BACKGROUNDS Two Models of Deformable Body continuum rigid-body spring deformation expressed in terms of field variables assembly of rigid-bodies connected by spring Distinct Element Method (DEM) simple treatment of failure: breakage of spring non-rigorous determination of spring constants breakage of spring deformable body particle modeling DEM analysis spring constant need to be determined in terms of material parameters
MATHEMATICAL INTERPLETATION OF RIGID-BODY SPRING MODEL Non-Overlapping Functions for Discretization ordinary FEM u 1 DEM u 1 u 2 u 2 smooth but overlapping functions are used for discretization characteristic functions of domain are used for discretization
PARTICLE DISCRETIZATION FOR FUNCTION AND DERIVATIVE Discretization of Function f(x) in terms of {ϕ (x)} form of discretization f d ( x) = f ϕ ( x) Discretization of Derivative f,i (x) in terms of {ψ (x)} form of discretization g d i ( x) = g ψ i ( x) different from {ϕ } E f ({f }) = (f ( x) f V minimize E f d ( x)) 2 ds derivative of {ϕ } is not bounded but integratable f = V 1 fϕ ds ϕ ds V g i β ( ) β ψ ϕ ds = f 1 ψ ds V V,i β optimal coefficients for given Voronoi blocks {ϕ } and Delaunay triangles {ψ }
1-D PARTICLE DISCRETIZATION x n x n+1 mother points f d ( x) = f ϕ ( x) average value taken over interval for ϕ is coefficient of characteristic function ϕ df dx ( x) = g( x) = g ψ ( x) y n-1 y n y n+1 average slope of interval for ψ is coefficient of characteristic function ψ middle point of neighboring mother points
2-D PARTICLE DISCRETIZATION dual domain decomposition Voronoi blocks for function Delaunay triangles for derivative function and derivative are discretized in terms of sets of non-overlapping characteristic functions, such that function and derivative are uniform in Voronoi blocks and Delaunay triangles,
COMPARISON OF PARTICLE DISCRETIZATION WITH ORDINARY DISCRETIZATION f 1 f 2 f 3 + + = f 1 ordinary discretization with linear function f 1 f 2 f 3 + + = f 2 x 1 f 3 particle discretization x 2 x 3 particle discretization of derivative derivative of particle discretization coincides with slope of plane which is formed by linearly connecting Voronoi mother points
PARTICLE DISCRETIZATION TO CONTINUUM MECHANICS PROBLEM Conjugate Functional J(u, σ) = V σ ij ε ij 1 2 σ ij c ijkl σ kl ds u σ i ij = = u i σ ij ϕ ψ hypo-voronoi for displacement Delaunay for stress: coefficients are analytically obtained by stationarizing J FEM-β: FEM with Particle Discretization stiffness matrix of FEM-β coincides with stiffness matrix of FEM with uniform triangular element including rigid-body-rotation, FEM-β gives accurate and efficient computation for field with singularity
FAILURE ANALYSIS OF FEM-β stiffness matrix of FEM-β [k [k [k 11 21 31 ] ] ] [k [k [k 12 22 32 ] ] ] [k [k [k 13 23 33 ] ] ] strain energy due to relative deformation of Ω 1 and Ω 2 through movement of Ω 3 for indirect interaction for direct interaction Ω 1 x 1 ( T k ] [k ]) [ ji = ij direct [k12] = [k12 ] + [k indirect 12 ] x 2 Ω 2 Ω 3 x 3 cut two springs of direct and indirect interaction together or separately, according to certain failure criterion of continuum
FAILURE MODELING BY BREAKING SPRINGS region with reduced strain energy broken edge P O Φ 2 Φ 3 C R J-integral 0.000520 0.000510 0.000500 0.000490 Analytical FEM-beta FEM A Φ 1 Q B 0 20000 40000 60000 80000 100000 0.000480 Number of Elements relative error of J-integral is around a few percents
EXAMPLE PROBLEM Simulation of Crack Growth Check of Convergence displacement strain/strain energy Pattern of Crack Growth can small difference in initial configuration cause large difference in crack growth? uniform tension
CONVERGENCE OF SOLUTION 0.006 0.006 0.004 0.004... ξ =2.09... Best Performance ηd ηd 0.002 0.002 0.000 2.0 2.1 2.2 2.3 2.4 ξ 0.000 10 5 10 4 NDF 10 3 10 2 displacement norm w.r.t. mesh quality displacement norm w.r.t. NDF displacement norm: η d = B u û B û 2 2 dv dv
CHECK OF CRACK GROWTH pattern of crack growth ε yy evolution of normal strain distribution
EXAMPLE: PLATE WITH 3 HOLES simulation of crack growth: crack stems from holes case a case b slight difference in location of 3 rd hole
DIFFERENCE IN CRACK GROWTH: DISTRIBUTION OF NORMAL STRAIN case a case b
SIMULATION OF BRAZILIAN TEST Brazilian test
SIMULATION OF FOUR POINT BENDING TEST four point bending test σ 11 0.01 0.00-0.01
FOUR POINT BENDING WITH IDEARLY HOMOGENEOUS MATERIALS σ 11 0.01 0.00-0.01 FEM-β puts two source of local heterogeneity, 1) mesh quality for particle discretization and 2) crack path along Voronoi boundary. An ideally homogeneous material which is modeled with best mesh quality sometimes fail to simulate crack propagation.
CONCLUDING REMARKS Particle Discretization discretization scheme using set of non-overlapping characteristic functions Continuum Mechanics Problem essentially same accuracy as FEM with uniform strain applicable to non-linear plasticity Failure Analysis simple but robust treatment of failure Monte-Carlo simulation for studying local heterogeneity effects on failure - candidates of failure patterns are pre-determined by spatial discretization