Experimental Fracture Mechanics of Functionally Graded

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Experimental Fracture Mechanics of Functionally Graded

Transcript Of Experimental Fracture Mechanics of Functionally Graded

Experimental Mechanics (2010) 50:845–865 DOI 10.1007/s11340-010-9381-z
Experimental Fracture Mechanics of Functionally Graded Materials: An Overview of Optical Investigations
C.-E. Rousseau · V.B. Chalivendra · H.V. Tippur · A. Shukla

Received: 30 March 2010 / Accepted: 7 June 2010 / Published online: 15 July 2010 © Society for Experimental Mechanics 2010

Abstract The experimental efforts towards understanding the fracture behavior of continuously graded Functionally Graded Materials (FGMs) using full-field optical methods are reviewed. Both quasi-static and dynamic fracture investigations involving mode-I and -II conditions are presented. FGM configurations with crack planes perpendicular to, parallel to, and inclined to the direction of compositional gradation are discussed. Different strategies adopted by various investigators to develop polymer-based FGM systems for experimental mechanics studies are also described in this overview. Major theoretical developments that have predated and paralleled the experimental studies have been presented as well. Finally, the paper notes a few potential new directions where further contributions are possible.
This is the 5th in a series of featured review articles to celebrate the 50th anniversary of Experimental Mechanics. These articles serve to touch on both areas of mechanics where the journal has contributed extensively in the past and emergent areas for the future.
C.-E. Rousseau · A. Shukla (SEM Fellow) University of Rhode Island, Kingston, RI, USA
C.-E. Rousseau e-mail: [email protected]
A. Shukla e-mail: [email protected]
V.B. Chalivendra (SEM member) University of Massachusetts, Dartmouth, MA, USA e-mail: [email protected]
H.V. Tippur (B, SEM member)
Auburn University, Auburn, AL, USA e-mail: [email protected]

Keywords Functionally graded materials · CGS · Photoelasticity · DIC · Fracture Mechanics
Introduction
The concept of Functionally Graded Materials (FGMs) arose from the early attempts to meet stringent material requirements pertaining to aerospace vehicles facing thermal cycles between elevated and cryogenic temperatures. The demands of achieving adequate toughness, resilience, compliance, and rigidity goals simultaneously motivated the early investigations. This goal is most efficiently met by assembling different materials with those desirable properties, while ensuring seamless and smooth spatial variation from one material to the other, thereby circumventing the deleterious effects such as residual stresses, planes of weakness, or stress concentration present in case of discrete interfaces. The gradual variation in material properties in the resulting composite has been indeed shown to improve failure performance, while preserving the intended thermal, tribological, and structural benefits of combining dissimilar materials [1]. Practical examples of such designs include the gradual amalgamation of tungsten and zirconia in ratios of 80:20, 60:40, 40:60, and 20:80, respectively. Other systems include titanium–titanium boride, titanium–zirconia, tungsten–silicon, and other metal–ceramic combinations. A gradual variation of porosity by the infusion of voids is also considered for lightweight structural applications. The concept, however innovative as it may appear, can be abundantly seen in nature. Some common examples include crosssectional microstructures found in bamboo shafts and animal bones. Although most works on FGMs to date

