Chapter map

1. 1. Points
2. 2. Displacements
3. 3. New bond
4. 4. Stretch
5. 5. Bond law
6. 6. Force magnitude
7. 7. Pair direction
8. 8. Critical stretch
9. 9. Damage
10. 10. Horizon
11. 11. Surface
12. 12. Energy
13. 13. Refinement
14. 14. Conclusion

Visit the How Mathematics Works hub for the wider series and its connections to education, reasoning, science, computing, design and careers.

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1. Points

Here the geometry becomes operational. Set x0=0,x1=1.

A concrete situation

Original bond length=1.

What the mathematics does not license

State horizon. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Draw pair. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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2. Displacements

This step deserves a careful pause. Set u0=0,u1=.1.

A concrete situation

New positions are 0 and 1.1.

What the mathematics does not license

Keep directions. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Record data. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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3. New bond

The useful question is not only how to calculate, but what the calculation assumes. Subtract new positions.

A concrete situation

Current length=1.1.

What the mathematics does not license

Use absolute length. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Record length. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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4. Stretch

A small example makes the mechanism visible. Compute length change/original length.

A concrete situation

Stretch=.1.

What the mathematics does not license

Dimensionless. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Record stretch. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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5. Bond law

This is where modelling choices begin to affect evidence. Choose teaching c=20.

A concrete situation

Force magnitude=c s.

What the mathematics does not license

Not a universal constitutive law. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Record law. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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6. Force magnitude

The central mathematical idea is concrete. Multiply 20 by .1.

A concrete situation

Magnitude=2.

What the mathematics does not license

Keep units declared. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Record force. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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7. Pair direction

Here the geometry becomes operational. Assign opposite directions.

A concrete situation

Pair force sum=0.

What the mathematics does not license

Central symmetric bond. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Record vectors. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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8. Critical stretch

This step deserves a careful pause. Choose sc=.08.

A concrete situation

Current .1 exceeds threshold.

What the mathematics does not license

Illustrative brittle rule. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Record comparison. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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9. Damage

The useful question is not only how to calculate, but what the calculation assumes. Remove broken bond later.

A concrete situation

Its future force contribution is zero.

What the mathematics does not license

State irreversibility. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Record state. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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10. Horizon

A small example makes the mechanism visible. Add another point inside delta.

A concrete situation

Neighbour integral gains a term.

What the mathematics does not license

Use volume weights. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Record contribution. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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11. Surface

This is where modelling choices begin to affect evidence. Remove neighbours on one side.

A concrete situation

Neighbourhood becomes incomplete.

What the mathematics does not license

Apply eligible correction. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Record effect. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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12. Energy

The central mathematical idea is concrete. Integrate bond work.

A concrete situation

Damage releases stored energy.

What the mathematics does not license

Calibrate fracture energy. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Record ledger. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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13. Refinement

Here the geometry becomes operational. Change spacing and horizon deliberately.

A concrete situation

Separate numerical from nonlocal length effects.

What the mathematics does not license

Hold calibration clear. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Record table. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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14. Conclusion

This step deserves a careful pause. State horizon,law,volumes,damage and validation.

A concrete situation

Worked stretch=.1 and force=2.

What the mathematics does not license

Two-point teaching bond. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Write conclusion. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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A seven-day practice lab

1. Day 1: define the decision, observation unit, X variables and Y or reference condition.
2. Day 2: reproduce the smallest worked example and check every dimension, unit and sign.
3. Day 3: build a transparent baseline and write what it assumes.
4. Day 4: fit the main method inside a training-only pipeline; record every parameter.
5. Day 5: challenge one assumption with a deliberately difficult dataset or operating regime.
6. Day 6: compare validation metrics with the practical workload created by errors.
7. Day 7: present one supported conclusion, one unresolved uncertainty and one safe next experiment.

Questions for a student, parent or mentor

• Which part of the model is a mathematical definition, and which part is a choice?
• What small calculation would catch a sign, scale or dimension error?
• What evidence comes from training, validation and the untouched test?
• What benign condition could look like a fault or prediction failure?
• How would the answer change for another response, asset, school project or time period?
• What should a human review before acting on the score?

