x + δx
x
Deformed
Undeformed
δx
u + δu
u
δu
δx
Similar equations apply for the x 1 and x 3 components of
the force and acceleration. The set of three equations can be
written simply using the summation convention
∂
∂
∂
∂
σ
ρ
ji
j
i
i
t
x
f
t
u
t
t
( , )
( , )
( , ) .
x
x
x
+
=
2
2
(46)
Here the fact that the stresses, forces, and displacements can
vary in both space and time is explicitly written. Alternatively,
because the stress tensor is symmetric, we can write
∂
∂
∂
∂
σ
ρ
ij
j
i
i
t
x
f
t
u
t
t
( , )
( , )
( , ) .
x
x
x
+
=
2
2
(47)
Note that the force in the i direction is obtained by summing
over the faces j of the block. If the partial derivative with
respect to x i is denoted by a comma, Eqn 47 becomes
σ ij,j (x, t) + f i (x, t) = ρ
∂
∂
2
2
u
t
t
i ( , ) .
x
(48)
This equation, called the equation of motion, is satisfied
everywhere in a continuous medium. It expresses Newton’s
second law, F = ma, in terms of surface and body forces. The
acceleration results from the body force and σ ij,j , the divergence of the stress tensor. A stress field that does not vary with
position has no divergence, and hence produces no force. It is
interesting to note that the divergence of the stress tensor gives
rise to a force, which is a vector, just as the divergence of a
vector yields a scalar (Section A.6.3).
An important form of the equation of motion describes a
body at equilibrium, whose acceleration is zero, so the divergence of the stress tensor exactly balances the body forces
σ ij, j (x, t) = −f i (x, t).
(49)
This equation of equilibrium must be satisfied for any static
elasticity problem, such as finding the stresses due only to
gravity.
Another important form, if no body forces are applied, is
σ ij, j (x, t) = ρ
∂
∂
2
2
u
t
t
i ( , ) .
x
(50)
This is called the homogeneous equation of motion, where
“homogeneous” refers to the lack of forces, as in the terminology of linear equations (Section A.4.4). This equation describes
seismic wave propagation except at a source, such as an earthquake or an explosion, where a body force generates seismic
waves.
2.3.8 Strain
If stresses are applied to a material that is not rigid, points
within it move with respect to each other, and deformation
Fig. 2.3-11 Geometry showing how deformation arises from the relative
displacement δu between two points originally separated by δx.
results. The strain tensor describes the deformation resulting
from the differential motion within the body.
Consider an element of solid material within which displacements u(x) have occurred. If a point originally at x is displaced
by u (Fig. 2.3-11), we describe the displacement of a nearby
point originally at x + δx by expanding the components of the
displacement vector in a Taylor series,
u i (x + δx) ≈ u i (x) +
∂
∂
u
x
i
j
( )
x δx j = u i (x) + δ u i ,
(51)
so that the relative displacement near x, δu i , is to the first order
δ
δ
u
u
x
x
i
i
j
j
=
( )
,
∂
∂
x
(52)
where the partial derivatives are evaluated at x.
Although we are interested in deformation that distorts
the body, there can also be a rigid body translation or a rigid
body rotation, neither of which produces deformation. To distinguish these effects, we add and subtract ∂u j /∂x i to Eqn 52
and then separate it into two parts
δ
δ
δ
u
u
x
u
x
x
u
x
u
x
x
i
i
j
j
i
j
i
j
j
i
j
=
+
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
+
−
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
1
2
1
2
∂
∂
∂
∂
∂
∂
∂
∂
= (e ij + ω ij )δx j .
