Basic Equations: Differential Form - Mechanics of Structures: Variational and Computational Methods - page 17

Using summation convention notation, this may be written as

1 f i

1 + E σ ij − E δ ij σ kk

= j

ij = with δ ij =

0 f i = j

and

σ = σ + σ + σ = σ + σ + σ (1.33a)

kk xx yy zz x y z

More generally, this is often placed in the form

= F ijkl σ kl (1.33b)

ij

6 matrix E − 1 .
From Eq. (1.32) [or Eq. (1.33)], the stresses can be found by inversion as functions of
strains

where F ijkl is the index notation form of the 6 ×

σ

− νν ν.

x x

1

− νν.

σ

y y

ν 1 0

.

σ

z z

ν ν 1 − ν

E

... = ... ... ... . ... ... ... ... (1.34a)

( 1 + ν)( 1 − 2 ν)

.

1 − 2 ν

τ xy 00 γ xy

2

.

1 − 2 ν

0 0 0

τ xz γ xz

2

.

00 1 − 2 ν
2

τ yz γ yz

σ = E (1.34b)

In index notation, this can be written as

E ν E

σ ij =

ν) δ ij kk + + ν) ij = λδ ij kk +

2 µ ij (1.35a)

( 1 + ν)( 1 − 2 ( 1

or

σ ij = E ijkl kl (1.35b)

where λ = E ν/ [ ( 1 + ν)( 1 − 2 ν) ] and µ = G = E / [2 ( 1 + ν) ] are the Lame's 12 constants, and

kk = xx + yy + zz = x + y + z .

It is sometimes convenient, for example, in soil mechanics, to express the material
law in terms of the bulk modulus K . To do so, define the deviatoric stress and strain
components

1
3 σ kk δ ij

σ ij = σ ij −

(1.36)

1
3 kk δ ij

ij = ij −

12 Gabriel Lame (1795-1870) was a French engineer who, after graduation from the Ecole Polytechnique, worked
at a Russian railroad institute and helped establish a new engineering school in St. Petersburg. He introduced and
applied curvilinear coordinates, and contributed to number theory, applied mechanics, and thermodynamics. In
1852, he co-authored (with Clapeyron) the first topic on the theory of elasticity.