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29 Οκτ 2013 (πριν από 3 χρόνια και 6 μέρες)

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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

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

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

.

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

or

displacements

-
the

inertia

forces

are

introduced
.

If

the

or

displacements

are

applied

very

slowly,

the

inertia

forces

can

be

neglected

and

a

static

analysis

can

be

justified
.

Dynamic

analysis

-

extension

of

static

analysis
.

Equation

of

motion

can

be

expressed
: