2. numerical modeling (1).ppt

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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 = f ij (x1, x2, a and boundary conditions)



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)



Definitions

Boundary Value Problem

Stress, displacement and mixed

TractionForce per unit area on a surface

Cauchy’s formula

Ti=ijn j



How to solve a BVP

Constitutive

Linear-elastic

EquilibriumQuasi-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, 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



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



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



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] + 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



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 inrock 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 errorHow 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 ElementMethod (BEM)

Discrete Element

Method (DEM)

Finite DiffferenceMethod (FDM)

From Becker



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



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



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



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



Which method best for fractures?

Capturing the 1/r 1/2 crack tip singularity

Fracture propagation



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



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



Complicated fracture geometry

Boundary Element is hands down the best



Poly3d

IGEOSS

3D

Complex fracturesLinear 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 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



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.



Displacement discontinuity

method

Constant

displacements

along each element

Better for bodies

with cracks

incorporates the

singularity indisplacement across

the crack



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.



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 anypoint 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.

2. numerical modeling (1).ppt

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