Chapter map

1. 1. Contact geometry
2. 2. Elastic law
3. 3. Relative speed
4. 4. Damping
5. 5. Normal total
6. 6. Pair balance
7. 7. Acceleration
8. 8. Time step
9. 9. Friction proposal
10. 10. Friction cap
11. 11. Contact state
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. Contact geometry

A small example makes the mechanism visible. Set radii .50 and .40 m, centre distance .89 m.

A concrete situation

Overlap=.01 m.

What the mathematics does not license

Use declared particle shapes. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Draw the contact normal. 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. Elastic law

This is where modelling choices begin to affect evidence. Use k_n=10000 N/m.

A concrete situation

Elastic force=100 N.

What the mathematics does not license

Linear law is illustrative. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Record stiffness and units. 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. Relative speed

The central mathematical idea is concrete. Use closing speed .20 m/s.

A concrete situation

Contact is approaching.

What the mathematics does not license

State sign convention. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Resolve normal velocity. 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. Damping

Here the geometry becomes operational. Use c_n=50 N s/m.

A concrete situation

Damping magnitude=10 N.

What the mathematics does not license

Damping opposes relative motion. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Record direction. 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. Normal total

This step deserves a careful pause. Add compatible compressive terms.

A concrete situation

Normal magnitude=110 N.

What the mathematics does not license

Do not add incompatible signs. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Record pair 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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6. Pair balance

The useful question is not only how to calculate, but what the calculation assumes. Apply equal and opposite forces.

A concrete situation

Internal force sum=0.

What the mathematics does not license

Include moments separately. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Check 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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7. Acceleration

A small example makes the mechanism visible. Use particle mass 2 kg.

A concrete situation

Magnitude=55 m/s squared.

What the mathematics does not license

Other forces omitted. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Record assumption. 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. Time step

This is where modelling choices begin to affect evidence. Use dt=.001 s.

A concrete situation

Velocity increment=.055 m/s.

What the mathematics does not license

Explicit teaching update. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Label time level. 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. Friction proposal

The central mathematical idea is concrete. Set tangential trial=50 N.

A concrete situation

Compare with Coulomb cap.

What the mathematics does not license

History law must be declared. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Record trial. 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. Friction cap

Here the geometry becomes operational. Use mu=.40.

A concrete situation

Cap=44 N.

What the mathematics does not license

Based on chosen normal force. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Record slip 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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11. Contact state

This step deserves a careful pause. Trial exceeds cap.

A concrete situation

Contact slides.

What the mathematics does not license

Reset tangential history consistently. Keeping this boundary clear makes the result easier to audit and transfer.

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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12. Energy

The useful question is not only how to calculate, but what the calculation assumes. Compute elastic and damping work.

A concrete situation

Separate stored from dissipated energy.

What the mathematics does not license

Friction also dissipates. A trustworthy workflow records the choice before seeing the final test result.

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

A small example makes the mechanism visible. Reduce size and time step together.

A concrete situation

Bulk response should be compared.

What the mathematics does not license

Calibration may change. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Record convergence. 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 is where modelling choices begin to affect evidence. State geometry, laws, integration and validation.

A concrete situation

Worked overlap=.01, normal=110, friction=44.

What the mathematics does not license

Two-particle teaching contact. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Write narrow claim. 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

The discrete element method represents a granular body as separate particles whose motion follows Newton’s laws while contacts supply forces and moments. The mathematics connects geometry, contact detection, constitutive force laws, friction, time integration and bulk behaviour that emerges from many local interactions.

Consider two circular particles with radii 0.50 m and 0.40 m whose centres are 0.89 m apart. Their geometric overlap is 0.50+0.40-0.89=0.01 m. In a soft-contact model, that overlap is a numerical measure used to calculate compression; it is not literal material interpenetration.

Choose an illustrative linear normal stiffness k_n=10,000 N/m. The elastic normal force magnitude is k_n times overlap, so it is 10,000 times 0.01=100 N.

