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# Strain+Tensor+Readings - e ‘L 3“".11 SEC 5—4...

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Unformatted text preview: e ‘L. 3“".11 SEC. 5—4. MATHEMATICAL DEFINITION OF STRAIN position.3 As this element is deformed. the force on the top plane reaches a __ ﬁnal value oh dx (1:. The total displacement of this force for a small defor- mation of the element is y (1y; see Fig. 5-6(b). Therefore. since the external L work done on the element is equal to the internal recoverable elastic strain energy. 1 l 1 dU mm = E T dx dz X y (1y = 3 Ty (Ix (1y (I: = Ery (IV W—i w—a ‘— (5'2) average force distance where (IV is the volume of the inﬁnitesimal element. By recasting Eq. 5-2. the strain-energy density for shear becomes , w.) M = ( 3—5) M : g 6—3) By using Hooke's law for shear stresses. 7 = Cy. Eq. 5-3 maybe recast as w.) M = ( \$5) = 51 (M) or Ushw, =J I:er (5-5) vol 20 Note the similarity of Eqs. 5—2—5-5 to Eqs. 3-12—3—15 for elements in a state of uniaxial stress. Applications of these equations are given in Chapters 6. 14. and 18. h." B GENERALIZED CONCEPTS OF I L STRAIN AND HOOKE’S LAW 54. Mathematical Deﬁnition of Strain3 - Since strains generally vary from point to point, the deﬁnitions of strain . _' must relate to an inﬁnitesimal element. With this in mind. consider an Nensional strain taking place in one direction, as shown in Fig. 5—7(a). Soine points like A and B move to A ’and B ’. respectively. During straining. mm A experiences a displacement u. The displacement of point B is 2This assumption does not make the expression less general. T1115 and the next section can be omitted Without loss 01 cuntmuny m the text. 173 174 CH. 5 GENERALIZED HOOKE‘S LAW. PRESSURE VESSELS Fig. 5-7 One» and two—dimensional strained elements in intial and final positions. u + Au. since in addition to the rigid—body displacement u, common to the whole element it, a stretch Au takes place within the element. On this basis. the deﬁnition of the extensional or normal strain is4 , Au du 8 — 1:13., 1; — a; (5'6) If a body is strained in orthogonal directions, as shown for a two— dimensional case in Fig. 5‘7(b). subscripts must be attached to e to differ entiate between the directions of the strains. For the same reason‘ it is also necessary to change the ordinary derivatives to partial onesTherefore, if at a point of a body. 11, v. and "w are the three displacement components occurring. respectively. in the x, y, and 1 directions of the coordinate axes. the basic deﬁnitions of normal strain become Bu 2m 6w (5-7) 4A more fundamental deﬁnition of extensional strain. more amenable to the more gen- eral concepts of stretching or extending, can be expressed using Fig. 5-7(c). as , D'C' _ DC 8‘ : Alli!” T (5‘68) where the vectorial displacements of points C and D are u(« : CC’ and up = DD ’. For the small defomiations considered here. Eq. 5-6a reduces to Eq 5h Also see Sections 11-11 and l 1—l2. vr-J ‘0'. ' 'o-.. 5Tb anive at equal to the 1 that strain :[This defor: stresses; see jgkk‘, The deﬁi FCOEq. 5-9: In Eqsi 5-9 a hie since it sequences 0: ‘ ‘ In exam displacemer If; it, and w mdependem _ among Sui 8 to one for i these equati texts on the “-E pNote that double subscripts analogously to those of stress can be used for .. Illese strainsThus. (5-8) _§_- share one of the subscripts designates the direction of the line element. .‘_ ad the other the direction of the displacement. Positive signs apply to iglongations. . In addition to normal strains. an element can also experience a shear gain as shown in the x y plane in Fig 5 7(c) This inclines the sides of the ”armed element 1n relation to the x and the y axes Since v is the dis- - - ment in the y direction as one moves in the x direction ()v/(ix 1s the . 1 of the initially horizontal side of the infinitesimal element Similarly vertical side tilts through an angle (ht/6y. On this basis. the initially _ut angle CDE is reduced by the amount (iv/ax + {in/(9y. Therefore. for I] angle changes. the deﬁnition of the shear strain associated with the '1'1 tycoordinates 1s 51?;- .