Overview
Continuum Mechanics is all about using linear algebra, with some calculus thrown in, to describe the deformations
(strains)
in objects and relate them to the resulting
stresses. This is represented by the
popular figure to the right showing an object in both its undeformed and deformed states. Continuum Mechanics provides the
tool box of methods needed to accomplish all the nuts-and-bolts calculations of structural analysis, whether it is performing
coordinate transformations, applying
material derivatives,
or extracting
principal stresses.
From this basic foundation, continuum mechanics expands into
equilibrium balances,
constitutive models that relate
material deformations to the
stresses
generated, and the
1st and 2nd Laws of Thermodynamics,
which set limits on the behavior of the constitutive models.
Upon completing these advanced topics, the
Navier-Stokes
equations can actually seem logical. And if it (ever) becomes
intuitively obvious why the
Second Piola-Kirchhoff stress
is the derivative of the
Helmholtz free energy with respect
to the
Green strain tensor, well then, you've graduated.
Notation and Conventions
It is common in continuum mechanics to represent scalars with regular, normal-weight variables. For example, mass and entropy
are represented by \(m\) and \(s\), respectively (although entropy is sometimes represented by \(\eta\)).
Vectors, tensors,
matrices, etc are represented by bolded variables such as
\({\bf v}\) for velocity. Furthermore,
vectors are represented by lowercase bold variables as just shown
for velocity, while higher-rank quantities, such as
strain tensors, are represented by uppercase bold variables, \({\bf E}\).
Of course, exceptions exist. Examples include the use of \(\boldsymbol{\sigma}\)
for
stress and \({\bf X}\) for the vector of
coordinates of a point on an object in the undeformed state.
