Part 01 Basic Elasticity Student

Solid Mechanics

Dr.NGUYEN T Hoang 1

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SOLID MECHANICSSOLID MECHANICS

Dr. NGUYEN The Hoang Dr. NGUYEN The Hoang

Email: Email: [email protected]@yahoo.com

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WHO AM I ?

Educational and professional background:Educational and professional background:

• PhD in Solid Mechanics and Material Science, University of Poitiers/ENSMA (Ecole Nationale Superieure de Mecanique et d’Aerotechnique), France, 2005

• Master in Engineering Mechanics, LMPM/University of Poitiers/ENSMA, France, 2002

• Bachelor of Engineering in Materials and Structures for Aeronautics and Transportation Vehicles, ENSMA, France, 2002

• Bachelor in Aeronautical Engineering, First Class Honours, HoChiMinh City University of Technology, Vietnam, 2001

• Deputy Head of Department of Aeronautical Engineering, HoChiMinh City University of Technology, Vietnam, 2005-2007

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How we work together ?In class:In class:

• I lecture in English (100%)

• At any time, you can raise questions (in English): raise your hand and STOP me

• I will explain/discuss about your questions both in English (90%) and Vietnamese (10% - in the cases of difficult issues / easy-misunderstood)

• I will explain all the issues, but not demonstrate all (due to the limited time of the course…and some already available in the Ref.books)

• You should prepare homeworks before next session

By group:By group:

• You should work in group �� benefits from your classmates’knowledge & experiences �� gain time!

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What will be covered in the course

Lectured topics include: Lectured topics include:

• Foundations of elasticity

• Basic equations for trusses, beams, plane elasticity and plates

• Multibody systems

• Basics of Finite Element Method (FEM)

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[2]1 exercise on calculating

deflection of a beam

under simple loading

4Beam analysis

1. Shear & moment diagrams

2. Normal & shear stresses

3. Deflection

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[2]Introduction on Project 1 of

“bridge model” FEM

simulation

4Trusses

1. Definition and examples

2. Methods of Joints

3. Compound trusses

4. Space frame

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[1]1-2 small exercies on

understanding of

stress-strain

relationship and/or

Mohr’s circle

4Basic Elasticity

1. Stresses - Mohr’s circle of stress

2. Equilibrium & boundary conditions

3. Strains & compatibility equations

4. Hooke’s law

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RefHome worksPeriodsContentsChapter

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[4](to be selected between

chapter 5 and 6)(2)Multibody system

1.Introduction

2.Kinematics

3.Equation of motion 6

[3](*) Project 1: “bridge model”

FEM simulation

(*) Project 2:

“connecting lug” FEM

simulation* Note: in the case of limited time, it

is to be selected between Project 1

and 2

6FEM

1.Applications of FEM

2.FEM procedure

3.FEM for Frames

4.FEM for Plates5

[1]Introduction of Project 2 on

“connecting lug” FEM

simulation

6Plates

1.Plate Theories

2.Stress-strain relationships

3.Deflection of a plate subjected to a distributed

transverse load and/or other loadings4

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Dr.NGUYEN T Hoang 4

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Schedule

5 (+6)4+54321Chapters

444444Periods

26.27/0722-Jul15-Jul8-Jul1-Jul24-Jun17-Jun10-Jun3-Jun27-MayMechanics

E987654321Week

** Lecture day = Thursday

** 1 lecture day = 1 morning session + 1 afternoon session

** 1 period = 45min.; 1 session = 2 periods = 90min.

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Your background

Others (technical)

Mathematics

Physics

Materials Science

Mechatronics

Others (economics/social)

Mechanical Eng.

Civil Eng.

%No.StudentsMajor

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Dr.NGUYEN T Hoang 5

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References

[1] Megson T.H.G., “Aircraft Structures for engineering students”,

Third Edition, Edward Arnold, 1999.

[2] Megson T.H.G., “Structural and stress analysis”, Butterworth-

Heineman, 2000.

[3] G. R. Liu, S. S. Quek, “The Finite Element Method: A Practical

Course”, Butterworth-Heinemann, 2003

[4] F. Amirouche, “Fundamentals of multibody dynamics : theory

and applications”, Prentice-Hall, 2006

Others: Interesting Websites

https://ecourses.ou.edu

http://emweb.unl.edu

http://www.ae.msstate.edu

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Introduction

ProcessesProcesses leading to fabrication of advanced engineering systems:

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Dr.NGUYEN T Hoang 6

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Introduction

SpecimensSpecimens

From research laboratory to industry application & knowledge diffusion

Carbon/Epoxy specimens

Metal alloy specimens

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IntroductionFrom research laboratory to industry application & knowledge diffusion

