2. numerical modeling
advertisement
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.
