This paper proposes a comprehensive framework, Theory–Simulation–Experimental Verification, for the elasto-aerodynamic analysis of elastic foil gas bearings (EFGBs). In contrast to many studies that approximate the foil structure using simplified two-dimensional models, the present work adopts a macro-element beam theory model that incorporates the actual 3D geometry, nonlinear elasticity, and frictional contact effects, and couples it directly with the Reynolds equation. To improve accuracy and robustness, the macro-beam results are validated against a fully coupled fluid–structure interaction (FSI) model developed in COMSOL Multiphysics. Emphasis is placed on quantifying the influence of foil thickness, clearance, and eccentricity, where the pressure distribution, foil deflection, and load capacity are obtained through the coupled solver. The results reveal that increasing foil thickness from 0.1 mm to 0.2 mm elevates the peak gas film pressure from 1.36 × 105 Pa to 1.97 × 105 Pa while simultaneously reducing displacement and pressure fluctuations, thereby enhancing bearing stability. Smaller clearances are shown to increase load capacity but also induce stronger oscillatory flow behavior, indicating a stiffness–stability trade-off. Additionally, prototype experiments with a 0.05 mm clearance confirm practical lift-off at 4300–7000 rpm under 10–30 N external loads, with measured torques of 0.18–0.30 N·m. By combining computational efficiency, 3D fidelity, and experimental validation, the proposed framework provides quantitative guidance for the design and optimization of EFGBs used in high-speed turbomachinery, such as aviation and compact energy systems, including turbine-based air-cycle refrigeration units and small gas-turbine rotors for unmanned aerial vehicles.

Abstract It has long been recognized that the theory of nonlinear elasticity provides a rich framework for a large variety of issues of interest to applied mathematicians. In particular, researchers with primary interest in nonlinear partial differential equations have been attracted to this area of continuum mechanics. However, the detailed theoretical background giving rise to the governing partial differential equations is not always familiar to non-specialists. The purpose of the present expository note is to attempt to alleviate this situation by describing a variety of nonlinear partial differential equations that have been found to govern the deformations of anti-plane shear and plane strain for isotropic incompressible hyperelastic solids in equilibrium.

Abstract 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 $${\mathbb {H}}^{{{\,\textrm{ZJ}\,}}}(\sigma )$$ H ZJ ( σ ) 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 $${\mathbb {H}}^{{{\,\textrm{ZJ}\,}}}(\sigma )$$ H ZJ ( σ ) .

Purpose The authors investigate the elasticity between VIX exchange traded products (ETPs)–including ETNs (VXX, XIV and TVIX) and ETFs (VIXY, SVXY and UVXY)–and VIX Futures. Design/methodology/approach This study applies quantile regression to uncover nonlinear elasticity dynamics in the daily price interactions between ETPs and Futures. Findings Employing decile regressions on the S&P 500 VIX Short-Term Total Return Index (SPVXSTR), the authors find that elasticity of VIX Futures to ETP prices is lower at market close but higher intraday, potentially due to liquidity differences, with peaks at the distribution’s extremes at close. VXX exhibits significantly higher elasticity than VIXY, likely due to its dominant, unhedged note structure, while XIV and SVXY show similar elasticity, and TVIX’s elasticity is half that of UVXY, reflecting its reduced leverage. These findings suggest that intraday liquidity amplifies futures responsiveness, with implications for hedging strategies during volatile closes and portfolio construction favoring dominant instruments such as VXX. Originality/value The linear relations between VIX ETPs and VIX Futures are well documented in the literature using mean-regression approaches, here estimated elasticities are assumed constant across the distribution of VIX Futures and ETPs. This study extends the analysis by employing quantile regression to capture quantile-specific elasticities, allowing for a more nuanced examination of inverse and leveraged products, where elasticity dynamics remain largely unexplored.

