A cracked structure made of two different elastic materials having a Griffith crack at the interface is analyzed when it is subjected to pure shear loading and ultrasonic loading. The waves generated by the applied load and the crack propagation resulted from the shear loading are investigated. Peri-ultrasound modeling tool is used for this analysis. A comparison between experimental results and numerical predictions shows a very good matching between the two. Furthermore, the increase in nonlinear ultrasonic response in presence of the interface crack could also be modeled by this technique. The computed results show that when the interface crack propagates, then it breaks the interface at one end of the crack and breaks the material with lower elastic modulus at the other end. The unique feature of this peridynamics-based modeling tool is that it gives a complete picture of the structural response when it is loaded—it shows how elastic waves propagate in the structure and are scattered by the crack, how the crack surfaces open up, and then how crack starts to propagate. Different modeling tools are not needed to model these various phenomena.

## Introduction

Interface cracks frequently appear in engineering structures during its production or when in service. Crack initiation at the interface can result in the failure of mechanical components made of multilayered materials [1,2]. Debonding of adhesive joints and decohesion of thin films from substrates are good examples of structures with interface cracks [3,4]. The weak bonding and stress concentration occurring at the interface with mismatched material properties are the main causes of interface cracks [59]. Thus, macroscopic cracks at the interface are created, and it plays an important role in any failure analysis of bimaterial structures.

Many research works use fracture mechanics principles to understand crack growth. Most common approaches for studying crack growth phenomenon require calculating strain energy release rate and stress intensity factor at the crack tip [10].

With our current state of knowledge, the key challenge that still remains today is the proper understanding of the true nature of material behavior at the crack tip. It is now established that the existence of the oscillatory behavior of the singular stress field at the tip of an interface crack predicted by some analytical models are not realistic. Achenbach et al. [11] introduced a cohesive model which allowed the near crack tip region to yield under high stresses. This approach completely removes the singularities at the crack tips. This solution is antisymmetric for the case of remote shear loading. Antisymmetric solution predicts overlapping of crack faces. Knowles and Sternberg [12] argued that the solution is not antisymmetric for the case of remote shear load, and the crack opens smoothly at least in the neighborhoods of both crack tips which is consistent with the finite element analysis [13]. Measurement of the displacement field around an interface crack in a bimaterial thin plate was conducted by Chiang and Hua [14] to see what really happens under pure shear loading.

Numerous contributions have been made after the work by Williams [15] on cracks at the interface of a bimaterial structure. Williams' solution predicted unrealistic oscillating singularity in the stress field near a crack tip. Readers can find detailed discussions on analytical solutions and its assumptions on boundary condition along the crack faces in various publications [1618]. Note that all analytical solutions failed for interfacial cracks loaded remotely in pure shear. For this reason, numerical analysis of the displacement field at the interface of a bimaterial structure is of interest to many authors.

Nonlocal theories are often used to solve problems that classical continuum mechanics cannot handle. Therefore, peri-ultrasound modeling tool that uses a bond-based version of nonlocal peridynamic theory [19] is an ideal tool for modeling crack propagation and detection. This analysis helps us to understand the mixed mode behavior at the interface. The damage was incorporated in the elastic constitutive model [20], and thus it can handle the spontaneous crack growth. It is generally a difficult task to identify and extract the failure modes for the shear deformation in a bimaterial structure. The primary objective of this paper is to extend an original peridynamic model to two dissimilar materials where the material properties are different on two sides of the interface. The crack is placed along the interface. The bimaterial is subjected to a pure shear load. The peri-ultrasound modeling technique [21] is utilized for investigation of the wave propagation and the nonlinear behavior of the bimaterial structure in presence of a crack at the interface. The basics of the peri-ultrasound modeling and the excitation function are described in Sec. 2. The excitation function is identical to the one considered in Ref. [21]. However, the problem solved here is new. In Ref. [21], a crack in a homogeneous material was considered, and the effect of the crack thickness was investigated, while in this paper an interface crack that has no reliable analytical solution is investigated.

