Three-dimensional Elasticity Equations Research Articles

Gradient preserving Operator Inference: Data-driven reduced-order models for equations with gradient structure

Hamiltonian Operator Inference has been introduced in Sharma et al. (2022) to learn structure-preserving reduced-order models (ROMs) for Hamiltonian systems. This approach constructs a low-dimensional model using only data and knowledge of the Hamiltonian function. Such ROMs can keep the intrinsic structure of the system, allowing them to capture the physics described by the governing equations. In this work, we extend this approach to more general systems that are either conservative or dissipative in energy, and which possess a gradient structure. We derive the optimization problems for inferring structure-preserving ROMs that preserve the gradient structure. We further derive an a priori error estimate for the reduced-order approximation. To test the algorithms, we consider semi-discretized partial differential equations with gradient structure, such as the parameterized wave and Korteweg–de-Vries equations, and equations of three-dimensional linear elasticity in the conservative case and the one- and two-dimensional Allen-Cahn equations in the dissipative case. The numerical results illustrate the accuracy, structure-preservation properties, and predictive capabilities of the gradient-preserving Operator Inference ROMs.

On the Importance of the Recovery Procedure in the Semi-Analytical Solution for the Static Analysis of Curved Laminated Panels: Comparison with 3D Finite Elements.

The manuscript presents an efficient semi-analytical solution with three-dimensional capabilities for the evaluation of the static response of laminated curved structures subjected to general external loads. A two-dimensional model is presented based on the Equivalent Single Layer (ESL) approach, where the displacement field components are described with a generalized formulation based on a higher-order expansion along the thickness direction. The fundamental equations are derived from the Hamiltonian principle, and the solution is found by means of Navier's approach. Then, an efficient recovery procedure, derived from the three-dimensional elasticity equations and based on the Generalized Differential Quadrature (GDQ) method, is adopted for the derivation of the three-dimensional solution. Some examples of investigation are presented, where the numerical predictions of refined three-dimensional Finite-Element-based models are matched with a high level of accuracy. The model is validated for both straight and curved panels, taking into account different lamination schemes and load shapes. Furthermore, it is shown that the numerical solution to the elasticity problem in the recovery procedure is determining and accurately predicting the three-dimensional static response of the doubly-curved shell solid.

Asymptotic long-wave model for a high-contrast two-layered elastic plate

The paper is concerned with the derivation of asymptotically consistent equations governing the long-wave flexural response of a two-layered rectangular plate with high-contrast elastic properties. In the general case, the plate is under dynamic and variable surface, volume, and edge forces. Performing the asymptotic integration of the three-dimensional (3D) elasticity equations in the transverse direction and satisfying boundary conditions on the faces and interface, we derived the sequence of two-dimensional (2D) differential equations with respect to required functions in the first two approximations. The eight independent restraints for the generalized displacements and stress resultants are considered to formulate the 16 independent variants of boundary conditions. One of the main results of the paper is the Timoshenko–Reissner type equation capturing the effect of the softer layer and taking into account the in-plane deformation induced by the edge forces. Comparative calculations of natural frequencies were carried out based on our and alternative models.

Variable-stiffness curved laminated-beams by curvilinear fibers with arbitrarily layup – Vibrational features by sine-based higher-order beam model with renewed-constitutive relations and improved-kinematics

The vibration characteristics of variable-stiffness based composite curved beams with general layups are examined using a higher-order beam theory that accounts for three-dimensional structural response features. The variable stiffness is spatially considered in the laminated beam by introducing the curvilinear fibres-based layers. The structural model is here represented by a sine-based shear deformable model considering zig-zag function for improving in-plane response and higher order polynomial for capturing through-thickness stretching effect. The appropriate constitutive relationships for laminated composite curved beam with arbitrary layup producing stiffness coupling effects are deduced from three-dimensional elasticity equations. The beam governing equations are then formed applying Hamilton’s principle and the free vibration features are predicted employing the eigenvalue analysis. The effectiveness of the curved laminated beam element developed here is tested against the available analytical solutions in the literature. A detailed numerical analysis by opting for different structural parameters like curved beam angle, thickness, ply-angle, layup, and beam end condition are made to investigate the vibration characteristics of general layup constant stiffness laminated curved beams and by including additional parameter, curvilinear fibre angle variation from centre-to-edge, for variable stiffness laminates.

