Traction-free Boundary Conditions Research Articles

Hamiltonian formalism for bistable-multilayered plates under non-mechanical stimuli

Bistable multilayered plates have attracted significant attention as morphing and adaptive structures, renowned for their unprecedented exceptional performance. However, accurately predicting both their shape and internal stresses poses formidable challenges due to their inherently geometrically nonlinear nature. For the first time, this paper introduces a Hamiltonian formalism aimed at achieving high-resolution analysis of nonlinear multilayered rectangular plates subjected to non-mechanical stimuli, including hygro-thermo-electro-magneto-elastic responses. A canonical system is strategically developed to compute membrane behaviors, yielding a set of dual differential equations for in-plane field variables. This method elegantly decouples out-of-plane variables from the full-state vector, leading to an analytical solution in the membrane problem via the technique of symplectic eigen-expansion. To predict the bending behaviors, the first variation of the Hamiltonian energy density function, expressed as a power series of the transverse deflection, ensures flexural equilibrium. The power series effectively captures all admissible out-of-plane deformations, enabling a smooth transition between linear and nonlinear plate responses, including bifurcation and limit points. The validity and accuracy of the proposed method are rigorously assessed through convergence tests and satisfaction of traction-free boundary conditions. The complete solution is evaluated across various configurations, ranging from monostability to bistability. It is cross-verified with high-fidelity finite-element methods (FEM), showing excellent agreements in both deformations and stress resultants. This research presents a physics-based methodology that unveils a parametric interplay between in-plane and out-of-plane field variables, curvatures, geometrically nonlinear strains, and inelastic external loadings. The framework serves as an efficient approach for comprehensively analyzing adaptive and morphing structures.

Adaptive deep homogenization theory for periodic heterogeneous materials

We present an adaptive physics-informed deep homogenization neural network (DHN) approach to formulate a full-field micromechanics model for elastic and thermoelastic periodic arrays with different microstructures. The unit cell solution is approximated by fully connected multilayers via minimizing a loss function formulated in terms of the sum of residuals from the stress equilibrium and heat conduction partial differential equations (PDEs), together with interfacial traction-free or adiabatic boundary conditions. In comparison, periodicity boundary conditions are directly satisfied by introducing a network layer with sinusoidal functions. Fully trainable weights are applied on all collocation points, which are simultaneously trained alongside the network weights. Hence, the network automatically assigns higher weights to the collocation points in the vicinity of the interface (particularly challenging regions of the unit cell solution) in the loss function. This compels the neural networks to enhance their performance at these specific points. The accuracy of adaptive DHN is verified against the finite element and the elasticity solution respectively for elliptical and circular cylindrical pores/fibers. The advantage of the adaptive DHN over the original DHN technique is justified by considering locally irregular porous architecture where pore–pore interaction makes training the network particularly slow and hard to optimize.

A robust five-unknowns higher-order deformation theory optimized via machine learning for functionally graded plates

This article presents a new kind of higher-order deformation theory, called Parametric Higher-order Deformation Theory (PHDT), for the static analysis of functionally graded plates (FGPs). The novelty of the PHDT is the use of strain shape functions that are calibrated by a set of tuning parameters to approximate 3D results along the plate thickness. The tuning parameters are assumed to vary with side-to-thickness ratios and power-law indexes. In contrast to higher-order shear deformation theories (HSDTs), the PHDT is not mathematically constrained to satisfy the traction-free boundary condition on the bottom plate’s surface. The proposed plate model is based on a 5-unknown HSDT previously presented by one of the authors. The governing equations are derived from the principle of virtual works, and Navier-type closed form solutions have been obtained for simply supported FGPs subjected to bisinuisoidal transverse pressure. A general methodology that uses genetic algorithms to determine the optimal tuning parameters of PHDTs for FGPs with various side-to-thickness ratios and power-law indexes is presented. The accuracy of the PHDT is assessed by comparing the results of numerical examples with a 3D elasticity solution, HSDTs reported in the literature, and the well-known Carrera Unified Formulation. The results show that quasi-3D displacement and stress distribution are obtained using a set of tuning parameters to form adaptable strain shape functions that are optimized for the given structural problem.

