Principle Of Minimum Potential Energy Research Articles

Analysis on the buckling performance of embedded co-cured composite sandwich plates under uniaxial compression condition

The research aims at exploring the buckling behavior of embedded co-cured composite sandwich structure (ECCSS) under static load. In accordance with the research objective, an analytical model, which is based on the interlayer stress continuous theory to formulate governing differential equations for predicting the critical buckling loads of ECCSS, has been established. The theoretical solution that satisfies the boundary conditions is obtained using the principle of minimum potential energy. The results obtained by ANSYS software have been used to corroborate the correctness of the analytical model. In addition, effects of various structural parameters (such as ply angles and skin-to-core thickness ratio) on buckling behavior of ECCSS have been investigated.

A six node refined mixed finite element model for the analysis of fiber reinforced polymer composite beams

A refined 6-node two-dimensional mixed finite element has been developed to analyze laminated composite using the minimum potential energy principle. This model uses four primary variables, two displacements u, w and two transverse stresses σz and τxz at each node. In the formulation process, fundamental elasticity relations, that is, displacement-strain relations, stress–strain relations, and the equilibrium equations, have been explicitly satisfied. The transverse stress terms have been invoked using the displacement-strain relation and then the stress–strain relation. The refinement in the model has been performed using the equilibrium equation in the x and z direction. This formulation is an improvement over the Desai and Ramtekkar (2002) model and contributes to the computational mechanics domain of laminates analysis. Refinement in the formulation has been performed by satisfying the equilibrium equation explicitly at the nodes. This refinement helps in getting better results with a fewer number of elements. Results obtained through the model has been compared with the elastic solution given by Pagano (1969) and the mixed finite element method by Desai and Ramtekkar (2002) model. Results show excellent agreement with the elastic solution and more economical as only 2/3rd number of the element as compared to Desai and Ramtekkar (2002) model is required for the converged results.

On the adhesive nanocontact of a graded coating

This paper analyzes the adhesive nanocontact properties between a rigid cylinder and an exponentially graded coating-substrate structure. While adhesion between the indenter and the graded coating is accommodated by Johnson–Kendall–Roberts adhesive model, surface effects are addressed by the full version of Steigmann–Ogden surface mechanical theory. In the presence of surface effects alone, the fundamental Flamant solution of the graded coating-substrate structure is determined first, with the aid of Fourier integral transforms. The displacement boundary condition under the rigid cylinder and the static equilibrium equation lead to dual Fredholm integral equations of the first kind. The adhesion is subsequently introduced in terms of the principle of minimum potential energy, by which both the adhesive nanocontact length and pressure are uniquely determined. In order to clarify the effects of adhesion and surface elasticity, parametric studies are carried out with respect to adhesive energy density, surface material constants, shear modulus gradation of the coating and indenter size. Surface effects tend to reduce contact length, maximum contact pressure and subsidence displacement. On the other hand, adhesion increases these three properties. Nonetheless, both effects result in higher pull-off forces. In general, the relation between adhesion and surface elasticity can be qualified as competitive. The current study systematically reveals the feature of adhesive nanocontact of a graded coating-substrate system in the presence of both adhesion and surface effects. It helps the community to extend adhesive nanocontact mechanics from homogeneous to graded materials and structures.

The peridynamics-based finite element method (PeriFEM) with adaptive continuous/discrete element implementation for fracture simulation

The classical finite element method has been successfully applied to many engineering problems but not for the cases with space discontinuity. In our previous work, a peridynamics-based finite element method was presented according to the principle of minimum potential energy, which enables discontinuity. As a continuation of the previous work, on the one hand, we derive the peridynamics-based finite element formulation from a new perspective, i.e., the principle of virtual work. On the other hand, we propose an adaptive continuous/discrete element conversion technique, thus the cracks could be described explicitly without increasing the computational cost significantly. Finally, numerical results are executed to verify the proposed method, including the computational cost, the influence of discretization strategy and critical stretch on crack paths, and the comparison of predicted crack paths with experimental results. Numerical results show the efficiency and accuracy of the proposed method.

