# U.S. South America Workshop. Mechanics and Advanced Materials Research and Education. Rio de Janeiro, Brazil. August 2 6, Steven L.

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1 Computational Modeling of Composite and Functionally Graded Materials U.S. South America Workshop Mechanics and Advanced Materials Research and Education Rio de Janeiro, Brazil August 2 6, 2002 Steven L. Crouch Department of Civil Engineering University of Minnesota

2 Research Group Jianlin Wang Sonia Mogilevskaya Yun Huang Lisa Gordeliy Benoît Legros Hamid Sadraie

3 Fiber-Reinforced Composite Materials 2-D model y matrix unit cell x interphases fiber

4 Standard Numerical Methods Finite element method Boundary element method Finite element mesh (after Wacker et al., 1998)

5 Our Approach Direct boundary integral method Approximation of the unknowns by Fourier Series or Spherical Harmonics Complex (for plane problems) or real variables formalism

6 Direct Boundary Integral Method u, n u s,σ, σ n s s Governing differential equations + Boundary conditions Enrico Betti n Fundamental solution Fundamental solution (e.g. point force in plane) L Integral identities (e.g. reciprocal theorem) Boundary Integral Equation

7 Fourier Series On the Propagation of Heat in Solid Bodies, 1807 Jean Baptiste Joseph Fourier f ( x) = 1 2 m= 1 a0 + an cosmx + bn sin mx m= 1 sin mx, cosmx [ 0, 2π ] z j R j θ j τ a complete orthogonal system over [ c ϕ ( x) + c ϕ ( x) + c N ϕ ( x) ] 0 f ( x) N N

8 Spherical Harmonics William Thomson (Lord Kelvin) T&T Treatise on Natural Philosophy (1867) Y + n Surface harmonics ( θ, ϕ ) = A P ( cosθ ) n m m m { A cosmϕ + B sin mϕ} T ( cosθ ) n n n m= 1 n n A complete orthogonal system Over the unit sphere (Lyapunov, 1899) Peter Guthrie Tait f ( θ, ϕ ) ( θ, ϕ ) = n=0 Y n

9 Algorithm (perfect bond between the constituents) 1. Fix number of terms in Fourier series 2. Solve linear algebraic system (error δ1) 3. Estimate an error for each inclusion (error δ2) 4. Increase number of terms in Fourier series by some value 5. Steps 2-4 repeated until error δ2 is met 6. Displacements, stresses and strains calculated in the matrix and the inclusions

10 Error Estimation Use the displacements at the boundaries as the unknowns to form a system of equations Calculate stresses at the boundaries Compare the stresses at a number of uniformly distributed points t 2 t 1 max t= t 1,..., t K { ε } t δ 2 t K

11 Numerical Example Multiple cracks and circular inhomogeneities in an infinite domain subjected to uniaxial tension in the x direction; contours of σ xx

12 Imperfect Interface Models Spring-type interface Partial debonding p Γ j p R j p z j p Γ j p R j p z j debonding p p µ j ν j p p µ j ν j Explicit presence of interphase layers p Γj1 p R j 1 p z j 1 p Γ p j0 Rj 0 p z j 0 p p µ j 0 ν j0 p p µ j1 ν j1 p Γ jn p R jn p Rj 0 p z j p p µ j 0 ν j 0 p Γ j0 p µ j (r) p ν j (r)

13 Numerical Example σ 0 material ν E/ Ematrix inclusion y matrix compl.co x stiff co δ = 10 ; δ 2 = terms

14 Numerical Results 2.5 perfect bond stiff coating 2.0 interphase compliant coating interphase 1.5 σ eff 1.0 matrix inclusion matrix

15 Inclusion with Interface Crack (Toya, 1974) σ 0 y ϕ µ,ν µ', ν ' 2α x

16 Computed radial and shear stresses (open circles) compared with analytical solution (solid lines); N=180 σ rr /σ µ ' = 44.2GN / m, ν ' = 0.22, µ = 2.39GN / m α = ϕ = 30 σ rθ /σ , ν ' = Angle, (a) θ Angle, (b) θ

17 Computed radial and shear displacement discontinuities (open circles) compared with analytical solution (solid lines); N= u r 0.10 / a, u θ 0.10 / a, Angle, (a) θ Angle, (b) θ

18 Example debonding of single inclusion σ = σ yy 0 y µ,ν x µ ' = µ, ν ' = ν Smooth interface: Stippes, Wilson, and Krull (1966) Rough interface: Hussain and Pu (1971)

19 Radial stress in zone of contact for smooth inclusion: solid line is analytical solution; open circles are computed results /σ 0 σ rr Angle,θ

20 Circumferential stress for smooth inclusion: solid line is analytical solution; open circles are computed results /σ 0 σ θθ Angle, θ -1.0

21 Comparison of computed radial (a) and shear (b) stresses for rough inclusion: solid lines are results from Fourier series approach /σ 0 σ rr 0.3 σ rθ /σ Angle, θ (a) Angle, θ (b)

22 Modeling evolving damage Initial attempt: Increment loading Use Mohr-Coulomb criterion σ r θ c σ rr tanφ ; σ rr T φ σ rθ c T c σ rr φ (c is cohesion; is angle of friction; T is tensile strength) Allow slip, separation (cracking); prohibit overlapping of displacement discontinuities during iteration

