Four Linear Continuum Theories - Continuum Mechanics of Anisotropic Materials - page 137

Table 6.2 The isotropic elastic constants expressed in terms of certain pairs of other isotropic
elastic constants
l m E n k

▪ ▪

l , m mð 3 l þ 2 mÞ l 3 l þ 2 m

3

2 ðl þ mÞ

l þ m

▪ ▪

l , n lð 1 2 nÞ lð 1 nÞð 1 2 nÞ lð 1 nÞ

3 n 3 n n

▪ ▪

l

l , k 3 ð k lÞ

2

9 k ð k lÞ
3 k l 3 k l

▪ ▪

m E

m , E ð Þm

ðE 3 mÞ

2 m E ð E 2 mÞ

3 ð 3 m E Þ

2 m

▪ ▪

mn
1 2 n

2

m , n 2 m (1 þ n ) 2 þ nÞ

3 ð 1 2 nÞ

mð 1

▪ ▪

m , k 3 k 2 m

3

9 km
3 k

3 k 2 m
6 k

þ m þ 2 m

▪ ▪

E
ð 1 þ nÞð 1 2 nÞ

n E

2 ð 1 þ nÞ

E
3 ð 1 2 nÞ

E , n

▪ ▪

E , k

3 k ð 3 k E Þ
9 k

3 kE
9 k E

ð 3 k E Þ
6 k E

▪ ▪

n , k

3 kn /(1 þ n ) 3 k ð 1 2 nÞ

2

3 k (1 2 n )

ð 1 þ nÞ

to (5.11N), a result algebraically equivalent to (5.11N) with D replaced by E then
follows:

T ¼ lð tr E Þ 1 þ 2 m E : (6.24)

For an isotropic linear elastic material there are just two independent elastic
constants. These two constants are represented, for example, by the Lam ´ moduli l

and . Another set of isotropic elastic constants in common use are the Young's

modulus E , the shear modulus G (

m

¼m ), and Poisson's ratio n , where the three

constants are related by 2 G (1

E so that only two are independent. Any single
isotropic elastic constant can be expressed in terms of any two other isotropic elastic
constants as documented by Table 6.2 , which contains expressions for most of the
usual isotropic elastic constants in terms of different pairs of the other isotropic elastic
constants. A frequently employed isotropic elastic constant is the bulk modulus k ,
which represents the ratio of an applied mean hydrostatic stress, - p

þ n ¼

)

(tr T )/3, to a
volumetric strain. Recall that tr E represents the volumetric strain per unit volume.
The relationship between volumetric strain per unit volume and the mean hydrostatic
stress, - p

¼

¼ (tr T )/3, is obtained by taking the trace of ( 6.24 ), thus -3 p ¼ (3 l þ 2 m )

tr E . The bulk modulus k is then given by the following different representations,

p
tr E ¼ l þ

2 3 ¼ E

k

nÞ ;

(6.25)

3 ð 1 2