NSF Workshop on Probability & Materials: From Nano-to-Macro Scale
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NSF Workshop on Probability & Materials: From Nano-to-Macro Scale. STOCHASTIC FRACTURE OF FUNCTIONALLY GRADED MATERIALS. Sharif Rahman The University of Iowa Iowa City, IA 52245. January 2005. OUTLINE. Introduction Fracture of FGM Shape Sensitivity Analysis Reliability Analysis

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NSF Workshop on Probability & Materials: From Nano-to-Macro Scale

STOCHASTIC FRACTURE OF

FUNCTIONALLY GRADED MATERIALS

Sharif Rahman

The University of Iowa

Iowa City, IA 52245

January 2005


Outline
OUTLINE Scale

  • Introduction

  • Fracture of FGM

  • Shape Sensitivity Analysis

  • Reliability Analysis

  • Ongoing Work

  • Conclusions


Introduction
INTRODUCTION Scale

  • The FGM Advantage

Fracture Toughness

Thermal Conductivity

Temperature Resistance

Compressive strength

Metal Rich

CrNi Alloy

Ceramic Rich

PSZ

Ilschner (1996)

FGMs avoid stress concentrations at sharp material interfaces and can be utilized as multifunctional materials


Introduction1

Micro- Scale

Scale

Macro-

Scale

Local

Elastic Field

Averaged

Elastic Field

Effective

Elasticity

Homogenization

Volume fraction, Porosity, etc.

Elastic Modulus,

Poisson’s Ratio,etc.

ceramic

Eceramic

metal

Emetal

INTRODUCTION

  • FGM Microstructure and Homogenization


Introduction2

Tensile Properties Scale

Applied Stress

Fracture

Toughness

Crack Size

and Shape

Material Resistance

Crack

Driving

Force

>

Geometry of

Cracked Body

Temperature

Loading Rate

Radiation

Loading

Cycles

Fatigue

Properties

INTRODUCTION

  • Objective

    Develop methods for stochastic fracture-mechanics analysis of functionally graded materials

Work supported by NSF (Grant Nos: CMS-0409463; DMI-0355487; CMS-9900196)


Fracture of fgm

FRACTURE OF FGM


Fracture of fgm1
FRACTURE OF FGM Scale

  • J-integral for FGM

  • J-integral for Two Superimposed States 1 & 2

Superscript 1  Actual Mixed-Mode State

Superscript 2  Auxiliary State with SIF = 1


Fracture of fgm2
FRACTURE OF FGM Scale

  • New Interaction Integral Methods

Method I:

Homogeneous

Auxiliary Field

Method II:

Non-Homogeneous

Auxiliary Field

Both isotropic (Rahman & Rao; EFM; 2003) and

orthotropic (Rao & Rahman, CM; 2004) FGMs can be analyzed


Fracture of fgm3
FRACTURE OF FGM Scale

  • Example 1 (Slanted Crack in a Plate)

Plane Stress Condition

L=2, W=1

=0.3

(N = 370)

Gradation Direction


  • Governing and Sensitivity Equations

Need a numerical method

(FEM) to solve these two

equations for

V(x)



x

x



SHAPE SENSITIVITY ANALYSIS


SHAPE SENSITIVITY ANALYSIS Scale

  • Performance Measure

  • Shape Sensitivity


SHAPE SENSITIVITY ANALYSIS Scale

  • Sensitivity of Interaction Integral Method

  • Method I : Homogeneous Auxiliary Field

  • Method II : Non-Homogeneous Auxiliary Field

Rahman & Rao; CM; 2004 and Rao & Rahman, CMAME; 2004


Shape sensitivity analysis

x Scale2

L

2b

2b

x1

2a

L

W

W

SHAPE SENSITIVITY ANALYSIS

  • Example 2 (Plate with an Internal Crack)

