The only nonzero compatibility equation for plane strain is given by @ 2 exy @ 2 ex @ 2 ey þ ¼ 2 @y2 @x2 @x@y
(12:3:6)
Using Hooke’s law in this result gives r2 (sx þ sy ) þ
Ea r2 T ¼ 0 1 n
(12:3:7)
Again note the additional thermal term in this relation when compared to the isothermal result given by (7.1.7). The additional terms in both (12.3.5) and (12.3.7) can be thought of as thermal body forces that contribute to the generation of the stress, strain, and displacement fields. Relations (12.3.5) would be used for the displacement formulation, while (12.3.4) and (12.3.7) would be incorporated in the stress formulation. The boundary conditions for the plane strain problem are normally specified for either the stresses Txn ¼ sx nx þ txy ny ¼ (Txn )s Tyn ¼ txy nx þ sy ny ¼ (Tyn )s
(12:3:8)
or the displacements u ¼ us (x, y) v ¼ vs (x, y)
(12:3:9)
where (Txn )s , (Tyn )s , us , and vs are the specified boundary tractions and displacements on the lateral surfaces. Note that these specified values must be independent of z and the temperature field must also depend only on the in-plane coordinates; that is, T ¼ T(x, y). It should be recognized that using Hooke’s law (12.3.3) in the traction boundary conditions (12.3.8) will develop relations that include the temperature field.
12.3.2 Plane Stress The fundamental starting point for plane stress (and/or generalized plane stress) in the x,yplane is an assumed stress field of the form sx ¼ sx (x, y), sy ¼ sy (x, y), txy ¼ txy (x, y) sz ¼ txz ¼ tyz ¼ 0
(12:3:10)
As per our previous discussion in Section 7.2 this field is an appropriate approximation for bodies thin in the z direction (see Figure 7-3). The thermoelastic strains corresponding to this stress field come from Hooke’s law:
Thermoelasticity
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