Stress, Strain, and Constitutive Relations (pptx)

Stress, Strain, and Constitutive Relations


0,0 MF

WRy

“ Does member A behave as same as member B ?

“Which one will fail first as W increased?”

Reason of Failure

Material failure :- fracture, tearing, rupture.Deform :- temporary and permanent deform


Reason of Failure

Material failure :- fracture, tearing, rupture.Deform :- temporary and permanent deform

Stress Concept“There is one to one relationship for many materials between the applied load and the motion”

k is the parameter that define its characteristic

“This is not TRUE!”

Why?

Stress ConceptEuler(1757) used force intensity “Stress”

A

F

Augustine Louis Cauch (1823-1827)

“Stress is force acting over an oriented area at any point in a body”

This means stress depending on coordinate system?

http://upload.wikimedia.org/wikipedia/commons/7/7f/Augustin_Louis_Cauchy.jpg

Stress Concept

Difference orientation exhibit difference effect

))((0

,lim directionfacexxxx

A dA

df

A

f

Orientation of A denoted by normal unit vector n

If n is in the same direction with fx

If n is perpendicular with fx

Normal stress

Shear stress

Stress ConceptShear stress in x-y plane

yxy

x

y

x

A dA

df

A

f

0lim

Stress assiciated with 2 directions

“Stress is TENSOR”

Stress Concept

Normal stress + Tensile

Shear stress + on positive face - on negative face

In equilibrium

yxxy

Otherwise rotation of particle occurs

00 xzyyzxM xyyxz

Stress Concept

Mathematically

zzzyzx

yzyyyx

xzxyxx

z

z

z

y

z

x

y

z

y

y

y

x

x

z

x

y

x

x

directionface

dA

df

dA

df

dA

df

dA

df

dA

df

dA

dfdA

df

dA

df

dA

df

))((

Stress ConceptAs defined, stress is a tensor that independent of the coordinate system. We still need to solve stress relative to a particular coordinate system

The same stress can be defined relates to a particular coordinate system.

zzzyzx

yzyyyx

xzxyxx

zzzyzx

yzyyyx

xzxyxx

Stress Concept

Glue is stronger in shear than tension

Example

Conventional coordinate

0, xyxx A

F

Glued surface coordinate

??, xyxx

Practical Coordinate for Biomechanics System

Circulatory and Pulmonary systems

• Arteries• Capillaries• Veins• Bronchioles


Cell systemSaccular aneurysmUrinary bladder


Left Ventricle

Prolate Spheroidal Coordinate

Homework Assignment

Do research about prolate spheroidal and its application in biomechanic


Toroidal Coordinate

Homework Assignment

Do research about prolate spheroidal and its application in biomechanic

Stress Concept Summary

• Stress concept is mathematical• May be computed at each point in a continuum body with respect

to a coordinate systemThere are 9 components, but only 6 independent in 3-D (3 normal and 3 shear)

• Stress in different coordinate can be related by transformation relations

Stress Transformation

If we virtually cut the plane into


Azy

Ahypz

yyhyp

yadj

xopp

hypadj

hypopp

sec)(

seccos

/cos

sin

sinsintan

costancossec0

AA

AAAF

xyyy

yxxxxxx


2sin2cos22

2

2cos12sin

2

2cos1

sincossin2cos 22

xyyyxxyyxx

xx

yyxyxxxx

yyxyxxxx

2sin2cos22 xy

yyxxyyxxxx

Stress TransformationSimilar to y’ direction

sintancostan

cossinsec0'

AA

AAAF

xyyy

xyxxxyy

2cos2sin2 xy

xxyyxy

Stress TransformationAbove cutting plane doesn’t isolate the stress components on y’ surface, then

cossin

sintancostansec0

AA

AAAF

yxyy

xyxxyxx

2cos2sin2 xy

xxyyyx


sincos

costansintansec0'

AA

AAAF

yxyy

xyxxyyy

2sin2cos22 xy

xxyyyyxxyy


All above equations are 2D relationship between 2 coordinate systems that share common origin

Stress definition is not unique, it determined by coordinate system

• We just need to solve it once, then related to others by transformation relations

• Stress independent of material

Stress TransformationExercise

Consider 2D state of stress in the figure below.

