This paper studies bounded hereditary load transfer in a perfectly bonded two-phase solid in which an elastic constraint acts in parallel with a creep-active fractional-Maxwell phase. Compatibility and equilibrium reduce the two-phase problem to a scalar Caputo equation governed by the memory order α , characteristic time τ , and stiffness invariant κ = K v ∕ ( K e + K v ) . The reduction has the transfer structure of a fractional standard linear solid; the contribution is therefore not a new hereditary kernel but a phase-resolved mechanics interpretation that maps κ to geometry and instantaneous phase stiffnesses and distinguishes physical total load from an auxiliary effective input. We establish a unique continuous mild solution and a unique absolutely continuous Caputo solution, derive the Mittag–Leffler response, and prove boundedness, concavity and monotone phase-load transfer. The deformation ceiling is fixed by κ , whereas the approach depends jointly on α , τ and κ through τ eff = τ ∕ ( 1 − κ ) 1 ∕ α . A Laplace-domain construction for compatible finite Fourier loads gives the physical total-load transfer function and the sharp bound 1 ≤ | H N ( iω ) | ≤ 1 ∕ ( 1 − κ ) . A reproducible 168-case sensitivity sweep shows that logarithmic observation times improve local conditioning in 147 cases, but not universally. An illustrative concrete-filled steel-tube calculation interpolates two non-trivial selected observations by construction, with zero residual degrees of freedom, and is explicitly treated as an instantiation rather than as predictive validation. The model thus provides a bounded, inspectable analytical benchmark for constrained hereditary solids.
Thermal disturbances induced by explosions or fires can rapidly alter the stress and temperature fields in surrounding rock masses, triggering strongly coupled thermo-hydro-mechanical processes that accelerate rock degradation, weaken structural integrity, and may ultimately threaten tunnel stability, posing a significant risk to long-term operational safety. In this work, the thermo-hydro-mechanical response of a cylindrical tunnel embedded in saturated soil subjected to thermal loading is investigated. Conventional thermoelastic models may not fully capture the finite-speed heat propagation, memory-dependent thermal transport, and delayed deformation behavior of soils under transient thermal shocks. Therefore, a fractional-order three-phase-lag generalized thermoelastic model incorporating strain relaxation effect is proposed to describe the coupled thermo-hydro-mechanical responses of saturated soils. The governing equations are solved using the Laplace transform method to obtain the corresponding solutions. The influence of fractional-order parameter, thermal relaxation time, strain relaxation time, and the magnitude of thermal loading is systematically examined. Results show that fractional-order and relaxation parameters significantly affect the evolution of temperature, pore water pressure, and displacement fields, while thermal loading mainly amplifies their magnitudes near the tunnel wall without changing the overall distribution patterns. The proposed model provides insight into the coupled heat and mass transfer under transient thermal shocks and offers guidance for mitigating stress concentration and improving tunnel stability.
It turns out that the arsenal of methods used to study non-equilibrium quantum field theory can be used to describe dissipation in classical mechanical systems. This paper aims at properly connecting these two worlds.
We present a numerical method for eigenfrequency optimization in two-dimensional and three-dimensional linearized elasticity, based on necessary optimality conditions where the optimal stiffness tensor saturates the upper Hashin–Shtrikman bound. The approach, grounded in the homogenization method, exploits recently derived explicit Hashin–Shtrikman bounds and corresponding optimal microstructures in three space dimensions for mixtures of two non-void elastic phases, enabling an efficient computational realization of the optimality criteria framework. It is capable of identifying global optimizers in most cases. Benchmark examples, including cantilever and bridge problems, demonstrate that true composite designs are obtained as optimal. A penalization procedure is then applied to recover mostly classical designs, with only a slight decrease in the objective functional. Finally, the proposed method is compared with the solid isotropic material with penalization method and is shown to yield improved performance.
In this paper, based on the C–Y surface energy elasticity theory, the axisymmetric frictional nanocontact problem of a finite-thickness elastic layer bonded to a rigid substrate is investigated. By introducing Love’s strain function and applying the Hankel integral transform, integral-form solutions for the stress and displacement fields of the finite-thickness elastic layer subjected to uniformly distributed pressure and friction are obtained. Numerical results demonstrate that surface effect, finite thickness, and friction jointly influence the nanocontact behavior of the elastic layer. Surface effect significantly suppresses the normal displacement and stress levels inside the contact area, leading to a pronounced hardening trend of the elastic layer. Decreasing the layer thickness enhances the constraint imposed by the rigid substrate, thereby reducing the surface normal displacement, which also stiffens the layer. Meanwhile, an increase in the friction coefficient also significantly suppresses the overall normal displacement. As a result, a generalized nanohardness is introduced to characterize the nanocontact property of the finite-thickness elastic layer; one can see that the nanohardness increases with increasing bulk surface energy density, decreasing layer thickness, and increasing friction coefficient. These results provide a basis for practical frictional nanocontact analysis and nanoindentation characterization of elastic layers.
