Shock and Vibration

Volume 2019, Article ID 9528978, 9 pages

https://doi.org/10.1155/2019/9528978

## Wave Dispersion and Propagation in Linear Peridynamic Media

School of Mechanical Engineering, Jiangsu University, Zhenjiang 212013, China

Correspondence should be addressed to Zhenying Xu; moc.361@77uxyhz

Received 25 December 2018; Revised 18 April 2019; Accepted 13 May 2019; Published 9 June 2019

Academic Editor: Toshiaki Natsuki

Copyright © 2019 Xiaolong Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We detail the linear peridynamic wave equation with a nonlocal integral form based on the linear peridynamic and dynamic theory. Wave dispersion in an infinite maraging steel material is obtained by analyzing the linear peridynamic wave equation. The dispersion curves, group velocity, phase velocity, and other wave parameters of the shear and longitudinal waves in an infinite media are obtained using numerical methods. We obtained the optimal calculation parameters by analyzing the weight function, horizon, mesh size, and other numerical calculation parameters on the dispersion curve. We simulated the propagation of waves in an infinite media by applying these parameters into the peridynamic wave equation. We conclude that the wave model can generate waves that propagate in all directions with initial loads. The wavefront is an ellipsoid.

#### 1. Introduction

Ultrasonic waves with unidirectionality and penetration properties are widely used in nondestructive testing [1–4]. In order to study the propagation of ultrasonic waves, the community usually uses finite element analysis methods to solve elastic dynamic equations in continuous media [5–8]. However, the derivative function does not exist at the defect when there is a defect (discontinuity) in the media [3, 4]. It is necessary to remesh the grid to make the defect outside of the boundary [9]. This method can simulate ultrasonic wave propagation in a media with the defect; however, the stresses at the crack tip are mathematically singular because of their undefined spatial derivatives. To study ultrasonic wave propagation in the media, we used the integral equation to describe the wave displacement change with location and time—named “the peridynamic method” [10, 11]. This method falls into the category of nonlocal models [12, 13] because particles separated by a finite distance can interact with each other [14, 15]. The peridynamic theory of solid mechanics has been proposed as a means of treating discontinuous media via a mathematical model that does not require a smooth distribution of mass or differentiability of the deformation [10]. The starting point of the theory is that the internal forces acting on a material point are determined via interactions between the point and all others within a finite distance from it. The peridynamic model uses integrals rather than differential equations so that the mathematical structure does not break down when a discontinuity occurs. Rather, the fracture is treated as a natural outcome of deformation that emerges according to the equations of motion and the constitutive model [16].

The theory of peridynamic waves was first proposed by Silling [10] in 2000, with a study of the propagation of linear stress waves and a discussion of wave dispersion. In 2003, Silling et al. [17] discussed the propagation of waves in a discontinuities bar. Later, Zimmermann [18] explored many features of the peridynamic theory including certain aspects of wave motion, material stability, and numerical solution techniques. In 2007, Silling et al. [19] gave the relationship between stress, displacement, and location under small deformation conditions making it easier to solve the microelastic peridynamic wave equation. Silling [20] gave the linearization theory of peridynamic in 2010. This theory paves the way for the study of linear elastic waves, making the peridynamic wave equation easier to solve. Bažant et al. [21] systematically studied the peridynamic stress wave in 2016 and gave the bond-based peridynamic stress wave theory and dispersion analysis. In the same year, Silling [22] studied solitary waves in elastic media and found that the solitary wave velocity is greater than that of the elastic wave. Butt et al. [23] conducted a detailed study of the peridynamic stress wave under linear elastic conditions in 2017. They then analyzed the state-based peridynamic wave dispersion and studied the mesh size, horizon, and weight function influence on dispersion.

In this paper, we gave the wave propagation equation a nonlocal integral form based on the research of linear peridynamic theory. The dispersion matrix [10] of the wave is given from the linear peridynamic theory, and the dispersion of the longitudinal and shear waves in the infinite media is analyzed. The phase velocity and group velocity of the waves in the infinite peridynamic media are given. The effects of weight function, horizon, and mesh size on the dispersion are analyzed. The peridynamic simulation is used in the infinite media to analyze the difference between the shear wave and the longitudinal wave. By analyzing the wave model, we conclude that the wavefront of the single-point excited wave is an ellipsoid.

#### 2. Basic Equations of Linear Peridynamic Media

The continuum peridynamic wave theory is in contrast to the classical continuum theory. Any material point in the reference configuration is acted upon by forces due to the deformation of all the material points within the neighborhood of finite radius centered at . The radius is called the horizon, and the material points within this neighborhood of in the reference configuration are called the family of .

