elasticity

Elasticity 

Constitutive Relation 

53 

Stanford Rock Physics Laboratory - Gary Mavko 

The descriptions of stress and strain individually are 

independent of the material. 

The connection between stress and strain is a 

description of the particular material being studied. 

This connection is called the constitutive relation or 

the stress-strain relation. 

An elastic material is one where the stress is a unique 

function of the strain and vice versa. An elastic solid 

has the following properties: 

The displacements and strains are independent of 

the history of loading 

When we remove the applied loads, the body 

returns to a unique relaxed state.

strain 

Elasticity 

54 


Example of an Elastic Medium: there is a unique 

relation between stress and strain. 

stress 

Inelastic medium: non-unique relation that may 

depend on rate or loading history. 

strain 

stress

Elasticity 

55 


Linear Elastic Material: Hookean Solid 

A special case is when the stresses and strains are 

linearly related: 

σij = Σ cijkl εkl kl 

stiffnesses 

(moduli) 

εij = Σ Sijkl σkl kl 

compliances 

Since there are 9 stress components and 9 strain 

components, there are 81 elastic constants. 

However, because of the symmetry of stress and 

strain, plus some other thermodynamic conditions 

there are at most 21 independent constants. 

c ijkl 

S ijkl 

The elastic stiffness and compliances are 

exactly equivalent. They are tensor inverses of each 

other.

Elasticity 

56 


Isotropic Linear Elastic Solid 

For an isotropic material, only two unique 

constants are needed. 

c ijrs = λδ ij δ rs + µ δ ir δ js + δ is δ jr 

in simpler form: 

σ ij = λδ ij ε kk +2µε ij 

where λ and µ are the Lamé constants. µ is also 

known as the shear modulus.

Elasticity 

57 


Linear Isotropic Elasticity: the Bulk Modulus 

Hooke’s Law 

σij = λεααδij +2µεi or 

σxx = λεαα +2µεxx σyy = λεαα +2µεyy By adding the first three equations: 

σ 0 = 1 

3 σ αα = 

σ zz = λε αα +2µε zz 

σ xy =2µε xy 

σ yz =2µε yz 

σ xz =2µε xz 

σαα =3λ +2µ εαα 

For convenience we define 

K= 

3λ +2µ 

3 

σ0 =Kεαα 

3λ +2µ 

3 

ε αα 

Here, K is the bulk modulus and gives the ratio of 

hydrostatic stress to volumetric strain.

Elasticity 

σ xx = λε αα +2µε xx 

σ yy = λε αα +2µε yy 





58 


Linear Isotropic Elasticity: the Shear Modulus 

In a simple shear experiment, only a single shear 

strain εxy is non-zero. 

ε xy 

Then Hooke’s law states: 

σ xy 


µ is the shear modulus, which gives the ratio of 

shear stress to shear strain. 

y 

x

σ xx = λε αα +2µε xx 

σ yy = λε αα +2µε yy 





Elasticity 

59 


Linear Isotropic Elasticity: the Young’s Modulus 

In a uniaxial stress experiment, 

only the axial normal stress 

is non-zero. 

Then from Hooke’s law we can derive: 

σ xx = 

But it is convenient to define 

E= 

σxx 

µ 3λ +2µ 

λ + µ 

σxx =Eεxx 

µ 3λ +2µ 

λ + µ 

ε xx 

σ xx 

This is Young’s modulus, which gives the ratio of 

axial stress to axial strain in a uniaxial stress state. 

y 

x 

z

Elasticity 

60 


Linear Isotropic Elasticity: the Poisson’s Ratio 

Similarly, in the uniaxial stress state we can derive 

the ratio of lateral strain to axial strain: 

ε yy =– 

And it is convenient to define Poisson’s ratio: 

ν = 

λ 

2 λ + µ ε xx 

λ 

2 λ + µ 

ε yy =–νε xx 

Ideally ν must lie within the limits –1 ≤ ν ≤ 

Practically it is almost always between 

when or 

We can interpret this as a fluid state or as an 

incompressible state. 

