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Patrizio Neff

Publications and source records attributed to Patrizio Neff.

At least 19 recordsLinked to original sources
arXiv2026

A globally defined polyconvex isotropic energy satisfying the true-stress-true-strain monotonicity condition (TSTS-M++)

Polyconvexity is a standard ingredient in the variational existence theory of finite elasticity, whereas true-stress-true-strain monotonicity (TSTS-M++) requires a positive incremental Cauchy-stress response. These two constitutive restrictions are independent, and Wollner, Holzapfel and Neff left open whether a compressible isotropic energy defined on the whole of $\mathrm{GL}^{+}(3)$ can satisfy both. We give an explicit affirmative answer. For every $\mu>0$ and $k>0$, the stored-energy function $W_k(F)=\frac{\mu}{2k}\bigl[\exp\bigl(k(\lVert F\rVert^2+3J^{-1}+J-7)\bigr)-1\bigr]$, $J=\det F$, is polyconvex and strictly rank-one convex. Its Cauchy-stress response satisfies TSTS-M++ globally if and only if $k\ge 1/(8\sqrt{3})$. In this regime every symmetric Cauchy stress corresponds to a unique positive-definite stretch, while the reference stretch is stress free with positive infinitesimal shear and bulk moduli. Stress bijectivity has the strictly smaller sharp threshold $k_{\mathrm{B}}\approx 0.00827233304$: at equality the stress map is a global homeomorphism with a nondifferentiable inverse, and above it the map is a global $C^{\infty}$ diffeomorphism. Thus, for $k_{\mathrm{B}}\le k<1/(8\sqrt{3})$, the map $V\mapsto\sigma(V)$ remains globally bijective while TSTS-M++ fails at finite strain. In the TSTS-M++ regime, every prescribed inner radius of a finite plane-strain annulus with a traction-free outer wall has a unique radial equilibrium, and its inner pressure increases smoothly and strictly from zero to infinity.

math.AP↗
arXiv2026

Polyconvexity for Cosserat nonlinear elasticity and nonlinear couple-stress theory

We study a class of nonlinear elastic energies whose constitutive structure is naturally expressed in terms of a stretch variable but is not directly covered by the standard polyconvex theory formulated in the deformation gradient. The problem is lifted by introducing an independent microrotation $ \overline R \in {\rm SO}(3)$ and the relative stretch $ \overline U = \overline R ^T{\rm D}\varphi$. The curvature variable $ \overline R ^T\operatorname{Curl} \overline R $ controls the full first-order variation of the rotation field and yields strong compactness of minimizing sequences. The identities $\operatorname{Cof}( \overline R^T{\rm D}\varphi) = \overline R^T\operatorname{Cof}{\rm D}\varphi$ and $\det( \overline R^T{\rm D}\varphi)=\det{\rm D}\varphi$ then allow the minors of the lifted stretches to be identified through the weak continuity of the corresponding minors of the deformation gradients. We prove existence in two regimes. The first allows convex dependence on $( \overline U ,\operatorname{Cof} \overline U ,\det \overline U )$ and assumes separate cofactor coercivity. The second depends only on $( \overline U ,\det \overline U )$ and requires no independent cofactor bound. For the constrained model, the weakly closed condition $ \overline R ^T{\rm D}\varphi\in\operatorname{Sym}^{+}(3)$, together with $\det{\rm D}\varphi>0$, implies $ \overline R =R:=\operatorname{polar}({\rm D}\varphi)$ for every admissible pair. The lifted problem is therefore equivalent to a deformation problem containing the curvature of the polar factor. The resulting model is a selectively rotationally regularized nonlinear elastic model of couple-stress type, rather than a purely first-gradient Biot model.

math.AP↗
arXiv2026

Monotonicity of isotropic tensor functions on the set of symmetric matrices: completing Rodney Hill's generalization of the Chandler Davis convexity theorem

