Peridynamics
Peridynamics is a non-local formulation of continuum mechanics that is oriented toward deformations with discontinuities, especially fractures.
[5] 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.
[1] The characteristic feature of peridynamics, which makes it different from classical local mechanics, is the presence of finite-range bond between any two points of the material body: it is a feature that approaches such formulations to discrete meso-scale theories of matter.
[1] 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.
Since partial derivatives do not exist on crack surfaces[1] 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.
[6] For example, in peridynamics, there is no need for a separate crack growth law based on a stress intensity factor.
[7] In the context of peridynamic theory, physical bodies are treated as constituted by a continuous points mesh which can exchange long-range mutual interaction forces, within a maximum and well established distance
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.
These pairwise bonds have varying lengths over time in response to the force per unit volume squared, denoted as[3]
[3] In this formulation of peridynamics, the kernel is determined by the nature of internal forces and physical constraints that governs the interaction between only two material points.
It guarantees the conservation of linear momentum in a system composed of mutually interacting particles.
The condition is satisfied if and only if the pairwise force density vector has the same direction as the relative deformed ray-vector.
[1] Due to the necessity of satisfying angular momentum conservation, the condition below on the scalar valued function
Following application of linear momentum balance, elasticity and isotropy condition, the micro-modulus tensor can be expressed in this form[1]
If the critical value is exceeded, the bond is considered broken, and a pairwise force of zero is assigned for all
This function provides a more detailed description of how the intensity of pairwise forces is distributed over the peridynamic horizon
If one wants to reflects the fact that most common discrete physical systems are characterized by a Maxwell-Boltzmann distribution, in order to include this behavior in peridynamics, the following expression for
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.
[11] Damage is incorporated in the pairwise force function by allowing bonds to break when their elongation exceeds some prescribed value.
In particular, this assumption implies that any isotropic linear elastic solid is restricted to a Poisson ratio of 1/4.
[5] The growing interest in peridynamics[6] 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,[15][16][17] wave dispersion,[18][19] intra-granular fracture.
[20] These phenomena can be described by appropriately adjustment of the peridynamic horizon radius, which is directly linked to the extent of non-local interactions between points within the material.
[21] 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,[22][23] geo-material failure,[24] rocks fragmentation,[25][26] and so on.
In biology, peridynamics has been used to model long-range interactions in living tissues,[27] cellular ruptures, cracking of bio-membranes,[28] and more.
[6] Furthermore, peridynamics has been extended to thermal diffusion theory,[29][30] enabling the modeling of heat conduction in materials with discontinuities, defects, inhomogeneities, and cracks.
