Minimum Total Potential Energy Research Articles

Base functions effect in stability finite element analysis of compressed plate

Geometric nonlinear solution of a compressed plate is presented in this paper. Basic assumptions are specified and incremental conditional equations are derived from the variational principle of minimum of total potential energy. Full Newton-Raphson procedure, in which the stiffness matrix is updated at every equilibrium iteration, has been applied for solution. The importance of modifications of base functions for solving geometric nonlinear problems is analysed. The solved example is presented, the differences are compared and explained.

Nonlinear and post-buckling responses of FGM plates with oblique elliptical cutouts using plate assembly technique

The aim of this study is to obtain the nonlinear and post-buckling responses of relatively thick functionally graded plates with oblique elliptical cutouts using a new semi-analytical approach. To model the oblique elliptical hole in a FGM plate, six plate-elements are used and the connection between these elements is provided by the well-known Penalty method. Therefore, the semi-analytical technique used in this paper is known as the plate assembly technique. In order to take into account for functionality of the material in a perforated plate, the volume fraction of the material constituents follows a simple power law distribution. Since the FGM perforated plates are relatively thick in this research, the structural model is assumed to be the first order shear deformation theory and Von-Karman\'s assumptions are used to incorporate geometric nonlinearity. The equilibrium equations for FGM plates containing elliptical holes are obtained by the principle of minimum of total potential energy. The obtained nonlinear equilibrium equations are solved numerically using the quadratic extrapolation technique. Various sets of boundary conditions for FGM plates and different cutout sizes and orientations are assumed here and their effects on nonlinear response of plates under compressive loads are examined.

Bending analysis of sandwich shell panels with exponentially graded core

This paper presents the bending analysis of sandwich shell panels having exponentially graded core with top and bottom facesheets which are composed of pure ceramic and pure metal respectively. The layerwise finite element formulation is developed by using a displacement field according to first-order shear deformation theory for each layer that maintains continuity at the layer interface. The governing differential equation for the present problem is obtained using the principle of minimum total potential energy. The correctness of the presented finite element formulation is established by validating the results obtained with the benchmark. Next, the bending responses for exponentially graded sandwich shell panels (E-FGM-SW) are obtained. The effect of parameters like curvature ratio and material properties on the bending responses of exponentially graded sandwich shell panels is investigated. It has been shown here that present layerwise finite element formulation is simple, accurate and widely applicable.

Free vibration analysis of functionally graded beams with variable cross-section by the differential quadrature method based on the nonlocal theory

This paper attempts to investigate the free vibration of functionally graded material beams with nonuniform width based on the nonlocal elasticity theory. The theoretical formulations are established following the Euler–Bernoulli beam theory, and the governing equations of motion of the system are derived from the minimum total potential energy principle using the nonlocal elasticity theory. In addition, the Differential Quadrature Method (DQM) is applied, along with the Chebyshev-Gauss-Lobatto polynomials, in order to determine the weighting coefficient matrices. Furthermore, the effects of the nonlocal parameter, cross-section area of the functionally graded material (FGM) beam and various boundary conditions on the natural frequencies are examined. It is observed that the nonlocal parameter and boundary conditions significantly influence the natural frequencies of the functionally graded material beam cross-section. The results obtained, using the Differential Quadrature Method (DQM) under various boundary conditions, are found in good agreement with analytical and numerical results available in the literature.

Size-dependent static and dynamic analysis of Reddy-type micro-beams by strain gradient differential quadrature finite element method

Abstract This paper proposes a strain gradient differential quadrature finite element method to analyze the size-dependent static and dynamic behaviour of Reddy-type micro-beams. This element has 6 of freedom per node and avoids the exploitation of shape functions. A sixth-order differential quadrature-based geometric mapping scheme is constructed to realize the higher-order continuity requirements of kinematic variables. And then, it is combined with the minimum total potential energy principle to derive the motion equation of a generic element. Afterwards, several numerical examples are provided to establish the validity of the developed element. Finally, we utilize this method to analyze the static bending, free vibration, and linear buckling characteristics of uniform and stepped micro-beams. Numerical results show that the current element has prominent convergence and adaptability advantages over the classical shape function-based element. Besides, the size-dependence of vibration and critical buckling mode shapes of micro-beams is demonstrated in the graphical form for the first time.

