Fracture Propagation as a Variational Problem

Learning objectives

By the end of this lesson, you should be able to:

Explain why fracture can be formulated as an energy minimization problem.
Understand the role of displacement \(u\), crack set \(\Gamma\), and damage field \(d\).
Explain the difference between local crack criteria and global variational fracture.
Write the basic Griffith energy functional.
Understand how phase-field fracture approximates sharp-crack variational fracture.
Recognize the role of irreversibility in fracture propagation.

1. Why think about fracture variationally?

In many mechanics problems, the equilibrium state of a body is obtained by minimizing an energy functional.

For example, in linear elasticity, the displacement field \(u\) can be found by minimizing the total potential energy:

\[ \Pi(u) = \int_{\Omega} \psi_e(\varepsilon(u)) \, d\Omega - W_{\text{ext}}(u) \]

where:

\(\Omega\) is the body,
\(u\) is the displacement field,
\(\varepsilon(u)\) is the strain tensor,
\(\psi_e\) is the elastic strain energy density,
\(W_{\text{ext}}\) is the work done by external forces.

The equilibrium displacement satisfies

\[ u = \arg\min \Pi(u) \]

This means the body chooses the displacement field that minimizes total potential energy.

Fracture can be viewed similarly, but with one additional unknown:

The crack itself is also unknown.

So in fracture mechanics, we do not only solve for displacement \(u\). We also solve for the crack geometry.

2. Unknowns in a fracture problem

In a classical sharp-crack description, the main unknowns are:

\[ u : \Omega \setminus \Gamma \rightarrow \mathbb{R}^n \]

and

\[ \Gamma \subset \Omega \]

where:

\(u\) is the displacement field,
\(\Gamma\) is the crack set,
\(\Omega \setminus \Gamma\) is the uncracked part of the body,
\(n = 2\) or \(3\) is the spatial dimension.

The crack \(\Gamma\) is not known ahead of time. It must be determined as part of the solution.

This makes fracture difficult because the domain itself changes as the crack grows.

3. Griffith's variational idea

Griffith's theory says that fracture is governed by an energy balance.

A crack may grow if the decrease in elastic energy is large enough to pay for the creation of new crack surface.

The total energy can be written as

\[ \Pi(u,\Gamma) = \int_{\Omega \setminus \Gamma} \psi_e(\varepsilon(u)) \, d\Omega + G_c |\Gamma| - W_{\text{ext}}(u) \]

where:

\(\psi_e\) is the elastic strain energy density,
\(G_c\) is the critical fracture energy,
\(|\Gamma|\) is the measure of the crack surface,
\(W_{\text{ext}}\) is the external work.

In 2D, \(|\Gamma|\) represents crack length.

In 3D, \(|\Gamma|\) represents crack surface area.

The fracture problem can then be written as

\[ (u,\Gamma) = \arg\min \left[ \int_{\Omega \setminus \Gamma} \psi_e(\varepsilon(u)) \, d\Omega + G_c |\Gamma| - W_{\text{ext}}(u) \right] \]

This is the variational form of Griffith fracture.

4. Physical interpretation of each energy term

The functional

\[ \Pi(u,\Gamma) = \int_{\Omega \setminus \Gamma} \psi_e(\varepsilon(u)) \, d\Omega + G_c |\Gamma| - W_{\text{ext}}(u) \]

contains three main parts.

Elastic strain energy

\[ \int_{\Omega \setminus \Gamma} \psi_e(\varepsilon(u)) \, d\Omega \]

This is the energy stored in the deforming solid.

If a crack grows, the body becomes more compliant. As a result, the stored elastic energy may decrease.

Fracture surface energy

\[ G_c |\Gamma| \]

This is the energy required to create new crack surface.

If the crack grows, \(|\Gamma|\) increases, so this term increases.

External work

\[ W_{\text{ext}}(u) \]

This is the work done by external loads.

Depending on the loading condition, external work may either increase or decrease the total potential energy.

5. The energetic competition

Fracture is controlled by a competition between two effects.

Crack growth releases elastic energy

When a crack extends, the body becomes softer. This can reduce the stored elastic energy.

This effect favors crack growth.

Crack growth creates new surfaces

Creating new crack surfaces requires energy.

This effect resists crack growth.

Therefore, crack growth occurs when the energy released by the body is greater than or equal to the energy required to create new surfaces.

In symbolic form:

\[ \text{energy released} \geq \text{fracture energy required} \]

or

\[ G \geq G_c \]

where:

\(G\) is the energy release rate,
\(G_c\) is the critical fracture energy.

6. Incremental fracture evolution

Fracture is usually not solved as a single one-step minimization problem.

Instead, the loading is applied incrementally.

At load step \(n\), we solve for the new displacement and crack state:

\[ (u_n,\Gamma_n) = \arg\min \Pi_n(u,\Gamma) \]

subject to the condition that cracks cannot heal:

\[ \Gamma_{n-1} \subseteq \Gamma_n \]

This condition means:

The new crack set must contain the old crack set.

A crack may grow, but it cannot disappear.

This is called crack irreversibility.

7. Why irreversibility is essential

Without irreversibility, the energy minimization problem may predict unphysical crack healing.

For example, imagine a material that cracks under high load. If the load is later reduced, a purely energetic minimization could decide that the cracked state is no longer favorable.

