Principle Of Minimum Potential Energy Research Articles

Analytical model for the nonlinear buckling responses of the confined polyhedral FGP-GPLs lining subjected to crown point loading

This paper focuses on the nonlinear buckling responses of the confined functionally graded porous (FGP) lining with polyhedral shapes reinforced by graphene platelets (GPLs). The FGP-GPLs polyhedral lining is surrounded tightly and rigidly by a medium. It is assumed that the interface is smooth between the lining and pipeline. The deformation of the lining may be represented by an admissible displacement expression when a point load is applied at the crown position. Afterward, the nonlinear governing equilibrium equations are obtained explicitly by combining the thin-walled shell theory and the principle of minimum potential energy. Solving the nonlinear governing equilibrium equations yields the critical buckling load. A comparison study is strictly completed with the other closed-form solutions when the polyhedral lining reduces to a circular one. In addition, an improvement factor is quantified to understand the effect of polyhedral shapes on the bending stiffness. Finally, the effects of polyhedral shapes, porosity coefficient, weight fraction, and geometric parameters of the GPLs are explored in terms of buckling load, hoop force, and bending moment.

Resonant Actuation Based on Dynamic Characteristics of Bistable Laminates

Bistable or multi-stable structures have found broad applications in the fields of adaptive structures, flow control, and energy harvesting devices due to their unique nonlinear characteristics and strong local stability behavior. In this paper, a theoretical model based on the principle of minimum potential energy and the Rayleigh–Ritz method is established to study the dynamic characteristics of a bistable unsymmetric laminate with a fixed center. Numerical results of this theoretical model were obtained and verified by an FEA model using ABAQUS. The nonlinear dynamic characteristics and the structural response under different levels of external excitation were investigated and verified by experiments. The realization conditions of single-well vibration and cross-well vibration of bistable laminates were determined, with which the actuation strategies can be optimized for targeting modal frequencies of bistable laminates.

Determination of asymptotic field coefficients for multi-material antiplane V-notches by the weak form quadrature element method

Weak form quadrature element analysis (QEA) of multi-material antiplane V-notches is conducted. Angular eigenfunctions are introduced into the inner subdomains where stress singularity exists, while the outer domain is divided into several quadrature elements according to geometric boundary conditions and material properties. Algebraic equations are established and coefficients are solved for after introducing the minimum potential energy principle. The influence of the first eight terms on improvement of far field stress and displacement estimations is demonstrated. As compared to the computational costs of other methods, much fewer degrees of freedom are needed in the present formulation to achieve the same accuracy of results, exhibiting the merits of QEA of fracture problems.

Programming metastable transition sequences in digital mechanical materials

The realization of unconventional computing methods in soft matter has inspired the creation of mechanological systems capable of processing information. These systems often utilize bistable mechanisms that serve as mechanically abstracted bits to perform digital logic operations. Yet, the input of such operations often requires manual control of complex cyclic loading to reach a desired configuration. This research presents a method to design digital mechanical materials that can enter a programmable sequence of metastable configurations through simple displacement-controlled inputs. Interactions between serially connected bistable units enable the transition and reset behavior of mechanical bits. An analytical model using the principle of minimum total potential energy of a single bit is presented to articulate systematic ways to tune the critical force thresholds and displacements of metastable transition sequences. This work elucidates how the application of a prescribed displacement allows for novel resetting behavior that enables simultaneous transitions of multiple bits. By combining the principles of deterministic transition sequences and mechanical computing, a mechanical analog to digital conversion (ADC) material system is introduced that digitizes a continuous force stimulus into binary configurations that perform computations within the mechanical material based on the applied load. The mechanical ADC establishes a foundation for the integration of environmental sensing, information processing, and response in a soft material system.

Vibration and acoustic radiation control of a panel with piezoelectric oscillators.

In this paper, the vibration of a thin plate system with piezoelectric sheets attached and its acoustic radiation are modeled theoretically. Then the results are compared with those of finite element simulation, and the results are relatively consistent. Finally, sound absorption and isolation experiments at non-intrinsic frequencies of the composite plate are conducted in a standing wave tube. Based on the principle of minimum potential energy, the variation equation of a thin plate with a piezoelectric sheet attached are listed, and the impedance matrix of the acoustic radiation of the plate coupled with each order of the cavity modal is calculated. After the modal displacement response of the plate is obtained, the absorption coefficient and sound insulation volume of the composite plate can be calculated. Through theoretical calculations and finite element simulations, it is found that the plate has a modal reorganization phenomenon at the peak of sound insulation, and the modal response of each order increases at the peak of sound absorption. After that, the sound absorption and isolation experiments were conducted in the standing wave tube, and the control effect of the piezoelectric shunt oscillator at the non-intrinsic frequency of the plate was improved by adding negative capacitance.

