2. numerical modeling (1).ppt
Transcript of 2. numerical modeling (1).ppt
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Elastic Theory of
Fractures
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Idealization of fracture for
mechanical analysis
Infinite length in x3 direction
Shape is constant in x3 direction
Homogeneous, isotropic and linear elastic
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Stress tensor
Stress tensor at any point depends on
Position
Geometry of crack
Traction on crack faces
Remote state of stress
ij = f ij (x1, x2, a and boundary conditions)
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Displacements depend on
Position
Crack geometry
Traction on crack facesRemote stress
Elastic moduli for stress boundary-value
problemui=gi(x1,x2,a,m,n and boundary conditions)
E=2m (1+n)
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Definitions
Boundary Value Problem
Stress, displacement and mixed
TractionForce per unit area on a surface
Cauchy’s formula
Ti=ijn j
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How to solve a BVP
Constitutive
Linear-elastic
EquilibriumQuasi-static
Compatibility
Can combine with constitutive relations to get
harmonic form for first stress invariant
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Solving the system in 2D
3 equations
2 equilibrium
1 compatibility
3 unknowns
Plane strain: 11, 12, 22
Boundary conditions for cracks
Stresses must match the far-field at x1 or x2 -> ∞
Stresses must match crack-face tractions tractions at
x1=0+, |x2|≤a
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Airy’s stress function
U=U(x1, x2, a, r 11,
r 12,
r 22,
c11,
c12)
If U has the following relations, the equilibriumconditions are satisfied
Substitute these into
compatibility and getbiharmonic for U
11 2U
x2
2 , 11 2U
x1 x2
, 22 2U
x12
4
U 0
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Making the Airy’s stress function
(even more) complex
Muskhelishvili: The Airy stress
function can be expressed as two
functions of the complex variable
Z ?
Re[ ] ? Im[ ] ?
Why? To make finding solutions
easier.
U ( z ) 12Re[ z ( z ) ( z )]
Nikoloz
Muskhelishivili
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Using the complex Airy’s
functions
Take derivatives of the Airy’s stress functions to
get stresses
Use constitutive relations to get strainsThen find and to match boundary conditions
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Westergaard function
H. M. Westergaard
(1939): reduced the two
unknown functions to
one function, m , for a
crack using symmetry
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The stress function
m(z) = Am[(z2-a2)1/2-z] + BmzDI (11
r -11c) 1/2(11
r +22r )
Am= -iDII = -i(12r -12
c ) Bm= 0
-iDIII -i(13r -13
c) 23r -i13
r
First part:crack contribution
Second part: remote loadcontribution
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But aren’t there simpler equations
out there?
Simpler relations have been
developed for the stress fields near
crack tips.
The Westergaard function gives the
stress field everywhere including the
crack tips.
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Boundary Element
Method
•Becker 1992. The Boundary Element Method
in Engineering: A Complete course, Mc Graw
Hill
•Crouch and Starfield, 1990 Boundary Element
Method in Solid Mechanics with applications inrock mechanics and geological engineering,
Unwin Hyman
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Discretization
Deformation of each small bit within the
body is solved analytically
Putting the bits together relies on
computation power of modern processors
Consider influence of neighboring bits
Principle of superposition
Discretization introduces errorHow could you assess or minimize this error?
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Solving a BVP
Prescribe
Geometry
Boundary conditions (stress or displacement)
Constitutive properties
Solve for stress and displacement/strain
throughout the body
Solution must be true to prescribed conditions
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What are the different methods?
Finite Element Method
(FEM)
Boundary ElementMethod (BEM)
Discrete Element
Method (DEM)
Finite DiffferenceMethod (FDM)
From Becker
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Finite element method
Approximates the governing
differential equations by solving the
system of linear algebraic equations
Mesh the body into equantvolumetric or planar elements
Computationally expensive with fine
grids but has a sparse stiffness
matrix Handles heterogeneous materials
well
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Boundary element method
Governing differential equations are
transformed into integrals over
boundaries. These integrals are
expressed as a system of linearalgebraic equations.
Boundaries discretized into linear or
planar equal sized elements
Computationally cheaper than FEM(fewer elements) but has a full and
asymmetric matrix
Clunky for heterogeneous materials
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Discrete Element Method
Discretizes the body intoparticles in contact
Analyzes the contactmechanics between eachparticle
Computationally expensivewith many elements
Handles heterogeneity very
well Useful for specific problems
e.g. fault gouge,deformation bands
Caveat: only use whencontact mechanicsdominate the deformation
Does not incorporate stress
singularity at crack tips
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Finite Difference Method
Solves governing differential equations bydifferencing method
Mesh the body -- solves at internal points
Computationally cheap and easy to program Cannot accurately incorporate irregular
geometries or regions of stress concentration
Appropriate for contact problems,heat and fluidflow
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Which method best for fractures?
Capturing the 1/r 1/2 crack tip singularity
Fracture propagation
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Crack tip singularity
Finite Element?Special grid designed to
capture the 1/r 1/2 crack tipsingularity
awkward and expensive Boundary Element?Each element is a
dislocation
A series of equal length
dislocations automaticallyincorporates the r -1/2 cracktip singularity
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Fracture Propagation
Finite Element?Fracture must be
remeshed and thespecial crack tip
elements moved to anew location
awkward
Boundary Element? Add another element to
the tip of the fracture
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Complicated fracture geometry
Boundary Element is hands down the best
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Poly3d
IGEOSS
3D
Complex fracturesLinear elastic homogeneous rheology
Frictional faults
Nice user interface
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Flamant’s solution
Deformation within a
half space due to two
point loads
One normal
One shear
wikipedia
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Distributed load
Superpose Flamant’s
solution as you
integrate over the
distributed load
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Rigid Die problem
What are the tractions that couldproduce a uniform displacement?
Displacement along boundary
element i due to tractions on allother elements, j=1 to N
Bij is the matrix of influencecoefficients
Effects of discretization and
symmetry u y
i
( x
i
,0) B
ij
T y
i
j1
N
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Fictitious Stress Method
Based on Kelvin’s problem
A point force within an infinite elastic solid
Similar to Flamant’s
Can be used for bodies of any shapeLeads to constant tractions along each element.
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Displacement discontinuity
method
Constant
displacements
along each element
Better for bodies
with cracks
incorporates the
singularity indisplacement across
the crack
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Displacement discontinuity
method
Displacement has a 1/r singularity
A series of constant displacement elements
replicates the 1/r
1/2
stress singularity at thecrack tip.
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Numerical procedure
The stresses on
the ith element due
to deformation on
the jth element
A is the boundary
influence
coefficient matrix
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Numerical procedure
Sum the effects for
all elements
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Numerical procedure
If you know
displacements
(displacement boundary
value problem) the
solution is found quickly.
If you have a mixed or
stress boundary value
problem, you need to
invert A to find the
displacements
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Numerical procedure
Once you know
displacements and stresses
on all elements, you can find
the displacements at anypoint within the body.
Flamant’s solution
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Frictional slip
|t|=c-m
Inelastic deformation
Converge to solution
Penalty Method
Direct solver
Apply a shear and normal stiffness to elements to
prevent interpenetration (e.g. Crouch and Starfield, 1990)Complementarity Method
Apply inequalities
Implicit solver (e.g. Maerten, Maerten and Cooke, 2010)
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Convergence for frictional slip
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What about 3D elements
Cominou and Dundurs developed angular
dislocation.
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Boundary integral method
Uses reciprocal theorem (Sokolnikoff) to solve
for unknown boundary conditions.