Modeling of nonlinear interactions between guided waves ...

Ultrasonics 74 (2017) 106–123

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Modeling of nonlinear interactions between guided waves and fatiguecracks using local interaction simulation approach

http://dx.doi.org/10.1016/j.ultras.2016.10.0010041-624X/� 2016 Elsevier B.V. All rights reserved.

⇑ Corresponding author.E-mail address: [email protected] (C.E.S. Cesnik).

Yanfeng Shen, Carlos E.S. Cesnik ⇑Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI 48109, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 1 June 2016Received in revised form 26 August 2016Accepted 2 October 2016Available online 6 October 2016

Keywords:Ultrasonic guided wavesFatigue crackLISADamage detectionStructural health monitoringNonlinear ultrasonicsContact model

This article presents a parallel algorithm to model the nonlinear dynamic interactions between ultrasonicguided waves and fatigue cracks. The Local Interaction Simulation Approach (LISA) is further developed tocapture the contact-impact clapping phenomena during the wave crack interactions based on the penaltymethod. Initial opening and closure distributions are considered to approximate the 3-D rough crackmicroscopic features. A Coulomb friction model is integrated to capture the stick-slip contact motionsbetween the crack surfaces. The LISA procedure is parallelized via the Compute Unified DeviceArchitecture (CUDA), which enables parallel computing on powerful graphic cards. The explicit contactformulation, the parallel algorithm, as well as the GPU-based implementation facilitate LISA’s high com-putational efficiency over the conventional finite element method (FEM). This article starts with the the-oretical formulation and numerical implementation of the proposed algorithm, followed by the solutionbehavior study and numerical verification against a commercial finite element code. Numerical casestudies are conducted on Lamb wave interactions with fatigue cracks. Several nonlinear ultrasonicphenomena are addressed. The classical nonlinear higher harmonic and DC response are successfullycaptured. The nonlinear mode conversion at a through-thickness and a half-thickness fatigue crack isinvestigated. Threshold behaviors, induced by initial openings and closures of rough crack surfaces, aredepicted by the proposed contact LISA model.

� 2016 Elsevier B.V. All rights reserved.

1. Introduction

Fatigue cracks may exist in a broad range of engineering mate-rials and are considered precursors to catastrophic failures. Effec-tive detection of fatigue cracks at their early stages is of criticalimportance and particular interest [1]. However, unlike gross dam-age, the fatigue cracks are barely visible in their closed state,imposing considerable difficulty for the conventional ultrasonictechniques which are only sensitive to open cracks [2]. On theother hand, nonlinear ultrasonic techniques have shown muchhigher sensitivity to incipient structural changes with distinctivenonlinear features, such as higher/sub harmonic generation, DCresponse, mixed frequency modulation response (sidebandeffects), and various frequency/amplitude dependent thresholdbehaviors [3,4]. The integration of nonlinear ultrasonic techniquesinto guided wave based interrogation procedures is drawingincreasing attention from the Structural Health Monitoring(SHM) and Non-destructive Evaluation (NDE) communities,

because such a practice inherits both the sensitivity from nonlinearNDE techniques and the large-area inspection capability from SHMguided waves [5].

In order to achieve the optimum design of nonlinear guidedwave based SHM system and the effective interpretation of thesensing signals, efficient computational models are needed. Manytheoretical work on nonlinear guided wave phenomena inducedby material nonlinearity have been conducted. Deng investigatedthe generation and accumulation of higher harmonics of Lambwaves using analytical formulations [6,7]. Srivastava and di Scaleastudied the existence of antisymmetric or symmetric Lamb wavesat nonlinear higher harmonics via modal analysis approach and themethod of perturbation [8]. Ohm and Hamilton derived a time-domain evolution equation for nonlinear Rayleigh waves indispersive media [9]. Muller et al. investigated the second har-monic generation of Lamb waves in nonlinear elastic plates usingthe analytical solution through the use of a perturbation method[10]. Manktelow et al. used a finite-element based perturbationanalysis to study the wave propagation in nonlinear periodic struc-tures [11]. On the other hand, the simulation of localized nonlineardynamics induced by fatigue cracks is also a challenging task; the

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Y. Shen, C.E.S. Cesnik / Ultrasonics 74 (2017) 106–123 107

3-D rough crack surface features and complex mode conversioncharacteristics of guided waves further add to the difficulty.

To efficiently predict the crack induced nonlinear guided wavesensing signals, an analytical model based on the exact Rayleigh-Lamb solutions has been developed, simulating multi-modal Lambwaves interacting with a breathing crack [12]. But it is only a 1-Drepresentation of the interaction phenomena and requires theinput of multiple nonlinear interaction coefficients from othercomputational modules. To fully capture the 3-D nonlinear dynam-ics, finite element method (FEM) for contact-impact problems hasbeen widely investigated, maturing over the years [13]. But thehandling of massively large matrices inevitably result in heavy costof computer resource and computational time even for explicitsolving schemes. Moreover, the proper choice of contact parame-ters to achieve solution convergence and accuracy is always foundto be challenging and tedious, which requires extensive modelingexperience. Time domain Boundary Element Method (BEM) hasbeen explored as an efficient approach to solving Contact AcousticNonlinearity (CAN) by considering it as a boundary-type nonlinearproblem, in which the nonlinear effect is contained in the bound-ary conditions only [14]. Current literature on BEM mainlyaddresses the interaction between contact cracks and bulk waves,yet modeling efforts on guided waves are still limited. FiniteDifference (FD) method has also been reported for contact-impact analysis of crack surfaces with much less computationalburden compared to conventional FEM [15]. The formulations pre-sented in these studies mainly address the wave crack interactionin homogeneous materials. To solve wave propagation in heteroge-neous materials, the Local Interaction Simulation Approach (LISA),based on FD formulation and Sharp Interface Model (SIM), has beendeveloped. 1-D, 2-D, and 3-D LISA formulations were first devel-oped for isotropic heterogeneous materials executed on Connec-tion Machines [16–18]. LISA underwent considerable progressduring the past decade, with its application in metallic structures[19,20], extension to general anisotropic materials [21–23], cou-pled field capabilities [24], hybridization with other numericalmethods [25,26], and execution on powerful Graphics ProcessingUnits (GPU) with Compute Unified Device Architecture (CUDA)technology [27–29].

Pioneer research on including nonlinear effects into LISA formu-lation has been conducted using a spring model in 1-D, 2-D, and 3-D cases [30–32]. This spring model formulation was originally tar-geted towards solving nonlinear waves induced by hysteresismaterial properties, not towards Contact Acoustic Nonlinearity(CAN). Although, to certain degree, it can captured the local nonlin-earity, but the possible stick-slip contact scenario was not consid-ered. Furthermore, the fact that rough contact surfaces havedistributed microscopic initial openings and closures was alsonot considered. Alternative approach other than the spring modelis desired to improve current nonlinear LISA solutions to addressCAN. It should be pointed out that this study is not the first attemptto model contact phenomena using LISA. A simplified treatmentwas proposed by Martowicz et al. where LISA was deployed tomodel the nonlinear vibro-acoustic wave interaction in crackedaluminum plates [33].

In this study, a 3-D LISA scheme is developed to simulate thenonlinear interactions between guided waves and fatigue cracks.A Coulomb friction model is integrated to capture the stick-slipcontact scenario. Distributed initial openings and closures betweenthe crack surfaces are considered to better approximate the natureof fatigue cracks. The numerical scheme is implemented usingCUDA and executed on GPU, achieving high computational effi-ciency. After the introduction of the theoretical formulation, theimplementation of the 3-D contact LISA is discussed, followed bysolution behavior study and numerical verifications. The numericalcase studies address several interesting aspects in wave crack

interaction: (1) generation of higher harmonics and DC response;(2) nonlinear mode conversion at various fatigue crack types;and (3) threshold behaviors introduced by initial crack conditions.

2. Theoretical formulation

This section illustrates the theoretical fundamentals and formu-lation of the proposed LISA modeling technique to simulate thenonlinear interactions between guided waves and fatigue cracks.The penalty method for contact problems is discussed first. Theimportant features included in the new model is introduced, suchas the state gap function and stick-slip contact formulation. Finally,the LISA formulation with contact capability is derived.