A method-specific audit before deployment

Peridynamics is a nonlocal continuum theory in which a material point interacts with other points inside a finite neighbourhood called its horizon. The equation of motion uses an integral of pair or state interactions instead of spatial derivatives of stress.

Consider two one-dimensional points initially one unit apart, at x0=0 and x1=1. Let their displacements be u0=0 and u1=0.1, so their new separation is 1.1.

A simple bond stretch is the change in bond length divided by original length. Here the stretch is (1.1-1)/1=0.1.

For a teaching linear bond law with micromodulus c=20 in compatible units, use force magnitude c times stretch. The magnitude is 20 times 0.1=2.

The force on the left point is directed right and the force on the right point is directed left. The pair sums to zero, illustrating internal linear-momentum balance for a symmetric central bond.

If the chosen critical stretch is 0.08, this teaching bond exceeds it because 0.1 is larger than 0.08. A brittle damage rule could then remove the bond contribution on later steps.

Did You Know? The same integral equation can remain meaningful at a crack because it does not require the displacement field to possess the spatial derivatives used by classical local elasticity.

The horizon sets the nonlocal interaction length. Making it smaller changes the model as well as the discretisation, so a convergence study must distinguish point spacing from horizon size.

Bond-based peridynamics is simpler but restricts the class of classical elastic responses it can reproduce. State-based formulations allow more general interactions and constitutive behaviour.

Particle volumes or quadrature weights multiply discrete interaction terms. A force density, total force and acceleration have different units, and silently omitting volume factors changes the model.

Near a free surface, a point has fewer neighbours inside its horizon. Surface effects or correction procedures may be needed if the chosen calibration assumed a full neighbourhood.

Damage as broken bonds offers a natural crack indicator, but the critical stretch and fracture energy must be calibrated consistently with horizon, discretisation and material behaviour.

Students can draw a horizon, list every interacting pair once, calculate stretch and equal-opposite forces, then remove one bond and observe how the force network changes.

Applications include dynamic fracture, impact, composites, fatigue research and coupled diffusion-mechanics. Nonlocality does not guarantee correct crack paths without material calibration and resolution.

For parents and students, the transferable idea is that nearby points can communicate across a finite distance, allowing a crack to emerge when enough local connections fail.

A responsible peridynamic report states the horizon, force or state law, discretisation, particle volumes, boundary treatment, damage criterion, time integrator, energy and momentum checks, convergence and validation.

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Frequently asked questions

What is peridynamics?

A nonlocal continuum theory whose equation of motion integrates interactions over a finite neighbourhood.

What is the horizon?

The maximum interaction distance around a material point in the chosen model.

Why can cracks emerge naturally?

The integral equation does not require displacement derivatives at a discontinuity, and damaged interactions can cease carrying force.

What is bond stretch?

The relative change in the distance between two interacting points.

Does nonlocality guarantee correct fracture?

No. Constitutive calibration, horizon, discretisation, boundaries, damage and validation remain essential.

What must be reported?

Horizon, interaction law, particle volumes, damage, boundaries, time integration, conservation, convergence and validation.

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Primary source and fact-check note

S. A. Silling, “Reformulation of Elasticity Theory for Discontinuities and Long-Range Forces,” Journal of the Mechanics and Physics of Solids 48, 2000. Definitions and core claims in this article were checked against this primary publication. Calculations shown here were independently recomputed. Examples are educational illustrations; they are not reported industrial performance results.

Useful next reading

• Phase-field method
• Finite element method
• Arbitrary Lagrangian–Eulerian method
• Numerical analysis hub
• How Mathematics Works

Keep the options open. Mathematics education supports clearer questions, transferable problem-solving skills and informed choices across science, engineering, computing, AI, statistics, design and many other careers. It does not guarantee intelligence, income, admission or employment; those outcomes depend on far more than one subject or model.

Method audit: representation, transfer and evidence

Neighbour audit

List every point and volume inside each horizon.

Bond audit

Check stretch, direction, symmetry and constitutive units.

Damage audit

Relate critical stretch to fracture energy and irreversibility.

Limit audit

Separate point refinement, horizon change and local-limit claims.