## Peri-Ultrasound Modeling

For developing the peri-ultrasound modeling tool, we follow Ref. [22], where the peridynamic model was introduced. The equation of motion at a point x in the reference configuration at time t is given by
$ρu¨(x,t)=∫Hxf(u(x′,t)−u(x,t),x′−x)dVx′+b(x,t)$
(1)
where $Hx$ denotes a neighborhood of $x$, $u$ is the displacement vector field, $b$ is a prescribed body force density field, $ρ$ is the mass density in the reference configuration, and $f$ is a paired function whose value is the force density per unit volume that the particle located at $x′$ (in the reference configuration) exerts on the particle located at point $x$ (also in reference configuration). The relative position $ξ$ of these two particles in the reference configuration is given by $ξ=x′−x$ and their relative displacement $η$ by $η=u(x′,t)−u(x,t)$. Note that $η+ξ$ is the relative position vector between two particles in the deformed configuration. The interaction force between two particles is called a bond. A degree of nonlocality was defined for nonlocal interaction between particles in peridynamic model based on the concept of horizon
$f(η,ξ)=0∀η whenever |ξ|>δ$
(2)

where ξ is shown in Fig. 1, and $η$ is relative displacement. It simply means that there is no interaction between $x$ beyond this horizon (see Fig. 1). The pairwise force interaction $f$ is required to be antisymmetric. The function $f$ which plays a fundamental role in the peridynamic theory has dimensions of force per unit volume squared

$f(−η,−ξ)=−f(η,ξ)∀η,ξ$
(3)

which assures conservation of angular momentum. Equation (3) means the force vector between two particles is parallel to their current relative position vector.

A material is said to be micro-elastic if the pairwise force function can be derived from a scalar micropotential $w$
$f(η,ξ)=∂w∂η(η,ξ)∀η,ξ$
(4)
The micropotential is the energy in a single bond and has dimensions of the energy per unit volume squared. The energy per unit volume in the body at a given point is therefore given by
$W=12∫Hxw(η,ξ)dVξ$
(5)
The factor $1/2$ appears because each endpoint of a bond owns only half of the energy of the bond. Peridynamic body is composed of a micro-elastic material. A linearized peridynamic mode for a micro-elastic material takes the form
$f(η,ξ)=C(ξ)η∀η,ξ$
(6)
where $C$, the micromodulus function for the material, is a second-order tensor given by
$C(ξ)=∂f∂η(0,ξ)∀ξ$
(7)
A micro-elastic material is said to be a proportional material if and only if pairwise force function is proportional to the stretch, $s(ξ,η)$
$s(ξ,η)=|η+ξ|−|ξ||ξ|$
(8)

$C(ξ)=cξ⊗ξ|ξ|3, i.e., Cij(ξ)=cξiξj|ξkξk|32$
(9)

In a Cartesian coordinate frame, $c$ denotes a constant. For example, in two dimension, $c=9E/2πεδ3$ where $E$ is the elastic modulus. $δ$ and $ε$ are the radius of horizon and the thickness of the structure, respectively. The determination of $c$ is discussed in detail in Ref. [23].

For peri-ultrasound modeling, the excitation wave packet is constructed by taking a sinusoidal carrier wave that is multiplied by the Gaussian function
$A(t)=a sin(ωt)e−pt2$
(10)

Equation (10) represents an oscillating motion that decays exponentially. To duplicate a typical experiment, the following values are taken—amplitude a = 59.3 × 106, angular frequency ω = 2.2305 × 106 rad/s, and p = 9.147 × 1010; p is a parameter that controls the width of the Gaussian function and t is time. Note that pt2 should be dimensionless.

Local continuum theory-based analytical framework and predictive tool for the simulation of guided Lamb wave propagation in a plate and its interaction with damages are available in the literature [24]. The peri-ultrasound computing tool that is developed here uses a new approach to specify the exciting waveform. Here, we set the frequency of the carrier wave in the wave packet. This can be modified easily in the peri-ultrasound tool. Number of cycles of the signal within the wave packet is defined. Gaussian shaped envelope of the wave packet is used here. The developed tool box has several choices for the envelope such as rectangular, triangular, tapered, and so on. The propagation distance is then set for the simulation. The carrier signal is taken as a sine function in this analysis. This can be changed to a cosine function as well if the user wants it. Figure 2 shows the input exciting signal with the central frequency of 355 kHz and having five peaks.

