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physics

Peridynamics

Peridynamics is a physics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Peridynamics rather than just read about it. In short: Peridynamics is a non-local formulation of continuum mechanics that is oriented toward deformations 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.

Peridynamics — main illustration
Peridynamics — illustration

Key takeaways

  • Peridynamics belongs to physics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Peridynamics to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Peridynamics from memory before moving on to harder problems.

Reference excerpt

Peridynamics is a non-local formulation of continuum mechanics that is oriented toward deformations 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 ( 1 / 3 {\displaystyle 1/3} for plane stress and 1 / 4 {\displaystyle 1/4} for plane strain in two-dimensional configurations; 1 / 4 {\displaystyle 1/4} 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

… excerpt ends here. Continue reading the full article.

Illustrations

Peridynamics: 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.
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: (a) Kinematics of material body Ω t {\displaystyle \Omega _{t}} within peridynamic theory. (b) Representation of peridynamic horizon of x {\displaystyle {\bf {x}}} .
(a) Kinematics of material body Ω t {\displaystyle \Omega _{t}} within peridynamic theory. (b) Representation of peridynamic horizon of x {\displaystyle {\bf {x}}} .
Peridynamics: Pictorial representation of some widely used micro-modulus function c ( ξ , δ ) = c ( 0 , δ ) k ( ξ , δ ) {\displaystyle c({\bf {\xi }},\delta )=c({\bf {{0},\delta )k({\bf {\xi }},\delta )}}} .
Pictorial representation of some widely used micro-modulus function c ( ξ , δ ) = c ( 0 , δ ) k ( ξ , δ ) {\displaystyle c({\bf {\xi }},\delta )=c({\bf {{0},\delta )k({\bf {\xi }},\delta )}}} .
Peridynamics: Representation of peridynamic pairwise force function f ( ξ , η ) {\displaystyle {\bf {f}}(\xi ,\eta )} with bond-breaking function μ ( s , t ) {\displaystyle \mu (s,t)} ; after the critical stretch value s 0 {\displaystyle s_{0}} is exceeded, the bond is considered broken and no force exists between the two involved material points.
Representation of peridynamic pairwise force function f ( ξ , η ) {\displaystyle {\bf {f}}(\xi ,\eta )} with bond-breaking function μ ( s , t ) {\displaystyle \mu (s,t)} ; after the critical stretch value s 0 {\displaystyle s_{0}} is exceeded, the bond is considered broken and no force exists between the two involved material points.
Peridynamics: A ductile fracture of an Al-Mg-Si alloy
A ductile fracture of an Al-Mg-Si alloy

Worked examples

Example 1 — a first encounter with Peridynamics

Start with the simplest possible case. Write down what Peridynamics claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Peridynamics before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Peridynamics ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Peridynamics

In research
Peridynamics appears in physics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Peridynamics in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Peridynamics is common in secondary-school and first-year university syllabi. It links to neighbouring topics Continuum mechanics, Fracture mechanics, so understanding it makes those chapters shorter.
In everyday life
Look for Peridynamics outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Peridynamics in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Peridynamics means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Peridynamics out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Peridynamics in simple terms?

Peridynamics is a non-local formulation of continuum mechanics that is oriented toward deformations 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 i…

Why does Peridynamics matter?

Because it connects several physics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Peridynamics?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Peridynamics.

Tags

  • Continuum mechanics
  • Fracture mechanics

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