Motivated by classical constitutive inequalities in isotropic nonlinear elasticity theory, we investigate the monotonicity of isotropic tensor functions of the form \[ \Sigma_f\colon\mathrm{Sym}(n)\to\mathrm{Sym}(n)\,,\quad \Sigma_f(Q^T\mathrm{diag}(\lambda_1,\dotsc,\lambda_n)\, Q) = Q^T\mathrm{diag}(f(\lambda_1,\dotsc,\lambda_n))\, Q \quad\forall\;Q\in\mathrm{O}(n) \] with a vector function $f=(f_1,\dotsc,f_n)\colon\mathbb{R}^n\to\mathbb{R}^n$ which is symmetric, i.e.\ satisfies \[ f_i(\lambda_{\pi(1)},\dotsc,\lambda_{\pi(n)}) = f_{\pi(i)}(\lambda_1,\dotsc,\lambda_n) \] for any permutation $\pi\colon\{1,\dotsc,n\}\to\{1,\dotsc,n\}$, where $\mathrm{Sym}(n)$ denotes the space of symmetric $n\times n$ matrices, $\mathrm{O}n$ is the orthogonal group and $\mathrm{diag}(\lambda_1,\dotsc,\lambda_n)$ is the diagonal matrix with diagonal entries $\lambda_1,\dotsc,\lambda_n\in\mathbb{R}$. We prove that vector-monotonicity of $f$ on $\mathbb{R}^n$ is equivalent to matrix-monotonicity of the induced isotropic tensor function $\Sigma_f$ on $\mathrm{Sym}(n)$. Our results generalize the Chandler Davis theorem for convex scalar isotropic functions and are obtained independently of Hill's original proof of this equivalence. We also discuss simple invertibility conditions for isotropic matrix functions. We conclude by showing that injectivity of the Cauchy stress $V\mapsto\sigma(V)$, continuous differentiability, and positive definiteness of $\mathrm{sym}\,\mathrm D\sigma(1\!\!\!\:1)$ in the natural state imply the strong Baker-Ericksen inequalities.

math.AP↗
arXiv2026

Polyconvexity implies Hill's inequality in ${\rm SL}(2)$

For compressible nonlinear isotropic elasticity it is well known that rank-one convexity, polyconvexity and the monotonicity of the Cauchy stress tensor with respect to the logarithmic stretch tensor (the true-stress-true-strain monotonicity, TSTS-M$^+\!$) are independent constitutive conditions which should, however, all together be satisfied for a physically meaningful description of idealized elastic materials. In the incompressible case, TSTS-M$^+\!$ turns into Hill's inequality since the Cauchy stress $\boldsymbol{\sigma}$ reduces to the Kirchhoff stress $\boldsymbol{\tau}$. Hill's inequality requires then monotonicity of the Kirchhoff stress in terms of the logarithmic stretch tensor evaluated for incompressible response. In this paper we clarify how the a priori independent notions of Legendre-Hadamard ellipticity (LH), polyconvexity and Hill's inequality are nevertheless intimately connected. More precisely, by providing several alternative proofs, we show that both LH-ellipticity (rank-one convexity) and polyconvexity imply the weak Hill inequality in the incompressible two-dimensional case.

math.AP↗
arXiv2026

Concurrent enforcement of polyconvexity and true-stress-true-strain monotonicity in incompressible isotropic hyperelasticity: application to neural network constitutive models

The design of physics-augmented neural networks (PANNs) for the purposes of constitutive modeling has received considerable attention as of late for a variety of material behaviors. Here, we revisit the classical framework of isotropic incompressible hyperelasticity in light of recent advances in the study of constitutive inequalities. We show that polyconvexity implies true-stress-true-strain monotonicity for a large class of incompressible strain-energy functions. The resulting elastic law obeys the physically reasonable Legendre-Hadamard (or ellipticity) condition as well as the notion of increasing stress with increasing strain. These results then inform the architecture of four distinct PANNs which are subsequently calibrated to three different sets of experimental data each. We show that different PANN parametrizations - satisfying the same constitutive constraints a priori - have varying approximation power for the description of material behavior. Moreover, even when distinct parametrizations perform comparatively well within the calibration regime, they show pronounced differences in extrapolation. This observation motivates a critical discussion about the predictive power of PANNs which also has implications for the modeling of more complex material behavior by virtue of neural networks.

math-ph↗
arXiv2026

Testing Hooke-like isotropic hyper-/hypo-elastic material models under finite simple shear deformations