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focus on thermal properties and elastic wave channeling, their ability to resist fracture is also of significant interest. Therefore, the aim of this paper is to review the state-of-the-art in this specific area of experimental fracture mechanics aided by full-field optical methods. The approach taken here is one wherein material variation in elastic modulus plays the principal role is influencing fracture properties. It is implicit, however, that other factors such as variations in fracture toughness will also play a role in guiding the behavior of the crack. For instance, Jin and Batra [2] expounded on a theory where they ascribed prominence to that parameter. Other factors such as density would also affect FGM fracture, especially during crack propagation. Nevertheless, most studies tend to emphasize Young’s modulus variations, as done throughout this manuscript.
A variety of processing methods have emerged over the years for preparing FGMs. For instance, Sasaki and Hirai [3] have fabricated FGMs using chemical vapor deposition. Bishop et al. [4] employed powder metallurgy. Chu et al. [5] made use of slip casting technique, whereas Sampath et al. [6] utilized plasma spraying. These methods largely influence the microstructure and hence macro-mechanical responses of FGMs. An additional differentiation regarding stepwise (or discrete) [7] versus continuous compositional variations [8] can also be made. The former category encompasses homogeneous materials that are layered in a stepwise fashion until the desired profile is attained whereas in the latter the same is achieved by a continuous variation of composition on a macroscale. Parameswaran and Shukla [9] also developed continuous graded materials using a gravity assisted casting process and characterized these FGMs for Young’s modulus and fracture toughness. In this work, the discussion is limited to the response of those with continuous variation of properties. It should be emphasized that continuity in the context of FGMs is used in the macroscopic sense. Indeed, most optical techniques are not capable of performing below the millimeter scale, whereas the particulate composites used here, have representative volume elements limited to the tens of micrometer range. Therefore, the continuity assumption relates solely to the overall gradation profile of the material.
It is further noted that despite the several fabrication techniques and proposed FGM systems, as listed above, researchers have not performed specific experiments involving these real materials. Instead, the focus has been on model materials fabricated in small quantities. These model materials involved irradiated plastics, thermosets, polymeric materials, sometimes interspersed with particulates. Compared to real materials,

they are far more cost effective, yet also hold the promise of implementation within real structures.
Theoretical interest in the fracture mechanics of nonhomogeneous materials predates its recent revival in the context of functionally graded materials. These include, for instance, the contributions of Atkinson and List [10] and Delale and Erdogan [11]. Several refinements to the theory and the proposed solutions have succeeded for nearly two decades, unaided by experimentation. The latter was impeded generally by the complexities associated with preparation of such materials in a consistent and controlled fashion. Recent advances in this regard have naturally let significant experimental mechanics studies to emerge. Amongst those, fracture experiments performed on FGMs will be limited to those conducted by means of optical tools are reviewed in this overview.
This paper will first proceed with an evolutionary account of the theory of fracture of nonhomogeneous materials essential for interpreting optically measured crack tip fields. Optical investigations performed respectively, with cracks perpendicular, and cracks parallel to the gradient directions are then addressed. This is followed by a description of dynamic experiments, solely for cracks parallel to the gradient direction but resulting in pure mode-I and mixed-mode behaviors. Throughout the paper, several materials and optical systems are described chronologically as applied to FGM research. Finally, the paper concludes with a section listing several areas where new experimental research could lead to a better understanding of the fracture behavior in FGMs, and a summary of contributions to the field.

Crack-Tip Stress Fields

Much of the experimental work towards understanding the behavior of graded materials weakened by cracks has risen from the need to first verify the theoretical formulations guiding the field. A common aspect of this theoretical cluster, dating from the 1970’s, uses an exponential variation of elastic properties. That is, the shear modulus is often expressed as:

μ(x, y) = μ0e(βx+γ y),

(1)

where β and γ define the spatial variations of the material along the x- and y- directions, respectively. This specific formulation, featuring an exponential gradation, is by far the most common, owing to the relative ease of manipulation of the compatibility equation. Other situations involving, for instance, linear variations would present a far more daunting task. The

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coordinates x and y are universally described as being aligned in directions parallel to and normal to the crack face, respectively, with the crack tip as the origin (see Fig. 1). In situations where material inertia is important as in dynamic crack growth cases, the mass density is made to follow an identical variation. (It should be noted that the variations in Poisson’s ratio have been found less important (e.g., Delale and Erdogan [11]).
The above mentioned researchers have offered one of the earliest solutions for a crack embedded within a nonhomogeneous planar solid subjected to arbitrary, quasi-static, boundary tractions using the method of integral equations. It revealed the standard square root stress singularity at the crack tip as in the homogeneous counterparts. Specific equations were established for obtaining stress intensity factor for the case of a uniformly pressurized crack. Differences between plane stress and plane strain assumptions were found to be minimal. Experimental validation of this specific solution is difficult and is yet to be attempted.
Using Williams’ eigenfunction expansion technique [12], Eischen [13] derived the quasi-static stress and inplane displacement fields, (ux, uy), for the case of a nonhomogeneous material with elastic properties described by a Maclaurin series expansion. The first few terms of displacement expressions are as follows:

ux = KI μ0

r gI (θ ) + KII

2π x

μ0

+ ux0 − ω0r sin (θ )