(53)
The ω ij term corresponds to a rigid body rotation without
deformation. To see this, note that because ω ij is antisymmetric
(ω ij = −ω ji ), the diagonal terms are zero, and there are only three
independent components. We can then form a vector ω with
components
ω k = ε stk ω st /2,
(54)
where ε stk is the permutation symbol (Eqn A.3.39). Using the
identity
2.3 Stress and strain 47
x
Deformed
Undeformed
δx
u + δu
u
δu
δx
Similar equations apply for the x 1 and x 3 components of
the force and acceleration. The set of three equations can be
written simply using the summation convention
∂
∂
∂
∂
σ
ρ
ji
j
i
i
t
x
f
t
u
t
t
( , )
( , )
( , ) .
x
x
x
+
=
2
2
(46)
Here the fact that the stresses, forces, and displacements can
vary in both space and time is explicitly written. Alternatively,
because the stress tensor is symmetric, we can write
∂
∂
∂
∂
σ
ρ
ij
j
i
i
t
x
f
t
u
t
t
( , )
( , )
( , ) .
x
x
x
+
=
2
2
(47)
Note that the force in the i direction is obtained by summing
over the faces j of the block. If the partial derivative with
respect to x i is denoted by a comma, Eqn 47 becomes
σ ij,j (x, t) + f i (x, t) = ρ
∂
∂
2
2
u
t
t
i ( , ) .
x
(48)
This equation, called the equation of motion, is satisfied
everywhere in a continuous medium. It expresses Newton’s
second law, F = ma, in terms of surface and body forces. The
acceleration results from the body force and σ ij,j , the divergence of the stress tensor. A stress field that does not vary with
position has no divergence, and hence produces no force. It is
interesting to note that the divergence of the stress tensor gives
rise to a force, which is a vector, just as the divergence of a
vector yields a scalar (Section A.6.3).
An important form of the equation of motion describes a
body at equilibrium, whose acceleration is zero, so the divergence of the stress tensor exactly balances the body forces
σ ij, j (x, t) = −f i (x, t).
(49)
This equation of equilibrium must be satisfied for any static
elasticity problem, such as finding the stresses due only to
gravity.
Another important form, if no body forces are applied, is
σ ij, j (x, t) = ρ
∂
∂
2
2
u
t
t
i ( , ) .
x
(50)
This is called the homogeneous equation of motion, where
“homogeneous” refers to the lack of forces, as in the terminology of linear equations (Section A.4.4). This equation describes
seismic wave propagation except at a source, such as an earthquake or an explosion, where a body force generates seismic
waves.
2.3.8 Strain
If stresses are applied to a material that is not rigid, points
within it move with respect to each other, and deformation
Fig. 2.3-11 Geometry showing how deformation arises from the relative
displacement δu between two points originally separated by δx.
results. The strain tensor describes the deformation resulting
from the differential motion within the body.
Consider an element of solid material within which displacements u(x) have occurred. If a point originally at x is displaced
by u (Fig. 2.3-11), we describe the displacement of a nearby
point originally at x + δx by expanding the components of the
displacement vector in a Taylor series,
u i (x + δx) ≈ u i (x) +
∂
∂
u
x
i
j
( )
x δx j = u i (x) + δ u i ,
(51)
so that the relative displacement near x, δu i , is to the first order
δ
δ
u
u
x
x
i
i
j
j
=
( )
,
∂
∂
x
(52)
where the partial derivatives are evaluated at x.
Although we are interested in deformation that distorts
the body, there can also be a rigid body translation or a rigid
body rotation, neither of which produces deformation. To distinguish these effects, we add and subtract ∂u j /∂x i to Eqn 52
and then separate it into two parts
δ
δ
δ
u
u
x
u
x
x
u
x
u
x
x
i
i
j
j
i
j
i
j
j
i
j
=
+
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
+
−
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
1
2
1
2
∂
∂
∂
∂
∂
∂
∂
∂
= (e ij + ω ij )δx j .
(53)
The ω ij term corresponds to a rigid body rotation without
deformation. To see this, note that because ω ij is antisymmetric
(ω ij = −ω ji ), the diagonal terms are zero, and there are only three
independent components. We can then form a vector ω with
components
ω k = ε stk ω st /2,
(54)
where ε stk is the permutation symbol (Eqn A.3.39). Using the
identity
2.3 Stress and strain 47