General Elasticity Theory Research Articles

Application of multi-optimizer neural network in predicting the time-variant response of the dynamic systems

Due to high utility of the sandwich structures in the vast spectrum of applications, this research focuses on the practical solution to protect them from damaging consequences of thermal shocks through parametric assessment of its time-variant thermoelastic performance. To this end, current research proposes a novel approach by taking the advantage of coupling the exactitude of the analytical solutions with the agility of the machine-learning methods to predict the accurate results by lower computational efforts. Case study of this research is a sandwich sector plate with functionally-graded carbon-nanotubes reinforced (FG-CNTR) nanocomposite face-layers exposed to the thermal shock loading. As another novelty, this study considers a radial-variant elastic substrate as a mechanism of resistance against the destructive consequences of thermal shock loading. As all machine-learning based regressors require the set of datapoints, this study employs harmonic format of the differential quadrature approach (HDQA) to find the system’s response toward thermal shock in the background of general elasticity theory at definite design- points. Laplace transform integrated with the extended model of Dubner and Abate’s scheme to translate the time-variant response of the system from time to Laplace domain and vice versa. Validity of the solution used in this survey is confirmed through comparing its outcomes with those of the published articles. As a valuable finding, it is strongly suggested to enhance the system’s thermal shock resistance by using the substrates which their reaction’s dependency to the radial coordinate is expressed by higher-order polynomial estimator functions.

Wave Propagation in Timoshenko–Ehrenfest Nanobeam: A Mixture Unified Gradient Theory

Abstract A size-dependent elasticity theory, founded on variationally consistent formulations, is developed to analyze the wave propagation in nanosized beams. The mixture unified gradient theory of elasticity, integrating the stress gradient theory, the strain gradient model, and the traditional elasticity theory, is invoked to realize the size effects at the ultra-small scale. Compatible with the kinematics of the Timoshenko–Ehrenfest beam, a stationary variational framework is established. The boundary-value problem of dynamic equilibrium along with the constitutive model is appropriately integrated into a single function. Various generalized elasticity theories of gradient type are restored as particular cases of the developed mixture unified gradient theory. The flexural wave propagation is formulated within the context of the introduced size-dependent elasticity theory and the propagation characteristics of flexural waves are analytically addressed. The phase velocity of propagating waves in carbon nanotubes (CNTs) is inversely reconstructed and compared with the numerical simulation results. A viable approach to inversely determine the characteristic length-scale parameters associated with the generalized continuum theory is proposed. A comprehensive numerical study is performed to demonstrate the wave dispersion features in a Timoshenko–Ehrenfest nanobeam. Based on the presented wave propagation response and ensuing numerical illustrations, the original benchmark for numerical analysis is detected.

On the dynamic response of bi-directional functionally graded nanobeams under moving harmonic load accounting for surface effect

This paper presents an investigation of the dynamic behavior of bi-directionally functionally graded (BDFG) micro/nanobeams excited by a moving harmonic load. The formulation is established in the context of the surface elasticity theory and the modified couple stress theory to incorporate the effects of surface energy and microstructure, respectively. Based on the generalized elasticity theory and the parabolic shear deformation beam theory, the nonclassical governing equations of the problem are obtained using Lagrange’s equation accounting for the physical neutral plane concept. The material properties of the beam smoothly change along both the axial and thickness directions according to power-law distribution, accounting for the gradation of the material length scale parameter and the surface parameters, i.e., residual surface stress, two surface elastic constants, and surface mass density. Using trigonometric Ritz method (TRM), the trial functions denoting transverse, axial deflections, and rotation of the cross sections of the beam are expressed in sinusoidal form. Then, with the aid of Lagrange’s equation, the system of equations of motion are derived. Finally, Newmark method is employed to find the dynamic responses of BDFG subjected to a moving harmonic load. To validate the present formulation and solution method, some comparisons of the obtained fundamental frequency and dynamic response with those available in the literature are performed. A parametric study is performed to extensively explore the impact of the key parameters such as the gradient indices in both directions, moving speed, and excitation frequency of the acting load on the dynamic response of BDFG nanobeams. The obtained results can serve as a guideline for assessing the multi-functional and optimal design of micro/nanobeams acted upon by a moving load.

