[PPT] - Computational Modeling of Composite and Functionally Graded PowerPoint Presentation

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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

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Research Group

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

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x y

unit cell interphases fiber matrix

Fiber-Reinforced Composite Materials

2-D model

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Finite element

method

Boundary element

method

Finite element mesh (after Wacker et al., 1998)

Standard Numerical Methods

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Our Approach

Direct boundary integral method Approximation of the unknowns by

Fourier Series or Spherical Harmonics

Complex (for plane problems) or real

variables formalism

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Direct Boundary Integral Method

Fundamental solution Governing differential equations + Boundary conditions Integral identities (e.g. reciprocal theorem) Boundary Integral Equation Fundamental solution (e.g. point force in plane)

s n

s n s n u

u σ σ , , ,

L

Enrico Betti 1823-1892

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Fourier Series

a complete orthogonal system over

j

R

j

z

τ

j

θ

1768-1830 Jean Baptiste Joseph Fourier

[ ]

π 2 ,

mx mx cos , sin

∑ ∑

∞ = ∞ =

+ + =

1 1

sin cos 2 1 ) (

m n m n

mx b mx a a x f

On the Propagation of Heat in Solid Bodies, 1807

[ ]

) ( ) ( ) ( ) (

2 2 1 1 ∞ →

→ + + −

N N N

x c x c x c x f ϕ ϕ ϕ …

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Spherical Harmonics

Peter Guthrie Tait 1831-1901 William Thomson (Lord Kelvin) 1824-1907 ‘T&T’ Treatise on Natural Philosophy (1867)

( ) ( )

{ }

( )

∑

=

+ + =

n m m n m n m n n n n

T m B m A P A Y

1

cos sin cos cos , θ ϕ ϕ θ ϕ θ

Surface harmonics

( ) ( )

∑

∞ =

= , ,

n n

Y f ϕ θ ϕ θ

A complete orthogonal system Over the unit sphere (Lyapunov, 1899)

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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

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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

1

t

2

t

K

t

{ }

1

2 ,...,

max

K

t t t t

ε δ

=

≤

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xx

σ

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

Numerical Example

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Spring

Spring-

type interface

type interface

p j

R

p j

z

p j

µ

p j

ν

p j

Γ

p j

R 0

p j

z 0

p j 0

µ

p j0

ν

p j

R 1

p j

z 1

p j1

Γ

p j0

Γ

p j1

µ

p j1

ν

p j

R 0

p j

z

p j 0

µ

p j0

ν

p jn

R

p jn

Γ

p j0

Γ

) (r

p j

µ ) (r

p j

ν

p j

R

p j

z

p j

µ

p j

ν

p j

Γ

Partial debonding

Partial debonding

Explicit presence of interphase layers

Explicit presence of interphase layers

Imperfect Interface Models

debonding debonding

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x y

σ

x y

σ

x y

σ

6 – 9 terms

3 2 6 1

10 ; 10

− −

= = δ δ

20.0 0.2647 stiff co 0.067 0.2647 compl.co 1.0 0.2647 matrix 38.9 0.2537 inclusion E/

ν

material

matrix

E

Numerical Example

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eff

σ

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 0.0 0.5 1.0 1.5 2.0 2.5

inclusion matrix

interphase interphase

matrix

perfect bond stiff coating compliant coating

Numerical Results

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ϕ

σ ν µ, y ' , ' ν µ x α 2

Inclusion with Interface Crack (Toya, 1974)

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/σ σ rr

30 60 90 120 150 180 210 240 270 300 330 360

1.0
0.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5

/σ σ θ

r 30 60 90 120 150 180 210 240 270 300 330 360

1.0
0.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5

θ

Angle, (a)

θ

Angle, (b)

Computed radial and shear stresses (open circles) compared with analytical solution (solid lines); N=180

30

35 . ' , / 39 . 2 , 22 . ' , / 2 . 44 '

2 2

= = = = = = ϕ α ν µ ν µ m GN m GN

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3

10 , /

−

∆ a ur

30 60 90 120 150 180 210 240 270 300 330 360 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18

3

10 , /

−

∆ a uθ

30 60 90 120 150 180 210 240 270 300 330 360 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18

θ

Angle, (a)

θ

Angle, (b)

Computed radial and shear displacement discontinuities (open circles) compared with analytical solution (solid lines); N=180

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Example — debonding of single inclusion

Smooth interface: Stippes, Wilson, and Krull (1966) Rough interface: Hussain and Pu (1971)

x y

σ σ =

∞ yy

ν ν µ µ = = ' , ' ν µ,

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/σ σ rr −

θ

Angle,

5 10 15 20 25 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Radial stress in zone of contact for smooth inclusion: solid line is analytical solution; open circles are computed results

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/σ σθθ

θ

Angle,

15 30 45 60 75 90

1.0
0.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Circumferential stress for smooth inclusion: solid line is analytical solution; open circles are computed results

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/σ σ rr −

θ

Angle, (a)

5 10 15 20 25 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

/σ σ θ

r

θ

Angle, (b)

5 10 15 20 25 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Comparison of computed radial (a) and shear (b) stresses for rough inclusion: solid lines are results from Fourier series approach

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Modeling evolving damage

Initial attempt: Increment loading Use Mohr-Coulomb criterion (c is cohesion; is angle of friction; T is tensile strength) Allow slip, separation (cracking); prohibit overlapping of displacement discontinuities during iteration