Doing experimentsDoing experiments

Traction machine

+ Environmental effects

+ Static/Fatigue tests

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Dr.NGUYEN T Hoang 7

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Introduction

Experimental resultsExperimental results

Damages in composite materials


Microscope observation

100µm00 layer

00 layer

900 layer

Fracture surfacesFracture surfaces

Material properties Material properties

lawlaw

crack

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microscopic

DUCTILE

BRITTLE

EXPERIMENTS IN FRACTUREDuctile Fracture

� ductile fracture surfaces also appear rough and irregular

� surface consists of many microvoids (lỗ khí) and dimples (lõm)

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Introduction

Publications

•• Internal presentations Internal presentations

•• Conferences, workshopsConferences, workshops

•• International JournalsInternational Journals

•• Industry reports Industry reports

Physical Sciences and Engineering

Life Sciences

Health Sciences

Social Sciences and Humanities


Mach 2

EU Project: Supersonic

Concorde’s successor

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Introduction

Typical FEM stress contours on deformed wing

Stress analysis and optimization

FEM-simulation is a powerful tool to reduce costs in product development.

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Introduction

Corresponding static deformation on wing/nacelles

Wall pressure on wing/nacelles

• Three-dimensional Navier-Stokes static coupling

simulation, • Aerodynamic structured grid is made of

approximately 7 million nodes, distributed in 91 blocks

• Computation was carried out in parallel on 15

processors

Aeroelasticity and Structural Dynamics


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Introduction

Aeroelasticity and Structural Dynamics

Impact of a bird

Sea landing of a commercial aircraft


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Dr.NGUYEN T Hoang 10

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Introduction

Full aircraft FEM development, verification, validation and analytical application

Analysis Tools•NASTRAN, ANSYS,

ABAQUS FEM solvers

•StressCheck•PATRAN, FEMAP, Pro-

Engineer pre- and post-

processors

•PC-based software

supported by high-end PC platforms for multiple

users simultaneously

Full system modeling and simulation


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1.1. Basic elasticityBasic elasticity1.1. Stress1.1. Stress

1.2. Equilibrium1.2. Equilibrium

1.3. Boundary Conditions1.3. Boundary Conditions

1.4. Principle stresses1.4. Principle stresses

1.5. Mohr1.5. Mohr’’s Circle of Stresss Circle of Stress

1.6. Strains1.6. Strains

1.7. Compatibility equations1.7. Compatibility equations

1.8. 1.8. HookeHooke’’ss LawLaw

1.9. Problems1.9. Problems

Content of chapterContent of chapter

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(STRESS CONCENTRATION FACTORS)

Stress distribution near a hole in a plateloaded in tension.

Geometric irregularities (changes

in cross-sections) are a must in

most of machine components:

Shoulders for bearings, Key slots

for mounting gears and pulleys,

threads, and splines. Any change

in cross-section alters the stress

distribution and increases the

stress.

Discontinuities are called stress raisersand areas where they occur are called stress concentration

Crack Tip StressCrack Tip Stress

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1. Basic elasticity1.1 Stress

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Stress matrixStress matrix

[ ]xx xy xz

yx yy yz

zx zy zz

σ τ τ

σ = τ σ τ τ τ σ

Or: in the form of vectorOr: in the form of vector

[ ]

xx

yy

zz

xy

yz

xz

σ σ σ

σ = τ

τ

τ

Direct

stresses

Shear

stresses

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Average normal

stress

Compression

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Average normal stressTraction / tension

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1.1.1. 1.1.1. Direct / Normal Stress

Direct Stress = Applied Force (P)

Cross Sectional Area (A)

� Units (SI): N/m2 (Pa), kPa, MPa, GPa

� US units: Force (P) in pounds (lb) or kilopounds (kip); Cross section (A) in square inches (in2) � Stress: pounds per square inch (psi) or kilopound per square inch (ksi)

� http://www.convertunits.com/SI-units.php

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1.1.2. Shear Stress

� Shear stresses are produced by equal

and opposite parallel forces not in line.

� The forces tend to make one part of the material slide over the other part.

� Shear stress is tangential to the area

over which it acts.

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1.1.2 Shear StressExample: A BEAM

Suppose: uniform / section

shear load:

shear stress:

Equilibrium

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1.1.2. Shear StressEx: Some Engineering Applications

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1.1.2. Shear StressEx: Engineering Application

F: shear force

(internal force)

P = P’

Shear stress

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Example 1Example 1Problem:

• 2 members are joined by a glue at

angle θ� Which stresses generated on inclined interface plane?� Calculate the stresses?

