Symmetry of Stress Tensor

16 From Fig. 2.1 the resultant force exerted on the positive X\ face is

L a 11AX2AX3 a 12AX2 AX3 a 13AX2 AX3 J similarly the resultant forces acting on the positive X2 face are

L a21AX3AX1 a22AX3AX1 a23AX3AX1 J

17 We now consider moment equilibrium (M = Fxd). The stress is homogeneous, and the normal force on the opposite side is equal opposite and colinear. The moment (AX2/2)a31 AX1AX2 is likewise balanced by the moment of an equal component in the opposite face. Finally similar argument holds for a32 .

is The net moment about the X3 axis is thus

M = AXi(o12AX2AX3) - AX2(0-21 AX3AXi) which must be zero, hence o12 = o21.

19 We generalize and conclude that in the absence of distributed body forces, the stress matrix is symmetric,

20 A more rigorous proof of the symmetry of the stress tensor will be given in Sect. 6.3.2.1.

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