-.3r _ 11“ (5-9) " arrive at this expression. it is assumed that tangents of small angles are - 1-1 to the angles themselves in radian measure. A positive sign for the .- strain applies when the element is deformed. as shown in Fig. 5—7(c). ' deformation corresponds to the positive directions of the shear s;see Fig. 1- —4.) f The deﬁnitions for the shear strains for the xz and yz planes are similar 1-:q59- - —-a——“3+ﬂ‘— — _a_w+a_v (510) ’sz ’YZX ax az ’sz _ 11y 6y az - - Eqs. 5-9 and 5-10. the subscripts on y can be permuted.This is permissi- since no meaningful distinction can be made between the two ' 'llcnces of each alternative subscript 1|! examining Eqs 5 7 5— 9. and 510. note that these six strain- ‘aCCment equations depend only on three displacement components 9 and 10. Therefore. these equations cannot be independent. Three * ndent equations can be developed showing the interrelationships ' g Eu 8H 8 .7“ y}: and y The number of such equations reduces 0116 for a two— dimensional case The derivation and application of ”equations known as the equations ofmmparibilm are given in . whit; on the theory of elasticity. SEC, 54. MATHEMATICAL DEFINITION OF STRAIN 175 176 CH. 5 GENERALIZED HOOKE‘S LAW. PRESSURE VESSELS 5-5. Strain Tensor ______________________.______———————————— The normal and the shear strains deﬁned in the preceding section together express the strain tensor. which is highly analogous to the stress tensor already discussed. lt is necessary. however. to modify the relations for the shear strains in order to have a tensor. an entity that must obey certain laws of transformation.5 Thus. the physically attractive deﬁnition of the shear strain as the change in angle y is not acceptable when the shear strain is a component of a tensor. Heuristically. this may be attributed to the following. In Fig. 5—8(a‘). positive y“. is measured from the vertical direction.The same positive y“. is measured from the horizontal direction in Fig. 5-8(b). In Fig. 5-8(c). the same amount of shear deformation is shown to consist of two er/Z‘s. The deformed elements in Figs. 5-8(a) and (b) can be obtained by rotating the element in 5-8(c) as a rigid body through an angle of yu_/2.The scheme shown in Fig. 5-8(c) is the correct one for deﬁning the shear—strain component as an element of a tensor. Since in this deﬁnition the element is not rotated as a rigid body. the strain is said to be pure or irrutatitmal. Following this approach. one redefines the shear strains as _ ‘ Yty _ :Y_\\_ en ~ 8n ‘ .2— ~ 7 r— v. 8).: = 8“ = ’2‘; = 3- (5—11) _ - L _ L 8” —— 8“- —— 7 —— 2 From these equations. the strain tensor in matrix representation can be assembled as follows: 'ny Yr 3 e r —— —— 2 2 EAT 8 l \' 6X: 7;: a y L. E e). \ 87“, a“. (5 — l 2 i) 2 2 e . \ a 3 v e . - 2 2 V The strain tensor is symmetric. Mathematically. the notation employed in the last expression is particularly attractive and has wide acceptance in continuum mechanics (elasticity. plasticity. rheology. etc.). Just as for the stress tensor. using indicial notation, one can write 5,, for the strain tensor. n is beyond the scope of this text. A better apprecia— ‘Rigorous discussion of this questio dy of Chapter 1 l. where strain transformation for a tion of it will develop. however. after stu two‘dimensional case is considered. Fig. 5-8 Shear deformations. The transfc Chapter 1 l - The sim , deﬁned in 5 tion of the 2 Q The rea mechanical are applic: However. c strains give meats (10 m 525-6. G. ~f01 7 In this sect and strain z _ previously {ions is reft applicable ( the same pi plex for ani properties i three ortho have nine ii ‘ the next SC< _ Crystalline r be as large Isotropic m2 _ the develo] ”A. P. Bore Wiley, 1935): l. 1956): L. F M Cliffs. NJ: pm,“ 5 ANALYSIS OF DEFORMATION Forces applied to solids cause deformation, and forces applied to liquids - cause ﬂow. Often, the major objective of an analysis is to find the defor» mation or flow. It is our objective in this chapter to analyze the deformation _ . of solid bodies in such a way as to be relevant to the state of stress in those- bodies. 