104
3 Continuum Mechanics and Nonlinear Elasticity
where we have used the symmetry properties of σ to obtain n · σ = σ T · n =
σ · n. Solving this eigenvalue problem in 3D gives three principal stresses σ i and
their corresponding principal directions n i . Since σ is symmetric, the eigenvalues
(principal stresses) are real and the eigenvectors (principal directions) are mutually
orthogonal.
3.5 Balance Laws
Any continuum must satisfy certain fundamental principles, regardless of whether
it is solid, liquid, or gas. These include balance laws for mass, momentum (linear
and angular), and energy, as well as the entropy inequality.
3.5.1 Balance of Mass
In classical (nonrelativistic) mechanics, the total mass contained in a closed system,
i.e., a system that does not exchange matter with its surroundings, must remain
constant. For some problems, living tissues can be treated as closed systems. At
other times, however, this assumption can be far off the mark. For example, water
can flow into or out of soft tissues as they deform, and solid mass can be added
through growth. In cases like these, it is best to treat tissues as open systems that can
exchange mass with their surroundings. For now, however, we consider a continuum
that does not gain or lose mass, and the equation for mass conservation is called a
continuity equation.
Continuity in 1D
Consider a rectangular element with initial length dX and cross-sectional area dA 0 .
During deformation, these quantities become dx and dA. If the mass dm of the
element is conserved, then we must have
d
dt
(dm) = 0.
(3.124)
In 1D, the densities ρ 0 and ρ represent the mass of the element per unit initial
and current length, respectively. Then, the mass can be written dm = ρ 0 dX =
ρ dx. Like any variable, both ρ 0 and ρ can be expressed in either material or spatial
form, but here we stipulate ρ 0 = ρ 0 (X, t) and ρ = ρ(x, t). Substituting dm =
ρ 0 (X, t) dX into (3.124) gives
d
dt
ρ 0 (X, t) = 0
(3.125)
3 Continuum Mechanics and Nonlinear Elasticity
where we have used the symmetry properties of σ to obtain n · σ = σ T · n =
σ · n. Solving this eigenvalue problem in 3D gives three principal stresses σ i and
their corresponding principal directions n i . Since σ is symmetric, the eigenvalues
(principal stresses) are real and the eigenvectors (principal directions) are mutually
orthogonal.
3.5 Balance Laws
Any continuum must satisfy certain fundamental principles, regardless of whether
it is solid, liquid, or gas. These include balance laws for mass, momentum (linear
and angular), and energy, as well as the entropy inequality.
3.5.1 Balance of Mass
In classical (nonrelativistic) mechanics, the total mass contained in a closed system,
i.e., a system that does not exchange matter with its surroundings, must remain
constant. For some problems, living tissues can be treated as closed systems. At
other times, however, this assumption can be far off the mark. For example, water
can flow into or out of soft tissues as they deform, and solid mass can be added
through growth. In cases like these, it is best to treat tissues as open systems that can
exchange mass with their surroundings. For now, however, we consider a continuum
that does not gain or lose mass, and the equation for mass conservation is called a
continuity equation.
Continuity in 1D
Consider a rectangular element with initial length dX and cross-sectional area dA 0 .
During deformation, these quantities become dx and dA. If the mass dm of the
element is conserved, then we must have
d
dt
(dm) = 0.
(3.124)
In 1D, the densities ρ 0 and ρ represent the mass of the element per unit initial
and current length, respectively. Then, the mass can be written dm = ρ 0 dX =
ρ dx. Like any variable, both ρ 0 and ρ can be expressed in either material or spatial
form, but here we stipulate ρ 0 = ρ 0 (X, t) and ρ = ρ(x, t). Substituting dm =
ρ 0 (X, t) dX into (3.124) gives
d
dt
ρ 0 (X, t) = 0
(3.125)