Euler-Lagrange Equations and Governing PDEs

Learning objectives

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

• Explain how governing PDEs arise from an energy functional.
• Derive the Euler-Lagrange equations for displacement and phase-field damage.
• Understand the difference between strong form and weak form.
• Interpret the displacement equilibrium equation.
• Interpret the phase-field damage evolution equation.
• Identify the role of natural and essential boundary conditions.
• Connect the variational phase-field fracture energy to finite element implementation.

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1. Why Euler-Lagrange equations matter

In phase-field fracture, we usually start from an energy functional.

For a brittle elastic solid, a common phase-field fracture energy is

where:

• \(u\) is the displacement field,
• \(d\) is the phase-field damage field,
• \(g(d)\) is the degradation function,
• \(\psi_e\) is the elastic strain energy density,
• \(G_c\) is the critical fracture energy,
• \(\ell\) is the phase-field length scale,
• \(W_{\text{ext}}\) is the external work.

The unknowns \(u\) and \(d\) are found by minimizing the energy:

subject to the irreversibility condition:

The Euler-Lagrange equations are the differential equations that come from this minimization problem.

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2. Stationarity of the energy

A minimum of the energy must satisfy stationarity with respect to admissible variations.

For displacement variations \(v\), we require:

For damage variations \(q\), we require:

Here:

• \(v\) is a virtual displacement or displacement test function,
• \(q\) is a virtual damage variation or damage test function.

In words:

At equilibrium, small admissible changes in \(u\) and \(d\) should not decrease the total energy.

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3. Kinematics and strain

For small-strain elasticity, the strain tensor is

The elastic strain energy density for an isotropic linear elastic material is

where \(\lambda\) and \(\mu\) are Lamé parameters.

The undamaged stress is

The degraded stress is

for the simplest isotropic degradation model.

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4. Displacement variation

The elastic part of the energy is

Taking the variation with respect to \(u\), using the test function \(v\), gives

Since

we get

Define

Then

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5. External work variation

Assume the external work has the form

where:

• \(b\) is a body force,
• \(\bar t\) is a prescribed traction,
• \(\Gamma_N\) is the Neumann boundary.

Then the variation is

Because the total potential energy contains \(-W_{\text{ext}}\), the displacement stationarity condition becomes

This is the weak form of mechanical equilibrium.

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6. Strong form of displacement equilibrium

The weak form is

Using integration by parts, this corresponds to the strong form

with boundary conditions

and

where:

• \(\Gamma_D\) is the Dirichlet boundary,
• \(\Gamma_N\) is the Neumann boundary,
• \(n\) is the outward unit normal.

The stress is degraded by damage:

Therefore, as \(d\) approaches 1, the material loses stiffness.

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7. Damage variation

Now take the variation of the energy with respect to the damage field \(d\).

The phase-field energy is

Only the first two terms depend on \(d\).

Using a damage test function \(q\), the damage variation is

The stationarity condition is

This is the weak form of the damage equation.

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8. Strong form of the damage equation

Starting from the weak form

integrate the gradient term by parts:

If the natural damage boundary condition is used,

then the boundary term vanishes.

The strong form becomes

This is the governing PDE for the damage field.

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9. Interpreting the damage PDE

The strong damage equation is

Each term has a physical meaning.

Crack-driving term

This term couples damage to elastic strain energy.

For the common degradation function

we have

So the crack-driving term is usually negative when strain energy is positive. This encourages damage growth.

Local fracture resistance term

This penalizes nonzero damage.

It resists spreading damage everywhere.

Gradient regularization term

This smooths the damage field and controls the width of the diffused crack.

Together, these terms define how the crack evolves spatially.

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10. Complete coupled strong form

For the simplest isotropic AT2 phase-field fracture model, the coupled governing equations are:

with

and

Boundary conditions:

Damage irreversibility:

or incrementally:

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11. Complete coupled weak form

Find \(u\) and \(d\) such that for all test functions \(v\) and \(q\):

and

with

and

This weak form is what we usually implement in finite element codes.

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12. Why weak form is preferred in FEM

The strong form contains second derivatives, such as

and divergence of stress,

These require high smoothness if solved directly.

