arXiv subjects

Ionel-Dumitrel Ghiba

Publications and source records attributed to Ionel-Dumitrel Ghiba.

At least 19 recordsLinked to original sources

arXiv2026

Nonlinear Kirchhoff--Love shell models derived from the Ciarlet--Geymonat energy: modelling and existence of minimizers

Starting from a three-dimensional model based on the Ciarlet--Geymonat energy, we derive nonlinear shell models within the classical elasticity theory of compressible isotropic materials. The Neo-Hookean term involving the norm of the deformation gradient leads to an energy depending on the first, the second, and the third fundamental forms of the deformed midsurface. The coefficients appearing in the resulting shell models depend on the classical Lamé coefficients of the three-dimensional material, on the thickness of the shell, and on the mean and Gaussian curvatures of the reference configuration. This shows that the behavior of the shell is influenced not only by the elastic coefficients but also by the initial geometry of the three-dimensional thin body. Since a purely asymptotic derivation may lead to nonlinear terms for which the lower semicontinuity of the resulting functionals is not clear, we combine the asymptotic reduction through the thickness with positive-weight quadrature rules for selected purely volumetric contributions. After deriving the models, we establish variational well-posedness in the sense of existence of minimizers. More precisely, we prove coercivity and weak lower semicontinuity of the reduced functionals and establish the existence of minimizers in appropriate Sobolev spaces. A key ingredient is a polyconvexity concept for shells together with compensated compactness results identifying the weak limits of the area-weighted mean and Gaussian curvatures. An important consequence of the convexity analysis is that Models I and II require no additional restriction on the thickness beyond the local geometric regularity condition ensuring the regularity of the shell parametrization. Only Model III, in which the quadratic volumetric contribution is treated by a Taylor expansion, requires an additional explicit thickness condition in our existence result.

math.AP↗

arXiv2026

Polyconvexity for incompressible inversion-symmetric energies of Valanis-Landel type

{Let $λ_i = ν_i(F)$ denote the three singular values of the deformation gradient $F \in {\rm GL}^+(3)$.} We consider the family of incompressible isotropic energies $ W_ψ(F)=\sum_i ψ\left(|\!\logλ_i|\right)$ with $ψ:[0,\infty)\mapsto\mathbb{R}$. Set $g(s)=ψ\left({\rm arcosh}\frac{s}{2}\right)$ for all $s\geq2$. If $g$ has a convex and non-decreasing extension $\bar g$ to $[0,\infty)$, then $W_ψ$ is the restriction to ${\rm SL}(3)$ of the explicit polyconvex function \ {$ F\mapsto\sum_i \bar g\left(ν_i(F+{\rm Cof} F)\right). $} The proof uses the identity $ν_i(F+{\rm Cof} F)=λ_i+λ_i^{-1}$, up to permutation, on ${\rm SL}(3)$ and Ball's convexity theorem for functions of the singular values. We also give a direct proof along rank-one lines contained in ${\rm SL}(3)$ and derive a convenient one-dimensional differential sufficient condition. In particular, $ F\mapsto\sum_i e^{\log^2\!λ_i} $ is rank-one convex on ${\rm SL}(3)$ and possesses the stated polyconvex extension. A simple-shear computation shows that scalar convexity in $\log λ_i$ alone is insufficient; the quadratic Hencky energy $\sum_i \log^2λ_i$ is not rank-one convex on ${\rm SL}(3)$.

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}φ$. 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 $\operatorname{Cof}( \overline R^T{\rm D}φ) = \overline R^T\operatorname{Cof}{\rm D}φ$ and $\det( \overline R^T{\rm D}φ)=\det{\rm D}φ$ 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}φ\in\operatorname{Sym}^{+}(3)$, together with $\det{\rm D}φ>0$, implies $ \overline R =R:=\operatorname{polar}({\rm D}φ)$ 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 \[ Σ_f\colon\mathrm{Sym}(n)\to\mathrm{Sym}(n)\,,\quad Σ_f(Q^T\mathrm{diag}(λ_1,\dotsc,λ_n)\, Q) = Q^T\mathrm{diag}(f(λ_1,\dotsc,λ_n))\, Q \quad\forall\;Q\in\mathrm{O}(n) \] with a vector function $f=(f_1,\dotsc,f_n)\colon\mathbb{R}^n\to\mathbb{R}^n$ which is symmetric, i.e.\ satisfies \[ f_i(λ_{π(1)},\dotsc,λ_{π(n)}) = f_{π(i)}(λ_1,\dotsc,λ_n) \] for any permutation $π\colon\{1,\dotsc,n\}\to\{1,\dotsc,n\}$, where $\mathrm{Sym}(n)$ denotes the space of symmetric $n\times n$ matrices, $\mathrm{O}n$ is the orthogonal group and $\mathrm{diag}(λ_1,\dotsc,λ_n)$ is the diagonal matrix with diagonal entries $λ_1,\dotsc,λ_n\in\mathbb{R}$. We prove that vector-monotonicity of $f$ on $\mathbb{R}^n$ is equivalent to matrix-monotonicity of the induced isotropic tensor function $Σ_f$ 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 $V\mapstoσ(V)$, continuous differentiability, and positive definiteness of $\mathrm{sym}\,\mathrm Dσ(1\!\!\!\:1)$ 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 $\boldsymbolσ$ reduces to the Kirchhoff stress $\boldsymbolτ$. 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↗

