CONTINUUM MECHANICS for ENGINEERS

ackground and definitions were given, along with a few specific cases
examined. This allows the inductive student to grasp the concepts presented.
Since only a few constitutive models were considered from a fundamental
basis, this section is devoted to a brief survey of constitutive equations. This
acts as an introduction for subsequent chapters, when various constitutive
models are discussed as applications of continuum mechanics.
A brief listing of some well-known constitutive equations is as follows:
(a) the stress-strain equations for a linear elastic solid assuming infinitesimal
strains,
σ ij = C ijkmε km
(5.12-1)
where the C ijkm are the elastic constants representing the properties of the
body. For isotropic behavior, Eq 5.12-1 takes the special form
σ ij = λδ ijε kk + 2µε ij
(5.12-2)
in which λ and µ are coefficients that express the elastic properties of the
material.
(b) the linear viscous fluid,
τ ij = K ijmnD mn
(5.12-3)
where τ ij is the shearing stress in the fluid and the constants K ijmn represent
its viscous properties. For a Newtonian fluid,
τ ij = λ * δ ijD kk + 2µ * D ij
where λ * and µ * are viscosity coefficients.
(c) plastic stress-strain equation,
P
dε= S dλ
(5.12-4)
(5.12-5)
where is the plastic strain increment, Sij the deviator stress, and dλ a
proportionality constant.
(d) linear viscoelastic differential-operator equations,
d P
εij
{P}S ij = 2{Q}η ij
σ ii = 3Kε ii
where {P} and {Q} are differential time operators of the form
ij
ij
(5.12-6a)
(5.12-6b)