Static, buckling, and free vibration responses of functionally graded carbon nanotube-reinforced composite beams with elastic foundation in non-polynomial framework

In this work, the static, buckling, and free vibration analysis of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beam resting on a Pasternak elastic foundation are studied. The secant function-based shear deformation theory (SFSDT) is used for this analysis. This theory fulfills the traction-free boundary conditions at the top and bottom surfaces of the beam, hence there is no need for a shear correction factor. Hamilton’s principle is used to determine the governing differential equations and boundary conditions whereas Navier’s solution technique is used for determining the closed-form solution. The analytical approach is used to examine the deflection, stresses, critical buckling load, and natural frequencies of the FG-CNTRC beam resting on the Pasternak elastic foundation including a shear layer and Winkler springs. To determine the material characteristics of FG-CNTRC beams, the Rule of the mixture is used. Uniform distribution (UD-beam), FG-X beam, FG-O beam, and FG-V beam are the different forms of CNT reinforcement distribution that are used in this study. Considering different span thickness ratios, the volume fraction and distribution of CNT, the Winkler spring, and the shear layer constant factors, all the structural responses are predicted. It is also observed that the present theory predicts the structural responses of the FG-CNTRC beam accurately when compared to other existing theories. A few new results are also included as the benchmark solutions for the new research.

Micromechanical analyses on bending of polysynthetically twinned single crystal of titanium aluminide

Micromechanical bending analyses via CPFEM of polysynthetically twinned (PST) single crystal of TiAl composing of one α2-phase lamella and six γ–phase lamellae have been performed. The results have demonstrated that a decrease in the domain size aspect ratio λD/λL or an increase in the volume fraction of α2 phase fα2 have an effect of increasing the induced bending moment and consequently bending resistance under both bending modes of hogging and sagging. The similar variations in λD/λL and fα2 results in a decreasing trends of maximum tensile and compressive initial yield stresses (normal bending stresses), obtained through small strain beam theory. Results have also shown the effects of imposed constraints on the bending behavior; traction-free boundary conditions result in activation of prismatic slips instead of pyramidal slips in α2-phase, leading to a lower observed bending moment for the variables λD/λL and fα2 considered. A scaling law is proposed that relates bending moment to domain size aspect ratio and bending angle.

Static and free vibration analysis of delaminated composite plates using secant function based shear deformation theory

In this work, the static and free vibration analysis of delaminated composite plates are studied. The secant function based shear deformation theory (SFSDT), is used for the analysis of multiple delaminated models. This theory exhibits a non-linear shear stress distribution and fulfills the traction-free boundary conditions at the top and bottom surfaces of the plates, hence there is no need for a shear correction factor. The finite element method, a numerical solution-based approach is used here considering an eight-noded isoparametric quadratic element. The investigation explores the effects of delamination size, position and stacking sequence on the flexural and dynamic characteristics of the composite plates. Additionally, other geometric factors such as delamination thickness and shape are investigated to understand their impact. The study aims to identify the optimal configurations that minimize the detrimental effects of delaminations. The results provide valuable insights into the static and free vibration behaviour of delaminated composite plates, contributing to design guidelines and optimization strategies for structurally sound and efficient composite structures. Overall, this research offers a comprehensive investigation of the static and free vibration analysis of delaminated composite plates, advancing knowledge in the field and facilitating improved design practices.

On the Impact of Mechanics on Electrochemistry of Lithium-Ion Battery Anodes

Models exploring electrochemistry-mechanics coupling in liquid electrolyte lithium-ion battery anodes have traditionally incorporated stress impact on thermodynamics, bulk diffusive transport, and fracture, while stress-kinetics coupling is more explored in the context of all solid-state batteries. Here, we showcase the existence of strong link between active particle surface pressure and reaction kinetics affecting performance even in liquid electrolyte systems. Traction-free and immobile particle surface mechanical boundary conditions are used to delineate the varying pressure magnitudes in graphite host during cycling. Both tensile and compressive stresses are generated in traction-free case, while a fixed surface subjects the entire particle to a compression state. Pressure magnitudes are nearly two to three orders of magnitude higher for the latter resulting in significant depression of open circuit potential and improvement of exchange current densities compared to stress-free state. The results demonstrate the need for incorporating stress-kinetics linkage in models and provide a rationale for putting battery electrodes under compression to improve kinetics.

A hyperbolic theory for the analysis of laminated shallow shells with double curvature

In this paper, higher-order closed-form analytical solutions to the static bending and free vibration problems of laminated composite shells with double curvature are obtained using a hyperbolic shear deformation theory. The current theory is a modification of the shape function provided by Soldatos [30] in his well-known hyperbolic theory. The distributions of transverse shear stresses through the thickness of the shell are precisely predicted by the current theory satisfying traction free boundary conditions at the top and the bottom surfaces of the shell. Hamilton's principle serves as the foundation for the development of equations of motion. Navier's method is used for the analysis of simply-supported laminated shells under static and free vibration conditions. Displacements, stresses, and natural frequencies are presented for different shells with double curvature. The results from past investigations are compared to verify the accuracy and efficacy of the present hyperbolic shell theory.

