composite properties and microstructure ii: strength

Composite Properties and Microstructure II:

Strength

Robert LiptonLouisiana State University

Composites: Where Mathematics Meets Industry

February 7, 2005IMA-Institute for Mathematics and its

Applications

Supported in part by: AFOSR and NSF

Structure and substructure - form and functionality - across multiple length

scales

Fiber reinforced epoxy Horizontal StabelizerBoeing 777Racing Sail

Bone

Multi-scale problem statementIt is supposed that the length scale of the composite microstructure is

significantly smaller than the length scale of the load.

To characterize failure initiation inside the composite it is required to assess the extreme field behavior at the length scale of the microstructure generated by macroscopic loading.

0

x

Microstructures considered in this presentation: Graded Locally Periodic Microstructure

Length scales:

ε

0

x

X’

1Microstructure length scale relative to load length scale is denoted by

ε

Consider also the random two phase layered composite

Partition the plane into strips of unit width.Over strip each flip a coin: if heads strip is blackif tails strip is white.

θProbability of heads isProbability of tails is: θ−1

1

Random laminar microstructure inside L- shaped domain

1/40

For a given realization of theLayered random medium rescaleit by 1/40 and inlay it insidethe L-shaped domain

Study two complementary aspects of stress transfer

Assess extent of overstressed zones in composite structures due to reentrant corners, bolt holes, rivets and other stress risersLower bounds on maximum local fields inside random microstructure due to macroscopic loading

The first part of this talk outlines a mathematically rigorous methodology for bounding overstressed zones inside composite materials due to stress risers

Boundary value problem in microstructured elastic media

)(xuε

)(xεσ)(xεε

10 <<< εLength scale of microstructure

Elastic displacement field2/)( ,,

εεεε ijji uu +=Stress tensor

)()()( xxCx εεε εσ =

iCxC =)(ε

Strain tensor

Ci is the elasticity of ith material

Traction BC on ΓNand displacement BC on ΓD

fdiv =− )( εσ

gxn =• )(εσUu =ε ΓD

ΓN

yRVE

1Local elastic tensor

Constitutive relation

Equation of elastic equilibrium

Boundary conditions:

Homogenization theory;convergence of averages

2/)( ,,Mij

Mji

M uu +=ε

dxxudxxuS

M

S∫∫ → )()(ε

dxxdxxS

M

S

)()( ∫∫ → εεε

)()()( xxCx MEM εσ =fdiv M =− σ

dxxdxxS

M

S∫∫ → )()( σσε

For any subset S:

Homogenized constitutive relation

Traction BC on ΓNand displacement BC on ΓD

UuM =gxn M =• )(σ

y

1RVE:Formula for CE

Load unit RVE with homogeneous strainLocal elastic tensor in microstructure = C(y)Micro-problem in RVE for local strain fluctuationLocal strain fluctuation e(y) solves:

CE given by:

__ε

0)))()((( =+ εyeyCdiv

dyyeyCCQ

E ))()((∫ += εε

Stress Bound Part 1

Stress in composite =

Stress in i-th phase =

Set of interest ``S’’Consider quadratic invariants of the stress tensor, e.g., Von Mises stress, square of dilatational stress. These are written as:

Stress bound on S: For almost every x in S and for εsufficiently small,

εσ

)(xiεσ

),(sup))(( )( trymicrogeomexfx iSxi ∈≤∏ εσ

10, <<< ε

))(( xi∏ εσ

L. (2004) Journal of Mech. Phys. Solids. 52 pp. 1053-1069.

Computation of Stress Bound – Multi Scale Analysis: Step 1 - Up scaling

Replace local elastic properties with effective elastic property CE(x).Find homogenized stressFind homogenized strain

)(xMσ

)()()( xxCx MEM εσ =

fdiv M =− σ

)(xMε

y

1RVE:Step 2. Down scaling

Load unit RVE with homogenized strainLocal elastic tensor in microstructure = C(y)Micro-problem in RVE for local strain fluctuationLocal strain fluctuation e(x,y) solves:

Strain fluctuation in i-th phase:Stress bound given by:

