Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

718
ENERgy METHOdS
y
dV = dx dy dz
z
x
x
Strain-Energy Density for Uniaxial Normal Stress
Axial deformation of an elastic bar affords a simple introduction to the concept of strain
energy. For more complex situations, it is sometimes necessary to consider how stored
strain energy is distributed throughout a deformed body. In such cases, a quantity called
strain-energy density is convenient. Strain-energy density is defined as the strain energy
per unit volume of material.
Consider a small volume element dV in a linearly elastic bar subjected to an axial
load, as shown in Figure 17.6. The force acting on each x face of this element is dF x = σ x
dy dz. If this force is applied gradually, like the force P shown in Figure 17.2, then the force
on the element increases from zero to dF x while the element elongates by the amount
dδ x = ε x dx. By Equation (17.1), the work done by dF x can be expressed as
1
dW = 2
( σ dydz)
ε dx
x
x
Furthermore, by conservation of energy, the strain energy stored in the volume element
must equal the external work:
dy
σ x
dx
dz
σ x
FIGURE 17.6 Volume element
in uniaxial tension.
1
dU = dW = 2
( σ dydz)
ε dx
The volume of the element is dV = dx dy dz. Thus, the strain energy in the volume
element can be expressed as
dU = 1
2 σ x ε x dV
Note that strain energy must be a positive quantity because the normal stress and the
normal strain always act in the same sense, either both positive (i.e., tension and elongation)
or both negative (i.e., compression and contraction).
The strain-energy density u can be determined by dividing the strain energy dU by the
volume dV of the element:
x
x
dU
1
u = = x x
dV 2
σ ε
(17.3)
If the material is linearly elastic, then σ x = Eε x and the strain-energy density can be
expressed solely in terms of stress as
σ x 2
u = (17.4)
2E
σ
σ x = Eε
x
or strain as
Eε u = 2
x
(17.5)
2
σ x
u Strain-energy
ε x
=
density
FIGURE 17.7 Strain-energy
density for elastic materials.
ε
Equations (17.4) and (17.5) have a straightforward geometric interpretation: Both
are equal to the triangular area below the stress–strain curve for a linear elastic material
(Figure 17.7). For materials that are not linearly elastic, the strain-energy density is still
equal to the area under the stress–strain curve; however, the area under the curve must be
evaluated by numerical or other methods.