Computational Modeling of Composite and Functionally Graded - PowerPoint PPT Presentation

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

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

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

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

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

Direct Boundary Integral Method Governing differential equations + Boundary conditions σ , σ s u n u , , s n s Enrico Betti Fundamental solution n Integral identities L (e.g. reciprocal theorem) Fundamental solution 1823-1892 (e.g. point force in plane) Boundary Integral Equation

Fourier Series Jean Baptiste Joseph Fourier On the Propagation of Heat in Solid Bodies, 1807 ∞ ∞ 1 ∑ ∑ = + + f ( x ) a a cos mx b sin mx 0 n n 2 = = m 1 m 1 sin mx cos , mx a complete orthogonal system over [ ] π 0 , 2 [ ] − ϕ + ϕ + ϕ → f ( x ) c ( x ) c ( x ) … c ( x ) 0 1 1 2 2 N N → ∞ N τ R 1768-1830 j θ z j j

Spherical Harmonics Peter Guthrie Tait ‘T&T’ Treatise on Natural Philosophy William Thomson (1867) (Lord Kelvin) Surface harmonics ( ) ( ) θ ϕ = θ Y , A P cos n n n { } n ( ) ∑ + ϕ + ϕ θ m m m A cos m B sin m T cos n n n = m 1 1831-1901 A complete orthogonal system Over the unit sphere 1824-1907 (Lyapunov, 1899) ∞ ( ) ( ) ∑ θ ϕ = θ ϕ f , Y , n = n 0

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

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 t uniformly distributed points 2 t 1 { } ε ≤ δ max t K t 2 = t t ,..., t 1 K

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

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

Numerical Example σ σ σ material ν E/ E 0 0 0 matrix inclusion 0.2537 38.9 matrix 0.2647 1.0 y y y compl.co 0.2647 0.067 stiff co 0.2647 20.0 x x x − − δ = δ = 6 3 10 ; 10 1 2 6 – 9 terms

Numerical Results 2.5 perfect bond stiff coating 2.0 compliant coating interphase interphase 1.5 σ eff matrix matrix inclusion 1.0 0.5 0.0 0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0 8.8 9.6 10.4 11.2

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

Computed radial and shear stresses (open circles) compared with analytical solution (solid lines); N=180 µ = ν = µ = ν = 2 2 ' 44 . 2 GN / m , ' 0 . 22 , 2 . 39 GN / m , ' 0 . 35 α = ϕ = � 30 6.5 6.5 6.0 6.0 5.5 5.5 5.0 5.0 4.5 4.5 4.0 4.0 σ rr / σ σ θ / σ 0 r 0 3.5 3.5 3.0 3.0 2.5 2.5 2.0 2.0 1.5 1.5 1.0 1.0 0.5 0.5 0.0 0.0 30 60 90 120 150 180 210 240 270 300 330 360 30 60 90 120 150 180 210 240 270 300 330 360 -0.5 -0.5 θ θ Angle, Angle, -1.0 -1.0 (a) (b)

Computed radial and shear displacement discontinuities (open circles) compared with analytical solution (solid lines); N=180 0.18 0.18 0.16 0.16 0.14 0.14 0.12 0.12 0.10 0.10 ∆ − − 3 ∆ 3 u r / a , 10 u θ / a , 10 0.08 0.08 0.06 0.06 0.04 0.04 0.02 0.02 0.00 0.00 30 60 90 120 150 180 210 240 270 300 330 360 30 60 90 120 150 180 210 240 270 300 330 360 Angle, θ Angle, θ (a) (b)

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

Radial stress in zone of contact for smooth inclusion: solid line is analytical solution; open circles are computed results 0.7 0.6 0.5 − σ rr / σ 0 0.4 0.3 0.2 0.1 0.0 0 5 10 15 20 25 θ Angle,

Circumferential stress for smooth inclusion: solid line is analytical solution; open circles are computed results 3.0 2.5 2.0 1.5 σ θθ / σ 0 1.0 0.5 0.0 0 15 30 45 60 75 90 Angle, θ -0.5 -1.0

Comparison of computed radial (a) and shear (b) stresses for rough inclusion: solid lines are results from Fourier series approach 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 − σ rr / σ σ θ / σ 0 r 0 0.3 0.3 0.2 0.2 0.1 0.1 0.0 0.0 0 5 10 15 20 25 0 5 10 15 20 25 Angle, θ θ Angle, (a) (b)

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

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 …

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

A Single Inclusion Close to the Boundary µ = ν = µ = ν = = 1 . 0 , 0 . 3 ; 100 . 0 , 0 . 3 , R / d 0 . 99 matrix matrix inc inc 89 terms σ ∞ σ − σ ( ) / Contours of 1 2 xx

40 Regularly Distributed Inclusions µ = ν = µ = ν = 1 . 0 , 0 . 15 ; 10 . 0 , 0 . 35 matrix matrix inclusion inclusion 103~117 terms σ ∞ σ − σ ( ) / Contours of 1 2 xx

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

Finite Domain with Circular Boundary p = 1.0 0 2.00 2 1.50 1.00 0.50 0.00 3 1 p = 1.0 -0.50 1 -1.00 Distribution of -1.50 4 σ − σ 1 2 -2.00 -2.00 -1.50 -1.00 -0.50 0 1.50 2.00 p = 1. 0

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

Effective (macroscopic properties) 3.00 γ = γ = 50 10 , 000 ( rigid inclusion ) 2.5 0 γ = µ / µ i 0 γ = 5 . 0 2.0 0 E eff /E 0 1.5 0 γ = 2 . 0 γ = 1 . 0 1.00 Labuz & γ = 0 . 5 Carvalho 0.50 γ = 0 . 2 (1996) γ = 0 ( hole ) 0.0 0 0.0 0 .1 0 .2 0 .3 0.4 0 .5 0.6 Fiber volume ratio

Effective Properties (epoxy matrix, E-glass fiber) y = µ = R 8 . 5 m ; V 50 % fiber f C B = µ h 1 . 0 m b = ν = E 84 GPA ; 0 . 22 fiber fiber D A x = ν = E 4 GPA ; 0 . 34 Matrix matrix = E 4 , 6 , 8 , 12 GPA ; int erphase ν = 0 . 34 int erphase

Variation of Effective Young’s Modulus µ µ µ µ E m m m m h = 1. 0 h = 0.5 h = 0. 1 h = 0. 01 inter (GPA) 4 12.09 12.09 12.09 12.09 6 13.68 12. 86 12.24 12. 10 8 14.67 13.29 12.32 12.11 12 15.84 13.76 12.40 12. 12

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

5,000 inclusions of random sizes and elastic properties under a uniaxial stress at infinity σ σ ∞ = ; Contours of 1 . 0 xx xx