Strain gradient differential quadrature Kirchhoff plate finite element with the C2 partial compatibility

This paper constructs a four-node Kirchhoff plate element considering dilatation, deviatoric stretch and rotation gradient effects to address the general boundary value problems of size-dependent isotropic thin micro-plates. This element benefits from the merits of differential quadrature method (DQM) and finite element method (FEM) and possesses four nodal displacement parameters at each node, i.e., deflection, its two first partial derivatives and one second mixed partial derivative with respect to two in-plane coordinates. To guarantee the C2 partial compatibility among neighboring elements, we establish a novel DQ-based geometric mapping scheme relating the deflection values at Gauss-Lobatto quadrature points to the displacement parameters at four nodes. By applying the DQ rule, the Gauss-Lobatto quadrature rule and the developed mapping scheme, the total potential energy of a generic gradient-elastic Kirchhoff plate element is represented as a function of nodal displacement parameters. The element formulation is derived using the minimum total potential energy principle. Several numerical examples are provided to demonstrate the validity of the proposed method and explore the static bending, free vibration and critical buckling behavior of thin micro-plates. It is validated that the size-dependence of vibration and critical buckling mode shapes of thin micro-plates can be observed in some cases.

Using orthotropic viscoelastic representative elements for C1 -continuous zigzag dynamic response assessment of sandwich FG circular plates with unevenly damaged adhesive layers

In the present analysis, dynamic responses of sandwich plates with uniformly or nonuniformly damaged adhesive layers are compared with those of the perfectly bonded plates. In this regard, a direction-dependent (orthotropic) Kelvin–Voigt viscoelastic representative element whose coefficients vary in the radial direction is employed to model the irregular/damaged adhesive layer. It is the first time that the effects of the nonuniform stiffness/viscoelasticity distribution of the adhesive layer on the dynamic behaviors are evaluated, especially for circular plates. The core or face sheets may be made from homogenous or heterogeneous materials. To extract accurate results, a C1 -continuous zigzag plate theory with interlaminar transverse stress continuity is employed. The resulting governing equations are extracted based on the principle of minimum total potential energy and solved by a Kantorovich-type power series solution in conjunction with the fourth-order Runge-Kutta numerical time integration scheme. Results reveal that the influence of the uniformly and unevenly damaged layer on the dynamic responses of the damaged sandwich plates is especially remarkable when rigid layers are utilized. Moreover, online health monitoring based on the deviation of the dynamic amplitude with respect to a predefined target is more efficient than using the indirect changes in the natural frequencies and phase shifts.

Static analysis of two-directional functionally graded cylindrical panels under the effect of symmetric loads using finite element method (FEM)

This paper offers the linear analysis of the static behavior of two directional functionally graded(2D-FG) cylindrical panels under the effect of internal symmetric loads. The mechanicalproperties of the cylindrical panel are given to be changed simultaneously through the thicknessand longitudinal directions as a function to the volume fraction of the constituents by a simplepower-law distribution. Based on Sander’s first order shear deformation shell theory (FSDT), theequations of motion for (2D-FG) panels are derived using the principle of minimum totalpotential energy (MPE). The finite element method (FEM) as an effective numerical tool isutilized to solve the equations of motion. The model has been compared with those available inthe literature and it observed good correspondence. The influences of the material variationalong the thickness and longitudinal directions, geometrical parameters, boundary conditionsand load parameters on the panel deformation are studied in detail.

Modification of the Time-Effective Moduli of Viscoelastic Bodies

The problem on constructing new time-effective characteristics of a linear viscoelastic body is considered. The initial medium is modeled as a viscoelastic composite. One its part has properties determined by time-effective moduli of the Lagrangian type, but the other part has properties determined by moduli of the Castigliano type. This model makes it possible to employ the methods of mechanics of composite materials, for example, to formulate the effective Voigt and Reuss moduli and to construct iterative transformations narrowing the Voigt and Reuss fork. These transformations are constructed in such a way that, at each iteration, the inequalities following from the minimum total potential energy principle and the theorem of complementary work are satisfied for the effective moduli. It is shown that, at each moment of time t, the sequences of iteratively transformed Voigt and Reuss moduli converge to the same limit, equal to the geometric mean of their initial values. By the example of the problem on bending of a viscoelastic plate, the approximate solutions obtained on the basis of the new time-effective characteristics found, are compared with an analytical solution. Their good agreement points to a high accuracy of the approximate solutions.