Mechanism workshop: calculate, challenge and transfer

Horizon map

List neighbours for several points.

Stretch table

Calculate tensile and compressive bonds.

Pair balance

Verify equal-opposite forces.

Volume test

Change quadrature weights and units.

Surface test

Compare full and truncated horizons.

Damage test

Break bonds above a threshold.

Energy ledger

Track stored and released bond energy.

Refinement

Vary spacing and horizon separately.

Student explanation

Explain cracks as failed connections.

Parent explanation

Relate nonlocal support to a network.

Evidence notebook: reconstruct every step

Evidence notebook 1: Points

Step 1 isolates points. Set x0=0,x1=1. The checked result is: Original bond length=1. Keep units, point or particle labels, grid or parameter coordinates, neighbourhood rules, signs, time level and precision beside every number. The safeguard is: State horizon. Draw pair. Change one input, recompute the local mechanism and the relevant invariant, and explain whether the difference comes from transfer, horizon, rational basis, neighbour operator, pressure solve, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 2: Displacements

Step 2 isolates displacements. Set u0=0,u1=.1. The checked result is: New positions are 0 and 1.1. Keep units, point or particle labels, grid or parameter coordinates, neighbourhood rules, signs, time level and precision beside every number. The safeguard is: Keep directions. Record data. Change one input, recompute the local mechanism and the relevant invariant, and explain whether the difference comes from transfer, horizon, rational basis, neighbour operator, pressure solve, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 3: New bond

Step 3 isolates new bond. Subtract new positions. The checked result is: Current length=1.1. Keep units, point or particle labels, grid or parameter coordinates, neighbourhood rules, signs, time level and precision beside every number. The safeguard is: Use absolute length. Record length. Change one input, recompute the local mechanism and the relevant invariant, and explain whether the difference comes from transfer, horizon, rational basis, neighbour operator, pressure solve, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 4: Stretch

Step 4 isolates stretch. Compute length change/original length. The checked result is: Stretch=.1. Keep units, point or particle labels, grid or parameter coordinates, neighbourhood rules, signs, time level and precision beside every number. The safeguard is: Dimensionless. Record stretch. Change one input, recompute the local mechanism and the relevant invariant, and explain whether the difference comes from transfer, horizon, rational basis, neighbour operator, pressure solve, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 5: Bond law

Step 5 isolates bond law. Choose teaching c=20. The checked result is: Force magnitude=c s. Keep units, point or particle labels, grid or parameter coordinates, neighbourhood rules, signs, time level and precision beside every number. The safeguard is: Not a universal constitutive law. Record law. Change one input, recompute the local mechanism and the relevant invariant, and explain whether the difference comes from transfer, horizon, rational basis, neighbour operator, pressure solve, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 6: Force magnitude

Step 6 isolates force magnitude. Multiply 20 by .1. The checked result is: Magnitude=2. Keep units, point or particle labels, grid or parameter coordinates, neighbourhood rules, signs, time level and precision beside every number. The safeguard is: Keep units declared. Record force. Change one input, recompute the local mechanism and the relevant invariant, and explain whether the difference comes from transfer, horizon, rational basis, neighbour operator, pressure solve, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 7: Pair direction

Step 7 isolates pair direction. Assign opposite directions. The checked result is: Pair force sum=0. Keep units, point or particle labels, grid or parameter coordinates, neighbourhood rules, signs, time level and precision beside every number. The safeguard is: Central symmetric bond. Record vectors. Change one input, recompute the local mechanism and the relevant invariant, and explain whether the difference comes from transfer, horizon, rational basis, neighbour operator, pressure solve, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 8: Critical stretch

Step 8 isolates critical stretch. Choose sc=.08. The checked result is: Current .1 exceeds threshold. Keep units, point or particle labels, grid or parameter coordinates, neighbourhood rules, signs, time level and precision beside every number. The safeguard is: Illustrative brittle rule. Record comparison. Change one input, recompute the local mechanism and the relevant invariant, and explain whether the difference comes from transfer, horizon, rational basis, neighbour operator, pressure solve, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 9: Damage