2.1. Penalty method for contact problems

Penalty method is initially investigated to solve constrainedoptimization problems and have been adopted as one of the pri-mary approaches to simulate contact problems in FEM. It approx-imates a constrained problem by an unconstrained one whosesolution ideally converges to the solution of the original con-strained problem. This convergence is achieved by punishing theviolation of these constrains.

Fig. 1 illustrates the principle of using penalty method toapproximate contact continuum mechanics. From a strict dynam-ics perspective, the kinematic axiom enforces the impenetrabilitycondition, i.e.

b1 \ b2 ¼ / ð1Þwhich means the two bodies b1 and b2 cannot penetrate each other.When they are in contact with each other, the material points ontheir boundaries may coalesce during the motion of the bodies.The contact surface c is defined as

c ¼ @b1 \ @b2 ð2ÞIf b1 and b2 are not in contact, they can be treated as separate sys-tems. When the two bodies are in contact, they will interact witheach other through the contact compressional tractions, i.e., no ten-sile tractions can occur:

ta � na 6 0 ð3Þwhere ta is the Cauchy traction vector with respect to boundary @baand na is the outward unit normal vector at this boundary location.

Utilizing the penalty method, the impenetrability condition isweakened; a small amount of penetration d is allowed to enablethe mathematical formulation. This penetration can be easily iden-tified as the measurement of violation against the impenetrabilitycondition and is penalized by introducing a contact stiffness thattends to minimize this violation. When an appropriate contactstiffness is reached, the amount of the penetration approacheszero, which makes the numerical solution converges towards thephysical contact phenomena.

2.2. Adding contact dynamics into local interaction simulationapproach

LISA is based on the FD method. It approximates the partial dif-ferential elastodynamic equations with finite difference quotientsin the discretized temporal and spatial domains. The coefficientsin LISA iterative equations (IEs) depend only on the local physicalmaterial properties. The Sharp Interface Model (SIM) enforces thestress and displacement continuity between the neighboring com-putational cells and nodes. Therefore, changes of material proper-ties in the cells surrounding a computational node can be capturedthrough these coefficients. A typical computational node and its

Fig. 1. Using penalty method to approximate contact dynamics.

108 Y. Shen, C.E.S. Cesnik / Ultrasonics 74 (2017) 106–123

surrounding cells/nodes are shown in Fig. 2. The final IEs deter-mine the displacements of a certain node at current time stepbased on the displacements of its eighteen neighboring nodes atprevious two/three time steps, depending on whether materialdamping is considered. For details of the derivation for the LISAIEs, the readers are referred to [22].

A typical contact procedure during wave crack interaction canbe categorized into four consecutive situations: (1) pre-contactstate, where the contact surfaces are separated from each otherand are subjected to free boundary conditions; (2) contact initia-tion, where contact counterparts meet, which triggers the exertionof contact forces; (3) in-contact motion, where the crack surfacesmove together in a stick-slip pattern, with interactive contactforces; (4) contact pair separation, where the contact counterpartsleave each other, releasing the contact interactions and regainingthe free boundary conditions. It can be noticed that the boundarycondition of the contact surfaces alters between free and con-strained situations.

To satisfy the alternating boundary conditions at the contactsurfaces, special treatment of the computational grid is needed.

Fig. 2. A generic node C at location (I, J, K) w

In conventional LISA technique and the previous nonlinear springmodel, the structural discretization is continuous [30,31]. Here, adiscontinuous mesh to model the crack surfaces is used. The struc-tural discretization is carried out using the commercial FEM soft-ware ANSYS 14. Then, a model converter is programed usingCUDA C to convert the nodal connectivity and material allocationfrom the format of FEM to that of LISA. During such a conversionprocedure, three additional efforts are made to prepare the LISAcomputational grid for the contact analysis, including contact pairrecognition, normal direction detection, and auxiliary air cell addi-tion. Fig. 3a shows the discontinuous mesh with separate nodesalong the contact surface. A pair of nodes located on the counter-part of the crack surface is designated as a contact pair. The contactforces are acted interactively on the contact pair nodes. Fig. 3bshows a generic contact node and its normal direction with respectto the contact surface. It should be noted that since LISA uses astructured mesh; there are six normal directions appearing inpairs, i.e., þx;�x; þy;�y; þz;�z. In practical practice of LISA,curved surfaces can be approximated by fitting the structuredmesh to the target geometry using dense discretization [34].

ith eighteen nearest neighboring nodes.

Fig. 3. Special treatment on the computational grid: (a) using discontinuous mesh to model contact surfaces; (b) detecting the normal direction at a generic contact node; (c)adding auxiliary air cells and contact forces to satisfy the alternating boundary conditions.


Fig. 3c illustrates the addition of auxiliary air cells to the LISA com-putational grid surrounding the contact nodes to satisfy a freeboundary condition when the contact surfaces are separated fromeach other and immersed in air. It can be seen that five auxiliarynodes and four air cells are added to each contact node. ComparingFig. 3c with Fig. 2, one can realize that by adding the auxiliary cells,a special contact surface node is brought to the unified LISA com-putation representation with eighteen neighboring nodes. Itshould be noted that the air cells on the contact surfaces are notimplemented during structural discretization in ANSYS but aregenerated during the model conversion procedure. The materialproperties for air cells are set by multiplying the structural mate-rial properties with a severe reduction factor of 1� 10�12. Whenthe crack surfaces come into contact, penalty contact forces areacted on the contact nodes to constrain the in-contact motion. Inthis way, the modeling of the alternating boundary condition canbe achieved.

To incorporate contact forces into the LISA formulation, con-sider the elastodynamic equations:

@lðSklmnwm;nÞ þ f k ¼ q€wk; k; l;m;n ¼ 1;2;3 ð4Þwhere Sklmn is the stiffness tensor, w is the displacement field, f isthe body force, and q denotes the material density. From Fig. 3c,the contact forces can be converted into equivalent body forceterms in Eq. (4). Following the standard LISA derivation procedure,Eq. (4) can be transformed using customary FD transformations forsecond-order space and time derivatives. Sharp Interface Model isused to enforce the displacement and stress continuity at interfacesof material inhomogeneity. Linear combination of the equationsleads to the final LISA iterative equations for the three displacementcomponents. Detailed LISA IE derivation procedure has beenreported in the literature (e.g. [22]), and is omitted here. The finalIEs of displacements with body force terms can be expressed in ageneral form as

Wtþ1k ¼ wðWt

i ;Wtj ;W

tk;W

t�1k Þ þW f

k ð5Þ

where the resulting displacement Wtþ1k , in k direction at the target

time step t þ 1, depends on itself and its neighboring nodes at theprevious two time steps. For the purpose of this study, the focus

is on the body force induced displacement term W fk at the contact

node ðI; J;KÞ, which is given by

W fk ¼ v

8

Xa;b;c¼�1

f Iþa;Jþb;Kþck ¼ v � �f I;J;Kk ð6Þ

where v ¼ Dt2=�q, with Dt being the time step and �q being the aver-age density of all the eight cells surrounding node ðI; J;KÞ, and �f I;J;Kk

denotes the average body force over the volume enclosed by theeight neighboring cells.

Fig. 4 shows the treatment of cells surrounding the crack tip. Fora general crack tip node O, it has eight neighboring cells. However,it is also exposed to air between the crack surfaces. Thus, the phys-ical crack tip computation is assumed to be the superposition oftwo complementary configurations with two air cells added onthe left and right corner of the crack tip node, respectively. Sincethe air cells only possess severely diminished material properties,the computational setup can be further combined to a finalequivalent configuration, where the material properties of the fourcrack surface cells reduce to half of their original values. And thefinal computational points A, B, and C take the averagedisplacement value of A+, B+, C+ and A�, B�, C�, for instanceWA

k ¼ 12 ðWAþ

k þWA�k Þ. Therefore, such a treatment brings a crack

tip node to the general computational configuration with eightneighboring cells and eighteen surrounding nodes. The only differ-ence is that the crack surface neighboring cells take only half oftheir original material properties values. All these assumptions ofsuperposition and equivalent configuration are made possibledue to the fact that the final LISA IEs are linear combinations ofthe displacement contributions. The material property adjustmentand nodal identification are completed in the model conversionprocedure. Similar crack tip treatment concept was proposed in aprevious finite difference study for crack edge analysis [35].

