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Computer model of the necking of an aluminum rod under tension. Colors indicate temperature increase due to plastic heating. Calculation performed with the Emu computer code using peridynamic state-based framework.

Peridynamics is a non-local formulation of continuum mechanics that is oriented toward deformations with discontinuities, especially fractures. Originally, "bond-based" peridynamic has been introduced [1], wherein, internal interactions forces between a material point and all the other ones with which it can interact, are modeled as a central forces field. This type of forces field can be imagined as a mesh of bonds connecting each point of the body with every other interacting points within a certain distance which depends on material property, called "peridynamic horizon". Later, to overcome bond-based framework limitations for the material Poisson’s ratio [2][3] ( for plane stress and for plane strain in two-dimesnional configurations; for three-dimensional ones), "state-base" peridynamics, has been formulated [4]. 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 its interaction zone.

The characteristic feature of peridynamics, which make it differs from classical local mechanics, is the presence of finite-range bond between any two points of the material body: it is a feature that approach such formulations to discrete meso-scale theories of matter.

Etymology

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The term "peridynamic," 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," a noun, is a shortened form of the phrase peridynamic model of solid mechanics [1].

Purpose

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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 [1]. 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 [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 holds 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 [5]. For example, in peridynamics, there is no need for a separate crack growth law based on a stress intensity factor.

Definition and basic terminology

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(a) Kinematics of material body within peridynamic theory. (b) Representation of peridynamic horizon of .

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 : 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" that a material point 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 [2].Consider a material body represented by , where can be either 1, 2, or 3. The body has a positive density . Its reference configuration at the initial time is denoted by . It's important to note that the reference configuration can either be the stress-free configuration or a specific configuration of the body chosen as a reference. In the context of peridynamics, every point in interacts with all the points within a certain neighborhood defined by , where and represents a suitable distance function on . This neighborhood is often referred to as in the literature. It is commonly known as the "horizon"[6][7] or the "family" of [2][8].

The kinematics of is described in terms of its displacement from the reference position, denoted as . Consequently, the position of at a specific time is determined by . 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:

where denotes the Euclidean norm.

The interaction between any and 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 (material-dependent) properties. It describes how the internal forces depend on the deformation. It's worth noting that the dependence of on has been omitted here for the sake of simplicity in notation. Additionally, an external forcing term, , is introduced, which results in the following equation of motion, representing the fundamental equation of peridynamics.

where the integral term is the sum of all of the internal and external per-unit-volume forces acting on :

The vector valued function is the force density that exerts on . This force density depends on the relative displacement and relative position vectors between and . The dimension of is .

Bond-based peridynamics

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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. For brevity sake, the following quantities are defined and so that

Actio et reactio principle

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For any and belonging to the neighborhood , the following relationship holds: . 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

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For any and belonging to the neighborhood , the following condition holds: . This condition arises from considering the relative deformed ray-vector connecting and as . The condition is satisfied if and only if the pairwise force density vector has the same direction as the relative deformed ray-vector. In other words, for all and , where is a scalar-valued function.

Hyperelastic material

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

or, equivalently, by Stokes' theorem

,

and, thus,

In the equation above is the scalar valued potential function in [1]. Due to the necessity of satisfying angular momentum conservation, the condition below on the scalar valued function follows

Integrating both sides of the equation, the following condition on is obtained [1]

,   

for a scalar valued function. The elastic nature of is evident: the interaction force depends only on the initial relative position between points and and the modulus of their relative position,, in the deformed configuration at time . Applying the isotropy hypothesis, the dependence on vector can be substituted with a dependence on its modulus ,

Bond forces can, thus, be considered as modeling a spring net that connects each point  pairwise with .

Linear elastic material

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if , the peridynamic kernel can be linearised around :

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

where, . Following application of linear momentum balance, elasticity and isotropy condition, the micro-modulus tensor can be expressed in this form [1]

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

   

Expressions for the peridynamic kernel

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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

and where the scalar function is defined as follow[6]

with

The constant is referred to as the "micro-modulus constant," and the function serves to indicate whether, at a given time , the bond stretch associated with the pair has surpassed the critical value . If the critical value is exceeded, the bond is considered "broken," and a pairwise force of zero is assigned for all .

After a comparison between the strain energy density value obtained under isotropic extension respectively employing peridynamics and classical continuum theory framework, the physical coherent value of micro-modulus can be found[9]

where is the material bulk modulus.

