Minimum Total Potential Energy Research Articles

Buckling and vibration analysis of axially functionally graded nanobeam based on local stress- and strain-driven two-phase local/nonlocal integral models

This paper presented an efficient finite element formulation for analysis of the buckling and free vibration of the axial functionally graded (AFG) Euler-Bernoulli nanobeam based on minimum total potential energy principle with linear constraints. The stress- and strain-driven two-phase local/nonlocal integral models are utilized to address small scale effects. The material properties of the AFG nanobeam vary exponentially along the axial direction. The nonlocal integral constitutive equation can be equivalently transformed into a differential equation with two constitutive boundary conditions. The governing equation and standard boundary conditions are derived via Hamilton's principle. The weak form associated with total potential energy is derived and the constitutive boundary conditions can be converted to external forces at the boundaries. By employing a numerical solution methodology on the basis of the finite element method (FEM) together with Lagrangian multiplier method (LMM), a unified finite element formulation based on stress- and strain-driven two-phase local/nonlocal elasticity is constructed to obtain the critical buckling load and vibration frequency. Meanwhile, the numerical results obtained through generalized differential quadrature method (GDQM) validate the efficiency and accuracy of the present results. The influence of gradient index, nonlocal parameter and local phase parameter on buckling and free vibration of AFG Euler-Bernoulli nanobeam is investigated in detail under different nonlocal models and boundary conditions. It is demonstrated that the gradient index has more effect on mode shapes than the buckling load and vibration frequency between the local stress- and strain-driven two-phase local/nonlocal models.

A new model for spatial rods incorporating surface energy effects

A new non-classical model for spatial rods incorporating surface energy effects is developed using a surface elasticity theory. A variational formulation based on the principle of minimum total potential energy is employed, which leads to the simultaneous determination of the equilibrium equations and complete boundary conditions. The newly developed spatial rod model contains three surface elasticity constants to account for surface energy effects. The new model recovers the classical elasticity-based Kirchhoff rod model as a special case when the surface energy effects are not considered. To illustrate the new spatial rod model, two sample problems are analytically solved by directly applying the general formulas derived. The first one is the buckling of an elastic rod of circular cross-section with fixed-pinned supports, and the other is the equilibrium analysis of a helical rod deformed from a straight rod. An analytical formula is derived for the critical buckling load required to perturb the axially compressed straight rod, and two closed-form expressions are obtained for the force and couple needed in deforming the helical rod. These formulas reduce to those based on classical elasticity when the surface energy effects are suppressed. For the buckling problem, it is found that the critical buckling load predicted by the current new model is always higher than that given by the classical elasticity-based model, and the difference between the two sets of predicted values is significantly large when the radius of the rod is sufficiently small but diminishes as the rod radius increases. For the helical rod problem, the numerical results reveal that the force and couple predicted by the current model are, respectively, significantly larger and smaller than those predicted by the classical model when the rod radius is very small, but the difference is diminishing with the increase of the rod radius.

On the gradient-enhanced damage model for a hyperelastic material

An approach for modeling damage in the material in a geometrically nonlinear setting is studied. We consider a gradient-enhanced damage approach where the damage variable is taken as an independent variable, and the fluxes conjugated to the damage gradient are computed. The concept of maximum dissipation is used to derive the global Kuhn–Tucker conditions. The total free energy function consists of two nonlocal terms corresponding to referential gradient of the nonlocal damage variable and a penalty term to bridge the local damage variable and the nonlocal damage variable. Based on the principle of minimum total potential energy, a coupled system of Euler–Lagrange equations is obtained and solved in weak form. These nonlinear system of equations is solved by a standard Newton–Raphson-type solution scheme. The regularization capabilities, as well as the accuracy and efficiency of the proposed damage models are demonstrated by some standard benchmark examples.