2rπ gxII (θ )

+ r {C2 (κ + 1) cos (θ ) + D2 (κ + 1) sin (θ )} 2μ0

+ O r3/2 + . . . ,

(2)

uy = KI μ0

r gI (θ ) + KII

2π y

μ0

+ uy0 + ω0r cos (θ )

2rπ gyII (θ )

+ r {C2 (κ − 3) sin (θ ) − D2 (κ + 1) cos (θ )} 2μ0

+ O r3/2 + . . . ,

(3)

where μ0 is the shear modulus at the crack tip. For generalized plane stress conditions, applicable to most experimental cases, κ = (3 − ν)/(1 + ν). Functions of θ are identical to those describing the behavior of homogeneous materials [14]. In the above, one can readily recognize the characteristics of a homogeneous crack associated with the first two powers of r. The influence of nonhomogeneity appears in the subsequent terms. By inference, the first two terms of the stress field are also devoid of nonhomogeneity. Only the local material properties pertaining to the crack tip position enters the equations. These do not, however, preclude the need for the higher order terms, as local plasticity, triaxiality, and singularity inherent to the immediate vicinity of the crack tip often prevents extraction of experimental (optical) data within that region. Thus, this solution can only be embraced with caution in experimental cases. , Using exponential variation of Young’s modulus, Parameswaran and Shukla [15] developed the asymptotic expansion of quasi-static crack-tip stress fields for a crack aligned with the direction of property variation. On similar lines, Chalivendra et al. [16] derived mixed-mode crack-tip stress fields for a crack inclined to the exponential elastic gradation. Jain et al. [17] continued analytical studies in FGMs by developing

Fig. 1 Propagating crack orientation with respect to direction of property variation in FGM

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both mode-I and mixed crack-tip field equations for linearly varying property variation. Recently, Chalivendra [18, 19] developed quasi-static crack-tip field equations for a crack in orthotropic nonhomogeneous solids using scaled coordinates approach proposed by Krenk [20]. For brevity, the authors consider a typical case of linear variation of Young’s modulus at an angle to property gradation as shown in Fig. 1 and the variation is expressed by:

E (X, Y) = E0 (1 + β X + γ Y) ,

(4)

where Eo is Young’s modulus value at the crack tip, β and γ are nonhomogeneity coefficients dependent on property gradation angle ϕ shown in Fig. 1. The compatibility equation in terms of Airys stress function for plane stress conditions takes the following form:

(1 + β X + γ Y)2 ∇2 ∇2 F − 2β (1 + β X + γ Y)

· ∂ ∇2 F − 2γ (1 + β X + γ Y) · ∂ ∇2 F

∂X

∂Y

+2 β2 + γ 2 ∇2 F − 2 (1 + ν)

· β2 ∂2 F + γ 2 ∂2 F − 2βγ ∂2 F = 0,

(5)

∂Y2

∂ X2

∂ X∂Y

where

∇2

=

∂2 ∂ X2

+

∂2 ∂Y2

.

It

may

be

observed

from

the

above equations that if nonhomogeneity coefficients

are set to zero, the equation reverts to the standard

biharmonic equation for homogeneous materials. The

nonhomogeneity coefficients make the solution process

complex and the solution for equation (5) is obtained

through an asymptotic analysis coupled with Wester-

gaard’s stress function approach [21]. The details for

the derivation of the solution of Airys stress func-

tion and the crack field equations can be found in

Jain et al. [17]. The normal stress in the y-direction

being the most prominent, the focus will be briefly

shifted towards it in an attempt to gain insight into the

stress field characteristics of FGMs. The equation is

described as:

1
σyy =
n=0

{Pn} + y {Pn} − y {Rn} + y {Qn}

− y2

y2

+ {Sn} − y {Sn} + β 2 {P0} − 2 {R0}

− y2 2

y2 {Q0} − 2 {S0} + γ

y2 2 {P0}

y2

y2

y2

− 2

{R0} + 2

{Q0} − 2

{S0} .

