Convexity in strain energy functions

Convexity of strain energy functions
Why convexity matters
Important nuance: full convexity is often too strong
- Hierarchy
Common convexity notions in hyperelasticity
How we can develop a convex strain energy function
Example: small-strain linear elasticity
Example: a polyconvex finite-strain form
Practical caution

Convexity of strain energy functions

In continuum mechanics, the strain energy density function ($W$) maps a deformation measure to stored elastic energy. Convexity refers to the shape of this function: loosely, if $W$ is convex, then any linear interpolation between two strain states does not produce energy above the interpolation of their energies.

For a scalar variable ($x$), convexity means

\begin{equation} W(\theta x_1 + (1-\theta)x_2) \le \theta W(x_1) + (1-\theta)W(x_2), \qquad 0\le \theta \le 1. \end{equation}

For multivariable constitutive models, this generalizes through the Hessian being positive semidefinite.

Why convexity matters

Material stability A convex energy tends to prevent nonphysical material instabilities. If the energy is non-convex, the material may admit multiple local minima, which can correspond to localization, phase-like switching, or mathematically unstable responses.
Well-posedness of boundary value problems: Convexity helps ensure existence and uniqueness of solutions in elasticity problems. Without it, equilibrium problems may become ill-posed or admit multiple solutions.
Numerical robustness: In finite element simulations, convex strain energy functions are generally more stable and easier to solve. Non-convexity often causes convergence issues, mesh sensitivity, or pathological bifurcations.
Physical admissibility: A constitutive law should give positive energy growth under increasing deformation and avoid unphysical softening unless such behavior is intentionally modeled.

Important nuance: full convexity is often too strong

For finite elasticity, ordinary convexity in the deformation gradient (F) is usually stronger than needed and may exclude realistic nonlinear behavior. Instead, weaker notions are often used:

Rank-one convexity
Quasiconvexity
Polyconvexity

These are especially important in hyperelasticity.

Hierarchy

\begin{equation} \text{Convexity} \implies \text{Polyconvexity} \implies \text{Quasiconvexity} \implies \text{Rank-one convexity} \end{equation}

The reverse implications generally do not hold.

Common convexity notions in hyperelasticity

1. Convexity in (F)

$W(F)$ is convex if

\begin{equation} W(\theta F_1 + (1-\theta)F_2) \le \theta W(F_1) + (1-\theta)W(F_2). \end{equation}

This is mathematically convenient but often too restrictive for real materials.

2. Rank-one convexity

A weaker condition tied to ellipticity:

\begin{equation} W(F + t\, a\otimes n) \end{equation}

must be convex in ($t$) for all vectors $(a,n)$. This is related to the Legendre–Hadamard condition and wave propagation/stability.

3. Polyconvexity

A very useful practical condition. (W(F)) is polyconvex if it can be written as a convex function of $F$, $\operatorname{cof}F$, and $\det F$:

\begin{equation} W(F)=\hat W(F,\operatorname{cof}F,\det F) \end{equation}

with $(\hat W)$ convex in its arguments.

This is widely used because it allows realistic constitutive modeling while still supporting existence theorems.

How we can develop a convex strain energy function

1. Choose appropriate strain variables

Sometimes the energy is not convex in one measure but becomes convex in another.

Examples:

logarithmic strain
principal stretches ($\lambda_i$)
invariants $(I_1, I_2, I_3)$
$(F,\operatorname{cof}F,\det F)$ for polyconvexity

A smart choice of variables can make convexity easier to enforce.

2. Build the energy as a sum of convex terms

If $(W_1, W_2,\dots)$ are convex, then

\begin{equation} W = \sum_i \alpha_i W_i, \qquad \alpha_i \ge 0 \end{equation}

is also convex.

For example,

\begin{equation} W(F)=a|F|^2 + b|\operatorname{cof}F|^2 + c\,\phi(\det F) \end{equation}

can be polyconvex if (a,b,c \ge 0) and (\phi) is convex on ((0,\infty)).

