Use of reciprocity considerations for the two-dimensional BEM analysis of wave propagation in an elastic half-space with applications to acoustic emission

Use of reciprocity considerations for the two-dimensional BEM analysis of wave propagation in an elastic half-space with applications to acoustic emission Irene Arias ∗and Jan D. Achenbach Center for Quality Engineering and Failure Prevention, Northwestern University, Evanston, IL 60208, USA Abstract A simple numerical treatment of the infinite boundary in the BEM analysis of twodimensional wave propagation problems in elastic half-spaces is proposed to avoid the spurious reflections of non-decaying Rayleigh waves introduced by the truncation of the boundary. The proposed method exploits the knowledge of the far-field asymptotic behavior of the solution to adequately correct the BEM displacement system matrix for the truncated problem to account for the contribution of the omitted part of the boundary. The reciprocal theorem of elastodynamics is used for a convenient computation of this contribution exclusively in terms of the boundary integrals of the original BEM system. The method is applied to the study of the acoustic emission from nucleating and propagating surface-breaking and buried cracks in a two-dimensional elastic half-space. It is shown to be particularly advantageous since it allows for an accurate calculation of the generated signal even when the observation point is located far from the acoustic emission source. ∗Corresponding author. Present Address: Graduate Aeronautical Laboratories, California Institute of Technology, 1200 E. California Blvd. MC: 205-45, Pasadena, CA 91125, USA. Tel: +1-626-395-4757, Fax: +1-626-449-2677. Email addresses: ia[email protected] (Irene Arias), [email protected] (Jan D. Achenbach). Preprint submitted to Elsevier Science 29 December 2003 1 Introduction The boundary element method (BEM) is ideally suited for the numerical analysis of problems of wave scattering by flaws such as cracks, cavities, and inclusions in elastic media. It has been successfully applied to the solution of twoand three-dimensional problems in the frequency domain and for unbounded or partially bounded elastic bodies. The most important feature of this numerical method is that it only requires the discretization of the boundary rather than the entire domain. In addition, the radiation condition at infinity is naturally included in the formulation. Therefore, it is particularly well suited for unbounded domains. However, the analysis of wave propagation in two-dimensional elastic half-spaces presents some difficulties, since the boundary is infinite. In elastodynamics, the BEM formulation for a half-space is usually stated in terms of full-space – rather than half-space – Green’s functions, and thus the discretization over the boundary of the half-space is needed in order to enforce the appropriate boundary conditions. Obviously, the infinite surface of the half-space has to be truncated for computational purposes. The simple truncation of the boundary produces spurious reflections of elastic waves at the limits of the computational domain. This approach can lead to accurate results in the three-dimensional case, since all types of waves exhibit geometrical attenuation and thus their reflections are not significant provided the discretization is extended far enough. However, in the two-dimensional case, Rayleigh waves propagate along the surface of the half-space without attenuation. Consequently, the spurious reflections are generally significant, independently of the extent of the discretization. In the frequency domain approach, this issue is typically addressed by adding a small amount of damping to the material. More sophisticated approaches include the infinite boundary element technique, first proposed in [1]. This technique maps the omitted part of the boundary, which is infinite, into a finite region. The behavior of the displacements and the tractions in the infinite region is modeled through decay functions suitable for each particular problem. Infinite elements have been used to analyze problems of wave propagation involving Rayleigh-waves [2,3]. In all cases, the resulting integrals over the infinite element require special numerical integration schemes and are particulary involved for the case of oscillatory kernels. In previous work [4], we have proposed a simple correction method for the truncation of the infinite boundary which allows the undamped Rayleigh waves to escape the computational domain without producing spurious reflections from its limits. This method is described here for a symmetric case of a half-space and applied to the numerical analysis of acoustic emission from the nucleation and propagation of cracks. The method exploits the knowledge of the asymptotic behavior of the solution – here Rayleigh surface waves are assumed to dominate the far-field solution – to adequately correct the BEM displacement system matrix for the truncated problem to account for the contribution of the omitted part of the boundary. The reciprocity theorem of 2 elastodynamics allows for a convenient computation