Step 9 isolates damage. Remove broken bond later. The checked result is: Its future force contribution is zero. Keep units, point or particle labels, grid or parameter coordinates, neighbourhood rules, signs, time level and precision beside every number. The safeguard is: State irreversibility. Record state. Change one input, recompute the local mechanism and the relevant invariant, and explain whether the difference comes from transfer, horizon, rational basis, neighbour operator, pressure solve, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 10: Horizon

Step 10 isolates horizon. Add another point inside delta. The checked result is: Neighbour integral gains a term. Keep units, point or particle labels, grid or parameter coordinates, neighbourhood rules, signs, time level and precision beside every number. The safeguard is: Use volume weights. Record contribution. Change one input, recompute the local mechanism and the relevant invariant, and explain whether the difference comes from transfer, horizon, rational basis, neighbour operator, pressure solve, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 11: Surface

Step 11 isolates surface. Remove neighbours on one side. The checked result is: Neighbourhood becomes incomplete. Keep units, point or particle labels, grid or parameter coordinates, neighbourhood rules, signs, time level and precision beside every number. The safeguard is: Apply eligible correction. Record effect. Change one input, recompute the local mechanism and the relevant invariant, and explain whether the difference comes from transfer, horizon, rational basis, neighbour operator, pressure solve, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 12: Energy

Step 12 isolates energy. Integrate bond work. The checked result is: Damage releases stored energy. Keep units, point or particle labels, grid or parameter coordinates, neighbourhood rules, signs, time level and precision beside every number. The safeguard is: Calibrate fracture energy. Record ledger. Change one input, recompute the local mechanism and the relevant invariant, and explain whether the difference comes from transfer, horizon, rational basis, neighbour operator, pressure solve, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 13: Refinement

Step 13 isolates refinement. Change spacing and horizon deliberately. The checked result is: Separate numerical from nonlocal length effects. Keep units, point or particle labels, grid or parameter coordinates, neighbourhood rules, signs, time level and precision beside every number. The safeguard is: Hold calibration clear. Record table. Change one input, recompute the local mechanism and the relevant invariant, and explain whether the difference comes from transfer, horizon, rational basis, neighbour operator, pressure solve, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 14: Conclusion

Step 14 isolates conclusion. State horizon,law,volumes,damage and validation. The checked result is: Worked stretch=.1 and force=2. Keep units, point or particle labels, grid or parameter coordinates, neighbourhood rules, signs, time level and precision beside every number. The safeguard is: Two-point teaching bond. Write conclusion. Change one input, recompute the local mechanism and the relevant invariant, and explain whether the difference comes from transfer, horizon, rational basis, neighbour operator, pressure solve, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Failure cases and responsible recovery

Horizon is just mesh size

It is a model length as well as a numerical parameter. Build a small counterexample, restore the missing condition and recalculate before making the claim.

Volume weights omitted

Discrete force units change. Build a small counterexample, restore the missing condition and recalculate before making the claim.

Pair counted inconsistently

Momentum and energy ledgers fail. Build a small counterexample, restore the missing condition and recalculate before making the claim.

All peridynamics is bond-based

State-based formulations are more general. Build a small counterexample, restore the missing condition and recalculate before making the claim.

Surface deficit ignored

Boundary calibration changes. Build a small counterexample, restore the missing condition and recalculate before making the claim.

Critical stretch copied

Fracture energy depends on model and horizon. Build a small counterexample, restore the missing condition and recalculate before making the claim.

Broken bond healed silently

Damage history becomes ambiguous. Build a small counterexample, restore the missing condition and recalculate before making the claim.

Crack picture as proof

Material validation is still required. Build a small counterexample, restore the missing condition and recalculate before making the claim.

Verified case study: fourteen steps from model to decision

1. Define the model

Write the governing balance, variables, units, domain and boundary or initial data.

2. Choose the representation

Name material points, nonlocal neighbourhoods, CAD basis functions or fluid particles.

3. State the transfer or interaction

Write the grid basis, bond law, rational map or neighbour operator.

4. Calculate a tiny case

Use the smallest nontrivial example with labelled coordinates.

5. Check an invariant

Verify mass, momentum, partition of unity, pair balance, energy, volume or uniform-state preservation as eligible.