The degenerate forced Duffing oscillator, characterized by zero linear stiffness and nonzero cubic stiffness, which is commonly known as the Ueda oscillator, has long been regarded as a canonical benchmark in nonlinear dynamics and chaos theory. In his seminal work, Ueda demonstrated the emergence of chaos at a dimensionless unit forcing frequency. In this study, it is shown that chaotic behaviour also arises at ultra-low forcing frequencies. This phenomenon is investigated by constructing basins of attraction in parameter space using the cardinal number measure, and the presence of chaos is further corroborated by bifurcation diagrams and the largest Lyapunov exponent analysis. The observed effect has potential applications in seismic and vibration engineering, as it indicates that long-period ground motions (e.g. near-fault earthquakes) may trigger chaotic responses and elevated acceleration levels in seismically isolated systems; nonlinear energy harvesting, where it opens the possibility of designing broadband nonlinear harvesters capable of efficiently operating under ultra-low-frequency excitation; and other applications, including nonlinear climate dynamics and ecological systems.
This paper proposes the application of the λ-Neumann Monte Carlo methodology to estimate the expected value and variance of the solution to the bending problem of an Euler–Bernoulli beam resting on a stochastic Pasternak foundation. Uncertainty is modeled in the foundation’s stiffness coefficients ( κ W and κ P ) via parameterized stochastic processes. The methodology proposed in this work differs from the usual approach, as it is developed based on theoretical results regarding the existence and uniqueness of the realizations. The consistency of the numerical approximations is ensured through rigorous analysis of the well-posedness of the underlying stochastic boundary value problem. This theoretical result guides the choice of approximation spaces via the Galerkin method, leading to realizations consistent with this result. Consequently, statistically consistent estimators of the expected value and variance of the transverse displacement stochastic processes are obtained. The Monte Carlo simulation method is used to evaluate the performance of the proposed methodology through two numerical examples.
This study investigates the mechanical behavior of oblique pantographic sheets composed of two fiber arrays interconnected by pivots with torsional stiffness. To characterize the macroscopic response of these lattice structures, we employ asymptotic homogenization techniques, introducing a small parameter ϵ defined as the ratio of the unit cell’s characteristic length to the domain’s overall dimension. When the pivot’s rotational stiffness scales as ϵ 2 p for an integer p , we derive different classes of effective first- or second-gradient (strain-gradient) continuum models for p = − 1 (rigid connection), p = 0 and p = 1 (compliant pivots at orders zero and two, respectively), and p = 2 (no contribution). The homogenized constitutive equations are derived in tensor form. These models capture the essential mechanical effects of pivot rotational resistance through shear-strain energy contributions. In each case, the effective elasticity coefficients of the homogenized models are expressed explicitly in terms of the microstructural properties—specifically, the fibers’ mechanical characteristics and the pivots’ torsional stiffness. For second-gradient continua, we investigate the well-posedness of different boundary value problems within the framework of anisotropic Sobolev spaces. We further establish the coercivity of the homogenized strain-energy function, thereby ensuring the uniqueness of the solutions. Quadratic energy expressions at the bisector reveal D 2 symmetry and, for balanced fibers, D 4 symmetry. Finally, we validate our findings with numerical results from elongation-bias tests at different skew angles, comparing the strain-energy contributions—flexural, shear, and elongation—between the continuum and discrete-frame models.
In the advanced high-performance mechanical systems, such as those in aerospace, automotive powertrains, and precision instrumentation, demand components that can simultaneously provide structural support and mitigate harmful vibrations. However, achieving an optimal balance between load-bearing capacity and vibration suppression remains a fundamental challenge, as these two functions often impose conflicting material and structural requirements. To address this inherent trade-off, this study proposes a novel metamaterial that integrates a DNA-inspired double-helix architecture with a spiral configuration. The static mechanical performance and dynamic transmissibility of topological variants, with strut side lengths of 2.0–2.8 mm, are systematically investigated. The results reveal a fundamental trade-off: increasing the strut thickness enhances the specific stiffness but shifts the fundamental resonant frequency from 51.65 Hz to 88.57 Hz, which significantly narrows the low-frequency isolation bandwidth. Experimental validation was conducted on a Selective Laser Sintering fabricated prototype, which confirmed the numerical predictions with a frequency deviation of less than 1.6%. Furthermore, a comparative analysis against a solid block is conducted to introduce and evaluate the concept of mass efficiency. The 2.0-mm topology achieves a substantial weight reduction of approximately 38.91% while achieving a maximum vibration attenuation of 45 dB, whereas the equivalent solid block provides negligible isolation. A Pareto optimization analysis demonstrates that the proposed spiral structure functions as a tunable platform, allowing for the customized balancing of static stability and dynamic filtering efficiency for lightweight applications.