Let the relative closing speed be 0.20 m/s and choose a normal damping coefficient c_n=50 N s/m. With a convention in which positive closing speed adds compressive resistance, the damping magnitude is 50 times 0.20=10 N and the total compressive contact magnitude is 110 N.

The two particles receive equal and opposite normal forces. If one particle has mass 2 kg and no other forces are included in the teaching step, its acceleration magnitude is 110/2=55 m/s². Over dt=0.001 s, a forward-Euler velocity increment would have magnitude 0.055 m/s.

Suppose a tangential spring proposes 50 N while the Coulomb coefficient is 0.40. The friction cap is μF_n=0.40 times 110=44 N, so the tangential force is limited to 44 N and the contact is sliding under this simplified rule.

Did You Know? A DEM simulation can begin with only particle-level laws yet produce force chains, dilation, shear bands and angle-of-repose behaviour at larger scales. Those outcomes still depend on calibration, particle shape, boundaries and resolution.

Contact detection is a geometric algorithm with physical consequences. Missing a neighbour removes a force, while double-counting a pair violates the intended balance. Efficient codes use cell lists, trees or bounding volumes and then perform an exact narrow-phase check.

Normal stiffness controls overlap and the fastest contact oscillation. Making stiffness very large may reduce overlap but also demands a smaller stable explicit time step. A stiffer model is not automatically a better model if the step is too large or the material law is wrong.

Tangential force needs a history variable because the contact frame can rotate while particles move. The stored tangential displacement must be projected consistently, capped by friction and reset or transferred when contact ends.

Rolling resistance, cohesion, nonspherical particles and breakage are additional model choices. Each changes which forces or moments are available, so a result should not be described as a generic prediction of all granular materials.

Boundary walls can be represented analytically or by particles. Wall roughness, friction, speed and compliance influence flow and stress, especially in small containers where boundary layers occupy a large fraction of the domain.

Students can audit a two-particle calculation by drawing the contact normal, computing overlap, resolving relative velocity, calculating elastic and damping forces, applying the friction cap and checking equal-opposite force and moment ledgers.

Applications include soil mechanics, powder handling, mining, pharmaceuticals, additive manufacturing and rock fracture. A useful simulation validates the outputs that matter—such as flow rate, force distribution or bulk stress—rather than relying on a plausible animation.

For parents and students, the transferable lesson is that mathematics can connect a tiny rule at one contact to a pattern created by thousands or millions of interactions.

A responsible DEM report states particle shapes and sizes, density, contact search, normal and tangential laws, damping, friction, rolling or cohesion options, boundaries, integrator, time step, overlap levels, calibration, convergence and validation.

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Frequently asked questions

What is the discrete element method?

A particle-based numerical method that advances separate bodies under contact and external forces.

What does overlap mean?

In a soft-contact DEM, geometric overlap is a numerical deformation measure used by the contact law.

Why is the time step often small?

Contact stiffness creates fast oscillations that an explicit integrator must resolve.

How is friction limited?

A common model caps tangential force by a coefficient times compressive normal force.

Does a realistic animation validate DEM?

No. Contact parameters, boundaries, resolution, bulk observables and independent evidence must be checked.

What must be reported?

Particles, contact search, force laws, friction, boundaries, integrator, step, calibration, convergence and validation.

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Primary source and fact-check note

P. A. Cundall and O. D. L. Strack, “A Discrete Numerical Model for Granular Assemblies,” Géotechnique 29, 1979. 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

• Material point method
• Peridynamics
• Smoothed particle hydrodynamics
• 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: what must be measured

Geometry ledger

Record radii, centres, contact normal, overlap and particle shape.

Force ledger

Separate elastic, damping, friction, rolling and cohesive components.

Motion ledger

Apply equal-opposite pair forces and integrate each particle.

Evidence ledger

Check overlap, energy, bulk response, refinement and experiment.