Use of Higher Order Plate Theory in dynamic analysis of SSFS and CSFS thick rectangular plates in orthogonal polynomials


 
 
 This study applied polynomial expressions as displacement and shear deformation functions in the free-vibration study of thick and moderately thick isotropic rectangular plates. Rectangular plates with two different edge conditions investigated in this work are: one with simple supports at three of its edges and with no support at the other edge denoted with the acronym (SSFS) and a rectangular plate with simple supports at opposite edges while the other opposite edges has a fixed support at one edge and no support at the other edge, this is denoted with the acronym (CSFS). The total potential energy of the plate was derived using the general theory of elasticity. The general governing equation of the plate was derived by minimizing the total potential energy equation of the plate. Edge conditions of the SSFS and CSFS plates were met and substituted into the general governing equation to obtain a linear expression which was solved to generate fundamental natural frequency function for the plates with various span-depth proportion (m/t) and planar dimensions proportion (n/m). The results obtained from this research were found to agree favourably with the results of similar problems in the literature upon comparison.
 
 

A mixed variational framework for higher-order unified gradient elasticity

The higher-order unified gradient elasticity theory is conceived in a mixed variational framework based on suitable functional space of kinetic test fields. The intrinsic form of the differential and boundary conditions of equilibrium along with the constitutive laws is consistently established. Various forms of the gradient elasticity theory, in the sense of stress or strain gradient models, can be retrieved as particular cases of the introduced generalized elasticity theory. The proposed stationary variational principle can effectively realize the nanoscopic structural effects while being exempt of restrictions typical of the nonlocal gradient elasticity model. The well-posed generalized gradient elasticity theory is invoked to study the mechanics of torsion and the torsional behavior of elastic nano-bars is analytically examined. The closed-form analytical formulae of the size-dependent shear modulus of nano-sized bar is determined and efficiently applied to reconstruct the shear modulus of SWCNTs with dissimilar chirality in comparison with the numerical simulation data. A practical approach to calibrate the characteristic lengths associated with the higher-order unified gradient elasticity theory is introduced. Numerical results associated with the torsion of higher-order unified gradient elastic bars are demonstrated and compared with the counterpart size-dependent elasticity theories. The conceived generalized gradient elasticity theory can beneficially characterize the nanoscopic response of advanced nano-materials.

Mathematical model of brush seals for gas turbine engines: A nonlinear analytical solution

Presented herein is a mathematical model employing differential equations formulation for brush seals used in gas turbine engines. These components are used to seal the bearing chamber from the environment and reduce the loss of lubricant in the atmosphere, ensuring a MTBR long enough to have required the change the seals only during the engine overhaul operation. The model assumes a single curved bristle loop in the form of a curved-bridge beam subjected to the influences of complex external loads (static and dynamic). Further, a model for clustered bristles is proposed. Specifically, the static forces acting on the curved-bridge beam include the weight of the oil capillary attached to the beam, the weight of the beam itself, the capillary force developed between the surfaces of the bristles in the brush and the temperature gradient. The dynamic forces include the leakage oil pressure and the rotation of the shaft. This complex loading induces a nonlinear large deflection on the curved-bridge beam. Also, the temperature gradient present on the bristles during the gas turbine engine operation generates a change in the geometry of the beam and in the magnitude of the forces acting on the bristles modeled as beams. In the present model, the weights are assumed as uniformly distributed forces on the surface of the beam while the capillary forces and the force generated by the rotating shaft are considered to be non-uniform. The equation expressing the curvature of the beam under general loading force is developed and one can choose the appropriate method of solving the generated differential equation after the expression of the general force is defined. Hence, the ordinary differential equation describing the nonlinear large deflection of the curved-bridge beam will be derived using general nonlinear elasticity theory.

Об осесимметричном нагружении пластины на локальной опоре в рамках теории Родионовой – Титаева – Черных

The problem of an axisymmetric bending of a plate with constant thickness under its own weight is considered. The plate has a circular support. The problem is solved in an axisymmetric statement using a nonclassical shell theory of Rodionova-Titaev-Chernykh (RTCh), which takes plate compression in thickness into account. The solution for this theory is obtained using Godunov's orthogonal sweep method. This solution is compared with the solution obtained using the general three-dimensional theory of elasticity, implemented in an open-source package Code_Aster using axisymmetric finite elements. The motivation for this study is the description of a stress-strain state of some variants of primary mirrors of large optical telescopes under the action of gravity. The obtained results characterizing the optical quality of the mirror surface are: the peak value (PV) and the root-mean square (RMS) of its displacement. A parametric study was carried out, i.e., the thickness of the plate and the half-width of the support were varied. The two methods were compared. It is shown that, as the plate thickness increases or the half-width of the support decreases, non-physical behaviour of the mirror surface takes place within the limits of the nonclassical theory of RTCh. A criterion of its applicability is therefore proposed.