A comparative study of scalable multilevel preconditioners for cardiac mechanics

In this work, we provide a performance comparison between the Balancing Domain Decomposition by Constraints (BDDC) and the Algebraic Multigrid (AMG) preconditioners for cardiac mechanics on both structured and unstructured finite element meshes. The mechanical behavior of myocardium can be described by the equations of three-dimensional finite elasticity, which are discretized by finite elements in space and yield the solution of a large scale nonlinear algebraic system. This problem is solved by a Newton-Krylov method, where the solution of the Jacobian linear system is accelerated by BDDC/AMG preconditioners. We thoroughly explore the main parameters of the BDDC preconditioner in order to make the comparison fair. We focus on: the performance of different direct solvers for the local and coarse problems of the BDDC algorithm; the impact of the different choices of BDDC primal degrees of freedom; and the influence of the finite element degree. Scalability tests are performed on Linux clusters up to 1024 processors, and we conclude with a performance study on a realistic electromechanical simulation.

Electro-mechanical analysis of functionally graded graphene reinforced composite laminated plate with macro fiber composite actuator

Graphene is considered to be a desirable reinforcement material for composite materials due to its exceptional mechanical properties. However, there is a lack of research on the interlaminar stress analysis of graphene reinforced composite plates bonded with a macro fiber composite (MFC) actuator subjected to electro- mechanical loads. Inaccurate descriptions of interlaminar shear deformations can significantly affect the mechanical performance of the MFC piezoelectric graphene reinforced composite plate due to the electro-mechanical coupling feature and abrupt change in material properties at the interfaces of neighboring layers. To address this issue, this study proposes an attractive plate model for the analysis of MFC piezoelectric laminated plate. The three-dimensional (3D) elasticity equations and the Reissner mixed variational theorem (RMVT) are employed to improve the precision of interlaminar shear stresses. The 3D elasticity solutions and results obtained from other models are used to evaluate the proposed model. Furthermore, this study thoroughly investigates the effect of several significant parameters on the deformations and stresses of the MFC piezoelectric plates with reinforcements.

Free Vibration Analysis of Functionally Graded Graphene-Reinforced Composite-Laminated Plates

Graphene is recognized as one of the most alluring reinforcements for composite materials due to its exceptional mechanical capabilities. However, the existing models disregarding the compatible requirements of interlaminar stresses in the literature may meet severe challenges for the dynamic analysis of functionally graded graphene-reinforced composite (FG-GRC) laminated thick plates. If transverse shear deformations cannot be described accurately, the prediction of natural frequencies for FG-GRC laminated plates will be seriously and directly affected. Accordingly, an attractive plate theory in conjunction with the analytical approach and finite-element formulation is proposed for free vibration analysis of FG-GRC plates. An improved transverse shear stress field is acquired in terms of the three-dimensional (3D) elasticity equations and the Reissner mixed variational theorem (RMVT). The performance of the suggested model was evaluated using 3D elasticity solutions and the results obtained from other models. Numerical results showed that the suggested model can produce promising results for the thick laminated and sandwich plates. Additionally, a thorough investigation was conducted to study the effects of several graphene parameters, including volume fractions, distribution patterns, span-to-thickness ratios, aspect ratios, boundary conditions, and stacking sequences, on the dynamic responses of FG-GRC laminated plates.

An analytical method for free vibration analysis of multi-layered transversely isotropic cylindrical shells

This work presents an effective analytical method based on displacement potential functions (DPF) for solving 3-D free vibration problem of thick and multi-layered transversely isotropic cylindrical shells with simply supported end boundary conditions. By using the DPF method, the three-dimensional elasticity equations are simplified and decoupled into two linear partial differential equations of fourth and second order as governing differential equations. These equations are analytically solved using the separation of variables method in terms of fields that exactly satisfy end boundary conditions and the continuity of a closed cylinder in the hoop direction. One of the advantages of the present work is the analysis of thick cylindrical shells using an exact method without any approximations and simplifying assumptions in the distribution of stress or strain along the thickness of the shells. The analysis is extended to multi-layered cylindrical shells by applying appropriate boundary conditions at the interfacial surfaces of the layers. Numerical comparisons to other analytical works shows the accuracy of the present work and also due to the lack of study for multi-layered transversely isotropic cylindrical shells, 3-D finite element analysis has been performed for comparison. In addition, various more involved problems are studied and solved analytically for single and multi-layered shells of transversely isotropic material and the effect of various parameters such as shell dimensions, mechanical properties, number of layers and other parameters on shell behavior are evaluated. The results of the present study can be used as benchmark solutions for other studies.