Following the same approach [10] the micro-modulus constant can be extended to , where is now a "micro-modulus function." This function provides a more detailed description of how the intensity of pairwise forces is distributed over the peridynamic horizon . Intuitively, the intensity of forces decreases as the distance between and increases, but the specific manner in which this decrease occurs can vary.

The micro-modulus function is expressed as

where the constant is obtained by comparing peridynamic strain density with the classical mechanical theories [11]; is a function defined on with the following properties (given the restrictions of momentum conservation and isotropy)[12]

where is the Dirac Delta function.

Widely employed Peridynamics Micromodulus
Pictorial representation of some widely used micro-modulus function .

Cylindrical micro-modulus

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The simplest expression for the micro-modulus function is

,

where : is the indicator function of the subset , defined as

Triangular micro-modulus

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It is characterized by to a be a linear function [13]

Normal micro-modulus

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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 can be utilized [14]

Quartic micro-modulus

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In the literature one can find also the following expression for the function[12]

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

Damage

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Representation of peridynamic pairwise force function with bond-breaking function ; after the critical stretch value is exceeded, the bond is considered broken and no force exists between the two involved material points.

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 braking is specified inside the expression of peridynamic kernel, by the function

If the graph of versus bond stretching is plotted, the action of bond braking function in fracture formation is clear. However not only abrupt fracture can be modeled in peridynamic framework and more general expression for can be employed[6].

State-based peridynamics

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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[2].

To address this lack of generality, the idea of "peridynamic states" was introduced. This 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 [4].

Mathematically the equation of the internal and external force term

used in the bond-based formulations is substituted by

where is the force vector state field.

A general m-order state is a mathematical object similar to a tensor, with the exception that it is [4]

  • 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

where is a Riemann-integrable function on , and is called "deformation vector state field" and is defined by the following relation

thus is the image of the bond under the deformation

such that

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

It can be proved [4] that balance of linear momentum follow from the definition of , while, if the constitutive relation is such that

the force vector state field satisfy balance of angular momentum.

Applications

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A ductile fracture of an Al-Mg-Si alloy

The growing interest in peridynamics [5] 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 [5]. 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. It has also been applied to study advection-diffusion phenomena in multi-phase fluids [31] and to construct models for transient advection-diffusion problems [32]. With its versatility, peridynamics has been used in various multi-physics analyses, including micro-structural analysis [33], fatigue and heat conduction in composite materials [34] [35], galvanic corrosion in metals [36], electricity-induced cracks in dielectric materials, and more.

