MCEN 5023/ASEN 5012 Chapter 5
Constitutive Equations and Linear Elasticity
Fall, 2006
Linear Elasticity
Constitutive Equations
Stresses
Strains
Forces
Displacements
Linear Elasticity
Constitutive Equations Constitutive equations characterize material properties: Stress Strain Voltage Current Temperature Heat flux
Linear Elasticity
Hookes Law
f = kx
Generalized Hookes Law For infinitesimal small strain:
= Ee
Linear Elasticity
Generalized Hookes Law Reduction of total number of constants in Dijkl
1. Symmetry of stress and strain tensors
ij = ji
ekl = elk
Linear Elasticity
Generalized Hookes Law
Reduction of total number of constants in Dijkl 2. Energy Potential W
W = ij d eij
Linear Elasticity
Generalized Hookes Law Matrix Form
Linear Elasticity
Generalized Hookes Law Reduction of total number of constants in Dijkl 3. Orthotropic Materials
The material that has three orthogonal planes of symmetry
Collagen Molecule ( ~1nm)
Minerals in Holes
Minerals Between Molecules (~10nm)
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Linear Elasticity
Generalized Hookes Law 3. Orthotropic Materials
Linear Elasticity
Generalized Hooks Law 4. Transversely Isotropic Materials 2
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Linear Elasticity
Generalized Hooks Law 5. Isotropic Materials 2 components
Youngs Modulus Poissons Ratio
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Linear Elasticity
Constitutive Model for Linear Elastic Isotropic Materials
E G= 2(1 + )
1 e11 = [ 11 ( 22 + 33 )] E 1 e22 = [ 22 ( 11 + 33 )] E 1 e33 = [ 33 ( 11 + 22 )] E
e12 = 1 12 2G 1 e23 = 23 2G 1 e13 = 13 2G 1 xy = xy G 1 yz = yz G 1 xz = xz G
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Linear Elasticity
Constitutive Model for Linear Elastic Isotropic Materials
1 + ij kk ij eij = E E
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Linear Elasticity
Constitutive Model for Linear Elastic Isotropic Materials
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Linear Elasticity
Constitutive Model for Linear Elastic Isotropic Materials
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Linear Elasticity
Constitutive Model for Linear Elastic Isotropic Materials
1 + eij = ij kk ij E E
ij = 2Geij +
(1 + )(1 2 )
ekk ij
(1 + )(1 2 )
, G, are called Lam Constants.
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Linear Elasticity
Constitutive Model for Linear Elastic Isotropic Materials
ij = 2Geij +
(1 + )(1 2 )
ekk ij
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Linear Elasticity
Constitutive Model for Linear Elastic Isotropic Materials
Five elastic constants:
Only two of them are independent
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Linear Elasticity
Linear Elasticity What is linear elasticity about? P P
undeformed deformed
X2
X2
X1
X1
Question: If we apply a force on a material, what are the stresses, strains and displacements?
Object: Linear Elastic Body (Mr. Potato; Machine elements; Human hard tissue) (E and v are given) Input
Boundary conditions (Applied force; Applied displacement ) Output Stresses, strains, displacements, at each material point (x1,x2,x3)
???
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Linear Elasticity
Linear Elasticity
Things we want: Stresses, strains, displacements, at each material point (x1,x2,x3)
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Linear Elasticity
Linear Elasticity
If we take displacements as basic unknowns:
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Linear Elasticity
Linear Elasticity
Displacements can be obtained by integration of strains. Things we want: Stresses, strains, at each material point (x1,x2,x3)
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Linear Elasticity
Linear Elasticity
If we take stresses as basic unknowns:
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Linear Elasticity
Linear Elasticity
Boundary conditions: Prescribed displacements Prescribed tractions
X2
s
X1
1 u
2 u
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Linear Elasticity
Linear Elasticity Boundary conditions: Example
c b
X2
X1
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Linear Elasticity
Linear Elasticity Boundary conditions: Example
c b
X2
X1
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Linear Elasticity
Linear Elasticity Boundary conditions: Boundary conditions due to symmetry
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Linear Elasticity
Linear Elasticity Boundary conditions: Boundary conditions due to symmetry
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Linear Elasticity
Boundary conditions: Boundary conditions due to symmetry
Nanoindentation: three-sided pyramidal tips are most often used.
Berkovich tip (included angle, 142.3)
Cube corner tip (included angle, 90)
Indentation Impression
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Linear Elasticity
Nanoindentation:
1000 900
Force
800 700
Force (uN)
600 500 400 300 200 100 0 0 20 40 60 80
Time
g in d oa L
Indentation Depth (nm)
Loading: Elastic + Plastic Unloading: Elastic
Un loa din g
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Linear Elasticity
Boundary conditions: Boundary conditions due to symmetry
Top Views
3D perspective
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Linear Elasticity
Summary of Equations of Linear Elasticity
Kinematics: Strain Compatibility: Naviers: Constitutive: Equilibrium: Stress Compatibility (B-M)
Eij = eij =
1 (ui, j + u j ,i ) 2
eij ,kl + ekl ,ij eik , jl e jl ,ik = 0
Gui , jj + ( + G )u j , ji + f i = 0
1 + ij kk ij eij = E E
ij = 2Geij + ekk ij
ij , j + f i = 0
2 ij +
1 ,ij = ij X k ,k (X i , j + X j ,i ) 1 + 1
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Linear Elasticity
Methods for solving linear elasticity problem
Compatibility
Displacement
Kinematics
Strain
Boundary Conditions
Constitutive
Stress
S-SF Rel.
Stress Function
Equilibrium
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Linear Elasticity
Two Principles
1. Principle of Superposition
T1 T2 T1 T2
=
X2 X2
+
X2
X1
X1
X1
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Linear Elasticity
Two Principles
2. St. Venants Principles: Statically equivalent systems of forces produce the same stresses and strains within a body except in the immediate region where the loads are applied.
P = qL
If characteristic length of the area where a force is acting on is L, the dimension of immediate region is ~L.
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Linear Elasticity
Example: Simple Tension
Gauge Length
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Linear Elasticity
Example: Simple Tension
x2 x3 x1
Gauge Length, L0
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Linear Elasticity
Example: Simple Tension
x2 x3 x1
ij = 2Geij +
(1 + )(1 2 )
ekk ij
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Linear Elasticity
Example: Simple Tension
x2 x3 x1
ij = 2Geij +
(1 + )(1 2 )
ekk ij
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Linear Elasticity
Example: Simple Tension
x2 x3 x1
1 + ij kk ij eij = E E
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Linear Elasticity
Example: Simple Tension
x2 x3 x1
1 + ij kk ij eij = E E
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