Non-linear mechanics of elastic Bernoulli-Euler curved beams is investigated within 4-D spacetime geometric framework. Large displacements of such structures are addressed by novel rate elasticity methodology, avoiding finite deformation approaches and linearization of nonlinear kinematic relations common in literature. The variational rate virtual power principle (RVPP) is formulated and ensuing notion of effective stress rate leveraged. A rate form of nonlinear elasticity for planar curved beams is consequently established, and application of the RVPP yields the consistent elastic stiffness operator governing the structural evolution problem. Straightforward and computationally efficient incremental procedure is implemented in the brand new automatic program named REBUV. An enhanced version of the Riks-Crisfield arc-length method is developed by exploiting the derived expression of the stiffness operator and is successfully applied to the investigated highly nonlinear benchmark problems. Comparison with analytical finite deformation approaches and established computational methods is carried out, supporting effectiveness of the presented approach. This paper provides significant extension to curved beams of a methodology proved capable of solving problems and limitations affecting conventional analytical and computational approaches.
An internal thermo-elasticity is introduced in the continualized model for mass-in-mass metamaterial in the continualized equation for attached masses. This thermo-elasticity alters the strain wave behavior more significantly than that introduced previously in the continualized equation for the masses of the main chain. The main result is that the band gap is not described by the dispersion analysis. Immediate disappearance of the band gap at non-zero internal thermo-elasticity is not observed in numerical simulations of the boundary harmonic excitation of the waves. It is obtained an approximated dispersion equation for small internal thermo-elasticity which accounts for gradual disappearance of the band gap. Another result concerns arising of two-mode wave due to internal thermo-elasticity for excitation frequencies above the band gap. It is shown, that this analytical solution is realized in numerical boundary excitation of the strain waves.
The objects of consideration are thin, linearly elastic, Kirchhoff–Love-type, open, circular cylindrical shells having a functionally (transversally) graded macrostructure and a tolerance-periodic microstructure in the circumferential direction. At the same time, the shells have constant geometrical, elastic and inertial properties in the axial direction. The aim of this contribution is to formulate and discuss a new, mathematical, averaged, non-asymptotic model for the analysis of selected dynamic problems for such shells. This, so-called, general (or extended) tolerance model will be derived by applying a certain extended version of the well-known tolerance modelling technique. The starting equations are the well-known governing equations of linear Kirchhoff–Love theory of thin, elastic cylindrical shells. For the functionally graded shells under consideration, the starting equations have highly oscillating, non-continuous and tolerance-periodic coefficients in the circumferential direction, whereas equations of the proposed model have continuous and slowly changing coefficients in this direction. Moreover, some of these coefficients depend on a microstructure size. It means that the proposed model makes it possible to study the effect of a microstructure size on the shell dynamics (the length-scale effect). The main similarities and differences between the general tolerance model derived here and the corresponding, known, standard tolerance model derived by means of the classical tolerance modelling technique will be discussed. It will be shown that in the framework of both the general and standard tolerance models not only the fundamental cell-independent, but also the new, additional, cell-dependent free vibration frequencies can be derived and analysed.
