A
PPLIED
M
ECHANICS
Lecture
10
Slovak University of Technology
Faculty of Material Science and Technology in Trnava
FUNDAMENTALS OF CONTINUUM MECHANICS
The fundamental equations of structural mechanics
:
stress

strain
relationship
contains
the
material
property
information
that
must
be
evaluated
by
laboratory
or
field
experiments
,
the
total
structure,
each
element,
and
each
infinitesimal
particle
within
each
element
must
be
in
force
equilibrium
in
their
deformed
position
,
displacement compatibility conditions must be satisfied.
If all three equations are satisfied at all points in time,
other conditions will automatically be satisfied.
E
xample

at
any
point
in
time
the
total
work
done
by
the
external
loads
must
equal
the
kinetic
and
strain
energy
stored
within
the
structural
system
plus
any
energy
that
has
been
dissipated
by
the
system
.
Virtual
work
and
variational
principles
are
of
significant
value
s
in
the
mathematical
derivation
of
certain
equations
.
FUNDAMENTALS OF CONTINUUM MECHANICS
The linear stress

strain relationships contain the material property
constants, which can only be evaluated by laboratory or field
experiments.
The mechanical material properties for most common material, such
as steel, are well known and are defined in terms of three numbers:
modulus of elasticity
E
,
Poisson’s ratio
n
,
coefficient of thermal expansion
a
.
Simplification

materials
are considered
isotropic (equal properties
in all directions) and homogeneous (
the
same properties at all points
in the solid).
Real materials have a
nisotropic
properties
, which may be different in
every element in a structure.
FUNDAMENTALS OF CONTINUUM MECHANICS
All stresses are by definition in units of force

per

unit

area.
In matrix notation, the six independent stresses
:
From equilibrium
of element:
The six corresponding engineering strains
:
FUNDAMENTALS OF CONTINUUM MECHANICS
MATERIAL PROPERTIES

Anisotropic materials
Material
p
roperties
:
Anisotropic
materials

The
most
general
form
of
the
three
dimensional
strain

stress
relationship
for
linear
structural
materials
subjected
to
both
mechanical
stresses
and
temperature
change
can
be
written
in
the
following
matrix
form
FUNDAMENTALS OF CONTINUUM MECHANICS
MATERIAL PROPERTIES

Anisotropic materials
I
n symbolic matrix form
Basic energy principles require that the
C
matrix for linear material
be symmetrical. Hence,
d
=
Cf
+
T
a
C

compliance matrix

the most fundamental definition of the material
properties
T

temperature increase

in reference to the temperature at zero stress
,
a

matrix indicates the strains caused by a unit temperature increase.
FUNDAMENTALS OF CONTINUUM MECHANICS
MATERIAL PROPERTIES

Orthotropic materials
Orthotropic
materials

t
he
shear
stresses,
acting
in
all
three
reference
planes,
cause
no
normal
strains
.
C

9
independent material constants,

3
independent thermal expansion
coefficients
This type of material property is very common

rocks, concrete, wood
,
many fiber
reinforced materials exhibit orthotropic behavio
u
r.
FUNDAMENTALS OF CONTINUUM MECHANICS
MATERIAL PROPERTIES

Isotropic materials
Isotropic
materials

equal
properties
in
all
directions,
the
most
commonly
used
approximation
to
predict
the
behavior
of
linear
elastic
materials
.
For isotropic materials: Young's modulus
E,
Poisson's ratio n
need to be defined.

shear modulus
FUNDAMENTALS OF CONTINUUM MECHANICS
MATERIAL PROPERTIES
–
Plane strain isotropic materials
Plane
strain
isotropic
materials

