Simulation of crack induced nonlinear elasticity using the combined finite-discrete element method

Highlights

• FDEM provides an explicit method to modeling crack induced nonlinear elasticity.

• The nonlinear elasticity phenomenon is very sensitive to crack surface contact area.

• Both normal and tangential contacts contribute to the nonlinear behavior of a solid.

Abstract

Numerical simulation of nonlinear elastic wave propagation in solids with cracks is indispensable for decoding the complicated mechanisms associated with the nonlinear ultrasonic techniques in Non-Destructive Testing (NDT). Here, we introduce a two-dimensional implementation of the combined finite-discrete element method (FDEM), which merges the finite element method (FEM) and the discrete element method (DEM), to explicitly simulate the crack induced nonlinear elasticity in solids with both horizontal and inclined cracks. In the FDEM model, the solid is discretized into finite elements to capture the wave propagation in the bulk material, and the finite elements along the two sides of the crack also behave as discrete elements to track the normal and tangential interactions between crack surfaces. The simulation results show that for cracked models, nonlinear elasticity is generated only when the excitation amplitude is large enough to trigger the contact between crack surfaces, and the nonlinear behavior is very sensitive to the crack surface contact. The simulations reveal the influence of normal and tangential contact on the nonlinear elasticity generation. Moreover, the results demonstrate the capabilities of FDEM for decoding the causality of nonlinear elasticity in cracked solid and its potential to assist in Non-Destructive Testing (NDT).

Introduction

Non-Destructive Testing (NDT) for detection and quantification of defects in solids (e.g. crack, delamination, debonding, pore and inter-granular contact) is of significant industrial and academic importance in many areas [1], [2], [3], [4], [5]. Probing and imaging using ultrasonic waves is a leading tool for such applications and there is a need for continued development of robust ultrasonic techniques in NDT [2], [3]. Among the many ultrasonic wave applications in NDT, a major class of them employ the principles of wave reflection, transmission or scattering. While extremely useful, these linear ultrasonic techniques are less capable of accurately detecting contact-type defects, and are also less sensitive to micro or closed cracks [2], [3], [4]. Since defects can behave nonlinearly under sufficient excitation, the Nonlinear Elastic Wave Spectroscopy (NEWS) methods have shown remarkable potential for defect detection and characterization [2], [3], [4], [6], [7], [8], [9], [10]. The nonlinear methods often involve exciting a solid with an ultrasonic signal of certain frequency and generating an output frequency spectrum consisting of harmonics and subharmonics of the exciting frequency. These effects, often referred to as Contact Acoustic Nonlinearity (CAN) [10], are mainly induced by clapping and frictional contacts of the defects [2], [3], [11]. Because of their high sensitivity, nonlinear ultrasonic approaches have seen an increasing interest in the NDT community over the past decades [1], [2], [4], [9], [12], [13].

A large number of experimental studies have been conducted using NEWS techniques for various types of defect detection in solid materials such as composite plates, metals, concretes and rocks, and have demonstrated successes in the field of NDT [6], [7], [8], [14], [15], [16], [17], [18], [19], [20], [21]. However, to date, the underlying microscopic mechanism of nonlinear techniques for defect detection is still poorly understood [4]. Numerical simulations, which are capable of providing detailed analyses of the nonlinear behavior at a level of spatial and temporal resolution not accessible experimentally, are therefore necessary for decoding the complicated mechanism associated with the nonlinear ultrasonic techniques. In addition to the ease of implementation, the numerical approaches are indispensable also because they can provide dedicated comparisons of nonlinear indicators with experimental results, and thus link measured macroscopic events to defect internal parameters (both physical and geometric) and in this way, a complete characterization of the defects can be achieved [2].

Simulation of ultrasonic wave propagation in solids with defects by considering the nonlinearity introduced by them is challenging, and has been the object of study for two decades [2], [3]. The numerical approaches used in this regard mainly include Finite-Difference Time-Domain (FDTD) and Finite Element Method (FEM). For example, Sarens et al. [22] implemented a three-dimensional finite difference, staggered grid simulation to model the contact nonlinear acoustic generation in a composite plate containing an artificial defect; Marhenke et al. [23] used FDTD simulations to cross-validate the simulated interference effects resulting from multiple ultrasonic reflections within the delamination layers with the laboratory experiments. In general, the FDTD is easy to implement; however, it has many restrictions, e.g. the defects in the FDTD simulations are usually restricted to rectangular shapes [2]. As a more flexible alternative, FEM is widely used in crack-wave interaction simulations. In particular, Kawashima et al. [24] used a FEM model to study CAN in which Rayleigh waves were employed to detect surface cracks; Blanloeuil et al. [12], [25] studied the nonlinear scattering of ultrasonic waves by closed cracks subjected to CAN; to investigate the clapping and friction induced nonlinearity in solids containing cracks, Van Den Abeele and colleagues [2], [3], [11], [26] implemented a series of comprehensive normal and tangential constitutive models into FEM to control the nonlinear behavior of crack surfaces. Among the many numerical simulations, some of them use hypothetical defects in which artificial nonlinear stress-strain relations are introduced into special elements to represent defects [24]; others employ physical defects by splitting the computational nodes along defects, and then the normal and tangential contact stresses, which are calculated based on the relative distances between the corresponding Gauss points located on the defect surfaces, are applied to the same Gauss point pairs as boundary conditions for the bulk material simulation [2], [3], [4], [5], [11], [12], [22], [23], [25], [26], [27], [28], [29]. The former only captures the defect behavior in an approximate manner; the latter may be difficult to explicitly realize complicated scenarios such as defects with irregular shapes and especially, the interactions between many defects of different types.

From a computational mechanics point of view, a solid with defects is essentially a combination of continua (bulk material) and discontinua (interaction between defect surfaces). Considering this, a numerical tool that has the capability of handling continua and discontinua simultaneously would be helpful. Fortunately, a recently developed numerical method – the combined finite–discrete element method (FDEM) [30], [31], [32], [33], which merges finite element-based analysis of continua with discrete element-based transient dynamics, contact detection and contact interaction solutions of discontinua, provides a natural solution for such simulation. To date, a systematic application of FDEM in NDT is not available in the literature. The goal of this paper is to introduce FDEM to the NDT community, and to demonstrate its power on simple problems.

A simple FDEM realization of a solid plate with a crack is presented in Fig. 1 where the solid (excluding the crack since it is considered as void) is discretized into finite elements to capture the motion and deformation of the bulk material, and the finite elements along the two sides of the crack also behave as discrete elements to track the normal and tangential interactions between crack surfaces. By employing FDEM, the system can be explicitly described and particularly, the contacts along the sides of the defects can be uniformly processed using well-developed discrete element method (DEM)-based algorithms.

The focus of the current work is to demonstrate the applicability of FDEM to the simulation of crack induced nonlinear elasticity in solids, and to introduce the approach as another alternative for NDT numerical based analysis. A comparison of the simulated results with laboratory experiments is beyond the scope of the present paper and thus it is left for future work. In the following sections, we first provide a brief introduction to the theories of FDEM. Then we illustrate the numerical model setup and present how the normal (clapping) and tangential (friction) contact may influence the nonlinear behavior of a cracked solid. The applicability of FDEM for NDT simulation is demonstrated, and the corresponding conclusions are drawn.