The weak form reduces the derivative requirements by integration by parts.

For example,

only requires first derivatives of \(d\) and \(q\).

This is why standard \(C^0\) finite elements can be used.

In FEniCS, this corresponds naturally to expressions like:

dot(grad(d), grad(q))*dx

rather than explicitly computing:

laplacian(d)

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13. Tension-compression split in the PDEs

The simple model degrades the full elastic energy:

However, this can cause damage under compression.

To avoid this, we split the energy:

and degrade only the tensile part:

Then the stress becomes

and the damage equation becomes

Now only tensile energy drives fracture.

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14. History field version

To enforce irreversibility approximately, many implementations use a history field:

Then the damage equation becomes

For

we have

Therefore,

This form is common in staggered phase-field fracture implementations.

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15. Linearized damage equation with history field

Using the history field form:

Expand the first term:

Group terms involving \(d\):

This is a linear elliptic PDE for \(d\) if \(\mathcal{H}\) is known.

This is one reason the staggered method is convenient:

1. solve \(u\),
2. update \(\mathcal{H}\),
3. solve a linear damage PDE for \(d\),
4. repeat.

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16. Weak form of the history-field damage equation

Starting from

multiply by a test function \(q\) and integrate:

This is one of the most common weak forms used in simple phase-field fracture codes.

In FEniCS-like notation, it resembles:

a_d = ((Gc/l) + 2*H)*d*q*dx + Gc*l*dot(grad(d), grad(q))*dx
L_d = 2*H*q*dx

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17. Damage bounds

The phase-field variable should satisfy

and irreversibility requires

Therefore, the complete bound is

The history-field method helps prevent healing, but it does not always strictly enforce the lower bound \(d_n \geq d_{n-1}\) in every numerical setting.

A more rigorous approach is to solve the damage problem as a bound-constrained variational inequality.

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18. Variational inequality form

With irreversibility, the damage minimization problem becomes constrained.

At load step \(n\), find \(d_n\) such that

and the energy cannot be reduced by any admissible damage variation.

This leads to a variational inequality rather than a simple equality.

Symbolically,

for all admissible \(q\) satisfying

This is more mathematically faithful to irreversible fracture.

However, it requires solvers that can handle bound constraints.

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19. Essential and natural boundary conditions

For the displacement field, common boundary conditions are:

Essential displacement boundary condition

This is imposed directly in finite element codes.

Natural traction boundary condition

This appears naturally in the weak form through the external work term.

For the damage field, the common natural boundary condition is:

This usually appears automatically from the weak form if no explicit damage boundary condition is imposed.

Sometimes, we impose \(d=1\) to prescribe an initial crack.

For example:

or initialize \(d\) using a smooth crack profile.

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20. Common implementation strategy

A typical staggered algorithm is:

Given d_{n-1}

For load step n:

    1. Apply displacement or force increment.

    2. Initialize d_n = d_{n-1}.

    3. Repeat until convergence:

        a. Solve mechanical equilibrium for u_n:

           div(sigma) + b = 0

        b. Compute tensile energy psi_plus.

        c. Update history field:

           H = max(H_old, psi_plus)

        d. Solve damage equation for d_n.

        e. Enforce bounds:

           d_n = max(d_n, d_{n-1})
           d_n = min(d_n, 1)

    4. Save u_n, d_n, reaction force, energy, etc.

This staggered approach is popular because each subproblem is easier than the fully coupled monolithic problem.

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21. FEniCS-style weak form sketch

A simple FEniCS-like damage equation with history field may look like:

# Trial and test functions
d = TrialFunction(Vd)
q = TestFunction(Vd)

# Known history field H
a_d = ((Gc/l) + 2*H)*d*q*dx + Gc*l*dot(grad(d), grad(q))*dx
L_d = 2*H*q*dx

The displacement weak form may look like:

u = Function(Vu)
v = TestFunction(Vu)

eps = sym(grad(u))
sigma0 = lambda_*tr(eps)*Identity(dim) + 2*mu*eps
sigma = g(d_old)*sigma0

F_u = inner(sigma, sym(grad(v)))*dx - dot(b, v)*dx - dot(tbar, v)*ds

In a nonlinear solve, one would compute the Jacobian:

J_u = derivative(F_u, u, du)

The exact implementation depends on:

• degradation function,
• tension-compression split,
• boundary conditions,
• monolithic or staggered scheme,
• irreversibility method.