SH Waves in a Half-Space Model of Functionally Gradient Piezoelectric Semiconductors with Initial Stresses

In terms of piezoelectric class (6 mm) crystal semiconductor, the propagation of shear surface waves on the half-space surface under initial stresses is studied. The initial stresses and material parameters are hypothesized to exponentially vary along direction only. Besides, the velocity equations of SH waves can be obtained based on the boundary conditions and the kinematic equations of the initially stressed graded piezoelectric semiconductor material, as well as the traction-free boundary conditions. Based on the numerical results, we explore how the semiconductivity, initial stresses as well as material gradient index impact the dispersive curves.

Finite element study of V-shaped crack-tip fields in a three-dimensional nonlinear strain-limiting elastic body

We study the crack-tip fields in a three-dimensional (3D) plate with a V-shaped crack employing Rajagopal’s theory of elasticity with the strain-limiting effect. The classical linear elastic fracture mechanics (LEFM) theory bears a well-documented inconsistency, i.e., for a static crack problem under the traction-free boundary condition, the LEFM theory predicts singular crack-tip strains. Such results are inconsistent with the constitutive relation derived using the assumption of uniform bound for the strain values. Instead, Rajagopal’s recent theory of elasticity provides implicit constitutive relations between the linearized strain and Cauchy stress in which one can impose an a-priori upper bound for strains. We formulate the strain-limiting theory comprehensively in this article. The linearized strain is expressed as a nonlinear function of the Cauchy stress. The equation for the equilibrium of the linear momentum under a special type of nonlinear constitutive relation reduces to a second-order quasilinear elliptic boundary value problem (BVP). Using a Picard-type linearization and a continuous Galerkin-type finite element technique, we solve several BVPs with tensile and shear displacement loading. The numerical results for the strains, the stresses, the stress intensity factor (SIF), and strain energy density (SED) are obtained for different Poisson values, strain-limiting parametric values, and the V-shaped angles. The growth of the near-tip strains is far less than that of stresses, and the results for both the SIF and SED are also consistent with the results from LEFM. Our work demonstrates that the framework of Rajagopal’s theory of elasticity can provide a basis for developing physically meaningful and mathematically well-posed BVPs to study evolution of cracks, damage, nucleation, or failure in elastic bodies.

Effects of graphene-platelets reinforcement on the free vibration, bending, and buckling of porous functionally-graded metal-ceramic plates

This paper presents a first investigation of the mechanical behavior of porous functionally-graded material (FGM) plates reinforced with graphene platelets (GPls) nanomaterial. As a main purpose, we extend the research on GPls incorporation from single-component matrix materials to porous metal-ceramic FGMs. Four different through-thickness distributions of GPls and porosities were investigated. The effective elastic modulus of GPls-reinforced FGM was calculated using the Halpin-Tsai homogenization model, and the effect of porosities on elastic properties was considered by an exponential model that is more accurate than the common rule of mixture. A recent hyperbolic higher-order plate theory that satisfies the traction-free boundary conditions and a parabolic variation of shear stresses was used to model kinematics. The influence of various parameters was studied after method validation, and it was revealed that GPls not only can improve the stiffness and mechanical response of FGM plates, but can also eliminate the inherent transverse anisotropy of FGMs and uncouple the static response when dispersed in proper distribution pattern. This effect will result in zero mid-plane extension in bending conditions, in a similar behavior to isotropic plates.

Visco-Elastic Interface Effect on the Dynamic Stress of Symmetrical Tunnel Embedded in a Half-Plane Subjected to P Waves

A semi-analytical method is developed to study the visco-elastic interface effect on the dynamic stress of a non-circular symmetrical tunnel in a half-plane subjected to P waves, and the interaction between the visco-elastic interface around the tunnel and the traction free boundary at the half-plane is analyzed. The complex variable function method combined with the wave function expansion method is introduced to obtain the theoretical expressions of waves around the non-circular tunnel. To satisfy the traction-free boundary condition at the half-plane, the large circle assumption is applied. The expanded coefficients are determined by using appropriate boundary conditions with stiffness parameters and viscosity coefficients of the interface. Selected numerical results are presented, and the interface effect on the dynamic stress under different embedded depths is discussed. It is found that the interacting effect of the visco-elastic interface and the traction-free boundary is quite related with the wave frequency. Comparison with existing results is also given to validate the semi-analytical method.

