22 CHAPTER 1 Basic Elasticity
ofthelineelementis
ε=lim
L→ 0L
L
ThechangeinlengthoftheelementOAis(O′A′−OA)sothatthedirectstrainatOinthexdirection
isobtainedfromtheequation
εx=O′A′−OA
OA
=
O′A′−δx
δx(1.16)
Now,
(O′A′)^2 =
(
δx+u+∂u
∂xδx−u) 2
+
(
v+∂v
∂xδx−v) 2
+
(
w+∂w
∂xδx−w) 2
or
O′A′=δx√(
1 +
∂u
∂x) 2
+
(
∂v
∂x) 2
+
(
∂w
∂x) 2
whichmaybewrittenwhensecond-ordertermsareneglected
O′A′=δx(
1 + 2
∂u
∂x)^12
Applyingthebinomialexpansiontothisexpression,wehave
O′A′=δx(
1 +
∂u
∂x)
(1.17)
inwhichsquaresandhigherpowersof∂u/∂xareignored.SubstitutingforO′A′inEq.(1.16),wehave
Itfollowsthatεx=∂u
∂xεy=∂v
∂yεz=∂w
∂z⎫
⎪⎪
⎪⎪
⎪⎪
⎬
⎪⎪
⎪⎪
⎪⎪
⎭
(1.18)
The shear strain at a point in a body is defined as the change in the angle between two mutually
perpendicular lines at the point. Therefore, if the shear strain in thexzplane isγxz, then the angle
betweenthedisplacedlineelementsO′A′andO′C′inFig.1.15isπ/ 2 −γxzradians.