We use the homotopy analysis method (HAM) to find analytical solutions to the Vakhnenko equation and the Korteweg-de Vries-Burgers (KdV-Burgers) equation, two well-known (2+1)-dimensional nonlinear evolution equations. These solutions are essential for describing complex physical processes in a variety of fields, including nonlinear optics, fluid dynamics, and plasma physics. The convergence of the HAM-based solution is demonstrated using the squared residual error technique. The HAM-based technique shows a strong match with the exact solution to the problems.
This study demonstrates, via a homogenized continuum model, that monolayered pantographic waveguides with nearly inextensible flexural elements support the propagation of rarefaction solitary waves, whose crests correspond to complete cell closure. To validate the continuum approach and explore key parametric effects, time-dependent simulations are conducted at a discrete scale examining the influence of the total number of cells, the applied displacement rate, and the extensional stiffness of the flexural elements on solitary wave propagation. The interaction of solitary waves is also investigated. Results reveal that, for an odd number of cells, the solitary waves emerge from the collision relatively unaltered, whereas for an even number of cells, the region between the two crests experiences extreme compression, culminating in complete cell closure and the formation of two propagating tails.
Additive manufacturing enables creation of complex geometries. Particularly, the material extrusion (MEX) based three-dimensional printing technique, often used for thermoplastic polymers, extrudes molten material through a nozzle to form a part layer-by-layer. This process creates a complex internal structure leading to nonlocal elastic properties often called mechanical metamaterials. In the case of incorporating viscous properties, due to the wide distribution of time scales within their complex internal structures, fractional calculus offers a promising approach to incorporate the whole history for accurately describing the rheological behavior of such materials with an inherent structure. However, its adoption within the scientific community remains limited due to the unconventional notation and complex properties of the fractional operators. This study aims at providing a more accessible overview of some basic fractional rheological models, highlighting their ability to provide a unified approach to the characterisation of polymeric metamaterials. The inverse analysis is achieved by generating various fictitious datasets and fitting them to the three models. We compare three rheological models to predict the material's response. Our findings show that even the simplest fractional rheological model effectively predicts the time-dependent behavior in many cases within one-dimensional linear viscoelasticity.
We present a theoretical framework for designing an "exoskeleton" composed of distributed micromechanisms attached to a continuous elastic structure. Rigid and deformable microcomponents impose local kinematic constraints on the host continuum, effectively altering its elastic response. Enabled by advances in additive manufacturing, such micromechanism textures can impart special higher-order gradient elasticity to a soft substrate while bridging classical and soft robotics. Using Hamilton's principle, we derive the governing equations and show that the elastic energy stores contributions depending on first-and second-order spatial derivatives of displacement. A simple three-point linkage mechanism is illustrated: it yields first-and second-order strain-gradient potentials via internal springs. The kinetic energy similarly exhibits novel mixed spatiotemporal inertia terms. Numerical examples confirm that the discrete microstructure homogenizes to a continuum with the targeted higher-gradient stiffness. This approach offers a new design paradigm for metamaterials and soft robots, allowing precise control of elasticity and deformation through integrated micromechanical actuation.
Many results on micromorphic media can be found in the literature, where the equations of motion and the energetic foundations of the Eringen micromorphic continuum have been well established. The present paper has been devoted to the continuum modeling of a finite number of interacting, separated continua (microdomains) at the microscale, in which the strain energy has been formulated through a generalization of the Cauchy-Green deformation tensor, resulting in a degenerate metric at the considered scale, and a (6 & times; 6) Green-Lagrange strain tensor. The equilibrium equations have been obtained by systematically applying the method of virtual power. For one of the first time, boundary layer conditions appear in micromorphic mechanics. The paper concludes with a discussion on the number of constitutive parameters, shown to coincide, in number, with those of classical (Cauchy) elasticity, together with the recovery of micropolar continua as a special case and wide spectrum of applications of the proposed framework. Further details concerning the algebraic expressions of the tensors involved are provided in the Appendices.
The Corvis tonometer is widely used to measure intraocular pressure in the cornea and contact lens. In this study, based on the results of finite element simulations from a previous study, the methodology for generating two types of metamodels is elaborated in detail, and the challenges associated with each is examined. In addition, the sensitivity of each metamodel category to its inputs is analyzed, and the influence of each parameter is thoroughly evaluated. The results show that, even under significant variations in the inputs, the changes in this parameter remain minimal, making it a worthy measure for estimating the projected pressure.
