Stress Equilibrium
Consider a rectangular block (with dimensions dx, dy, and unity) loaded along its edges
by stresses that vary in x and y. X and Y are body forces per unit volume.
Sum the forces in x and y to establish equilibrium:
⎡
⎤
⎛ ∂σ yx ⎞
⎡
⎤
⎛ ∂σ xx ⎞
⎟ dxdy ⎥ − σ xx dy + ⎢σ yx dx + ⎜
⎟dxdy ⎥ − σ yx dx + Xdxdy = 0
⎢σ xx dy + ⎜
⎝
⎠
∂
x
∂
y
⎣
⎦
⎝
⎠
⎣
⎦
In terms of force per unit area (dx dy), we thus have:
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∂σxx ∂σyx
+
+ X = 0 ∂x
∂y
A similar treatment in the y direction gives:
∂σ xy ∂σ yy
+
+ Y = 0 ∂x
∂y
These are the equations for equilibrium. A gradient in normal stress in one coordinate
direction is always balanced out by a gradient in shear stress in the other coordinate
direction.
These equations hold everywhere in a body that is in static equilibrium (no
accelerations at any point).
These equations are approximately correct in a body in quasi-static equilibrium
(accelerations are small enough that the corresponding forces (=ma) are very small
compared to typical body forces or stress gradients).
If the only body force is due to gravity and y is vertical: X = 0 and Y = -ρg
NOTE: “Most problems in structural geology are better posed by comparing an initial state with gravitation
loading, to a final state with gravitation loading plus the appropriate tectonic forces. In this context one
should ignore the contributions of gravitational loading to the strain and displacement fields.” Pollard and
Fletcher (2005), p.312.