Thermo-Elastic Creep Analysis and Life Assessment of Thick Truncated Conical Shells with Variable Thickness

A semi-analytical method is presented to investigate time-dependent thermo-elastic creep behavior and life assessment of thick truncated conical shells with variable thickness subjected to internal pressure and thermal load. Based on the first-order shear deformation theory (FSDT), equilibrium equations and boundary conditions are derived using the minimum total potential energy principle. To the best of the researcher’s knowledge, in previous studies, thermo-elastic creep analysis of conical shell with variable thickness based on the FSDT has not been investigated. Norton’s law is assumed as the material creep constitutive model. The multilayered method is proposed to solve the resulting equations, which yields an accurate solution. Subsequently, the stresses at different creep times can be obtained by means of an iterative approach. Using Robinson’s linear life fraction damage rule, the creep damages of conical shells are determined and Larson–Miller parameter (LMP) is employed for assessing the remaining life. The results of the proposed approach are validated with those of the finite element method (FEM) and good agreement was found. The results indicate that the present analysis is accurate and computationally efficient.

Constructing 3D Self-Supporting Surfaces with Isotropic Stress Using 4D Minimal Hypersurfaces of Revolution

This article presents a new computational framework for constructing 3D self-supporting surfaces with isotropic stress. Inspired by the self-supporting property of catenary and the fact that catenoid (the surface of revolution of the catenary curve) is a minimal surface, we discover the relation between 3D self-supporting surfaces and 4D minimal hypersurfaces (which are 3-manifolds). Lifting the problem into 4D allows us to convert gravitational forces into tensions and reformulate the equilibrium problem to total potential energy minimization, which can be solved using a variational method. We prove that the hyper-generatrix of a 4D minimal hyper-surface of revolution is a 3D self-supporting surface, implying that constructing a 3D self-supporting surface is equivalent to volume minimization. We show that the energy functional is simply the surface’s gravitational potential energy, which in turn can be converted into a surface reconstruction problem with mean curvature constraint. Armed with our theoretical findings, we develop an iterative algorithm to construct 3D self-supporting surfaces from triangle meshes. Our method guarantees convergence and can produce near-regular triangle meshes, thanks to a local mesh refinement strategy similar to centroidal Voronoi tessellation. It also allows users to tune the geometry via specifying either the zero potential surface or its desired volume. We also develop a finite element method to verify the equilibrium condition on 3D triangle meshes. The existing thrust network analysis methods discretize both geometry and material by approximating the continuous stress field through uniaxial singular stresses, making them an ideal tool for analysis and design of beam structures. In contrast, our method works on piecewise linear surfaces with continuous material. Moreover, our method does not require the 3D-to-2D projection, therefore it also works for both height and non-height fields.

Compressive instability of open section nanocomposite struts using a layerwise theory

The application of carbon nanotubes (CNTs) to improve postbuckling characteristics of T-section nanocomposite strut is the subject of this research. A solution procedure according to the layerwise theory along with the first-order shear deformation theory (FSDT) by taking into account the von Karman geometrical nonlinearity is proposed. Nonlinear governing equations are developed in regard to the minimum total potential energy principle, and solved by the aid of the Rayleigh–Ritz method in conjunction with Newton–Raphson method. A comprehensive parametric study is conducted to provide an insight into effects of CNT volume fraction and dispersion, geometrical parameters and boundary conditions on the bifurcation points and paths, in-plane displacement, normal stress and strain, and bending moment of T-section nanocomposite struts. A three dimensional finite element analysis using the ABAQUS commercial software is carried out to verify the obtained results, which shows close agreement. Results indicate that dispersion of CNTs plays a dominant role in bifurcation behavior of nanocomposite struts. The results also show that volume fraction of CNTs has a significant effect on the load-carrying capacity of open section nanocomposite struts. The results of this research shed light into using ultra-high-strength and light-weight open section nanocomposite struts for civil and aerospace applications.