1
,
13
,
23
and
13
,
23
are
zero,
matrix
is
reduced
to
a
3
3
array
.
For the case of plane strain

the displacement and strain in the
3

direction are zero. Poisson's ratio
n

approaches 0,5
.
The normal stress in the 3

direction is
The stress

strain relationship
FUNDAMENTALS OF CONTINUUM MECHANICS
MATERIAL PROPERTIES
–
Plane stress isotropic materials
Plane
stress
isotropic
materials
–
3
,
13
,
23
are
zero,
matrix
is
reduced
to
a
3
3
array
.
The stress

strain relationship
FUNDAMENTALS OF CONTINUUM MECHANICS
MATERIAL PROPERTIES
–
Fluid

like materials
Fluid

like
materials

isotropic
materials,
which
have
a
very
low
shear
modulus
compared
to
their
bulk
modulus,
materials
are
referred
to
as
nearly
incompressible
solids
.
The pressure

volume relationship for a solid or a fluid
where

bulk modulus of the material.
The volume change e is equal to
1
+
2
+
3
, and the hydrostatic
pressure
indicates equal stress in all directions.
FUNDAMENTALS OF CONTINUUM MECHANICS
EQUILIBRIUM AND COMPATIBILITY
EQUILIBRIUM
AND
COMPATIBILITY
Equilibrium
equations

set
the
externally
applied
loads
equal
to
the
sum
of
the
internal
element
forces
at
all
joints
or
node
points
of
a
structural
system
;
They
are
the
most
fundamental
equations
in
structural
analysis
and
design
.
The
exact
solution
for
a
problem
in
solid
mechanics
requires
that
the
differential
equations
of
equilibrium
for
all
infinitesimal
elements
within
the
solid
must
be
satisfied
.
FUNDAMENTALS OF CONTINUUM MECHANICS
EQUILIBRIUM AND COMPATIBILITY
Fundamental
equilibrium
equations
The
3
D
equilibrium
of
an
infinitesimal
element
The
body
force

X
i
,
is
per
unit
of
volume
in
the
i

direction
and
represents
gravitational
forces
or
pure
pressure
gradients
.
FUNDAMENTALS OF CONTINUUM MECHANICS
EQUILIBRIUM AND COMPATIBILITY
Stress
resultant
–
forces
and
moments
For
a
finite
size
element
or
joint
a
substructure
or
complete
structural
system
the
following
six
equilibrium
equations
must
be
satisfied
Compatibility
requirements
For
continuous
solids

defined
strains
as
displacements
per
unit
length
.
To
calculate
absolute
displacements
at
a
point,
we
must
integrate
the
strains
with
respect
to
a
fixed
boundary
condition
.
A
solution
is
compatible
if
the
displacement
at
all
points
is
not
a
function
of
the
path
.
Therefore,
a
displacement
compatible
solution
involves
the
existence
of
a
uniquely
defined
displacement
field
.
FUNDAMENTALS OF CONTINUUM MECHANICS
EQUILIBRIUM AND COMPATIBILITY
Strain
displacement
equations
T
he
small
displacement
fields
u
1
,
u
2
and
u
3
are
specified
.
The
consistent
strains
can
be
calculated
directly
from
the
following
well

known
strain

displacement
equations
FUNDAMENTALS OF CONTINUUM MECHANICS
EQUILIBRIUM AND COMPATIBILITY
Definition
of
rotation
R
otation
of
a
horizontal
line
may
be
different
from
the
rotation
of
a
vertical
line

following
mathematical
equations
are
used
to
define
rotation
of
the
three
axes
FUNDAMENTALS OF CONTINUUM MECHANICS
EQUILIBRIUM AND COMPATIBILITY
Dynamic
equilibrium
R
eal
physical
structures
behave
dynamically
when
subjected
to
loads
or
displacements

the
additional
inertia
forces
are
introduced
.
If
the
loads
or
displacements
are
applied
very
slowly,
the
inertia
forces
can
be
neglected
and
a
static
load
analysis
can
be
justified
.
Dynamic
analysis

extension
of
static
analysis
.
Equation
of
motion
can
be
expressed
:
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