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Wikipedia / Peridynamics

Peridynamics is a non-local formulation of continuum mechanics that is oriented toward deformationss with discontinuities, especially fractures. Originally, bond-based peridynamic was introduced, wherein, internal interaction forces between a material point and all the other ones with which it can interact, are modeled as a central force field. This type of force field can be imagined as a mesh of bonds connecting each point of the body with every other interacting point within a certain distance which depends on a material property, called the peridynamic horizon. Later, to overcome bond-based framework limitations for the material Poisson's ratio for plane stress for plane strain for three-dimensional ones), state-base peridynamics, has been formulated. Its characteristic feature is that the force exchanged between a point and another one is influenced by the deformation state of all other bonds relative to its interaction zone.

The characteristic feature of peridynamics, which makes it different from classical local mechanics, is the presence of finite-range bonds between any two points of the material body: it is a feature that approaches such formulations as discrete meso-scale theories of matter.

Etymology

The term peridynamic, as an adjective, was proposed in the year 2000 and comes from the prefix peri-, which means all around, near, or surrounding; and the root dyna, which means force or power. The term peridynamics, as a noun, is a shortened form of the phrase peridynamic model of solid mechanics.

Purpose

A fracture is a mathematical singularity to which the classical equations of continuum mechanics cannot be applied directly. The peridynamic theory has been proposed with the purpose of mathematically models fractures formation and dynamic in elastic materials. It is founded on integral equations, in contrast with classical continuum mechanics, which is based on partial differential equations. Since partial derivatives do not exist on crack surfaces and other geometric singularities, the classical equations of continuum mechanics cannot be applied directly when such features are present in a deformation. The integral equations of the peridynamic theory hold true also on singularities and can be applied directly, because they do not require partial derivatives. The ability to apply the same equations directly at all points in a mathematical model of a deforming structure helps the peridynamic approach to avoid the need for the special techniques of fracture mechanics like xFEM. For example, in peridynamics, there is no need for a separate crack growth law based on a stress intensity factor.

Definition and basic terminology

: the peridynamic horizon radius. This perspective approaches much more to molecular dynamics than macroscopic bodies, and as a consequence, is not based on the concept of stress tensor (which is a local concept) and drift toward the notion of pairwise force exchanges within its peridynamic horizon. With a Lagrangian point of view, suited for small displacements, the peridynamic horizon is considered fixed in the reference configuration and, then, deforms with the body. can be either 1, 2 or 3. The body has a positive density. The reference configuration can either be the stress represents a suitable distance in the literature. It is commonly known as the horizon or the family.

The kinematics. Furthermore, for each pair of interacting points, the change in the length of the bond relative to the initial configuration is tracked over time through the relative strain, which can be expressed as:

denotes the Euclidean norm.

is referred to as a bond. These pairwise bonds have varying lengths over time in response to the force per unit volume squared, denoted as

This force is commonly known as the pairwise force function or peridynamic kernel, and it encompasses all the constitutive, is introduced, which results in the following equation of motion, representing the fundamental equation of peridynamics:

The vector valued function is the force density. The dimension.

Bond-based peridynamics

so that

Action and reaction principle

. This expression reflects the principle of action and reaction, commonly known as Newton's third law. It guarantees the conservation of linear momentum in a system composed of mutually interacting particles.

Angular momentum conservation

. This condition arises from considering the relative deformed ray-vector . The condition is satisfied if and only if is a scalar-valued function.

Hyperelastic material

An hyperelastic material is a material with constitutive relation such that:

or, equivalently, by Stokes' theorem

and, thus,

is the scalar valued potential function. Due to the necessity of satisfying angular momentum follows

is obtained

a scalar valued function. The elastic. Applying the isotropy,

Linear elastic material

then, a second-order micro-modulus tensor can be defined as

is the identity tensor. Following application of linear momentum balance, elasticity and isotropy condition, the micro-modulus tensor can be expressed in this form

Therefore, for a linearised hyperelastic material, its peridynamic kernel holds the following structure

Expressions for the peridynamic kernel

The peridynamic kernel is a versatile function that characterizes the constitutive behavior of materials within the framework of peridynamic theory. One commonly employed formulation of the kernel is used to describe a class of materials known as prototype micro-elastic brittle (PMB) materials. In the case of isotropic PMB materials, the pairwise force is assumed to be linearly proportional to the finite stretch experienced by the material, defined as

so that

where

is defined as follow

with

is referred to as the micro-modulus constant. If the critical value is exceeded, the bond is considered broken.