(6)

The variables P, Q, R, and S in the above equation are associated with powers of distance from the crack tip, r, with unknown coefficient whose numerical values can only be retrieved experimentally or numerically. These functions are commonplace in homogeneous analysis [21], with P0 and R0 exhibiting the well-known squareroot singularity for modes I and II, respectively, and their coefficients representing the stress intensity factors KI and KII.
Clearly, the prominent terms that involve singular characteristics are devoid of the influence of nonhomogeneity. The term involving the T-stress also remains unaltered. Only those beyond r1/2 power term are affected by the material nonhomogeneity. It is finally observed that mode mixity will not be a factor unless material variation is asymmetric with respect to the crack direction.
A more experimentally adaptable problem was later studied by Erdogan and Wu [22]. Indeed a surface crack normal to a boundary has been frequently the subject of experimental investigation under various configurations such as modified compact tension (MCT), single-edge notch tension (SENT), under three- or four-point, or uniaxial loading conditions, amongst others. The specific analytical solution includes the inherent material properties variations, and the stress intensity factor is found to be directly coupled with the elastic modulus μ(x). The predictive behavior of FGMs with variations of a decade between increasing and decreasing Young’s modulus gradients reveals a doubling in the value of normalized stress intensity factor.
Using the asymptotic analysis, theoretical crack tip expressions were first obtained for both exponential and linear variation of mechanical properties by Parameswaran and Shukla [23] for all three modes of dynamic fracture in graded materials. Chalivendra et al. [24] developed higher order mixed-mode crack tip stress field equations for a crack propagating at an angle to the property gradation direction. Subsequently, Shukla and Jain [25] and Lee [26] obtained crack-tip stress field equations using displacement potentials for a crack propagating in FGMs.
For a typical case of exponential variation of shear modulus, Lamé’s constant and mass density considered by Parameswaran and Shukla [15], the equations of motion takes the form:

∇2 + β ∂ − β κ − 1 ∂ω = κ − 1 ρ0 ∂2

∂X

κ + 1 ∂Y κ + 1 μ0 ∂t2

∇2ω + β ∂ω − β 3 − κ ∂ = ρ0 ∂2ω ,

(7)

∂X

κ − 1 ∂Y μ0 ∂t2

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where κ = (3 − 4ν) for plane strain, and the relation for plane stress given by κ = (3 − ν) / (1 + ν); ν is the Poisson’s ratio; = (∂u/∂ X) + (∂v/∂Y) is the dilatation and ω = (∂v/∂ X) − (∂u/∂Y) is the rotation; μ0 and ρ0 are the shear modulus and mass density at the origin; β is the nonhomogeneous factor. The above equations would reduce to the classical 2-D wave equations of dilatation and rotation by assigning β a value of zero. Due to nonhomogeneity, these equations lose their classical form and remain coupled in two fields
and ω, through the nonhomogeneity parameter β. Again, using an asymptotic approach, they provided a direct expression for the first stress invariant, though individual stress components may be derived from their formulation. The expression is provided for the −1/2, 0, and 1/2 powers of r, and is reproduced here for the exponential case, when the elastic variation is along the crack orientation:

σxx + σyy = 2 (λc + μc) · eβx

A0rl−1/2 cos θ2l + A1

+ A2r1/2 cos θl − β A0r1/2 cos 3θl

l

2 4αl2 l

2

+

2β αs

B0rs1/2 cos θs ,

(8)

(k + 2) αl2 − αs2

2

where

rl;s = x2 + αl2;s y2 ,

αl =

1 − ρ0c2 , μ0 (κ + 2)

tan θl;s = αl;s y , x

αs =

1 − ρ0c2 , μ0

λc and μc are the local (crack-tip) values of the Lamé’s constant and shear modulus, respectively; ρ is the material density, c is the crack speed, and κ is the ratio of the Lamé’s constant to the shear modulus. Note that just as in the work of Erdogan and Wu [22], the exponent characterizing the material variation appears outside the expansion, as an external factor. The asymptotic expansion itself follows the homogeneous model.
Recognizing that cracks could experience highly transient conditions during growth, Jain and Shukla [27] and Chalivendra and Shukla [28] obtained transient stress and out-of-plane displacement fields, respectively, for cracks propagating in FGMs. Recently, Chalivendra [29] has also derived transient crack-tip out-of-plane displacement field equations for cracks propagating along a gradually varying curved path in FGMs, and Lee et al. [30] has developed thermomechanical stress fields for propagating cracks in graded materials. In favor of brevity, these equations are not reproduced here due their extended nature.