A common volumetric choice is

\begin{equation} \phi(J)=J-\ln J \quad \text{or} \quad \phi(J)=(J-1)^2, \qquad J=\det F. \end{equation}

3. Enforce positive-definite Hessian

For energies written in terms of a strain vector (\varepsilon), compute

\begin{equation} H = \frac{\partial^2 W}{\partial \varepsilon^2}. \end{equation}

If ($H$) is positive semidefinite over the admissible strain range, then ($W$) is convex in that variable.

This is the direct route for small-strain elasticity, where

\begin{equation} W(\varepsilon)=\frac12 \varepsilon : \mathbb{C} : \varepsilon \end{equation}

is convex if the elasticity tensor ($\mathbb{C}$) is positive definite on symmetric strains.

4. Use polyconvex templates

For finite-strain hyperelasticity, one of the safest approaches is to start with known polyconvex forms.

Typical structure:

\begin{equation} W(F)=W_{\text{iso}}(F) + W_{\text{vol}}(J) \end{equation}

or better,

\begin{equation} W(F)=\hat W(F,\operatorname{cof}F,J). \end{equation}

Examples of polyconvex ingredients:

$(|F|^p), (p\ge 1)$
$(|\operatorname{cof}F|^q), (q\ge 1)$
convex penalties in (J)

This is more reliable than fitting arbitrary invariant-based polynomials.

5. Restrict parameters during calibration

Even if the form is theoretically capable of convexity, fitted parameters may destroy it.

So when calibrating, impose constraints such as:

shear modulus $(>0)$
bulk modulus $(>0)$
coefficients of convex terms $(\ge 0)$
Hessian positivity over the strain domain of interest
ellipticity constraints

This is especially important in data-driven constitutive identification.

6. Use input-convex neural networks or constrained surrogates

For ML-based constitutive models, convexity can be embedded architecturally.

Examples:

Input Convex Neural Networks (ICNNs) enforce convexity with respect to selected inputs
constrained spline or basis expansions with nonnegative coefficients
energy models with automatic differentiation and Hessian regularization

For mechanics, one may define

\begin{equation} W = W(\text{invariants}) \end{equation}

and constrain the network so that ($W$) is convex in those invariants or polyconvex through special parameterization.

This is particularly useful for hyperelastic surrogate modeling and constitutive discovery.

Example: small-strain linear elasticity

\begin{equation} W(\varepsilon)=\frac{\lambda}{2}(\operatorname{tr}\varepsilon)^2 + \mu\, \varepsilon:\varepsilon \end{equation}

Convexity requires the stiffness to be positive definite, which implies

\begin{equation} \mu > 0, \qquad 3\lambda + 2\mu > 0 \end{equation}

in 3D isotropic elasticity.

These are the familiar stability restrictions.

Example: a polyconvex finite-strain form

A simple candidate:

\begin{equation} W(F)=a|F|^2 + b|\operatorname{cof}F|^2 + c(J-1)^2 - d\ln J \end{equation}

with $J=\det F$, and suitable positive constants. More carefully, one often uses volumetric terms that blow up as $(J\to 0^+)$, such as

\begin{equation} W_{\text{vol}}(J)=\frac{\kappa}{2}(J-1)^2 - \kappa \ln J \end{equation}

or related convex-in-($J$) penalties over ($J>0$).

This helps ensure resistance to interpenetration and compression collapse.

Practical caution

Convexity is not always desirable globally in the strictest sense.

Why?

Some real materials exhibit buckling, softening, phase transitions, or instability.
LCEs, hydrogels, and other smart materials may have non-convex free energies due to microstructural reorientation or multiphysics coupling.
In such cases, non-convexity may be physically meaningful.

So the right goal is often not “make everything fully convex,” but rather:

ensure local stability in the operating regime,
preserve ellipticity where needed,
use polyconvexity or constrained forms for numerical robustness,
allow controlled non-convexity only when it represents real physics.