A framework for the dimensional reduction of transient thermo-structural problems is presented in which the cross-sectional heat diffusion is encoded as an autonomous evolution system for the thermal stress resultants. A through-the-thickness temperature ansatz satisfying the essential boundary conditions is postulated and the heat equation is projected via a weighted-residual Galerkin method, yielding coupled first-order evolution equations for the thermally induced membrane force and bending moment. The resulting 1D model requires no resolved cross-sectional temperature computation at runtime, yet preserves the diffusive time scales of the parent 2D problem; the structural response follows from Euler–Bernoulli beam theory driven by the evolved thermal resultants. The projection procedure is methodologically general; its accuracy is demonstrated here for a homogeneous rectangular beam under mixed Dirichlet–Neumann thermal conditions, for which an exact 2D Fourier series solution is derived independently. A three-way numerical validation against the exact solution and a high-fidelity 2D finite element reference confirms that the reduced 1D model reproduces deformationfields to within a few percent, although the reconstructed cross-sectional stress underestimates the peak thermal eigenstress by approximately 43
Periodic concrete-rubber layered foundations can act as mechanical filters that open low-frequency band gaps and thus provide seismic and vibration isolation. Their engineering performance, however, depends on how sensitive these band gaps are to unavoidable material variability. This study presents a tolerance analysis of a one-dimensional transfer-matrix model of a periodic two-layer cell under longitudinal excitation. Using the Floquet-Bloch approach, the boundaries of the first two band gaps and the frequency of maximum spatial attenuation were identified. A parametric sweep with ± 20
In this paper, dynamic crack growth in auxetic and non-auxetic cracked honeycomb structures subjected to thermal shock is investigated. The honeycomb core is modeled as an equivalent homogeneous orthotropic medium based on its effective thermoelastic properties. Auxetic and non-auxetic configurations are generated by varying the interior cell angle, and their dynamic fracture responses are examined for different crack and specimen geometries. The coupled thermoelastic governing equations are solved using the extended finite element method and the Newmark time-integration scheme. Mixed-mode stress intensity factors are evaluated by the interaction integral method, and the crack-growth direction and speed are predicted using an anisotropic crack-growth criterion. Two representative cracked honeycomb structures are studied: a rectangular honeycomb plate with an edge crack and a honeycomb annular disc with a curved crack. The results show that the honeycomb cell angle has a significant influence on the transient temperature field, mixed-mode SIFs, crack-tip temperature, crack-growth speed, and final crack trajectory. Non-auxetic honeycombs generally exhibit larger crack-driving forces, higher Mode I SIFs, faster crack propagation, and greater crack-path deviation. In contrast, auxetic honeycombs reduce the local crack-opening tendency through geometry-induced stress redistribution, resulting in lower crack-growth speeds and improved resistance to thermally induced fracture. These findings highlight the importance of auxeticity and cell-level geometry in controlling dynamic thermal-fracture behavior in honeycomb structures.
Nematic liquid crystal elastomers (LCEs) are well-known for their actuation properties and are thus very suitable for the construction of artificial muscles. The distinguishing features of nematic LCEs are possible due to their molecular configuration and the orientation of the mesogens. The simulation of LCEs is very challenging, since a very large amount of elements is usually needed and also several variables might be necessary for mimicking their unusual behavior. Improving the numerical methods for dealing with the simulation of nematic LCEs is essential. For this purpose, we describe and implement a multilevel numerical procedure for modeling the reorientation of the mesogens. The algorithm is implemented in the dynamic numerical framework with distinct drilling degrees of freedom for the rotations of the mesogens and the bulk elastomer. In each time step, the drilling degrees of freedom are obtained from an elementwise system of equations. In this regard a new invariant free energy density is introduced for the sake of preventing the singularity of the tangent. Afterwards, the rotations at the local nodes are included in the global system of equations by static condensation. We compare results obtained by the multilevel approach with the ones from the monolithic procedure for the case of a virtual experiment, in which Dirichlet and Neumann boundary conditions on the orientation mapping are applied.
This tribute is presented in recognition of Professor Mark Kachanov’s 80th birthday. A leading authority in the mechanics of materials, Professor Kachanov has made substantial contributions to the development of micromechanics, the theory of effective properties, contact mechanics, the electromechanics of piezoelectric materials, and cross‑property relations. The article outlines key milestones of his scientific career and highlights the enduring impact of his research.