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

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

Quasiconvex relaxation of planar Biot-type energies and the role of determinant constraints

We derive the quasiconvex relaxation of the Biot-type energy density $\lVert\sqrt{\operatorname{D}φ^T \operatorname{D}φ}-I_2\rVert^2$ for planar mappings $φ\colon\mathbb{R}^2\to \mathbb{R}^2$ in two different scenarios. First, we consider the case $\operatorname{D}φ\in\textrm{GL}^+(2)$, in which the energy can be expressed as the squared Euclidean distance $\operatorname{dist}^2(\operatorname{D}φ,\textrm{SO}(2))$ to the special orthogonal group $\textrm{SO}(2)$. We then allow for planar mappings with arbitrary $\operatorname{D}φ\in\mathbb{R}^{2\times 2}$; in the context of solid mechanics, this lack of determinant constraints on the deformation gradient would allow for self-interpenetration of matter. We demonstrate that the two resulting relaxations do not coincide and compare the analytical findings to numerical results for different relaxation approaches, including a rank-one sequential lamination algorithm, trust-region FEM calculations of representative microstructures and physics-informed neural networks.

math.AP↗

arXiv2024

Hypo-elasticity, Cauchy-elasticity, corotational stability and monotonicity in the logarithmic strain

We combine the rate-formulation for the objective, corotational Zaremba-Jaumann rate \begin{align} \frac{{\rm D}^{\rm ZJ}}{{\rm D} t} [σ] = \mathbb{H}^{\rm ZJ}(σ).D, \qquad D = {\rm sym} {\rm D} v\,, \end{align} operating on the Cauchy stress $σ$, the Eulerian strain rate $D$ and the spatial velocity $v$ with the novel \enquote{corotational stability postulate} (CSP)\begin{equation} \Bigl\langle \frac{{\rm D}^{\rm ZJ}}{{\rm D} t}[σ], D \Bigr\rangle > 0 \qquad \forall \, D\in{\rm Sym}(3)\setminus\{0\} \end{equation} to show that for a given isotropic Cauchy-elastic constitutive law $B \mapsto σ(B)$ in terms of the left Cauchy-Green tensor $B = F F^T$, the induced fourth-order tangent stiffness tensor $\mathbb{H}^{\rm ZJ}(σ)$ is positive definite if and only if for $\widehatσ(\log B):=σ(B)$, the strong monotonicity condition (TSTS-M$^{++}$) in the logarithmic strain is satisfied. Thus (CSP) implies (TSTS-M^{++}) and vice-versa, and both imply the invertibility of the hypo-elastic material law between the stress and strain rates given by the tensor $\mathbb{H}^{\rm ZJ}(σ)$. The same characterization remains true for the corotational Green-Naghdi rate as well as the corotational logarithmic rate, conferring the corotational stability postulate (CSP) together with the monotonicity in the logarithmic strain tensor (TSTS-M^{++}) a far reaching generality. It is conjectured that this characterization of (CSP) holds for a large class of reasonable corotational rates. The result for the logarithmic rate is based on a novel chain rule for corotational derivatives of isotropic tensor functions.

math.AP↗

arXiv2024

A natural requirement for objective corotational rates -- on structure preserving corotational rates