Analysis of Shear Stagger Deformation of Existing Shield Tunnel below Induced by Quasirectangular Shield Tunneling

To use urban underground space more efficiently is a research hotspot of urban underground development. Compared with conventional circular tunnels, quasirectangular tunnels have the characteristics of larger space utilization and higher economic benefits, gradually used in urban rail engineering in recent years. But the ground disturbance induced by the quasirectangular tunneling needs further researches. This paper derives the calculation formula of the additional stress of existing shield tunnel below caused by the quasirectangular shield tunneling by mirror image method and Mindlin solution. The existing shield tunnel is simplified as an elastic foundation short beam connected by tensile springs and shear springs. This paper establishes the energy-deformation coupling equation by principle of minimum potential energy so that the disturbance induced by quasirectangular tunneling to the existing shield tunnel below can be analyzed. The FEM is established with the actual engineering as the background. It is by comparing the numerical simulation results and field monitoring data that the theoretical prediction formula is verified. The research results show that the theoretical calculation results consist highly with numerical simulation results and field monitoring data, verifying practicality of formulas in this paper. Calculation results of shear stagger model are more accurate, which can be also used to analyze the staggered and splayed amount between segments. As the tunnel construction proceeds, uplift deformation, segment stagger, shear force, and segment splay of existing shield tunnel gradually increase, which are symmetrical about the central axis of new tunnel. After the shield machine passes through the central axis of the existing shield tunnel for 20 meters, stable disturbance to the existing shield tunnel occurs. Stagger amount and shear force between segments reach the minimum at the position with maximum uplift deformation of existing shield tunnel and reach the maximum at inflection point of the uplift deformation curve.

A deep learning energy-based method for classical elastoplasticity

The deep energy method (DEM) has been used to solve the elastic deformation of structures with linear elasticity, hyperelasticity, and strain-gradient elasticity material models based on the principle of minimum potential energy. In this work, we extend DEM to elastoplasticity problems involving path dependence and irreversibility. A loss function inspired by the discrete variational formulation of plasticity is proposed. The radial return algorithm is coupled with DEM to update the plastic internal state variables without violating the Kuhn–Tucker consistency conditions. Finite element shape functions and their gradients are used to approximate the spatial gradients of the DEM-predicted displacements, and Gauss quadrature is used to integrate the loss function. Four numerical examples are presented to demonstrate the use of the framework, such as generating stress–strain curves in cyclic loading, material heterogeneity, performance comparison with other physics-informed methods, and simulation/inference on unstructured meshes. In all cases, the DEM solution shows decent accuracy compared to the reference solution obtained from the finite element method. The current DEM model marks the first time that energy-based physics-informed neural networks are extended to plasticity, and offers promising potential to effectively solve elastoplasticity problems from scratch using deep neural networks.

An interval spline finite point method for size-dependent mechanical behaviors of defective functionally graded material nanobeam

In this paper, a novel method, i.e., interval spline finite point method is presented to investigate the mechanical behavior of functionally graded material nanobeam with defects, which can provide some guidance for the design of nanoscale devices. The size-dependent behavior of the functionally graded material nanobeam is considered, and the defects of nanobeam is taken into account and simulated by weakening/strengthening the stiffness of the nanobeam, where the material properties are considering as interval parameters. Combing the interval mathematic and the Timoshenko beam theory, the governing equation of the defective functionally graded material nanobeam is derived based on principle of minimum potential energy and variational principle. To solve the high order equilibrium equation with interval parameters, the interval spline finite point method is proposed, where the B-spline function is adopted as basic function to guarantee the accuracy. Moreover, to obtain the mechanical response of the nanobeam efficiently, a gradient-based vertex strategy is presented. Finally, two numerical examples are given and the effectiveness of the proposed method are validated by comparison with the existing methods.

Mathematical model of fluid flow along a straight through filled with a distributed flow from above

In a one-dimensional approximation and in the absence of friction forces, a mathematical model has been developed for the steady flow of an incompressible fluid along a straight line, for example, a drainage gutter, into which a distributed flow flows from above. The boundary condition missing for solving the system of equations of momentum and continuity is determined using the principle of minimum potential energy. For a rectangular chute, equations are obtained that allow one to calculate the distributions of the layer thickness and fluid velocity along the length of the chute with a plug at one end and without a plug, slope and without slope to the horizon, depending on the intensity of the incoming flow, the size of the chute, and the density of the liquid. This model, by means of a simple recalculation, can also be extended to the flow in a trough of a different cross-sectional profile. The results of the study can be applied to the calculation of external drainage systems.