This paper presents a finite element model developed to analyze the role of cubic elastic anisotropy in the strain energy of face-centered cubic (FCC) metal decahedral nanoparticles. The particle can be viewed as an assembly of five single-crystalline tetrahedral domains containing a central wedge disclination. Anisotropic elastic properties are explicitly incorporated by aligning the crystal axes of each domain with the 〈100〉-type crystallographic directions of the cubic lattice. Residual stresses are simulated using a disclination approach, which involves the kinematic closure of the angular gap between tetrahedra followed by the activation of bonded contact along the interfaces. Parametric finite element simulations are performed for a range of FCC metals with the anisotropy ratio A varying from 1.21 (Al) to 3.84 (Pb). The results reveal that for a given value of the Poisson’s ratio, the stored strain energy tends to decrease as A increases. For the isotropic case (A = 1), the finite element solution agrees within ∼ 5
Dissimilar welds used in high-temperature power plant components develop complex residual stresses, and the usefulness of post-weld heat treatment (PWHT) in relieving these stresses remains insufficiently understood. This study investigates the influence of PWHT on the relaxation of residual stress in a Grade 91–Inconel 82–316LN weldment using a combined experimental and numerical approach. The generalised plane strain (GPS) approach is employed for the numerical modelling. Through-thickness residual stresses are quantified using the deep hole drilling (DHD) and compared with finite element simulations incorporating elastic–plastic–creep behaviour. Results show that PWHT leads to partial stress relaxation, with compressive stresses developing in the Grade 91 region while tensile stresses remain in the Inconel 82 and 316LN regions. A stress reduction of up to 200 MPa is observed in the weld metal, whereas the 316LN side exhibits only marginal change. In the Inconel 82 region, both DHD measurements and GPS predictions indicate a similar stress trend, with an approximate reduction of 100 MPa after PWHT. Overall, the results demonstrate that PWHT leads to only partial stress relaxation in dissimilar metal welds, and that appropriate representation of out-of-plane constraint through the GPS formulation is essential for accurately predicting residual stress evolution. These findings underline the importance of incorporating reliable residual stress assessment in the life evaluation of dissimilar metal welds operating under high-temperature service conditions.
This paper develops a distributed-order fractional photo-thermoelastic diffusion model for semiconductor plates subjected to ultrafast laser heating with variable thermal conductivity. The present formulation introduces continuous weight functions into both phase lags of the dual-phase-lag heat conduction law, so that each fractional derivative is integrated over a distribution of orders rather than evaluated at a single fixed order. This construction encodes the full spectrum of thermal relaxation mechanisms from ultrafast carrier–lattice energy exchange to slow moisture-assisted redistribution within a single unified framework that contains all classical, single-fractional, and dual-fractional thermoelastic theories as recoverable limiting cases. The generalized heat equation is coupled with carrier diffusion–recombination, moisture transport, and linear thermoelasticity, and the variable thermal conductivity is incorporated through the Kirchhoff transformation. Five physically motivated weight functions are investigated: Dirac delta (recovering the dual-fractional model as a special case), Gamma-type, Uniform, Gaussian, and Bi-modal distributions. The resulting system of distributed-order fractional integro-differential equations is discretized using Gauss–Legendre quadrature for the order-domain integrals, the L1/L1-2 formulas for the Caputo fractional derivatives, selected according to whether the quadrature-node order lies below or above unity, and second-order central finite differences in space; a rigorous convergence analysis establishes the overall truncation error as O(h_α ^2G + Δ t^2-α _max + h^2) , which is confirmed numerically. Parametric studies on a silicon plate reveal three principal effects of the distributed-order formulation: (i) a memory spectrum smoothing mechanism in which the superposition of multiple relaxation scales attenuates the thermoelastic wave amplitude by 50– 60% relative to the dual-fractional reference; (ii) a phase modification effect that elongates the oscillation wavelength of the displacement and stress fields due to dispersive spreading across the memory spectrum; and (iii) a non-trivial interaction between variable thermal conductivity and the memory spectrum, whereby the continuous distribution partially masks the VTC-induced spatial redistribution of thermal energy. These findings demonstrate that distributed-order fractional models capture qualitatively different physics from their fixed-order counterparts and provide a more faithful description of multiscale thermal memory in laser-excited semiconductors.
A coupled problem “chemistry–mechanics–diffusion” is discussed in an elastic–viscoelastic framework. The competition between chemical and mechanical influences on the chemical reaction is demonstrated and studied. A chemical reaction between an elastic solid constituent and a diffusing constituent is considered. As a result of the reaction, the elastic constituent transforms into a viscoelastic one, which is described by the standard linear solid model (SLSM). The reaction is localized at the reaction front and is accompanied by volume expansion which, in turn, leads to stresses affecting the velocity of the front propagation. Reaction front kinetics is described based on the notion of a chemical affinity tensor. A plane strain problem with a planar reaction front is considered. The system of equations, describing the joint processes of the propagation of the reaction front and the relaxation of stresses generated by the transformation and external strains is derived. The restrictions on stresses at which the front can propagate are found. To cover the range of changes in the viscosity coefficient, the elastic and relaxation limits are introduced for the SLSM. The various regimes of the front propagation and stress relaxation are distinguished. Special attention is paid to the discussion of impossibility of unblocking the blocked front during further relaxation of stress. Finally, a quasi-equilibrium mode of the front propagation is considered. A characteristic number similar to the Damköhler number is introduced to compare the effect of the rate of the diffusion and the reaction rate on the propagation velocity of the front. In general, the paper demonstrates a variety of coupled behaviors of the reaction front and relaxing stresses, depending on the elastic parameters of solid constituents, the viscosity coefficient of the transformed material, the chemical energies of the constituents of the reaction and external strains.
