Local balance of mass, momentum and energy - Biomechanics: Concepts and Computation - page 191

and in column/matrix notation:

T ∼ ,

balance of mass ρ =− ρ tr( D ) =− ρ ∼

balance of momentum

T

T

∼ σ + ρ q

∼ = ρ ∼ = ρ

∼ .

In a typical Lagrangian description, the field variables are considered to be a func-
tion of the material coordinates x 0 , being defined in the reference configuration,
and time t . It can be stated that the balance laws have to be satisfied for all x 0
within the domain V 0 and for all points in time.
Because of the physical relevance (for solids) of the deformation tensor F the
balance of mass will usually not be used in the differential form as given above,
but rather as

ρ 0
det( F )

ρ 0
det( F ) .

ρ = and also ρ = (11.21)

The gradient operator ∇ (and also ∼ ) in the balance of momentum equation, built
up from derivatives with respect to the spatial coordinates, can be transformed
into the gradient operator ∇ 0 (and ∼ 0 ) with respect to the material coordinates,
see Section 9.6 . The balance of momentum can then be formulated according to
F − T

· ∇ 0 · σ + ρ q = ρ a = ρ ˙ v , (11.22)

and also

F − T T

∼ 0 T σ + ρ q

∼ = ρ

∼ = ρ ˙ ∼ . (11.23)

In a typical Eulerian description, the field properties are considered to be a func-
tion of the spatial coordinates

x , indicating locations in the current configuration,
and time t . It can be stated that the balance laws have to be satisfied for all

x within

the domain V and for all points in time.
To reformulate the balance laws the material time derivative is 'replaced' by the
spatial time derivative, see Section 9.4 . For the mass balance this yields
δρ
δ t +

· ∇ ρ =− ρ =− ρ ∇·

v tr( D ) v , (11.24)

and so

δρ

δ t + ∇·

( ρ v ) = 0. (11.25)

In column/matrix notation:
δρ
δ

T T ∼ ,

t + ∼

∼ ρ =− ρ tr( D ) =− ρ ∼ (11.26)