See also

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References

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  1. ^ a b c d e f g Silling, S.A. (January 2000). "Reformulation of elasticity theory for discontinuities and long-range forces". Journal of the Mechanics and Physics of Solids. 48 (1): 175–209. doi:10.1016/S0022-5096(99)00029-0.
  2. ^ a b c d Madenci, Erdogan; Oterkus, Erkan (2014). Peridynamic theory and its applications. New York, NY: Springer. pp. 19–43. ISBN 978-1-4614-8464-6.
  3. ^ Macek, Richard W.; Silling, Stewart A. (November 2007). "Peridynamics via finite element analysis". Finite Elements in Analysis and Design. 43 (15): 1169–1178. doi:10.1016/j.finel.2007.08.012. ISSN 0168-874X.
  4. ^ a b c d Silling, S. A.; Epton, M.; Weckner, O.; Xu, J.; Askari, E. (2007-08-08). "Peridynamic States and Constitutive Modeling". Journal of Elasticity. 88 (2): 151–184. doi:10.1007/s10659-007-9125-1. ISSN 0374-3535.
  5. ^ a b c Javili, Ali; Morasata, Rico; Oterkus, Erkan; Oterkus, Selda (November 2019). "Peridynamics review". Mathematics and Mechanics of Solids. 24 (11): 3714–3739. doi:10.1177/1081286518803411. ISSN 1081-2865.
  6. ^ a b c Silling, S.A.; Askari, E. (June 2005). "A meshfree method based on the peridynamic model of solid mechanics". Computers & Structures. 83 (17–18): 1526–1535. doi:10.1016/j.compstruc.2004.11.026.
  7. ^ Ren, Huilong; Zhuang, Xiaoying; Cai, Yongchang; Rabczuk, Timon (2016-12-21). "Dual-horizon peridynamics: Dual-horizon peridynamics". International Journal for Numerical Methods in Engineering. 108 (12): 1451–1476. doi:10.1002/nme.5257.
  8. ^ Chen, Ziguang; Bakenhus, Drew; Bobaru, Florin (November 2016). "A constructive peridynamic kernel for elasticity". Computer Methods in Applied Mechanics and Engineering. 311: 356–373. doi:10.1016/j.cma.2016.08.012.
  9. ^ Silling, S.A.; Askari, E. (June 2005). "A meshfree method based on the peridynamic model of solid mechanics". Computers & Structures. 83 (17–18): 1526–1535. doi:10.1016/j.compstruc.2004.11.026.
  10. ^ Bobaru, Florin; Duangpanya, Monchai (September 2010). "The peridynamic formulation for transient heat conduction". International Journal of Heat and Mass Transfer. 53 (19–20): 4047–4059. doi:10.1016/j.ijheatmasstransfer.2010.05.024.
  11. ^ Chen, Zhiyong; Woody Ju, J.; Su, Guoshao; Huang, Xiaohua; Li, Shuang; Zhai, Lianjun (July 2019). "Influence of micro-modulus functions on peridynamics simulation of crack propagation and branching in brittle materials". Engineering Fracture Mechanics. 216: 106498. doi:10.1016/j.engfracmech.2019.106498.
  12. ^ a b Huang, Dan; Lu, Guangda; Wang, Chongwen; Qiao, Pizhong (June 2015). "An extended peridynamic approach for deformation and fracture analysis". Engineering Fracture Mechanics. 141: 196–211. doi:10.1016/j.engfracmech.2015.04.036.
  13. ^ Ha, Youn Doh; Bobaru, Florin (March 2010). "Studies of dynamic crack propagation and crack branching with peridynamics". International Journal of Fracture. 162 (1–2): 229–244. doi:10.1007/s10704-010-9442-4. ISSN 0376-9429.
  14. ^ Kilic, Bahattin (2008). "Peridynamic Theory for Progressive Failure Prediction in Homogeneous and Heterogeneous Materials". {{cite journal}}: Cite journal requires |journal= (help)
  15. ^ Agwai, Abigail; Guven, Ibrahim; Madenci, Erdogan (September 2011). "Predicting crack propagation with peridynamics: a comparative study". International Journal of Fracture. 171 (1): 65–78. doi:10.1007/s10704-011-9628-4. ISSN 0376-9429.
  16. ^ Lipton, Robert (October 2014). "Dynamic Brittle Fracture as a Small Horizon Limit of Peridynamics". Journal of Elasticity. 117 (1): 21–50. doi:10.1007/s10659-013-9463-0. ISSN 0374-3535.
  17. ^ Silling, S. A.; Weckner, O.; Askari, E.; Bobaru, F. (March 2010). "Crack nucleation in a peridynamic solid". International Journal of Fracture. 162 (1–2): 219–227. doi:10.1007/s10704-010-9447-z. ISSN 0376-9429.
  18. ^ Coclite, G. M.; Dipierro, S.; Fanizza, G.; Maddalena, F.; Romano, M.; Valdinoci, E. (March 2023). "Qualitative Aspects in Nonlocal Dynamics". Journal of Peridynamics and Nonlocal Modeling. 5 (1): 1–19. doi:10.1007/s42102-021-00064-z. ISSN 2522-896X.
  19. ^ Seleson, Pablo; Parks, Michael L.; Gunzburger, Max; Lehoucq, Richard B. (January 2009). "Peridynamics as an Upscaling of Molecular Dynamics". Multiscale Modeling & Simulation. 8 (1): 204–227. doi:10.1137/09074807X. ISSN 1540-3459.