The effect of spatial distribution of randomly-placed particles in a representative composite volume on the thermoelastic effective properties and local stress and strain distribution is analyzed. Quantitative assessment is performed using both the full-field finite element analyses and the mean-field interaction model, known also as a “cluster” model. The latter model is developed in the multi-family setting enabling one to study the mean stress and strain separately for each inclusion in the representative unit cell. The particles are assumed to be spherical and of equal size, while considered examples differ by the volume fraction of inclusions and the particle spatial layout, and in particular the mean nearest-neighbour distances.
Heat and mass transfer in semi-circular cavities remains a central focus in thermal engineering research. The present study investigates transport phenomena in a porous semi-circular enclosure filled with Boger fluid, with particular emphasis on the role of nanolayer thermal conductivity. A centrally embedded heated step is introduced as a thermal barrier to capture the influence of localized heating on flow dynamics. The governing partial differential equations are nondimensionalized using similarity transformations and solved numerically through the finite element method (FEM). The analysis demonstrates that key parameters–including solvent fraction, nanolayer thickness, time relaxation ratio, particle size, Darcy number, buoyancy ratio, Rayleigh number, Brownian diffusion, thermophoresis, Eckert number, and Schmidt number strongly affect flow circulation, thermal and concentration fields, and local Nusselt number variation. It is observed that changes in the solvent fraction and time relaxation ratio indirectly influence material dispersion and transport within the fluid. Increasing nanolayer thickness enhances the heat transfer rate in both the temperature profile and the local Nusselt number, whereas larger particle radius exhibit the opposite effect. Higher Brownian motion and thermophoresis parameters intensify the flow field, temperature, and concentration distributions, but reduce the local Nusselt number. Moreover, larger time variations yield stronger flow circulation, elevated temperature fields, and enhanced concentration distributions compared to smaller time values. The centrally heated step-shaped obstacle is shown to provide significantly more effective heat and mass transfer relative to the cold and adiabatic cases.
In this study an analytical study of thermoelastic damping in piezothermoelastic fiber-reinforced composite (PTFRC) Kirchhoff plate resonators considering size dependency is carried out through generalized heat conduction. Effective characteristics of the composite medium are computed based on a combination of rule of mixture (RM) and strength of materials (SM) approach; thereby the coupling of the mechanical-electromagnetic and thermal characteristics of the PTFRC can be realistically modeled. Modified couple stress (MCS) theory and dual phase lag (DPL) conduction are employed in formulating the governing equations. Based on numerical calculations, the influence of various factors like fiber volume fraction, phase lags, and size dependency on the damping effect and frequency shift of the composite can clearly be observed. The potential to optimize the dynamics behavior of the composite through careful material design can be inferred from the current findings. The proposed approach offers a strong base for application oriented studies in multifunctional composites and may prove to be useful in developing smart sensors, micro/nano electromechanical systems (MEMS/NEMS) resonators, energy harvesters, and other adaptive structures.
The paper demonstrates the ability of beam lattices to reduce the amplitude of a wave generated by the external high-pressure impulse. Therefore such lattices are good candidates for the design of protective layers. The suggested lattice design is based on the comprehensive study of basic phenomena in a mass-spring chain with partly breakable springs. The protection is provided by a specific lattice layout architecture with sacrificial beam elements. The distributed damage caused by their failure weakens the lattice material. However, it remains intact, and the pressure applied to one boundary of the layer diminishes at the opposite boundary. The maximum amplitude is defined by the interaction between the waves of three different types: the leading wave propagating in the undamaged lattice, the phase transition wave following the leading wave, and the unloading wave propagating in the damaged lattice toward the phase transition front. Two phase transition zones may be generated, and their influence on the protection effect is examined.
