Page 224 - Intro to Tensor Calculus

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Conservation of Linear Momentum

Let R denote a region in space where there exists a material volume with density % having surface
i
tractions and body forces acting upon it. Let v denote the velocity of the material volume and use Newton’s
second law to set the time rate of change of linear momentum equal to the forces acting upon the volume as
in (2.3.5). We find
ZZZ ZZ ZZZ
δ j ij j
%v dτ = σ n i dS + %b dτ.
δt R S R
j
Here dτ is an element of volume, dS is an element of surface area, b are body forces per unit mass, and σ ij
are the stresses. Employing the Gauss divergence theorem, the surface integral term is replaced by a volume
integral and Newton’s second law is expressed in the form
ZZZ
j j ij
%f − %b − σ ,i dτ =0, (2.3.16)
R
j
where f is the acceleration from equation (1.4.54). Since R is an arbitrary region, the equation (2.3.16)
implies that
j
j
σ ij ,i + %b = %f . (2.3.17)
This equation arises from a balance of linear momentum and represents the equations of motion for material
in a continuum. If there is no velocity term, then equation (2.3.17) reduces to an equilibrium equation which
can be written
j
σ ij ,i + %b =0. (2.3.18)
This equation can also be written in the covariant form

si
g σ ms,i + %b m =0,

which reduces to σ ij,j + %b i = 0 in Cartesian coordinates. The equation (2.3.18) is an equilibrium equation
and is one of our fundamental equations describing a continuum.

Conservation of Angular Momentum

The conservation of angular momentum equation (2.3.6) has the Cartesian tensors representation

ZZZ ZZ ZZZ
d
%e ijk x j v k dτ = e ijk x j σ pk n p dS + %e ijk x j b k dτ. (2.3.19)
dt R S R
Employing the Gauss divergence theorem, the surface integral term is replaced by a volume integral to obtain

ZZZ
d ∂
e ijk % (x j v k ) − e ijk %x j b k + p (x j σ pk ) dτ =0. (2.3.20)
R dt ∂x
Since equation (2.3.20) must hold for all arbitrary volumes R we conclude that

d ∂σ pk
e ijk % (x j v k )= e ijk %x j b k + x j + σ jk
dt ∂x p