On uncertainty modeling of thermoelastic vibration for porous nanosandwich beams with gradient core based on nonlocal higher order beam model

In the present paper, the size-dependent random thermoelastic vibration analysis of porous nanosandwich beams with gradient core are demonstrated based on the nonlocal higher order beam model. The nonlocal elasticity theory incorporation with the higher order shear deformation theory is used to describe the kinematic and constitutive relations. The governing equations are obtained from minimum potential energy principle. The finite element method is used to solve the governing equations. Three types of the FG sandwich nanobeams with three types of porous cores are taken into account, and the influences of these structures and porosity types on the deterministic and stochastic behavior of sandwich beams are thoroughly discussed. Initially convergence and validation study is preformed to validate the present theoretical model by comparing the present results with the several previous works and Monte Carlo Simulation (MCS). Further, a comprehensive parametric study is performed to demonstrate the influence of the length to thickness ratio, degree of uncertainty in the material properties, nonlocal parameter, porosity distribution type, porosity volume fraction, temperature change, and gradation of material on the deterministic and stochastic behaviors of the FG nanosandwich beams. The developed model and computed results may be useful in the precise and reliable design of the components of nanodevices and systems.

Formation Mechanism and Modeling Method of Wrinkling Defects in Variable Angle Tow Steering Fiber Placement

Variable angle tow steering technology is capable of manufacturing complex aviation parts with a trajectory of intricate curvature planned based on stress or profile characteristics, which greatly improves the forming efficiency, design flexibility and mechanical properties of composite structures. In view of the forming defects such as buckling and wrinkles caused by the lateral bending of fiber prepreg tow, a theoretical buckling model based on the Rayleigh Ritz method, the principle of minimum potential energy and the viscoelastic foundation is established, in which the adhesion coefficient is characterized by the degree of intimate contact to introduce process parameters. On the basis of the contact mechanics analysis, the distribution of the compaction pressure and bending stress is studied to improve the theoretical model, and the critical buckling load and the minimum radius of the tow under the normal and tangential contact conditions are determined precisely. Finally, the finite element models of compaction and variable angle steering placement are proposed, and the theoretical model and simulation model are verified by corresponding trials. It is demonstrated that defects can be effectively suppressed through optimizing process parameters.

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

Weak form quadrature element methods (QEM) are chosen for determination of asymptotic field coefficients for in-plane V-notch problems. The entire domain is divided into two parts the potential energies of which are computed separately. Stress and displacement descriptions are introduced into the sector domain around the notch tip. Coefficients are extracted from the algebraic equations directly after introducing the minimum potential energy principle. The effect of higher order terms on the local stress and displacement descriptions near V-notch tip is analyzed. High adaptivity and fast convergence of the QEM are demonstrated. Comparisons of several benchmark example results and those in the literature are made to validate the present formulation.

Simulation of membrane deployment accounting for the nonlinear crease effect based on absolute nodal coordinate formulation

With the advantages of lightweight and high stowing efficiency, the origami membrane has a broad application prospect in the space structure. As an improvement of the existing deployment simulations, the behavior of crease is considered and both static and dynamic simulation are conducted in this paper. The origami membrane is modeled as a flexible multibody system and its deployment is simulated based on the absolute nodal coordinate formulation. The nonlinear effect of the crease to resist deployment is integrated into the multibody system via the virtual work principle. The facet is modeled by the thin shell element and the crease is constructed as a nonlinear torsional spring with specific position and gradient constraints. The behavior of the crease is parameterized based on the experimental data. The modeling approach is verified by the numerical/experimental reference of the Z-folding structure in the existing literature. For the multi-crease origami membrane model considering the crease effect, a form-finding method to obtain the initial configuration of the flexible origami is proposed based on the principle of minimum potential energy. The deployment of the classical Miura-ori unit membrane structure is analyzed. The magnitude of driving force is estimated based on the force analysis of rigid deployment. Based on the simulation result of flexible deployment, the driving force is planned to achieve a stable deployment with a constant increase speed in the deployment ratio. This work is expected to provide a reference for the design of space membrane structure, the estimation of driving force, and optimizing the deployment strategy of an origami membrane.

Size-Dependent Buckling and Post-Buckling Analysis of the Functionally Graded Thin Plate Al-Cu Material Based on a Modified Couple Stress Theory.

Size-dependent functionally graded material thin plate buckling and post-buckling problems are considered using the framework of the MCST (Modified Couple Stress Theory). Based on modified couple stress theory and power law, the post-buckling deflection and critical buckling load of simply supported functionally graded material thin plate are derived using Hamilton’s minimum potential energy principle. The analysis compares the simulation results of linear buckling and nonlinear buckling. Innovatively, a power-law distribution with scale effects is considered. The influences of scale effect parameters l and power-law index parameters k on buckling displacement, load, and strain energy of plates have been investigated. In this article, it is found that the critical buckling displacement, critical buckling load, and buckling strain energy increase with increases in the power-law index parameters k. The membrane energy decreases as the power-law index parameter increases. If the upper and lower layers are swapped, the opposite result is obtained. In comparison, the scale effect parameter is more influential than the power-law exponent. The critical buckling displacement in the x-direction is not affected by scale effects. The critical buckling load, the membrane energy, and buckling strain energy increase as the scale effect parameter increases. Scale effects increase material stiffness compared with traditional theory, and the power-law index parameters affect FGM properties such as elastic modulus, Poisson’s ratio, density, etc. Both scale effects parameters and power-law index parameters have important effects on the mechanical behavior of materials.