A good number of ultrasonic measurement systems use contact-type transducers, like surface mounted lead zirconate titanates (PZTs) [25] to excite the structure and get the response at a distance. The applied voltage produces vibratory motion of the transducer that generates displacement on the contact surface. A stacked device can produce a displacement $Δh=n×d33×V$. Here, $n$ is the number of stacks, and $d33$ is piezoelectric constant with unit (meter/Volt); it is the electric displacement component for $V=1$ V and n = 1 in the direction of polling axis [26]. The schematic of PZT is shown in Fig. 3. The input displacement signal for the peridynamic simulation is expressed in the form of a function of time $D(t)$. This function controls the movement of particles around the PZT particle within its horizon. The input function D(t) can be constructed by multiplying the wave packet function A(t) by a calibration coefficient as follows:

$D(t)=c¯A(t)$
(11)

where the carrier wave $A(t)$ is a function of time, and the calibration coefficient $c¯$ is used to match the simulated result amplitude with the experimental data if necessary. A schematic of radial excitation of particles is shown in Fig. 4.

Normalized spectral plots are obtained by taking fast Fourier transform of the horizontal velocity time history that can be written as
$Fj=∑k=0N−1Vje−i2πkj/N,j=0,...,N−1$
(12)

where $F$ is the fast Fourier transform, $V$ is the velocity, and $N$ is the total number of steps.

From the normalized spectral plot of the received signal, the degree of material nonlinearity can be measured by extracting a feature called sideband peak count (SPC) [27]. Material nonlinearity increases with the initiation of fatigue cracks. Figure 5 illustrates the concept of SPC. Different peaks in the left diagram of Fig. 5 indicate waves propagating through the medium with different frequency. If the material is nonlinear, then waves of different frequency interact with one another and generate additional peaks due to frequency modulation. These additional peaks have small amplitudes and appear on one or both sides of the main peaks. Small peaks around the main peaks are called the sideband peaks. Number of these sideband peaks is an indirect measure of the degree of material nonlinearity [27]. Strength or energy associated with the sidebands can also be an indicator of the degree of material nonlinearity [2831].

To quantify the extent of material nonlinearity in a damaged structure, the received signal can be analyzed in the following manner. The normalized SPC is defined as the ratio of the number of sideband peaks above a moving threshold to the total number of sideband peaks above the zero line. For example, one can set 0.1 as the threshold value for identifying the dominant peaks. Any peak above 0.1 times or 10% of the highest peak value can be counted as a dominant peak, while all peaks smaller than 10% of the highest peak are then counted as sideband peaks. The moving threshold is then increased from 0% to 10% of the highest peak value. The peri-ultrasound tool counts the number of sideband peaks above the moving threshold and divides that number by the total number of sideband peaks (above the zero threshold value but below the 10% of the highest peak value) as shown below:
$SPCth=NpeaksthNtotal$
(13)

## Problem Statement

Figure 6 shows the problem geometry or the model used for the computation. A square domain having several thousand particles distributed uniformly in x and y directions in one layer is analyzed. In z or the thickness direction, the model is only one particle deep. This model solves plane strain problem where all motions are confined in one plane.

How the bond constants are assigned to the particles in the region close to the interface affects the material behavior. Bond constants can be assigned using a weight function. The weight function can be defined based on the magnitude of the initial bond length in the reference configuration. The weight function is written as follows:
$c=αc1+βc2 where α=|ξa|/|ξ| and β=|ξb|/|ξ|$
(14)

$|ξ|=|ξa|+|ξb|$
(15)
Explicit Verlet time integration scheme (Δt = 0.014 μs and 10,000 time steps) is used to find the velocity and displacement of every particle. The position Verlet integration method can be implemented as
$un+1/2=un+(Δt2)u˙nu˙n+1=u˙n+(Δt)u¨n+1/2un+1=un+1/2+(Δt2)u˙n+1$
(16)

## Numerical Results and Discussion

The grid spacing of 0.5 mm was decreased to increase the number of material points from about 65,000 to 258,000. It is shown in Fig. 7 that with almost 300% increase of the material points the solution does not change significantly. Therefore, it can be stated that 65,000 material points are sufficient for this analysis.