1 2 

0 ≤ ν ≤ 1 2 

ν → 1 µ → 0 K → ∞ 

2

Elasticity 

Uniaxial Strain 

61 


Consider the case when displacements occur only in 

the x direction: 

This can occur in the laboratory by putting rigid 

lateral constraints, or in a plane P-wave. 

Then we can derive: 

σxx = λ +2µ εxx 

So we can define a new modulus M for this 

experiment that we will call the uniaxial strain 

modulus, or the “P-wave” modulus. 

M= K+ 4µ = λ +2µ 

3 

This gives the ratio of axial stress to axial strain in 

a uniaxial strain situation. 

x 

z

Mineral Density 

Quartz 2.6500 

Calcite 2.7100 

Dolomite 2.8700 

Clay (kaolinite) 1.5800 

Muscovite 2.7900 

Feldspar (Albite) 2.6300 

Halite 2.1600 

Anhydrite 2.9800 

Pyrite 4.9300 

Siderite 3.9600 

gas 0.00065000 

water 1.0000 

oil 0.80000 

densities in g/cm 3 

moduli in GPa 

velocities in km/s 

Elasticity 

62 


Examples of Elastic Moduli 

Young's Modulus 

95.756 

84.293 

116.57 

3.2034 

100.84 

69.010 

37.242 

74.431 

305.85 

134.51 

0.0000 

0.0000 

0.0000 

Bulk Modulus 

36.600 

76.800 

94.900 

1.5000 

61.500 

75.600 

24.800 

56.100 

147.40 

123.70 

0.00013000 

2.2500 

1.0200 

Shear Modulus 

45.000 

32.000 

45.000 

1.4000 

41.100 

25.600 

14.900 

29.100 

132.50 

51.000 

0.0000 

0.0000 

0.0000 

Vp 

6.0376 

6.6395 

7.3465 

1.4597 

6.4563 

6.4594 

4.5474 

5.6432 

8.1076 

6.9576 

0.44721 

1.5000 

1.1292 

Vs 

4.1208 

3.4363 

3.9597 

0.94132 

3.8381 

3.1199 

2.6264 

3.1249 

5.1842 

3.5887 

0.0000 

0.0000 

0.0000 

Poisson's Ratio 

0.063953 

0.31707 

0.29527 

0.14407 

0.22673 

0.34786 

0.24972 

0.27888 

0.15417 

0.31876 

0.50000 

0.50000 

0.50000 

C.1

λ+2µ /3 

Eµ 

3(3µ–E) 

λ 1+ν 

3ν 

µ 21+ν 

3 1–2ν 

λ 

µ 3λ+2µ 

λ+µ 

9K K–λ 

3K–λ 

9Kµ 

3K+µ 

1+ν 1–2ν 

ν 

2µ(1+ν) 

3K(1–2ν) 

Elasticity 

K–2µ /3 

µ E–2µ 

(3µ–E) 

3K 3K–E 

9K–E 

µ 2ν 

1–2ν 

3K ν 

1+ν 

63 

λ 

2 λ+µ 

λ 

3K–λ 

3K–2µ 

23K+µ 

E/(2µ)–1 

3K–E 

6K 


Relationships Among Elastic Constants in an Isotropic Material 

K E λ ν M µ 

- 

- 

- 

- 

E 

3 1–2ν 

- 

- 

- 

- 

- 

- 

Eν 

1+ν 1–2ν 

- 

- 

- 

- 

λ+2µ 

3K–2λ 

K+4µ /3 

µ 4µ–E 

3µ–E 

3K 3K+E 

9K–E 

λ 1–ν 

ν 

µ 2–2ν 

1–2ν 

3K 1–ν 

1+ν 

E 1–ν 

1+ν 1–2ν 

- 

3K–λ /2 

- 

- 

3KE 

9K–E 

λ 1–2ν 

2ν 

- 

3K 1–2ν 

2+2ν 

E 

2+2ν

elasticity

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