5.1 DEFORMATION If we pull a rubber band, it stretches. If we compress a cylinder, it shortens. If we bend a rod, it bends. If we twist a shaft, it twists. See Fig. 5.1. Tensile stress causes - tensile strain. Shear stress causes shear strain. This is common sense. To express ‘ ' these phenomena quantitatively, it is necessary to deﬁne measures of strain. Consider a string of an initial length L0. If it is stretched to a length L, as '. . ' shown in Fig. 5.1(a), it is natural to describe the change by dimensionless ratios such as L/Lo, (L — Lo)/Lo, and (L — L0)/L. The use of dimensionless ratios L eliminates the absolute length from consideration. It is commonly felt that these . ratios, and not the lengths L0 or L, are related to the stress in the string. This expectation can be veriﬁed in the laboratory. The ratio L/Lo is called the stretch ratio and is denoted by the symbol A. The ratios I L~L0 E_L~1.0 Lo’ L e: (5.14) are strain measures. Either of them can be used, although numerically, they are different. For example, if L = 2, and L0 = 1, we have A = 2, e = 1, and e’ = 5 We shall have reasons (to be discussed later) also to introduce the measures L2 — L3 2L2 ’ L2- 1.3, e = 2L3 e = (5.1—2) IfL = zandL0 =1,wehavee = gande = %. ButifL = 1.01 andLo = 1.00, then e t 0.01, s i 0.01, e i 0.01, and e' i 0.01. Hence, in infinitesimal elongations, 112 Sec. 5.1 Deforma‘ (E m (1 Figure 5.1 (c) Twistin all of these strain mea they are different. The preceding 5 mations. For exampl ends, as shown in Fi; top will be shortened, strains are related to To illustrate she When the shaft is tv shown in Fig. 5.1(d). It is more customary reasons for this will t The selection 01 strain relationship (i. if we pull on a string a curve of the tensile empirical formula rel: strain is simple becau: It was found that, for 1 in uniaxial stretching, where E is a constant stresses. Equation (5. be a Hookean material that is called a yield s. rces applied to liquids :is is to ﬁnd the defor- nalyze the deformation ' state of stress in those ider, it shortens. If we 1. Tensile stress causes non sense. To express easures of strain. :hed to a length L, as y dimensionless ratios f dimensionless ratios 1monly felt that these ass in the string. This L0 is called the stretch (5.1—1) numerically, they are = 2,: =1,ande’= )duce the measures (5.1—2) : 1.01 and L0 = 1.00, nitesimal elongations, 113 Sec. 5.1 Deformation (cl (d) Figure 5.1 Patterns of deformation. (a) Stretching. (b) Bending. (c) Twisting. (d) Simple shear. all of these strain measures are approximately equal. In finite elongations, however, they are different. The preceding strain measures can be used to describe more complex defor— _ mations. For example, if we bend a rectangular beam by moments acting at the ends, as shown in Fig. 5.1(b), the beam will deﬂect into an arc. The “fibers” on top will be shortened, and those on the bottom will be elongated. These longitudinal strains are related to the bending moment acting on the beam. To illustrate shear, consider a circular cylindrical shaft, as shown in Fig. 5.1(c). When the shaft is twisted, the elements in the shaft are distorted in a manner ' shown in Fig. 5.1(d). In this case, the angle a may be taken as a measure of strain. It is more customary, however, to take tan a or % tan a as the shear strain; the reasons for this will be elucidated later. The selection of proper measures of strain is dictated basically by the stress- Strain relationship (i.e., the constitutive equation of the material). For example, if we pull on a string, it elongates. The experimental results can be presented as 'a curve of the tensile stress (I plotted against the stretch ratio A or strain e. An empirical formula relating (r to e can then be determined. The case of infinitesimal strain is simple because the different measures of strain just presented all coincide. ‘ It was found that, for most engineering materials subjected to an infinitesimal strain In uniaxial stretching, a relation like 0’ = Ee (5.1—3) Where E is a constant called Young’s modulus, is valid within a certain range of stresses. Equation (5.1—3) is called Hooke’s law. A material obeying it is said to be a Hookean material. Steel is a Hookean material if (r is less than a certain bound that is called a yield stress in tension. 