A material point at time has a force density acting on it, and the motion is given by the following equation [24]:where is the reference configuration of the body, is the density in the reference configuration, is the displacement, and is the body force density. Equation (2) is a function of displacement that represents the internal force density (per unit volume) exerting on by other body points. Since force vector state depends only on the deformation of the family of , and defining the vector to be a bond, we assume the following:

Under the assumption of a small deformation that does not need a small deformation gradient, the peridynamic equation (1) is accurately approximated by the following linear integrodifferential equation:where is a neighborhood of . As discussed in [20], the radius of is in general , where is the horizon. is a tensor-valued function which is named “the micromodulus function.”

Similar to the linear classical theory, the linear peridynamic theory concerns small deformations. However, the damage and fracture that may be included in a linearized model, as noted above, does not require small displacement gradients [20]. Under this assumption of smallness, the peridynamic equation of equilibrium reduces to a linear integral equation. The linearized theory is more compatible with implicit solvers than the full theory. It is also much easier to obtain theoretical results about the properties of the linear equations such as well posedness than those of the nonlinear model. The properties of waves are also much easier to analyze within the linearized theory.

#### 3. Wave Dispersion in Linear Peridynamic Media

The one-dimensional linear peridynamic wave model is popular because of its simplicity. Butt et al. [23] focused on the dispersion properties of a state-based linear peridynamic model. They derived the dispersion relation for one-, two-, and three-dimensional cases and specifically investigated the effect of the horizon, mesh size, and weighting function on the dispersion. Here, we focus on the dispersion characteristics of waves based on linear peridynamics. To study the peridynamic wave dispersion, we provide the general form of a plane wave:

Of these, is the vibration amplitude and is the displacement, with as the wavenumber and as the circular frequency. When the external force density in (4) is zero, we can obtain the following equation by substituting (5) into (4):

Here, is a bond between two material points. Changing the form of equation (6) leads to

The Euler transform in (7) is as follows:

The micromodulus function is an even function, and is an odd function and integrated into the asymmetric domain . Thus, we can get the following:

Thus, we conclude from (8) and (9) that

We conclude from equation (10) that the dispersion of the peridynamic wave is only related to the micromodulus function, density, and horizon. These parameters are inherent properties of the material.

The micromodulus function can be written as

Of these, , is the bulk modulus, is the shear modulus, and is the weight function. The weight function and horizon selection are under intense investigation (equation (11)). If the selected horizon and weight function are consistent with the properties of the material, then the simulation results are close to the experimental results. Otherwise, a large error will occur. Equations (10) and (11) can lead to the following:

From (11) and (12), we found that is a matrix (dispersion matrix). This matrix first appeared in [10].

This work assumes that a wave propagates along the wave vectors in a Cartesian coordinate system. At this time, the stress wave propagates along , which can be seen as a one-dimensional rod model. Thus, we can obtain . The expansion of (13) is

When we study a longitudinal wave propagating along , we can obtain only the displacement of . Equation (15) leads to the following:

Equation (16) shows that the longitudinal wave along the direction is related to the material parameter ; thus, we can obtain the longitudinal wave dispersion equation by simplifying (16) to be

Similarly, the displacement of the horizontal shear wave propagating in the direction only has . A combination of (14) and (15) shows that the dispersion is related to the material parameters . The dispersion equation is as follows:

Similarly, as the displacement of the vertical shear wave propagates along , only . Combination of (14) and (15) shows that the dispersion is related to the material parameters . The dispersion equation is as follows:

Equations (11) and (12), and the symmetry of the spatial integral indicate that the micromodulus function matrix in an infinite media is a symmetric matrix; the values on the diagonal are equal. Therefore, (18) and (19) indicate the same shear wave dispersion equation with only different vibration directions.

##### 3.1. Longitudinal Wave Dispersion

This peridynamic method was used to simulate the wave dispersion in maraging steel. The following terms were used: material density , bulk modulus , shear modulus , mesh size , and horizon . If there was a vibration at the center of the infinite media, then the longitudinal wave propagates along within the infinite media, and we can obtain the discrete equation from dispersion equation (17) aswhere

The literature [5] shows that the relationship between the wavenumber, phase velocity, and circular frequency is as follows:

The group velocity can be found from the phase velocity via formula (24):

Figures 1–3 plots the dispersion curve, phase velocity, and group velocity, respectively, from (20)–(24).