23 Issues Crack initiation and propagation are problems: If no crack is present then no stress raiser exists; Small crack produces locally high stresses crack grows too much using tensile stress criterion Cannot calculate stress intensity factors Better to integrate stresses over a characteristic length? (What should this be?) Work is continuing

24 Effect of Free Boundary u, n u s,σ, σ n s s single inclusion Melan s fundamental solution (point force in a half-plane) L n Just few results were available and they were contradictory FS = FS + M K FS ad

25 A Single Inclusion Close to the Boundary µ matrix = 1.0, ν = 0.3; µ = 100.0, ν = 0.3, R / d = matrix inc inc terms ( σ σ )/ σ xx Contours of 1 2

26 40 Regularly Distributed Inclusions µ matrix = matrix inclusion inclusion 1.0, ν = 0.15; µ = 10.0, ν = 103~117 terms 0.35 ( σ σ )/ σ xx Contours of 1 2

27 200 Randomly Distributed Inclusions ( σ σ )/ σ xx Contours of 1 2

28 Finite Domain with Circular Boundary 2.00 p = p = p =1. 4 Distribution of σ 1 σ 2

29 Finite Domain with Convex Polygonal Boundaries D A C B Embed a domain of interest in a fictitious circular domain Apply load at the boundary of the circle to satisfy (in a least squares sense) boundary conditions on the physical domain

30 Effective (macroscopic properties) 3.00 γ = 10,000 ( rigid inclusion) γ = γ = µ i /µ 0 γ = 5.0 E eff /E γ = 2.0 Labuz & Carvalho (1996) γ = 1.0 γ = 0.5 γ = Fiber volume ratio γ = 0 ( hole)

31 Effective Properties (epoxy matrix, E-glass fiber) 0.34 ; 12 8, 6, 4, 0.34 ; ; 84 int int = = = = = = erphase erphase matrix Matrix fiber fiber GPA E GPA E GPA E ν ν ν m h V m R f fiber µ µ % ; 8.5 = = = y D C A B x b

32 Variation of Effective Young s Modulus E inter (GPA) µ m µ m µ m µ m h = 1. 0 h = 0.5 h = 0. 1 h =

33 Fast Solvers V. Rokhlin, 1985 L. Greengard and V. Rokhlin, 1987 Data information 10,000 inclusions with 0.5 filling ratio Computation time (1.5GHz CPU) Direct method 1.5 months Single-level FMA 6 hours Multi-level FMA 2 hours

34 5,000 inclusions of random sizes and elastic properties under a uniaxial stress at infinity σ =1.0 ; Contours of σ xx xx

35 Comparison of the Algorithm Complexity direct algorithm single-level fast multipole algorithm multi-level fast multipole algorithm 10 5 CPU time in seconds number of degrees of freedom

36 Modeling of Graded Composite Materials 1.0 µ particle /µ matrix = 10; Contours of σ yy 1.0

37 Modeling of Graded Composite Materials (continued) 1.0 µ particle /µ matrix = 10; Contours of σ yy 1.0

38 Linear Viscoelasticity; Boltzmann model E ve E e σ η ve σ ε e ε ve One dimensional representation Stress Constant strains applied Strain Constant stresses applied Time Time Relaxation curve Creep curve

39 Example - Two inclusions and two holes C D E 1,ν 1 E 2,ν2 Stress (MPa) C-ANSYS C-Boundary integral D-ANSYS D-Boundary integral Time (second) Ee,Eve,ν, η 0.6 Computational costs Displacement (mm) C-ANSYS C-Boundary integral D-ANSYS D-Boundary integral Boundary integral method: 8 minutes, terms in Fourier series ANSYS: 11 hours, 20,375 elements Time (second)

40 25 Elastic Inclusions in a Viscoelastic Plane d y G i, ν i σ 0 r i G, G ve, ν, γ (= θ λ = θ µ ) x σ 0

41 Some Results 2.0 γ t / = t / γ = 0.01 t / γ = yy 0 σ /σ x / d σ yy / σ 0 on the line y = 0

42 25 Spherical Cavities in y-z Plane z y x

43 . 0 Contours of σ / σ (0) near yy yy cavity # z y

44 Transient Heat Conduction in Composite Materials T ( x, t) Integral Identity = α t 0 Ω T ( s, τ ) G( s, τ ) n Method of Solution G( s, τ ) T ( s, τ ) dsdτ n T(x,t) is the temperature at point x at time t G(s,t) is the Green s Function Analytical space integration Approximation of temperature and flux on the boundary in Fourier series Laplace transform in time to solve boundary integral equations Verification of the results Results for one disc and one cavity agree with solution by Carslaw and Jaeger (Conduction of Heat in Solids, 1946)

45 Future Work Microcontinuum models Objectives Extend existing continuum models to account for microscopic space scale and strain gradient effects (nonlocal constitutive behavior) Examine well-established microcontinuum theories (e.g. Mindlin s microstructure theory) Develop a computational basis for modeling micro- and macroscopic behavior of materials with microstructure Benefits Incorporate size effects Address boundary layer effects Obtain more realistic results for critical regions of high deformation gradients

46 Other Future Work Continue to work on 3D Loosening of Inclusions Viscoelasticity Transient thermoelasticity Functionally graded materials?

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