2.3 Stress and strain 49
x 2
x 1
x 2
x 1
x 2
x 1
x 1
x 2
x 1
(a)
> 0, u 2 = 0
∂u 1
∂x 1
(b)
< 0, u 2 = 0
∂u 1
∂x 1
(c)
> 0,
=
= 0
∂u 1
∂x 2
∂u 1
∂x 1
∂u 2
∂x 2
(d)
> 0,
∂u 1
∂x 2
> 0
∂u 2
∂x 1
x 2
(e)
< 0,
∂u 1
∂x 2
> 0
∂u 2
∂x 1
dx 3
dx 1
dx 2
dx 3
∂u 3
∂x 3
dx 1
∂u 1
∂x 1
dx 2
∂u 2
∂x 2
Fig. 2.3-13 Change in volume of a small block of material with faces
normal to the coordinate axes, due to the principal strains. The fractional
change in volume is the dilatation, the sum of the principal strains.
In assuming that material is elastic, we also assume that the
displacements from an unstrained initial state are small. This
assumption, known as infinitesimal strain theory, is generally
valid for seismic waves. For example, a body wave may have a
displacement on the order of 10 microns, and a wavelength on
the order of 10 km. Expressing all quantities in meters, the resulting strain is about (10 −5 /10 4 ) = 10 −9 , certainly small enough
for infinitesimal theory to be valid. However, for strains greater
than about 10 −4 , the linear relation between stress and strain
fails. This occurs in regions of the earth’s mantle under very
high pressure, or when rocks break during an earthquake
(Section 5.7.2).
The stress and strain for a linearly elastic material are related
by a constitutive equation called Hooke’s law,
σ ij = c ijkl e kl ,
(64)
written here using the summation convention. The constants
c ijkl , the elastic moduli, describe the properties of the material.
To understand how the elastic moduli affect the equation of
motion, we write the constitutive equation (64) using the fact
that the strains are derivatives of the displacement,
Fig. 2.3-12 Some possible strains for a two-dimensional element.
x 2
x 1
x 2
x 1
x 2
x 1
x 1
x 2
x 1
(a)
> 0, u 2 = 0
∂u 1
∂x 1
(b)
< 0, u 2 = 0
∂u 1
∂x 1
(c)
> 0,
=
= 0
∂u 1
∂x 2
∂u 1
∂x 1
∂u 2
∂x 2
(d)
> 0,
∂u 1
∂x 2
> 0
∂u 2
∂x 1
x 2
(e)
< 0,
∂u 1
∂x 2
> 0
∂u 2
∂x 1
dx 3
dx 1
dx 2
dx 3
∂u 3
∂x 3
dx 1
∂u 1
∂x 1
dx 2
∂u 2
∂x 2
Fig. 2.3-13 Change in volume of a small block of material with faces
normal to the coordinate axes, due to the principal strains. The fractional
change in volume is the dilatation, the sum of the principal strains.
In assuming that material is elastic, we also assume that the
displacements from an unstrained initial state are small. This
assumption, known as infinitesimal strain theory, is generally
valid for seismic waves. For example, a body wave may have a
displacement on the order of 10 microns, and a wavelength on
the order of 10 km. Expressing all quantities in meters, the resulting strain is about (10 −5 /10 4 ) = 10 −9 , certainly small enough
for infinitesimal theory to be valid. However, for strains greater
than about 10 −4 , the linear relation between stress and strain
fails. This occurs in regions of the earth’s mantle under very
high pressure, or when rocks break during an earthquake
(Section 5.7.2).
The stress and strain for a linearly elastic material are related
by a constitutive equation called Hooke’s law,
σ ij = c ijkl e kl ,
(64)
written here using the summation convention. The constants
c ijkl , the elastic moduli, describe the properties of the material.
To understand how the elastic moduli affect the equation of
motion, we write the constitutive equation (64) using the fact
that the strains are derivatives of the displacement,
Fig. 2.3-12 Some possible strains for a two-dimensional element.