This review provides a structured and comprehensive overview of the current state of fracture mechanics for functionally graded granular (meta)materials. The conceptual framework that situates functionally graded materials (FGMs) within the wider class of metamaterials is first introduced, followed by a classification of relevant microstructural architectures and gradient types (compositional, porous and microstructural). Emphasis is placed on damage and fracture phenomena peculiar to graded systems—including Mode I/II/III, mixed-mode, impact, fatigue, thermal and creep fractures—and on how spatial variations of elastic and fracture properties modulate crack initiation, propagation and arrest. Physical and mathematical modeling approaches are critically examined, spanning classical first-gradient formulations, higher-order strain-gradient theories and micromorphic frameworks, and highlights recent advances in phase-field and multiscale numerical strategies that capture gradient-induced toughening and size effects. Experimental and numerical evidence consistently demonstrates that tailored gradient architectures can markedly enhance damage tolerance, produce rising R-curves and increase impact resistance, whereas gradient orientation and interface smoothness crucially determine crack paths and failure modes. In conclusion, outstanding challenges are identified (rigorous multiscale coupling, standardized in-situ characterization, and design rules compatible with additive manufacturing) and a roadmap is proposed for integrating modeling, fabrication and testing to translate graded metamaterial concepts into reliable structural applications.
This study investigates the planar transient dynamics of a linearized string interacting with a moving mass. Unlike conventional models that idealize the moving system as a point mass or particle, the mass is modeled more realistically as a finite body that maintains dual contact with the string at closely spaced points. This modeling choice naturally incorporates the rotational inertia of the moving mass into the formulation. For a prescribed motion of the moving mass along the string, the coupled governing equations are derived in terms of the Dirac delta function. In the limiting case where the two contact points approach at the mass center, the formulation reduces to an equivalent monopole-dipole representation. The governing equations are discretized using the Galerkin method and integrated in time via a Runge-Kutta scheme. The formulation is first applied to a stretched string subjected to a uniformly moving mass. Particular attention is given to the trajectory paradox reported in earlier studies for moving point masses near a boundary. Using the present two-point contact model, this paradox is examined in terms of trajectory at the front contact, the separation between the actual and virtual centers of mass, and the variation in contact-point spacing induced by abrupt responses near the boundary. Subsequently, a new system, in which a string is vertically hung and carries a uniformly moving mass, is analyzed. Trajectory convergence is demonstrated for both ascending and descending motions. Similar to the trajectory paradox, a jump-like response appears near the upper fixed boundary when the mass ascends at higher velocities. At these velocities, the two-point contact model yields divergent results across the entire domain, indicating limitations in its applicability. This behavior highlights the need for future studies that incorporate nonlinear effects to more accurately capture the observed dynamics.
The fracture initiation likelihood of numerous offset interfacial cracks in a structure that comprises a functionally graded magneto-electro-elastic (FGMEE) layer, which is sandwiched between two distinct functionally graded material (FGM) layers of equal thickness, is examined in this study. One of the offset cracks is located centrally at the interface between the FGMEE strip and one FGM layer, while the other two are positioned symmetrically along the interface of the FGMEE and another FGM layer. The cracks are idealized as impermeable and are under the influence of in-plane magnetic and electric, and anti-plane mechanical loadings. The governing partial differential equations are reduced to a set of first-kind integral equations of singular type with the help of an integral transform approach. Owing to the complexity of the kernel, a numerical solution based on the Chebyshev polynomial is employed. At last, the expressions of stress intensity factor (SIF) are obtained at the crack tips. A comparative assessment is used to validate the findings. The study highlights, through detailed graphical illustration, the consequence of strips' width ratio, non-homogeneity parameter ratio, interfacial crack length ratios, and applied electric-magnetic loadings on the normalized SIFs.