2.2.1. Equations for stick contact motionThere are several possible contact scenarios with three possible

contact directions and stick-slip motions. The stick contact motionis discussed first, where the crack surfaces moves together alongwith each other without relative slip motion during contact.Then, the formulation for slip contact motion is given in the nextsection.

According to Section 2.1, the penalty method states that inorder to punish the relative penetration d, the contact force can

be casted into the form of F ¼ kCd, where kC is the contact stiffness.When solving for the tangential contact force in stick motion, d canbe generalized to relative tangential violation. Between two con-tact surfaces, three contact forces may exist: a normal directioncomponent and two tangential direction components. In a localcontact coordinate system (XN � XS1 � XS2) shown in Fig. 5, the nor-mal contact force can be written as

Fig. 4. Treatment of cells surrounding the crack tip.

Fig. 5. A general local contact coordinate with respect to contact surfaceorientation.


FN ¼ HðdNÞKNkCNdN

DxNDxS1DxS2 ð7Þ

whereHðdNÞ is a control parameter taking the value of either one orzero, corresponding to in-contact and separate situations, respec-

tively; KN is the normal contact stiffness multiplier; kCN is the nor-mal contact stiffness/modulus; dN is the contact conditionviolation (normal penetration in this case); DxN; DxS1; DxS2 arethe spatial cell sizes along three local contact coordinate axes. Theterm dN=DxN can be interpreted as the normal contact strain. Theterm DxS1DxS2 represents the effective area on which the contactstress acts. Thus, the meaning of Eq. (7) is straight forward. For anormal contact case, the penetration can be evaluated by

dN ¼ ðWnþ;tN �Wn�;t

N Þ � bugap ð8Þ

whereWnþ;tN andWn�;t

N stand for the normal direction displacementsof a contact pair on the contact surfaces with nþ and n� as normaldirections at time step t. An initial distributed gap function bugap ateach contact pair is considered to capture the distributed initialcrack status. It is clear that in order to exert contact force only dur-ing contact, HðdNÞ is governed by

HðdNÞ ¼ 1; dN > 00; dN 6 0

(ð9Þ

The tangential contact force FS1 in XS1 direction is given by

FS1 ¼ HðdNÞKSkCS1dS1

DxNDxS1DxS2 ð10Þ

whereKS is the tangential contact stiffness multiplier; kCS1 is the tan-gential contact stiffness/modulus; dS1 is the tangential violation inXS1 direction; The term dS1=DxN can be interpreted as the engineer-ing tangential contact strain. The tangential violation dS1 for this sit-uation take the following form

dS1 ¼ ðWnþ;tS1 �Wn�;t

S1 Þ � dS1 C ð11ÞIt can be noticed that tangential violation dS1 involves an initial

condition term dS1 C which indicates the relative tangential dis-placements at the instant of contact initiation. This initial condi-tion term should be updated during the time marchingprocedure whenever the crack surfaces come into contact

dS1 C ¼ Wnþ;t cS1 �Wn�;t c

S1 ð12Þwhere the superscript t c means the time step when contact wasinitiated.

Similarly, for XS2 direction, the tangential contact force FS2 isgiven by

FS2 ¼ HðdNÞKSkCS2dS2

DxNDxS1DxS2 ð13Þ

where kCS2 is the corresponding tangential contact stiffness/modulusand dS2 is calculated by


S2 Þ � dS2 C ð14Þ

dS2 C ¼ ðWnþ;t cS2 �Wn�;t c

S2 Þ ð15ÞUp to this point, a normal contact force and two tangential con-

tact components have been expressed in the general local contactcoordinate system. For the purpose of numerical implementationand code generation, the contact force in X1 � X2 � X3 coordinatecan be expressed using a unified formulation which embraces allthree possible contact surface orientations shown in Fig. 6. Theunified expressions of the contact forces are

F1 ¼ HðdNÞ XX �KNkC11d1Dx1

Dx2Dx3 þ YX �KSkC66d1Dx2

Dx1Dx3

�þZX �KSkC55

d1Dx3

Dx1Dx2

�F2 ¼ HðdNÞ XY �KSkC66

d2Dx1

Dx2Dx3 þ YY �KNkC22d2Dx2

Dx1Dx3

�þZY �KSkC44

d2Dx3

Dx1Dx2

�F3 ¼ HðdNÞ XZ �KSkC55

d3Dx1

Dx2Dx3 þ YZ �KSkC44d3Dx2

Dx1Dx3

�þZZ �KNkC33

d3Dx3

Dx1Dx2

�ð16Þ

where XX; YX; ZX, etc. are case selection parameters which takethe value of one, negative one, or zero. These parameters are givenin Table 1 for various contact surface orientations corresponding toFig. 6. It should be noted that these parameters not only select thecontact case but also determine the contact force direction. Forinstance, when a contact surface normal direction is x1þ, the corre-sponding parameter is �1, denoting a contact force pointing

Fig. 6. Contact scenarios with contact surfaces oriented in three different directions.

Table 1Case selection parameters for the unified formulation.

Contact surfacenormal direction

XX XY XZ YX YY YZ ZX ZY ZZ

x1þ �1 �1 �1 0 0 0 0 0 0x1� +1 +1 +1 0 0 0 0 0 0x2þ 0 0 0 �1 �1 �1 0 0 0x2� 0 0 0 +1 +1 +1 0 0 0x3þ 0 0 0 0 0 0 �1 �1 �1x3� 0 0 0 0 0 0 +1 +1 +1


towards the surface and satisfy the contact constraint in Eq. (3). Itcan be noticed that there are totally three columns of expressionsin Eq. (16) and each of them corresponds to a contact scenario inFig. 6. For instance, when the case selection parameters are set tothe first two rows of Table 1, only the first column of Eq. (16) is acti-vated. In this case, d1 ¼ dN evaluated from Eq. (8), d2 and d3 corre-spond to the tangential violations evaluated from Eqs. (11) and (14).

After obtaining the contact forces at the contact nodes, the con-tact forces are converted into the equivalent body forces. Accordingto the definition of body force, the contact force per unit volume isevaluated, i.e.,

�f k ¼ Fk=ðDx1Dx2Dx3Þ ð17Þwhere Dx1Dx2Dx3 is the nominal volume that encloses each contactnode. Substituting Eqs. (17) and (16) into Eq. (6) yields

Wf stick1 ¼HðdNÞvðXXKNkC11d1g

2x þYXKSkC66d1g

2y þZXKSkC55d1g

2z Þ

Wf stick2 ¼HðdNÞvðXYKSkC66d2g

2x þYYKNkC22d2g

2y þZYKSkC44d2g

2z Þ

Wf stick3 ¼HðdNÞvðXZKSkC55d3g

2x þYZKSkC44d3g

2y þZZKNkC33d3g

2z Þ

ð18Þ

where gx ¼ 1=Dx1; gy ¼ 1=Dx2; gz ¼ 1=Dx3. Thus, the displacementcontributions from contact forces are obtained for stick contactmotion. However, slip contact motion may also exist and will beaddressed in the following section.

2.2.2. Equations for slip contact motionIn this study, the stick-slip contact motion is considered by

introducing the Coulomb friction model shown in Fig. 7. When

Wf slip1 ¼ HðdNÞv XXKNkC11d1g

2x þ YXlDK

NkC22d2g2y

d1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðd1Þ2 þ ðd3Þ2

q þ ZXl

0B@Wf slip

2 ¼ HðdNÞv XYlDKNkC11d1g

2x


q þ YYKNkC22d2g2y þ ZYl

0B@Wf slip

3 ¼ HðdNÞv XZlDKNkC11d1g

2x


q þ YZlDKNkC22d2g

2y ffiffiffi

ðdq

0B@

the tangential contact force is under a critical value FSmax, the

crack surfaces move following the stick contact motion as dis-cussed in Section 2.2.1 and the tangential force grows linearlywith the tangential violation dS. When the tangential contactforce exceeds the critical value, the crack surfaces will undergorelative slip motion.