A conic programming approach to the wrinkling of pneumatic membranes using convex potentials

This paper presents a novel conic programming approach for modeling the wrinkling of isotropic hyperelastic membranes subject to nonlinear loading and finite strains, employing convex potentials derived from tension field theory. The incompressible neo-Hookean strain energy is recast as a minimization problem over a set of cones, with a semidefinite constraint on the deformed surface metric. To address pneumatic membranes, a linearized volumetric potential is introduced, reestablishing convexity for follower forces, and Boyle's law is expressed as a minimization problem over the exponential cone.The primal–dual interior point method solves the total potential energy minimization problem, with convergence verified using the analytical solution of a sheared planar membrane. The proposed model, applied to examples of increasing complexity, reveals intriguing membrane behaviors rarely discussed in existing literature. The method is shown to be robust, even when handling highly wrinkled membranes, which are challenging to address using direct nonlinear equilibrium analysis or nonlinear interior point methods.Implementation using high-level automatic code generation tools (FEniCS) results in concise, extensible code. The findings point to several avenues for future research, such as exploring complex material models, mesh refinement, dynamics, and parametric design. Additionally, the linearization process implies potential applicability of the methods to nonlinear problems beyond membrane mechanics.

Modeling and analysis of radial throw of micro-milling tool based on tool-attachment errors in high-speed rotating

In the process of micro-milling complex structures, radial throw is likely to cause lower dimensional/form accuracy, which will severely affect the performance of the components. Therefore, it is necessary to investigate the source contributing to the radial throw. According to the kinematic process of the radial throw arising from the attachment errors of the micro milling cutters, a mathematical model considered the attachment errors and the variable section geometry of the micro milling cutters was established to predict the radial throw on the basis of the principle of minimum total potential energy and the Timoshenko beam theory. In comparison to the mathematical model, the proposed finite element model (FEM) presetting the attachment errors shows the effect of modal vibration and the variation of radial throw with attachment errors. To further investigate the environmental factors influencing the radial throw, a measurement method based on the laser measurement system is presented to achieve the on-machine measurement of runout, and a specific extraction method of attachment errors was proposed to help validate the mathematical model. The difference between the results of the mathematical model, FEM, and the experiment is not exceeding 10 % within 3 μm of the maximum runout. Radial throw varies strongly with attachment errors, and the high rotation speed will aggravate the system instability caused by the attachment errors. The main contributor among experimental factors to the variation of runout is the form accuracy of micro milling cutters, accounting for 4.7 % at the highest. This method can provide a theoretical basis for the sources of unbalanced cutting force analysis and the improvement of dimensional/form accuracy.

Elasticity and stability of corrugated conical shells with diverse orthotropy

Curved thin-walled structures have attracted significant attention within engineering communities thanks to their remarkable properties, including high load-bearing capacity, tailorable directional stiffness, adaptability, and lightweight characteristics. The geometric intricacies of these structures not only influence their mechanical behavior but also provide a vast design space for exploring new frontiers in structural development. This study investigated a unique form of curved thin-walled structure, namely the corrugated annular conical shell, which exhibits diverse degrees of cylindrical orthotropy across various corrugation patterns. An analytical model was formulated based on the minimum total potential energy principle and first-order shear deformable shell theory to predict the non-monotonous force-deflection relationships and non-uniform stress distributions in corrugated conical shells when axially loaded under compression. Within this novel framework, the local undulation shape, corrugation pitch, and amplitude emerged as influential determinants of the responses of corrugated shells, intrinsically intertwined with the polar and shear orthotropy ratios. The accuracy of our analytical model was verified by comparing its results to those obtained using the finite element method and experimental measurements. A machine learning method was employed to classify the stability behaviors of corrugated annular conical shells and derive comprehensive design maps for the first time. Corrugated annular conical shells hold enticing potential for applications as adaptive spring units or building blocks for more sophisticated architectural metamaterials.

Effect of graphene reinforcement distribution on energy release rate and vibration of thermally pre/post-buckled delaminated composite plates

The nonlinear thermal stability responses of functionally graded graphene-reinforced composite (FG-GRC) laminated plates with single or multiple through-the-width delaminations, are investigated in the article. The possibility of delamination growth is evaluated using the three-dimensional crack tip element (3D-CTE) method, which determines the energy release rate (ERR) at the delamination edge. Furthermore, the influence of delamination configurations and the types of graphene reinforcement distribution patterns, on the free vibration of FG-GRC delaminated plates in thermally pre/post-buckled regimes is evaluated. The von Karman geometrical nonlinearity is adopted in a solution based on the layerwise third-order shear deformation theory (TSDT). The nonlinear equilibrium equations derived by the minimum total potential energy principle are solved using the Ritz method in conjunction with the Newton–Raphson iterative procedure. A three-dimensional finite element model is also developed using ABAQUS to corroborate the accuracy of the theoretical results. Parametric studies reveal that while the FGX graphene distribution pattern improves the bending stiffness of delaminated composite plates more than FGA, the ERR at the near surface delamination edge in FGX pattern is twice as high as that in FGA. This implies that the possibility of delamination propagation in FGX plates can be higher than FGA. Additionally, while the fundamental frequency of the FGX plate with and without delamination exceeds the frequency of other graphene distribution patterns at the reference temperature, its natural frequency is the lowest among all patterns in the post-buckled regime.