A hybrid CPU–GPU solver based on three-dimensional spectral Chebyshev technique for determining the dynamic behavior of thick sandwich panels

In this study, static and vibration behavior of a curved thick sandwich-structured composite composed of a honeycomb core layer and two face-sheets reinforced with carbon nanotubes is investigated. The governing equations are derived using three-dimensional elasticity equations and Hamilton’s principle. The problem domain is discretized following Gauss–Lobatto sampling approach. To numerically solve the governing equations, a spectral method based on Chebyshev polynomials is used and a CPU–GPU based hybrid solver is developed. The presented approach is validated by comparing the predicted natural frequencies and deformation of the structure to those obtained from finite element analysis. The results show that in-house CPU–GPU based spectral solution approach decreases the computational duration remarkably and predicts the dynamic and static behavior of the system accurately. Since composite materials offer great flexibility in tailoring the dynamic behavior of the structure, an optimization study is also performed to benefit from the developed computationally efficient solver. In this study, the CNT orientations are selected as design variables to maximize the fundamental frequency of the structure. Based on the analyzed cases for different thickness ratios, curvature amounts, volume content, and distributions of CNTs, it is shown that the vibration behavior of the structure can be significantly tailored.

Spectral and Evolution Analysis of Composite Elastic Plates with High Contrast

We analyse the behaviour of thin composite plates whose material properties vary periodically in-plane and possess a high degree of contrast between the individual components. Starting from the equations of three-dimensional linear elasticity that describe soft inclusions embedded in a relatively stiff thin-plate matrix, we derive the corresponding asymptotically equivalent two-dimensional plate equations. Our approach is based on recent results concerning decomposition of deformations with bounded scaled symmetrised gradients. Using an operator-theoretic approach, we calculate the limit resolvent and analyse the associated limit spectrum and effective evolution equations. We obtain our results under various asymptotic relations between the size of the soft inclusions (equivalently, the period) and the plate thickness as well as under various scaling combinations between the contrast, spectrum, and time. In particular, we demonstrate significant qualitative differences between the asymptotic models obtained in different regimes.

The Solvability of the 3-D Elastic Wave Equations in Inhomogeneous Media

In this research, the three-dimensional elastic wave equations with variable coefficients (i.e. propagate through inhomogeneous media) are solved with the application of the Fourier transform in the spatial coordinates. The wave equation is coupled variable coefficients PDEs whose solutions may have significant in engineering applications. The method utilizes the second order ODE as the baseline for obtaining the complete solution. The solution of second order ODEs is expressed in one integration because the variable coefficients are broken down into several functions and resulted in first order reduction. Moreover, the coupled equations are performed by the order reduction of the higher order ODEs into the second order. The extended procedure for integral equation is implemented for the solutions from the transformed wave equations to generate the explicit expression. It is shown that the proposed method of integral evaluation is resulted in finding the roots of polynomials. Hence, it is concluded that the solvability of the elastic wave equations is ensured by the proposed method.

Determination of three-dimensional thermo-mechanical behavior of bidirectional functionally graded rectangular plates with the theory of elasticity

In this study, the thermo-mechanical behavior of rectangular plates functionally graded (FG) in two directions along the plane is investigated with three-dimensional (3D) elasticity equations. In numerical analysis, the finite difference method (FDM) is used and the directional variation of the derivatives of the material properties is taken into account. The thermo-mechanical behavior of the plates is investigated for different compositional gradient exponents and it is emphasized that the variation of the compositional gradient exponent in both directions significantly changed the thermal stress distribution and the results are confirmed.

A linear two-dimensional mathematical model for thin two-layer plates with partial shear interaction, with a view towards application to laminated glass

This work presents a consistent derivation, from three-dimensional linear elasticity, of a two-dimensional mathematical model describing the bending and in-plane stretching behaviours, under a general system of quasi-static distributed loads and prescribed support displacements, of thin two-layer plates with partial shear interaction. This layerwise model is specifically tailored for the requirements posed by the analysis of laminated glass plates commonly used in building structures (consisting of two thin glass layers bonded together by an adhesive interlayer). Our approach, based on Podio-Guidugli’s method of internal constraints, avoids mutually contradictory assumptions (not uncommon in the literature on structural mechanics) and yields a complete two-dimensional characterisation of displacement, strain and stress fields that exactly satisfy the field equations of three-dimensional linear elasticity and the boundary conditions at the end faces. The choice of generalised variables is designed to bring to light the following fundamental conclusion and physical insight: the resulting two-dimensional boundary value problem is a combination of the equations of Kirchhoff and Mindlin plates (with specified rigidities).Two examples illustrate the application of the proposed model: (i) the cylindrical bending of plate strips and (ii) a family of problems with Navier-type analytical solution. The solutions exhibit continuity across the whole range of zero, partial and full interaction between the layers. Moreover, the Navier-type solutions are consistently in close agreement with the results of three-dimensional finite element analyses. On the contrary, analogous results previously reported in the literature exhibit considerable deviations. An explanation for these discrepancies is discussed in detail.

