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

where P is some function of δ. The work done on the bar must equal the change in energy
of the material, 3 and this energy change is termed the strain energy U because it involves
the strained configuration of the material. If δ is expressed in terms of axial strain (δ = L e)
and P is expressed in terms of axial stress (P = Aσ), Equation (a) becomes
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THEORIES OF FAILuRE
e
2
2
W = U = ∫ ( σ) ( A)
( L)
de = AL∫
σ de
0
e
0
(b)
where σ is a function of e. (See Figure 15.8c.) If Hooke’s law applies then,
e = σ/ E de = dσ/
E
and Equation (b) becomes
or
U
AL 2
= ⎛ d
E
∫ σ σ
⎝ ⎜ ⎞ σ
⎠
⎟
0
U
=
σ 2 2
⎛ ⎞
AL
⎝
⎜
2E ⎠
⎟
(c)
Equation (c) gives the elastic strain energy (which is, in general, recoverable) 4 for axial
loading of a material obeying Hooke’s law. The quantity in parentheses, σ 2 2/(2E), is the
elastic the strain energy u in tension or compression per unit volume, or the strain-energy
density, for a particular value of stress σ below the proportional limit of the material.
Thus,
1 1
2
E
2
u = σe = σ = e
(15.1)
2 2E
2
For shear loading, the expression would be identical except that σ would be replaced by τ,
e by γ , and E by G.
The concept of elastic strain energy can be extended to include biaxial and triaxial
loadings by writing the expression for strain-energy density u as 1/2σe and adding the energies
due to each of the stresses. Since energy is a positive scalar quantity, the addition is the
arithmetic sum of the energies. For a system of triaxial principal stresses σ p1 , σ p2 , and σ p3 ,
the total elastic strain-energy density is
1
u = [ σ p1ep1 + σ p2ep2 + σ p3ep3]
(d)
2
When the generalized Hooke’s law expressions for strains in terms of stresses from Equation
(13.16) of Section 13.8 are substituted into Equation (d), the result is
1
u = { σ p1[ σ p1 − v( σ p2 + σ p3)] + σ p2[ σ p2 − v( σ p3 + σ p1)] + σ p3[ σ p3 − v( σ p1 + σ p2)]}
2 E
3
Known as Clapeyron’s theorem, after the French engineer B. P. E. Clapeyron (1799–1864).
4
Elastic hysteresis is neglected here as an unnecessary complication.