Following the practice of Section 2.2.1, the equations will befirst derived in the general local contact coordinate system(XN � XS1 � XS2). The critical condition between stick and slip con-tact motions can be expressed asffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðFS1Þ2 þ ðFS2Þ2

q> lSF

N ð19Þ

where lS is the static friction coefficient of the material. For numer-ical implementation, Eq. (19) can be written as

KSffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkCS1dS1Þ

2 þ ðkCS2dS2Þ2

q> lSK

NkCNdN ð20Þ

When this condition is satisfied, the slip contact motion will betriggered. In such a case, the normal contact force will not be influ-enced and still take the same form as that in Section 2.2.1. However,the tangential contact forces will be the sub-components of thedynamic friction force developed between the contact surfaces, i.e.ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðFS1Þ2 þ ðFS2Þ2

q¼ lDF

N ð21Þ

where lD is the dynamic friction coefficient. Thus, the normal andtangential contact forces can be expressed as

FN ¼ HðdNÞKNkCNdN

DxNDxS1DxS2

FS1 ¼ lDHðdNÞKNkCNdN

DxNDxS1DxS2

dS1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðdS1Þ2 þ ðdS2Þ2

qFS2 ¼ lDHðdNÞKNkCN

dN

DxNDxS1DxS2

dS2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðdS1Þ2 þ ðdS2Þ2

qð22Þ

It should be noted that, compared with the stick contact motion,the estimation of tangential violations dS1 and dS2 also changes andthe tangential violations need to be updated for every time stepduring the slip contact motion. The judging criterion in Eq. (20)also need to be tested for each time step. In this case, the tangentialviolations become the relative motion between two consecutivetime steps and can be written as


S1 Þ � ðWnþ;t�1S1 �Wn�;t�1

S1 ÞdS2 ¼ ðWnþ;t

S2 �Wn�;tS2 Þ � ðWnþ;t�1

S2 �Wn�;t�1S2 Þ

ð23Þ

After converting this local contact coordinate expressions intothe unified formulations in the computational coordinates, andsubstituting Eqs. (22) and (17) into Eq. (6), one arrives at the finaldisplacement components contributed by slip contact forces

DKNkC33d3g

2z


q1CA

DKNkC33d3g

2z


q1CA

d3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1Þ2 þ ðd3Þ2

þ ZZKNkC33d3g2z

1CAð24Þ

Fig. 7. Stick-slip contact motion governed by Coulomb friction model.

Fig. 8. Rough surfaces of a fatigue crack with initial openings and closures.


It should be noted that the contact LISA formulations developedin this study are targeted towards the simulation of ultrasonicwaves in solids for SHM and NDE applications. It is assumed thatthe global material stays within the linear elastic region and thelocalized contact surfaces being the only nonlinear source. It is alsosafe to assume that the material particle motions are small inamplitude by orders of magnitudes compared to the size of the dis-cretization and thus the relative slip motion of contact surfaceswill not distort the computational grid.

2.2.3. Capturing initial crack openings and closures using the gapfunction

The observation of fatigue cracks revealed that the micro struc-ture of the crack surfaces is roughwith random3-D distributed vari-ations in the order ofmicrometers [36]. Onemay notice thatmerelyfor the consideration of wave propagation, such micro geometrieswill not have any influence on the elastic waves, considering thewave length is orders ofmagnitude larger than themicro scale.How-ever, when the contact mechanism comes into play, such structuraldetails become important. As shown in Fig. 8, the rough surfaces of afatigue crackwill form initial openings and closures, distributed in arandompattern. The nonlinearity of the resultingwaves depends on

Wf slip1 ¼ HðdNÞv XXKNkC11ðd1 þ bu gapÞg2

x þ YXlDKNkC22d2g

2y

d1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðd1Þ2 þ ð

q0B@

Wf slip2 ¼ HðdNÞv XYlDK

NkC11d1g2x


q þ YYKNkC22ðd2 þ bu ga

0B@Wf slip

3 ¼ HðdNÞv XZlDKNkC11d1g

2x


q þ YZlDKNkC22d2g

2y q

0B@

the relative amplitude between thematerial particlemotion and thesizeof the initial openings/closures, i.e., in order to generate thenon-linear phenomena, the interrogating waves need to be strongenough to open and close the crack surfaces by overcoming the ini-tial crack openings and closures. Consequently, threshold nonlinearbehaviors may be introduced.

The micro scale structural details are beyond the modelingcapability of conventional numerical methods and will easily resultin a problem that is computationally prohibitive. In an effort tocapture the influence of the micro crack structures, a distributedgap function bugap is introduced at each contact pair. It is straightforward to assume a gap function with normal distribution. Butfrom a rigorously scientific perspective, this distribution shouldbe generated from the measurements of fatigued samples, whichis outside the scope of this paper. Here, the focus is to extend LISA’smodeling capability to address the initial crack conditions andnonlinear threshold behavior in a general fashion by

ugap ¼ Pðx1; x2; x3;WÞ ð25Þwhere x1; x2; x3 represent the location of the contact pair, and W isthe distribution function which may come from previous experi-ence or experimental measurements.

It should be noted that when the gap function takes a positivevalue, it represents the initial openings at the contact pairs. Onthe other hand, the physical meaning behind a negative gap valuecan be interpreted by relating it to an initial closure status, i.e., theinitial closure stress br can be converted to a negative gap functionby

bugap ¼brDxNKNkCN

ð26Þ

Since the crack closure stress is a static boundary conditionrather than a transient initial condition, it will thus not developinto a propagating wave field. Hence, when conducting the tran-sient dynamic solution, this initial negative gap is excluded fromthe computation of the normal direction contact force. On theother hand, the initial closure stress will influence the stick-slipmotion judgment and tangential force computation and shouldbe included in the tangential motion terms.

To achieve this, the normal penetration formulation (Eq. (8)) iskept and the stick-slip motion condition (Eq. (20)) remainsunchanged, but compensation is made on the negative gap in thenormal terms in the final displacement formulation. When initialclosure at a contact pair is detected, the following expressionsare invoked instead of Eqs. (18) and (24)

Wf stick1 ¼HðdNÞv½XXKNkC11ðd1þ bugapÞg2

x þYXKSkC66d1g2y þZXKSkC55d1g

2z �

Wf stick2 ¼HðdNÞv½XYKSkC66d2g

2x þYYKNkC22ðd2þ bu gapÞg2

y þZYKSkC44d2g2z �

Wf stick3 ¼HðdNÞv½XZKSkC55d3g

2x þYZKSkC44d3g

2y þZZKNkC33ðd3þ bu gapÞg2

z �ð27Þ

ffiffiffiffiffiffiffiffiffiffid3Þ2

þ ZXlDKNkC33d3g

2z


q1CA

pÞg2y þ ZYlDK

NkC33d3g2z


q1CA


þ ZZKNkC33ðd3 þ bu gapÞg2z

1CAð28Þ


It can be observed that all the normal terms are compensatedby adding bu gap, excluding the closure stress from dynamic normalcontact forces and including it in tangential components computa-tion. In this way, the micro initial openings and closures effects arecaptured using the contact LISA model.

2.3. Numerical implementation and parallel computing on GPU

After presenting the theoretical formulations of the contactLISA, the numerical implementation and code generation are illus-trated. Fig. 9 shows the implementation logic of the contact algo-rithm. The preprocessor of commercial FEM code ANSYS is usedto build the model geometry, generate the mesh, and allocatematerial properties. The nodal connectivity results from ANSYSare then input into the model converter to obtain the LISA grid con-nectivity and cell material properties. In the meanwhile, the modelconverter detects the contact pairs by looking for co-located nodesin the mesh. The surface normal direction of each contact node isfound out for case selection according to Table 1. During this pro-cedure, a non-contact node will identify its eighteen neighboringnodes and the material properties of eight surrounding cells, whilea contact node will be facilitated with five auxiliary nodes and fourair cells (as shown in Fig. 3). For a non-contact node, its motion isgoverned by the conventional linear LISA formulation, i.e., the w

Fig. 9. Contact modeling im

term in Eq. (5). And the nodal displacement result Wt¼Mþ1k at the

target time step t ¼ M þ 1 can be computed based on the displace-ment results of its eighteen neighboring nodes at the previous twotime steps at t ¼ M and t ¼ M � 1. For a contact node, its motion isgoverned by Eq. (5) containing both the conventional linear LISA

formulation w and the contact force contribution Wf ;t¼Mk . To calcu-

late Wf ;t¼Mk , the penetration dN and tangential violations dS1; dS2 are

evaluated by Eqs. (8), (11), and (14). If penetration is detected, i.e.,dN > 0, then the contact computational configuration is initiated

(H ¼ 1; Wf ;t¼Mk – 0), else the computational configuration uses a

crack open status, where the contact force contribution is zero

(H ¼ 0; Wf ;t¼Mk ¼ 0).

