Continuum mechanics is a powerful framework for describing physical processes by modeling matter as a continuous medium, allowing for the analysis of deformable objects and phenomena that vary in space and time.
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welcome to the first video of a series
of videos on Continuum mechanics in this
first video I will give an overview of
Continuum mechanics or more generally
Continuum physics and explain why
continum mechanics is so powerful for
describing physical processes in our daily
daily
life most of you are probably familiar
with the term mechanics mechanics is a
branch of physics in which the forces
acting on objects and the resulting
motion of these objects are studied in
classical mechanics the objects are
typically assumed to be rigid that is
they may move in Space over time but do
not deform up on acting
forces here you can see an exemplary
animation of some moving particles that
do not change their shape over
time in continum mechanics objects are
not anymore assumed to be rigid but
instead their shape may change over time
as you can see in this exemplary
simulation of a solid that deforms under external
external
forces so does this mean that classical
mechanics is the study of non-deformable
objects and continum mechanics is the
study of deformable objects well yes you
could say that but there is a little bit
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it
any object that is modeled within the
framework of continum mechanics occupies
a certain domain in this
three-dimensional space this domain is
often denoted by Capital Omega if a
point x with its three components being
X1 X2 and X3 lies within the domain we
say that X is an element of Omega if not
we say that X is not element of Omega
points that are right at the border of
the domain get a special treatment we
say that they are element of the
boundary of Omega which is denoted by partial
partial
Omega The crucial Assumption of
Continuum mechanics is that the domain
Omega is filled by a so-called Continuum
but what does this mean it means that at
every Point within the domain the
physical state of mattera can be defined
through a set of variables one example
of such a state variable is the
temperature capital
T because the temperature is defined at
each point in our domain we can define a
function T of X this function which
gives us for any point in the domain the
temperature at this point is called the temperature
temperature
field because the temperature at each
point is a scalar value the temperature
field is an example for a so-called scalar
scalar
field besides the temperature other
state variables can be likewise
described by Fields another example of a
field is the displacement field U of X
in contrast to the temperature field the
displacement field gives us at each
point a vector with the three components
U1 U2 and u3 U1 describes the
displacement in X1 Direction U2 in X2
Direction and u3 in X3 Direction each of
these components U1 U2 and u3 can be
interpreted as Scala fields and together
they form a so-called Vector field a
vector field can best be illustrated by
picking a few points in the domain and
drawing the corresponding vectors at
these points here I'm illustrating an
exemplary displacement field the
displacement tells us to which point
each point in the domain is moving when
the meta is deformed again the
displacement field is a field
so the displacement is defined at
infinitely many points in the domain
only for illustration purposes a finite
number of vectors is shown here other
fields used to describe the state of
mattera are the strain field Epsilon of
X and the stress field Sigma of X these
two fields are neither scalar Fields nor
Vector Fields but so-called tensor
Fields illustrating a tensor field is
not straightforward and will be covered
in a future video there are many more
fields that are used in continum
mechanics to characterize the state of
mattera more examples include the
velocity field the pressure field the
electric or magnetic potential field and many
many
more it highly depends on the physical
problem under consideration which of
these fields should be chosen as state
variables the fact that the state
variables are described by by fields in
continum mechanics that is by functions
of space and not for example as a set of
scalar values is the most critical
difference to classical mechanics or discrete
discrete
mechanics in fact fields are so
omnipresent in continum mechanics that a
physicists would also call continum
mechanics a field
theory note that we have here assumed
that the fields are functions of space
only for time dependent problems the
fields can further be considered as
functions of time this means that
continum mechanics can be leveraged to
model physical processes in which the
physical quantities that we are
interested in vary in space and time
this is applicable for every physical
process that we encounter in our
everyday life which explains why Contin
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continue Mechanics for practical
applications can be divided into two
branches solid mechanics and fluid
mechanics solid mechanics is concerned
with the deformation of solid bodies
under external influences while fluid
mechanics is concerned with the flow of
liquids and
gases many Concepts in solid mechanics
and fluid mechanics are very similar
however the equations that are used to
describe the problems and the
computational methods for solving these
equations can be quite different in
solid mechanics one typically uses among
others Fields like the displacement
field The Strain field or the stress
field to corize the physical state of
the system in fluid mechanics Fields
like the velocity field or the pressure
field are more commonly
used note that our meta being a
Continuum does not necessarily mean that
the fields that describe the physical
state of the Continuum are continuous
there may be sudden changes of the state
variables in space examples could be
jumps in the displacement field which
can be interpreted as cracks in solid
mechanics or jumps in the velocity field
which can be interpreted as shocks in
fluid mechanics so the fields do not
need to be continuous but they must be
defined for all points in the physical domain
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when trying to understand what continum
mechanics is it is helpful to take a
look at some counter examples in solid
Mechanics for example we could model
each atom in the solid individually and
we could assume that these atoms
interact with each other that is they
attract or repel each other according to
certain rules and we could use this
model model to simulate the behavior of
a solid likewise in fluid mechanics we
could model each particle in the fluid
individually and used this model to
simulate a fluid flow both of these
examples are counter examples of Contin
mechanics because physical quantities
like the displacement or the velocity
are not described by Fields as in Contin
mechanics this is because the
displacement and the velocity are not
defined at any point in space but only
at those points where an atom or
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present we have learned that in continum
mechanics the physical state of matter
is described by different fields the
general goal of continum mechanics is to
find relations between those different
fields within the domain and to further
relate those fields to the physical
quantities at the boundary of the domain
this is why problems in continum
mechanics are typically described by
so-called boundary value problems here
I'm showing an exemplary boundary value
problem in Continuum solid mechanics if
you're not familiar with all the
specific terms in the boundary value
problem don't worry for now in this
series of videos on continum mechanics
we will go through all of them step by
step for now I just want you to notice
that the boundary value problem consists
of two different types typ of equations
some equations describe relations
between different fields over the domain
Omega and other equations relate those
fields to information that is given at
the boundary partial
Omega note that many terms in the
boundary value problem depend on X to
get a more compact notation the
dependence of the fields on X is often
not explicitly indicated in continum
mechanics in this solid mechanics
example example the boundary value
problem relates the displacement field
to The Strain field to the stress field
and to physical quantities at the
boundary given some material properties
and sufficient information at the
boundary like for example applied
displacements or boundary forces the
boundary value problem can be solved to
determine the displacement field over the
solid that's it for this video I hope
you could get some idea of what continer
mechanics is detailed concepts are
planned for future videos please
subscribe to the channel if you'd like
me to produce more videos for this series
series bye
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