Worked evidence notebook: reconstruct every number

Evidence notebook 1: Contact geometry

Set radii .50 and .40 m, centre distance .89 m. Overlap=.01 m. Keep every unit, direction, coordinate, matrix entry, sign convention and rounding decision visible. Use declared particle shapes. Draw the contact normal. Then change one input, recompute the affected mechanism and explain whether the difference comes from geometry, constitutive choice, kernel support, conditioning, boundary data, time integration or numerical precision. Preserve the calculation so another reader can reproduce the narrow conclusion.

Evidence notebook 2: Elastic law

Use k_n=10000 N/m. Elastic force=100 N. Keep every unit, direction, coordinate, matrix entry, sign convention and rounding decision visible. Linear law is illustrative. Record stiffness and units. Then change one input, recompute the affected mechanism and explain whether the difference comes from geometry, constitutive choice, kernel support, conditioning, boundary data, time integration or numerical precision. Preserve the calculation so another reader can reproduce the narrow conclusion.

Evidence notebook 3: Relative speed

Use closing speed .20 m/s. Contact is approaching. Keep every unit, direction, coordinate, matrix entry, sign convention and rounding decision visible. State sign convention. Resolve normal velocity. Then change one input, recompute the affected mechanism and explain whether the difference comes from geometry, constitutive choice, kernel support, conditioning, boundary data, time integration or numerical precision. Preserve the calculation so another reader can reproduce the narrow conclusion.

Evidence notebook 4: Damping

Use c_n=50 N s/m. Damping magnitude=10 N. Keep every unit, direction, coordinate, matrix entry, sign convention and rounding decision visible. Damping opposes relative motion. Record direction. Then change one input, recompute the affected mechanism and explain whether the difference comes from geometry, constitutive choice, kernel support, conditioning, boundary data, time integration or numerical precision. Preserve the calculation so another reader can reproduce the narrow conclusion.

Evidence notebook 5: Normal total

Add compatible compressive terms. Normal magnitude=110 N. Keep every unit, direction, coordinate, matrix entry, sign convention and rounding decision visible. Do not add incompatible signs. Record pair force. Then change one input, recompute the affected mechanism and explain whether the difference comes from geometry, constitutive choice, kernel support, conditioning, boundary data, time integration or numerical precision. Preserve the calculation so another reader can reproduce the narrow conclusion.

Evidence notebook 6: Pair balance

Apply equal and opposite forces. Internal force sum=0. Keep every unit, direction, coordinate, matrix entry, sign convention and rounding decision visible. Include moments separately. Check ledger. Then change one input, recompute the affected mechanism and explain whether the difference comes from geometry, constitutive choice, kernel support, conditioning, boundary data, time integration or numerical precision. Preserve the calculation so another reader can reproduce the narrow conclusion.

Evidence notebook 7: Acceleration

Use particle mass 2 kg. Magnitude=55 m/s squared. Keep every unit, direction, coordinate, matrix entry, sign convention and rounding decision visible. Other forces omitted. Record assumption. Then change one input, recompute the affected mechanism and explain whether the difference comes from geometry, constitutive choice, kernel support, conditioning, boundary data, time integration or numerical precision. Preserve the calculation so another reader can reproduce the narrow conclusion.

Evidence notebook 8: Time step

Use dt=.001 s. Velocity increment=.055 m/s. Keep every unit, direction, coordinate, matrix entry, sign convention and rounding decision visible. Explicit teaching update. Label time level. Then change one input, recompute the affected mechanism and explain whether the difference comes from geometry, constitutive choice, kernel support, conditioning, boundary data, time integration or numerical precision. Preserve the calculation so another reader can reproduce the narrow conclusion.

Evidence notebook 9: Friction proposal

Set tangential trial=50 N. Compare with Coulomb cap. Keep every unit, direction, coordinate, matrix entry, sign convention and rounding decision visible. History law must be declared. Record trial. Then change one input, recompute the affected mechanism and explain whether the difference comes from geometry, constitutive choice, kernel support, conditioning, boundary data, time integration or numerical precision. Preserve the calculation so another reader can reproduce the narrow conclusion.