Modeling Method for Pose Error of Machine Tool Linear Feed System Based on the Principle of Minimum Potential Energy

Modeling Method for Pose Error of Machine Tool Linear Feed System Based on the Principle of Minimum Potential Energy

FREE VIBRATION ANALYSIS OF ORTHOTROPIC RECTANGULAR PLATES USING THE RITZ NUMERICAL METHOD

The main objective of this paper is to contribute to a better understanding of the dynamic response of FRP rectangular plates through modal analysis. The Ritz’s approximation method originated from the principle of minimum potential energy and was chosen to formulate the Eigenvalue problem. The natural frequencies and mode shapes are determined for the first 36 free vibration modes by assuming the double Fourier series functions for the transverse displacement. The most common Beam functions are used as a functional basis. Results are shown for the specific plate problem of Angle ply rectangular plate clamped at one edge, supported at two adjacent edges, and accessible at another edge (CSSF). It is observed that maximum natural frequencies are obtained under the clamped boundary of the plate on all edges of (CCCC) for the rectangular laminates.

Application of the p-version of FEM to hierarchic rod models with reference to mechanical contact problems

The formulation of a system of hierarchic models for the simulation of the mechanical response of slender elastic bodies, such as elastic rods, is considered. The present work is concerned with aspects of implementation and numerical examples. We use a finite element formulation based on the principle of minimum potential energy. The displacement fields are represented by the product of one-dimensional field functions and two-dimensional director functions. The field functions are approximated by the p-version of the finite element method. Our objective is to control both the model form errors and the errors of discretization with a view toward the development of advanced engineering applications equipped with autonomous error control procedures. We present numerical examples that illustrate the performance characteristics of the algorithm.

Stability of a bounded liquid layer on a rotating horizontal plane

The paper is devoted to the study of the stability of the relative equilibrium of a bounded liquid layer on a flat solid rotating base. A uniform gravity field is oriented perpendicular to the solid surface and presses the droplet against it. The equilibrium shape and its perturbations are axisymmetric. The free surface is simply-connected. The analysis is performed both for the case of the free contact line and for the case of the fixed one. The results obtained by these two models are compared, and the effect of the input parameters on the stability is investigated. It is established that the second model is in better accordance with empirical data. Unlike the first one, it allows the possibility of a zero height of the layer at the center at certain values of the contact angle and determines the negative effect of a low wettability of the solid substrate on the stability of the droplet. The minimum potential energy principle is used as a stability criterion. In this process all physically admissible small variations of a free surface shape are considered. An equilibrium state is supposed to be stable if and only if it corresponds to a minimum potential energy on the set of allowable virtual displacements, which is more restricted when the contact line is fixed.

Flexural analysis of I-section beams functionally graded materials

This paper aims to present the flexural behavior of thin functionally graded (FG) I-beams. The characteristics of metal-ceramic materials are defined using a power-law function dependent on volume fraction. The bending-torsion equations for this problem are derived based on Vlasov's theory for thin-walled beams and the principle of minimum total potential energy. All geometric properties are expressed according to the functional graded power index law. A nonlinear algebraic system is obtained, and the deflection of the structure is numerically derived. To confirm the precision and effectiveness of the suggested method, a standard benchmark test case is implemented, that concerns the flexural analysis of an FGM I-section beam according to power index and aspect ratio parameters.

Application of novel refined mixed finite elementmethod in the analysis of composite laminated beams

This paper shows the application of refined finite element model developed by authors, which incorporate all the elastic equation that are, equilibrium equation, constitutive equation and kinematic equation to be satisfied explicitly at nodes. The present method considers the equilibrium equation for the refinement of the approximation function. Authors used the concept of mixed finite element model by choosing stress and displacement as the primary variable, along with the equilibrium equation as a refinement on approximation function. Ramtekkar [1] in his model restrict the arbitrary choosing of stress function by selecting the stress variation from displacement derived space, by imposing the elastic relations on the variation and then used the Minimum Potential Energy Principle to obtain the governing differential equation. In present work, authors satisfy the equilibrium equation at nodes also by considering the body forces as variables in the formulation which enables the model to satisfy the equilibrium equation explicitly at the node. The formulation is validated by the elastic beam problem solved by Pagano [2] for all three cases. Results obtained are very accurate and also shows faster convergence over the Ramtekkar [1] model.

First-order and second-order wrinkling of thin elastic film laminated on a graded substrate

Wrinkling of the film-on-substrate structure is widely used in stretchable electronics, and the literature has shown that the substrate with gradient modulus has the advantage of maintaining the structure's integrity. Hence, it is necessary to have a comprehensive understanding of the evolution of the wrinkles in the film-on-elastic-graded-substrate structure. Firstly, a unified theoretical framework is established. Secondly, by using the energy method and the principle of minimum potential energy, the wrinkling behaviour of this structure is investigated, and the analytical expressions of the critical buckling strain, wavelength and wrinkling amplitude of this structure are obtained. Then, based on the energy increment method and Maxwell stability criterion, the post-wrinkling behaviour of this structure is studied, and the theoretical formulations for the critical buckling strain, wavelength and wrinkling amplitude are obtained. Finally, numerical examples are carried out to verify the correctness of the proposed formulations. Through the numerical results, it is found that by modulating the applied strain and decay rate of the exponential substrate, the wrinkling pattern of the film-on-exponential-substrate structure can be transferred from a first-order wrinkling pattern to a second-order wrinkling pattern. Moreover, by tuning the applied strain and characteristic length of the bilayer substrate, the wrinkling pattern of the film-on-bilayer-substrate structure can be determined. These findings are useful for guiding the design of stretchable electronics based on the film-on-elastic-graded-substrate structure.

