A crack in a viscoelastic functionally graded material layer embedded between two dissimilar homogeneous viscoelastic layers – antiplane shear analysis

A crack in a viscoelastic functionally graded material (FGM) layer sandwiched between two dissimilar homogeneous viscoelastic layers is studied under antiplane shear conditions. The shear relaxation modulus of the FGM layer follows the power law of viscoelasticity, i.e., = ₀ exp (y/h) [t₀ exp (y/h) /t]^q, where h is a scale length, and ₀,t ₀,, and q are material constants. Note that the FGM layer has position-dependent modulus and relaxation time. The shear relaxation functions of the two homogeneous viscoelastic layers are =₁(t ₁/t)^q for the bottom layer and =₂(t ₂/t)^q for the top layer, where ₁ and ₂ are material constants, and t ₁ and t ₂ are relaxation times. An elastic crack problem of the composite structure is first solved and the `correspondence principle' is used to obtain stress intensity factors (SIFs) for the viscoelastic system. Formulae for SIFs and crack displacement profiles are derived. Several examples are given which include interface cracking between a viscoelastic functionally graded interlayer and a viscoelastic homogeneous material coating. Moreover, a parametric study is conducted considering various material and geometric parameters and loading conditions.     

Antiplane crack analysis of a functionally graded material by a BIEM

Elastostatic analysis of an antiplane crack in a functionally graded material (FGM) is performed by using a hypersingular boundary integral equation method (BIEM). An exponential law is applied to describe the spatial variation of the shear modulus of the FGM. A Galerkin method is applied for the numerical solution of the hypersingular traction BIE. Both unidirectional and bidirectional material gradations are investigated. Stress intensity factors for an infinite and linear elastic FGM containing a finite crack subjected to an antiplane crack-face loading are presented and discussed. The influences of the material gradients and the crack orientation on the stress intensity factors are analyzed.     

Dynamic stress intensity factor of a functionally graded material under antiplane shear loading

Summary The dynamic response of a finite crack in an unbounded Functionally Graded Material (FGM) subjected to an antiplane shear loading is studied in this paper. The variation of the shear modulus of the functionally graded material is modeled by a quadratic increase along the direction perpendicular to the crack surface. The dynamic stress intensity factor is extracted from the asymptotic expansion of the stresses around the crack tip in the Laplace transform plane and obtained in the time domain by a numerical Laplace inversion technique. The influence of graded material property on the dynamic intensity factor is investigated. It is observed that the magnitude of dynamic stress intensity factor for a finite crack in such a functionally graded material is less than in the homogeneous material with a property identical to that of the FGM crack plane.     

Equivalent Homogeneous Variable Depth Beams for Cracked FGM Beams; Compliance Approach

We explore the compliance approach for K _I evaluation of a cracked three point beam (TPB) of functionally graded material (FGM). We suggest an equivalent beam of variable height for cracked FGM beams which includes the homogeneous beam for the linear, quadratic and cubic variations in elastic modulus along the beam span. The efficacy of equivalent beams is demonstrated numerically on a 250 × 41 × 6 mm FGM beam with modulus varying from 210 – 315 GPa under 5000 N load.     

Orthotropic index for bone

An orthotropic index (OI) is proposed to indicate the existence of a preferred material direction in each of the symmetry planes of an orthotropic material such as bone. Currently, this function is performed by the anisotropy ratio (AR) of any two Young’s moduli or compressive (A _c ) and shear (A _s ) anisotropy factors comprised of complicated functions of the elastic constants. The OI incorporates the four independent engineering constants (the shear modulus and Poisson’s ratio in addition to the two Young’s moduli) in each symmetry plane into a single index. The OI thus improves upon the AR by reflecting orthotropy in a more holistic sense and upon the AR, A _c and A _s by taking on a unique value (zero) only when the material is in fact isotropic.     

Interface cracking between functionally graded coatings and a substrate under antiplane shear

Coating technology plays a significant role in a number of applications such as high temperatures, corrosion, oxidation, wear, and interface. In this paper, we investigate the interface cracking between ceramic and/or functionally graded coatings (FGM coatings) and a substrate under antiplane shear. Four coating models are considered, namely single layered homogeneous coating, double layered piece-wise homogeneous coating, single layered FGM coating and double layered coating with an FGM bottom coat. Mode III stress intensity factors (SIFs) are calculated for the different coating models. In the case of μ_L > μ₀ where μ₀ is the shear modulus of the substrate and μ_L the shear modulus of the material at the surface of the coating, it is found that the single layered FGM coating reduces SIF slightly, whereas the coating system with a top homogeneous layer and a thin FGM bottom layer reduces SIF significantly. In the case of μ_L < μ₀ the SIF is found to be larger for the FGM coatings than for the homogeneous coatings. The FGM coating, however, may still be superior to homogeneous coatings in this case as FGM coatings usually provide better bonding strength between the coating and substrate. Finally, the applicability of the SIF concept in the fracture of FGM coatings is discussed. Large modulus gradients in thin coatings may seriously restrict the application of SIFs as the SIF-dominant zone may fall into the crack tip nonlinear deformation and damage zone. The same argument exists for some interphase models in interface crack solutions.     