of this contribution involving the same element integrals that form the original BEM system. The proposed method is easy to implement and, in the present case, it comes at essentially no additional cost as compared to the simple truncation of the boundary. The accuracy of the solution provided by the proposed model depends on the accuracy of the assumption that Rayleigh waves strongly dominate at the end points of the computational domain. However, once the computational domain is extended far enough from the source region for this assumption to hold, then, unlike the simply truncated solution, the corrected solution is accurate everywhere in the computational domain and for all computed times. In situations where the region of interest extends far beyond the source region where waves are generated, as is commonly the case in acoustic emission configurations, the proposed method reduces the extent of the computational domain. In this paper, the quantitative characterization of AE events is explored computationally for a selected set of examples. The acoustic emission from nucleating surfacebreaking and buried cracks is studied, and the generated surface motions in a twodimensional homogeneous, isotropic, linearly elastic half-space are analyzed by the boundary element method (BEM) in the frequency domain. The numerical approach considers defects of finite sizes, and computes the surface disturbances both in the near and far fields. Only cracks normal to the surface are considered here. However, the presented computational approach can also be used to address more general configurations and geometries. Acoustic emissions (AEs) are transient stress waves within solids radiated from localized sudden changes in the stress state. They are usually associated with damage events in the material, such as crack nucleation and growth, plastic activity, or various debonding and fracture mechanisms in composite materials. This wave motion propagates through the solid and eventually produces disturbances in the surface, which in principle can be detected. Thus, acoustic emission can be used to monitor the damage activity in specimens, provided that the generated signals are large enough relative to the noise level. Acoustic emission techniques have been used in various situations, from analyzing the development of texture in martensitic materials [5], to the monitoring of corrosion processes and welds in pressure vessels and bridges in service. Acoustic emission has also been applied to monitoring the fatigue crack growth in laboratory tests. A particularly fertile field of application is the analysis of damage in composite materials containing at least a brittle phase [6]. Acoustic emission techniques are quite different from other nondestructive evaluation (NDE) methods since internal flaws developing or evolving in all the sample can be detected by taking measurements in a limited region of the surface of the sample. Furthermore, AE techniques allow for continuous monitoring of components while they are in use, and damage events are active. This contrasts with conventional nondestructive techniques which analyze the integrity of components after all events 3 have occurred. Nevertheless, AE as a nondestructive testing technique presents several problems. Due to reflections with boundaries and other features of the sample, the recorded signals depend on the overall structure. It is very difficult to establish a direct correspondence between a disturbance on the surface and the particular event that caused it. Moreover, the detection threshold is often hindered by the background noise, and weak AE events may remain unnoticed. This is particularly important since the AE signals cannot be enhanced. Another drawback of AE testing is its irreproducibility; once an event has occurred and the associated disturbance has decayed, it is not possible to record it again. As noted in [7], a more fundamental difference between AE techniques and conventional NDE methods is the fact that the former detects the evolution of damage, rather than the level of damage itself. While in conventional NDE techniques the detection threshold in limited by the absolute size of the defect, detectability of AE signals depends on the rate of defect growth. Due to this inherent feature, AE is particularly useful for brittle materials, while for ductile materials significant defect growth may remain unnoticed [7,8]. Due to the above mentioned difficulties to quantitatively interpret acoustic emissions, a statistical approach has often been followed. Cumulative counts of AE events are often used to characterize the different phases that lead to failure in laboratory fatigue tests [9,10]. However, increasing attention is being drawn to the waveforms of individual damage events. For instance, in [11], the characteristic signals during the initial micro-cracking, further growth, and coalescence of micro-cracks into localized cracks complement the statistical analysis of the process. Similar attention to the individual waveforms is present in other experimental studies [12]. Theoretical efforts have been made to quantitatively analyze