Peridynamic Theory and Its Applications

One of the underlying assumptions in the classical theory is its locality. The classical continuum theory assumes that a material point only interacts with its immediate neighbors; hence, it is a local theory. The interaction of material points is governed by the various balance laws. Therefore, in a local model a material point only exchanges mass, momentum, and energy with its closest neighbors. As a result, in classical mechanics the stress state at a point depends on the deformation at that point only. The validity of this assumption becomes questionable across different length scales. In general, at the macroscale this assumption is acceptable. However, the existence of long-range forces is evident from the atomic theory and as such the supposition of local interactions breaks down as the geometric length scale becomes smaller and approaches the atomic scale. Even at the macroscale there are situations when the validity of locality is questionable, for instance when small features and microstructures influence the entire macrostructure.

At any instant of time, every point in the material denotes the location of a material particle, and these infinitely many material points (particles) constitute the continuum. In an undeformed state of the body, each material point is identified by its coordinates, \( {{\mathbf{x}}_{(k) }} \) with \( (k=1,2,\ldots,\infty ) \), and is associated with an incremental volume, \( {V_{(k) }} \), and a mass density of \( \rho ({{\mathbf{x}}_{(k) }}). \) Each material point can be subjected to prescribed body loads, displacement, or velocity, resulting in motion and deformation. With respect to a Cartesian coordinate system, the material point \( {{\mathbf{x}}_{(k) }} \) experiences displacement, \( {{\mathbf{u}}_{(k) }} \), and its location is described by the position vector \( {{\mathbf{y}}_{(k) }} \) in the deformed state. The displacement and body load vectors at material point \( {{\mathbf{x}}_{(k) }} \) are represented by \( \mathbf{u}_{(k) }({{\mathbf{x}}_{(k) }},t) \) and \( \mathbf{b}_{(k) }({{\mathbf{x}}_{(k) }},t) \), respectively. The motion of a material point conforms to the Lagrangian description.

Within classical continuum mechanics, a material point can only interact with other material points in its nearest neighborhood. As depicted in Fig. 3.1, the material point \( k \) at location \( {{\mathbf{x}}_{(k) }} \) can only have interactions with the material points labeled as \( (k-1) \), \( (k+1) \), \( (k-m) \), \( (k+m) \), \( (k-n) \), and \( (k+n) \). These interactions are represented by “internal traction vectors.” For the material point \( k \) that is located on a surface whose unit normal is \( {{\mathbf{n}}^T}=({n_x},{n_y},{n_z}) \), the components of a traction vector, \( {{\mathbf{T}}^T}=({T_x},{T_y},{T_z}) \), are related to the Cauchy stress components as in which \( ({\sigma_{xx(k) }},{\sigma_{yy(k) }},{\sigma_{zz(k) }}) \) and \( ({\sigma_{xy(k) }},{\sigma_{xz(k) }},{\sigma_{yz(k) }}) \) are the normal and shear stress components, respectively.

The auxiliary parameters, C in Eq. 2.43 and A and B in Eq. 2.48, can be determined by using the relationship between the force density vector and the strain energy density, \( {W_{(k) }} \), at material point \( k \) given by Eq. 2.49 in the form,

Fiber-reinforced laminated composites are generally constructed by bonding unidirectional laminae in a particular sequence. Each lamina has its own material properties and thickness. As shown in Fig. 5.1, the fiber orientation angle, \( \theta \), is defined with respect to a reference axis, \( x \). Fiber direction is commonly aligned with the \( {x_1}- \) axis, and transverse direction is aligned with the \( {x_2}- \) axis. A unidirectional lamina is specially orthotropic. Thus, a thin lamina has four independent material constants of elastic modulus in the fiber direction, \( {E_{11 }} \), elastic modulus in the transverse direction, \( {E_{22 }}, \) in-plane shear modulus, \( {G_{12 }} \), and in-plane Poisson’s ratio, \( {\nu_{12 }} \).

Material damage in peridynamics (PD) is introduced through elimination of interactions (micropotentials) among the material points. It is assumed that when the stretch, \( {s_{(k)(j) }} \), between two material points, \( k \) and \( j \), exceeds its critical value, \( {s_c} \), the onset of damage occurs. Damage is reflected in the equations of motion by removing the force density vectors between the material points in an irreversible manner. As a result, the load is redistributed among the material points in the body, leading to progressive damage growth in an autonomous fashion.

The peridynamic (PD) equation of motion is an integro-differential equation, which is not usually amenable for analytical solutions. Therefore, its solution is constructed by using numerical techniques for spatial and time integrations. The spatial integration can be performed by using the collocation method of a meshless scheme due to its simplicity. Hence, the domain can be divided into a finite number of subdomains, with integration or collocation (material) points associated with specific volumes (Sect. 7.1). Associated with a particular material point, numerical implementation of spatial integration involves the summation of the volumes of material points within its horizon. However, the volume of each material point may not be embedded in the horizon in its entirety, i.e., the material points located near the surface of the horizon may have truncated volumes. As a result, the volume integration over the horizon may be incorrect if the entire volume of each material point is included in the numerical implementation. Therefore, a volume correction factor is necessary to correct for the extra volume. A volume correction procedure required for such a case is described in Sect. 7.2.

This chapter provides solutions to many benchmark problems and comparisons with those of classical continuum mechanics, i.e., analytical or finite element analysis. Failure is not allowed in the construction of peridynamic solutions. These benchmark problems concern primarily structures with simple geometries and under simple quasi-static and dynamic loads.

In Chap. 8, peridynamic solutions of many benchmark problems were presented and compared with the classical theory in the absence of failure prediction. This chapter presents solutions to various problems while considering failure initiation and propagation. When available and suitable, the peridynamic (PD) predictions are compared with the finite element analysis (FEA) solutions.

This chapter concerns the peridynamic modeling of contact between two bodies due to an impact event. The impactor can be either rigid or deformable, and the target body is deformable. The interpenetration of bodies must be prevented between the bodies during the analysis. The treatment of contact due to a rigid impactor is different than that of a deformable impactor; Silling (2004) implemented two different techniques in the EMU code. The following sections will describe how interpenetration between the two bodies can be prevented while modeling contact due to a rigid or a deformable impactor. Also, applications are presented to well-known contact events such as the impact of two flexible bars, a rigid cylinder impacting a rectangular plate, and the Kalthoff and Winkler (1988) experiment. The peridynamic solutions to these problems are obtained by developing specific FORTRAN programs, which are available on the website http://extras.springer.com.

The PD theory provides deformation, as well as damage initiation and growth, without resorting to external criteria since material failure is invoked in the material response. However, it is computationally more demanding compared to the finite element method. Furthermore, the finite element method is very effective for modeling problems without damage. Hence, it is desirable to couple the PD theory and FEM to take advantage of their salient features if the regions of potential failure sites are identified prior to the analysis. Then, the regions in which failure is expected can be modeled by using the PD theory and the rest can be analyzed by using FEM.

The peridynamic (PD) theory can be applied to other physical fields such as thermal diffusion, neutronic diffusion, vacancy diffusion, and electrical potential distribution. This paves the way for fully coupling various field equations and deformation within the framework of peridynamics using the same computational domain.

This chapter concerns the derivation of the coupled peridynamic (PD) thermomechanics equations based on thermodynamic considerations. The generalized peridynamic model for fully coupled thermomechanics is derived using the conservation of energy and the free-energy function. Subsequently, the bond-based coupled PD thermomechanics equations are obtained by reducing the generalized formulation. These equations are also cast into their nondimensional forms. After describing the numerical solution scheme, solutions to certain coupled thermomechanical problems with known previous solutions are presented.