Nonlinear analysis of perforated composite plates under compression by experimental, semi-analytical and numerical methods

The purpose of this paper is to investigate the effects of different cutout shapes on the nonlinear behavior of perforated composite plates using experimental, numerical and semi-analytical methods and to present their capabilities. For analysis, the composite plates are subjected to a compressive load and different out-of-plane boundary conditions are applied to the lateral edges. Investigations are performed by various methods such as experimental tests, finite element method and a semi-analytical technique. For laboratory tests, a fixture has been designed and built that can create different boundary conditions on the lateral edges of perforated plates and can also test laminates up to 6 mm thick. On laminates, three types of openings are created in square, rectangular and elliptical shapes and the perforated laminates are tested with two types of boundary conditions, including clamped on all four edges (CCCC) and simply on two lateral edges and clamped on two ends (SSCC). These experiments are also simulated in the Abaqus finite element package, taking into account the considerations raised and with the nonlinear Riks method. Also, with similar assumptions in experimental and numerical methods, a semi-analytical method has been developed to compare the results and capabilities of different methods in nonlinear analysis of perforated laminates with different boundary conditions. In the last method, the displacement fields are approximated using Legendre functions and assuming the first-order shear deformation theory. Then, the nonlinear system of equilibrium equations governing the perforated plates is obtained through the principle of minimum total potential energy and is solved by the common method of quadratic extrapolation technique. Finally, the capabilities, advantages and disadvantages of each method, especially the fixtures made for this research, are discussed.

Bending information and dynamic stability of the laminated axisymmetric nanoplate on the linear and torsional elastic substrate based on zigzag theory

In the current work, for the first time, bending and vibration information of the axisymmetric thick annular nanoplate made of laminated composite layers is studied. The structure is placed on Winkler-Pasternak substrate with elastic and tortional coefficients. Initial stress as an external load is also considered in the composite structure. For modeling the mechanical deformation in layer-wise composite materials, zigzag theory that its functions themselves are continues in the interface of layers is presented. By coupled quasi-3D new refined theory (Q3D-NRT) and nonlocal strain gradient theory are used for obtaining the nonclassical governing equations a general boundary condition of the current composite nanoplate. Also, for obtaining the static and dynamic equations of the current nanostructure, Hamilton’s, and Minimum total potential energy principles are used, respectively. Via differential quadrature method with weighting factors and nth-order derivative, the nonclassical governing equations with the aid of size-dependent boundary conditions are solved. By comparing the current results with the outputs of previous reports, the accuracy of the results is shown.

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.

In-plane buckling of flexibly bonded three-layer pinned-fixed half-sine shallow arches

In this paper, the nonlinear in-plane instability of symmetrically layered half-sine shallow arches with interlayer slip, which are soft-hinged supported on both ends, is studied. The equations of nonlinear equilibrium are derived with the principle of the minimum of total potential energy under the assumption that for each layer separately the kinematic hypotheses of the Euler–Bernoulli theory are valid. Based on these equations, analytical expressions for limit point buckling and bifurcation buckling are derived for a sine half-wave curved shallow arch subjected to a sine half-wave radial loading. In particular, closed-form solutions for the critical loads, the limit points, the bifurcation points, and the nonlinear pre- and post-buckling equilibrium paths are provided. To validate these relations, in two example problems the analytically obtained solutions are compared with the results of a finite element analysis with plane stress continuum elements, where the deflection and the interlayer slips in the pre- and post-buckling equilibrium path are examined. The derived relationships provide comprehensive insight into the in-plane (in-)stability of shallow arches with interlayer slip without the need for costly numerical parameter studies.

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.

