ABSTRACT
XO ′′ YO ′′ and are no longer mutually perpendicular. For small strains we may define
Šthe dilatational (or extensional) strain in the x-direction ( ) ( )
xdx zyxzydxx xxx
dxxx ∂ ∂=−+= → ξξξε ,,,,lim
and similarly the dilatational strain in the y-and z-directions, respectively, as
, y
∂= ξε . z z
zz ∂ ∂= ξε
Šthe shear strain in the y-direction of a point on the x-axis, given by the angle
α (see Figure 6.1) and denoted by yxε ( ) ( )
xdx zyxzydxx yyy
dxyx ∂ ∂=−+== → ξξξαε ,,,,lim
and similarly the shear strain in the x-direction of a point on the y-axis
y x
xy ∂ ∂== ξβε
The last two components are not independent, as ,2/ θπβα −=+ so that the decrease of the angle between dx and dy is introduced as a new parameter, called angle of shear, in the xy-plane and given the symbol xyγ according to
xyyxxy ∂ ∂+∂
∂=+=−= ξξεεθπγ 2
It is obvious that a rigid rotation of the deformed body about O allows α and β to be varied, so that we may always assume that .or xyyx εεβα == Consequently, we have
⎟⎟⎠ ⎞
⎜⎜⎝ ⎛
∂ ∂+∂
∂== yx
ξξγε 2 1
2 1 (6.2)
X
Y
0' α
β θ
Y'
X'
0 dx
dy (x,y,z)ξ y (x+dx,y,z)ξ y
ξ x ξ x(x+dx,y,z)
Similar results apply to the shear strains in the yz-and xz-planes. We thus have the following six independent parameters which define the state of strain in terms of the three displacement components:
ξε ∂= ∂ , ,y y
yy ∂ ∂= ξε zzz z
ξε ∂= ∂ (6.3)
⎟⎟⎠ ⎞
⎜⎜⎝ ⎛
∂ ∂+∂
∂== yx
ξξεε 2 1
, ⎟⎟⎠ ⎞
⎜⎜⎝ ⎛
∂ ∂+∂
∂== zy yz
zyyz ξξεε
2 1
, ⎟⎠ ⎞⎜⎝
⎛ ∂ ∂+∂
∂== xz
ξξεε 2 1
The strain components are positive if the displacement iξ corresponds to an axis which has the same sign as the position axis and negative if these signs are different. The matrix
(6.4)⎟⎟ ⎟
⎠
⎞
⎜⎜ ⎜
⎝
⎛ =
εεε εεε εεε
εˆ
is called the strain matrix of a given element in the continuous medium. It gives the representation of a second rank symmetric tensor ,iˆjε which allows us to write the relationship between the position vector components and those of the displacement vector
iξ as
jiji xεξ (6.5) where .,,, zyxji = Since it is a symmetrical matrix about the principal diagonal, three principal directions always exist, which are orthogonal to one another, with reference to which the matrix is diagonal (see Appendix II). This means that three orthogonal directions exist which remain orthogonal after deformation. The three strain components corresponding to these principal directions are called the principal strains and will be
denoted by the use of a single suffix, that is , and .x y zε ε ε When the principal axes are known, they are usually taken as the coordinate axes and the matrix becomes diagonal.