0111 114 Analysis of Deformation Chap. 5 Corresponding to Eq. (514), the relationship for a Hookean material sub. jected to an inﬁnitesimal shear strain is T = G tan a (51.4) where G is another constant called the shear modulus or modulus of rigidity. The range of validity of Eq. (5.1—4) is again bounded by a yield stress, this time in shear. The yield stresses in tension, in compression, and in shear are different in general. Equations (5.1-3) and (5.1—4) are the simplest of the constitutive equations. The more general cases will be discussed in Chapters 7, 8, and 9. Deformations of most things in nature and in engineering are much more complex than those just discussed. We therefore need a general method of treat~ ment. First, however, let us consider the mathematical description of deformation. Let a body occupy a space S. Referred to a rectangular Cartesian frame of reference, every particle in the body has a set of coordinates. When the body is deformed, every particle takes up a new position, which is described by a new set of coordinates. For example, a particle P, located originally at a place with coor- dinates (0., a2, a3), is moved to the place Q _with coordinates (x1, x2, x3) when the body moves and deforms. Then the vector PQ is called the displacement vector of the particle. (See Fig. 5.2.) The components of the displacement vector are, clearly, xi ‘ 01, X2 " 02, x3 ‘ a3. 03,X3 OZIXZ Figure 5.2 Displacement vector. If the displacement is known for every particle in the body, we can construct the deformed body from the original. Hence, a deformation can be described by a displacement ﬁeld. Let the variable (a1, a2, a3) refer to any particle in the original configuration of the body, and let (x1, x2, x;) be the coordinates of that particle when the body is deformed. Then the deformation of the body is known if x1, x2, x3 are known functions of ab a2, a3: (5.1-5) x. = xi(a1, a2, a3). Sec. 5.2 The Strair - This is a transforma mechanics, we assun .3 transformed into a n« to one; i.e., the func "' ' the unique inverse A rigid-body m are not directly relat consider the stretchin three neighboring po formed to the points h.» . .9 for every point in the .r; . , --1 The displaceme 6| '3 [if If a displacemei --"_ tion, we may write :3. .3: ._ If that displace: ,F . we write in 33' i5 _ In order that the trans {5‘ what conditions must b 33‘ Note: If the trans .1: functions x, (a,, a2, 11,) m 7' g lap/6a,] must not vanisf _ .1». (See Sec. 2.5.) .‘r . Ti _' 5.2 THE STRAIN 1 3-. The idea that the str ‘~ ' Robert Hooke (1635 ‘ explained it in 1678 2 ‘2‘; ,- .- or “The power of an: r. The meaning of this 3 pulled a rubber band )eforrnation Chap. 5 Hookean material sub— (5.1-4) todulus of rigidity. The 'ress, this time in shear. re different in general. constitutive equations. and 9. :ering are much more neral method of treat— ‘iption of deformation. lar Cartesian frame of tes. When the body is described by a new set y at a place with coor- :s (x,, x2, x3) when the displacement vector of em vector are, clearly, Displacement vector. ody, we can construct 1 can be described by particle in the original inates of that particle )dy is known if x,, x2, (5.1—5) .1' e.‘ . an 1....‘.1..';iM....