2. numerical modeling
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Transcript 2. numerical modeling
Elastic Theory of
Fractures
Idealization of fracture for
mechanical analysis
Infinite length in x3 direction
Shape is constant in x3 direction
Homogeneous, isotropic and linear elastic
Stress tensor
Stress tensor at any point depends on
Position
Geometry of crack
Traction on crack faces
Remote state of stress
ij = fij (x1, x2, a and boundary conditions)
Displacements depend on
Position
Crack geometry
Traction on crack faces
Remote stress
Elastic moduli for stress boundary-value
problem
ui=gi(x1,x2,a,m,n and boundary conditions)
E=2m (1+n)
Definitions
Boundary Value Problem
Stress, displacement and mixed
Traction
Force per unit area on a surface
Cauchy’s formula
Ti=ijnj
How to solve a BVP
Constitutive
Linear-elastic
Equilibrium
Quasi-static
Compatibility
Can combine with constitutive relations to get
harmonic form for first stress invariant
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
Airy’s stress function
U=U(x1, x2, a, r11, r12, r22, c11, c12)
If U has the following relations, the equilibrium
conditions are satisfied
2U
2U
2U
11 2 , 11
, 22 2
x 2
x1 x 2
x1
Substitute these into
compatibility and get
biharmonic for U
U 0
4
Making the Airy’s stress function
(even more) complex
Muskhelishvili: The Airy stress
function can be expressed as two
functions of the complex variable
U(z) Re[z (z) (z)]
1
2
Z ?
Re[ ] ? Im[ ] ?
Why? To make finding solutions
easier.
Nikoloz
Muskhelishivili
Using the complex Airy’s
functions
Take derivatives of the Airy’s stress functions to
get stresses
Use constitutive relations to get strains
Then find and to match boundary conditions
Westergaard
function
H. M. Westergaard
(1939): reduced the two
unknown functions to
one function, m , for a
crack using symmetry
The stress function
m(z) = Am[(z2-a2)1/2-z] + Bmz
DI
Am= -iDII =
-iDIII
(11r-11c)
1/2(11r+22r)
-i(12r-12c )
-i(13r-13c)
Bm=
First part:crack contribution
Second part: remote load
contribution
0
23r-i13r
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.
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 in
rock mechanics and geological engineering,
Unwin Hyman
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 error
How could you assess or minimize this error?
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
What are the different methods?
Finite Element Method
(FEM)
Boundary Element
Method (BEM)
Discrete Element
Method (DEM)
Finite Diffference
Method (FDM)
From Becker
Finite element method
Approximates the governing
differential equations by solving the
system of linear algebraic equations
Mesh the body into equant
volumetric or planar elements
Computationally expensive with fine
grids but has a sparse stiffness
matrix
Handles heterogeneous materials
well
Boundary element method
Governing differential equations are
transformed into integrals over
boundaries. These integrals are
expressed as a system of linear
algebraic 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
Discrete Element Method
Caveat: only use when
Discretizes the body into
contact mechanics
particles in contact
dominate the deformation
Analyzes the contact
Does not incorporate stress
mechanics between each
singularity at crack tips
particle
Computationally expensive
with many elements
Handles heterogeneity very
well
Useful for specific problems
e.g. fault gouge,
deformation bands
Finite Difference Method
Solves governing differential equations by
differencing 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 fluid
flow
Which method best for fractures?
Capturing the 1/r1/2 crack tip singularity
Fracture propagation
Crack tip singularity
Finite Element?
Special grid designed to
capture the 1/r1/2 crack tip
singularity
awkward and expensive
Boundary Element?
Each element is a
dislocation
A series of equal length
dislocations automatically
incorporates the r-1/2 crack
tip singularity
Fracture Propagation
Finite Element?
Fracture must be
remeshed and the
special crack tip
elements moved to a
new location
awkward
Boundary Element?
Add another element to
the tip of the fracture
Complicated fracture geometry
Boundary Element is hands down the best
Poly3d
IGEOSS
3D
Complex fractures
Linear elastic homogeneous rheology
Frictional faults
Nice user interface
Flamant’s solution
Deformation within a
half space due to two
point loads
One normal
One shear
wikipedia
Distributed load
Superpose Flamant’s
solution as you
integrate over the
distributed load
Rigid Die problem
What are the tractions that could
produce a uniform displacement?
Displacement along boundary
element i due to tractions on all
other elements, j=1 to N
Bij is the matrix of influence
coefficients
Effects of discretization and
symmetry
N ij
uy (x,0) B Ty
i
i
j1
i
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 shape
Leads to constant tractions along each element.
Displacement discontinuity
method
Constant
displacements
along each element
Better for bodies
with cracks
incorporates the
singularity in
displacement across
the crack
Displacement discontinuity
method
Displacement has a 1/r singularity
A series of constant displacement elements
replicates the 1/r1/2 stress singularity at the
crack tip.
Numerical procedure
The stresses on
the ith element due
to deformation on
the jth element
A is the boundary
influence
coefficient matrix
Numerical procedure
Sum the effects for
all elements
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
Numerical procedure
Once you know
displacements and stresses
on all elements, you can find
the displacements at any
point within the body.
Flamant’s solution
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)
Convergence for frictional slip
What about 3D elements
Cominou and Dundurs developed angular
dislocation.
Boundary integral method
Uses reciprocal theorem (Sokolnikoff) to solve
for unknown boundary conditions.