We test some Hooke-like isotropic hyper-/hypo-elastic material models under finite simple shear deformations (cf., Thiel et al. Int. J. Non-linear Mech. 112: 57--72, 2019) and show that (1) the components of the Cauchy stress tensor for any Cauchy/Green isotropic elastic material under left finite simple shear (LFSS) deformation are equal to the components of the rotated Cauchy stress tensor for the same material under right finite simple shear (RFSS) deformation; (2) for any Hill's linear isotropic hyperelastic material model based on a symmetrically physical (SP) strain measure, LFSS and RFSS deformations lead to Eulerian and Lagrangian pure shear stresses, respectively; (3) for any two-power Ogden's isotropic hyperelastic material model based on a SP strain function, LFSS and RFSS deformations lead to Eulerian and Lagrangian pure shear stresses, respectively; (4) for some Hooke-like isotropic hypoelastic materials with constitutive relations based on corotational stress rates under LFSS deformation, the behavior of the Cauchy stress tensor components as a function of the shear parameter is qualitatively similar to that for the same materials under simple shear deformation. In addition, we confirm the results of Lin (Lin R.C. ZAMP, 75: 191, 2024) showing that for some Hooke-like isotropic hypoelastic materials with constitutive relations based on corotational stress rates without initial stresses under RFSS deformation, the Cauchy stress tensor components coincide with those for the Hencky isotropic hyperelastic material.

math.AP↗
arXiv2026

Analytical solutions for long cylindrical shells under radial deformations based on the isotropic relaxed micromorphic continuum

This study presents a closed-form analytical solution for the elastostatic response of long cylindrical shells composed of microstructured materials within the framework of the isotropic relaxed micromorphic continuum. The formulation accounts for microstructural effects by introducing an independent micro-distortion tensor field in addition to the classical displacement field. Under the assumptions of axisymmetric deformation and plane strain conditions, the governing equilibrium equations reduce to a coupled system of ordinary differential equations in the radial coordinate. By introducing suitable auxiliary variables, the system is reformulated into a non-homogeneous modified Bessel equation, which admits an exact analytical solution. Explicit expressions are derived for the radial displacement field and the non-zero components of the micro-distortion tensor. Numerical examples are presented to illustrate the influence of material parameters and the characteristic length on the displacement. The results demonstrate that the relaxed micromorphic model predicts deviations from classical elasticity where microstructural effects are more pronounced. The obtained solution provides valuable physical insight into the mechanics of cylindrical shells and serves as a benchmark for validating numerical implementations of relaxed micromorphic models.

math.AP↗
arXiv2026

A structure-preserving discretisation of SO(3)-rotation fields for finite Cosserat micropolar elasticity

We introduce a new method, dubbed Geometric Structure-Preserving Interpolation ($\Gamma$-SPIN) to preserve physics-constraints inherent in the material parameter limits of the finite-strain Cosserat micropolar model. The method advocates to interpolate the Cosserat rotation tensor using geodesic elements, which maintain objectivity and correctly represent curvature measures. At the same time, it proposes relaxing the interaction between the rotation tensor and the deformation tensor to alleviate locking effects. This relaxation is achieved in two steps. First, the regularity of the Cosserat rotation tensor is reduced by interpolating it into the N\'ed\'elec space. Second, the resulting field is projected back onto the Lie-group of rotations. Together, these steps define a lower-regularity projection-based interpolation. The construction allows the discrete Cosserat rotation tensor to match the polar part of the discrete deformation tensor. This ensures stable behaviour in the asymptotic regime as the Cosserat couple modulus tends to infinity, which constrains the model towards its couple-stress limit. We establish the consistency, stability, and optimality of the proposed method through several benchmark problems. The study culminates in a demonstration of its efficacy on a more intricate curved domain, contrasted with outcomes obtained from conventional interpolation techniques.

math.NA↗
arXiv2025

Cosserat micropolar and couple-stress elasticity models of flexomagnetism at finite deformations

We propose geometrically nonlinear (finite) continuum models of flexomagnetism based on the Cosserat micropolar and its descendent couple-stress theory. These models introduce the magneto-mechanical interaction by coupling the micro-dislocation tensor of the micropolar model with the magnetisation vector using a Lifshitz invariant. In contrast to conventional formulations that couple strain-gradients to the magnetisation using fourth-order tensors, our approach relies on third-order tensor couplings by virtue of the micro-dislocation being a second-order tensor. Consequently, the models permit centrosymmetric materials with a single new flexomagnetic constant, and more generally allow cubic-symmetric materials with two such constants. We postulate the flexomagnetic action-functionals and derive the corresponding governing equations using both scalar and vectorial magnetic potential formulations, and present numerical results for a nano-beam geometry, confirming the physical plausibility and computational feasibility of the models.

