )" notation here simply denotes different coordi- nate systems; it does not imply differentiation as used in many courses on differential equations.] In some cases, it may be more natural to compute one set of components, say uxx , uxy , , Uw whereas in other cases, it may be more useful to compute another set of components, say u~, U~y,... , cfzz. A good example of this need is the case of a rectangular structure that consists of two members that are glued together on a 45° angle (Fig. 2.7). Because glue is stronger in shear than in extension (empirically compare removing a Post- it® note by applying a normal versus a shear force), it is useful to know how much of the applied force f results in shear versus normal stresses at the glued interface; that is, we would like to know the values of u~ and ~y, which are computed relative to (0; e;, e;, e;). Yet, from Figure 2.7, it is clearly easier to enforce eqUilibrium relative to (0; ex, ey , ez ); that is, assuming the force is applied uniformly over the surface area on which it acts, it is easy to show (see Section 3.3.2) that 'i.F= 0 yields Uxx = ffA and uxy = 0 on a cross section with an outward unit normal n = ex that cuts through the glued 1-1 X ~I 1-1 ~, p axx y' y x' FIGURE 1-1 ~cr'~ x 2.7 Free-body diagrams of the same structure cut along two cross sections: one with an outward unit normal in the direction of the applied force (which is natural for solving the equilibrium problem) and one with an outward unit normal to the glued surface (which exposes stresses that act thereon and are important with regard to possible debonding). In each case, the components are identified as O(face)(direction). Note: although we could denote stresses with respect to an x' face and x' direction as a,.:,.:, we prefer to denote them as a'"" for convenience. 54 2. Stress, Strain, and Constitutive Relations region. It is clear, therefore, that multiple coordinate systems can be useful even in the same problem. Fortunately, we shall discover in the Section 2.3 that the desired values (J~ and (J~y can be determined directly from the more easily computed values (Jxx = f/A and (Jxy = 0; that is, we will not need to solve the equilibrium problem for each coordinate system of interest. Inasmuch as coordinate systems are introduced for convenience, many different coordinate systems prove useful in the wide variety of problems that fall within the domain of biomechanics. For problems in the circula- tory and pulmonary systems, for example, the nearly circular nature of the arteries, capillaries, veins, and bronchioles render cylindrical-polar coordinate systems very useful. For problems involving certain cells, saccular aneurysms, the urinary bladder, and so forth, spherical coor- dinates are very useful. For problems in cardiac mechanics, particularly for the left ventricle, prolate spheroidal coordinates are useful. For prob- lems in developmental cardiology, toroidal coordinates are conven- ient. Indeed, the list goes on and on, including more complex coordinate systems. Fortunately, regardless of the coordinate system, our notation (J(face)(direction) will hold; that is, we seek measures that describe the intensity of the force relative to both the oriented area on which the force acts and the direction of the applied force. In cylindrical coordinates (r, e, z), we have (Fig. 2.8) (Jrr (JrO [ [(J] = (J6r (J66 (2.11) (Jzr (Jzo and likewise for spherical coordinates (r, e, I/J), we have (Fig. 2.9) [(J] = (Jrr [ (JOr (JrO (JOO (Jr~ (Jo~, 1 (2.12) (Jt/lr (Jt/lO (J~~ each at every point p. It is important to review and understand that which is represented in these figures. Independent of the specific coordinate system, a J-D state of stress is one in which only one component of stress (e.g., (Jxx) is nonzero relative to the prescribed coordinate system; a 2-D state of stress is one in which four com- ponents of stress (e.g., (Jw (JyY' (Jxy, and (Jyx, three of which are independent because (Jxy = (Jyx) may be nonzero relative to the chosen coordinate system; a 3-D state of stress is one in which all nine components (six of which are independent) may be nonzero in general. In summary, the concept of stress is a mathematical one. Stress may be computed at each point in a continuum body; when resolved with respect to a coordinate system, there are nine components at each point, although  2.2. Concept of Stress 55 cylinder (TOr./././././././ FIGURE 2.8 Components of stress relative to a cylindrical coordinate system, again using the standard notation O"(face)(direction). / sphere / / / / / / / \ \ \ \ \ \ \ (T88 \ FIGURE 2.9 Normal components of stress relative to a spherical coordinate system, again denoting the components as O"(face)(direction). As an exercise, add the shearing components. 