The variation of the horizontal displacement u (which is the y-component of displacement, see Fig. 6) between the two crack faces is shown in Fig. 8. Note that y direction is perpendicular to the crack surface. The crack runs in the x-direction along the interface of the bimaterial structure. Figure 8 shows that the predicted results are in very good agreement with the experimental data. This matching is better than the matching between the experimental and analytical results [32]. Thus, the reliability of the developed peri-ultrasound modeling scheme is verified from the good matching between the predicted results and the experimental data. The computed results are also compared with the available closed-form global solution [33,34]. Clearly, the analytical solution is not very accurate.

The variation of the vertical displacement v (which is the x-component of displacement, parallel to the crack, see Fig. 6) between two crack surfaces along the crack axis (x-axis) is then computed for the bimaterial specimen subjected to pure shear. The experimental results [32] for the crack opening displacement in the sliding direction (parallel to the crack surface) were not close to the analytical solution and were not reliable. Hence, peri-ultrasound predicted results were compared with the available analytical solution only [32]. Figure 9 shows that the predicted result is in good agreement with the analytical solution away from the crack tips. There is a discrepancy between analytical solution and peri-ultrasound predictions near the crack tips. Analytical solution has two parts. First, the global solution describes the far field behavior, and the asymptotic solution describes the region near the crack tip. The relative sliding displacement approaches zero at the crack tips for the peri-ultrasound solution, which is more realistic than the analytical solution that predicts nonzero value at the crack tip.

The interaction between elastic waves and crack is the driving force for the crack propagation. This driving force determines in which direction crack should propagate. Figure 10 shows that the crack propagates from the upper crack tip into the material with lower elastic modulus. On the other crack tip, the crack shows a self-similar propagation along the interface. It should be noted here that no additional assumption for opening and closing of crack faces was made in this model. This has an advantage over other finite element method) or extended finite element method-based simulation techniques, since such assumptions are required for finite element method/extended finite element method analyses.

Figure 11 shows the wave motion generated by the transducer in the cracked bimaterial structure at different times starting from 14 μs and ending at 126 μs with an interval of 14 μs. Wave transmission through the interface, and wave reflections from the crack, interface, and boundaries are clearly visible in these images. The shadow generated behind the crack is also clearly visible in these images. Figure 12 shows the displacement time history at the receiver position in presence and absence of the crack. Significant decrease in the signal amplitude at the receiver position due to the presence of the crack is obvious in both Figs. 11 and 12.

Figure 13 compares the normalized spectral plots generated from the velocity time histories. Figure 14 shows the SPC variation as a function of the threshold value generated from the spectral plots of Fig. 13. SPC is an indication of the degree of the material nonlinearity. The higher value for the SPC implies higher nonlinearity [27]. Therefore, Fig. 14 indicates that the material nonlinearity is slightly increased when the interface crack is considered. This observation is consistent with our expectation that the damaged structure should be more nonlinear than the undamaged structure.

## Conclusions

In this paper, a cracked structure made of two different elastic materials having a Griffith crack at the interface is analyzed. The structure is subjected to pure shear loading and ultrasonic transducer induced loading. Peri-ultrasound modeling tool is used for this analysis. Pure shear loading is considered because an interface crack subjected to pure shear is yet an unsolved problem analytically. The classical analytical solution of an interface crack shows unrealistic oscillating stress singularity near the crack tips. The nonlocal theory-based peri-ultrasound modeling tool appears to produce a better solution. The crack opening displacement predicted by this modeling tool showed a closer match with the experimental data compared to the analytical solution. Furthermore, slight increase in the nonlinear ultrasonic response in presence of the interface crack could also be modeled by this technique. This investigation shows that peri-ultrasound modeling tool works like a universal tool for material modeling because without any special consideration it can help us to visualize how elastic waves propagate in a cracked structure, how the crack surfaces open up, and how crack starts to propagate along an interface or into a material.

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