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1.1.3. Safety factor1.1.3. Safety factor

A: initial cross areaUltimate tensile strength

Working

load/design

load

Ultimate normal load

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Example 1 (cont.)Example 1 (cont.) Problem:


angle θ• Ultimate stresses:� Determine range of angles θ, if

safety factors: shear stress =4.27,

normal stress=5.28

U

U

22 MPa

11MPa

σ =

τ =

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ExampleExample Problem:


angle θ• Ultimate stresses:� Determine range of angles θ, if

safety factors: shear stress =4.27,

normal stress=5.28

U

U

22 MPa

11MPa

σ =

τ =

36

Example 2

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1.1. Basic elasticityBasic elasticity

1.2. Equilibrium 1.2. Equilibrium

Forces applied:

+ Surface forces

+ Body forces

(gravitational,

inertial �per

unit of volume �

X, Y, Z )

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1.2. Equilibrium 1.2. Equilibrium

Taking moments about an axis

through the centre of the element parallel to the zaxis

(Forces) equlibrium of the

element in directions x,y,z:

The equations of equilibrium must be

satisfied at all interior points in a

deformable body under a 3D force system.

Eqs. (1)

Eqs. (2)

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EqsEqs (1) demonstration(1) demonstration

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1. Basic elasticity1.2. Equilibrium: Plane stress

structural components are

fabricated from thin material sheet

Plane stress (2D case)

Equilibrium (2D plane stress)

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1.3. Boundary Conditions1.3. Boundary Conditions

Equilibrium (3D) �� ONLY 3 equations for 6 unknowns of stresses �

“Statically Indeterminate problems”

� we NEED: BOUNDARY CONDITIONS

Cosines:

l = dy/ds

m = dx/ds

3D

2D

Summations of forces in the

directions X,Y give:

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Demonstration:Demonstration:

2D

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1.4. Principal Stresses1.4. Principal Stresses

Plane stress (2D)

forces are ignored

We want to

find stresses on plane (AB)..

..Element (ECD) is in

equilibrium

Direct stress

Shear stress

(1.8)

(1.9)

Both vary withθθθθ

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DemonstrationDemonstrationPlane stress (2D)

..Element (ECD) is in equilibrium

Direct stress

Shear stress

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� MAX or MIN

Student do by themselves

2 solutionsShear stress = 0 (comparing with

Equation 1.9)and / 2θ θ + π

τ

(1.8)

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maximum or major principal stress

2 solutions

minimum or minor principal stress

Shear stress = 0

2 principle stresses (σI, σII) on 2 perpendicular principal

planes (on which shear stresses =0)

τ

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And how about maximum SHEAR stress ??? �� students answer

2 SOLUTIONS:

Remark:

NB: the planes of maximum shear stress are inclined at 45" to the principal planes

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1.5. Mohr1.5. Mohr’’s Circle of Stresss Circle of Stress

The state of stress at a point in a deformable body may

be determined graphically by Mohr's circle of stress

�Q1

� Q2

� centre C

� rotate Q1

angle 2 θ�

( )ngiven , ?θ⇒ σ τ

( )nQ ,′ σ τ

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1.5. MOHR1.5. MOHR’’S CIRCLE OF STRESSS CIRCLE OF STRESS

RADIUS:

CENTRE (C):

Principle stresses:

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MAX/MIN. NORMAL STRESSES

�� Where it is?

MAX./MIN. SHEAR STRESSES

�� Where they are ???

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EXAMPLEEXAMPLE

Direct stresses of 160 N/mm2, tension, and 120 N/mm2,

compression, are applied at a particular point in an elastic

material on two mutually perpendicular planes.

The principal stress in the material is limited to 200 N/mm2,

tension.

Calculate:+ the allowable value of shear stressat the point on the given planes

+ the value of the other principal stress

+ the maximum value of shear stress

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EXAMPLE Direct stresses of 160 N/mm2, tension, and 120 N/mm2,

compression, are applied at a particular point in an elastic

material on two mutually perpendicular planes.

The principal stress in the material is limited to 200 N/mm2,

tension.

UNKNOWN: θ…MOHR CIRCLE ?

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EXAMPLE

�OσT

� OP1 = 160

� OP2 = -120

� C: MIDPOINT OF P1P2

� OB = 200

� MOHR’S CIRCLE: WITH RADIUS = CB

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1. Basic elasticity1.6. STRAINS

Longitudinal / direct strains direct

stresses σchanges in lengthchanges in length

Shear strainsshear stresses changes in angle changes in angle

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1.6.1 Direct / Normal Strain1.6.1 Direct / Normal Strain

� loads applied to a body � deformation will occur � dimension

change

/ Lε = δ

A bar � subjected to axial tensile

loading force, then tensile strain is:

NB:

� strain is dimensionless

� Compressive strain = - /L

� Strain is positive for an increase in dimension and negative for a reduction in dimension.