Axisymmetric bending vibration analysis of bidirectional functionally graded rotating micro-disk under thermo-mechanical loading

The free vibration behavior of a bidirectional functionally graded rotating micro-disk that is subjected to uniform transverse pressure and high-temperature thermal loading has been studied. The micro-disk is functionally graded along the radial and thickness directions. The problem is mathematically formulated using two distinct but interrelated steps within the framework of Kirchhoff plate theory and modified couple stress theory. The first step determines the time-invariant deformed configuration of the micro-disk under centrifugal, pressure, and thermal loading using minimum potential energy principle. The second step determines the free vibration behavior of the micro-disk in the neighborhood of the deformed configuration using Hamilton’s principle. The solutions of the governing equations for both these steps are obtained using the Ritz method. The mathematical model is successfully validated with various reduced problems. The numerical results for the first four axisymmetric bending vibration modes are presented to investigate the effects of wide range of parameters such as rotational speed, applied pressure, thermal loading, size-dependent thickness, volume fraction indices, and radius ratio. The mode shapes of vibration are illustrated through surface and contour plots.

Accurate mechanical buckling analysis of couple stress-based skew thick microplates

We investigate, for the first time, the elastic buckling of skew thick microplates under combined in-plane shear and compressive loading within the framework of modified couple stress theory. The displacement field of the microplates is described by a two-variable refined shear deformation theory, wherein the transverse deflection is partitioned into bending and shear components, resulting in a simple and universal elastic buckling model. The Euler-Lagrange equation is used to obtain the equations governing the motion of the skew thick microplates. Obtaining an analytical buckling solution for general boundary supported skew microplates is challenging; therefore, a C1-type four-node 32-DOF differential quadrature finite element is developed. The element stiffness and geometric stiffness matrices are derived according to the minimum potential energy principle. Significant parametric studies are conducted with different geometrical dimensions, boundary edges, in-plane loadings, and material length scale parameters. The buckling characteristics of skew microplates are investigated using a combination of unequal biaxial compression loads and in-plane compression and shear loads. Numerical results imply that the buckling modes are influenced by the combination of size effects and skew angle and not by the buckling loads alone.

Multiobjective Optimization Design of Truss Antenna

During the design and manufacturing process of the truss antenna, the surface accuracy of the truss antenna is inherently affected by tolerance. An appropriate optimal design of the truss antenna structure is important to improve surface accuracy. In order to receive the optimal design of the truss structure, this paper adopts the multiobjective optimization algorithm based on an approximate model to optimize the tolerance model with random error. Firstly, considering the influence of the processing and assembly errors of the members on the surface accuracy of the structure, the equilibrium state equation of the truss is established by the principle of minimum potential energy. Then, the relationship between the tolerance and the surface accuracy is obtained by the Monte Carlo method. For improving the computing efficiency of the Monte Carlo method, an approximate model of the truss antenna unit is established, where the rod length tolerance is set as the design variable, and the truss surface accuracy and processing cost are set as the objective functions. Finally, tolerance optimization is carried out by using the multiobjective genetic algorithm. The results indicate that the Pareto solution is obtained with an error less than 10%. Moreover, a set of solutions of the tolerance are obtained which can meet different antenna design requirements. And the results show that the influence of the web rod is significantly greater than that of the bottom rod on the surface accuracy of the structure.

Isogeometric algorithm for one-step inverse forming of sheet metal

The isogeometric one-step inverse forming method aims to predict the shape of blanks and the formability of stamping parts using accurate geometry without meshing. In this paper, the discrete surface Ricci flow algorithm in computational conformal geometry is introduced as an unfolding algorithm for one-step inverse forming of isogeometric membrane elements. A combination of discrete surface Ricci flow algorithm and elastic iteration is proposed to predict the initial solution. It is proved applicable, even for the extremely curled parts which have negative angles and vertical walls. In addition, we propose a method that the external force of the punch is equivalent to the control point by means of Greville abscissae. A nonlinear equation system is established based on the principle of minimum potential energy. The Newton–Raphson iteration is used to calculate the blank shape and the thickness distribution of the stamping parts. The examples of square box, flower box, and L shape prove that the algorithm can predict the contour of the initial configuration and the formability of sheet metal stamping.