Evidence notebook 10: Friction cap

Use mu=.40. Cap=44 N. Keep every unit, direction, coordinate, matrix entry, sign convention and rounding decision visible. Based on chosen normal force. Record slip force. Then change one input, recompute the affected mechanism and explain whether the difference comes from geometry, constitutive choice, kernel support, conditioning, boundary data, time integration or numerical precision. Preserve the calculation so another reader can reproduce the narrow conclusion.

Evidence notebook 11: Contact state

Trial exceeds cap. Contact slides. Keep every unit, direction, coordinate, matrix entry, sign convention and rounding decision visible. Reset tangential history consistently. Record state. Then change one input, recompute the affected mechanism and explain whether the difference comes from geometry, constitutive choice, kernel support, conditioning, boundary data, time integration or numerical precision. Preserve the calculation so another reader can reproduce the narrow conclusion.

Evidence notebook 12: Energy

Compute elastic and damping work. Separate stored from dissipated energy. Keep every unit, direction, coordinate, matrix entry, sign convention and rounding decision visible. Friction also dissipates. Record ledger. Then change one input, recompute the affected mechanism and explain whether the difference comes from geometry, constitutive choice, kernel support, conditioning, boundary data, time integration or numerical precision. Preserve the calculation so another reader can reproduce the narrow conclusion.

Evidence notebook 13: Refinement

Reduce size and time step together. Bulk response should be compared. Keep every unit, direction, coordinate, matrix entry, sign convention and rounding decision visible. Calibration may change. Record convergence. Then change one input, recompute the affected mechanism and explain whether the difference comes from geometry, constitutive choice, kernel support, conditioning, boundary data, time integration or numerical precision. Preserve the calculation so another reader can reproduce the narrow conclusion.

Evidence notebook 14: Conclusion

State geometry, laws, integration and validation. Worked overlap=.01, normal=110, friction=44. Keep every unit, direction, coordinate, matrix entry, sign convention and rounding decision visible. Two-particle teaching contact. Write narrow claim. Then change one input, recompute the affected mechanism and explain whether the difference comes from geometry, constitutive choice, kernel support, conditioning, boundary data, time integration or numerical precision. Preserve the calculation so another reader can reproduce the narrow conclusion.

Misconceptions, failure modes and recovery

Overlap is physical penetration

It is a soft-contact numerical deformation measure. Build the smallest counterexample, restore the missing condition and repeat the calculation before making a broader claim.

Every contact is linear

Hertzian, cohesive and other laws may be appropriate. Build the smallest counterexample, restore the missing condition and repeat the calculation before making a broader claim.

Damping always adds

It opposes relative motion under a declared sign convention. Build the smallest counterexample, restore the missing condition and repeat the calculation before making a broader claim.

Friction has no memory

Tangential displacement history commonly matters. Build the smallest counterexample, restore the missing condition and repeat the calculation before making a broader claim.

Stiffer is always better

Stable explicit time steps become smaller. Build the smallest counterexample, restore the missing condition and repeat the calculation before making a broader claim.

Round particles fit every material

Shape and breakage can control bulk behaviour. Build the smallest counterexample, restore the missing condition and repeat the calculation before making a broader claim.

Walls are neutral

Boundary roughness and motion affect stress and flow. Build the smallest counterexample, restore the missing condition and repeat the calculation before making a broader claim.

Animation is validation

Bulk outputs need calibration and independent evidence. Build the smallest counterexample, restore the missing condition and repeat the calculation before making a broader claim.

A responsible eight-step decision path

1. Define the grains

Choose size, shape, density and dimensional model.

2. Define contact

State normal, tangential, damping and friction laws.

3. Build geometry

Detect candidate neighbours and exact contacts.

4. Integrate motion

Advance translation and rotation with a stable step.