On the complete solution of the three-dimensional solid space problems based on a novel curvilinear elasticity representation

A solution scheme is presented for the three-dimensional elasticity theory in the curvilinear cylindrical coordinate, without introducing any new physically dubious potential function. Exact analytical solutions for the essential 3-D solid full- and half-space problems are developed based on the introduced elasticity approach. The Kelvin's classical problem of concentrated force acting in the interior of an infinite solid space, as well as the Boussinesq's half-space problem of concentrated force normal to the free surface of a semi-infinite 3-D solid are treated. Furthermore, the three-dimensional non-axisymmetric problem of semi-infinite solid space under concentrated tangential force on its surface (the so-called Cerruti's problem) is revisited. It is demonstrated that unlike most of the conventional solutions in the literature for the aforementioned problems, the present approach is capable of obtaining the complete solution without needing to combine with other approaches. Evidently, the completeness issue associated with any new method of potentials is not present for the established direct elasticity approach. At the end, an extension of the approach to a nonhomogeneous elastic media is presented and, as an application, the Boussinesq's problem for a nonhomogeneous half-space is solved analytically. The developed representation may serve as an efficient replacement of the original three-dimensional Navier's elasticity equations for solving different 3-D elastic problems in the future.

On the natural frequency investigation of the thick hollow cylinder with 2D-FGM Mori-Tanaka scheme

ABSTRACT In this paper, we conduct the natural frequency analysis for a 2D-FGM (functionally graded material) finite-length, thick-walled cylinder by the graded finite element method. The material properties gradations are considered along with the thickness and the longitudinal axis of the cylinder according to the Mori-Tanaka scheme and power-law volume fraction distributions. Therefore, the graded properties are implemented by user subroutine UMAT in ABAQUS/Standard™. The eigenvalue problem is solved using three-dimensional elasticity equations, and the current technique has been confirmed by earlier research. The major purpose of this study is to see how FGM configurations affect the vibration behaviour of the cylinder. The variation of the cylinder's first natural frequencies for various thickness-to-outer-radius and length-to-outer-radius ratios is presented, taking into account varied material distributions. According to the findings, raising the power exponents, particularly the radial one, increases the natural frequencies.

A computational framework to obtain nonlinearly elastic constitutive relations of special Cosserat rods with surface energy

We present a computational framework to obtain nonlinearly elastic constitutive relations of special Cosserat rods with surface energy. The framework allows the rod’s cross-section to have arbitrary shape and the rod material to obey arbitrary three-dimensional material model for its bulk volume and arbitrary surface energy model for its lateral surface. The kinematics of the Helical Cauchy–Born rule is used to first construct a family of six-parameter (corresponding to the six strain measures of rod theory) helical rod configurations in which the generated three-dimensional strain field is uniform in the rod’s arc-length coordinate. This uniformity allows the three-dimensional nonlinear equations of elasticity to reduce to just the rod’s cross-sectional plane wherein the cross-section’s boundary also experiences traction due to the inherent surface energy in the rod’s lateral surface. The solution of this cross-sectional problem yields the induced in-plane displacement and out-of-plane warping in the cross-section for arbitrarily prescribed strain measures of the rod. We then derive expressions for the induced internal contact force, moment and stiffnesses of the rod in terms of the solution of the deformed cross-section and the prescribed strains of the rod. A finite element formulation is presented to solve the nonlinear cross-sectional deformation problem and further obtain the induced internal contact force, moment and stiffnesses. The presented framework will be useful for modeling of nanorods where surface energy plays a dominant role. Several numerical examples are presented using the presented framework to illustrate the effect of surface energy parameters on the deformation of the nanorod’s cross-section and the nanorod’s bending, torsional, extensional and shearing stiffnesses. We also derive improved analytical formulas for the extensional and bending stiffnesses of isotropic rectangular nanorods in their reference state. These formulas reduce to the existing widely used formulas for a special choice of material parameters, i.e., when the surface Poisson’s ratio and the bulk Poisson’s ratio match thus highlighting the limitation of the existing formulas.