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Instead, they are presented in an abridged form as they are utilized directly in the analysis of experimental data. The reader is thus referred to a later section on dynamic mixed-mode crack propagation.
The studies pertaining to the mode-III fracture of FGMs have also been numerous (e.g., [22, 31–37]) and have proven useful to understand the structure of crack tip fields in FGMs. The latest effort in the arena is attributed to Kubair et al. [38] who have provided an asymptotic solution to the problem of an inelastic FGM whose property variation is describable by a power law. Just as is the case for the other two modes of fracture, the stress variation is again shown to have the standard square-root of r singularity and retain a form identical to that of the homogeneous case. The second term in the expansion invokes the material property gradation. However, unlike the other solutions, the coefficient of the second term is fully defined in an unusually dependent relation to the first term. As mode-III problems are difficult to simulate experimentally, verification of this and other solutions is yet to be undertaken.
Quasi-static Experimental Investigations
Crack Perpendicular to the Direction of Gradation
Material system I
Butcher et al. [8] have published the first set of fracture experiments performed on FGMs with a continuous compositional gradation. The reader is directed to the original publication for details necessary to permit duplication of the material. Briefly, the method is based on a gravity assisted scheme wherein solid A-glass (soda-lime) spheres of mean diameter of ∼40 μm are mixed into a slow curing epoxy. The mixture is poured into a mold for preparing uniform thickness (5.5 mm) sheets of FGM. Gravity draws the heavy glass particles to the bottom of the mold, whereas the lighter epoxy rises to the surface. Concurrent gelation of the composite guards against complete segregation of the two constituents, resulting in a three-fold continuous, and semi-linear variation in elastic modulus (2,800 MPa < E < 8,500 MPa) over a range of 40 to 50 mm. Additional physical properties of these materials are listed in Table 1. For comparative purposes, those of other materials systems discussed in this paper are also presented in the table.
For conducting experiments, specimens 125 mm long and 24 mm high were manufactured from cast sheets, with gradation along the largest dimension of the specimens. Edge cracks of length 5.5 mm (150 μm width)

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Table 1 Comparison of physical properties of FGM material systems

Material system
I II III

Young’s modulus E (MPa)
2,800–8,500 160–250
3,600–5,200

Poisson’s ratio ν –
0.37–0.33 0.45 0.33–0.43

Fracture toughness KIc √ (MPa m)
1,400–2,200 1.8–0.8 470–800

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Density ρ (kg/m3)
1,150–1,850 960
1,200–950

Gradation range (mm)
40 150 250

were introduced at different locations along the length of the specimens, perpendicularly to the gradation, and the specimens were loaded in a three-point bend configuration. Also, the specimens were made specularly reflective in the region of interest by depositing an aluminum film onto the specimen surface to perform interferometric measurements.
Experimental method I: description of interferometer
The optical method of reflection-mode coherent gradient sensing [39], was used to map crack tip deformations. The schematic in Fig. 2 illustrates the experimental set up.
A collimated beam of light is incident on the specimen surface through a beam splitter. The reflected light beam (or the object wave front) carries the information about the surface non-planarity as a result of deformations near the crack tip. The angular deflections of

the light rays relative to the optical axis are then measured as interference fringes by a wave front shearing apparatus consisting of two Ronchi gratings of identical pitch with grating lines parallel to the y-direction (horizontal) for wave front shearing in the x-direction (vertical, coincident to the plane of the crack). The gratings are physically separated along the optical axis by a distance , to shear the wave fronts laterally in discrete directions depending on the grating pitch and the wave length of the light used. These wave fronts are collected by a positive lens of a camera and diffraction spots are displayed on the back focal plane of the lens. By filtering either the +1 or −1 diffraction spots, the optical fringe patterns representing contours of constant (∂w/∂ x) can be imaged. Mathematically, the governing equation of the technique is expressed by,
∂w = Np , N = 0, ±1, ±2 . . . , (9) ∂x 2