5. Check pair balance

Audit forces, moments, work and dissipation.

6. Calibrate

Use eligible particle and bulk observations.

7. Refine

Change particle resolution and time step.

8. Validate

Test a separate experiment or operating condition.

Learner lab: build transferable understanding

Two-particle compression

Recalculate normal force for three overlaps. State a prediction, show the arithmetic, record a diagnostic, compare a baseline and finish with one justified conclusion plus one limit.

Damping sign

Reverse relative speed and check resistance direction. State a prediction, show the arithmetic, record a diagnostic, compare a baseline and finish with one justified conclusion plus one limit.

Friction transition

Sweep trial tangential force across the Coulomb cap. State a prediction, show the arithmetic, record a diagnostic, compare a baseline and finish with one justified conclusion plus one limit.

Time-step test

Halve dt and compare contact oscillation. State a prediction, show the arithmetic, record a diagnostic, compare a baseline and finish with one justified conclusion plus one limit.

Energy ledger

Separate elastic storage and damping loss. State a prediction, show the arithmetic, record a diagnostic, compare a baseline and finish with one justified conclusion plus one limit.

Shape test

Compare disks with clumps or polygons. State a prediction, show the arithmetic, record a diagnostic, compare a baseline and finish with one justified conclusion plus one limit.

Wall test

Change wall friction and observe bulk response. State a prediction, show the arithmetic, record a diagnostic, compare a baseline and finish with one justified conclusion plus one limit.

Calibration

Fit one property and validate another. State a prediction, show the arithmetic, record a diagnostic, compare a baseline and finish with one justified conclusion plus one limit.

Student explanation

Explain force chains from local contacts. State a prediction, show the arithmetic, record a diagnostic, compare a baseline and finish with one justified conclusion plus one limit.

Parent explanation

Relate many tiny rules to a bulk pattern. State a prediction, show the arithmetic, record a diagnostic, compare a baseline and finish with one justified conclusion plus one limit.

Deeper transfer: from classroom mechanism to responsible use

From one contact to bulk stress

A contact force is local, but engineers often need a continuum-scale stress. A representative-volume calculation combines branch vectors with contact forces and divides by volume. The averaging region, boundary contacts and dynamic contribution must be defined, because different choices can change the reported stress even when the particle state is identical.

Coordination and force chains

The coordination number counts active contacts per particle under a declared rule. Its average helps describe connectivity, while force-chain plots show that strong forces are often highly uneven. Neither metric alone proves stability; rattlers, anisotropy, fabric and contact-force distributions add distinct information.

Restitution and damping

Normal damping is often calibrated through a coefficient of restitution for a simple collision. That calibration depends on the contact law, effective mass and impact speed. Reusing one damping number across shapes, materials and velocity regimes should be justified rather than assumed.

Rotation and torque

Tangential force acts away from a particle centre and therefore produces torque. The angular acceleration equals torque divided by moment of inertia. Omitting rotation can make frictional rolling, shear resistance and energy dissipation physically inconsistent even when translational force balance looks correct.

Polydispersity and segregation

A distribution of particle sizes can improve packing realism and reduce artificial crystallisation. It can also produce segregation under vibration or flow. A convergence study must distinguish numerical refinement from a deliberate change in the physical size distribution.

Quasi-static versus dynamic

A slowly loaded test may still contain numerical inertia if loading is too fast or damping is excessive. Compare kinetic energy with elastic or boundary work and repeat at a slower rate before describing a result as quasi-static.

Calibration without overfitting

Contact parameters should be tied to measurements they can influence. Fit a small set against one experiment, then test a different stress path or flow condition. Matching every output by freely tuning many parameters can hide a wrong mechanism.

Career connection

Geotechnical engineers, process engineers, computational scientists and materials researchers use this mathematics to connect measurable bulk behaviour with local geometry and force. Students build transferable skills in vectors, differential equations, coding, dimensional analysis and evidence-based modelling.