δ

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1.6.2 Shear Strain1.6.2 Shear Strain

P Q

S R

F

D D’

A B

C C’

L

x

φ

Shear strain is the distortion produced by shear stress on

an element or rectangular block as above. The shear strain, (gamma) is given as:

= x/L = tan γ φ

γ

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� For small �

� Shear strain � change in right angle

� It is dimensionless and is measured in radians

φ γ φ=

1.6.2 Shear Strain (Cont.)1.6.2 Shear Strain (Cont.)

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1. Basic elasticity1.6.3. Strains: general case

DIRECT STRAINDIRECT STRAIN

perpendicular line

elements OA, OB and OC at a point Oin a deformable body

subjected

to forces

(at O)

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DemonstrationDemonstration

(direct strains)

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DemonstrationDemonstration

(direct strains)

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1.6.3. STRAIN: general case1.6.3. STRAIN: general case

DIRECT STRAINS SHEAR STRAINS

Eqs (1.18) and (1.20) are derived on the assumption that the displacementsinvolved are small. Normally these linearized equations are adequate for most types of structural problem but in cases where deflections are large,

for example types of suspension cable etc., the full, non-linear, large deflection equations must be employed.

components of displacement

(see demonstration in Ref.[1])

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the body remains continuous during the

deformation so that no voids are formed

three single-valued functions for displacement

the six strains are defined in terms of three

displacement functions then they must bear some

relationship to each other and cannot have arbitrary

values

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the six strains are defined in terms of three displacement

functions then they must bear some relationship to each other

and cannot have arbitrary values.

six equations of strain compatibility

the six equations of strain compatibility which must be

satisfied in the solution of three-dimensional problems in elasticity.

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1.8. HOOKE1.8. HOOKE’’S LAWS LAW

Robert Hooke, who in 1676 stated,

One-dimensional (1D) Hooke's Law

where F is the applied force, u is the deformation of the elastic body

subjected to the force F, and k is the spring constant (i.e. the ratio of

previous two parameters).

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1D LAW1D LAW

YoungYoung’’s moduluss modulus

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Stress-Strain Diagram: Ductile Materials

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Stress-Strain Diagram: Ductile Materials

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yield stress determination

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Stress-Strain Diagram: Brittle Materials

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Hooke’s Law: Modulus of Elasticity

• Below the yield stress

Elasticity of Modulus

or Modulus Youngs=

=

E

Eεσ

• Strength is affected by alloying,

heat treating, and manufacturing

process but stiffness (Modulus of

Elasticity) is not.

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Elastic vs. Plastic Behavior

• If the strain disappears when the

stress is removed, the material is

said to behave elastically.

• When the strain does not return

to zero after the stress is

removed, the material is said to

behave plastically.

• The largest stress for which this

occurs is called the elastic limit.

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Elastic Recovery

Strain

Str

ess

Loading

Unloading

Loading

Unloading

Reloading

elastic strainStrain

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Elastic and Plastic Strain

Str

ess

StrainPlastic

Elasticeeep

P

Total Strain

(e,S) e = ee + ep

ee =S

E

ep = e − ee

The 0.2% offset yield stress

is the stress that gives a plastic

(permanent) strain of 0.002

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E is the elastic modulus (also known as

Young’s modulus), and G is the shear modulus. The elastic and shear moduli are

material constants characterizing

the stiffness of the material.

ISOTROPIC

1D LAW1D LAW

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Poisson's Ratio (REVIEW)

If the load is in the x-direction,

then strain in the y- and z-

direction will be

εy = εz = -ν*εx

Poisson's ratio is the amount

of transverse contraction, or

negative strain, when strained

in a given direction.

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1. Basic elasticity1.8. HOOKE’S LAW

Generalized Hooke's Law (Anisotropic Form)3D LAW

Where:

C is the stiffness matrix,

S is the compliance matrix,

and S = C-1

= CONSTITUTIVE RELATIONS

Material Properties

6*6 = 36 constants

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metallic alloys and thermoset polymers are considered isotropic

material properties are independent of direction

2 independent variables (i.e. elastic constants) in their stiffness and

compliance matrices

Young's modulus E and the Poisson's ratio

alternative elastic constants K (bulk modulus) and/or G (shear modulus)

Hooke's Law �� Isotropic MATERIALS

ν

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Hooke's Law �� Isotropic MATERIALS (3D)

2 CONSTANTS: TO DETERMINE

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EXPLICITE RELATIONSHIPS (3D)EXPLICITE RELATIONSHIPS (3D)


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EXPLICITE RELATIONSHIPS (2D)EXPLICITE RELATIONSHIPS (2D)

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1. Basic elasticity1.9. PROBLEMS

Problem 9.a:Problem 9.a:

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Problem 9.b:Problem 9.b:

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Problem 9.c:Problem 9.c:

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