Page 271 - Intro to Tensor Calculus

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Plane Stress

An assumption of plane stress is usually applied to thin flat plates. The plate thinness is assumed to be
in the z−direction and loads are applied perpendicular to z. Under these conditions all stress components
with a subscript z are assumed to be zero. The remaining stress components are then treated as functions
of x and y.
In Cartesian coordinates the stress tensor is expressible in terms of its physical components and can be
represented by the matrix
   
σ 11 σ 12 σ 13 σ xx σ xy σ xz
 σ 21 σ 22 σ 23  =  σ yx σ yy σ yz  .
σ 31 σ 32 σ 33 σ zx σ zy σ zz
If we assume that all the stresses with a subscript z are zero and the remaining stresses are only functions of
x and y we obtain a state of plane stress. The constitutive equations simplify if we assume a state of plane
stress. These simplified equations are

1 ν E
e xx = σ xx − σ yy σ xx = [e xx + νe yy ]
E E 1 − ν 2
1 ν E
e yy = σ yy − σ xx σ yy = [e yy + νe xx ]
E E 1 − ν 2
ν
e zz = − (σ xx + σ yy ) σ zz =0 =(1 − ν)e zz + ν(e xx + e yy )
E (2.4.64)
1+ ν E
e xy = σ xy σ xy = e xy
E 1+ ν
e xz =0 σ yz =0
e yz =0. σ xz =0
For a state of plane stress the compatibility equations reduce to
2 2 2
∂ e xx ∂ e yy ∂ e xy
+ =2 (2.4.65)
∂y 2 ∂x 2 ∂x∂y
and the three additional equations

2 2 2
∂ e zz ∂ e zz ∂ e zz
=0, =0, =0.
∂x 2 ∂y 2 ∂x∂y
These three additional equations complicate the plane stress problem.
Airy Stress Function

In Cartesian coordinates we examine the equilibrium equations (2.4.25(b)) under the conditions of plane
strain. In terms of physical components we find that these equations reduce to

∂σ xx ∂σ xy ∂σ yx ∂σ yy ∂σ zz
+ + %b x =0, + + %b y =0, =0.
∂x ∂y ∂x ∂y ∂z
The last equation is satisfied since σ zz is a function of x and y. If we further assume that the body forces
~
are conservative and derivable from a potential function V by the operation %b = −gradV or %b i = −V ,i
we can express the above equilibrium equations in the form:

∂V
∂σ xx ∂σ xy
+ − =0
∂x ∂y ∂x
(2.4.66)
∂V
∂σ yx ∂σ yy
+ − =0
∂x ∂y ∂y