?,, xyyyxx

Principal Stress and Maximum Shear

Are there any coordinate system that normal or shear stresses are maximum or minimum?

0at minimum

90at maximum

xx

xx


The maximum/minimum normal stress can be obtained by

022cos22sin2

xyyyxxxx

d

d

22tan

2cos

2sin

yyxx

xyp

p

p

The maximum or minimum normal stress occurs when

yyxx

xyp

2tan

2

1 1


Substitute p into transformation relations, we will get principal normal stress

Hxy

p

2sin

2

2

2 xyyyxxH

2

2

minmax/ 22 xyyyxxyyxx

xx

How about minmax/yy


02

1

2

yyxx

xyxyxxyy

pxy HH

Associated shear stress always zero at principal normal stress


Where is the location of maximum shear stress?

022sin22cos2

xyxxyyxy

d

d

xy

xxyys

s

s

2

2tan2cos

2sin

The maximum or minimum normal stress occurs when

xy

xxyys

2tan

2

1 1


The differ angle between p and s is /4 or 45 degree

2

2

2 xyyyxx

sxym

The normal stress at s is not zero

If principal stress occurs at = 0 then

2yyxx

m

Stress Summary

Concept of StrainMechanic :- study of forces and associated motionDynamic :- study to motion like velocity and acceleration

MotionVelocity

Acceleration

Solid Biomechanics :- interest in displacement

Concept of StrainDisplacement vector ≡ the difference between where we are and where we were

XtXxtXu ),(),(

Concept of Strain

Uniform force and homogeneous material uniform stress

Will the displacement uniform and can be used to measure the response ?

At fixed end: displacement is zeroAt free end: displacement is maximum

However

StrainX

u

constant

Green’s Strain

• Strain expression difference in other coordinates• Green’s strain is quadratic

Strain Distribution

2

2

2

2

1

2

1

2

1

Z

u

Z

uE

Y

u

Y

uE

X

u

X

uE

ZZZZ

YYYY

xxXX

Strain DistributionIf displacement is very small, then nonlinear term can be neglect

Strain distributionConsider motion

0,1 yx uXu

0,0,1 xyyyxx

Strain distributionConsider motion

0, yx ukXu

kxyyyxx 2

1,0,0

Strain distribution• Above is linearized relationship, good for small deform, practically

difference• From linearized rigid body motion doesn’t raise strain, practically wrong,

especially in biomechanics • In practice, we use the interpolation function to estimate displacements

between measurement points• Interpolation functions can be used to design number and placement of

markers for measuring displacement• 3D strain of heart wall = 4 points• 2D strain of heart wall = 3 points

Strain Transformation

xyxxyy

xy

yyxyxxyy

yyxyxxxx

22

22

22

sincos2

cossin2

coscossin2sin

sincossin2cos

Principal Strain

xy

xxyys

xyyyxx

xy

yyxx

xyp

xyyyxxyyxx

yyxx

2tan

2

1

2

2/tan

2

1

22

1

2

2

minmax/

1

2

2

minmax/minmax/

Strain Based Experimental• Track motion by multiple markers constant displacement vectors strain• X-Ray, MRI, Laser Doppler can be used to track motion of surface or embedded

markers• Strain gages

Constitutive Behavior

Constitution ≡ internal make up of the material

We have different respond from• Different material :- rubber vs metal• Same major compose different internal :- metal alloy, tendor vs cornea• Same material different environment

Constitution relations ≡ mathematical relations that describe the respond of a material and applied load

