Calculate Constraint Stress in Matter.js

This article explains how to compute the mechanical stress acting across a rigid constraint connecting two bodies in the Matter.js 2D physics engine. Because Matter.js uses position-based dynamics rather than a force-based solver, it does not provide an out-of-the-box stress value for constraints. By analyzing the deviation between the constraint's rest length and its resolved length alongside the effective mass of the connected bodies, you can accurately derive the tension, compression, and mechanical stress occurring at any given frame.

Understanding Constraint Mechanics in Matter.js

Matter.js resolves constraints using relaxation techniques over multiple iterations, directly modifying body positions rather than accumulating contact impulses. In physical terms, mechanical stress (\(\sigma\)) is defined as internal force (\(F\)) divided by the cross-sectional area (\(A\)) of the member:

\[\sigma = \frac{F}{A}\]

To obtain mechanical stress, you must first calculate the magnitude of the normal force (tension or compression) acting along the constraint axis.

Method: The Effective Mass and Positional Deviation Approach

In a position-based system, a constraint is stretched or compressed by a displacement offset (\(\Delta x\)) away from its target rest length (\(L_0\)). By combining this positional error with the system's reduced mass (effective mass) and the engine's time-step, you can estimate the force required to maintain equilibrium.

1. Determine Anchor World Coordinates

Calculate the absolute world positions of the constraint's anchor points, taking body rotation and position into account:

const pointA = constraint.bodyA 
  ? Matter.Vector.add(constraint.bodyA.position, constraint.pointA)
  : constraint.pointA;

const pointB = constraint.bodyB 
  ? Matter.Vector.add(constraint.bodyB.position, constraint.pointB)
  : constraint.pointB;

2. Calculate Positional Deviation

Compute the current Euclidean distance between the two points and determine the change relative to the resting length:

const delta = Matter.Vector.sub(pointB, pointA);
const currentLength = Matter.Vector.magnitude(delta);
const separation = currentLength - constraint.length; // Positive = Tension, Negative = Compression

3. Calculate the Effective Mass

The reduced mass (\(\mu\)) represents the combined inertial resistance of the two bodies:

\[\frac{1}{\mu} = \frac{1}{m_A} + \frac{1}{m_B}\]

If one body is static, its inverse mass is zero (\(1/\infty = 0\)).

const invMassA = constraint.bodyA ? constraint.bodyA.inverseMass : 0;
const invMassB = constraint.bodyB ? constraint.bodyB.inverseMass : 0;
const effectiveMass = (invMassA + invMassB > 0) ? 1 / (invMassA + invMassB) : 0;

4. Derive Force and Mechanical Stress

Using the engine's time step (\(\Delta t\), in seconds), compute the instantaneous constraint force, and then divide by the assumed cross-sectional area:

const dt = engine.timing.lastDelta / 1000; // Convert milliseconds to seconds
const stiffness = constraint.stiffness; // Defaults to 1 for rigid constraints

// F = (effectiveMass * acceleration) = m_eff * (separation / dt^2) * stiffness
const force = (effectiveMass * (Math.abs(separation) / (dt * dt))) * stiffness;

// Stress = Force / Area
const crossSectionalArea = 0.05; // Defined structural cross-section in meters squared
const stress = force / crossSectionalArea;

Complete Implementation Example

Attach a listener to the afterUpdate event to monitor stress dynamically on each simulation tick:

Matter.Events.on(engine, 'afterUpdate', () => {
  const bodyA = constraint.bodyA;
  const bodyB = constraint.bodyB;

  const pA = bodyA ? Matter.Vector.add(bodyA.position, constraint.pointA) : constraint.pointA;
  const pB = bodyB ? Matter.Vector.add(bodyB.position, constraint.pointB) : constraint.pointB;

  const currentDist = Matter.Vector.magnitude(Matter.Vector.sub(pB, pA));
  const displacement = currentDist - constraint.length;

  const invMassA = bodyA ? bodyA.inverseMass : 0;
  const invMassB = bodyB ? bodyB.inverseMass : 0;

  if (invMassA + invMassB === 0) return; // Both bodies are static

  const reducedMass = 1 / (invMassA + invMassB);
  const dt = (engine.timing.lastDelta || 16.666) / 1000;

  // Approximate internal normal force
  const normalForce = (reducedMass * Math.abs(displacement) * constraint.stiffness) / (dt * dt);

  // Define cross-sectional area (e.g., thickness * depth)
  const crossSectionArea = 10; 
  const mechanicalStress = normalForce / crossSectionArea;

  const stressType = displacement >= 0 ? 'Tension' : 'Compression';

  if (mechanicalStress > 50000) {
    // Break constraint if stress exceeds structural yield strength
    Matter.Composite.remove(engine.world, constraint);
  }
});

Using this method, rigid constraints with high stiffness will produce realistic internal forces reflecting external loads, gravity, and centrifugal movement.