This study investigates the transient thermomechanical response of a solid circular cylinder containing microscopic voids when exposed to two simultaneous thermal excitations: a gradually increasing ramp-type thermal shock applied at the outer surface, and an internal pulsed heat source whose intensity decays exponentially with radial penetration into the material. The cylinder is assumed rigidly fixed at its outer boundary, while the void field obeys a mixed surface condition that allows partial venting of pore pressure. The theoretical framework uniquely combines three advanced physical concepts: spatio-temporal nonlocality of the Klein–Gordon type, a higher-order three-phase-lag heat conduction model capturing finite thermal wave speeds, and full coupling between temperature, displacement, and void dynamics. The governing equations are solved using a hybrid analytical–numerical method. Laplace transformation reduces the system to ordinary differential equations, whose solutions are expressed in terms of modified Bessel functions. The unknown coefficients are determined by applying the transformed boundary conditions: a ramped temperature rise at the surface, zero radial displacement, and the mixed void condition. Numerical inversion of Laplace transforms yields time-domain results. Key findings show that increasing the ramp rise time significantly reduces peak temperature and stress levels, avoiding unphysical singularities. A higher attenuation coefficient confines heating near the surface, protecting the cylinder interior. An impermeable surface suppresses void expansion near the boundary, altering local stress distributions. Simultaneous activation of both spatial and temporal nonlocality reduces hoop stress by up to thirty-five percent, providing intrinsic damping. This work offers direct engineering guidelines for designing lightweight porous components in aerospace, nuclear, and biomedical applications under severe thermal shocks with mechanical constraints.
This work examines coupled thermoelastic responses in a fiber-reinforced solid containing a spherical cavity, formulated within the Green–Naghdi type III (GNIII) theory. The basic equations are solved numerically using the finite element method with quadratic interpolation for the field variables. The cavity surface is subjected to an exponentially decaying pulsed heat flux while remaining mechanically constrained. Graphical results are used to assess how fiber reinforcement and the characteristics of the thermal pulse modify the resulting thermal and mechanical fields. A comparison between reinforced and homogeneous (unreinforced) configurations clarifies the role of reinforcement in reshaping temperature, displacement, and stress distributions. The study offers insights that can inform the design and performance optimization of fiber-reinforced media for thermoelastic engineering applications.
Classical Fourier heat conduction predicts infinite thermal wave speed and fails to capture memory effects observed in ultrafast and microscale thermal processes. To address these challenges, this study develops a unified hierarchical thermoelastic framework based on the Modified Moore–Gibson–Thompson (MMGTn) model. The formulation systematically extends MMGT1 to arbitrary order n, enabling multi-stage damping and enhanced control over thermal wave propagation and memory depth. Structural analogies with electrical and mechanical systems are employed to provide physical insight and parameter interpretation. Furthermore, fractional time derivatives are incorporated to capture anomalous heat conduction effects. The coupled thermoelastic response of a semi-infinite strip under thermal shock is analyzed using Laplace transform techniques and numerical inversion. Comparisons with CV, GN, and fractional models show that MMGTn achieves better control of thermal wave speed, stress concentration, and memory effects. This unified framework offers a robust and interpretable tool for multiscale thermal analysis, with applications ranging from bio-thermoelastic modeling to laser treatment, RF ablation, and advanced material design.
Vibrations play an extremely important role in most industrial branches, in transport, civil engineering, in multibody systems. Damping is a property that appears in these systems and determines, in a decisive way, good functioning. In this context, a proper modeling of the systems, which then provides a description of the phenomena that appear and the possibility of theoretical studies in the design phase is necessary in most engineering applications. Classical analysis methods for the study of damping use in the first instance viscoelastic models, which, however, provide approximate results, valid only under certain conditions. The Rayleigh damping model is more suitable but also presents quite limitations and sometimes provides results far from reality. The best method for systems with several degrees of freedom is the Caughey factorization which meets a series of advantages and proves useful in applications. The least squares approach is suggested in the paper as a way to build the damping matrix approximation. Thus, the benefits of modal decoupling can be applied to real-world computations.
