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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. 56(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. 52. the strainenergy density for shear becomes , w.) M = ( 3—5) M : g 6—3)
By using Hooke's law for shear stresses. 7 = Cy. Eq. 53 maybe recast as
w.) M = ( $5) = 51 (M)
or
Ushw, =J I:er (55)
vol 20 Note the similarity of Eqs. 5—2—55 to Eqs. 312—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. 57 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
(57) 4A more fundamental deﬁnition of extensional strain. more amenable to the more gen
eral concepts of stretching or extending, can be expressed using Fig. 57(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. 56a reduces to Eq 5h Also see Sections 1111 and l 1—l2. vrJ ‘0'. ' 'o.. 5Tb anive at equal to the 1 that strain :[This defor:
stresses; see jgkk‘, The deﬁi
FCOEq. 59: In Eqsi 59 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. (58) _§_ 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“ (59) " arrive at this expression. it is assumed that tangents of small angles are
 11 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. 59 and 510. 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 55. 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. 58(b). In Fig. 58(c). the same amount of shear deformation is
shown to consist of two er/Z‘s. The deformed elements in Figs. 58(a) and
(b) can be obtained by rotating the element in 58(c) as a rigid body
through an angle of yu_/2.The scheme shown in Fig. 58(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. 58 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 5256. 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 (5.13) 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: (5.15) 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 rigidbody 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— (5.14) 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). (5.19) 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.) .‘ﬂl . . .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 rigidbody 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: ,
iiiI
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. (521) ‘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, (5.24) ” ; _
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 (521) a...
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