Analytical Vs Numerical
Analysis in Solid Mechanics
Dr. Arturo A. Fuentes
Created by:
Krishna Teja Gudapati
Solid Mechanics
Solid
Mechanics
is
a
collection
of
mathematical
techniques
and
physical
laws
that
can
be
used
to
predict
the
behavior
of
a
solid
material
when
subjected
to
loading
.
Engineers
and
scientists
use
solid
mechanics
for
a
wide
range
of
applications,
including
:
Mechanical
Engineering
Geo

Mechanics
Civil Engineering
Manufacturing Engineering
Biomechanics
Materials Science
Microelectronics
Nanotechnology
To know more about solid mechanics visit:
http://www.engr.panam.edu/~afuentes/mechmat.htm
Defining a Problem in Solid
Mechanics
Regardless
of
the
field,
the
general
steps
in
setting
up
a
problem
in
solid
mechanics
are
always
the
same
:
1
.
Decide
what
you
want
to
calculate
2
.
Identify
the
geometry
of
the
solid
to
be
modeled
3
.
Determine
the
loading
applied
to
the
solid
4
.
Decide
what
physics
must
be
included
in
the
model
5
.
Choose
(and
calibrate)
a
constitutive
law
that
describes
the
behavior
of
the
material
6
.
Choose
a
method
of
analysis
7
.
Solve
the
problem
Choosing a Method of Analysis
Once
you
have
set
up
the
problem,
you
will
need
to
solve
the
equations
of
motion
(or
equilibrium)
for
a
continuum,
together
with
the
equations
governing
material
behavior,
to
determine
the
stress
and
strain
distributions
in
the
solid
.
Several
methods
are
available
for
this
purpose
.
Analytical solution (or) Exact solution
: There is a good
chance that you can find an exact solution for:
1.
2D (plane stress or plane strain) linear elastic solids,
particularly under static loading.
2
.
2
D
viscoelastic
solids
3
.
3
D
linear
elasticitity,
usually
solved
using
transforms
.
4
.
2
D
(plane
strain)
deformation
of
rigid
plastic
solids
(using
slip
line
fields)
Choosing a Numerical Analysis
Method
Numerical
Solutions
:
are
used
for
most
engineering
design
calculation
in
practice
.
Numerical
techniques
include
1
.
The
finite
element
method
–
This
is
the
most
widely
used
technique,
and
can
be
used
to
solve
almost
any
problem
in
solid
mechanics
.
2
.
The
boundary
integral
equation
method
(or
boundary
element
method)
–
is
a
more
efficient
computer
technique
for
linear
elastic
problems,
but
is
less
well
suited
to
nonlinear
materials
or
geometry
.
3
.
Free
volume
methods
–
Used
more
in
computational
fluid
dynamics
than
in
solids,
but
good
for
problems
involving
very
large
deformations,
where
the
solid
flows
much
like
a
fluid
.
4
.
Atomistic
methods
–
used
in
nanotechnology
applications
to
model
material
behavior
at
the
atomic
scale
.
Molecular
Dynamic
techniques
integrate
the
equations
of
motion
(Newton’s
laws)
for
individual
atoms
;
Molecular
static's
solve
equilibrium
equations
to
calculate
atom
positions
.
Complex Bio

Mechanical
Example
Let us consider a human leg bone (Femur) with the
following mechanical properties that are taken from an
average healthy human being
Mass Density:
0.237 g/cm^3
Poisson’s Ratio:
0.3
Mod. Of Elasticity:
17*10^10 dyn/cm^2
=17 Gpa
Force applied:
4482216.2 dyn=100 lb
Mesh size:
50%
Thermal coefficient of expansion: 0.000027 /c
Simple Example of an
Analytical Solution
Simple Example of an
Numerical Solution
Taking Finite Element Method/Analysis (F.E.A) by using ALGOR Software
The same problem defined in Analytical Solution and with assuming the missing data
Dimensions taken
Adding Loads and
Boundary conditions
Stresses
Strains
Femur Models Taken for the
Finite Element Analysis
Simple cylinder
1st Approx. of Femur
Bone
2
nd
Approx. of Femur
Bone
Cylinder with layers
Imported Approx. from
a 3d scanner
F.E.A Structure with Loads and
Boundary Conditions Applied
Simple Cylinder
First Approx. Bone
Second Approx. Bone
Cylinder with Layers
Imported Approx. from
a 3d Scanner
Stress Results
Simple Cylinder
First Approx. Bone
Second Approx. Bone
Cylinder with Layers
Imported Approx. from
a 3d Scanner
Strain Results
Simple Cylinder
First Approx. Bone
Second Approx. Bone
Cylinder with Layers
Imported Approx. from
a 3d Scanner
Nodal Displacement Results
Simple Cylinder
First Approx. of Bone
Second Approx. of Bone
Cylinder with Layers
Imported Approx. from
a 3d Scanner
FEM Resources
To know more about FEM visit:
http://www.engr.panam.edu/~afuentes/fea.htm
References
Karim Khan, 2001. Physical Activity and Bone Health.
John D. Curry, 1996. Bones Structure and mechanics.
Bourne, Geoffrey H., The biochemistry and physiology of bone.
Kardestuncer, H., 1987. Finite Element Handbook, McGraw

Hill, New
York.
Nikishkov, G.V., 1998. Introduction to the Finite Element Method,
unpublished lecture notes, University of Arizona, Tucson, AZ.
Segerlind, L. J., 1984. Applied Finite Element Analysis, John Wiley and
Sons, New York.
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