)(xMε

0)))(),()((( =+ xyxeyCdiv Mε),( yxei

)))(),()(((sup),()( xyxeyCtrymicrogeomexf Mi

iy

i ε+= ∏

Stress Bound Part 2

}))((,,{ txwhereVx i >∈ ∏ εσOver stressed region in i-th phase =

)(tiελVolume of over stressed region in i-th phase =

|)})({| )( txf i ≥ txf i ≥)()(= Volume of set where

|})({|)(lim )(0 txft i

i ≥≤→ε

ε λStress bound:

L. (2004) Journal Mech. Phys. Solids. 52 pp. 1053-1069

Stress assessment in shaft with L shaped cross section filled with random laminate

and subjected to torsion loadβ =shear compliance in phase 1α= shear compliance in phase 2α < βθ=probability of phase 2Homogenized in-plane stress σM = (σ1

M, σ2M)

)))(1(/()( αβθααβθγα −−+−=

)))(1(/())(1( αβθααβθγβ −−+−−−=

21

22

2)1( |)(||)(|)1()( xxxf MM σσγα ++=

21

22

2)2( |)(||)(|)1()( xxxf MM σσγ β ++=

Direct comparison between the homogenized stress and rigorous bounds on overstressed

zones inside each elastic materialUp scaling only:Homogenized stress: σM

Level line of |σM|2

Up scaling + Down scalingGives stress bound andEncodes effects of stress fluctuationat the level of the microgeometry

Here: α=2, β=10, θ=.33L. (2004) SIAM J. Appl. Math. Vol 65., pp. 475—493.

Optimization of graded microstructure: fiber reinforced shaft subject to torsion

Cylindrical shaft with X-shaped cross section subject to torsion loading. Reinforce with locally layered material. Here relative layer thicknesses can change as well as layer orientation.

Solution of design Problem via inverse homogenization.

Effective stiffness is explicit function of local layer orientation and material 1 area fraction.Stress bound explicit function of local layer orientation and material 1 area fractionSo design variables are local layer orientation and area fraction of material 1

Use the effective stiffness and stress bound to identify a locally layered microstructure with the desired properties

Goal: Design for maximum torsional rigidity while minimizing the effect of the stress concentrations at the reentrant corners.

L. and M. Stueibner (2004) International Journal for Numerical Methods in Engineering, Submitted.

Formulation of design problem

})({ 21)(min dxdxfRigidity pi

Designs∫∫+− λ

5.0=β - Compliance in shear for compliant material

25.0=α - Compliance in shear for stiff material

θ p = 1,2= local area fraction of compliant material.

Cdxdx ≤−∫∫ 21)1( ϑResource constraint:

Design 1. Density distribution of compliant material for a shaft optimized for rigidity only

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1The Final Density Distribution

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Area of stiff material = 60%

Rigidity = 0.82

Design 2. Density distribution of compliant material for shaft optimized for rigidity subject to constraint on f1

Area of stiff material = 60%

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Rigidity = 0.61

Design 3. Density distribution of compliant material for ashaft optimized for rigidity subject to constraint on f2

Area of stiff material = 60%

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9Rigidity = 0.62

Contour plots for f1 Designs 1 and 2

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0.6

0.3

0.1

0.3

0.61

1.83.6

0.5

1

1.5

2

2.5

3

3.5

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0.1

0.30.6 0.

13.

6

1.8

10.6

0.3

0.1

0.5

1

1.5

2

2.5

3

3.5

Design 1 Design 2

Contour plots for f2 Designs 1 and 3

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0.1

0.1

0.30.6

11.8

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0.1

0.04 0.1

0.20.4

0.7

11.4

0.04

0.2

0.4

0.6

0.8

1

1.2

Design 3Design 1

Consider a sample of composite subjected to a uniform thermal gradient.

In this part of the talk we developlower bounds for the maximum Temperature gradient generated at the level of the microgeometry.

A realization of a random medium

One and two point correlationfunctions

One point correlation Two point correlation

r

Probability that both endsof a segment lies in greenphase when thrown into

composite randomly

Probability of a pointlying inside the green phase

(Volume fraction of the green phase)

One and Two-point Correlation Functions

These statistical functions can be computed using techniques from image analysis. Berryman 1988.

ObjectivesConsider the ensemble of composite samplescharacterized by the same one and two point correlations.

Subject each sample to the same imposedtemperature gradient.

Find a lower bound on the maximum temperature gradient

inside the samples that is given in terms of the one and two point statistics.

