Page 40 - Aircraft Stuctures for Engineering Student

5
1 .I Stress-strain relationships 25
lateral strains

CX
Ey = -u- (1.41)
E’ E
in which u is a constant termed Poisson’s Ratio.
For a body subjected to direct stresses ux, uv and uz the direct strains are, from
Eqs (1.40). (1.41) and the principle of superposition (see Chapter 4, Section 4.9)
1
E, = E [cr, - u(uy + uz)]
1
Ey = E [ay - u(ax + a=)] (1.42)

Suppose now that, at some arbitrary point in a material, there are principal strains
and corresponding to principal stresses aI and uII. If these stresses (and strains)
are in the direction of the coordinate axes x and y respectively, then T~~ = y,, = 0 and
from Eq. [ 1.34) the shear strain on an arbitrary plane at the point inclined at an angle
8 to the principal planes is
y = (E~ - qI) sin 28 (1.43)
Using the relationships of Eqs (1.42) and substituting in Eq. (1.43) we have

1
y = - [(aI - uaII) - (oII - uaI)] sin 28
E
or

( 1.44)

Using Eq. (1.9) and noting that for this particular case rxy = 0, a.x = oI and cy = aI,
27 = (aI - aII) sin 28
from which we may rewrite Eq. (1.44) in terms of 7 as

(1.45)

The term E/2( 1 + u) is a constant known as the rnodirlus of rigidily G. Hence
y = r/G

and the shear strains yyv, -yxz and -/,. are expressed in terms of their associated shear
stresses as follows

(1.46)

Equations (1.46), together with Eqs (1.42), provide the additional six equations
required to determine the 15 unknowns in a general three-dimensional problem
in elasticity. They are, however, limited in use to a linearly elastic isotropic body.