Fig. 2 Schematic of functionally graded beam and reflection CGS optical set-up used for mapping crack tip deformations [40]

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where N denotes the fringe orders, p is the grating pitch (25 μm), and is the grating separation distance (43 mm), resulting in an approximate sensitivity of 0.008o per half-fringe.
Designating the specimen thickness as B, the out-ofplane displacement w is related to the in-plane stresses as follows:

2w = − ν σx + σy . (10)

B

E

Therefore, the stress formulations posed in the previous section can be used to extract fracture parameters such as stress intensity factors KI and KII. Note that for an accurate estimation of stress intensity factors using 2D linear elastic K-dominant theories the data in the range of 0.4 < r/B < 0.7 and 90◦ < θ < 135◦ [39] is preferable.

Analysis of interferometric data
The experiments of Butcher et al. [8] featuring cracks normal to the direction of gradation were quickly extended by Rousseau and Tippur [40], who also com-

pared fracture responses of homogeneous and bimaterial specimens with those of FGMs. Cracks were placed at locations ξ = 0, 0.17, 0.33, 0.58, 0.83, and 1 along the graded region of the specimen. The variable ξ represents the fraction of distance from the most compliant location (ξ = 0: pure epoxy, corresponding to elastic modulus E1) to the stiffest location (ξ = 1: 52% volume fraction of spherical beads in the epoxy matrix, corresponding to elastic modulus E2). Homogeneous experiments were conducted, respectively, at both of these extremes, while the bimaterials were generated by conjoining independent specimens with properties E1 and E2.
Interferograms for two FGMs, a bimaterial, and a homogeneous material are presented in Fig. 3. A pronounced fringe asymmetry relative to the crack is evident for the bimaterial. It is far weaker, yet perceivable in case of FGMs, in the form of a fringe tilt toward the compliant region (left side). This asymmetry occurs in spite of the symmetric loading used in these experiments.
In this work, the analysis of the optical data considered solely the singular term which depended only on

Fig. 3 Crack tip interference

representing contours for

FGMs with ξ equal to (a)

0.33, (b) 0.58, (c) bimaterial

(V f = 0,0 and 0.5), and (d)

(a)

(b)

homogeneous glass/epoxy

composite (V f = 0.5) [40]

(c)

(d)

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the local properties at the crack tip, and did not include the influence of the gradation parameters. Companion finite element analyses were executed to evaluate stress intensity factors that agreed with the experimental results closely, thereby reinforcing the notion of dominance of singularity in cases where the crack crosses the path of the gradation. The results also showed that for such cracks, the stress field becomes inherently mixed mode with material deformation heavily shouldered by the compliant regions of the specimen.
These experiments also highlighted the superiority in performance of FGMs when compared to comparable bimaterials. As illustrated by Fig. 4, for similar magnitudes of loading, the energy release rates for FGMs, normalized by the local fracture toughness at the crack tip, fall consistently below those of bimaterials by at least a factor of 3. However, the apparent deficiency of the bimaterials cannot be attributed to interfacial weakness, as FGMs with cracks located at ξ = 0 and 1 could only be achieved by bonding, thereby the insertion of an artificial interface identical to that of the bimaterials. Also, comparative assessment between FGMs, which can also be inferred from Fig. 4, suggests that more favorable conditions exist when the crack is located on the stiffer side of the gradient.
As cracks extend in such FGMs, the material asymmetry gives rise to crack kinking, as shown in Fig. 5. Based on assumption of local crack homogeneity in FGMs, as formulated by Gu and Asaro [41], predictions of kink angle were plotted and compared with numerical as well as experimental measurements. All three techniques compared favorably showing a greater

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(a)

(b)

(c)

(d)

Fig. 5 Crack kinking angle in FGMs at ξ equal to (a) 0.17, (b) 0.58, and (c) 1.00; (d) variation of kink angle at all ξ locations for FGM with a/W = 0.25 [40]

kinking tendency as a crack is situated closer to the compliant end of the material.
Understanding of crack kinking in FGMs was further advanced by Abanto-Bueno and Lambros [42] by inserting an inclined crack with respect to the direction of gradation. However, before discussing their contribution, the model FGM material they have developed and the experimental technique they have used are first described in the following.