3-D analytical solution of non-homogeneous transversely isotropic thick closed cylindrical shells

This paper presents an effective analytical method based on displacement potential functions (DPF) for solving 3D static problem of thick and multilayer transversely isotropic cylindrical shells with simply supported end boundary conditions. By using the DPF method, the three-dimensional elasticity equations are simplified and decoupled into two linear partial differential equations of fourth and second order as governing differential equations. The governing equations are solved by the separation of variable method in terms of fields that exactly satisfy end boundary conditions and the continuity of a closed cylinder in the hoop direction. The analysis covers a straightforward solution process for handling problems on multilayered cylindrical shells of transversely isotropic material, adopting all boundary and continuity conditions. Extensive sets of general radial loads located on the inner and outer faces of the cylindrical shell may be stated and examined with in a systematic manner. Comparisons are performed to other existing analytical results for one and multilayered cylindrical shells, and show excellent agreement for different materials, thicknesses and aspect ratios of the shell. In addition, various more involved problems are studied and solved analytically for single and three-layered shells of transversely isotropic material with different sets of radial loading functions at the outer and inner shell surfaces. The results of the present study can be used as benchmark solutions for other studies.

On governing equations for a nanoplate derived from the 3D gradient theory of elasticity

The paper is concerned with the asymptotically consistent theory of nanoscale plates capturing the spatial nonlocal effects. The three-dimensional (3D) elasticity equations for a thin plate are used as the governing equations. In the general case, the plate is acted upon by dynamic body forces varying in the thickness direction, and by variable surface forces. The thickness of the plate is assumed to be greater than the characteristic micro/nanoscale measure and much smaller than the in-plane characteristic dimension (e.g., the wave or deformation length). The 3D constitutive equations of gradient elasticity are used to link the fields of nonlocal stresses and strains. Using the asymptotic approach, a sequence of relations for stresses and displacements in the form of polynomials in the transverse coordinate with coefficients depending on time and in-plane coordinates was obtained. The asymptotically consistent 2D differential equation governing vibration (or static deformation) of a plate accounting for both transverse shears and the spatial nonlocal contribution of the stress and strain fields was derived. It was revealed that capturing nonlocal effects in all directions leads to an increase in the correction factor compared with the well-known 2D theories based on kinematic hypotheses and the Eringen-type gradient constitutive equations. The effect of the internal length scales parameters on free low-frequency vibrations and displacements of a plate is discoursed.

Numerical models of earthquake mechanism

The tectonic creep and its variation after particular earthquakes are studied by the Stokes equation. The stress state of the region is modelled according to a hypothesis of plate tectonics in which the lithosphere of the region is laterally compressed across the Eurasian, Indian, and Arabian plates. The 1966 Tashkent (Uzbekistan) earthquake and the 1976 Gazli (Uzbekistan) earthquake are selected as examples to study different models of earthquake focal mechanisms. Based on the specifics of the geodynamic formulation, the three-dimensional equations of moment elasticity and hydromechanics are reduced to two-dimensional equations for averaged stresses, displacements, and displacement velocities. The two-dimensional equations are solved by boundary integral equations. The stresses can be useful in zoning maps. The vertical velocities obtained from the creep model of the earth's crust can serve as additional data to Central Asia's horizontal velocities from GPS measurements.

Static and dynamic responses of simply supported sandwich plates using non-polynomial zigzag theory

The static, free vibration and transient responses of simply-supported laminated composites and sandwich plates are analytically derived using a non-polynomial zigzag theory. This mathematical model is displacement-based with five field variables and uses a non-polynomial mathematical function to introduce the non-linearity of transverse shear stresses through-thickness, thus eliminating the use of a large number of higher-order terms in the kinematic field. The zigzag kinematics is implemented by the inclusion of auxiliary variables defined at the interfaces of the plates with a piecewise linear function of the thickness coordinate. The dynamic governing equations of equilibrium are derived with Hamilton’s principle that generates five coupled partial differential equations. The Navier’s solution technique with the Newmark’s time integration scheme is adopted to find the solutions of the coupled equations. The responses in the form of displacements, stresses, natural frequencies, and displacement–time responses are obtained by solving several standard problems in the literature. The transverse shear stresses are accurately obtained by integrating the three-dimensional equilibrium equations of elasticity. The forced-vibration analysis of the laminated composites and sandwich plates are presented in detail by considering various forms of time-dependent loads that also includes the various forms of blast loads. The present results agree closely with the published results in the literature for both laminated composites and sandwich plates.