Static and free vibration analysis of sandwich shells with double curvature considering the effects of transverse normal strain

Higher-order closed-form solutions for the static and free vibration problems of sandwich shells with double curvature are presented in the current study based on a new hyperbolic shell theory considering the effects of transverse normal strain. A theory involves six unknowns and satisfies traction-free boundary conditions at the top and the bottom surfaces of the shell. The theory does not require a shear correction factor to account for the strain energy due to the shear deformation effect. A shell consists of three layers, wherein the top and the bottom layers, i.e. face sheets are made up of hard material and the middle layer, i.e. core is made up of soft material. The governing equations and associated boundary conditions of the theory are produced by employing Hamilton’s principle. Semi-analytical closed-form solutions for the static and free vibration problems are produced by the Navier technique for simply supported boundary conditions of the shell. The present results are compared with results that have already been published to confirm the accuracy and efficacy of the current higher-order hyperbolic shell theory.

Effects of transverse normal strain on the deformation of laminated and sandwich arches under the action of concentrated force

It is well-known that the concentrated force develops high-stress concentration at the point of application which is one of the important parameters to be considered while designing the laminated composite structures under the action of concentrated force. The literature available shows that the study on static deformation of laminated sandwich shallow arches under the action of the concentrated force is limited. In the present study, static deformation of laminated sandwich arches under the action of concentrated loading is investigated using higher-order arch theory considering the effects of transverse normal strain. An exponential type higher-order arch theory is developed in this study which satisfies the traction-free boundary conditions at the top and the bottom surfaces of the arch using constitutive relations. Governing equations are derived within the framework of the principle of virtual work. An analytical solution for the static deformation of simply supported laminated and sandwich shallow arches are obtained using Navier’s technique. The effects of the lamination scheme, radius of curvature and aspect ratio on the deflection, and stresses of shallow arches are evaluated. The present results are compared with previously published results wherever possible for the verification of the present theory.

Isogeometric Analysis for Free Vibration of Functionally Graded Plates Using a New Quasi-3D Spectral Displacement Formulation

This paper presents a novel quasi-three-dimensional shear deformation theory called the spectral displacement formulation (SDF) for analyzing the free vibration of functionally graded plates. The SDF expresses the unknown displacement field as a unique form of the Chebyshev series in the thickness direction. By increasing the truncation number in the Chebyshev series, the bending analysis results can approach the three-dimensional elasticity solution and satisfy the traction-free boundary conditions without requiring a shear correction factor. The SDF is an extension of the classical plate theory, thereby naturally avoiding the shear-locking phenomenon. These characteristics enable the SDF to apply to plates of arbitrary thickness while maintaining accuracy. The nonuniform rational B-spline-based isogeometric approach is employed to enhance the applicability of this theory to free vibration analysis of functionally graded plates with complex geometries and different boundary conditions. Numerical examples are presented to demonstrate the accuracy and reliability of the proposed method in analyzing the free vibration of functionally graded plates.

Analysis of higher modes of Rayleigh-type waves in fiber-piezo thermoelastic composite structure

The study presents a theoretical modeling and analysis for the purpose of designing of acoustic devices and sensors with high performance. The propagation of Rayleigh-type waves in a piezoelectric substrate guided with fiber-reinforced layer is studied. Lord-Shulman (L-S) theory is used for thermoelasticity for both layer and substrate. Direct Sturm-Liouville method is employed to obtain the dispersion relation using traction free, thermally insulated, electrically open and short boundary conditions. The effect of different material parameters (e.g., reinforce, thickness ratio, and amplitude of thermal displacement) on the velocity (phase and group) of the considered wave are shown graphically by taking suitable numerical examples. The effect of specific loss factor is presented through graph. The present study may be utilized for better optimization of surface acoustic wave devices and sensors.