We investigate objective corotational rates satisfying an additional, physically plausible assumption. More precisely, we require for \begin{equation*} \frac{{\rm D}^{\circ}}{{\rm D} t}[B] = \mathbb{A}^{\circ}(B).D \end{equation*} that $\mathbb{A}^{\circ}(B)$ is positive definite. Here, $B = F \, F^T$ is the left Cauchy-Green tensor, $\frac{{\rm D}^{\circ}}{{\rm D}t}$ is a specific objective corotational rate, $D = {\rm sym} \, {\rm D} v$ is the Eulerian stretching and $\mathbb{A}^{\circ}(B)$ is the corresponding induced fourth order tangent stiffness tensor. Well known corotational rates like the Zaremba-Jaumann rate, the Green-Naghdi rate and the logarithmic rate belong to this family of ``positive'' corotational rates. For general objective corotational rates $\frac{{\rm D}^{\circ}}{{\rm D} t}$ we determine several conditions characterizing positivity. Among them an explicit condition on the material spin-functions of Xiao, Bruhns and Meyers (2004). We also give a geometrical motivation for invertibility and positivity and highlight the structure preserving properties of corotational rates that distinguish them from more general objective stress rates. Applications of this novel concept are indicated.

math.AP↗

arXiv2024

A constitutive condition for idealized isotropic Cauchy elasticity involving the logarithmic strain

Following Hill and Leblond, the aim of our work is to show, for isotropic nonlinear elasticity, a relation between the corotational Zaremba-Jaumann objective derivative of the Cauchy stress $σ$, i.e. \begin{equation} \frac{{\rm D}^{\rm ZJ}}{{\rm D} t}[σ] = \frac{\rm d}{{\rm d}{t}}[σ] - W \, σ+ σ\, W, \qquad W = {\rm skew}(\dot F \, F^{-1}) \end{equation} and a constitutive requirement involving the logarithmic strain tensor. Given the deformation tensor $F ={\rm D} φ$, the left Cauchy-Green tensor $B = F \, F^T$, and the strain-rate tensor $D = {\rm sym}(\dot F \, F^{-1})$, we show that \begin{equation} \label{eqCPSdef} \begin{alignedat}{2} \forall \,D\in{\rm Sym}(3) \! \setminus \! \{0\}: ~ \langle{\frac{{\rm D}^{\rm ZJ}}{{\rm D} t}[σ]},{D}\rangle > 0 \quad &\iff \quad \log B \longmapsto \widehatσ(\log B) \;\textrm{is strongly Hilbert-monotone} &\iff \quad {\rm sym} {\rm D}_{\log B} \widehat σ(\log B) \in{\rm Sym}^{++}_4(6) \quad \text{(TSTS-M$^{++}$)}, \end{alignedat} \tag{1} \end{equation} where ${\rm Sym}^{++}_4(6)$ denotes the set of positive definite, (minor and major) symmetric fourth order tensors. We call the first inequality ``corotational stability postulate'' (CSP), a novel concept, which implies the \textbf{T}rue-\textbf{S}tress \textbf{T}rue-\textbf{S}train strict Hilbert-\textbf{M}onotonicity (TSTS-M$^+$) for $B \mapsto σ(B) = \widehat σ(\log B)$, i.e. \begin{equation} \langle \widehatσ(\log B_1)-\widehatσ(\log B_2),{\log B_1-\log B_2} \rangle> 0 \qquad \forall \, B_1\neq B_2\in{\rm Sym}^{++}(3) \, . \end{equation} In this paper we expand on the ideas of Hill and Leblond, extending Leblonds calculus to the Cauchy elastic case.

math.AP↗

arXiv2024

The Biot stress -- right stretch relation for the compressible Neo-Hooke-Ciarlet-Geymonat model and Rivlin's cube problem

The aim of the paper is to recall the importance of the study of invertibility and monotonicity of stress-strain relations for investigating the non-uniqueness and bifurcation of homogeneous solutions of the equilibrium problem of a hyperelastic cube subjected to equiaxial tensile forces. In other words, we reconsider a remarkable possibility in this nonlinear scenario: Does symmetric loading lead only to symmetric deformations or also to asymmetric deformations? If so, what can we say about monotonicity for these homogeneous solutions, a property which is less restrictive than the energetic stability criteria of homogeneous solutions for Rivlin's cube problem. For the Neo-Hooke type materials we establish what properties the volumetric function $h$ depending on ${\rm det}\, F$ must have to ensure the existence of a unique radial solution (i.e. the cube must continue to remain a cube) for any magnitude of radial stress acting on the cube. The function $h$ proposed by Ciarlet and Geymonat satisfies these conditions. However, discontinuous equilibrium trajectories may occur, characterized by abruptly appearing non-symmetric deformations with increasing load, and a cube can instantaneously become a parallelepiped. Up to the load value for which the bifurcation in the radial solution is realized local monotonicity holds true. However, after exceeding this value, monotonicity no longer occurs on homogeneous deformations which, in turn, preserve the cube shape.