Active deformations in biological tissues, such as those driven by growth or contraction, are commonly modeled through a multiplicative decomposition of the deformation gradient. While this framework is well established in forward problems, the inverse task of reconstructing the original configuration from a deformed state remains largely unexplored. We study three inverse formulations aimed at recovering the reference configuration: algebraic removal of the active tensor (growth removal formulation), reversed application of the full deformation sequence (reverse path formulation), and a variational approach enforcing equilibrium under a reversed active input (inverse formulation). Analytical results show that the first two methods fail to recover the undeformed state due to incompatibility and noncommutativity. The variational formulation, though mechanically consistent, does not ensure reversibility either. Numerical simulations on twenty cases with random small-magnitude active tensors reveal a wide range of reconstruction errors depending on the structure of the active field. We attribute these discrepancies to geometric incompatibilities and path dependence in elastic relaxation. Our results highlight fundamental limitations of inverse reconstruction and suggest that recovering a true reference configuration from shape data may require additional constraints, regularization, or data-driven inference.
The accumulation of microdamage renders bone tissues more vulnerable to excessive fatigue loading, which could potentially lead to fragility fractures. There is a need for an effective and noninvasive approach for predicting damage-prone regions. The literature lacks investigation into quasi-brittle damage-informed remodelling, which is hypothesized to accurately capture the damage state. The present study aims to formulate a strain-driven microdamage-informed remodelling framework to accurately depict the quasi-brittle nature of bone and to predict damage and adaptation state of a two-dimensional proximal human femur. Additionally, the potential of the proposed approach was assessed under the effects of the key remodelling parameters. Results imply that insufficient magnitude of remodelling, rapid stimulus diffusion, excessively fast remodelling, and inadequate diffusion coefficient promote substantial damage accumulation. The choice of adequate parameters reduces the femur fracture risk. The proposed model can serve clinicians as a predictor of likely femur fracture regions in elderly populations.
The Duhem model has been widely used to describe hysteretic behaviours, particularly those observed in piezoelectric materials. We explore the thermomechanical basis of the Duhem model. In a conservative system, the time rate of the dependent variable (e.g., stress) is related to the time rate of the independent variable (e.g., strain) through the second derivative of the Helmholtz free energy. To account for hysteresis in a nonconservative system, Duhem augmented the expression of the time rate of the dependent variable for a conservative system by adding a term featuring a piecewise continuous and differentiable function representing dissipation and leading to permanent changes in the state variables. We call this Duhem's irreversibility function or simply Duhem's function. As an example of application, we show how the Duhem model is equivalent to classical elastoplasticity with isotropic and kinematic hardening, with a judicious choice of Duhem's function. To illustrate this example, we numerically simulate the cyclic loading of a nonlinear elastic material with linear hardening. This work shows how, after more than a century from its conception and without knowledge of the specific system (e.g., the decomposition into elastic and plastic strain), the Duhem model constitutes a viable phenomenological approach to the modelling of hysteresis.
The aging phenomena are often associated with the diffusion of certain particles which then activate internal chemical reactions. In this paper, the system of such particles is modelled with an damaging fluid and a hemi-variational method is proposed in order to describe damage and deformation of a dam-shaped two-dimensional body, where a key point is the introduction of the coupling between the damage and the damaging fluid concentration. Another important key point of the present model is the presence of energetic thresholds for damage activation, which are assumed to be different in tension (lower) and in compression (higher). In this work, the body is subjected not only to the self-weight of the two-dimensional dam body and to the left-hand side water pressure, and therefore to the distributed external loads, dual of the displacement field, but also to the dual of the concentration of the mentioned damaging fluid, called the external distributed damaging fluid influx pressure. The parametric analyses are carried out in terms of diffusivity and damage-concentration coupling. The influx pressure drives the incoming flow of the damaging fluid, which is coupled with the damage variable, which is induced to evolve until the failure event, which characterises the lifetime of the structure. In order to evaluate the reasonableness of the model two cases are analysed: in the first one the body is modelled having a rectangular shape. In the second one a realistic trapezoidal dam shape is considered. Both have the same area and height in order to compare the results. This comparison yields the following intuitive considerations. First of all, at the beginning of the time history, i.e., at the time at which the structure is assumed to be built, the tensile state, evaluated by the positive part of the trace of the deformation tensor, of the rectangular structure, is much higher then that of the trapezoidal one. The evolution of damage is faster in the rectangular model and, as expected, the present trapezoidal shapes for the design of the dams possess a longer lifetime and are valid not only for the structural response at the time of construction but also, within the presented model, for a better aging performance. It is worth noting that this is a standard result for dam engineers, and justifies the shapes that are used in the present dam design.
We review and compare mathematical procedures aimed at rigorously deriving fluid-dynamic descriptions of classical particle systems in the large-scale limit of Hamiltonian dynamics. Despite the longstanding nature of this program, its fundamental problems remain open, with only a few notable exceptions that we briefly highlight.
An asymptotic analysis is conducted on a two-dimensional heterogeneous Canham-Helfrich flexoelectric biomembrane with constitutive laws rooted in thermodynamic principles. In the linearized Canham-Helfrich theory, the biomembrane's elasticity is characterized by a fourth-order energy density. For a Canham-Helfrich lipid bilayer membrane with protein inclusions, periodic homogenization yields four distinct effective coefficients. Numerical simulations illustrate the dependence of these homogenized coefficients on the volume fraction and externally applied electric field. This theoretical & computational approach provides a framework that enhances biomembrane behavior understanding under complex loading and complex microstructures, with promising applications in advanced materials modeling and simulation.
