25 Elastic Displacements and Waves
269
The strain tensor ε is symmetrical,
ε xy = ε yx ,
ε xz = ε zx ,
ε yz = ε zy .
(336)
We are now in a position where we can precisely formulate the three basic assumptions of the linear theory of elasticity for the three-dimensional case.
1. In the Newtonian term for inertia, the total time derivative for the momentum P of
the moving masses is replaced with the partial derivative of the material velocity v.
2. The spatial distributed mass density ρ is considered as a constant parameter.
3. Hooke’s law is valid, in other words the stress tensor σ is directly proportional to
the tensor of relative strain ε at position x and point of time t . This results in the
equations
d
dt
P = ρ
∂
∂t
v ,
ρ
= const. ,
σ(x, t)= C · · ε(x, t) .
(337)
Equations (337) formally look exactly like the corresponding Eqs. (67), for the onedimensional case. This however should not distract the reader from the fact that
Hooke’s law is now far more complicated. The quantities σ and ε are tensors of
the second order that we have to describe as matrices with three rows and columns.
Hooke’s tensor C, the tensor of the coefficients of elasticity, then has to be a fourth
order tensor. Both points in C · · ε should point towards the fact that a double sum
must be taken. This relationship can be written as
σ ik =
3
r =1
3
s=1
C ikrs ε rs .
(338)
For the following, we will adopt the so-called Einsteinian summation convention,
with which many equations can be easier formulated. Above all doubly occurring
subscripts, 1–3 will be summed up without expressively writing the summation
symbol . For Eq. (338), we therefore simply write
σ ik = C ikrs ε rs .
(338a)
For a comparison with other representations of the theory of elasticity, we draw the
reader’s attention to a notation introduced by W. Voigt. In Voigt’s notation, two
subscripts are joined into one, which then goes from 1 to 6. The the stress–strain
relation (338) according to Voigt reads
σ I =
6
K =1
C I K ε K
Voigt’s notation
(339)
269
The strain tensor ε is symmetrical,
ε xy = ε yx ,
ε xz = ε zx ,
ε yz = ε zy .
(336)
We are now in a position where we can precisely formulate the three basic assumptions of the linear theory of elasticity for the three-dimensional case.
1. In the Newtonian term for inertia, the total time derivative for the momentum P of
the moving masses is replaced with the partial derivative of the material velocity v.
2. The spatial distributed mass density ρ is considered as a constant parameter.
3. Hooke’s law is valid, in other words the stress tensor σ is directly proportional to
the tensor of relative strain ε at position x and point of time t . This results in the
equations
d
dt
P = ρ
∂
∂t
v ,
ρ
= const. ,
σ(x, t)= C · · ε(x, t) .
(337)
Equations (337) formally look exactly like the corresponding Eqs. (67), for the onedimensional case. This however should not distract the reader from the fact that
Hooke’s law is now far more complicated. The quantities σ and ε are tensors of
the second order that we have to describe as matrices with three rows and columns.
Hooke’s tensor C, the tensor of the coefficients of elasticity, then has to be a fourth
order tensor. Both points in C · · ε should point towards the fact that a double sum
must be taken. This relationship can be written as
σ ik =
3
r =1
3
s=1
C ikrs ε rs .
(338)
For the following, we will adopt the so-called Einsteinian summation convention,
with which many equations can be easier formulated. Above all doubly occurring
subscripts, 1–3 will be summed up without expressively writing the summation
symbol . For Eq. (338), we therefore simply write
σ ik = C ikrs ε rs .
(338a)
For a comparison with other representations of the theory of elasticity, we draw the
reader’s attention to a notation introduced by W. Voigt. In Voigt’s notation, two
subscripts are joined into one, which then goes from 1 to 6. The the stress–strain
relation (338) according to Voigt reads
σ I =
6
K =1
C I K ε K
Voigt’s notation
(339)