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22. Strong form versus weak form summary

Concept	Strong form	Weak form
Mechanical equilibrium	\(\nabla\cdot\sigma + b = 0\)	\(\int_\Omega \sigma:\varepsilon(v)d\Omega = W_{\text{ext}}(v)\)
Damage equation	\(g'(d)\psi_e + G_c(d/\ell - \ell\Delta d)=0\)	\(\int_\Omega g'(d)\psi_e q + G_c(dq/\ell + \ell\nabla d\cdot\nabla q)d\Omega=0\)
Derivative order	Higher	Lower
FEM suitability	Less direct	Natural
Boundary terms	Must be specified explicitly	Natural BCs appear automatically

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23. Common mistakes

Mistake 1: Using full elastic energy under compression

If the full elastic energy drives damage, cracks may grow under compression.

Use a tension-compression split when modeling tensile brittle fracture.

Mistake 2: Forgetting irreversibility

Without irreversibility, cracks can heal during unloading.

Always enforce:

Mistake 3: Choosing mesh size too large compared to \(\ell\)

The damage profile must be resolved.

A typical requirement is:

or

Mistake 4: Confusing strong and weak forms

The strong form contains \(\Delta d\), but the weak form contains \(\nabla d \cdot \nabla q\).

In FEM, implement the weak form.

Mistake 5: Missing residual stiffness

If \(g(1)=0\), the stiffness matrix may become singular.

Use a small residual stiffness \(k\):

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24. Conceptual summary

Starting from the energy functional

stationarity with respect to \(u\) gives mechanical equilibrium:

stationarity with respect to \(d\) gives the damage PDE:

The weak forms are the forms used in finite element implementation.

The complete model must also include:

• boundary conditions,
• irreversibility,
• damage bounds,
• tension-compression split if needed,
• a numerical solution strategy.

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25. Key terms

Term	Meaning
Euler-Lagrange equation	Governing equation obtained from stationarity of an energy functional
Strong form	PDE form of the governing equations
Weak form	Integral form used in FEM
Test function	Virtual variation used to derive weak form
Stationarity	Condition that first variation of energy is zero
Mechanical equilibrium	Balance of internal and external forces
Damage PDE	Governing equation for phase-field crack evolution
Natural boundary condition	Boundary condition arising automatically from integration by parts
Essential boundary condition	Boundary condition imposed directly
History field	Variable storing maximum crack-driving energy
Variational inequality	Constrained variational statement for irreversible damage

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26. Suggested references

1. Griffith, A. A. (1921). The phenomena of rupture and flow in solids. Philosophical Transactions of the Royal Society of London. Series A.
2. Francfort, G. A., & Marigo, J.-J. (1998). Revisiting brittle fracture as an energy minimization problem. Journal of the Mechanics and Physics of Solids.
3. Bourdin, B., Francfort, G. A., & Marigo, J.-J. (2000). Numerical experiments in revisited brittle fracture. Journal of the Mechanics and Physics of Solids.
4. Bourdin, B., Francfort, G. A., & Marigo, J.-J. (2008). The variational approach to fracture. Journal of Elasticity.
5. Miehe, C., Hofacker, M., & Welschinger, F. (2010). A phase field model for rate-independent crack propagation. Computer Methods in Applied Mechanics and Engineering.
6. Pham, K., Amor, H., Marigo, J.-J., & Maurini, C. (2011). Gradient damage models and their use to approximate brittle fracture. International Journal of Damage Mechanics.
7. Borden, M. J., Verhoosel, C. V., Scott, M. A., Hughes, T. J. R., & Landis, C. M. (2012). A phase-field description of dynamic brittle fracture. Computer Methods in Applied Mechanics and Engineering.

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The equations above describe propagation once a phase-field fracture model is chosen. The next conceptual issue is whether the same model correctly predicts the birth of cracks in uncracked bodies.