---
title: "Dynamic Crack Branching in a Stretched Strip Interactive Lab | Sandbox Physics"
description: "Why does a fast crack split or wander instead of running straight? A fast crack gets more energy than one straight path can absorb."
canonical_url: "https://sandboxphysics.com/labs/dynamic-crack-branching/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/dynamic-crack-branching/"
last_modified: "2026-10-29"
---

Fracture mechanics · Dynamic phase field

# Dynamic Crack Branching in a Stretched Strip

**Why does a fast crack split or wander instead of running straight?** A fast crack gets more energy than one straight path can absorb. Above a threshold load the extra energy feeds side branches, so the crack splits; frozen variations in strength push it sideways, so even a single crack wanders.

Drive an antiplane crack through a stretched strip. Low loads give a steady straight crack, high loads split it, and frozen disorder roughens its path.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

Karma and Lobkovsky mode-III phase-field model in scaled units on a square lattice of spacing 0.5, strip half-width 15, damping 2 and no extra double-well barrier. At this spacing the lattice raises the fracture energy about 1.26 times above the continuum value, so loads are shown against both. Branching thresholds depend on strip width, spacing and seed; this is a qualitative reproduction, not the published figure.

Tip speed over shear wave speed

$0.25$

Load over lattice toughness

$1.24$

Broken columns with two branches

$0\%$

Lateral spread of the path

$0$

Both plots come from the simulated lattice. Lengths are in units of the process-zone size and time in units of that size over the shear wave speed.

Save the run and read the model

The JSON lists the settings, seed, model version, speed trace, path statistics and every broken site of the run.

The model and the strip geometry follow Karma and Lobkovsky, 2004. Their published branching map used a wider strip and finer lattice than this page. [Read the source](https://arxiv.org/abs/cond-mat/0401056)

[Read the fracture course: from Griffith energy to phase-field cracks and branching](https://sandboxphysics.com/stories/fracture-mechanics/)

Physics tutorial

## A fast crack has energy to spare

**Background** In a long stretched strip the stored energy per unit length ahead of the crack is fixed by the grips. A moving crack that dissipates less than that amount speeds up until the dissipation near its tip catches up.

**Why it matters** Karma and Lobkovsky used a dynamic phase-field model to show that above a threshold load the tip no longer stays a single straight crack. It oscillates, splits into branches or wanders.

Start with the essentials

- **Focus question**: Why does a fast crack split or wander instead of running straight?
- **One-sentence intuition**: Energy balance fixes how much is dissipated per unit length; the tip chooses its shape. Above threshold, two branches dissipate the energy that one straight crack cannot.

### Core mathematical model

#### Karma–Lobkovsky energy

$$
E=\int\left[\frac{\rho}{2}\dot u^{2}+\frac{\kappa}{2}|\nabla\phi|^{2}+h\,f(\phi)+\frac{\mu}{2}\,g(\phi)\left(|\nabla u|^{2}-\epsilon_c^{2}\right)\right]dx\,dy
$$

Antiplane displacement $u$ carries kinetic and elastic energy. The phase field $\phi$ is one in intact material and zero where it is broken. The double well $f$ and the softening function $g$ shape the broken state.

#### Driving force of the strip

$$
G=\frac{\mu\Delta^{2}}{W},\qquad G_c=2\gamma
$$

Grips at $y=\pm W$ move by $\pm\Delta$. Far ahead of the tip each unit length stores this energy; far behind, none. The Lab reports loads against the continuum and the lattice toughness.

#### Speed just above threshold

$$
v\,(G-G_c)=\chi^{-1}\int\left(\frac{\partial\phi}{\partial t}\right)^{2}d\mathbf x
$$

Energy released per unit time by the moving tip equals the rate of dissipation in the process zone. The Lab checks this balance on its steady cracks to within a few percent.

### Common difficulties

#### Branching is not just noise

**Typical misconception** A crack branches only because the material is imperfect.

**Better mental model** In this model a perfectly uniform strip still branches above a threshold load. Disorder changes the details, not the existence of the instability.

#### What this model leaves out

**Typical misconception** The thresholds shown here match a real material.

**Better mental model** Karma and Lobkovsky mode-III phase-field model in scaled units on a square lattice of spacing 0.5, strip half-width 15, damping 2 and no extra double-well barrier. At this spacing the lattice raises the fracture energy about 1.26 times above the continuum value, so loads are shown against both. Branching thresholds depend on strip width, spacing and seed; this is a qualitative reproduction, not the published figure.

### Run the experiment

1. 01

   #### A steady crack

   Choose Steady straight crack and watch the speed trace in the data row.

   **What to observe:** The crack accelerates, then runs straight at an almost constant speed.

2. 02

   #### Feed it more energy

   Choose Two symmetric branches, or drag a grip outward to raise the load.

   **What to observe:** At high load the tip splits into two branches that run side by side.

3. 03

   #### Add frozen disorder

   Choose Disorder roughens the path and compare the path record with earlier runs.

   **What to observe:** Random variations in the critical strain push the crack sideways; the lateral spread grows with disorder strength.

Physical conclusion

Ask how much energy the crack must dissipate per unit length and whether one straight tip can do that. When it cannot, look for oscillation, branching or a wandering path.

## Related experiments and articles

Fracture mechanics · Reading path

### Fracture mechanics from Griffith’s energy balance to branching cracks

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Fast cracks and branching in a dynamic phase-field model**](https://sandboxphysics.com/stories/fracture-mechanics/fast-cracks-and-branching/)
2. [Article **How frozen disorder roughens a fast crack path**](https://sandboxphysics.com/stories/fracture-mechanics/disorder-and-rough-crack-paths/)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)