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Two-Dimensional Elasticity 121

In matrix form, we write

 ε   ∂∂x 0 
/
x
u
     
=
/
 ε  =  0 ∂∂y    , or ε = Du (4.8)
y
v
   ∂∂y ∂∂  x  
/
/
 γ xy   
From this relation, we know that, if the displacements are represented by polynomials,
the strains (and thus stresses) will be polynomials of an order that is one order lower than
the displacements.
4.2.5 Equilibrium Equations
In plane elasticity, the stresses in the structure must satisfy the following equilibrium
equations:
∂σ x + ∂τ xy + f = 0
∂x ∂y x
(4.9)
∂τ xy + ∂σ y + f = 0
∂x ∂y y

where f and f are body forces (forces per unit volume, such as gravity forces). In the FEM,
x
y
these equilibrium conditions are satisfied in an approximate sense.
4.2.6 Boundary Conditions
The boundary S of the 2-D region can be divided into two parts, S and S (Figure 4.5). The
t
u
boundary conditions (BCs) can be described as
u = u, v = v, on S u
t x = t x , t y = t y , on S t (4.10)

in which t and t are tractions (stresses on the boundary) and the barred quantities are
x
y
those with known values.
4.2.7 Exact Elasticity Solution

The exact solution (displacements, strains, and stresses) of a given problem must satisfy
the constitutive relations, equilibrium equations, and compatibility conditions (structures

t y
p
t x
y
S t
S u
x

FIGURE 4.5
Boundary conditions for a structure.