Bending and free vibration analyses of ribbed plates with a hole based on the FSDT meshless method

A meshless method based on the first-order shear deformation (FSDT) theory and moving least-squares (MLS) approximation for bending and free vibration analyses of the ribbed plate with a hole is proposed. Firstly, a ribbed plate with a hole is regarded as a composite structure of a plate (with a hole) and ribs. Nodes are used to discretize the plate and ribs. Secondly, the displacement fields of the plate and ribs are established based on the FSDT and MLS, and their potential and kinetic energy functional are derived. The diffraction law is used to treat the discontinuity related to the hole on the plate. Thirdly, the total potential and kinetic energy of the ribbed plate is obtained by superposing the potential and kinetic energy of the plate and ribs and introducing the compatibility condition. Finally, the equations governing the elastic bending and free vibration of the ribbed plate with a hole are obtained according to the principle of Minimum Potential Energy and Hamilton’s Principle. The boundary conditions are enforced by the full transformation method. Thereafter, several ribbed plate with hole examples with different holes, rib arrangements, boundary conditions and load conditions are calculated. The results are compared with those given by ABAQUS and literature. The results show that the effectiveness and accuracy of the proposed method in analyzing the static and dynamic problems of the ribbed plate with a hole.

Thermal Buckling and Vibration Analysis of SMA Hybrid Composite Sandwich Beams

This work studies the buckling and free vibration behavior of Shape Memory Alloy Hybrid Composite (SMAHC) sandwich beams under a thermal environment. The sandwich beams consist of layers reinforced with SMAs and a FGM core, and they are simply supported at both ends. The higher order theory is combined with the Minimum Potential Energy principle or Hamilton principle to derive the governing equations of the thermal buckling and thermal vibration problems, respectively. The material properties of the beam are assumed as temperature-independent (TID) or temperature-dependent (TD). In the last case, two different types of thermal distribution are considered, namely a uniform and a linear distribution. The results based on the proposed formulation are verified against the reference literature, with a very good matching. A parametric study checks for the influence of different effective parameters such as thickness-to-length ratios, volume fraction powers, initial strain, volume fraction of SMA wires, and temperature distribution on the overall mechanical response of the selected structural member, with useful insights from a design standpoint.

Size-Dependent Buckling Analysis of Microbeams by an Analytical Solution and Isogeometric Analysis

This paper proposes an analytical solution and isogeometric analysis numerical approach for buckling analysis of size-dependent beams based on a reformulated strain gradient elasticity theory (RSGET). The superiority of this method is that it has only one material parameter for couple stress and another material parameter for strain gradient effects. Using the RSGET and the principle of minimum potential energy, both non-classical Euler–Bernoulli and Timoshenko beam buckling models are developed. Moreover, the obtained governing equations are solved by an exact solution and isogeometric analysis approach, which conforms to the requirements of higher continuity in gradient elasticity theory. Numerical results are compared with exact solutions to reveal the accuracy of the current isogeometric analysis approach. The influences of length–scale parameter, length-to-thickness ratio, beam thickness and boundary conditions are investigated. Moreover, the difference between the buckling responses obtained by the Timoshenko and Euler–Bernoulli theories shows that the Euler–Bernoulli theory is suitable for slender beams.

Pseudo-dynamic analysis of reinforced slope with anchor cables

The anchor cable is an excellent reinforcement device for enhancing slope performance subjected to earthquakes. Thus, this work combines the minimum potential energy principle with the pseudo-dynamic method and develops a novel approach to determining a reinforced slope safety factor (SF). Specifically, the mobilized shear stress on the slip surface is solved utilizing the moment equilibrium equation of the sliding mass. Moreover, the proposed scheme introduces the elastic foundation coefficient characterizing the soil’s compressibility. Subsequently, the potential energy equation is derived. Finally, the SF is calculated as the ratio of the anti-sliding moment to the sliding moment for evaluating the slope stability. The developed method is evaluated on four examples, with the results highlighting that our method is efficient and accurate. Meanwhile, we examine the influence of the variation of the axis forces on the SF when the foundation coefficient is smaller, revealing that the depth of the sliding mass is negatively correlated with the SF, while the landslide area and length of the sliding mass are positively correlated with the SF. Furthermore, this work also studies the influences of amplification factor, earthquake acceleration factor, anchored force, cohesion, and internal friction angle on the SF. Overall, this work provides suggestions for the design and construction management of reinforced slopes with anchor cables.