Elasticity theory in general relativity

The general relativistic theory of elasticity is reviewed from a Lagrangian, as opposed to Eulerian, perspective. The equations of motion and stress–energy–momentum tensor for a hyperelastic body are derived from the gauge–invariant action principle first considered by DeWitt. This action is a natural extension of the action for a single relativistic particle. The central object in the Lagrangian treatment is the Landau–Lifshitz radar metric, which is the relativistic version of the right Cauchy–Green deformation tensor. We also introduce relativistic definitions of the deformation gradient, Green strain, and first and second Piola–Kirchhoff stress tensors. A gauge-fixed description of relativistic hyperelasticity is also presented, and the nonrelativistic theory is derived in the limit as the speed of light becomes infinite.

Higher order mixture nonlocal gradient theory of wave propagation

The higher order mixture nonlocal gradient theory of elasticity is conceived via consistent unification of the higher order stress‐ and strain‐driven mixture nonlocal elasticity and the higher order strain gradient theory. The integro‐differential constitutive law is established applying an abstract variational approach and appropriately replaced with the equivalent differential condition subject to nonclassical boundary conditions. The introduced higher order elasticity theory provides, as special cases, a variety of generalized elasticity theories adopted in nanomechanics to assess size effects in continua with nanostructural features. The well‐posed higher order mixture nonlocal gradient theory is elucidated and invoked to examine the flexural wave propagation. The closed‐form wave propagation relation between the phase velocity and the wave number is analytically derived. The determined wave propagation response and ensuing results are compared and calibrated with the pertinent molecular dynamic simulations. The demonstrated results of the phase velocity of the flexural wave propagation detect new benchmarks for numerical analyses. The proposed higher order size‐dependent elasticity approach can be profitably employed in rigorous analysis of pioneering nanotechnological devices.

Study of the Strain Response of Asphalt Pavements on Orthotropic Steel Bridge Decks through Field Testing and Numerical Simulation

Abstract A composite structural system consisting of an orthotropic steel bridge deck, waterproof prime, binding course, and wearing surface is complicated. To simplify the analysis of such systems, general elastic and continuous structure theory is applied to the asphalt pavement on steel bridge decks to determine the magnitude of the strain. Large differences in mechanical results have been previously obtained in assessments of steel bridge deck pavements. Few studies have focused on the mechanical state and time-dependent response of a realistic steel bridge deck wearing surface because of the difficulty of measuring the strain of the pavement on a steel deck of a long-span bridge that is in service. This study performed static and dynamic loading tests on an actual long-span steel bridge and analyzed the characteristics of the time-dependent strain response of the bridge deck wearing surface. The results demonstrate that the asphalt wearing surface exhibited significant loading time and rate-dependent characteristics under both static and dynamic loading conditions. The test analysis results also revealed that the axle load, vehicle speed, and longitudinal slope of the bridge had significant effects on the mechanical responses of the bridge deck wearing surface. Based on a comparison between the finite element numerical simulation results and the measurements obtained from testing the actual bridge, the magnitudes of the mechanical strain responses of the wearing surface on a steel bridge deck under the dynamic wheel load of a vehicle were analyzed assuming that the wearing surface is elastic and that the elastic modulus can be determined from a four-point bending test. However, the cumulative effect of plastic deformation must be considered when analyzing the fatigue damage of a permanent wearing surface.

Numerical Research on a Three-Dimensional Solid Element Based on Generalized Elasticity Theory

A finite element equation is established based on generalized elasticity theory by applying a virtual work principle. Then, a penalty function term is added to the virtual work equation by imposing rotation and displacement as independent variables. An 8-node element with full integration, an 8-node element with reduced integration, and a 20-node element with full integration are constructed using difference integration schemes and shape functions. The influences of structural degrees of freedom and the penalty parameter on convergence are analyzed via the three elements. It is shown that the 8-node element with reduction integration and the 20-node element with full integration are convergent, whereas the 8-node element with full integration is divergent. The scale effects of a slender beam, a short beam, a thin plate, and a medium-thick plate are numerically analyzed. Lastly, the scale effects of the frequencies that correspond to the bending mode, torsion mode, and tension-compression mode for a pretwisted plate are studied. It is found that the frequencies that correspond to the bending mode and torsion mode exert a scale effect, whereas the frequency that corresponds to the tension-compression mode does not. The essence of the scale effect is that the rotational deformation of the microstructure is amplified.