cond-mat.mtrl-sci↗
arXiv2025

In search of constitutive conditions in isotropic hyperelasticity: polyconvexity versus true-stress-true-strain monotonicity

The polyconvexity of a strain-energy function is nowadays increasingly presented as the ultimate material stability condition for an idealized elastic response. While the mathematical merits of polyconvexity are clearly understood, its mechanical consequences have received less attention. In this contribution we contrast polyconvexity with the recently rediscovered true-stress-true-strain monotonicity (TSTS-M${}^{++}\!$) condition. By way of explicit examples, we show that neither condition by itself is strong enough to guarantee physically reasonable behavior for ideal isotropic elasticity. In particular, polyconvexity does not imply a monotone trajectory of the Cauchy stress in unconstrained uniaxial extension which TSTS-M${}^{++}\!$ ensures. On the other hand, TSTS-M${}^{++}\!$ does not impose a monotone Cauchy shear stress response in simple shear which is enforced by Legendre-Hadamard ellipticity and in turn polyconvexity. Both scenarios are proven through the construction of appropriate strain-energy functions. Consequently, a combination of polyconvexity, ensuring Legendre-Hadamard ellipticity, and TSTS-M${}^{++}\!$ seems to be a viable solution to Truesdell's Hauptproblem. However, so far no isotropic strain-energy function has been identified that satisfies both constraints globally at the same time. Although we are unable to deliver a valid solution here, we provide several results that could prove helpful in the construction of such an exceptional strain-energy function.

math-ph↗
arXiv2025

Constitutive properties for isotropic energies in ideal nonlinear elasticity for solid materials: numerical evidences for invertibility and monotonicity in different stress-strain pairs

As a service for the solid mechanics community we gather in this paper constitutive properties of a collective list of isotropic elastic energies for compressible materials. Of interest to us are the invertibility and monotonicity of certain stress-strain pairs. The calculations are done numerically by our own evaluation algorithm and presented in a yes/no-table. Such an overview has been missing up to now. It is intended to expand the table with further energies as time goes on and updates will be found on arxive.

physics.class-ph↗
arXiv2025

A New Framework for Unidimensional Structures Based on Generalised Continua

The present work introduces a family of beam models derived from a three-dimensional higher-order elasticity framework. By incorporating three kinematic fields - the macroscopic displacement u, the micro-distortion tensor P, and the third-order tensor N - the study systematically explores three regimes: holonomic, semi-holonomic, and non-holonomic. These regimes correspond to varying levels of kinematic constraints, ranging from classical elasticity to a fully relaxed model. The holonomic case reduces to a higher-order Euler--Bernoulli beam model, while the semi-holonomic case generalises the Timoshenko beam model. The non-holonomic case provides a unified framework that naturally incorporates both dislocations and disclinations. Furthermore, the holonomic and semi-holonomic models are shown to emerge as singular limits of the non-holonomic model by increasing specific penalty coefficients. Simplified ordinary differential equation systems are derived for specific cases, such as pure traction and bending, illustrating the practical applicability of the models. The results highlight the hierarchical structure of the proposed framework and its ability to capture material defects in beam-like structures.

math.AP↗
arXiv2025

Two types of compressible isotropic neo-Hookean material models

In this contribution, we present a systematic study of the performance of two known types of compressible generalization of the incompressible neo-Hookean material model. The first type of generalization is based on the development of vol-iso neo-Hookean models and involves the additive decomposition of the elastic energy into volumetric and isochoric parts. The second simpler type of generalization is based on the development of mixed neo-Hookean models that do not use this decomposition. Theoretical studies of model performance and simulations of some homogeneous deformations have shown that when using volumetric functions $(J^q+J^{-q}-2)/(2q^2)$ ($J$ is the volume ratio, and $q\in \mathbb{R}$ is a parameter, $q\geq 0$) from the Hartmann-Neff family [Hartmann and Neff, Int. J. Solids Structures, 40: 2767-2791 (2003)] with parameter $q\geq 2$ (the preferred value is $q=5$), mixed and vol-iso models show similar performance in applications and have physically reasonable responses in extreme states, which is convenient for theoretical studies. However, contrary to vol-iso models, mixed models allow the use of a wider set of volumetric functions with physically reasonable responses in extreme states. A further feature of mixed models is simpler expressions for stresses and tangent stiffness tensors.

