Cavitation in Higher-Dimensional Elasticity

• Cavitation in higher-dimensional elasticity is the phenomenon where continuous deformations in elastic solids create singular internal voids under extreme conditions.
• Variational renormalization and null-Lagrangian regularization techniques enable the modeling of finite-energy cavitating minimizers beyond critical stretch thresholds.
• Advanced numerical methods such as punctured domain regularization, phase-field approaches, and shooting techniques offer practical solutions for simulating cavitation.

Cavitation in higher-dimensional elasticity refers to the phenomenon where an initially continuous deformation in an elastic solid leads to the formation of a singular internal void, often interpreted mathematically as the sudden creation of a nontrivial cavity or “hole.” In the modern variational and PDE frameworks, especially under radial symmetry, cavitation is characterized by mappings which fail to extend continuously to isolated points, with singular measures (such as Dirac masses in the distributional Jacobian) encoding the emergence of cavities. This behavior is central to both equilibrium and dynamical elasticity in dimensions $n \geq 2$, with qualitative and quantitative distinctions arising as spatial dimension increases.

1. Mathematical Foundations: Functional Setting and Energy Blow-Up

The standard reference configuration is the unit ball $B = \{x \in \mathbb{R}^n : |x| < 1\}$, and deformations are maps $u:B \to \mathbb{R}^n$ with positive determinant almost everywhere. The total elastic energy is given by

$E[u] = \int_B W(F(x))\,dx,$

where $F = \nabla u$ and $W$ is typically split as $W(F) = W_{\text{dir}}(F) + h(\det F)$. Here $W_{\text{dir}}$ depends on the principal stretches (isotropic part), and $h$ is a volumetric term satisfying growth and blow-up conditions as $\det F \to 0^+, \infty$:

• $h(d) \to \infty$ as $d \to 0^+$ or $d \to \infty$,
• $h'(d) \to -\infty$ as $d \to 0^+$, $h'(d) \to +\infty$ as $d \to \infty$, with $h$ convex and $W$ polyconvex.

Under these conditions, the critical result by Vodopyanov–Goldshtein–Reshetnyak asserts that any discontinuous $u$ with $\det \nabla u > 0$ a.e. must have infinite energy for $W$ of $n$-growth type. For radial deformations $u(x) = r(R)\frac{x}{R}$, $R = |x|$, a cavity at the origin ($r(0) = c > 0$) leads to a singularity:

• Near $R = 0$, $r(R)/R \to +\infty$, $r'(R) \to 0$, but the Jacobian determinant $\delta(R) = r'(R)[r(R)/R]^{n-1}$ stays bounded.
• Nonetheless, $\int_{0}^{R_0} R^{n-1}h(\delta(R))\,dR$ diverges as $R \to 0$, signaling infinite elastic energy for any cavitating map unless the cavity size vanishes (Negron-Marrero et al., 2021).

2. Variational Renormalization and Existence of Cavitating Minimizers

To obtain a finite-energy variational theory accommodating cavitation, a null-Lagrangian regularization is employed. One identifies, via integration by parts and energy flux identities (e.g., the Green-type identity), that the divergent energy contribution in cavitating configurations is purely a boundary term at the would-be cavity. Subtracting this via a null Lagrangian $N$ leads to a modified functional:

$$\hat{I}[r] = \lim_{\epsilon \to 0^+} \left\{ \int_{\epsilon}^1 R^{n-1}\Phi(r', r/R, ...)\,dR - \frac{\kappa(n-1)}{n}r(\epsilon)^n\ln\left( \frac{r(\epsilon)}{\epsilon} \right) \right\},$$

yielding a corresponding modified stored-energy density $\hat{\Phi}$, so the Euler–Lagrange equations are unchanged in the interior.

For the modified functional, the natural boundary condition at the cavity is the vanishing of the modified (renormalized) radial Cauchy stress:

$$\widehat{T}(r(0)) = \lim_{R \to 0} \widehat{T}(r(R)) = 0,$$

where $$\widehat{T}(r) = r^{1-n}\,\partial_1 \hat{\Phi}(r', r/R,...)$$. Finite-energy cavitating minimizers exist for boundary stretch $\lambda$ exceeding a critical threshold $\lambda_c$. For $\lambda < \lambda_c$ the minimizer is the homogeneous stretch; for $\lambda > \lambda_c$ the minimizer cavitates, as characterized by an algebraic stress-vanishing condition (Negron-Marrero et al., 2021).

3. Analytical and Numerical Approximation of Cavitating Solutions

Direct numerical simulation of cavitating solutions is precluded by the “repulsion property”: smooth or standard finite-element approximations fail, as their energies diverge when approximating a minimizer with singularities (cavities) (Negrón-Marrero et al., 2023). To overcome this, several methodologies are adopted:

• Punctured domain (core-radius) regularization: Cavities are approximated by excising small balls (radius $\epsilon$) around candidate flaw points, solving the PDEs on these perforated domains with suitable boundary integrals capturing the desired cavity volume and surface (Bresciani et al., 10 Mar 2025).
• Phase-field/Modica–Mortola regularization: A diffuse interface approach introduces a phase function $v \geq 0$ representing the cavity, coupled into the energy via penalizations of $|\nabla v|^\alpha$ and perimeter-like terms. $\Gamma$-convergence guarantees that, as $\epsilon \to 0$, solutions converge to the sharp-interface cavitating minimizer, with singularities captured in measure and perimeter terms (Negrón-Marrero et al., 2023).
• Shooting method for ODEs: In the case of radial symmetry, a shooting algorithm adjusts initial cavity sizes to satisfy both the boundary data and the singular inner free-stress condition, with convergence as inner radius $\epsilon \to 0$ (Negron-Marrero et al., 2021).