Transient thermal stress analysis of an edge crack in a functionally graded material

An edge crack in a strip of a functionally graded material (FGM) is studied under transient thermal loading conditions. The FGM is assumed having constant Young's modulus and Poisson's ratio, but the thermal properties of the material vary along the thickness direction of the strip. Thus the material is elastically homogeneous but thermally nonhomogeneous. This kind of FGMs include some ceramic/ceramic FGMs such as TiC/SiC, MoSi₂/Al₂O₃ and MoSi₂/SiC, and also some ceramic/metal FGMs such as zirconia/nickel and zirconia/steel. A multi-layered material model is used to solve the temperature field. By using the Laplace transform and an asymptotic analysis, an analytical first order temperature solution for short times is obtained. Thermal stress intensity factors (TSIFs) are calculated for a TiC/SiC FGM with various volume fraction profiles of the constituent materials. It is found that the TSIF could be reduced if the thermally shocked cracked edge of the FGM strip is pure TiC, whereas the TSIF is increased if the thermally shocked edge is pure SiC.     

The axisymmetric crack problem in a non-homogeneous interfacial region between homogeneous half-spaces

In this paper, the axisymmetric crack problem in a non-homogeneous interfacial region between two homogeneous half-spaces is considered. It is assumed that the shear modulus varies continuously between that of the two half-spaces; and the shear modulus for the interface region is approximated by = ₀ e^mz. By using Hankel transform technique the problem is reduced to a pair of singular integral equations. The solutions of the problem are obtained for different material combinations and loading conditions; and modes I and II stress intensity factors, and the direction of a probable crack growth are calculated.     

Intersonic shear crack growth along weak planes

Classical dynamic fracture theories predict the Rayleigh surface wave speed (c _R ) to be the limiting speed of propagation for mode-I cracks in constitutively homogeneous, isotropic, linear elastic materials subjected to remote loading. For mode-II cracks, propagating along prescribed straight line paths, the same theories, while excluding the possibility of crack growth in the speed regime between c _R and the shear wave speed, c _s , do not exclude intersonic (c _s <υ<c _l ) crack tip speeds. In the present study, we provide the first experimental evidence of intersonic crack growth in such constitutively homogeneous and isotropic solids, ever recorded in a laboratory setting. Intersonic shear dominated crack growth, featuring shear shock waves, was observed along weak planes in a brittle polyester resin under far-field asymmetric loading. The shear cracks initially propagate at speeds just above c _s and subsequently accelerate rapidly to the longitudinal wave speed (c _l ) of the solid. At longer times, when steady state conditions are attained, they propagate at speeds slightly higher than √2–c _s . The experimental results compare well with existing asymptotic theories of intersonic crack growth, and the significance of the preferred speed of √2–c _s is discussed. Received: 13 September 1999 / Reviewed and occerted: 19 November 1999     

An analytical approach for fracture and fatigue in functionally graded materials

The problem of brittle crack propagation and fatigue crack growth in functionally graded materials (FGMs) is addressed. The proposed analytical approach can be used to estimate the variation of the stress-intensity factor as a function of the crack length in FGMs. Furthermore, according to the Paris’ law, the fatigue life and the crack-tip velocity of crack propagation can be predicted in the case of fatigue crack growth. A comparison with numerical results obtained according to the Finite Element method will show the effectiveness of the proposed approach. Detailed examples are provided in the case of three-point bending beam problems with either a FGM interlayer, or a FGM external coating. A comparison is presented between two types of grading in the elastic modulus: a continuous linear variation in the FGM layer and a discrete approximation with a multi-layered beam and a constant Young’s modulus in each layer.     