AE events. One approach borrowed from seismology consists in the representation of the AE process by point-sources, analogously to the shear dipole point representation in laser generation of ultrasound [13]. The force dipole tensor provides a simple and convenient way to represent various types of fracture events, while retaining the fundamental physics [7]. This approach however does not account for the finite extent of the defect, and is therefore valid only in the far field. Besides, it assumes that all the stress changes in the fracture event occur simultaneously, i.e. it does not address the dynamic crack propagation. A more detailed asymptotic analysis including the finite size, the curvature of the crack front, and the crack propagation speed in an unbounded solid has been developed in [8]. In particular, this analysis showed that brittle events generate stronger acoustic emission signals than ductile crack propagation. Similar analysis of the stress-waves radiated from sudden activity at the crack tip in an infinite body has been reported in [14]. In [15] the disturbances generated by the fracture processes of a buried penny-shaped crack on the free surface of a half-space were analyzed. The present paper is organized as follows. The proposed approach to model the acoustic emission from nucleating and propagating surface-breaking and buried cracks in 4 σ 8 σ 8 Initial state Emission problem for a surface-breaking crack Emission problem for a buried crack σ 8 H(t) σ 8 H(t) a a d Fig. 1. Modeling approach to the nucleation of surface-breaking and buried cracks. an elastic solid is described in detail in Section 2. The elastic solid is modeled as a homogeneous, isotropic, linearly elastic half-space. Since only cracks normal to the surface are considered, symmetry arguments restrict the analysis to a quarter-space. In Section 3, the numerical technique for the treatment of the infinite surface of the quarter-space is formulated and its implementation in a frequency domain boundary element scheme is discussed. Selected numerical examples are presented in Section 4 to illustrate some features of the surface disturbances and the application of the presented computational method to a relevant field in NDE. 2 Modeling approach We consider a homogeneous, isotropic, linearly elastic half-space subject to a uniform tensile stress at infinity σ∞parallel to the surface (see Fig. 1). By virtue of linear superposition, the total field is decomposed into the incident and emitted fields. The uniform stress field can be understood as a static incident field. The nucleation of a crack is viewed as a sudden release of the corresponding traction on the crack faces. Consequently, it is analyzed by considering the field generated by the sudden application of a horizontal traction of −σ∞H(t) on the crack faces, H(t) being the Heaviside step function. Thus, similarly to the point-source representation, we assume that all the changes in the source occur simultaneously. We consider surface-breaking cracks, as well as buried cracks. Crack growth is modeled by suddenly releasing the asymptotic crack tip field [16], as illustrated in Fig. 2. Again, it is clear that this approach does not address the dynamic crack propagation phenomenon. Furthermore, the in5 σ 8 σ 8 KI Initial state Emission problem for a surface-breaking crack Emission problem for a buried crack H(t) σ 8 σ 8 KI KI B A σasympt. H(t) σasympt. a a d Fig. 2. Modeling approach to the propagation of surface-breaking and buried cracks. crement of the crack must be small compared to other dimensions of the problem for the asymptotic crack field solution to be valid. A singular traction quarter-point (STQP) element has been used to reproduce the r−1/2singularity of the asymptotic stress field at the crack tip. By displacing the mid-node of a quadratic boundary element with straight-line geometry to a quarter of its length and adequately modifying the element shape functions, the interpolated traction field in the element exhibits the appropriate asymptotic behavior at the crack tip [17]. The release of the stress in the crack faces is treated numerically by replacing the Heaviside step function by S-shaped functions Sα(t). The sharpness of this regularized step, which is controlled by the parameter α, is a simple model for the brittleness or ductility of the fracture process. The considered S-shaped functions are of the form (see Fig. 3) Sα(t) = 1 1 + e(t−t0)/α ,(1) where t0is a time shift. In the numerical examples presented in the following Section, the values adopted for the parameters are: α= 0.08 and t0= 0.43µs, which correspond to a brittle event. 