This paper examines the construction of continuous models for the dynamics of lattice meta-materials with long-range interactions. Lagrange lattices with nearest-neighbour and second-neighbour coupling, and with nearest-neighbour and third-nearest-neighbour coupling, are studied. Particular attention is paid to characteristic points on the dispersion curve, such as rotons, maxons, and stationary inflation points. Multipoint Padé approximants are employed for approximation purposes. Selecting the parameters of the fractional–rational function ensures that the dispersion curves of the discrete and continuum systems coincide at the characteristic points. The resulting partial differential equations have a low order in spatial variables. Importantly, these systems provide a good approximation of the discrete systems for any stiffness of coupling.
We apply the techniques of one-to-one mapping and analytic continuation to derive a closed-form solution to the anti-plane elasticity problem of a cracked anisotropic elastic elliptical inhomogeneity embedded in an infinite anisotropic elastic matrix subjected to uniform remote anti-plane shear stresses. The anisotropy of the elliptical inhomogeneity results in the finite crack and the elliptical interface being non-confocal. The mode III stress intensity factor at the crack tip is extracted from the solution, with an emphasis on a vanishingly thin inhomogeneity. When a particular condition on the ratio of the two remote anti-plane stresses is met, the remote loading will not induce any singular stresses at the crack tips and the stress field within the cracked inhomogeneity is uniform. Explicit general solutions for an arbitrary singularity located either in the matrix or the cracked inhomogeneity are derived.
We investigate the propagation of transverse waves and the associated energy transmission in one-dimensional third-gradient continua. Starting from a variational formulation for small transverse displacements, we derived the corresponding dispersion relation. Owing to the contribution of the third-gradient kinematics, such a relation takes the form of a sixth-order polynomial in the wavenumber. Furthermore, we obtained the expression of the energy flux averaged over the period, which enabled the definition of a transmission coefficient for third-gradient continua. In the present study, we examine in detail the dependence of the admissible wavenumbers and of the transmission coefficient on specific third-gradient parameters: the characteristic length of the third-gradient (defined as the ratio between the third-gradient and the second-gradient parameters) and the inertial–stiffness ratio (given by the ratio between the mass density and the second-gradient parameter). In addition, for the transmission coefficient, we analyze its sensitivity to variations of the third-gradient’s characteristic length and its dependence on other constitutive parameters, with particular attention to the role played by the third-gradient contributions compared to second-gradient configurations. This analysis highlights the influence of higher-order microstructural effects on both wave propagation and energy transport, thereby providing further insight into the mechanical response of generalized continua.
This study investigates the coupled thermal and chemical interactions in a nonlocal homogeneous isotropic thermoelastic diffusive thick circular plate subjected to axisymmetric heat supply. Both surfaces of the plate are assumed to be stress free. Employing Laplace and Hankel transform techniques, analytical expressions for the field quantities are obtained in transformed domain. The corresponding solution in the physical domain is obtained through a specially developed algorithm. Numerical simulations are presented graphically to illustrate behavior of components of displacement, stresses, temperature change, mass concentration and material potential. The influence of the nonlocal parameter and diffusion phenomena on various field variables is examined in detail. The results show that the classical (local) model predicts higher mechanical responses, while increasing nonlocal effects significantly reduce displacement and stress magnitudes. These findings demonstrate that nonlocal thermoelastic diffusion theory is essential for accurately modeling size-dependent behavior in advanced materials especially at micro and nano-scales.
Recently, the 40th anniversary of materials exhibiting negative characteristics was celebrated. The pioneering work began in 1984 when Herakovich [1] designed a layered composite with a negative Poisson’s ratio across the layers, followed in 1985 by inverse honeycomb designs [2, 3]. In 1986, Kolpakov and Rakin [4] introduced a layered composite with a negative coefficient of thermal expansion across the layers. Given two major periods of interest in composites with negative properties, one during the 1990s and 2000s, and another ongoing today, we revisit these pioneering layered composite models from a contemporary perspective. This paper demonstrates that although negative coefficients (Poisson’s ratio and thermal expansion coefficient) are rare in natural materials, they can be achieved in layered composites constructed from homogeneous material layers, the simplest composite structure designs. Using illustrative metamaterial models, we provide visual explanations of the micromechanical mechanisms underlying these negative-coefficient phenomena.
In the context of linear thermoelasticity theories, we study the plane deformation of an arbitrarily shaped nano-inclusion embedded in an infinite isotropic matrix under uniform remote in-plane heat flux. The nanoscale thermoelastic effects are modeled using the theories of interface heat conduction, interface thermoelasticity and interface tension. An efficient series-based algorithm is developed to determine the full thermoelastic field in the entire composite system. Numerical examples are presented to validate the feasibility of the present solution and to explore how the interface stretching stiffness and the interface residual tension influence the thermal stress distribution around the vertices of the inclusion for some common inclusion shapes including ellipse, triangle, square, pentagon and hexagon.