56 2. Stress, Strain, and Constitutive Relations only six components are independent relative to each 3-D coordinate system-three normals and three shears. Because coordinate systems can be related via transformation relations, the various components of stress can be related through transformation relations. Let us now derive these useful relations for Cartesian components. 2.3 Stress Transformations Consider a 2-D state of stress relative to either (0; ex, ey ) or (0; e~, e;) as shown in Figure 2.10. Because these figures merely represent the stresses that act at point p, we can cut either square part in order to represent com- ponents relative to both coordinate systems in a single figure. Anticipating the need to sum forces and moments to enforce equilibrium (of the parts), let the diagonally cut part be of uniform width ~z and length ~y along the vertical cut edge. Hence, the three exposed areas of interest are computed easily, as shown in the Figure 2.1l. From geometry, we have sin a = opp/hyp and cos a = adj/hyp, where opp == Ax and adj == ~y. Hence, hyp = ~y/cosa = ~yseca. Now, if we mUltiply FIGURE 2.10 General 2-D state of stress at the point p emphasizing again that the components are defined with respect to the orientation of the area over which they act (i.e., the face) and the direction of the applied force (i.e., the direction). Hence, different sets of components coexist at the same point. This allows us to make fictitious cuts that expose, on the same element, components relative to different coordinate systems.  2.3. Stress Transformations 57 T y at"" y' x' 1cr~ d y au au x ayy ayx ayy I- dx ·1 FIGURE 2.11 Detailed diagram of the fictitious element from Figure 2.10 with the 2-D components of stress isolated relative to two different Cartesian coordinate systems. Remembering that if a body is in equilibrium, then each of its parts are in equilibrium, we can therefore use a force balance to relate the components of stress for the two coordinate systems. Alpha is an arbitrary cutting angle. through by the width Llz, then we have the result that the area that 6 xx and a~y act over is given by Llz (hyp) = M sec a, with LlyLlz = M being the area over which axx and axy act. Similarly, ayy and ayx act over Llz( opp) = Llz(hyp) sin a = M sec asin a = M tan a. Now, we are ready to sum forces. Balanc- ing forces (Le., stresses multiplied by the areas over which they act) in the x' direction requires that we find the components in the x' direction. Clearly, the x-directed forces must be multiplied by cos a, whereas the y-directed forces must be multiplied by sin a to get the x' direction components. Hence, equilibrium yields L Fx' = 0 = a~M sec a - (a xxM)cos a - (a yxM tan a )cos a - (ayyM tan a) sin a -(axyM)sina or axx=axxcos 2 a+ 2 I '. 2 a, axysmacosa+ayysm (2.13) wherein we let axy = ayx from above, and we see that the M cancels through- out, thereby rendering the equation valid for arbitrarily chosen (small) dimensions about point p. It is important to realize, therefore, that the con- tinuum concept of stress actually represents an average force intensity within a small region (neighborhood) centered about the point of interest. 