This paper examines i-graded manifolds as semiformal homogeneity structures, comparing two polynomial filtrations from their local models. In finite dimensions, these are componentwise equivalent, yielding isomorphic graded completions; generally, one induces a finer topology. By the Batchelor-Gawedzki-type theorem (Kotov-Salnikov), every i-graded manifold over base M is noncanonically isomorphic to one associated with its canonical i-graded bundle (Batchelor-Gawedzki bundle). In finite dimensions, this is the formal neighborhood of the zero section with the induced homogeneity structure. The Kotov-Salnikov graded Borel lemma extends weight-k functions from the formal neighborhood to smooth ones of the same weight. Here, this generalizes to a Borel-Whitney theorem: homogeneity morphisms of formal neighborhoods lift to smooth homogeneity maps between Batchelor-Gawedzki bundles. Categorically, let Bibe the category of finite-dimensional i-graded vector bundles with homogeneity morphisms, and Mani the category of finite-dimensional i-graded manifolds. The functor F: Bi -> Mani sends bundles to formal neighborhoods of their zero sections. The graded Batchelor-Gawedzki and Borel-Whitney theorems imply F is full and surjective on objects.
To describe the dynamics of Hamiltonian systems with differential constraints, a Hamiltonian vector field is projected onto the tangent planes of a distribution using a symplectic structure to obtain a vector field whose phase flow preserves distributions. The possibility of implementing symplectic projection in degenerate cases is considered when the restriction of the symplectic structure to the tangent planes of a distribution is a degenerate 2-form. An application is presented for studying the systems with one nonintegrable constraint of general form. The method of symplectic projection is described in the framework of Dirac structures.
This study conducts a numerical analysis of unsteady natural convection (UNC), heat transfer (HT), and entropy generation (Sgen) in a square cavity occupied with thermally stratified air. The enclosure comprises a uniformly heated bottom wall, thermally stratified vertical sidewalls, and a cooled top wall. Simulations are performed utilizing the finite volume method (FVM) with an invariable Prandtl number (Pr) of 0.71 and an extensive range of Rayleigh numbers (Ra) varying from 101 to 108. The analysis includes flow visualization through streamlines and isotherm plots, temperature time series (TTS), spectral analysis, limit point and limit cycles analyses, and the largest Lyapunov exponent (lambda L), highlighting flow regime transitions. As Ra increases, the flow changeovers from a steady symmetric state to chaotic regime through a sequence of bifurcations: a pitchfork bifurcation (Ra approximate to 7 & times; 103 to 8 & times; 103), a Hopf bifurcation (Ra approximate to 2 & times; 106 to 3 & times; 106), and the onset of chaos (Ra approximate to 107 to 2 & times; 107). Critical Ra values are identified, signifying the shift from a steady symmetric state to chaotic state. Validation against benchmark results confirms the accuracy and reliability of the simulations. The evaluation of average Nusselt number (Nu) and Sgen demonstrates that heat transfer enhancement is accompanied by increased irreversibility, with a notable rise in Nu at Ra = 3 & times; 106. Specifically, Nu at the top wall increases from 21.973 (Ra = 107) to 38.524 (Ra = 2 & times; 107), while at the bottom wall it rises from 22.029 to 38.477, corresponding to an approximate 75% increase in HT for both surfaces. These findings reveal the intricate interplay between cavity geometry, thermal stratification, and convective dynamics, offering valuable
Bioinspired 3D chiral microfliers, mimicking wind-dispersed seeds, have garnered significant attention for distributed environmental monitoring and wireless sensor networks. Prior researches have mainly focused on parameters such as wing fold angle and center-of-mass position. However, the critical role of wing tilt angle (gamma) remains underexplored, creating a knowledge gap in precise descent behavior control. In this work, we combine blade element theory with computational fluid dynamics simulations to quantitatively analyze the effects of gamma on rotating frequency ( f ) and overall drag coefficient (Cn). Our findings reveal a non-monotonic relationship between gamma and both rotating frequency ( f ) and overall drag coefficient (Cn), with optimal performance achieved at intermediate wing tilt angles. These findings provide fundamental insights into microflier aerodynamics while offering practical guidelines for performance optimization. The established relationships enable customized design of descent behaviors, from prolonged hovering for environmental sensing to rapid descent for time-critical deployments.
Paul Germain (1983) founded micromorphic continuum mechanics on the principle of virtual power. He derived the strong formulation for equilibrium and natural boundary conditions without introducing any constitutive relations. In this paper, based on these results, we prove the corresponding potential and complementary energy theorems in the case of linearized deformation measures. More specifically, starting from convex volume deformation energies, we prove that natural equilibrium conditions imply minimality of the total energy for the class of first gradient micromorphic media.