Abstract We study free-discontinuity functionals in nonlinear elasticity, where discontinuities correspond to the phenomenon of cavitation. The energy comprises two terms: a volume term accounting for the elastic energy, and a surface term concentrated on the boundaries of the cavities in the deformed configuration that depends on their unit normal. First, we prove the existence of energy-minimizing deformations. While the treatment of the volume term is standard, that of the surface term relies on the regularity of inverse deformations, their weak continuity properties, and Ambrosio’s lower semicontinuity theorem for special functions of bounded variation. Additionally, we identify sufficient conditions for minimality by employing outer variations and applying the formula for the first derivative of the anisotropic perimeter.

A variant of the mathematical theory of deformation of multilayer nonlinearly elastic (according to Kauderer) plates of arbitrary constant thickness with non-symmetric structure in thickness has been constructed. The transverse load on the horizontal faces can be arbitrary static. The components of the stress-strain state (SSS) and the boundary conditions on the lateral surface are functions of three spatial coordinates. Spatial boundary value problems for multilayer plates are reduced to two-dimensional using three-dimensional equations of the theory of elasticity, the Reissner variational principle, and the expansion of the components of the SSS into infinite mathematical series by combinations of Legendre polynomials within each layer. This approach differs significantly from the approaches of other authors. The main dependencies, boundary conditions and systems of equilibrium differential equations with high-order partial derivatives with respect to the displacement components are derived. All dependencies and equations contain nonlinear terms. The new methodology for constructing a variant of the nonlinear theory makes it possible to accurately satisfy the boundary conditions on the horizontal faces of the plate and on the lateral surface, and to accurately satisfy the conditions of rigid conjugation of adjacent layers. The system of equilibrium equations has a high order. An analytical method for solving these systems is proposed and developed. The method is based on algebraic, differential and operator transformations of the initial systems. They are reduced to two convenient defining systems: one describes the vortex edge effect with a refinement of the SSS, and the other describes a refined internal SSS with a potential edge effect. The order of the systems of differential equations does not depend on the number of layers, but depends only on the number of retained terms in the mathematical seriess. The internal SSS is separated from the potential edge effect. By the method of order reduction, the determining systems are reduced to second-order differential equations. This significantly simplifies the solution of boundary value problems. General solutions for all components of the SSS were found through general solutions of second-order differential equations. For plates with non-symmetric structure, the equations of skew-symmetric and symmetric deformation are interconnected, unlike plates with a symmetric structure. Numerical results are presented for a two-layer linearly elastic plate under cylindrical bending.

Abstract We consider periodic homogenization of hyperelastic models incorporating incompressible behavior via the constraint $$\det (\nabla u)=1$$ det ( ∇ u ) = 1 . We show that the ’usual’ homogenized integral functional $$\int W_{\textrm{hom}}(\nabla u)\,dx$$ ∫ W hom ( ∇ u ) d x , where $$W_{\textrm{hom}}$$ W hom is the standard multicell-formula of non-convex homogenization restricted to volume preserving deformations, yields an upper bound for the $$\Gamma $$ Γ -limit as the scale of periodicity tends to zero.

Abstract A thermomechanical model has been developed and implemented to couple temperature, thermal expansion, contact, and wear using the finite element method. This three-dimensional, transient, and nonlinear model is unconditionally stable because of the use of an implicit solver. The weak form of the nonlinear heat transfer and elasticity equations has been derived, incorporating penalty contact, conduction, convection, radiation, and deformation. Several boundary conditions, including Dirichlet, Neumann, and Robin conditions, have been applied. We focus on mechanical train brakes due to their strong thermomechanical coupling effect. The simulation results have been validated against full-scale experimental data. Additionally, mesh sensitivity and time step sensitivity analyses are conducted to further assess the model’s accuracy. Friction heat is calculated at each time step through contact conditions, enabling the identification of hot spots and thermal cracks. The results demonstrate that this thermomechanical model is both efficient and robust. This model has been open-sourced, providing a powerful tool for advanced research in thermomechanical multiphysics analysis.