  20. ^ Behzadinasab, Masoud; Foster, John T. (April 2020). "A semi-Lagrangian constitutive correspondence framework for peridynamics". Journal of the Mechanics and Physics of Solids. 137: 103862. doi:10.1016/j.jmps.2019.103862.
  21. ^ Askari, E; Bobaru, F; Lehoucq, R B; Parks, M L; Silling, S A; Weckner, O (2008-07-01). "Peridynamics for multiscale materials modeling". Journal of Physics: Conference Series. 125: 012078. doi:10.1088/1742-6596/125/1/012078. ISSN 1742-6596.
  22. ^ Ni, Tao; Pesavento, Francesco; Zaccariotto, Mirco; Galvanetto, Ugo; Zhu, Qi-Zhi; Schrefler, Bernhard A. (July 2020). "Hybrid FEM and peridynamic simulation of hydraulic fracture propagation in saturated porous media". Computer Methods in Applied Mechanics and Engineering. 366: 113101. doi:10.1016/j.cma.2020.113101.
  23. ^ Zhou, Xiao-Ping; Wang, Yun-Teng; Shou, Yun-Dong (August 2020). "Hydromechanical bond-based peridynamic model for pressurized and fluid-driven fracturing processes in fissured porous rocks". International Journal of Rock Mechanics and Mining Sciences. 132: 104383. doi:10.1016/j.ijrmms.2020.104383.
  24. ^ Song, Xiaoyu; Khalili, Nasser (January 2019). "A peridynamics model for strain localization analysis of geomaterials". International Journal for Numerical and Analytical Methods in Geomechanics. 43 (1): 77–96. doi:10.1002/nag.2854. ISSN 0363-9061.
  25. ^ Panchadhara, Rohan; Gordon, Peter A.; Parks, Michael L. (March 2017). "Modeling propellant-based stimulation of a borehole with peridynamics". International Journal of Rock Mechanics and Mining Sciences. 93: 330–343. doi:10.1016/j.ijrmms.2017.02.006.
  26. ^ Zhou, Xiao-Ping; Wang, Yun-Teng (January 2021). "State-of-the-Art Review on the Progressive Failure Characteristics of Geomaterials in Peridynamic Theory". Journal of Engineering Mechanics. 147 (1). doi:10.1061/(ASCE)EM.1943-7889.0001876. ISSN 0733-9399.
  27. ^ Lejeune, Emma; Linder, Christian (August 2017). "Modeling tumor growth with peridynamics". Biomechanics and Modeling in Mechanobiology. 16 (4): 1141–1157. doi:10.1007/s10237-017-0876-8. ISSN 1617-7959.
  28. ^ Taylor, Michael; Gözen, Irep; Patel, Samir; Jesorka, Aldo; Bertoldi, Katia (2016-11-09). van Veen, Hendrik W. (ed.). "Peridynamic Modeling of Ruptures in Biomembranes". PLOS ONE. 11 (11): e0165947. doi:10.1371/journal.pone.0165947. ISSN 1932-6203. PMC 5102442. PMID 27829001.{{cite journal}}: CS1 maint: PMC format (link) CS1 maint: unflagged free DOI (link)
  29. ^ Bobaru, Florin; Duangpanya, Monchai (April 2012). "A peridynamic formulation for transient heat conduction in bodies with evolving discontinuities". Journal of Computational Physics. 231 (7): 2764–2785. doi:10.1016/j.jcp.2011.12.017.
  30. ^ Oterkus, Selda; Madenci, Erdogan; Agwai, Abigail (May 2014). "Peridynamic thermal diffusion". Journal of Computational Physics. 265: 71–96. doi:10.1016/j.jcp.2014.01.027.
  31. ^ Foster, John (2019). "Nonlocal and fractional order methods for near-wall turbulence, large-eddy simulation, and fluid-structure interaction". Technical report, University of Texas at Austin Austin United States <math>}.
  32. ^ Zhao, Jiangming; Chen, Ziguang; Mehrmashhadi, Javad; Bobaru, Florin (November 2018). "Construction of a peridynamic model for transient advection-diffusion problems". International Journal of Heat and Mass Transfer. 126: 1253–1266. doi:10.1016/j.ijheatmasstransfer.2018.06.075.
  33. ^ Buryachenko, Valeriy A. (October 2020). "Generalized effective fields method in peridynamic micromechanics of random structure composites". International Journal of Solids and Structures. 202: 765–786. doi:10.1016/j.ijsolstr.2020.06.022.
  34. ^ Hu, Y.L.; Madenci, E. (January 2017). "Peridynamics for fatigue life and residual strength prediction of composite laminates". Composite Structures. 160: 169–184. doi:10.1016/j.compstruct.2016.10.010.
  35. ^ Oterkus, Erkan; Madenci, Erdogan (2012-03-28). "Peridynamic analysis of fiber-reinforced composite materials". Journal of Mechanics of Materials and Structures. 7 (1): 45–84. doi:10.2140/jomms.2012.7.45. ISSN 1559-3959.
  36. ^ Zhao, Jiangming; Jafarzadeh, Siavash; Rahmani, Mohammad; Chen, Ziguang; Kim, Yong-Rak; Bobaru, Florin (September 2021). "A peridynamic model for galvanic corrosion and fracture". Electrochimica Acta. 391: 138968. doi:10.1016/j.electacta.2021.138968.

Further readings

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