J resistance behavior in functionally graded materials using cohesive zone and modified boundary layer models

This paper describes elastic–plastic crack growth resistance simulation in a ceramic/metal functionally graded material (FGM) under mode I loading conditions using cohesive zone and modified boundary layer (MBL) models. For this purpose, we first explore the applicability of two existing, phenomenological cohesive zone models for FGMs. Based on these investigations, we propose a new cohesive zone model. Then, we perform crack growth simulations for TiB/Ti FGM SE(B) and SE(T) specimens using the three cohesive zone models mentioned above. The crack growth resistance of the FGM is characterized by the J-integral. These results show that the two existing cohesive zone models overestimate the actual J value, whereas the model proposed in the present study closely captures the actual fracture and crack growth behaviors of the FGM. Finally, the cohesive zone models are employed in conjunction with the MBL model. The two existing cohesive zone models fail to produce the desired K–T stress field for the MBL model. On the other hand, the proposed cohesive zone model yields the desired K–T stress field for the MBL model, and thus yields J _R curves that match the ones obtained from the SE(B) and SE(T) specimens. These results verify the application of the MBL model to simulate crack growth resistance in FGMs.     

The Time-Averaged Elastodynamic J-Integral

The time-averaged path independent J-integral for a stationary crack subjected to time-harmonic elastic waves is introduced. It can be determined from remote fields providing an alternative approach to compute the stress intensity factors. The J-integral is evaluated for a semi-infinite crack impinged by a plane sheer wave at an oblique angle.     

The mode III axisymmetric crack problem in a non-homogeneous interfacial region between homogeneous half-spaces

In this study, the mode III axisymmetric crack problem in a non-homogeneous interfacial region between two homogeneous half-spaces is considered. The shear modulus of the interfacial layer is assumed to be μ₂(z)=μ₀ e_mz. It is also assumed that this shear modulus varies continuously between that of the two half-spaces. By using the Hankel transform technique the problem is reduced to a singular integral equation. The problem is solved for various material combinations, crack geometries and for three different sets of crack surface tractions, and the corresponding stress intensity factors are tabulated. This revised version was published online in August 2006 with corrections to the Cover Date.     

Sub-surface crack in an inhomogeneous half-plane: Wave scattering phenomena by BEM

Elastic wave propagation in a cracked inhomogeneous half-plane is studied herein with the aid of the Boundary Element Method (BEM). More specifically, inhomogeneity arises in the half-plane because of spatial dependence of its material parameters on the depth coordinate. Furthermore, conditions of plane strain are assumed to hold, while the external load is an incident time-harmonic pressure (P) wave. More specifically, inhomogeneity is restricted to the case where both shear modulus and density profiles are quadratic functions of depth, but vary proportionally to each other, while Poisson's ratio is fixed at one-quarter. This way, body wave speeds remain macroscopically constant and it becomes possible to recover appropriate fundamental solution using a functional transformation method in conjunction with the Radon transform. Subsequently, a non-hypersingular, traction-based BEM is developed for solution of this particular problem. The usual quadratic boundary elements are used for discretization of all surfaces, supplemented by special edge-type boundary elements to model crack-tips. The present methodology is validated against standard examples appearing in the literature, and is subsequently used to study boundary-value problems (BVP) involving a sub-surface crack in an inhomogeneous half-plane swept by time-harmonic P-waves. The results of this detailed parametric study demonstrate that surface wave-fields are sensitive to the degree of material inhomogeneity, to the characteristics of the incident wave and especially to the relative position of crack versus free-surface.     

Self-developing, glycerin-containing, thick-layer bichromated gelatin as a medium for recording volume holograms

The synthesis of layers of glycerin-containing, self-developing bichromated gelatin between 100 and 500 μm thick is described and the holographic characteristics of this light-sensitive material are discussed. Experimental data obtained by measuring the diffraction efficiency of holograms of two plane waves recorded using a symmetric system for layers of different thickness and various ammonium bichromate concentrations showed that the optimum layers for hologram recording are around 200 μm thick and have an ammonium bichromate concentration of 2–2.5% by weight of dry gelatin. The sensitivity of these layers is 5–10 J/cm². Pis’ma Zh. Tekh. Fiz. 25, 64–69 (March 12, 1999)     

Residual Stiffness of Cracked Cross-Ply Composite Laminates Under Multi-Axial In-plane Loading