A recurrent mistake in nonlinear elasticity: How a recent paper keeps the error alive

Mathematical and numerical assessment of Data-Driven Identification method applied to nonlinear elasticity

Stress boundedness and existence of radial minimizers in constrained nonlinear elasticity

A high-order immersed finite-difference discretization for solving linear and nonlinear elasticity problems

Learning homeomorphic image registration via conformal-invariant hyperelastic regularisation.

Abstract Let $$\mathbb {D}(u)$$ D ( u ) be the Dirichlet energy of a map u belonging to the Sobolev space $$H^1_{u_0}(\Omega ;\mathbb {R}^2)$$ H u 0 1 ( Ω ; R 2 ) and let $$\mathcal {A}$$ A be a subclass of $$H^1_{u_0}(\Omega ;\mathbb {R}^2)$$ H u 0 1 ( Ω ; R 2 ) whose members are subject to the constraint $$\det \nabla u = g$$ det ∇ u = g a.e. for a given g , together with some boundary data $$u_0$$ u 0 . We develop a technique that, when applicable, enables us to characterize the global minimizer of $$\mathbb {D}(u)$$ D ( u ) in $$\mathcal {A}$$ A as the unique global minimizer of the associated functional $$F(u):=\mathbb {D}(u)+ \int _{\Omega } f(x) \, \det \nabla u(x) \, \, \textrm{d}x$$ F ( u ) : = D ( u ) + ∫ Ω f ( x ) det ∇ u ( x ) d x in the free class $$H^1_{u_0}(\Omega ;\mathbb {R}^2)$$ H u 0 1 ( Ω ; R 2 ) . A key ingredient is the mean coercivity of F on $$H^1_0(\Omega ;\mathbb {R}^2)$$ H 0 1 ( Ω ; R 2 ) , which condition holds provided the ‘pressure’ $$f \in L^{\infty }(\Omega )$$ f ∈ L ∞ ( Ω ) is ‘tuned’ according to the procedure set out in [1]. The explicit examples to which our technique applies can be interpreted as solving the sort of constrained minimization problem that typically arises in incompressible nonlinear elasticity theory.

Abstract We study injectivity for models of Nonlinear Elasticity that involve the second gradient. We assume that $$\Omega \subset {\mathbb{R}}^n$$ is a domain, $$f\in W^{2,q}(\Omega ,{\mathbb{R}}^n)$$ satisfies $$|J_f|^{-a}\in L^1$$ and that f equals a given homeomorphism on $$\partial \Omega$$ . Under suitable conditions on q and a we show that f must be a homeomorphism. As a main new tool we find an optimal condition for a and q that imply that $$\mathcal {H}^{n-1}(\{J_f=0\})=0$$ and hence $$J_f$$ cannot change sign. We further specify in dependence of q and a the maximal Hausdorff dimension d of the critical set $$\{J_f=0\}$$ . The sharpness of our conditions for d is demonstrated by constructing respective counterexamples.

ABSTRACT Rotating structural components are commonly found in engineering structures and the natural world. This study presents a comprehensive investigation into the mechanical behavior and instability mechanisms of rotating soft tubes subjected to axial loads, with a particular emphasis on buckling phenomena. A theoretical framework based on the nonlinear elasticity and linear incremental theory is established, accompanied by finite element simulations to analyze and validate the nonlinear deformation behavior of rotating tubes. We systematically identify the critical factors contributing to instability, revealing two distinct instability mechanisms influenced by axial loads and rotation, and examine how the initial radius ratio affects the critical thresholds and associated modes. Furthermore, we examine how geometric parameters influence the stability and mechanical response of the tubes, offering insights for optimizing performance through dimensional tuning and providing practical guidance for the design and safe operation of rotating soft structures. The results provide a scientific basis for mitigating structural instability during rapid deformation by tailoring external loads, thereby contributing to optimizing performance and operational safety.