Prediction of machining deformation induced by turning residual stress in thin circular parts using ritz method

Workpiece deformation induced by surface residual stress during turning is a serious concern in the manufacturing of thin-walled parts. Prediction of machining deformations caused by residual stress is essential to explain deformation mechanism and design machining strategies. Although many studies have been focused on calculating those deformations using the finite element method (FEM), it is time-consuming and redundant. Based on energy equations considering residual stress fields developed by the three-dimensional elastic theory and the principle of minimum potential energy, a new analytic model was established to predict machining deformation of the thin circular parts. In this model, the modified Rayleigh-Ritz method and pseudo inverse method were used to solve energy equations under various machining conditions to obtain machining deformation. Corresponding experiments and existing methods were considered to validate the accuracy and efficiency of the proposed model. Based on the quantitative results of the proposed model, the effects of machining conditions on machining deformation were discussed comprehensively, which provides guidance for machining strategies. • The energy equations of the thin circular plates considering residual stress distributions in all directions is developed. • The improved Rayleigh–Ritz method associated with the pseudo inverse method are used to solve the machining deformation . • The influence factors of machining deformation induced by residual stresses are comprehensively analyzed.

Size-dependent nonlinear post-buckling analysis of functionally graded porous Timoshenko microbeam with nonlocal integral models

Strain-driven (ɛD) and stress-driven (σD) two-phase local/nonlocal integral models (TPNIM) are applied to study the size-dependent nonlinear post-buckling behaviors of functionally graded porous Timoshenko microbeams. The differential governing equations and boundary conditions of motion are derived on the basis of the principle of minimum potential energy and expressed in nominal form. The integral relation between local and nonlocal stresses is transformed unitedly into equivalent differential form with constitutive constraints for ɛD- and σD-TPNIMs. Bending deflection and cross-sectional rotation are derived explicitly with Laplace transformation for linear buckling. Taking into account boundary conditions and constitutive constraints, a general eigenvalue problem is obtained to determine linear buckling mode shape (LBMS) and nominal buckling load (NBL). Local and nonlocal LBMS based Ritz–Galerkin methods and general differential quadrature method (GDQM) based Newton–Raphson’s method are utilized to obtain the numerical solutions for linear and nonlinear NBLs. Numerical results show that nonlocal LBMS based Ritz–Galerkin method and GDQM would lead to same prediction for linear NBLs as analytical method, and nonlocal LBMSs based Ritz–Galerkin and GDQM based Newton–Raphson’s method would lead to exact same prediction for nonlinear post-buckling loads. However, local LBMS based Ritz–Galerkin method is failed to provide accurate prediction for both linear NBLs and nonlinear post-buckling loads, through the difference between local and nonlocal LBMSs is not significant. The influence of nonlocal parameters, porous distribution patterns and boundary conditions on the linear buckling and nonlinear post-buckling behaviors are investigated numerically.

Bending and buckling analysis of functionally graded Timoshenko nanobeam using Two-Phase Local/Nonlocal piezoelectric integral model

Previous studies indicate that nonlocal piezoelectric differential model would lead to an inconsistent bending response of Euler-Bernoulli nanobeam. In this paper, static bending and elastic buckling of functionally graded piezoelectric (FGP) Timoshenko nanobeam are studied through two-phase local/nonlocal piezoelectric integral model. The differential governing equations and standard boundary conditions are derived on the basis of the principle of minimum potential energy. The relations between general strain and nonlocal stress components are expressed as integral forms and then transformed equivalently into differential forms with constitutive boundary conditions. Several nominal variables are introduced to simplify the mathematical formulation. It shows that Timoshenko nanobeam based on purely nonlocal piezoelectric integral model would lead to ill-posed mathematical model and inconsistent size-dependent response with and without considering constitutive boundary conditions, respectively. The general differential quadrature method is applied to obtain the numerical results for bending deflections and buckling loads. The influence of nonlocal parameters, boundary conditions, functionally graded index and buckling order on the bending deflections and buckling loads is investigated numerically. A consistent softening response is obtained.