math.AP↗

arXiv2023

An essay on deformation measures in isotropic thin shell theories. Bending versus curvature

It has become commonplace for the stored energy function of any realistic shell model to align ``within first order" with the classical Koiter membrane-bending (flexural) shell model. In this paper, we assess whether certain extended Cosserat shell models are consistent with the classical linear Koiter model. In doing this, we observe that there are numerous reasons why a modified version of the classical Koiter model should be considered, a consensus reached not only by Koiter himself but also by Sanders and Budiansky, who independently developed the same theory during the same period. To provide a comprehensive overview of the strain measures employed in our Cosserat shell models, this paper presents them in a unified manner and compares them with the strain measures previously utilized in the literature. We show that all our new strain tensors either generalize (in the case of nonlinear constrained or unconstrained models) or coincide (in the case of the linear constrained model) with the strain tensors recognized as the ``best" or those possessing a well-defined geometric interpretation connected to bending or curvature.

math-ph↗

arXiv2023

Rayleigh waves in isotropic elastic materials with micro-voids

In this paper, we show that a general method introduced by Fu and Mielke allows to give a complete answer on the existence and uniqueness of a subsonic solution describing the propagation of surface waves in an isotropic half space modelled with the linear theory of isotropic elastic materials with micro-voids. Our result is valid for the entire class of materials admitting real wave propagation which include auxetic materials (negative Poisson's ration) and composite materials with negative-stiffness inclusions (negative Young's modulus). Moreover, the used method allows to formulate a simple and complete numerical strategy for the computation of the solution.

math.AP↗

arXiv2023

Explicit formula for the Gamma-convergence homogenized quadratic curvature energy in isotropic Cosserat shell models

We show how to explicitly compute the homogenized curvature energy appearing in the isotropic $Γ$-limit for flat and for curved initial configuration Cosserat shell models, when a parental three-dimensional minimization problem on $Ω\subset \mathbb{R}^3$ for a Cosserat energy based on the second order dislocation density tensor $α:=\overline{R} ^T {\rm Curl}\,\overline{R} \in \mathbb{R}^{3\times 3}$, $\overline{R}\in {\rm SO}(3)$ is used.

math.AP↗

arXiv2022

Linear constrained Cosserat-shell models including terms up to ${O}(h^5)$. Conditional and unconditional existence and uniqueness

In this paper we linearise the recently introduced geometrically nonlinear constrained Cosserat-shell model. In the framework of the linear constrained Cosserat-shell model, we provide a comparison of our linear models with the classical linear Koiter shell model and the "best" first order shell model. For all proposed linear models we show existence and uniqueness based on a Korn's inequality for surfaces.

math.AP↗

arXiv2022

A linear isotropic Cosserat shell model including terms up to $O(h^5)$. Existence and uniqueness

In this paper we derive the linear elastic Cosserat shell model incorporating effects up to order $O(h^5)$ in the shell thickness $h$ as a particular case of the recently introduced geometrically nonlinear elastic Cosserat shell model. The existence and uniqueness of the solution is proven in suitable admissible sets. To this end, inequalities of Korn-type for shells are established which allow to show coercivity in the Lax-Milgram theorem. We are also showing an existence and uniqueness result for a truncated $O(h^3)$ model. Main issue is the suitable treatment of the curved reference configuration of the shell. Some connections to the classical Koiter membrane-bending model are highlighted.

math.AP↗

arXiv2022

A geometrically nonlinear Cosserat (micropolar) curvy shell model via Gamma convergence

Using $Γ$-convergence arguments, we construct a nonlinear membrane-like Cosserat shell model on a curvy reference configuration starting from a geometrically nonlinear, physically linear three-dimensional isotropic Cosserat model. Even if the theory is of order $O(h)$ in the shell thickness $h$, by comparison to the membrane shell models proposed in classical nonlinear elasticity, beside the change of metric, the membrane-like Cosserat shell model is still capable to capture the transverse shear deformation and the {Cosserat}-curvature due to remaining Cosserat effects. We formulate the limit problem by scaling both unknowns, the deformation and the microrotation tensor, and by expressing the parental three-dimensional Cosserat energy with respect to a fictitious flat configuration. The model obtained via $Γ$-convergence is similar to the membrane {(no $O(h^3)$ flexural terms, but still depending on the Cosserat-curvature)} Cosserat shell model derived via a derivation approach but these two models do not coincide. Comparisons to other shell models are also included.

math.AP↗