When the crack surfaces come into contact, the in-contactmotion type needs to be determined between stick contact and slipcontact according to Eq. (20). For stick contact motion, the initialrelative tangential positions dS1 C and dS2 C need to be updated byEqs. (12) and (15), whenever the contact status changes fromout-of-contact to in-contact, i.e., ðdN;t¼M�1 6 0Þ \ ðdN;t¼M > 0Þ. Con-sequently, when such an update happens, the tangential violationsare also re-evaluated accordingly. If initial opening exists between a

contact pair (bu gap P 0), Eq. (18) is invoked to calculate Wf ;t¼Mk . If

initial closure is detected, Eq. (27) is used instead. For slip contact,

plementation process.

Fig. 10. Simplified model of ideal contact force eliminating penetration.


the tangential violations need to be updated and re-evaluated ateach calculation step according to Eq. (23), as long as the in-contactmotion type stays as slipmotion. If the initial opening existsbetween a contact pair ( bugap P 0), Eq. (24) is invoked to calculate

Wf ;t¼Mk . If initial closure is detected, Eq. (28) is utilized instead.

Finally, the contact force contribution Wf ;t¼Mk is added to the linear

LISA resultw to obtain the final displacement resultWt¼Mþ1k of a con-

tact node. The principle idea behind the algorithm is to judge thecomputational configuration (crack close/open and stick/slipmotion) based on the results of the previous two time steps andcompute the contact force contribution towards the displacementsat the target time step. Time marching procedure will progress thesolution to obtain the simulated time-space wave field.

In this study, the contact LISA algorithm was implementedusing CUDA technology and the computation was executed in par-allel on GPUs (NVIDIA GeForce Titan of 2688 CUDA cores with28,672 concurrent threads). There are two major characteristicsof the current contact LISA formulation that enables the computa-tion to be expedited. First, LISA is massively parallel. This isbecause the computation of a general node or a contact node onlydepends on the solutions of its eighteen neighboring nodes at theprevious two time steps. Thus, the behavior of each node is inde-pendent from the others at the target time step, i.e., the computa-tion of each node can be carried out individually in parallel.Second, the wave propagation simulation tasks usually requiredense discretization of the structure, resulting in a computation-ally intensive problem. GPUs, with their massive concurrent threadfeature, are suitable to handle such large size problems by dis-tributing the workloads among a large number of functional unitsand carry out highly efficient parallel computing. During the com-putation, the parameters are first established in the host memory(RAM). Then a copy of these parameters is sent to the device mem-ory (GPU global memory) for it to be processed. The computationof each node is assigned to a functional thread, i.e., each threadgathers the displacements of its eighteen neighboring nodes andone contact pair node (if identified as a contact node) at previoustwo time steps, process the material properties in the eight sur-rounding cells, and execute the kernel to compute the displace-ments of this node at the target time step. Since one of thebottlenecks of a CUDA program is the data transfer between thedevice memory and host memory, results are transferred fromthe GPU to the CPU only sporadically (every 20–30 steps depend-ing on the frequency of the propagating waves) to minimize suchdata transfer.

It should be pointed out that the workflow shown in Fig. 9involves the computational configurations for different contactscenarios such as contact or not as well as stick or slip motionand thus different equations are applied. However, the computa-tion on each GPU thread is kept unified by the same ‘‘case struc-ture” during CUDA programing, i.e., each computational nodewill go through the same selection procedure of the corresponding‘‘case” for each time step, which ensures the parallel execution onthe GPU threads. At the end of each time step, CUDA synchroniza-tion is executed to facilitate the concurrency of all the parallelthreads. Thus, the shortest computational time for each time stepdepends on the longest execution time of the thread taking themost computationally demanding equations.

It should also be noted that the inherent self-coupling ofdisplacement-dependent forcing terms is decoupled by utilizingthe results of previous time steps to judge the computational con-figuration instead of using current (target time step) displace-ments. Such practice has been widely adopted in contact analysisusing FEM [13]. Thus, there is no influence from this additionalterm on the model stability nor is there a relationship betweenthe contact stiffness and the critical time step.

3. Solution behavior study and numerical verification

In this section, a solution behavior study on the contact LISAformulation is presented. Guidelines for the proper choice of com-putational parameters for an accurate solution are given. Theoret-ical values of contact stiffness are derived first as the initial guessto feed into the model. The proper range of contact stiffness mul-tipliers is tested by observing the behavior of contact solutions.Then, a benchmark problem is simulated by the contact LISA modeland verified against the solution from ANSYS.

3.1. Proper choice of contact parameters

The values of contact stiffness have been found to directly influ-ence the accuracy of the results in penalty method based contactmodels [13]. In general, a low contact stiffness will produce aninaccurate solution with large contact surface penetration, violat-ing the impenetrability condition. With the increment of contactstiffness, the violation will become smaller and smaller, whichmakes the solution converge towards the physical contact condi-tion and arrive at an accurate region. However, when the contactstiffness exceeds a certain value, different adverse solution behav-iors may be observed in FEM and contact LISA. For the case of FEMsimulations, excessive contact stiffness may result in the ill-conditioning of the stiffness matrix, which usually causes solutionconvergence difficulty. For the contact LISA formulation, the con-tact surfaces may violently jump away from each other upon con-tact, deviating from the physics and accurate solution. Thus, it iscritical to choose proper values of contact stiffness in order toobtain accurate contact solutions. The proposed process starts witha theoretical contact stiffness using a simplified model as the initialpoint and adjust the stiffness multipliers to find the range ofparameters for accurate solutions.

Fig. 10 shows two elastic bodies with general stiffness values ofK1 and K2, respectively. The body on the left hand side is a fixedtarget, while the right hand side is a contactor. Due to body motion,a penetration d is introduced when no contact force is applied.Then two contact forces f with the same magnitude but oppositedirections are applied on the contact surfaces. The contact forceswith the ideal value will eliminate the penetration between thetwo bodies. The equilibrium equations satisfying this conditioncan be written as


K1d1 ¼ f

K2ðd� d1Þ ¼ f

�ð29Þ

Cancelling the term d1 in Eq. (29) yields

K1K2

K1 þ K2d ¼ f ð30Þ

By identification, the initial value of contact stiffness in LISA canbe estimated to be

kCii ¼14

Pa;b;c¼�1

~SCONTAii

� �14

Pa;b;c¼�1

~STARGEii

� �14

Pa;b;c¼�1

~SCONTAii þ 14

Pa;b;c¼�1

~STARGEii

i ¼ 1;2;3;4;5;6

ð31Þ

where ~SCONTAii and ~STARGEii represent the diagonal terms of the stiffness

matrix of the contactor and target materials. ~Sii ¼ Siiðiþ a;jþ b; kþ cÞ denotes the stiffness component in one of the eight cellssurrounding the point ði; j; kÞ depending on the choice of ða;b; cÞfrom ðþ1;�1Þ. Since for each contact node, four of its neighboringcells are auxiliary air cells with negligible stiffness contribution,thus the average effective stiffness need to be computed using thefour solid material cells. And this is the reason why the factor ofone quarter is chosen.

After the initial estimation of contact stiffness at each contact

pair, contact stiffness multipliers KN and KS are assigned to the cor-responding stiffness values. By adjusting these parameters, therange can be found where solutions are accurate with coherentcontact behavior.

0 1 2 30

0.1

0.2

0.3

0.4

0.5

Fig. 12. Convergence and contact behavior study: low contact stiffness results in large pstiffness causes contact surface jump.

Fig. 11. Schematic of model setup of solution behavior test for the proper choice ofcontact stiffness.