High-Precision Isogeometric Static Bending Analysis of Functionally Graded Plates Using a New Quasi-3D Spectral Displacement Formulation

A new quasi-three-dimensional (3D) shear deformation theory, called the spectral displacement formulation (SDF), is proposed for high-precision static bending analyses of functionally graded plates. The main idea is to expand unknown displacement fields into Chebyshev series of a unique form in the thickness direction; the truncation numbers are set to be adjustable to meet various application requirements. Specifically, 3D elasticity solutions and traction-free boundary conditions can be approached by increasing the number of Chebyshev bases. The SDF is also an extension of the classical plate theory and naturally avoids the shear locking problem, making it versatile for functionally graded material (FGM) plates of arbitrary thicknesses. The C1 continuity requirement for the discretization of the generalized displacements is conveniently fulfilled by the nonuniform rational B-splines (NURBS)-based isogeometric method. Numerical examples demonstrate the excellent performance of the proposed method for the displacement and stress analyses of functionally graded plates. The high precision and versatility of the present method have manifested its great potential applications in strain-based or stress-based reliability analysis, optimization design, fatigue analysis, and fracture analysis of FGM plates, and other related fields.

Hygro-thermo-mechanical analysis of sandwich shallow shells considering the effects of transverse normal strain

The static response of softcore doubly-curved sandwich shallow shells under hygro-thermo-mechanical loading is presented herein using a fifth-order shell theory which considers the effects of transverse normal strains and stresses at the same time. The present theory accounts for non-linear variations of displacements. The strain-displacement relations take into account only small curvatures following a shallow shell description. The present theory ensured the traction-free boundary conditions at the top and the bottom faces of the shell. The governing equations along with the natural and essential boundary conditions are derived using the principle of virtual work. The governing equations are solved by using Navier’s method. Three layered sandwich shallow shells of double curvature subjected to hygro-thermo-mechanical loading are analyzed. The accuracy and efficacy of the present theory are proved by applying it to the benchmark solutions available in the literature and then the present theory is extended for the analysis of softcore sandwich shells of double curvature. Since an exact 3D elasticity solution is not available for many such problems; this paper can be serving as a benchmark reference solution for future researchers working in this area.

Effects of Transverse Normal Stress on Hygrothermomechanical Analysis of Laminated Shallow Shells

Hygrothermomechanical analysis of doubly curved laminated composite shallow shells is presented in this paper using a two-dimensional higher-order shell theory considering the effects of transverse normal stress. The present theory accounts for the fifth-order expansion of the thickness coordinates in the in-plane displacements and the fourth-order expansion in the transverse displacement. The main focus of this work is to study the effects of both transverse normal stress and the fifth-order expansion of thickness coordinates on the hygrothermomechanical responses of doubly curved laminated shallow shells. The governing equations of the theory are derived based on the principle of virtual work and solved by using Navier’s method. The theory ensures traction-free boundary conditions using constitutive relations and does not require a shear correction factor. Interlaminar transverse shear stresses are recovered using the three-dimensional equilibrium equations of the theory of elasticity to ascertain the stress continuity at the layer interface. Three-dimensional elasticity solutions are taken as a basis for comparing the present results in thermal problems, whereas this study contributes some useful results to the hygrothermomechanical analysis of doubly curved laminated composite shallow shells.

Finite element study of V-shaped crack-tip fields in a three-dimensional nonlinear strain-limiting elastic body

We study the crack-tip fields in a three-dimensional (3D) plate with a V-shaped crack employing Rajagopal’s theory of elasticity with the strain-limiting effect. The classical linear elastic fracture mechanics (LEFM) theory bears a well-documented inconsistency, i.e., for a static crack problem under the traction-free boundary condition, the LEFM theory predicts singular crack-tip strains. Such results are inconsistent with the constitutive relation derived using the assumption of uniform bound for the strain values. Instead, Rajagopal’s recent theory of elasticity provides implicit constitutive relations between the linearized strain and Cauchy stress in which one can impose an a-priori upper bound for strains. We formulate the strain-limiting theory comprehensively in this article. The linearized strain is expressed as a nonlinear function of the Cauchy stress. The equation for the equilibrium of the linear momentum under a special type of nonlinear constitutive relation reduces to a second-order quasilinear elliptic boundary value problem (BVP). Using a Picard-type linearization and a continuous Galerkin-type finite element technique, we solve several BVPs with tensile and shear displacement loading. The numerical results for the strains, the stresses, the stress intensity factor (SIF), and strain energy density (SED) are obtained for different Poisson values, strain-limiting parametric values, and the V-shaped angles. The growth of the near-tip strains is far less than that of stresses, and the results for both the SIF and SED are also consistent with the results from LEFM. Our work demonstrates that the framework of Rajagopal’s theory of elasticity can provide a basis for developing physically meaningful and mathematically well-posed BVPs to study evolution of cracks, damage, nucleation, or failure in elastic bodies.