Analysis in Solid Mechanics

baconossifiedMechanics

Oct 29, 2013 (3 years and 11 months ago)

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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.