T c

rr rr r

≤ − ≤ σ φ σ σ θ ; tan

φ

rr

σ

θ

σ r

φ

c – c T

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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 …

Issues…

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s n

s n s n u

u σ σ , , , Melan’s fundamental solution (point force in a half-plane)

L

ad K M

FS FS FS + =

Effect of Free Boundary

Just few results were available and they were contradictory

single inclusion

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A Single Inclusion Close to the Boundary

Contours of

1 2

( ) /

xx

σ σ σ ∞ −

99 . / , 3 . , . 100 ; 3 . , . 1 = = = = = d R

inc inc matrix matrix

ν µ ν µ

89 terms

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40 Regularly Distributed Inclusions

Contours of

1 2

( ) /

xx

σ σ σ ∞ −

35 . , . 10 ; 15 . , . 1 = = = =

inclusion inclusion matrix matrix

ν µ ν µ

103~117 terms

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200 Randomly Distributed Inclusions

Contours of

1 2

( ) /

xx

σ σ σ ∞ −

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p =1.0 p =1.

2.00
1.50
1.00
0.50

1.50 2.00

2.00
1.50
1.00
0.50

0.00 0.50 1.00 1.50 2.00 p =1.0

1

4 1 2 3

Finite Domain with Circular Boundary

Distribution of

2 1

σ σ −

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Finite Domain with Convex Polygonal Boundaries

A B C D

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

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Effective (macroscopic properties)

) ( hole = γ

Labuz & Carvalho (1996)

0.0 .1 .2 .3 0.4 .5 0.6 0.0

0.50

1.00 1.5 2.0 2.5

3.00 Fiber volume ratio

2 . = γ 5 . = γ . 1 = γ . 2 = γ . 5 = γ

50 = γ

) ( 000 , 10 inclusion rigid = γ

Eeff /E0

/µ µ γ

i

=

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Effective Properties (epoxy matrix, E-glass fiber)

34 . ; 12 , 8 , 6 , 4 34 . ; 4 22 . ; 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 . 1 % 50 ; 5 . 8 = = = y D C A B x b

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(GPA) h = 1. 0 h = 0.5 h = 0. 1 h = 0. 01

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

inter

E

m µ

Variation of Effective Young’s Modulus

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Fast Solvers

V. Rokhlin, 1985
L. Greengard and V. Rokhlin, 1987
Data information

Data information

Computation time

Computation time (1.5GHz CPU) (1.5GHz CPU)

Direct method

2 hours

Multi-level FMA

10,000 inclusions with 0.5 filling ratio 6 hours

Single-level FMA

1.5 months

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5,000 inclusions of random sizes and elastic properties under a uniaxial stress at infinity ; Contours of

. 1 =

∞ xx

σ

xx

σ

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103 104 105 106 1.0 101 102 103 104 105 106 107

number of degrees of freedom

direct algorithm single-level fast multipole algorithm multi-level fast multipole algorithm

CPU time in seconds

Comparison of the Algorithm Complexity

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µparticle/µmatrix= 10; Contours of σyy

1.0 1.0

Modeling of Graded Composite Materials

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Contours of σyy

µparticle/µmatrix= 10;

1.0 1.0

Modeling of Graded Composite Materials (continued)

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ηve σ σ Ee εve εe Eve

Linear Viscoelasticity; Boltzmann model

Creep curve Relaxation curve

One dimensional representation Time Stress

Constant strains applied

Time Strain

Constant stresses applied

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Example - Two inclusions and two holes

D C

η ,ν ,E E

ve e

,

1 1,ν

E

2 2,ν

E

Computational costs

Boundary integral method: 8 minutes,

11-13 terms in Fourier series

ANSYS: 11 hours, 20,375 elements

2 4 6 8

Time (second)

2 4 6 8 10 12 14 16

Stress (MPa) C-ANSYS C-Boundary integral D-ANSYS D-Boundary integral

2 4 6 8

Time (second)

0.2 0.4 0.6

Displacement (mm) C-ANSYS C-Boundary integral D-ANSYS D-Boundary integral

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G, Gve, ν, γ (= θλ= θµ) x y σ 0 σ 0 d Gi, νi ri

25 Elastic Inclusions in a Viscoelastic Plane

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σyy/ σ0 on the line y = 0

16
8

8 16

x / d

0.0 0.5 1.0 1.5 2.0

σ /σ

t / = 0.005 t / = 0.01 t / = 1

γ γ γ

yy

Some Results …

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1 2 3 4 5 6 7 8 9 10 11 12 14 15 16 17 18 19 20 21 22 23 24 25 x y z

25 Spherical Cavities in y-z Plane

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0.2 .2 0.2 . 2 0. . 2 0.4 0.4 0.4 0.4 . . 4 4 0.6 6 . 6 0.6 . . 6 . 8 0.8 0.8 . 0.8 1 1 1.2 1 . 2 . 2 1.4 . 4 . 1.6 1.8 . 1.8

y z

Contours of near cavity #11

) (

/

yy yy σ

σ

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Transient Heat Conduction in Composite Materials

τ τ τ τ τ α dsd s T n s G n s T s G t x T

t

∫ ∫

Ω ∂

      ∂ ∂ − ∂ ∂ = ) , ( ) , ( ) , ( ) , ( ) , (

Method of Solution Integral Identity

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) T(x,t) is the temperature at point x at time t G(s,t) is the Green’s Function

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Future Work – Microcontinuum models

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 Objectives

Incorporate size effects
Address boundary layer effects
Obtain more realistic results for critical regions of high deformation

gradients

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