Estimation of the Strength of Plates with Cracks Based on the Maximum Stress Criterion in a Scale-Dependent Generalized Theory of Elasticity

The problem of the strength of a plate made of a brittle material with through mode I cracks is discussed. In contrast to the approach based on the singular solution of the classical theory of elasticity for a plane with a crack and on linear fracture mechanics, we propose to use nonsingular solutions obtained within the generalized elasticity theory and, as a result, to implement a method, conventional for the strength estimation of solids with stress concentration, based on the maximum stress criterion. The maximum stress is determined from a nonsingular solution of the generalized elasticity equations for a plane with a crack. The reported experimental results for plates with cracks under tension and bending confirm the solution obtained by the proposed method and allow it to be compared with a solution based on linear fracture mechanics. In fact, a new concept of fracture mechanics is put forward, which is free of singular solutions and allows the problems of fracture mechanics to be treated as problems of stress concentration. Comparison of the obtained analytical solutions with the experimental data has shown that the scale factor of generalized elasticity determines the critical state in fracture mechanics with no less accuracy than the critical stress intensity factor and therefore can be used as a fracture criterion. The resulting explicit nonsingular solutions allow the prediction of the stress concentration caused by a crack.

Dynamics study of a microsensor for perceiving unsafe acceleration and angular speed

A micro-trigger sensor that can perceive unsafe acceleration and angular speed is proposed in this study. This sensor can send an alarm message regarding the excessive speed of rotating machinery or activate the automobile airbag during emergencies. A mathematical and mechanical model with size-dependent is established through generalized elasticity theory and Euler–Bernoulli beam hypothesis. The governing equations of motion are deduced using Hamilton’s principle and solved using a modal superposition method. The influence of the degree of freedom (DOF) on the dynamics response is analyzed, and the result converges when the DOF is 5. Meanwhile, the dynamics response of the microcantilever under instantaneous impact and centrifugal force are evaluated along with the engineering background. Oscillation is observed in dynamics response results, but oscillation is not obvious when the deflection change rate is minimal. Furthermore, the scale effect of the dynamics response is examined, and the obvious scale effect is noted in the calculation results. Lastly, the trigger position of the circular ring is ascertained via the graphical method in cases of specified and unspecified trigger time.

Research on a micro-trigger sensor for overspeed alarm and emergency protection

A micro-trigger sensor for alarm or emergency protection under abnormal working conditions is proposed. The mathematical and mechanical models with scale-dependent are established using the Euler-Bernoulli beam hypothesis and the generalized elasticity theory. Meanwhile, governing equations are deduced using the principle of minimum potential energy. The design formulas of the trigger angular speed and acceleration for the micro-trigger sensor are established. The iteration method is used to solve the trigger angular speed that cannot be directly calculated by the design formula. The scale effects of the deflection of the microcantilever under angular speed and acceleration are analyzed by using generalized elasticity theory and classical elasticity theory, and significant scale effects are observed. Lastly, the control design of the micro-trigger sensor is investigated. It shows that the measurement range of the trigger angular speed and acceleration can be increased by adopting the two-circular ring co-regulation scheme.

On the figure of elastic planets I: gravitational collapse and infinitely many equilibria.

A classic problem of elasticity is to determine the possible equilibria of an elastic planet modelled as a homogeneous compressible spherical elastic body subject to its own gravitational field. In the absence of gravity, the initial radius is given and the density is constant. With gravity and for small planets, the elastic deformations are small enough so that the spherical equilibria can be readily obtained by using the theory of linear elasticity. For larger or denser planets, large deformations occur and the general theory of nonlinear elasticity is required to obtain the solution. Depending on the elastic model, we show that there may be parameter regimes where there exist no equilibrium or arbitrarily many equilibria. Yet, at most two of them are dynamically stable with respect to radial disturbances. In some of these models, there is a critical initial radius at which spherical solutions cease to exist. For planets with larger initial radii, there is no spherical solution as the elastic forces are not sufficient to balance the gravitational force. Therefore, the system undergoes gravitational collapse, an unexpected phenomenon within the framework of classical continuum mechanics.