Identify a realization that has the smallest maximum temperature gradient.

Setting

Cube filled with compositemade from two heat conductors.The volume fractions of eachis prescribed.

0=∆T

TnTn ∇•=∇• 21 σσ

Conductivities:Volume fractions:

σ1 > σ2θ1, θ2

Inside each phase

On interface

xET •−and is periodic on the cubeET =∇

First Moment –Effective Properties

Q2 = part of cube occupied by phase twoQ1 = part of cube occupied by phase one

Cube domain = Q = Q1 ∪ Q2

χ1 = 1, in Q1 , zero outside

χ2 = 1, in Q2 , zero outside

σ(x)=χ1σ1+χ2σ2

)()( xTxEe ∇=Σ σ

Earlier work bounds of Hashinand Shtrikman for effective properties

Hashin Shtrikman bounds on effective conductivityfor isotropic composites (1962).

eΣ≤−+−

+)(3

)(3

2122

21212 σσθσ

σσσθσ

)(3)(3

1211

12121 σσθσ

σσσθσ−+−

+≤Σe

Higher order moments of temperature gradient

Higher moments sensitive to high local thermal gradients

Damage initiation due to high local thermal gradients

rrT/1

1 ||∇χ ∞ ≥ r ≥ 2

rrT/1

2 ||∇χ ∞ ≥ r ≥ 2

Maximum temperature gradients

|})({||| maxlim1

/1

1 xTTQx

rr

r∇∇ =

∈∞→

χ

|})({||| maxlim2

/1

2 xTTQx

rr

r∇∇ =

∈∞→

χ

|})({||||| max)(xTT

QxQL

∇∇ =∈

∞

Optimal Lower Bounds Isotropic Case I

Hypothesis: isotropic effective thermal properties

For ∞ ≥ r ≥ 2

||3/)(

||2212

2/11

/1

1 ET rrr

θσσσσθχ−+

≥∇

The lower bound is attained by the Hashin Shtrikmancoated sphere assemblage with core of material oneand coating of material two.

(Lipton J.Appl.Phys.2004(96) 2821-2827)

Optimal Lower Bounds Isotropic Case II

For ∞ ≥ r ≥ 2

||3/)(

||1121

1/12

/1

2 ET rrr

θσσσσθχ−+

≥∇

The lower bound is attained by the Hashin Shtrikmancoated sphere assemblage with core of material twoand coating of material one.


Optimal Lower Bounds Isotropic Case III

||3/)(

||||1121

1)( ET QL θσσσ

σ−+

≥∇ ∞

The lower bound is attained by the Hashin Shtrikmancoated sphere assemblage with core of material twoand coating of material one.

(Lipton, J.Appl.Phys.2004(96) 2821-2827)

Anisotropic Mixtures and theTensor of Geometric Parameters

Two point correlation function of material one S1(r)Matrix of geometric parameters ``M’’, J.R. Willis 1977

M is a symmetric matrix and Trace{M}=1

∑≠

⊗=

021

21 ||)(ˆ1

k kkkkSM

θθEigenvalues of M: 0≤d1≤d2≤d3≤1

and d1+d2+d3=1

Optimal Lower Bounds Anisotropic Case I

For ∞ ≥ r ≥ 2

||)(

||32212

2/11

/1

1 Ed

T rrr

θσσσσθχ−+

≥∇

The lower bound is attained by the coated ellipsoid assemblage with core of material one and coating of material two with major axis aligned with the imposed temperaturegradient E.

θ1 and d1 , d2 , d3 and E are fixed

E


Optimal Lower Bounds Anisotropic Case II

For ∞ ≥ r ≥ 2

||)(

||11121

1/12

/1

2 Ed

T rrr

θσσσσθχ−+

≥∇

The lower bound is attained by the coated ellipsoid assemblage with core of material two and coating of material one with minor axis aligned with the imposed temperaturegradient E.


E


Optimal Lower Bounds Anisotropic Case III


||)(

||||11121

1)(

Ed

TQL θσσσ

σ−+

≥∇ ∞

The lower bound is attained by the coated ellipsoid assemblage with core of material two and coating of material one with minor axis aligned with the imposed temperaturegradient E.

E


composite properties and microstructure ii: strength

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