After a comparison between the strain energy density can be found

is the material bulk modulus.

Following the same approach is now a micro-modulus function increases, but the specific manner in which this decrease occurs can vary.

The micro-modulus function is expressed as

is obtained by comparing peridynamic strain density with the classical mechanical theories; with the following properties (given the restrictions of momentum conservation and isotropy)

is the Dirac delta function.

Cylindrical micro-modulus

The simplest expression for the micro-modulus function is

is the indicator function, defined as

Triangular micro-modulus

to a be a linear function

Normal micro-modulus

If one wants to reflects the fact that most common discrete physical systems are characterized by a Maxwell-Boltzmann distribution can be utilized

Quartic micro-modulus

function

Overall, depending on the specific material property to be modeled, there exists a wide range of expressions for the micro-modulus and, in general, for the peridynamic kernel. The above list is, thus, not exhaustive.

Damage

Damage is incorporated in the pairwise force function by allowing bonds to break when their elongation exceeds some prescribed value. After a bond breaks, it no longer sustains any force, and the endpoints are effectively disconnected from each other. When a bond breaks, the force it was carrying is redistributed to other bonds that have not yet broken. This increased load makes it more likely that these other bonds will break. The process of bond breakage and load redistribution, leading to further breakage, is how cracks grow in the peridynamic model.

Analytically, the bond breaking is specified inside the expression of the peridynamic kernel, by the function

can be employed.

State-based peridynamics

The theory described above assumes that each peridynamic bond responds independently of all the others. This is an oversimplification for most materials and leads to restrictions on the types of materials that can be modeled. In particular, this assumption implies that any isotropic linear elastic solid is restricted to a Poisson ratio of 1/4.

To address this lack of generality, the idea of peridynamic states was introduced. This framework allows the force density in each bond to depend on the stretches in all the bonds connected to its endpoints, in addition to its own stretch. For example, the force in a bond could depend on the net volume changes at the endpoints. The effect of this volume change, relative to the effect of the bond stretch, determines the Poisson ratio. With peridynamic states, any material that can be modeled within the standard theory of continuum mechanics can be modeled as a peridynamic material, while retaining the advantages of the peridynamic theory for fracture.

Mathematically the equation of the internal and external force term

used in the bond-based formulations is substituted by

is the force vector state field.

is a mathematical object similar to a tensor, with the exception that it is

in general non-linear;
in general non-continuous;
is not finite dimensional.

Vector states are states of order equal to 2. For so called simple material is defined as

is a Riemann-integrable is called deformation vector state field and is defined by the following relation

under the deformation

such that

which means that two distinct particles never occupy the same point as the deformation progresses.

It can be proved, while, if the constitutive relation is such that

the force vector state field satisfy balance of angular momentum.

Applications

The growing interest in peridynamics come from its capability to fill the gap between atomistic theories of matter and classical local continuum mechanics. It is applied effectively to micro-scale phenomena, such as crack formation and propagation, wave dispersion, intra-granular fracture. These phenomena can be described by appropriate adjustment of the peridynamic horizon radius, which is directly linked to the extent of non-local interactions between points within the material.

In addition to the aforementioned research fields, peridynamics' non-local approach to discontinuities has found applications in various other areas. In geo-mechanics, it has been employed to study water-induced soil cracks, geo-material failure, rocks fragmentation, and so on. In biology, peridynamics has been used to model long-range interactions in living tissues, cellular ruptures, cracking of bio-membranes, and more. Furthermore, peridynamics has been extended to thermal diffusion theory, enabling the modeling of heat conduction in materials with discontinuities, defects, inhomogeneities, and cracks. It has also been applied to study advection-diffusion phenomena in multi-phase fluids and to construct models for transient advection-diffusion problems. With its versatility, peridynamics has been used in various multi-physics analyses, including micro-structural analysis, fatigue and heat conduction in composite materials, galvanic corrosion in metals, electricity-induced cracks in dielectric materials and more.