Prediction of tunnelling induced ground movement in clay using principle of minimum total potential energy

The accurate prediction of tunnelling induced ground movement is of practical and theoretical significance. This paper proposes a mixed empirical-analytical method based on the principle of minimum total potential energy to predict the ground movement due to tunnelling in undrained clay. The proposed method considers the plastic behaviour of the soil. Based on the theory of curvilinear Euler beam, the proposed method considers the soil-lining interaction, which is simplified as a constant force in classical MSD methods. Moreover, the proposed method can deal with the deformation mechanism with multiple undetermined parameters. The tunnelling-induced ground movement can be found by minimizing the total potential energy of the concerned system composed of soil and lining. Four well-documented tunnelling projects are adopted to evaluate the applicability of the proposed method. The results reveal that our method provides better predictions of both surface and subsurface settlements above the tunnel than the other two semi-analytical methods. Moreover, our method excels the other two methods in predicting the tunnelling projects with large gap parameter values as it considers the plastic behaviour of the soil.

A numerical study on the installation configuration design of compliant vertical access riser (CVAR) in deepwater

In oil and gas transportation, the compliant vertical access riser (CVAR) system conducts economic and operational advantages due to its special configuration. In this paper, the static performance of the CVAR is studied by an efficient numerical method. According to the unique configuration of the CVAR, the mathematical model is established based on minimum total potential energy principle. The equilibrium equations are discretized by finite difference method and then solved by Newton-Raphson technique. The accuracy of the numerical method is verified by OrcaFlex software. Detailed parametric analysis is carried out based on practical engineering requirements. The effects of buoyancy section's configuration including buoyancy factor, buoyancy section's length and buoyancy section's location, and external environmental factors including wellhead offset distance, water depth and current velocity on installation shape and mechanical properties of the CVAR are analyzed. The research shows that the presented numerical method has good applicability and flexibility for the early installation configuration design of marine risers which have no touchdown section but only one touchdown point and are equipped with multiple buoyancy blocks subjected to forces of different directions and magnitudes, such as CVARs.

Nonlinear thermo-elastic stability of variable stiffness curvilinear fibres based layered composite beams by shear deformable trigonometric beam model coupled with modified constitutive equations

Nonlinear thermo-elastic buckling characteristics of composite variable stiffness beam with layers making use of curvilinear fibres under thermal environment is attempted here. The model is based on a shear deformable theory introducing trigonometric function, and considering von Kármán’s assumptions based geometrical nonlinear effect. The beam constitutive equation is modified according to the stress-free situation in the width direction of beam-Poisson’s effect in the formulation for predicting the behaviour of general lay-up composite beams. By the principle of minimum total potential energy, the governing equations in terms of incremental stiffness matrices are formed introducing general beam finite element. The global equilibrium equations formulated are solved for envisaging the post-buckling path through eigenvalue analysis iteratively, thus establishing the relationship of thermal temperature against moderate amplitude level of beam deflection. A systematic parametric analysis considering different lamina properties such as curvilinear fibre path angles and including lay-up sequences, thermal expansion coefficient, mixed laminate combining straight and curvilinear fibres-based layers is carried out on thermo-structural stability of curvilinear fibre-based beams. Also, the influence of geometric factors, flexible beam end support, and variation in thermal profile, etc. over the stability behaviour of beam is examined.

A numerical study on the installation configuration design of steep wave riser (SWR) in deep water

A new calculation procedure is presented for the structural analysis of a steep wave riser (SWR) system. The equilibrium equation of the SWR is established based on Minimum total potential energy principle. The coupled equations are discretized by the finite difference method and then solved by Newton-Raphson technique. The accuracy of the numerical model is verified by Orcaflex software. Traditional parametric analysis of buoyancy factor, buoyancy section’s length, buoyancy section’s location and ocean current velocity are performed. An idea of engineering evaluation of the SWR’s working condition from the perspective of hang-off angle, tension, and bending radius is proposed. Through comparison, it is found that the hang-off angle is the main parameter affecting the installation safety of the SWR. And the allowable working water depth and installation span are determined. In addition, the optimization idea of the SWR’s configuration based on S-shape coefficient and hang-off angle is put forward, and the optimal configuration of a specific SWR is analyzed.

