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

PositionCrack geometryTraction on crack facesRemote stressElastic moduli for stress boundary-value

problemui=gi(x1,x2,a,, and boundary conditions)

E=2 (1+)

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, r

12, r22, c

11, c12)

If U has the following relations, the equilibrium conditions are satisfied

Substitute these into compatibility and get biharmonic for U

€

σ11=∂2U∂x2

2 ,σ11=−∂2U

∂x1∂x2

,σ22 =∂2U∂x1

2

€

∇4U = 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 φ(z)+χ(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] + BmzI (11

r-11c) 1/2(11

r+22r)

Am= -iII = -i(12r-12

c ) Bm= 0

-iIII -i(13r-13

c) 23r-i13

r

First part:crack contribution Second part: remote load

contribution

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

Discretizes the body into particles in contact

Analyzes the contact mechanics between each particle

Computationally expensive with many elements

Handles heterogeneity very well

Useful for specific problems e.g. fault gouge, deformation bands

Caveat: only use when contact mechanics dominate the deformation

Does not incorporate stress singularity at crack tips

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

IGEOSS3DComplex fracturesLinear elastic homogeneous rheologyFrictional faultsNice 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

€

uy

i

(xi

,0) = Bij

Ty

i

j=1

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 in displacement across the crack

Displacement discontinuity method

Displacement has a 1/r singularityA 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

=c- 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.