Fig. 11 shows the model setup of solution behavior test. It con-sists of a 500-mm long, 20-mm wide, and 5-mm thick aluminumstrip. A 10-mm long through-thickness breathing crack is locatedin the center of the strip. A pair of in-phase line prescribed dis-placements (1 lm peak to peak value) were used to generate100-kHz 10-cycle tone burst S0 guided waves into the structure.They will propagate along the structure, interact with the breath-ing crack, bring the crack information with them, and are finallypicked up at the sensing point. The out-of-plane displacement at10 mm right after the crack was recorded and post-processed forthe convergence study.

The contact stiffness multipliers KN and KS were simultane-ously increased from 0 to 7 with a step of 0.2. The consecutivetwo solutions at the sensing point before and after an incrementwere compared. The difference between these two solutions weremeasured by the L2-norm based ratio

RL2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPN

j¼1 ½siðjÞ � si�1ðjÞ�2PNj¼1 s

2i�1ðjÞ

vuut ð32Þ

where si and si�1 represents the signal before and after the ith incre-ment of the multipliers. N is the time data length.

Fig. 12 shows the solution convergence study results as well asthe contact behaviors under various ranges of stiffness multiplier.The solution is assumed to have converged to the accurate solutionwhen the RL2 drops below 5%. Thus, the suggested range of the con-tact stiffness multiplier is between 0.8 and 3.4. It can be noticedthat the theoretical initial estimation of the contact stiffnessworked well in terms of solution convergence. In terms of contactbehavior, when the contact stiffness is low, large penetration willhappen between the contact surfaces, violating the impenetrabilitycondition and resulting in inaccurate solutions. With an appropri-ate contact stiffness, the contact surfaces interact with each othercoherently. When the contact stiffness becomes excessively large,contact surfaces may violently jump away from each other uponinteraction, which is not physically plausible. This jumping phe-nomena happens because the selected time step is not small

4 5 6 7

enetration; appropriate contact stiffness render coherent contact; excessive contact

0 100 200 300 400 500 600 700 800 900 1000

10-8

10-6

0 20 40 60 80 100 120 140 160 180 200-5

0

5

10x 10

-8

Fig. 13. Solutions within the converged contact stiffness range agree well with each other (K ¼ 0:8 and K ¼ 3:4): (a) time domain simulation signals; (b) frequency domainspectra.


enough to fully capture the interaction and stabilize the contactpair penetration behavior.

Fig. 13 presents the comparison between the results usingK ¼ 0:8 and K ¼ 3:4. It should be noted these two contact stiffnessmultipliers bounded the lower and upper limits of the suggestedcontact stiffness range for solution convergence and coherent con-tact behavior. The results agree well with each other in both thetime domain signal and the frequency domain spectrum. Heavydistortions in the time domain signals can be observed due tothe nonlinear interaction between guided waves and the breathingcrack. In the frequency domain spectra, one can notice the funda-mental excitation frequency component at f c as well as the nonlin-ear higher harmonics 2f c;3f c , and so on. The DC response at lowfrequency range can also be clearly identified. The differences inthe spectra only appeared at much higher frequencies.

3.2. Numerical verification against finite element solution

To verify the contact LISA model, a benchmark simulation forthe setup described in Fig. 11 is conducted and the solution com-pared with the result from ANSYS 14. The ANSYS contact modelused SOLID45 eight node structural element to discretize thegeometry and CONTA173/TARGE170 contact elements to modelthe crack surfaces. Element size of 1-mm for the in-plane andacross the thickness directions were used. The mesh size le waschecked against the shortest wavelength kmin possible at the thirdharmonics, i.e., le 6 kmin=20. A time step Dt of 0.125 ls was utilizedfor the time marching Newmark-beta integration to ensure theaccuracy up to the third harmonic frequency, i.e., Dt 6 1=20fmax.In the LISA model, the cell size was 1 mm in the in-plane directionand 0.5 mm across the thickness. The time marching step was0.06 ls obtained from the CFL number requirement [22]. The stiff-ness multiplier values were set to 1.4.

Fig. 14 shows the verification of the contact LISA model againstthe ANSYS solution. It can be noticed that the signals agree witheach other in time domain with merely slight differences. The fre-quency spectra shown in Fig. 14b also demonstrated that the fre-quency components of fundamental excitation, low frequency DCcomponent, second, third, even fourth higher harmonics compare

very well with each other. Differences only appeared at very highfrequency range, where the time step is sufficiently small to gener-ate accurate results or meaningful comparison between the twomethods. This numerical verification against commercial finite ele-ment code attested the capability and validity of the new contactLISA model.

Finally, the new contact LISA model also achieved much highercomputational efficiency over the conventional nonlinear FEMsimulation. Both computational tasks were conducted on an AsusESC2000 G2 workstation with a 2.00 GHz Intel Xeon E2-2650 pro-cessor, 32 GB of 1.60 GHz memory, and an Nvidia GeForce GTXTitan graphics processor with 2688 CUDA cores. The FEM simula-tion with 279,900 degrees of freedom took around 19 s for eachtime step, resulting in a total computational time of 8 h for 1500time steps. On the other hand, LISA simulation with 648,120degrees of freedom merely consumes around 0.043 s for each timestep, resulting in a total computational time of 2.15 min for 3000time steps. Thus, it is apparent that the new contact LISA modelis much more efficient than the conventional nonlinear FEM simu-lation, while achieving comparable results.

4. Numerical study on guided wave interaction with fatiguecracks

After laying out the theory behind the new contact LISA modeland verifying its implementation, numerical case studies are con-ducted on the nonlinear phenomena introduced by the interactionbetween guided waves and fatigue cracks.

4.1. Nonlinear higher harmonics and DC response: linear vs. nonlinearinteractions

This numerical case study aims to demonstrate the classicalnonlinear higher harmonic and DC component generation phe-nomena during the nonlinear interaction between guided wavesand a fatigue crack [37,38]. This situation is compared with thelinear interaction between guided waves and a notch, where thenotch surfaces will not come into contact. Again, the model setupdescribed in Fig. 11 is adopted. The only difference is that it

0 100 2000

0.5

1x 10

-4

0 50 100-5

0

5x 10

-7

(a) (b)

Fig. 15. Excitation signal: (a) time domain; (b) frequency domain spectrum.

0 100 200 300 400 500 600 700 800 900 1000

10-8

10-6

0 20 40 60 80 100 120 140 160 180 200-5

0

5

10x 10-8

FEM(ANSYS)Contact LISA

Fig. 14. Contact LISA solution compares well with the result from commercial FEM package ANSYS: (a) time domain simulation signals; (b) frequency domain spectra.


contains air cells to model the notch. A simulation case of wavepropagation in a pristine strip is also conducted to further demon-strate the signal characteristics induced by a notch and a fatiguecrack. A 10-cycle 100-kHz tone burst excitation is used to generatepure S0 Lamb mode into the structure. Fig. 15 shows the time traceof the excitation signal and its frequency spectrum.

Fig. 16 presents the simulation results, comparing the sensingsignals between the pristine, notch, and the fatigue crack cases. Itcan be noticed that the time trace of the notch case resemblesthe waveform of the pristine signal with amplitude drop and bal-anced positive and negative oscillatory amplitudes. In contrast,the time trace from the fatigue crack case is heavily distorted withimpact induced high amplitude sharp peaks and a low frequencymodulation. The waveform also shows zigzags, unlike the smoothpattern present in the notch crack case. The frequency spectra ofthese two signals show great difference too. It can be observed thatthe pristine and notch signal spectra show only the excitation fre-quency component at 100 kHz. No higher harmonics nor DC com-ponent exist. On the other hand, the fatigue crack signal presentsdistinctive nonlinear higher harmonics and DC component in itsspectrum. The generation of higher harmonics and DC componentis a classical nonlinear phenomenon and it serves as the basis formany nonlinear ultrasonic inspection methodologies. The newcontact LISA model is capable of simulating such nonlinear ultra-sonic signals.

4.2. Nonlinear mode conversion at various fatigue crack types

When guided waves interact with structural damage, modeconversion may happen, transforming one wave mode intoanother. The nonlinear contact-impact dynamics during waveinteraction with fatigue cracks will give rise to even more compli-cated mode conversion phenomena. Fig. 17 shows the model setuputilized to study such mode conversion effects at fatigue cracks. Itconsists of a 400-mm long, 200-mm wide, and 5-mm thick alu-minum plate. The excitation points are located on the top and bot-tom surfaces near the boundary to selectively generate symmetricand antisymmetric Lamb modes and minimize reflections. A 10-mm long fatigue crack resides in the center of the plate. In thisstudy, two types of fatigue crack configurations were investigated:a through-thickness crack and a half-thickness crack. The LISAmodel employed in this section adopts the in-plane cell size of0.5 mm and a through-thickness cell size of 0.25 mm. The timestep according to the Courant–Friedrichs–Lewy (CFL) condition is32.38 ns, which corresponds to a CFL number of 0.99.