Fig. 4 Variation of energy release rate with ξ for FGM with a/W = 0.3 normalized by the corresponding critical values of energy release rate [40]

Material system II
Lambros et al. [43] have devised a way of preparing continuous FGMs for experimental modeling without using different constituents. The base material used for this purpose is polyethylene co-carbon monoxide (also known as ECO plastic), which undergoes degradation when irradiated with ultraviolet (UV) light. The

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resulting molecular alteration results in the material becoming stiffer and stronger but more brittle. They prepared specimens by placing in a UV irradiation chamber. The shield was linked to a stepping motor that gradually but continuously displaced it, exposing the specimen underneath for durations ranging between 5 and 122 hours, depending on the desired type of gradation. They have achieved up to a 1.5 fold increase in elastic modulus, relative to the unaltered ECO having a value of E = 160 MPa.
Experimental method II: description of digital image correlation

The 2D Digital Image Correlation (DIC) method scrutinizes the material surface from which in-plane displacements and strain between two consecutive loading states can be derived. The technique, originally introduced by Peters and Ranson [44] and subsequently developed by Sutton and co-workers [45], is a fullfield measurement method. Briefly, it consists of either making use of surface texture, or introducing superficial random speckles onto the specimen being investigated. Subsequent comparison and monitoring of speckles between two states of stress yield displacements and hence strains.

Data analysis of DIC

The uniqueness of the work carried out by AbantoBueno and Lambros [42], and later by Oral et al. [46] includes the introduction of mode-mixity by successively varying the direction of material gradation with respect to the placement of the crack. Furthermore, in addition to the generalized maximum hoop stress criterion used by Rousseau and Tippur [40], they have appended additional terms corresponding to the influence of the T-stress, giving rise to the relation:

KI sin α + KII (3 cos α − 1)

− 16 T√2πrc sin α cos α = 0, (11)

3

2

which includes a parameter rc related to the microstructural length scale of the materials.
The FGM is incrementally loaded, and values of stress intensity factors, along with T-stress are obtained by inserting the DIC generated displacements into equations (2) and (3) and simultaneously solving the equations using a Newton–Raphson approach. A typical post-initiation and post-kinking image from their work for an FGM is shown in Fig. 6. In addition to the conspicuous crack opening, the decorated speckles are also clearly visible on the surface of the specimen.

Fig. 6 Representative digital images recorded during the fracture tests, with crack tip location identified by arrow [42, 46]
Based on the results they note that withholding or including the T-stress term in the data analysis could have as much as 8% error in fracture parameters in case of homogeneous materials, inferring the need for its inclusion in FGMs. They also note that in homogeneous materials, crack growth criteria are solely guided by the maximum hoop stress. In FGM, on the other hand, the region of lower fracture toughness seems to also promote kinking towards it. Thus, although the prediction proved accurate to a maximum of approximately 20%, an additional criterion based on local failure properties is necessary for establishing a comprehensive FGM crack kinking rules.
Crack Parallel to the Direction of Gradation
Using the same material and optical techniques employed in the study of crack kinking, Lambros et al. [43] and Li et al. [47] investigated the quasi-static fracture behavior of ECO-based FGMs with edge cracks in the direction of gradation when loaded in pure mode-I configuration. The optical images were collected at a rate of 60 frames/s and used in concert with the 2D DIC technique to obtain full-field displacements. The experimental boundary conditions and measurements were then supplied to a finite element model, wherein a modified J-integral formulated for FGMs [48] was used to retrieve energy release rate and stress intensity factor of the loaded material. This hybrid method produced considerable success, as it resulted in a maximum difference of 12% with the exact solution of Erdogan and Wu [22]. They also studied conditions leading to crack initiation, and concluded that it is primarily dictated by the ability of the crack to accumulate enough