For the regularized approaches, $\Gamma$-convergence and compactness properties hold in dimensions $n\geq 2$, with additional care needed in the Sobolev exponent regime ($p>n-1$ for classical degree/invertibility, $p > \lfloor n/2\rfloor$ for newer weak-limit methods) (Campbell et al., 9 Jun 2025, Bresciani et al., 10 Mar 2025).

4. Geometric, Topological, and Quantitative Criteria for Cavitation

Cavitation in higher dimensions is often characterized as the failure of a (Sobolev-regular or quasiconformal) mapping $f : B^n \setminus \{0\} \to \mathbb{R}^n$ to extend continuously at the origin, such that the image contains a non-degenerate internal boundary. This perspective aligns with the modulus approach:

• The conformal modulus of curve families is a quantitative test for the presence of a cavity.
• Classical $K$- and $L$-dilatation conditions are insufficient for detecting cavitation in $n>2$.
• Recent work introduces sharp criteria involving directional dilatations $Q_f$ (normal) and $D_f$ (angular), yielding necessary and sufficient integral characterizations for the occurrence of cavitation (Golberg et al., 21 Dec 2025).

A mapping cavitates at $0$ if

$$I_Q(f) = \int_{S^{n-1}} \left( \int_0^1 Q_f(tu) \frac{dt}{t} \right)^{1-n} d\sigma(u) > 0,$$

while continuous extendibility (no cavity) is characterized by divergence of an integral involving $D_f$.

5. Dynamic Cavitation: PDE Evolution, Shocks, and Energy Paradox

In the time-dependent (elastodynamic) context, cavitation corresponds to the spontaneous formation of growing voids, modeled by radial self-similar weak solutions to the elastodynamics equations:

$$y_{tt} - \operatorname{Div} S(\nabla y) = 0, \quad S(F) = \frac{\partial W}{\partial F}(F),$$

with $y(x, t) = w(R, t)\frac{x}{R}$.

For $d = 2, 3$, the dynamic initiation of cavitation is always accompanied by a precursor shock, and the critical stretch for dynamic cavitation matches the bifurcation threshold for equilibrium cavitation in the corresponding static problem (Miroshnikov et al., 2014). The “slic-solution” framework shows that the work needed to form a cavity restores global energy conservation by adding the energetic penalty of void creation, resolving the paradox of apparent energy loss in entropic weak solutions (Giesselmann et al., 2013).

6. Variational Models with Surface Energy and Sobolev Regimes ($p < n-1$)

Recent advances extend the admissible space for cavitating minimizers to weak $W^{1,p}$ limits of homeomorphisms, permitting $p < n-1$ provided $p > \lfloor n/2\rfloor$. The total energy functional is augmented by a perimeter (surface energy) term for the cavity:

$$E_{\text{total}}(f) = \int_{\Omega} \left( |Df|^p + \varphi(\det Df) \right) dx + \gamma\, P(A(f), \mathbb{R}^n),$$

where $A(f)$ is the cavitation set and $P$ denotes its perimeter. Lower semicontinuity and existence of minimizers are preserved, and the approach accommodates physically realistic scenarios of fracture and multiple or countably many cavities (Campbell et al., 9 Jun 2025).

7. Dimensional Dependence and Open Problems

In $n \geq 3$, all the above frameworks—variational, modulus, PDE evolution, and numerical—extend with dimension through explicit dependence on radial ODEs, perimeter (as the $(n-1)$-Hausdorff measure), and integrability exponents. No fundamentally new qualitative behavior arises in higher dimensions; however, modulus-based criteria are increasingly important for sharp detection of cavitation, and the combinatorics of multiple cavity configurations grows richer. Certain PDE and shock-structure issues, especially for $d \geq 4$, remain open, particularly the status of sonic connections in dynamical cavitation (Miroshnikov et al., 2014).

---

In summary, the study of cavitation in higher-dimensional elasticity has led to a mature mathematical framework unifying variational renormalization, topological and geometric criteria, dynamic PDE analysis, and rigorous regularization/numerical schemes. Continued work addresses thresholds for cavitation, energetic cost of void nucleation, interplay between dimension, regularity, and microstructure, and robust simulation approaches across all physically relevant settings (Negron-Marrero et al., 2021, Bresciani et al., 10 Mar 2025, Giesselmann et al., 2013, Golberg et al., 21 Dec 2025, Negrón-Marrero et al., 2023, Campbell et al., 9 Jun 2025, Miroshnikov et al., 2014).