Abstract On account of the ongoing trend towards lightweight construction, the use of dual-phase steels (DP steels) has gained high popularity. However, the underlying micromechanics, which give DP steel its prominent property of reaching high strengths while maintaining high formability, are also the cause of the difficulty in predicting springback after forming processes. The resulting geometrical change of a sheet metal part due to springback is mainly determined by the materials elastic properties and the residual stresses due to deformation. Therefore, the influence of the by now widely observed nonlinear elasticity has been considered in forming simulations and yielded significant improvement. However, an exact prediction is still not possible. Thus, to deepen the understanding of the elastic-plastic material behavior, the influence of another known property-altering phenomenon on the nonlinear elasticity of DP steel, the relaxation over time is analyzed. The results of the experimental relaxation tests show a slight but definitive influence on the elastic behavior of the investigated steels. The recorded unloading moduli are being furthermore used in a springback simulation to test their applicability and potential use. A final validation of the numerical results with an experimental plate bending test highlights the need for further investigation of the numerical adaptation of a relaxation dependent springback prognosis.

Abstract Most theories and applications of elasticity rely on an energy function that depends on the strains from which the stresses can be derived. This is the traditional setting of Green elasticity, also known as hyper-elasticity. However, in its original form the theory of elasticity does not assume the existence of a strain energy function. In this case, called Cauchy elasticity, stresses are directly related to the strains. Since the emergence of modern elasticity in the 1940s, research on Cauchy elasticity has been relatively limited. One possible reason for this is that for Cauchy materials, the net work performed by stress along a closed path in the strain space may be nonzero. Therefore, such materials may require access to both energy sources and sinks. This characteristic has led some mechanicians to question the viability of Cauchy elasticity as a physically plausible theory of elasticity. In this paper, motivated by its relevance to recent applications, such as the modeling of active solids, we revisit Cauchy elasticity in a modern form. First, we show that in the general theory of anisotropic Cauchy elasticity, stress can be expressed in terms of six functions, that we call Edelen-Darboux potentials. For isotropic Cauchy materials, this number reduces to three, while for incompressible isotropic Cauchy elasticity, only two such potentials are required. Second, we show that in Cauchy elasticity, the link between balance laws and symmetries is lost, in general, since Noether’s theorem does not apply. In particular, we show that, unlike hyperleasticity, objectivity is not equivalent to the balance of angular momentum. Third, we formulate the balance laws of Cauchy elasticity covariantly and derive a generalized Doyle–Ericksen formula. Fourth, the material symmetry and work theorems of Cauchy elasticity are revisited, based on the stress-work 1-form that emerges as a fundamental quantity in Cauchy elasticity. The stress-work 1-form allows for a classification via Darboux’s theorem that leads to a classification of Cauchy elastic solids based on their generalized energy functions. Fifth, we discuss the relevance of Carathéodory’s theorem on accessibility property of Pfaffian equations. Sixth, we show that Cauchy elasticity has an intrinsic geometric hystresis, which is the net work of stress in cyclic deformations. If the orientation of a cyclic deformation is reversed, the sign of the net work of stress changes, from which we conclude that stress in Cauchy elasticity is neither dissipative nor conservative. Seventh, we establish connections between Cauchy elasticity and the existing constitutive equations for active solids. Eighth, linear anisotropic Cauchy elasticity is examined in detail, and simple displacement-control loadings are proposed for each symmetry class to characterize the corresponding antisymmetric elastic constants. Ninth, we discuss both isotropic and anisotropic Cauchy anelasticity and show that the existing solutions for stress fields of distributed eigenstrains (and particularly defects) in hyperelastic solids can be readily extended to Cauchy elasticity. Tenth, we introduce Cosserat–Cauchy materials and demonstrate that an anisotropic three-dimensional Cosserat–Cauchy elastic solid has at most twenty four generalized energy functions.

Influences of inter-bubble interactions on ultrasonic cavitation dynamics and subharmonic emissions in viscoelastic media with nonlinear elasticity☆

An Unconditionally Energy Stable Gradient Flow for Phase Field Modelling of Structural Topology Optimization in Geometrically Nonlinear Elasticity