math.AP↗
arXiv2025

Propagation of Love waves in linear elastic isotropic Cosserat materials

We investigate the propagation of Love waves in an isotropic half-space modelled as a linear {elastic isotropic} Cosserat material. To this aim, we show that a method commonly used to study Rayleigh wave propagation is also applicable to the analysis of Love wave propagation. This approach is based on the explicit solution of an algebraic Riccati equation, which operates independently of the traditional Stroh formalism. The method provides a straightforward numerical algorithm to determine the wave amplitudes and speed{s}. Beyond its numerical simplicity, the method guarantees the existence and uniqueness of a subsonic wave speed, addressing a problem that remains unresolved in most Cosserat solids generalised {continua} theories. Although often overlooked, proving the existence of an admissible solution is, in fact, the key point that validates or invalidates the entire analytical approach used to derive the equation determining the wave speed. Interestingly, it is confirmed that the Love waves do not need the artificial introduction of a surface layer, as indicated in the literature.

math.AP↗
arXiv2025

Defects in unidimensional structures

In a previous work of the first authors, a non-holonomic model, generalising the micromorphic models and allowing for curvature (disclinations) to arise from the kinematic values, was presented. In the present paper, a generalisation of the classical models of Euler-Bernoulli and Timoshenko bending beams based on the mentioned work is proposed. The former is still composed of only one unidimensional scalar field, while the later introduces a third unidimensional scalar field, correcting the second order terms. The generalised Euler-Bernoulli beam is then shown to exhibit curvature (i.e. disclinations) linked to a third order derivative of the displacement, but no torsion (dislocations). Parallelly, the generalised Timoshenko beam is shown to exhibit both curvature and torsion, where the former is linked to the non-holonomy introduced in the generalisation. Lastly, using variational calculus, asymptotic values for the value taken by the curvature in static equilibrium are obtained when the second order contribution becomes negligible; along with an equation for the torsion in the generalised Timoshenko beam.

math-ph↗
arXiv2025

Rate-form equilibrium for an isotropic Cauchy-elastic formulation: Part I: modeling

We derive the rate-form spatial equilibrium system for a nonlinear Cauchy elastic formulation in isotropic finite-strain elasticity. For a given explicit Cauchy stress-strain constitutive equation, we determine those properties that pertain to the appearing fourth-order stiffness tensor. Notably, we show that this stiffness tensor $\mathbb{H}^{\text{ZJ}}(\sigma)$ acting on the Zaremba-Jaumann stress rate is uniformly positive definite. We suggest a mathematical treatment of the ensuing spatial PDE-system which may ultimately lead to a local existence result, to be presented in part II of this work. As a preparatory step, we show existence and uniqueness of a subproblem based on Korn's first inequality and the positive definiteness of this stiffness tensor. The procedure is not confined to Cauchy elasticity, however in the Cauchy elastic case, most theoretical statements can be made explicit. Our development suggests that looking at the rate-form equations of given Cauchy-elastic models may provide additional insight to the modeling of nonlinear isotropic elasticity. This especially concerns constitutive requirements emanating from the rate-formulation, here being reflected by the positive definiteness of $\mathbb{H}^{\text{ZJ}}(\sigma)$.

math.AP↗
arXiv2025

Global existence and uniqueness of weak solutions for a Willis-type model of elastodynamics

The existence and uniqueness of weak solutions is shown for a system related to the Willis model of elastodynamics. Both the whole space case and the case of a bounded smooth domain are studied. To this end the equations are reformulated as a linear symmetric hyperbolic system of first order and the existing theory for such systems is applied. If the initial and boundary data is regular enough, classical solutions are obtained. The possibility to transform the problem to a linear symmetric hyperbolic system hinges on a new symmetry condition on the Willis coupling tensor S, not yet considered in the literature. This condition demands that S is a totally symmetric third-order tensor.

math.AP↗