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energy, allowing it to exceed the local critical value at the crack tip. Finally, the monotonic increase in the magnitude of stress intensity factor or energy release rate with crack length, observed by these researchers, appears to be universal, as this also holds true for nonhomogeneous and bi-materials featuring crack normal to the gradient direction [40].
K-dominance in FGM
Experiments aimed at reaching a similar understanding of crack tip stress field when the plane of the latter lies in the direction of gradation were performed by Rousseau and Tippur [49]. The method of CGS, discussed previously, was used in their evaluation. Cracks were alternatively placed in the compliant and the stiff regions, respectively, and only a K-dominant formulation was used in the analysis of the interferograms. The latter are not reproduced here for they display similar demeanor to those of the homogeneous case presented in Fig. 3. Nevertheless, qualitative deviations from the homogeneous case are noticeable. When the crack is located on the stiff side of the gradient (decreasing gradient, or E2/E1 < 1), frontal fringe lobes are prone to expansion, while having the crack on the compliant side (increasing gradient, or E2/E1 > 1) has the opposite effect. These qualitative differences translate into differences in the measured value of the SIF. As seen in Table 1, for identical loading and geometry, the FGM with decreasing gradient registers nearly a 40% increase in SIF, when compared to its inverted gradient configuration, an FGM with a crack pointing toward the direction of decreasing gradient. However, one may not view the former as an inferior configuration since crack tip moduli of the two cases also differ by about 40%. On the other hand, when each FGM is compared to its homogeneous counterpart, it is clear that the FGM with increasing gradient is better than its counterpart, whereas, the FGM with decreasing gradient performs relatively poorly compared to the equivalent homogeneous material.

Further comparisons were performed between the experimental results, on the one hand, and finite element simulations of the experiments and the analytical solution of Erdogan and Wu [22], on the other hand. Whilst the latter two are identical, the experimental results underestimates them by about 20%. This points to the deficiency of analyzing crack tip fields using a K-dominant description for certain types of FGMs (Table 2).
Quasi-static crack growth experiments were also performed by the same authors, the results of which are shown in Fig. 7. Despite a lower crack tip toughness or resistance for the case corresponding to the crack residing on the compliant side, this specific FGM is capable of sustaining higher load before fracturing than the FGM having a crack on the stiff side of the specimen. Following initiation, no increase in load is necessary to sustain crack growth.
The issue of K-dominance in FGMs was further addressed by Abanto-Bueno and Lambros [50] during an evaluation of quasi-static crack growth in their ECO-based FGM. The specimen was subjected to pure mode-I loading and its surface strains measured by the DIC technique. All their evaluations were conducted for cracks parallel to the gradient direction, and located on the stiffer and more brittle side of the gradient (decreasing gradient, or E2/E1 < 1).
It must be noted that their R-curve differs from the afore-mentioned results reported by Rousseau and Tippur [49]. Indeed, whereas the latter reported no increase in loading to sustain propagation, the ECO material necessitates higher levels of energy to achieve continued growth, thereby resulting in a monotonically increasing R-curve. This difference is likely due to the difference, in this case increase, in ductility of ECO which is essentially absent in glass-filled epoxy FGMs.
Returning to the subject of K-dominance, AbantoBueno and Lambros [50] found only a limited region capable of justifying this assumption. Thus, only an area 4 mm distant from the crack tip (r/B = 0.13) and within a sweeping angle between 90◦ and 135◦ on either side of the plane of the crack could be used to provide

Table 2 Comparison of analytical, numerical, and measured stress intensity factors [49]

Method of calculation
FEA (plane stress, ν = 0.32 or 0.34)
Analytical (plane strain, ν unknown)
CGS; K-dominant description (plane stress, ν = 0.32 or 0.34)

FGM E2/E1 = 2.23 E0 = 4.8 GPa 1.27
1.28
1.06

Homogeneous E2/E1 = 1 E0 = 4.8 GPa 1.51
1.52
1.26

FGM E2/E1 = 0.41 E0 = 6.8 GPa 1.77
1.79
1.44

Homogeneous E2/E1 = 1 E0 = 6.8 GPa 1.46
1.45
1.20
FgmsCrackMaterialsFigVariation