Constitutive BehaviorConstitutive relations quantifying of molecules, cells, tissues, organs, biomaterial are important for analysis and design

General characteristics Solid like

Fluid like

Material’s response to applied load Linear :- metal, bones (small strain)

Nonlinear :- elastomers, soft tissues (large strain)

Constitutive BehaviorElastic behavior:

• Stress and strain plot identical during loading and unloading• The material recover its original size and shape when load removed• Metal exhibit “elastic” under small strain• Tissues and rubber exhibit only “nearly elastic” under many condition• The different due to moving the structural proteins within viscous,

protreogly can dominated ground substance matrix

Homogeneous: behavior of material is independent of the position

Metal and rubber like are often homogeneousFiber reinforced composite is not homogeneousSoft tissues are composite – elastin collagens, proteoglycans, and water – can be considered as homogeneous in some circumstances - skin, lung parenchyma, myocardium bones, and brain tissues (in some cases)Some circumstances that needed to accounting for heterogeneity – intima, media, adventitia in blood vessels, and cortia vs cancellous bone

Constitutive BehaviorIsotropic: behavior of material is dependent on its orientation

• Metals exhibit isotropic under small strain• Rubber exhibit isotropic under large strain• Tendon is not exhibit isotropic (depend on oriented of Type I Collagen)• Most tissues exhibit anisotropic, very difficult to quantify

Constitutive BehaviorSummary

oWe characterize response of material in terms of•linearity•elasticity•homogeneity•Isotropy

oNo material is satisfy all conditions aboveo The constitutive relations and material behaviors that describe them

depend on the condition of interest

Constitutive BehaviorSummary

o Strain level of interest dictate the constitutive behavior of material• We seldom design implant biomaterials to exceed their yield

under action of in-vivo loado Soft tissues behave differently depending on whether they are

hydrated, heated excessively, or exposed to certain mediations

“Constitutive relations describe material behavior, not the material itself”

Nonlinear, anisotropic, and inelastic

Constitutive Behavior“Nonlinearity, inelasticity, anisotropy, and heterogeneity are common characteristic of soft tissues”

Bones

•Near linear behavior•Anisotropic•Heterogeneous

Constitutive BehaviorFocus on primary class of material behavior as LEHI

L inearE lasticH omogeneousI sotopic

Hookean LEHI behavior

1 1[ ( )] , ,

21 1

[ ( )] , ,2

1 1[ ( )] , ,

2

xx xx yy zz xy xy

yy yy xx zz xz xz

zz zz xx yy yz yz

TE G

TE G

TE G

Hooke’s Law

Constitutive Behavior

Hooke’s law for Transverse Isotropy

Hooke’s Law is for isotropic material

Some materials are anisotropic – wood, skin, bone, tissues, and heart muscle

When material behave in one-direction different from all directions in an orthogonal plane – transversely isotropic

1 1( ) , , ,

21 1

( ) , , ,2

1 1( ), ,

2

xx xx yy zz xy xy

yy yy xx zz xz xz

zz zz xx yy yz yz

E E G

E E G

E E G

2(1 )

EG

There are 5 unknowns ???

Good for linear, elastic, and homogeneous

Hooke’s law for orthotropyOrthotropy – response of material differs in three orthogonal directions – artery, and bones

If linear, elastic, and homogeneous

3121

1 2 3 12

3212

2 1 3 13

13 23

3 1 2 23

1 1, ,

2

1 1, ,

2

1 1, ,

2

xx xx yy zz xy xy

yy yy xx zz xz xz

zz zz xx yy yz yz

E E E G

E E E G

E E E G

13 31 23 3212 21

1 2 1 3 2 3

, , ,E E E E E E

Reading assignment section 2.6.5

Mechanical Properties of Bone

Most soft tissues exhibit nonlinear behavior over large strain, teeth and bones tends to exhibit a linearly elastic behavior over small strain

Bones

Stress, Strain, and Constitutive Relations (pptx)

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