ABSTRACT
We seek to obtain a two-dimensional form of Hooke's law in index notation which is valid for plane strain. Thus we use Eq. (12.5) to eliminate T33 from Eq. (12.4a), yielding
I-v2 v(l+v) E11 =-e T11 - E T22
Rearranging terms, we get 1 + v vCl + v)
E11 = E T11 - E (T11 + T22) which may also be rewritten as
1-v 2 ' v
(a)
(b)
(12.6)
(12.7)
(12.8)
(12.9)
(12.10)
Finally, Eqs. (12.8) and (12.10) may be expressed in two-dimensional index notation as
(12.11)
where a, f3 are free to take on values 1 and 2 and 'Y is summed from 1 to 2. * To obtain the inverted form of Eq. (12.11), it is best to go back to the
three-dimensional isotropic Hooke's law, (12.12)
where A and /-L are the Lame constants. All strains on the right side of this equation containing the index 3 are by definition zero, and thus if we restrict i,j on the left side to 1,2, we immediately get
(12.13)
We see that Eqs. (12.11) and (12.13) are both in the same form as their threedimensional counterparts, with the former requiring the introduction of modified elastic constants. You should verify that a formal inversion of Eq. (12.11) does indeed yield Eq. (12.13) (see Problem 12.1). Since Ell> E22, EI2 are functions of Xt,X2 only, Eq. (12.13) directly confirms the fact that 1'11> 1'22, and 1'12 are also functions of XI>X2 only. Furthermore, Eq. (12.5) then gives 1'33 as a function of Xt,X2' Summarizing what we have learned about the functional forms of the stresses, we can say that
1'13 = 1'23 = 0 (12.14)
Inserting conditions (12.14) into the equations of equilibrium [Eqs. (9.2)], we then obtain
a1'l1 a1'12 B 0 -+-+ 1= aXI aX2 (12.15) a1'21 a1'22 B 0 -+-+ 2= aXI aX2
which in two-dimensional index notation is given as
or (12.16)
Note that the body-force component B3 has been set equal to zero, in accordance with previously stated condition 1 in Section 12.1. It is frequently the case that the body-force distribution Ba is conservative (i.e., B = -grad V), or in index notation,
where V(XI>X2) is a scalar potential. Equilibrium equations (12.16) then become a1'a{3 _ av = 0 aX{3 aXa
or l' a{3. {3 - v: a = 0 (12.18)
Next we consider traction-type boundary conditions on the lateral surface of a prism (such as shown in Fig. 12.3). On this surface we apply Cauchy's formula,
or in expanded form,
T~v) = T31 VI + T32 V2 + T33 V3 (12.19)
Setting T31 = T32 = 0 [Eqs. (12.14)), T~v) = 0 (condition 1 of Section 12.1) and V3 = 0 (see Fig. 12.3), we end up with the indicial equation
on C I (12.20) where, as shown in Fig. 12.3, s is a curvilinear coordinate on the boundary curve C enclosing the region at the cross section R.