58 2. Stress, Strain, and Constitutive Relations Recalling the trigonometric identities 2 1+ cos2a 2 1-cos2a cosa= 2 ' sm a= 2 ' sin2a = 2 sina cosa (2.14) Eq. (2.13) can be rewritten as I 1+cos2a I' 1-cos2a a xx =a xx 2 +axy sm2a+ayy 2 (2.15) or a xx + a yy a xx - a yy a ' _ ----'-"-. xx- 2 + 2 cos2a + a xsm2a y' (2.16) Given that Eq. (2.13) is a perfectly acceptable way to compute a~ from values of stress relative to (0; ex, ey ) for any a, one might ask: Why use the trigonometric identities to obtain the alternate form [Eq. (2.16)]? This is actually a good question, the answer to which comes from hindsight. Throughout this text, we must remember that even what may appear to be simple or obvious may have taken great thinkers many years to realize (e.g., nearly 150 years passed between Hooke's ideas on force to Cauchy's on stress). We will see below that Eq. (2.16) is extremely convenient in one particular application. It is also important to remember that we, as students, benefit from the many hours, days, indeed weeks or even years of thought by many which resulted in simplifications we have today. Forces in the y' direction (Fig. 2.11) can similarly be balanced, namely LFy' = 0 = a;yM sec a + (a xxM)sin a - (axyM)cosa - (a yyM tan a) cos a + (a yxM tan a ) sin a, or a;y = 2 sina cosa( a yy ; a xx ) + (cos 2 a - sin2 a)a xy- (2.17) Again, using trigonometric identities, we can rewrite this equation as a xy = a -a yy xx 2 I 2 sm a +axy cos 2a. (2.18) Remembering that if a body is in eqUilibrium, then all of its parts are in eqUilibrium, we often fictitiously cut a body into multiple different parts to expose, on cut surfaces, specific components of stress of interest. Because the selection of the oblique cutting plane in Figure 2.10 did not isolate a y' surface (Le., an area with outward unit normal e;), we must consider another free-body diagram that isolates a;y and a;x (Fig. 2.12). Doing so, we can again balance forces in x' and y'. This is left as an exercise; thus, show that given such a cut,  2.3. Stress Transformations 59 , (Tyx I I I +----1_ 'Yxx I ----~ I FIGURE 2.12 Alternate fictitious cut (cf. Fig. 2.11) to expose y/-face components of stress cr..2,Fx' = 0 ~ O";xMseca = - (O"xxM tana)cosa -(O"xyM tana)sina + (0" yyM) sin a + (0" yxM) cos a, or O";x = 2 sina cosa( O"yy ; 0" xx ) + (cos 2 a - sin 2 a)O" xy' (2.19) Again, using trigonometric identities, we have the alternate form 0" yy - 0" xx. 2 2 0" yx = sm a + 0" xy cos a, I 2 (2.20) which is the same as Eq. (2.18), as it should be (i.e., O"~y must equal O";x to satisfy the balance of angular momentum for a rectangular body cut paral- lel to x' and y'). Finally, show that.2,Fy' =0 ~ O";yMseca =(O"xxM tana)sina -(O"xyM tana)cosa + (O"yyM)cosa -(O"yxM)sina, or O"yy = 0" xx sm 2 a - 2 0" xy sma I. cosa + O"yy cos 2 a. (2.21) This equation can then be written as , _ 0" xx + 0"yy 0" yy - 0" xx O"yy- 2 + 2 cos2a-O"xysm2a. (2.22) 60 2. Stress, Strain, and Constitutive Relations Together, Eqs. (2.16), (2.18), (2.20), and (2.22) show that the components of a 2-D state of stress relative to one Cartesian coordinate system can be related to those of any other Cartesian system sharing a common origin. All that is needed is the angle a that relates the two coordinate systems; indeed, as a check, we see that at a= 0, the (0; x, y) and (0; x', y') coordi- nate systems coincide, and our transformations yield O'~ = O'xx at a = 0, and so on, as they should. Although it can be shown that similar transformation relations hold for 3-D states of stress and also for other coordinate systems, we will not go into the details here. Rather, the most important things to realize are that the concept of stress is defined at every point in a contin- uum body and that the components of the stress (tensor) are not unique; they are determined by the coordinate system of interest. Fortunately, one does not have to solve the equations of eqUilibrium to determine the value of each component of stress relative to each coordinate system. Rather, one only needs to solve equilibrium once (in terms of the coordinate system that is most convenient) and then to compute any related component of interest through the transformation relations. Because these derivations did not require us to specify the material, these relations are good for any solid or fluid as long as the continuum assumption is valid. We will thus use these transformations throughout this book. Example 2.2 Consider the 2-D state of stress in Figure 2.5. Find the values of stress cTxx,