CENN: Conservative energy method based on neural networks with subdomains for solving variational problems involving heterogeneous and complex geometries

We propose a conservative energy method based on neural networks with subdomains for solving variational problems (CENN), where the admissible function satisfying the essential boundary condition without boundary penalty is constructed by the radial basis function (RBF), particular solution neural network, and general neural network. Loss term is the potential energy, optimized based on the principle of minimum potential energy. The loss term at the interfaces has the lower order derivative compared to the strong form PINN with subdomains. The advantage of the proposed method is higher efficiency, more accurate, and less hyperparameters than the strong form PINN with subdomains. Another advantage of the proposed method is that it can apply to complex geometries based on the special construction of the admissible function. To analyze its performance, the proposed method CENN is used to model representative PDEs, the examples include strong discontinuity, singularity, complex boundary, non-linear, and heterogeneous problems. Furthermore, it outperforms other methods when dealing with heterogeneous problems.

A novel method to predict nanofilm morphology on arbitrary-topographical substrate

As physical properties of nanofilms are strongly influenced by their adhered substrates and presented morphologies, it is of vital importance to determine their attaching morphologies on substrates. However, the complex topography of substrate in the practical applications hinders the prediction of attached nanofilm morphology. Previous theoretical methods fail to analyze the complex topography, and molecular dynamics (MD) simulations are limited to small time and size scales. In order to predict the morphology of a nanofilm with an arbitrary size on a substrate with an arbitrary topography, a novel universal method is proposed by combining the mechanical analysis, discrete cosine transform, and multi-resolution analysis together, named multi-resolution discrete cosine transform (MRDCT) method. In this method, the attaching analysis for a complex-topographical substrate is transformed level by level into a sequence of simple mechanical analyses for a 2D-cosine-topographical substrate, of which a semi-analytical model is derived in advance based on the principle of minimum potential energy. Taking the advantages of the pre-derived mechanical model and the multi-level analysis, the MRDCT method has an extremely high efficiency, only about 10−4 of time cost of MD simulations, and moreover, it is not limited by the size scale of nanofilms. The results from both MD simulations and experiments highlight the competency of this method for various complex-topographical substrates. Furthermore, a general conformation criterion in terms of the substrate roughness is derived analytically based on the MRDCT method to assess whether the substrate can be fully conformed by the nanofilm. The proposed MRDCT method can not only be used in the mechanical analysis of film/substrate systems but also provide a new paradigm to solve other complex mechanical problems.

A new perspective on determination of the critical slip surface of three-dimensional slopes

In the traditional slope stability analysis, the shape and location of critical slip surface (CSS) are identified by an objective function with the minimum safety factor (SF). However, the method cannot identify the true CSS when the minimum SFs of different critical slip surfaces are equal or extremely close. In order to solve the issue, the idea of slip surface with the maximum sliding direction (SD) as the CSS is proposed in this paper. The shape of CSS is regarded as a rotational ellipsoid with 6 parameters, and the genetic algorithm (GA) is adopted to search for the location of CSS. The three-dimensional (3D) slope stability analysis is performed based on the minimum potential energy principle, which provides an analytical formulation of SD as well as SF without iteration. Two benchmark slopes and a practical engineering case are employed to verify the rationality of the proposed method. The parametric analysis reveals that the larger SD the smaller SF. The developed method is a useful technique support for determining the CSS and the shape of 3D slope.

Research on formation mechanism of the mining-induced horizontal fractures in rock strata separation in the Jurassic coalfield, Huang long, Shaanxi, China

To analyze the mining-induced mechanism of the horizontal fractures due to bed separation in Jurassic coalfield, China, the mineral composition identification test, triaxial compression tests, and overburden response were performed in this study. The mineral composition of rocks is composed of terrigenous sand (cuttings, quartz, and feldspar), terrigenous silt, clay, and a few interstitial materials in adjacent rock strata. The differences of mineral compositions are proved to be the essential factors through triaxial compression tests on rock samples, which found that the strength, Young’s modulus, Poisson’s ratio, and plasticity of adjacent have great differences in rock samples. For predicting the variation of the horizontal fractures, the overburden is supposed to be a thin cube plate. And the range of sizes is corresponding to the scope of the extracted coal seam. Then, based on the elasticity thin plate theory, w(x, y)=C0(1+cos(xπ/a)) (1+cos(yπ/a)) is proposed as the deflection function and the stress and deformation of rock strata are obtained by the variational method and minimum potential energy principle. Hence, a mathematical model suitable is proposed to analyze the variation of the horizontal fractures. The Yuanzigou coal mine is taken as an example, the theoretical results are verified by an exploration borehole video imager. In conclusion, the distribution characteristics of rock strata and the differences in these physical parameters are the essential factors causing horizontal fracture between adjacent rock strata. Besides, the relationship between horizontal fractures and roof water inrush was also discussed in this study.