3 Numerical technique The above described elastodynamic problems for a two-dimensional half-space are solved numerically by the direct frequency domain boundary element method with 6 time σ 8 σ σσ 8 time large small αα Fig. 3. Regularized S-shaped step. quadratic interpolations. The symmetry of the emission problems allows us to restrict the analysis to a quarter-space. In the following, we present a numerical technique which allows the undamped Rayleigh waves propagating along the infinite surface of the quarter-space to escape the computational domain without producing spurious reflections from its limits. It should be noted that the vertical boundary of the quarterspace is a fictitious boundary resulting from symmetry considerations. Therefore, no Rayleigh waves travel along this boundary and it can be truncated at a sufficiently large distance without loss of accuracy. The frequency domain boundary integral equation for a point ξon the boundary of the quarter-space Γ in the absence of body forces may be obtained from the reciprocal theorem of elastodynamics as cαβ(ξ)uβ(ξ, ω) = ZΓhu∗ αβ(ξ,x, ω)tβ(x, ω)−t∗ αβ(ξ,x, ω)uβ(x, ω)idΓ(x), α, β = 1,2, (2) where u∗ αβ and t∗ αβ are the full-space frequency domain elastodynamic fundamental solution displacement and traction tensors respectively [18]. Note that u∗ αβ(ξ,x, ω) and t∗ αβ(ξ,x, ω) represent the “β” component of the displacement and the traction on the boundary, respectively, at the point xdue to a unit time-harmonic load of angular frequency ωapplied at the point ξin the direction “α”. Also, uβ,tβare frequency domain displacements and tractions on the boundary, ωstands for the angular frequency and cαβ is called the jump coefficient given by: cαβ(ξ) =      1 2δαβ,if Γ is smooth at point ξ, cαβ,if Γ has a corner at point ξ, (3) where δαβ represents the Kronecker delta. The jump coefficient for corner points can be derived by an indirect approach as described in [18]. The integrals in Eq. (2) are interpreted in the sense of the Cauchy Principal Value. Let us denote as Γ1the part of the boundary corresponding to the crack face, where 7  Γ 8 Γ 0 Γ 2 ξ ξΝ Γ 1 2 x 1 x  Fig. 4. Schematic definition of the computational domain: Γ = Γ∞∪Γ0∪Γ1∪Γ2. tractions are applied, and Γ2the remaining part of the fictitious vertical boundary of the quarter-space that is discretized. Let Γ0be the part of the traction-free surface of the quarter-space which will be included in the discretization and Γ∞the remaining infinite part which will be omitted (see Fig. 4). In this case, Eq. (2) becomes cαβ(ξ)uβ(ξ, ω) + ZΓ∞ t∗ αβ(ξ,x, ω)uβ(x, ω)dΓ(x) +ZΓ0∪Γ1∪Γ2 t∗ αβ(ξ,x, ω)uβ(x, ω)dΓ(x) = ZΓ1∪Γ2 u∗ αβ(ξ,x, ω)tβ(x, ω)dΓ(x). (4) The three characteristic waves along the traction-free boundary, namely the longitudinal, transverse, and Rayleigh waves, all contribute to the displacement field. However, it is well known that body waves exhibit geometrical decay in the propagating direction, whereas Rayleigh waves in two dimensions do not. Body waves generated by a line source in an infinite medium present cylindrical wavefronts and, consequently, the amplitude of the disturbance decays as r−1/2,rbeing the distance from the source. Along the surface of a half-space, the amplitude of the body wave disturbances decays as r−1. By contrast, Rayleigh waves produced by a line source in a half-space present plane wavefronts and the amplitude of the corresponding disturbance does not decay with r. Therefore, the displacement far-field solution can be approximated by the Rayleigh surface wave component of the solution, thereby neglecting the contribution of the body waves. Hence, if the truncation points ξNis located far enough from the source region, then we can write for the infinite boundary Γ∞ ξ∈Γ∞:uα(ξ, ω)≈R(ω)uSR α(ξ, ω),(5) where Ris the unknown complex amplitude of the far-field Rayleigh wave and uSR α represents the frequency domain displacements corresponding to a unit amplitude time-harmonic Rayleigh surface wave of angular frequency ωpropagating along the surface of the quarter-space in the positive direction. The expressions for uSR αcan be found, for instance, in [19]. 8 Hence, Eq. (2) can be rewritten as: cαβ(ξ)uβ(ξ, ω) + R(ω)Iα(ξ, ω) +ZΓ0∪Γ1∪Γ2 t∗ αβ(ξ,x, ω)uβ(x, ω)dΓ(x) = ZΓ1∪Γ2 u∗ αβ(ξ,x, ω)tβ(x, ω)dΓ(x),(6) where Iα(ξ, ω) := ZΓ∞ t∗ αβ(ξ,x, ω)uSR β(x, ω)dΓ(x).(7) Note that in Eq. (6) the complex amplitude Ris unknown, but the integrand in Eq. (7) is known. Therefore, the integrals Iα(ξ, ω) may be approximated numerically. Here, however, we propose a more elegant approach based on the reciprocity theorem of elastodynamics by which the integral Iα(ξ, ω) is replaced by an integral over a finite domain. Let us consider a harmonic Rayleigh surface wave of frequency ωand unit amplitude propagating along the free surface of the corresponding half-space in the positive direction. For the computation of Iα(ξ, ω) we introduce a vertical fictitious boundary of infinite extent. Let us choose the Rayleigh surface wave as one elastodynamic state and the time-harmonic full-space fundamental solution of the same frequency ωas the other elastodynamic state. By virtue of the reciprocity theorem of elastodynamics, and after the limiting process of taking the observation point ξto the boundary, an integral representation may be derived for the quarterspace as: cαβ(ξ)uSR β(ξ, ω) = ZΓhu∗ αβ(ξ,x, ω)tSR β(x, ω)−t∗ αβ(ξ,x, ω)uSR β(x, ω)idΓ(x).