Bi-stability of pressurized electrically actuated flat micro-plates

Despite the usual belief that only curved micro-structures can behave bi-stably, recently, it has been substantiated that flat micro-structure can also experience this type of behavior. In this way, the present paper aims to investigate snapping behavior in pressurized 2-D flat micro-electro-mechanical plates with a rectangular shape. To this end, a size-dependent and geometric non-linear thin plate model based on the modified couple stress theory is employed. The governing equilibrium equations as well as their corresponding boundary conditions are obtained using the principle of the minimum total potential energy. Employing an efficient Galerkin's discretization procedure, the instability thresholds of the system are determined by vanishing the first and the second variations of the total potential energy expression of the system. The present findings for non-pressurized systems are verified by those published in previous studies. Furthermore, the results concerning the cases under the differential pressures are validated through direct comparison with those extracted from 3-D finite element simulations carried out in COMSOL Multiphysics commercial software. Results reveal that flat micro-plates may experience snap-through instability when they are subjected to some certain values of the differential pressure in the opposite direction of the electrical field. To find the conditions in which a flat micro-plate may face with the snapping behavior, a large number of systems with different material and geometric properties are analyzed. Employing this detailed parametric study, snapping criteria which provide the combination of the system properties and the level of the applied pressure required for prompting snap-through instability are then introduced.

Nonlinear stability analysis of piecewise actuated piezoelectric microstructures

The main objective of this research is to provide a general nonlinear model of adjustable piezoelectric microwires with the ability to tune the stability conditions. In order to increase the controllability and improve system characteristics, only a part of the substrate is electrostatically actuated and the piezoelectric voltage is also applied. The governing equation of equilibrium (EOE) is derived from the principle of minimum total potential energy. The influences of the surface layer, size dependency, piezoelectricity, and dispersion forces are also included simultaneously. To solve the nonlinear differential equation, a numerical method is implemented and the obtained results are validated with available experimental and numerical results. Afterward, a set of parametric studies is carried out to examine the coupled effects of piezo-voltage, length/position of non-actuated pieces, nonlinear curvature, and molecular forces on the microresonators. It is found that the beam deflection and the pull-in voltage have sensitive-dependence on the system behavior. Furthermore, the beam deflection can increase or decrease with consideration of different positions of non-actuated pieces. This research is expected to fill a gap in the state of the art of the piezoelectric microstructures and present relevant results that are instrumental in the investigation of advanced actuated microdevices.

3D stress analysis of generally laminated piezoelectric plates with electromechanical coupling effects

• Interlaminar stresses of laminated piezoelectric plates are obtained using analytical solution. • Electromechanical coupling effect of the piezoelectric plate is taken into account. • The solution is obtained using 3-D multi-term extended Kantorovich method. • Coupling influence on the edge effects of piezolaminated plates with various layups is investigated. In this paper, the interlaminar stresses of generally laminated piezoelectric (PZT) plates are presented. The electromechanical coupling effect of the piezoelectric plate is considered and the governing equations and boundary conditions are derived using the principle of minimum total potential energy. The solution procedure is a three-dimensional multi-term extended Kantorovich method (3DMTEKM). The objective of this paper is to study coupling influence on the edge effects of piezolaminated plates with finite dimensions and arbitrary lay-ups under uniform axial strain. These results can provide a benchmark for checking the accuracy of the other numerical methods or two-dimensional laminate theories. To verify the accuracy of the 3DMTEKM, special cases such as cross-ply or symmetric laminates are investigated and the results are compared with other analytical solutions available in the literature. Excellent agreement is achieved and then other numerical results are presented for general cases. Numerical examples imply on the singular behavior of interlaminar normal/shear stresses and electric field strength components near the edges of the piezolaminated plates. The coupling influence on the free edge effect with respect to the lay-ups of piezoelectric plate is studied in several examples.

