Elasticity_ Barber

22 1 Introduction
1.3.1 Lamé’s constants
It is often desirable to solve equations (1.58–1.60) to express σ xx in terms of
e xx etc. The solution is routine and leads to the equation
σ xx = Eν(e xx + e yy + e zz )
(1 + ν)(1 − 2ν)
+ Ee xx
(1 + ν)
(1.67)
and similar equations, which are more concisely written in the form
etc., where
and
λ =
σ xx = λe + 2µe xx (1.68)
Eν
(1 + ν)(1 − 2ν) = 2µν
(1 − 2ν)
(1.69)
e ≡ e xx + e yy + e zz ≡ e ii ≡ div u (1.70)
is known as the dilatation.
The stress-strain relations (1.66, 1.68) can be written more concisely in
the index notation in the form
σ ij = λδ ij e mm + 2µe ij , (1.71)
where δ ij is the Kronecker delta, defined as 1 if i=j and 0 if i≠j. Equivalently,
we can use equation (1.55) with
c ijkl = λδ ij δ kl + µ (δ ik δ jl + δ jk δ il ) . (1.72)
The constants λ, µ are known as Lamé’s constants. Young’s modulus and Poisson’s
ratio can be written in terms of Lamé’s constants through the equations
E =
µ(3λ + 2µ)
(λ + µ)
1.3.2 Dilatation and bulk modulus
; ν =
λ
2(λ + µ) . (1.73)
The dilatation, e, is easily shown to be invariant as to coördinate transformation
and is therefore a scalar quantity. In physical terms it is the local
volumetric strain, since a unit cube increases under strain to a block of dimensions
(1+e xx ), (1+e yy ), (1+e zz ) and hence the volume change is
∆V = (1 + e xx )(1 + e yy )(1 + e zz ) − 1 = e xx + e yy + e zz + O(e xx e yy ) . (1.74)
It can be shown that the dilatation e and the rotation vector ω are harmonic
— i.e. ∇ 2 e = ∇ 2 ω = 0. For this reason, many early solutions of elasticity
problems were formulated in terms of these variables, so as to make use
of the wealth of mathematical knowledge about harmonic functions. We now