6. Audit signs and units

Check force direction, volume factors, rational denominators and pressure-source convention.

7. Challenge a difficult region

Test cell crossing, a free surface, a crack, repeated knot or disordered neighbours.

8. Refine

Improve points, grid, basis, horizon or interaction radius consistently.

9. Separate errors

Distinguish modelling, transfer, quadrature, sampling, iteration and rounding.

10. Compare a baseline

Use an exact, manufactured, published or independently implemented reference.

11. Count cost

Record particles, neighbours, control variables, nonzeros, memory and elapsed time.

12. Validate

Compare application outputs with independent measurements or accepted benchmarks.

13. Communicate limits

State what the result supports and what remains untested.

14. Preserve evidence

Save equations, data, code, parameters, meshes, neighbour lists, logs and corrected arithmetic.

Learner lab: ten checks for transferable understanding

Lab question 1

Create a reproducible mini-investigation of horizons, bonds, stretch, nonlocal force, damage, energy and fracture. For task 1, state one prediction, calculate a transparent case, record units and conventions, compare an independent baseline, identify one failure mode, and explain the result once for a student and once for a parent. End with one practical next step and one claim the evidence does not support.

Lab question 2

Create a reproducible mini-investigation of horizons, bonds, stretch, nonlocal force, damage, energy and fracture. For task 2, state one prediction, calculate a transparent case, record units and conventions, compare an independent baseline, identify one failure mode, and explain the result once for a student and once for a parent. End with one practical next step and one claim the evidence does not support.

Lab question 3

Create a reproducible mini-investigation of horizons, bonds, stretch, nonlocal force, damage, energy and fracture. For task 3, state one prediction, calculate a transparent case, record units and conventions, compare an independent baseline, identify one failure mode, and explain the result once for a student and once for a parent. End with one practical next step and one claim the evidence does not support.

Lab question 4

Create a reproducible mini-investigation of horizons, bonds, stretch, nonlocal force, damage, energy and fracture. For task 4, state one prediction, calculate a transparent case, record units and conventions, compare an independent baseline, identify one failure mode, and explain the result once for a student and once for a parent. End with one practical next step and one claim the evidence does not support.

Lab question 5

Create a reproducible mini-investigation of horizons, bonds, stretch, nonlocal force, damage, energy and fracture. For task 5, state one prediction, calculate a transparent case, record units and conventions, compare an independent baseline, identify one failure mode, and explain the result once for a student and once for a parent. End with one practical next step and one claim the evidence does not support.

Lab question 6

Create a reproducible mini-investigation of horizons, bonds, stretch, nonlocal force, damage, energy and fracture. For task 6, state one prediction, calculate a transparent case, record units and conventions, compare an independent baseline, identify one failure mode, and explain the result once for a student and once for a parent. End with one practical next step and one claim the evidence does not support.

Lab question 7

Create a reproducible mini-investigation of horizons, bonds, stretch, nonlocal force, damage, energy and fracture. For task 7, state one prediction, calculate a transparent case, record units and conventions, compare an independent baseline, identify one failure mode, and explain the result once for a student and once for a parent. End with one practical next step and one claim the evidence does not support.

Lab question 8

Create a reproducible mini-investigation of horizons, bonds, stretch, nonlocal force, damage, energy and fracture. For task 8, state one prediction, calculate a transparent case, record units and conventions, compare an independent baseline, identify one failure mode, and explain the result once for a student and once for a parent. End with one practical next step and one claim the evidence does not support.

Lab question 9

Create a reproducible mini-investigation of horizons, bonds, stretch, nonlocal force, damage, energy and fracture. For task 9, state one prediction, calculate a transparent case, record units and conventions, compare an independent baseline, identify one failure mode, and explain the result once for a student and once for a parent. End with one practical next step and one claim the evidence does not support.

Lab question 10

Create a reproducible mini-investigation of horizons, bonds, stretch, nonlocal force, damage, energy and fracture. For task 10, state one prediction, calculate a transparent case, record units and conventions, compare an independent baseline, identify one failure mode, and explain the result once for a student and once for a parent. End with one practical next step and one claim the evidence does not support.