Continuous constitutive model for bimodulus materials with meshless approach

One of the difficulties in dealing with bimodular materials including composites, rock and asphalt-mixture material is the discontinuity of Young's modulus when the principal stress changes sign, i.e. from a tensile stress state to a compressive stress state. According to the general elastic theory proposed by Ambartsumyan, there are two kinds of domains in which the coefficients of elasticity are constant. The discontinuity of Young's modulus causes divergence in the computational procedure. In order to overcome this difficulty, two continuous modes for bimodular materials are proposed in this paper. The non-linear equilibrium equations have been formulated with a continuous constitute equation of stress and strain. The meshless finite block method is successful in solving the nonlinear problems for bimodular materials. The numerical solutions of the meshless finite block method in a strong form are obtained using an iterative technique. The degree of accuracy and convergence of the proposed technique is demonstrated by directly comparing the achieved results with the finite element method and analytical solutions.

Fracton-Elasticity Duality.

Motivated by recent studies of fractons, we demonstrate that elasticity theory of a two-dimensional quantum crystal is dual to a fracton tensor gauge theory, providing a concrete manifestation of the fracton phenomenon in an ordinary solid. The topological defects of elasticity theory map onto charges of the tensor gauge theory, with disclinations and dislocations corresponding to fractons and dipoles, respectively. The transverse and longitudinal phonons of crystals map onto the two gapless gauge modes of the gauge theory. The restricted dynamics of fractons matches with constraints on the mobility of lattice defects. The duality leads to numerous predictions for phases and phase transitions of the fracton system, such as the existence of gauge theory counterparts to the (commensurate) crystal, supersolid, hexatic, and isotropic fluid phases of elasticity theory. Extensions of this duality to generalized elasticity theories provide a route to the discovery of new fracton models. As a further consequence, the duality implies that fracton phases are relevant to the study of interacting topological crystalline insulators.

Curvature instability of membranes near rigid inclusions.

In multicomponent membranes, internal scalar fields may couple to membrane curvature, thus renormalizing the membrane elastic constants and destabilizing the flat membranes. Here, a general elasticity theory of membranes is considered that employs a quartic curvature expansion. The shape of the membrane and its deformation energy near a long rod-like inclusion are studied analytically. In the limit where one can neglect the end effects, the nonlinear response of the membrane to such inclusions is found in exact form. Notably, exact shape solutions are found when the membrane is curvature unstable, manifested by a negative rigidity. Near the instability point (i.e., at vanishing rigidity), the membrane is stabilized by the quartic term, giving rise to a different length scale and scale exponents for the shape and its energy profile than those found for stable membranes. The contact angle induced by an applied force at the inclusion provides a method to experimentally determine the quartic curvature modulus.

A numerical investigation and experimental verification of size effects in loaded bovine cortical bone.

In this paper, we present 2- and 3-dimensional finite element-based numerical models of loaded bovine cortical bone that explicitly incorporate the dominant microstructural feature: the vascular channel or Haversian canal system. The finite element models along with the representation of the microstructure within them are relatively simple: 2-dimensional models, consisting of a structured mesh of linear elastic planar elements punctuated by a periodic distribution of circular voids, are used to represent beam samples of cortical bone in which the channels are orientated perpendicular to the sample major axis, while 3-dimensional models, using a corresponding mesh of equivalent solid elements, represent those samples in which the canals are aligned with the axis. However, these models are exploited in an entirely novel approach involving the representation of material samples of different sizes and surface morphology. The numerical results obtained for the virtual material samples when loaded in bending indicate that they exhibit size effects not forecast by either classical (Cauchy) or more generalized elasticity theories. However, these effects are qualitatively consistent with those that we observed in a series of carefully conducted experiments involving the flexural testing of bone samples of different sizes. Encouraged by this qualitative agreement, we have identified appropriate model parameters, primarily void volume fraction but also void separation and matrix modulus by matching the computed size effects to those we observed experimentally. Interestingly, the parameter choices that provide the most suitable match of these effects broadly concur with those we actually observed in cortical bone.

Contribution to beam theory based on 3-D energy principle

This paper presents a general elastic beam theory, which is consistent with the principle of stationary three-dimensional potential energy. For the sake of simplicity we consider the case of a rectangular cross section. The series expansion of the displacement field up to fourth-order in h (dimension of the cross section) is defined by 45 unknowns. The first variation of the potential energy must be zero but we only impose that each term guarantees an error. By adding supplementary lateral boundary conditions and on two extremities end cross section of the beam, we finally arrive at a well posed system of unidimensional differential equations. A linear algebraic dependence with respect to 16 displacement fields allows us to reduce the unknown to 19 displacement fields. To our knowledge this work is the first contribution to this end when the beam problem is completely three-dimensional.