Frequency-wavenumber analysis is deployed to analyze thewave mode components propagating through the crack. Thefrequency-wavenumber analysis for 1-D wave propagation, by itsnature, is a 2-D Fourier transform signal processing technique onthe time-space wavefield data, which is in terms of time variablet and space vector x. It can transform the time-space wavefielduðt;xÞ into its frequency-wavenumber representation Uðf ;kÞ by

Uðf ;kÞ ¼Z 1

�1

Z 1

�1uðt; xÞe�jð2pft�k�xÞdtdx ð33Þ

where f and k being the frequency variable and wavenumber vec-tor, respectively. j denotes the imaginary unit. The details and com-prehensive examples of such technique can be found in Ref. [39]. Inthis study, it is realized by the 2-D FFT function in MATLAB. InFig. 17, the analysis of path A provides the information on the modetype incident to the fatigue crack before its interaction with thedamage, while path B is used to analyze the converted wave modesafter the interaction.

Fig. 18 shows the time and frequency domain representation ofthe excitation signal used in this study, as well as the frequency

0 50 100 150 200 250 300 350 400

10-10

10-8

10-6

0 20 40 60 80 100 120 140 160 180 200-1

-0.5

0

0.5

1x 10-7

PristineNotchFatigue crack

Fig. 16. The interaction of guided waves with a fatigue crack brings in distinctive nonlinear higher harmonics, while their interaction with a notch does not. (a) Time tracecomparison; (b) frequency spectrum comparison.

Fig. 17. Model of 5-mm thick aluminum plate, showing two wave paths forwavenumber analysis: path A for analysis before the interaction; path B for analysisafter the interaction.

00

500

1000

1500

2000

cf

0 100 200 300 400 5000

2

4

x 10-5

0 5 10 15 20-5

0

5x 10

-7

Fig. 18. Nonlinear response may bring in higher modes: (a) time domain excitation signamm thick aluminum plate showing possible wave modes at higher harmonic componen


wavenumber dispersion curves of Lamb waves in the 5-mm thickaluminum plate. A 250-kHz 5-cycle tone burst excitation is usedthat mainly generates the fundamental symmetric mode (S0) andantisymmetric mode (A0) into the structure. The complexity ofmode conversion at fatigue cracks arises from the generation ofhigher harmonics, which spans the high frequency range contain-ing higher wave modes. Thus, mode conversion will not only hap-pen between S0 and A0 wave modes at the fundamental excitationfrequency as the case of linear interaction, but also among A1, S1,S2, A2, etc. at the higher harmonic components as well. In this par-ticular example, mode conversion into A1 at 2f c may be possiblesince the second harmonic frequency is beyond A1’s cut-off fre-quency. At 3f c , mode conversion into A2, S1, and S2 become possi-ble. The mode conversion will involve even more modes with agreater number of higher harmonic components considered. Itshould be noted that the mode conversion not only depends on

200 400 600 800 1000

cf 2 cf 3 cf 4 cf

l; (b) frequency domain spectrum of the excitation; (c) wavenumber curves of the 5-ts.


possible wave modes that may exist at corresponding higher har-monic components, but also determined by the incident wavemode and the fatigue crack geometric features. In this study, S0or A0 waves are generated selectively into the structure and themode conversion characteristics at a through-thickness crack anda half-thickness crack are studied.

4.2.1. Wave interaction and mode conversion at a through-thicknessfatigue crack

Fig. 19 presents the results for S0 wave interaction and modeconversion at a through-thickness fatigue crack. The wave propa-gation snapshots of out-of-plane displacement were captured,showing S0 wave generation, propagation, and scattering at thefatigue crack. It can be observed that the transmitted main wavefront shows shadings due to the interference with the scatteredwave field. It was found that the scattered wave field possessesseveral wave components with different wavelengths and groupvelocities, which may stem from the higher harmonic componentsof the same wave mode or the converted wave modes. Thefrequency-wavenumber analysis of the time-space domain wavefield allows us to identify the wave modes participating in thepropagation. Since only the S0 mode was selectively generated,one can only find the S0 wave component in the frequency-wavenumber analysis of path A. On the other hand, thefrequency-wavenumber analysis of path B shows that higher har-monic components of S0 appeared at 500 kHz (2f c) and 750 kHz(3f c); S2 wave mode components were also found due to modeconversion at 750 kHz (3f c) and 1000 kHz (4f c), while no antisym-metric wave mode appeared from mode conversion. The rationalebehind such phenomena is that the through-thickness crack and S0wave motion are both symmetric with respect to the mid-plane ofthe plate, which will not introduce antisymmetric disturbances togenerate antisymmetric wave modes. The existence of convertedwave modes depends on the coupling of contact-impact tractiondistribution and the corresponding mode shapes. This also implies

Fig. 19. S0 wave interaction and mode conver

that the mode shape of S1 waves do not couple well with thecontact-traction distribution during the wave damage interaction.

The theoretical dispersion curves are obtained from semi-analytical finite element (SAFE) method. It can be noticed thatthe LISA results of the frequency-wavenumber data deviates fromthat of the theoretical dispersion curves obtained from SAFEmethod at high frequency ranges. It has been reported that LISAformulation would introduce numerical bending and shifts of thedispersion data [40], which may affect the higher harmonic gener-ation and wave damage interaction for the considered frequencyrange.

Fig. 20 presents the results for A0 wave interaction and modeconversion at a through-thickness fatigue crack. Guided wave gen-eration, propagation, and scattering from the crack are shown inthe snapshots from LISA simulation. The frequency-wavenumberanalysis of path A shows that the main incident wave componentis A0 mode. The frequency-wavenumber analysis of path B demon-strates that A0 mode did not appear at 500 kHz (2f c). Instead, S0waves from mode conversion showed up at 500 kHz. A0 and A1mode appeared at 750 kHz (3f c) and S0 mode was present at1000 kHz (4f c). The mode conversion phenomenon presents analternating participation pattern between symmetric and antisym-metric modes at higher harmonic frequencies. It can be concludedthat an antisymmetric incident wave mode will generate bothantisymmetric and symmetric modes at a through-thickness fati-gue crack. The rationale behind the alternating mode conversionis that, during one cycle of A0 wave interaction with the crack,the off-mid-plane contact-impact will happen twice, introducingtwo cycles of symmetric impact and one complete cycle ofantisymmetric impact.

4.2.2. Wave interaction and mode conversion at a half-thicknessfatigue crack

Fig. 21 presents the results of S0 wave interaction and modeconversion at a half-thickness fatigue crack. The wave propagation

0 500 10000

500

1000

1500

2000

0 500 10000

500

1000

1500

2000

sion at a through-thickness fatigue crack.

0 500 10000

500

1000

1500

2000

0 500 10000

500

1000

1500

2000

Fig. 20. A0 wave interaction and mode conversion at a through-thickness fatigue crack.

Fig. 21. S0 wave interaction and mode conversion at a half-thickness fatigue crack.


snapshots capture the wave generation, propagation, and interac-tion with the crack. It can be noticed that, compared with thethrough-thickness crack case, low frequency DC response becomequite obvious. This can be attributed to two reasons when theout-of-plane displacement of the wave fields are plotted: (1) the

mode shape of the converted A0 DC component has a much largerout-of-plane component than the generated S0 mode (with verysmall out-of-plane component); (2) the principle sensing devicesuch as laser vibrometry or piezoelectric sensors capture the quan-tity of out-of-plane velocity or strain rate. The application of the


time derivative will introduce a modification factor of ix, i.e.,

@ðAeixtÞ=@t ¼ ixAeixt for a general harmonic wave field. Thus thehigher harmonic components with large value of frequency x willbe amplified, while the DC component will be diminished. Thefrequency-wave number analysis of path A shows the incident S0mode. The frequency-wavenumber analysis of path B shows thatmode conversion happened between fundamental modes (S0 intoA0) at 250 kHz (f c), which was not observed in the through-thickness crack case. At 500 kHz (2f c), all possible wave modesshowed up. The mode conversion no longer show selective or alter-nating phenomena. S0 mode was converted to A0 and A1 mode.

