100
3 Continuum Mechanics and Nonlinear Elasticity
Equation (3.100) also is valid in 3D, and so the Cauchy stress vectors acting on
the element are given by
T =
df
dA
,
T i =
df i
dA i
(i not summed),
(3.108)
where dA and dA i are the areas of the corresponding faces on which the forces act.
Substitution into Eq. (3.107) yields
T dA − T 1 dA 1 − T 2 dA 2 − T 3 dA 3 + b (h dA/3) = ρa (h dA/3),
in which dV has been replaced by the volume of a tetrahedron with h being the
distance from the base (slanted face of area dA) to the opposite vertex. As h → 0,
this relation becomes
T dA = T i dA i .
(3.109)
Next, we note that the areas of the orthogonal faces represent the projected areas
of the slanted face on the three coordinate planes. Since the area vector of the n-face
is dA = n dA, the areas of the other faces are given by
dA i = dA · e i = dA n · e i = dA(e i · n).
(3.110)
Inserting this expression into Eq. (3.109) and dividing through by dA yield
T = T i (e i · n) = (T i e i ) · n
= n · (T i e i )
T
= n · (e i T i ).
The dyadic on the right side of the dot product is defined to be the Cauchy stress
tensor
σ = e i T i ,
(3.111)
and, therefore,
T = n · σ .
(3.112)
This relation is known as Cauchy’s stress formula. In terms of Cartesian components, generalizing Eqs. (3.105) to three dimensions yields
T i = σ ij e j ,
(3.113)
where the σ ij are stress components relative to the x i -axes. Substitution into (3.111)
gives
σ = σ ij e i e j ,
(3.114)
which identifies σ ij as the Cartesian components of the Cauchy stress tensor σ .
Thus, if the components of σ at a point are known relative to any set of orthogonal
3 Continuum Mechanics and Nonlinear Elasticity
Equation (3.100) also is valid in 3D, and so the Cauchy stress vectors acting on
the element are given by
T =
df
dA
,
T i =
df i
dA i
(i not summed),
(3.108)
where dA and dA i are the areas of the corresponding faces on which the forces act.
Substitution into Eq. (3.107) yields
T dA − T 1 dA 1 − T 2 dA 2 − T 3 dA 3 + b (h dA/3) = ρa (h dA/3),
in which dV has been replaced by the volume of a tetrahedron with h being the
distance from the base (slanted face of area dA) to the opposite vertex. As h → 0,
this relation becomes
T dA = T i dA i .
(3.109)
Next, we note that the areas of the orthogonal faces represent the projected areas
of the slanted face on the three coordinate planes. Since the area vector of the n-face
is dA = n dA, the areas of the other faces are given by
dA i = dA · e i = dA n · e i = dA(e i · n).
(3.110)
Inserting this expression into Eq. (3.109) and dividing through by dA yield
T = T i (e i · n) = (T i e i ) · n
= n · (T i e i )
T
= n · (e i T i ).
The dyadic on the right side of the dot product is defined to be the Cauchy stress
tensor
σ = e i T i ,
(3.111)
and, therefore,
T = n · σ .
(3.112)
This relation is known as Cauchy’s stress formula. In terms of Cartesian components, generalizing Eqs. (3.105) to three dimensions yields
T i = σ ij e j ,
(3.113)
where the σ ij are stress components relative to the x i -axes. Substitution into (3.111)
gives
σ = σ ij e i e j ,
(3.114)
which identifies σ ij as the Cartesian components of the Cauchy stress tensor σ .
Thus, if the components of σ at a point are known relative to any set of orthogonal