Mechanical Modeling of Material Behavior - Continuum Mechanics of Anisotropic Materials - page 11

analysis of beam bending and viscous fluid used, for example, in the model of flow
through a pipe. The use of the word “continuum” in describing these models stems
from the idea illustrated in Fig. 1.1 , namely that the model is a domain of Cartesian
space in the same shape of the object being modeled. It therefore has all the
properties necessary to use the analytical machinery of the calculus. In particular,
displacements, strains, velocities, and rates of deformation may be calculated.
These developments will be presented in the next and subsequent chapters. The
deformable continuum is the focus of this text because it is the primary class of
models employed in the study of solid and fluid at both the macro scale, at the
nanometer scale, and at scales in between.

1.8 Lumped Parameter Models

Lumped parameter models are extended rigid object models in which some of the
elements are assumed not to be rigid, but to respond in simplified specific ways. The
word “lumping” is used to imply that not all the properties are modeled exactly, but
in a somewhat approximate way. For example, in a lumped parameter model the
image of the object in Euclidean space, as shown in Fig. 1.1 , need not be an exact
model of the object, just a model that contains the features the modeler desires.
The mechanical concept of “Coulomb friction” is a “lumped” concept as it occurs in
the formula of the French engineer Charles Augustin de Coulomb (1736-1806).
The static friction formula of Coulomb is employed to express the force F necessary
to cause the motion of weight W resting upon a frictional horizontal surface as
F represents the coefficient of friction. The sources of what is

called “friction” between the surface and the weight W are varied and include,
among other things, the effect of surface adhesion, surface films, lubricants, and
roughness; these effects are “lumped” together in the concept of Coulomb friction
and expressed as a single coefficient,

¼ m W , where m

m

.
When linear springs and dashpots are used as elements in a model they are
“lumped” representations of an object's stiffness or damping. Their properties
describe the constitution of the element and are called constitutive properties.
The spring element is also called the Hookian model (Fig. 1.6a ) and is characterized
by an equation that relates its overall lengthening or shortening, x , to the force
applied to the spring, F , by a spring constant, k ; thus F

kx (Love, 1927 ). This
model is named after the English natural philosopher Robert Hooke (1635-1703).
The dashpot is called the viscous model or damper (Fig. 1.6b ) and is characterized
by an equation that relates the rate of its overall lengthening or shortening, d x /d t ,to
the force applied to the dashpot, F , by a damping constant,

¼

(d x /d t ).
Lumped parameter models employing springs and dashpots are used extensively
in the study of mechanical systems. Simple forms of these models are used to
explain the material response phenomena called creep and stress relaxation. Creep
is the increasing strain exhibited by a material under constant loading as the time
increases. A typical creep experiment on a specimen of material is performed by

; thus F ¼