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Special Issue Honoring Professor Fazil Erdogan’s Contributions to Mixed Boundary Value Problems of Inhomogeneous and Functionally Graded Materials

Mixed-Mode Dynamic Crack Growth in a Functionally Graded Particulate Composite: Experimental Measurements and Finite Element Simulations

[+] Author and Article Information

The Goodyear Tire and Rubber Company, Department of Mechanical Engineering,Auburn University, Auburn, AL 36849

Hareesh V. Tippur

Department of Mechanical Engineering, Auburn University, Auburn, AL 36849

The following parameters were used for convergence control (37): the half-step residual $tolerance=20$, ratio of the largest residual to the corresponding average force norm $(Rnα)=0.005$, and the ratio of the largest solution correction to the largest corresponding incremental solution value $(Cnα)=0.01$.

J. Appl. Mech 75(5), 051102 (Jul 10, 2008) (14 pages) doi:10.1115/1.2932095 History: Received May 24, 2007; Revised February 07, 2008; Published July 10, 2008

Abstract

Mixed-mode dynamic crack growth behavior in a compositionally graded particle filled polymer is studied experimentally and computationally. Beams with single edge cracks initially aligned in the direction of the compositional gradient and subjected to one-point eccentric impact loading are examined. Optical interferometry along with high-speed photography is used to measure surface deformations around the crack tip. Two configurations, one with a crack on the stiffer side of a graded sheet and the second with a crack on the compliant side, are tested. The observed crack paths are distinctly different for these two configurations. Furthermore, the crack speed and stress intensity factor variations between the two configurations show significant differences. The optical measurements are examined with the aid of crack-tip fields, which incorporate local elastic modulus variations. To understand the role of material gradation on the observed crack paths, finite element models with cohesive elements are developed. A user-defined element subroutine for cohesive elements based on a bilinear traction-separation law is developed and implemented in a structural analysis environment. The necessary spatial variation of material properties is introduced into the continuum elements by first performing a thermal analysis and then by prescribing material properties as temperature dependent quantities. The simulated crack paths and crack speeds are found to be in qualitative agreement with the observed ones. The simulations also reveal differences in the energy dissipation in the two functionally graded material (FGM) cases. $T$-stresses and hence the crack-tip constraint are significantly different. Prior to crack initiation, larger negative $T$-stresses near the crack tip are seen when the crack is situated on the compliant side of the FGM.

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Figures

Figure 1

(a) Schematic of the FGM specimen (darker shades represent stiffer materials), (b) material property variation along the width of the sample, and (c) variation of dynamic crack initiation toughness along the width of the sample

Figure 2

Two mixed-mode FGM test configurations: (a) crack on the compliant side of the sample with impact occurring on the stiff side (E1<E2) and (b) crack on the stiff side of the sample with impact occurring on the compliant side (E1>E2). Impact velocity (V)=5m∕s. (Shading is used to denote compositional gradation; darker shades represent stiffer material.)

Figure 7

Finite element discretization. (a) Overall view of the finite element discretization, (b) magnified view of the mesh showing Region 1 (continuum elements) and Region 2 (continuum and cohesive elements) and (c) enlarged view of the mesh near the interface of Regions 1 and 2.

Figure 8

Thermal analysis to apply graded material properties. (a) Nodal temperature results from thermal analysis, and (b) magnified view of the cohesive element region.

Figure 9

Stress intensity factors extracted from CGS interferograms by performing overdeterministic least-squares analysis on difference formulation of CGS governing equation (Eq. 5): (a) KI history and (b) KII history. The quality of least-squares fit for (c) E1<E2(t−ti=20μs) and (d) E1>E2(t−ti=−20μs).

Figure 10

Evolution of various energies in dynamic simulation for both FGM configurations: (a) kinetic energy and strain energy and (b) energy absorbed by cohesive elements

Figure 11

Effect of the initial slope of the TSL on (a) displacement and (b) on stress results in elastodynamic simulations on uncracked beams at a node along the lower edge at mid-span

Figure 12

Snapshots of σyy stress field at two different time instants: (a) 122μs and (b) 154μs for E1<E2 (crack initiation time=131μs), and (c) 121μs and (d) 171μs for E1>E2 (crack initiation time=133μs)

Figure 13

Snapshots of uv displacement field at two different time instants: (a) 122μs and (b) 154μs for E1<E2 (crack initiation time=131μs), and (c) 121μs and (d) 171μs for E1>E2 (crack initiation time=133μs)

Figure 14

Crack growth behavior in FGM sample under mixed-mode loading. Absolute crack length history from (a) experiments and (b) finite element simulations, ti is crack initiation time (ti=155μs for E1<E2 and 145μs for E1>E2 in experiments, and ti∼130μs for both E1<E2 and E1>E2 in simulations).

Figure 15

Nonsingular crack-tip stress histories: (a) variation of apparent T-stress with radial distance at a certain time instant before crack initiation and (b) T-stress history up to crack initiation for E1<E2 and E1>E2

Figure 3

Selected CGS interferograms representing contours of δw∕δX1 in FGM samples; (a) and (b) are for the case of E1<E2 and (c) and (d) are for the case of E1>E2. The time at which the images are taken after impact is indicated below each image. The current crack tip is indicated by an arrow.

Figure 4

Multiple fractured FGM specimens (right half) demonstrating experimental repeatability for (a) FGM with a crack on the stiffer side (E1<E2) and (b) FGM with a crack on the compliant side (E1>E2). Photograph showing fractured specimens for (c) FGM with a crack on the compliant side (E1<E2) and (d) FGM with a crack on the stiffer side (E1>E2). Impact point is indicated by letter “I” and initial crack tip by letter “C.”

Figure 5

(a) Schematic of FGM sample with linear material property variation, and (b) elastic modulus variation in graded samples (broken line denotes the crack tip location)

Figure 6

Details on cohesive element formulations: (a) undeformed and deformed configurations of the crack tip region. (b) Local and global coordinate systems used for a cohesive element. Prescribed TSL for (c) pure normal separation and for (d) pure tangential separation.

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