Human-like motion planning of robotic arms based on human arm motion patterns

AbstractRobots with human-like appearances and structures are usually well accepted in the human–robot interaction. However, compared with human-like appearances and structures, the human-like motion plays a much more critical role in improving the efficiency and safety of the human–robot interaction. This paper develops a human-like motion planner based on human arm motion patterns (HAMPs) to fulfill the human–robot object handover tasks. First, a handover task is divided into two sub-tasks, that is, pick-up and delivery, and HAMPs are extracted for these two sub-tasks separately. The resulting HAMPs are analyzed, and a method is proposed to select HAMPs that can represent the characteristics of the human arm motion. Then the factors affecting the duration of the movement primitives are analyzed, and the relationship between the duration of the movement primitives and these factors is determined. Based on the selected HAMP and the computed duration of the movement primitives, a human-like motion planning framework is developed to generate the human-like motion for the robotic arms. Finally, this motion planner is verified by the human–robot handover experiments using a KUKA IIWA robot. It shows that the resulting trajectories can correctly reflect the relative relationship between the joints in the human arm motion and are very close to the recorded human arm trajectories. Furthermore, the proposed motion planning method is compared with the motion planning method based on minimum total potential energy. The results show that the proposed method can generate more human-like motion.

Large Deflection Analysis of Clamped Thin Rectangular Isotropic Plate under the Action of a Uniform Distributed Load

This present paper deals with large deflection of clamped thin rectangular isotropic plate under uniformly distributed load. The analysis of clamped plates is of great important and its structural behavior has not been completely understood. Previous studies have made a great in road in the analysis of clamped plates through varied approaches mostly assuming trigonometric series as displacement field. A variational method based on the principle of minimization of total potential energy has been employed through assumed displacement field. Therefore, the von Karman differential equation was decoupled through direct integration retaining its non-linear character. The aim of this present studies is to provide solution to analysis structural behavior of clamped (CCCC) thin rectangular isotropic plate under uniform distributed. The analysis was accomplished through a theoretical formulation of shape function based on polynomial series and subsequently, the formulated shape function was used on Ritz energy method. The resulting functional was minimized to obtain a general amplitude equation of the form K1∆3+K2∆+K3. Where K1, K2 and K3 are coefficients of amplitude equation and ∆ is the deflection coefficient (factor). Newton-Raphson method was used to evaluate the deflection coefficient. Values of ∆ from Timoshenko and that from present study were compared with an aspect ratios ranging from 1.0 to 1.5 with an increment of 0.1. From results obtained, the average percentage difference for the pervious and present studies is 3.7646%, The percentage difference for the plate was within acceptable limit of 0.05 or 5% level of significance in statistics. It is shown from the comparison that good agreement exists between the present and previous works and that this approach provides simply, direct, straightforward and highly accurate solutions for this family of problems.

Folding behaviour of a deployable composite cabin for space habitats – Part 2: Analytical investigation

In the Part 1 of two companion papers, a deployable composite cabin (DCC) for space habitats was introduced and its folding behaviour was comprehensively studied. This Part 2 paper is dedicated to present an analytical model for rapidly predicting the folding behaviour of the DCC. Based on the classical Euler-Bernoulli beam model, the geometric and physical equations describing the folding behaviour of the DCC were established in this paper. By combining Maclaurin series expansion, orthogonal Chebyshev polynomials, Galerkin method and Harmonic balance method, analytical models for predicting the load-displacement curve and shape function of the DCC were presented. The analytical solution of compressive strength was derived from the principle of minimum total complementary potential energy, classical laminate theory and Tsai-Hill failure criterion. In order to validate the analytical model proposed in this paper, the prediction results were compared with experimental results and numerical simulation results in the Part 1, and the three were in good agreement. Finally, the effect of geometric parameters (i.e. arc length, axial length and thickness of the thin plate) on the folding behaviour of the DCC was further analyzed by using the analytical model.

Geometrically nonlinear large-deflection analysis of heated functionally graded composite panels with single and multiple perforations

In this article, the nonlinear flexural analysis of perforated functionally graded composite panel is performed under heat conduction and uniform/sinusoidal pressure. Here, functionally graded composite panel is modeled with single (1 × 1) and multiple (2 × 2, 3 × 3) perforations having identical surface area. A geometrically nonlinear mathematical model is adopted using the Green-Lagrange strain field via the higher-order shear-deformation mid-plane kinematics. The overall temperature-dependent elastic and thermal properties of the functionally graded (metal/ceramic) material are computed using the power-law function via Voigt’s homogenization scheme. The equilibrium equations are derived through the minimum total potential energy principle via isoparametric finite element approximations and solved further by adopting Picard successive technique. The convergence and validation of the model are carried out by performing mesh refinement and comparison tests, respectively. In adding, new numerical results for single/multiple perforated FGM panels are illustrated at various sets of parametric combinations, and compared in detail.