2L=2W=20, 2a=2, =0.3

Plane Stress Conditions

Gradation Direction


Fracture reliability

FGM System

Random Input

Failure Probability

Load

Fracture initiation and propagation

Material & gradation properties

Geometry

Failure Criterion

FRACTURE RELIABILITY


Fracture reliability1
FRACTURE RELIABILITY Scale

  • Multivariate Function Decomposition

  • Univariate Approximation

At most 1 variable in a term

  • Bivariate Approximation

At most 2 variables in a term

  • General S-Variate Approximation

At most S variables in a term


Fracture reliability2
FRACTURE RELIABILITY Scale

  • Reliability Analysis

  • Performance Function Approximations

Univariate

Bivariate

Terms with dimensions

2 & higher

Terms with dimensions 3 & higher


Fracture reliability3

Lagrange Scale

shape

functions

FRACTURE RELIABILITY

  • Lagrange Interpolation

  • Monte Carlo Simulation

Univariate

Approximation

Bivariate Approximation


Fracture reliability4

Scale2

1

(a)

FRACTURE RELIABILITY

  • Example 3 (Probability of Fracture Initiation)

Performance Function (Maximum Hoop Stress Criterion)

Gradation

Direction


Fracture reliability5
FRACTURE RELIABILITY Scale

  • Example 3 (Results)


Ongoing work

Micromechanics

Rule of Mixtures

Mori-Tanaka Theory

Self-Consistent Theory

Eshelby’s Inclusion Theory

Particle Interaction

Gradients of Volume Fraction

Nonhomogeneous

Random Field

Volume Fraction

Porosity

Nonhomogeneous

Random Field

Spatially-varying FGM

Microstructure

Stochastic Material Properties

Elastic Modulus

Poisson’s Ratio

Yield Strength

etc.

ONGOING WORK


Ongoing work1
ONGOING WORK Scale

  • Level-Cut Random Field for FGM Microstructure

Grigoriu (2003)

Homogeneous microstructure

Translation Random Field

Second-Moment Properties

volume fraction

two-point correlation function

Filtered Non-Homogeneous Poisson Field

Find probability law of Z(x) to match target statistics p1 and p11


Ongoing work2
ONGOING WORK Scale

  • Multi-Scale Model of FGM Fracture


Conclusions
CONCLUSIONS Scale

  • New interaction integral methods for linear-elastic fracture under mixed-mode loading conditions

  • Continuum shape sensitivity analysis for first-order gradient of crack-driving force with respect to crack geometry

  • Novel decomposition methods for accurate and computationally efficient reliability analysis

  • Ongoing work involves stochastic, multi-scale fracture of FGMs


References
REFERENCES Scale

  • Rao, B. N. and Rahman, S., “A Mode-Decoupling Continuum Shape Sensitivity Method for Fracture Analysis of Functionally Graded Materials,” submitted to International Journal for Numerical Methods in Engineering, 2004.

  • Rahman, S., “Stochastic Fracture of Functionally Graded Materials,” submitted to Engineering Fracture Mechanics, 2004.

  • Xu, H. and Rahman, S., “Dimension-Reduction Methods for Structural Reliability Analysis,” submitted to Probabilistic Engineering Mechanics, 2004.

  • Rahman, S. and Rao, B. N., “A Continuum Shape Sensitivity Method for Fracture Analysis of Isotropic Functionally Graded Materials,” submitted to Computational Mechanics, 2004.

  • Rao, B. N. and Rahman, S., “A Continuum Shape Sensitivity Method for Fracture Analysis of Orthotropic Functionally Graded Materials,” accepted in Mechanics and Materials, (In Press).

  • Rahman, S. and Rao, B. N., “Continuum Shape Sensitivity Analysis of a Mode-I Fracture in Functionally Graded Materials,” accepted in Computational Mechanics, 2004 (In Press).

  • Rao, B. N. and Rahman, S., “Continuum Shape Sensitivity Analysis of a Mixed-Mode Fracture in Functionally Graded Materials,” accepted in Computer Methods in Applied Mechanics and Engineering, 2004 (In Press).

  • Rao, B. N. and Rahman, S., “An Interaction Integral Method for Analysis of Cracks in Orthotropic Functionally Graded Materials,” Computational Mechanics, Vol. 32, No. 1-2, 2003, pp. 40-51.

  • Rao, B. N. and Rahman, S., “Meshfree Analysis of Cracks in Isotropic Functionally Graded Materials,” Engineering Fracture Mechanics, Vol. 70, No. 1, 2003, pp. 1-27.


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