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Stress, Strain, and Constitutive Relations
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
Stress, Strain, and Constitutive Relations
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?
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
Practical Coordinate for Biomechanics System
Cell systemSaccular aneurysmUrinary bladder
Practical Coordinate for Biomechanics System
Left Ventricle
Prolate Spheroidal Coordinate
Homework Assignment
Do research about prolate spheroidal and its application in biomechanic
Practical Coordinate for Biomechanics System
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
Stress Transformation
Azy
Ahypz
yyhyp
yadj
xopp
hypadj
hypopp
sec)(
seccos
/cos
sin
sinsintan
costancossec0
AA
AAAF
xyyy
yxxxxxx
Stress Transformation
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
Stress Transformation
sincos
costansintansec0'
AA
AAAF
yxyy
xyxxyyy
2sin2cos22 xy
xxyyyyxxyy
Stress Transformation
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
Principal Stress and Maximum Shear
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
Principal Stress and Maximum Shear
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
Principal Stress and Maximum Shear
02
1
2
yyxx
xyxyxxyy
pxy HH
Associated shear stress always zero at principal normal stress
Principal Stress and Maximum Shear
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
Principal Stress and Maximum Shear
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