__J-__..... -.. . In... I Sec. 5.2 The Strain 115 This is a transformation (mapping) from a,, a2, a; to x,, x2, x3. In continuum mechanics, we assume that deformation is continuous. Thus, a neighborhood is transformed into a neighborhood. We also assume that the transformation is one to one; i.e., the functions in Eq. (5.1—5) are single valued, continuous, and have the unique inverse 0. = a;(xi, x2, x3) (5.1—6) 7 for every point in the body. The displacement vector u is then defined by its components 141 = Xi " ai- (5.1—7) If a displacement vector is associated with every particle in the original posi— tion, we may write 2, (5.14;) If that displacement is associated with the particle in the deformed position, we write ui(a1, 02, 113) = xi(ar, 02, a3) — a“ u,(x1, x2, x3) = x, ~ a,(x,, x2, x3). (5.1-9) PROBLEM - In order that the transformation (5.1—6) be single valued, continuous, and differentiable, ‘ What conditions must be satisfied by the functions x,(a1, a2, (13)? Note: If the transformation is single valued, continuous, and differentiable, then the " ,,_ functions x,(a,, a,, (13) must be single valued, continuous, and differentiable, and the J acobian “I" lax/6a,) must not vanish in the space occupied by the body. The last statement is nontrivial. ,1 .' - (See Sec. 2.5.) .‘ﬂ-l . . .3311! STRAIN n. 1- ‘ The idea that the stress in a body is related to the strain was first announced by Robert Hooke (1635—1703) in 1676 in the form of an anagram, ceiiinosssttuv. He _ explained it in 1678 as Ut tensio sic vis, or “The power of any springy body is in the same proportion with the extension.” The meaning of this statement is clear to anyone who ever handled a spring or _ Pulled a rubber band. A rigid-body motion induces no stress. Thus, the displacements themselves are not directly related to the stress. To relate deformation with stress, we must consider the stretching and distortion of the body. For this purpose, let us consider three neighboring points P, P’, P” in the body. (See Fig. 5.3.) If they are trans- _ formed to the points Q, Q’, Q” in the deformed configuration, the change in area 116 Analysis of Deformation Chap, 5 ;_ Q Sec, 5.3 Strain Con e7 1 The difference betwet .3' ‘ several changes in the .-_.,. it: , iii-I leg- or as We deﬁne the strain t Figure 5.3 Deformation of a body. W3 “y. :m 2.." ﬁt, :12}. if“: and angles of the triangle is completely determined if we know the change in length of the sides. But the “location” of the triangle is undetermined by the change of _ so that the sides. Similarly, if the change in length between any two arbitrary points of ‘ '3"; the body is known, the new configuration of the body will be completely deﬁned, ' -_ ' except for the location of the body in space. The description of the change in ' _ distance between any two points of the body is the key to the analysis of defor- 12;, mation. {ii The strain tensor Consider an inﬁnitesimal line element connecting the point P(a1, 02, a,) to a :3 .- Green's strain tensor. 3 neighboring point P’(a, + dal, a2 + dag, a3 + dag). The square of the length (13., i strains and by Almans, of PP' in the original configuration is given by -- g; tensor. In analogy wit 2 _ 2 2 2 m . Lagrangian and e.~,- as (180 —' (1111 + (1112 + das. (52-1) ‘r That EU and 6,,“ When P and P' are deformed to the points Q(x,, x2, x3) and Q'(x, + dx,, x; + -.-- respectively, follows f‘ dx;, x, + dx,), respectively, the square of the length ds of the new element QQ' -' 3i 5 (52—11). The tensors (132 = dxi + dx§ + dxﬁ. (5.2—2) 1;? An immediate C( __ , :4,- _ 0 implies E” = e,,» = By EQS. (5.1 5) and (5.1—6), we have (:5, every line element fen 6x; 6a ﬂu ' and sufﬁcient conditio dx, = — dab do,- = “1 dxi' (52—3) * [£13. all components of the all] 6x; -' . it. Hence, on introducing the Kronecker delta, we may write :1 _ 3a, 3,, - u STRAIN COMPONENTS (13(2) 2 5.76111,- dd] = 817—— "—J' dx, dxm, (5.2-4) -” ; _ 6x, 6x,” _ If; 'I' If we introduce the di 6x 8x _ '. 2 _ _ __' _L * .F- (15 —- 5i] dxi dx/ —- 5,] 8a, 6a,, dill dam. (5.2‘5) —t t _ Deformation Chap. 5 lOW the change in length mined by the change of two arbitrary points of be completely deﬁned, iption of the change in o the analysis of defor- : point P(a,, a2, a,) to a square of the length dSo (52-1) a...
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