Fig. 22 presents the results for A0 wave interaction and modeconversion at a half-thickness fatigue crack. The wave propagationsnapshots also show obvious DC response during the wave damageinteraction. The frequency-wavenumber analysis of path A showsthe incident A0 Lamb wave. The frequency-wavenumber analysisof path B shows that after the interaction, the mode conversionis mainly between fundamental S0 and A0 modes (A0 into S0),while the converted higher order wave modes are not obvious.The wave components of A0 and mode converted S0 can be clearlyidentified at 500 kHz (2f c) and 750 kHz (3f c).

Comparing the case study of the two crack types, it was foundthat mode conversion at the through-thickness crack presents aspecial mode selection phenomenon for both S0 and A0 incidentwaves and a wave mode alternation effect for A0 incident wave.While the mode conversion at the half-thickness fatigue crack doesnot present such behavior. The mode conversion between funda-mental modes at the excitation frequency was observed at thehalf-thickness crack, but not in the through-thickness crack case.

4.3. Threshold behavior induced by initial openings and closures ofcrack surfaces

In reality, fatigue cracks always present complicated 3-D micro-scopic rough features, with initial openings and closures along the

Fig. 22. A0 wave interaction and mode conv

crack surfaces [36]. These microscopic features may induce thresh-old behaviors, which the ideal ‘‘breathing crack” models cannotcapture. In the current contact LISA effort, a distributed gap func-tion was introduced to depict the general threshold behaviors frominitial crack opening and closure, as discussed in Section 2.2.3.

In this section, a case study on two fatigue cracks with differentcrack surface features will be presented. The same model setup inFig. 11 was adopted, in which there are totally 209 contact pairs.Random normal distributions of gap values between the contactpairs to represent the rough crack surface condition was assumed(in practice, such distributions may be obtained from the observa-tion of microscopic images of fatigue cracks). Fig. 23a shows theinitial gap value distribution of an initially open crack, with allthe contact pairs set apart; Fig. 23b presents the initial gap valuedistribution of a partially closed crack, with negative gap valuesstanding for initial closure stresses through Eq. (26).

100-kHz and 10-cycle tone burst excitations were used to gen-erate S0 waves into the structure. The peak to peak excitationamplitude was increased from 10 lm to 150 lm with a step of10 lm. The nonlinearity of the sensing signals was measured bythe nonlinear index

Nindex ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAð2f cÞ þ Að3f cÞ þ Að4f cÞ

Aðf cÞ

sð34Þ

where Aðf cÞ, Að2f cÞ, Að3f cÞ, and Að4f cÞ denote the spectral amplitudeof the fundamental excitation frequency, the second, third, andfourth higher harmonics. The nonlinearity indices of the sensingsignals are traced against the excitation amplitude in Fig. 24.

Fig. 24a shows the nonlinearity trend in the initially open crackcase. It can be observed the nonlinearity trend can be clearly cate-gorized into three evolving stages: a linear region, a monotonicincrement region, and a saturation region. When the excitationamplitude is below a certain threshold, where the relative dis-placements between the crack surfaces cannot overcome the initial

ersion at a half-thickness fatigue crack.

Fig. 23. Normally distributed gap values as example cases to approximate 3-D rough crack surface condition: (a) initially open crack; (b) partially closed crack.

Fig. 24. Signal nonlinearity depends the relative relationship between excitation amplitude and initial gap condition at the crack surfaces: (a) initially open crack(corresponding to Fig. 23a); (b) partially closed crack (corresponding to Fig. 23b).


gaps to induce contact, no nonlinearity is present. This means thatthe wave damage interaction is linear. With increasing excitationamplitudes, contact-impact crack surface interactions start toform, and the nonlinearity of the sensing signal undergoes a mono-tonic increment. When the excitation amplitude exceeds a secondthreshold value, all the contact pairs are fully engaged in thecontact-impact interactions, and the nonlinearity of the sensingsignals plateaus, reaching a saturation region. It was found thatboth linear and nonlinear interactions between guided wavesand an initially open fatigue crack may happen. And the interactiontype falls into a threshold problem depending on the excitationamplitude and the initial gap values.

Fig. 24b shows the nonlinearity trend for a partially closed fati-gue crack. It was found that even very small excitation amplitudewould introduce nonlinearity into the sensing signal. The nonlin-earity is weak when the excitation amplitude is small, and itincreases monotonically with the excitation amplitude until reach-ing a threshold value, beyond which the nonlinearity goes into asaturation region.

Such threshold behaviors indicate that to take advantage ofnonlinear ultrasonics in structural inspections, one needs to gener-ate an interrogating wave field that is strong enough to trigger thecontact-impact interactions between crack surfaces. Below a cer-tain threshold value, the wave damage interaction may still staywithin the linear region, and one may not be able to obtain anyuseful nonlinear information for fatigue damage detection.Through this case study, it has been demonstrated that the newcontact LISA model could effectively capture the threshold behav-iors in nonlinear ultrasonics.

5. Concluding remarks and future work

In this study, a parallel algorithm to model the nonlineardynamic interactions between ultrasonic guided waves and fatigue

cracks was developed. This algorithm is based on the Local Interac-tion Simulation Approach (LISA) and penalty method. A Coulombfriction model was integrated to capture the stick-slip contactmotions between the crack surfaces. The LISA procedure was par-allelized via the Compute Unified Device Architecture (CUDA) andexecuted on powerful graphic cards. The explicit contact formula-tion, the parallel algorithm, as well as the GPU-based implementa-tion facilitated LISA’s superb computational efficiency over theconventional finite element method (FEM). Guidelines for theproper choice of contact parameters were given. The numericalverification against ANSYS showed good agreement between thenew contact LISA model and finite element method. It also demon-strated that the new contact LISA model achieved much highercomputational efficiency than the conventional FEM simulation.

Numerical case studies were conducted on Lamb wave interac-tion with fatigue cracks. Several nonlinear ultrasonic aspects wereaddressed. The classical nonlinear higher harmonic and DCresponses were successfully captured. It was found that the gener-ation of higher harmonic and DC component would only happenduring the nonlinear interaction between guided waves and fati-gue cracks, while the linear interaction between guided wavesand a notch will not introduce such nonlinear phenomena. Thenonlinear mode conversion at a through-thickness and a half-thickness fatigue crack was also investigated. It was found thatcomplex mode conversion phenomena involving in higher Lambmodes will happen during the nonlinear wave damage interac-tions. For the through-thickness crack case, symmetric incidentwave would only convert to symmetric modes, while antisymmet-ric incident wave would convert to both symmetric and antisym-metric modes, but in a selective mode and alternating pattern.For the half-thickness crack case, the symmetric and antisymmet-ric incident waves did not show any selective mode effect andwould convert to both major mode types. Threshold behaviors,induced by initial opening and closure of rough crack surfaces,


were depicted by the proposed contact LISA model. The numericalcase study showed that to take advantage of nonlinear ultrasonicsin structural inspections, one needs to generate an interrogatingwave field that is strong enough to trigger the contact-impactinteractions between crack surfaces. Below a certain thresholdvalue, the wave damage interaction may still stay within the linearregion, and one may not be able to obtain meaningful nonlinearinformation from the damage.

For future work, experimental validation and comparison of theproposed contact LISA model should be conducted. The computa-tion of nonlinear scattering coefficients should be considered.The same principle should be applicable to the simulation ofdelamination and impact damage in composite structures. TheLISA model should be extended to include nonlinear elasticity inorder to fully capture the influence of ultrasonic waves with finiteamplitude during wave damage interaction.

Acknowledgements

This work was sponsored by the National Rotorcraft TechnologyCenter (NRTC) Vertical Lift Rotorcraft Center of Excellence(VLRCOE) at the University of Michigan, with Mahendra J. Bhagwatas the program manager. Opinions, interpretations, conclusions,and recommendations are those of the authors and are not neces-sarily endorsed by the United States Government.

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