Elasto-plastic analysis of thermo-mechanically loaded functionally graded disks by an iterative variational method

The present work concentrates on the propagation of elastic-plastic front of functionally graded rotating disk under centrifugal and thermal load in post-elastic regime. The modeling of FGM is done for continuous distribution of ceramic and metal constituents along the radial direction by using power law variation of volume fraction. Minimum total potential energy principle in Hencky’s deformation theory of plasticity is used to obtain the governing equations in post-elastic regime. By using von-Mises yield criterion and assuming bilinear material behaviour, the solution of the unknown displacement field is achieved from the governing equation in an iterative manner. The solution algorithm is implemented by using MATLAB® computational simulation software. A comparative study is carried out for the present mathematical model for some reduced problems and good agreement is found. The effect of material parameters, disk geometries and types of temperature distribution field on the elasto-plastic performance of FG disk is investigated and some numerical results are presented in dimensionless form. The angular speeds and temperatures corresponding to yield initiation and plastic collapse are provided for different disk geometries, material parameters and temperature distributions. The effects of the problem parameters on the limiting angular speeds, limiting temperatures and the corresponding limit load factors are provided.

Strain gradient differential quadrature beam finite elements

In this paper, the superiorities of finite element method (FEM) and differential quadrature method (DQM) are blended to construct two types of beam elements corresponding to modified strain gradient -Bernoulli and Timoshenko beam models respectively. The two elements, being independent of shape functions and introducing three kinds of strain gradient effects, possess 3-DOFs (degrees of freedom) and 4-DOFs separately at each node. The Lagrange interpolation formula is employed to establish the trial functions of deflection and or rotation at Gauss-Lobatto quadrature points. To realize the inner-element compatibility condition, displacement parameters of quadrature points are converted into those of element nodes through a DQ-based mapping strategy. Total potential energy functional for each beam model is discretized in terms of nodal displacement parameters. The associated differential quadrature finite element formulations are derived by the minimum total potential energy principle. Specific expressions of element stiffness and mass matrices and nodal vector are provided. Numerical examples concerning with static bending, free vibration and buckling of macro/micro-beams are presented to demonstrate the availability of the proposed elements.

Stability of elastic column with spring supports at both clamped ends

This paper analyzes the large deflections and stability of an elastic column. The column is supported by a spring at each of its clamped ends, and is also subject to a conservative compressive terminal force. The equilibrium equation is derived from the principle of the minimum total potential energy, and its solution is expressed using Jacobian elliptic functions. The stability is determined with the help of the Jacobi criterion, and the resulting bifurcation diagrams are also determined.

Size-Dependent Green’s Function for Bending of Circular Micro Plates Under Eccentric Load

In this paper, a Green’s function is developed for bending analysis of micro plates under an asymmetric load. In order to consider the length scale effect, the modified couple stress theory is used. This theory can accurately predict the behavior of micro structures. A thin micro plate is considered and therefore the classical plate theory is utilized. The size dependent governing equilibrium equation of a circular micro plate under an eccentric load is obtained by using the minimum total potential energy principle. This equation is a partial differential equation and it is hard to solve it for an arbitrary loading. A transformation of the coordinate system is introduced to obtain the asymmetric exact solution for deflection of circular micro-plates. By using the obtained size dependent Green’s function, the bending behavior of microplates under arbitrary loads can be easily defined. The results are presented for different asymmetric loads. Also, it is concluded that the length scale has a significant effect on bending of micro plates.

A Soft Ring-Shaped Actuator for Radial Contracting Deformation: Design and Modeling.

Bioinspired movements such as elongation/shortening and bending are extensively studied in the field of soft robotics. However, radial contracting deformation has been overlooked, another typical deformation in nature, such as the constrictive deformation in the human esophagus and bladder. We present a novel soft ring-shaped actuator that generates radial contraction and examine its quasi-static performance under pressurization. This actuator consists of a soft ring-shaped body and a rigid casing. A single circular air chamber embedded concentrically inside the actuator is created through the lost-wax method. A theoretical model of the quasi-static state of the actuator's contracting deformation was formulated in light of the minimum total potential energy principle. Meanwhile, a finite-element method (FEM) model was constructed to stimulate the radial contraction of the actuator. In addition, experiments were conducted to identify the maximum input pressure and measure the deformations at the specific input pressure. When the actuators were pressurized under 20.0 kPa, axisymmetric contractions were achieved. The theoretical model and FEM models produced the predictions in good agreement with experimental results, with slight mean relative differences regarding peak points, 3.4% and 5.4%, respectively. In addition, pilot tests suggest that the ring-shaped actuator can achieve the key tasks of robotic manipulators, such as gripping and holding.