(8) Invoking the zero traction boundary conditions along Γ0and Γ∞, Eq. (8) becomes: ZΓ∞ t∗ αβ(ξ,x, ω)uSR β(x, ω)dΓ(x) = −cαβ(ξ)uSR β(ξ, ω) −ZΓ0∪Γ1∪Γ2 t∗ αβ(ξ,x, ω)uSR β(x, ω)dΓ(x) + ZΓ1∪Γ2 u∗ αβ(ξ,x, ω)tSR β(x, ω)dΓ(x). (9) Note that along the fictitious vertical boundary the quantities uSR βand tSR βdecay exponentially. Therefore, the infinite fictitious vertical boundary can be truncated without loss of accuracy, i.e. the integral along the infinite fictitious boundary can be extended to Γ1∪Γ2only. Eq. (9) allows us to compute the integrals Iα(ξ, ω), in terms of integrals over the bounded boundaries of the problem. This step constitutes the key to the present approach and provides a simple way of calculating the correction to account for the omitted part of the boundary represented by the second term in Eq. (6). Through Eq. (5), the solution is described asymptotically as a Rayleigh wave of unknown amplitude and phase. In order to eliminate the unknown Rin Eq. (6), the solution in the computational domain is matched to the far-field solution at the end 9 012345 −1.5 −1 −0.5 0 0.5 1 time (µs) vertical displacement (nm) ∆a = 5% ∆a = 10% ∆a = 15% 0246810 −1.5 −1 −0.5 0 0.5 1 time (µs) vertical displacement (nm) ∆a = 5% ∆a = 10% ∆a = 15% Fig. 10. Surface normal displacement due to the acoustic emission from the propagation of a buried crack (a= 1.0 mm and d= 0.5 mm) for different growth lengths ∆a, at distances of 4 mm (left) and 16 mm (right) from the plane of the crack. 0 1 2 3 4 5 6 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 time (µs) vertical displacement (nm) d = 1.0 mm d = 5.0 mm d = 10.0 mm Fig. 11. Surface normal displacement due to the acoustic emission from the nucleation of buried cracks of length a= 1.0 mm at a distance of 5.0 mm from the plane of the crack. The midpoints of the cracks are located at different depths dbeneath the surface. waveforms has been studied by considering the nucleation of crack of a specific size located at different distances beneath the surface. The corresponding surface disturbances at a distance of 5.0 mm from the plane of the crack are shown in Fig. 11. As pointed out, the relative significance of the Rayleigh wave component with respect to the body wave components decreases as the crack depth increases. In addition, it can be noted in Fig. 11 that the amplitude of the signals exhibit a maximum for an intermediate depth which suggests an angular dependency in the amplitude of acoustic emission signals. This has been further investigated by considering the variation along the surface of the half-space of the acoustic emission 16 0 1 2 3 4 5 6 7 8 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 time (µs) vertical displacement (nm) 0.5 mm 1.0 mm 1.5 mm 0 1 2 3 4 5 6 7 8 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 time (µs) vertical displacement (nm) 2.0 mm 4.0 mm 16.0 mm Fig. 12. Surface normal displacement due to the acoustic emission from the nucleation of a buried crack (a= 1.0 mm and d= 1.0 mm) at different distances from the plane of the crack. from a specific buried crack. Fig. 12 displays the normal surface displacements at different observation points due to the nucleation of a crack of length a= 1.0 mm located at a distance of d = 1.0 mm beneath the surface. It can be noted that as the observation point departs from the crack position, the amplitude increases, reaches a maximum, and then decreases in the far-field. This observation is consistent with the results on the angular variation of the amplitude of the acoustic emission reported in [8]. 5 Conclusions The acoustic emissions from the nucleation and propagation of surface-breaking and buried cracks have been calculated. A computational approach based on the boundary element method has been implemented. By the use of reciprocity considerations, a correction method for the truncation of the infinite boundary has been presented, which allows the undamped Rayleigh waves to escape the computational domain without producing spurious reflections from its limits. This technique produces numerical solutions which are highly accurate everywhere in the computational domain and for all computed times. This is particularly useful for studies of acoustic emission, since the observation point, which is often located far from the source region, can be brought close to the truncation point without loss of accuracy. It has been shown that in the limit of a small nucleating surface-breaking crack, the surface disturbances tend to those generated by a shear dipole at the surface, as has been pointed out in previous studies. We have analyzed the effect of the size of the nucleating crack and the length of the crack growth for both surface-breaking and 17 buried cracks. In addition, we have studied the effect of the buried depth in the surface disturbances originated form nucleating buried cracks. The analysis of the acoustic emission signals for nucleating buried cracks at different observation locations along the surface of the half-space has shown evidence of an angular dependence consistent with previous theoretical studies. Acknowledgements This paper is based upon work partially supported by the Federal Aviation Administration under Contract #DFTA 03-98-F-IA029, and partially supported by the Office of Naval Research under Contract N00014-89-J-1362. References [1] J. 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