In this paper, the Equivalent Constraint Model (ECM) together with a 2-D shear lag stress analysis approach is applied to predict residual stiffness properties of polymer and ceramic matrix [0/90 _n /0] cross-ply laminates subjected to in-plane biaxial loading and damaged by transverse and longitudinal matrix cracks. It is found that the longitudinal Young’s modulus, shear modulus and major Poisson’s ratio undergo large degradation as the matrix crack density increases, with Poisson’s ratio appearing to be the most affected by transverse cracking. In cross-ply laminates with thick 90° layer strip-shaped delaminations begin to initiate and grow from the tips of matrix cracks at the 0°/90° interface. These delaminations contribute to further stiffness degradation of such laminates, and hence have to be taken into account in failure analysis models. The thickness of the 90° layer plays an important role; the thicker the 90° layer, the bigger stiffness reduction suggesting a size (volume) effect at ply level. In SiC/CAS cross-ply laminates reduction in the longitudinal modulus occurs mainly due to transverse cracks, while the shear modulus appears to be the most affected by the presence of longitudinal cracks. The shear modulus reduction ratio predicted previously by a semi-empirical formula is, in the most of cases, within 10% of the current ECM/2-D shear lag approach value. In some cases, though, the error of the semi-empirical finite element expression can be as big as 20% since it fails to capture damage mode interaction.     

Crack Spacing Effect on the Brittle Fracture Characteristics of Semi-infinite Functionally Graded Materials with Periodic Edge Cracks

The effect of crack spacing on the brittle fracture characteristics of a semi-infinite functionally graded material (FGM) with periodic edge cracks is discussed. The incompatible eigenstrain induced in the material due to mismatch in the coefficients of thermal expansion is considered in the analysis. The nonhomogeneity of the material is simulated by an equivalent eigenstrain, whereby the problem is reduced to that of a cracked homogeneous material with incompatible and equivalent eigenstrains. A method is then formulated to calculate the stress intensity factor of periodic edge cracks in such a semi-infinite homogeneous medium and applied to calculate apparent fracture toughness of a semi-infinite FGM from its prescribed composition profile. Inverse calculation is also carried out to compute composition profile from prescribed apparent fracture toughness of the semi-infinite FGM. Numerical calculations are carried out for semi-infinite TiC/Al₂O₃ FGM and the results are shown in the figures.     

Comparison of elastic interaction of a dislocation and a crack for four bonding conditions of the crack plane

A comparison of elastic interaction of a dislocation and a crack for four bonding conditions of the crack plane was made. Four cases of single crystalline material, sliding grain boundary, perfectly bonded interface, and sliding interface were considered. The stress intensity factors arising from edge and screw dislocations and their image forces for the above four cases were compared. The stress intensity factor at a crack tip along the perfectly bonded interface arising from screw dislocation can be obtained from that in a single crystalline material if the shear modulus in the single crystalline material is replaced by the harmonic mean of both shear moduli in the bimaterial. The stress intensity factor at a crack tip along the sliding interface arising from edge dislocation in the bimaterial can be obtained from that along the sliding grain boundary in the single material if the μ/(1−ν) in the single material is substituted by the harmonic mean of μ/(1− ν) in the bimaterial where μ and ν are the shear modulus and Poisson's ratio, respectively. The solutions of screw dislocation near a crack along the sliding grain boundary and sliding interface are the same as that of screw dislocation and its mirror image. Generally, the effect of edge dislocation for perfectly bonded interface on the crack propagation is more pronounced than that for the sliding interface. The effect of edge dislocation on the crack propagation is mixed mode for the cases of perfectly bonded interface and single crystalline material, but mode I fracture for the cases of sliding interface and sliding grain boundary. All curves of Fx versus distance r from the dislocation at interface to the right-hand crack tip are similar to one another regardless of dislocation source for both sliding interface and perfectly bonded interface. The level of Fx for m=0 is larger than that for m=−1. This revised version was published online in August 2006 with corrections to the Cover Date.     

Crystal growth and elastic properties of In<Subscript>2</Subscript>Se<Subscript>3</Subscript>

Perfect α-In₂Se₃ single crystals have been grown, and ultrasound velocities, v _i (i = 1–7), have been measured in single-crystal α-In₂Se₃ in various directions for different polarizations. We have determined the components of its elastic tensor (C _ij ) and calculated its elastic characteristics: elastic compliance, Young’s modulus, shear modulus, linear and volume compressibilities, bulk modulus, and Poisson’s ratio.     

Acoustic emission from a surface-breaking crack under cyclic loading

Summary An isotropic, homogeneous, elastic half-space is subjected to a uniform stress system composed of a constant tensile stress with a superimposed cyclic tensile stress, both parallel to the free surface. The cyclic stresses are assumed to generate a surface-breaking crack of length l(t) which propagates normal to the surface. The unloading of the crack faces generates acoustic emission, which is primarily composed of surface waves. The elastodynamic reciprocity theorem for time-harmonic waves is used to determine the radiated system of transient and steady state surface waves. Dedicated to Professor Franz Ziegler on the occasion of his 70th birthday