But real cracks do not simply vanish when we unload the body.

Therefore, we impose

\[ \Gamma_{n-1} \subseteq \Gamma_n \]

in sharp-crack fracture.

In phase-field fracture, this becomes

\[ d_n \geq d_{n-1} \]

where \(d\) is the damage variable.

Equivalently, in rate form,

\[ \dot{d} \geq 0 \]

This means damage can only increase.

8. Local versus global fracture criteria

Classical fracture mechanics often uses local crack-tip criteria such as

\[ K_I \geq K_{IC} \]

or

\[ G \geq G_c \]

These are extremely useful when the crack geometry is known and the crack path is relatively simple.

However, in complex problems, the crack path may not be known in advance. Cracks may:

nucleate,
curve,
branch,
merge,
interact with boundaries,
interact with other cracks.

A variational formulation is more general because the crack path is determined by energy minimization.

Instead of prescribing the crack path, we allow the crack to evolve in the direction that reduces the total energy most favorably.

9. Energy release rate from the variational viewpoint

The energy release rate \(G\) can be understood as the decrease in potential energy per unit crack extension.

For a crack length \(a\), we define

\[ G = -\frac{d\Pi_{\text{elastic}}}{da} \]

where:

\(\Pi_{\text{elastic}}\) is the elastic part of the potential energy,
\(a\) is the crack length.

The crack grows when

\[ G = G_c \]

for stable quasi-static crack growth.

If

\[ G < G_c \]

then there is not enough energy to grow the crack.

If

\[ G > G_c \]

then crack growth may become unstable or dynamic, depending on the problem.

10. Why sharp-crack variational fracture is hard numerically

The sharp-crack variational formulation is elegant, but difficult to solve directly.

The main difficulty is that \(\Gamma\) is a discontinuity.

As the crack evolves, the computational domain changes:

\[ \Omega \setminus \Gamma \]

This creates several challenges:

The mesh may need to conform to the crack path.
Crack branching is hard to track explicitly.
Crack merging requires special algorithms.
Crack nucleation requires additional criteria.
The displacement field is discontinuous across the crack.
The crack surface measure \(|\Gamma|\) must be computed accurately.

Traditional numerical methods often require explicit crack tracking, remeshing, or enrichment functions.

Examples include:

cohesive zone methods,
extended finite element method,
remeshing-based LEFM,
interface element methods.

Phase-field fracture avoids explicit crack tracking by replacing the sharp crack with a smooth damage field.

11. Conceptual summary

Sharp-crack Griffith fracture:

\[ \Pi(u,\Gamma) = \int_{\Omega \setminus \Gamma} \psi_e(\varepsilon(u)) \, d\Omega + G_c|\Gamma| - W_{\text{ext}} \]

Phase-field fracture:

\[ \Pi_\ell(u,d) = \int_{\Omega} g(d)\psi_e(\varepsilon(u)) \, d\Omega + \int_{\Omega} G_c \left( \frac{d^2}{2\ell} + \frac{\ell}{2}|\nabla d|^2 \right) d\Omega - W_{\text{ext}} \]

The main idea is:

Fracture propagation can be treated as an energy minimization problem where the body chooses both the displacement field and the crack configuration.

In phase-field fracture, the crack configuration is represented by a smooth damage field \(d\), making the problem much easier to solve numerically.

12. Key terms

Term	Meaning
Variational problem	A problem formulated as minimization or stationarity of an energy functional
Crack set \(\Gamma\)	The sharp crack surface or crack line
Fracture energy \(G_c\)	Energy required to create a unit crack surface
Crack irreversibility	Condition that cracks can grow but cannot heal
Energy release rate \(G\)	Energy released per unit crack extension
Phase-field \(d\)	Smooth scalar variable representing fracture damage
Length scale \(\ell\)	Parameter controlling the width of the diffused crack
Degradation function \(g(d)\)	Function that reduces stiffness as damage grows
History field \(\mathcal{H}\)	Variable storing the maximum crack-driving energy
Staggered scheme	Alternating solution of displacement and damage
Monolithic scheme	Simultaneous solution of displacement and damage

13. Suggested references

Griffith, A. A. (1921). The phenomena of rupture and flow in solids. Philosophical Transactions of the Royal Society of London. Series A.
Francfort, G. A., & Marigo, J.-J. (1998). Revisiting brittle fracture as an energy minimization problem. Journal of the Mechanics and Physics of Solids.
Bourdin, B., Francfort, G. A., & Marigo, J.-J. (2000). Numerical experiments in revisited brittle fracture. Journal of the Mechanics and Physics of Solids.
Bourdin, B., Francfort, G. A., & Marigo, J.-J. (2008). The variational approach to fracture. Journal of Elasticity.
Miehe, C., Hofacker, M., & Welschinger, F. (2010). A phase field model for rate-independent crack propagation. Computer Methods in Applied Mechanics and Engineering.
Ambrosio, L., & Tortorelli, V. M. (1990). Approximation of functional depending on jumps by elliptic functional via Gamma-convergence. Communications on Pure and Applied Mathematics.

The variational formulation is powerful but hard to solve with explicit sharp cracks. This motivates replacing the sharp crack set by a smooth field in the next stage of the theory.