Principle Of Complementary Energy Research Articles

Examination of the complementary energy principle in modified couple stress theory and its application for analysis of size effects in the internal force field of functionally graded microbeams

The precise quantification of the internal force field in MEMS piezoresistive sensor micro structure is crucial for ensuring accurate signal output. This paper proposes and proves the complementary energy principle associated with modified couple stress theory (MCST) based on the linear elasticity assumption. And this principle is used to discuss and analyze whether there is a size effect in the internal force field of functionally graded (FG) micro beams for the MEMS piezoresistive sensor. The results show that the size effects exist in the internal forces and the stationary point coordinate x of the total moment for the micro beam. The size effects are related to the dimensionless material scale parameter l/h, the supported spring stiffness, the temperature and the support rotational movements. Intuitive explanations of the size effects are given for the internal force fields and its stationary point coordinates. Behaviors of the solutions agree with the published data when the MCST degenerates into the classical theory (CT).

Large deformation of cable networks with fiber sliding as a second-order cone programming

A new model for irreversible large deformations of fiber networks is developed. The fibers are considered as inextensible cables that slide relative to each other in the frictional junctions. This sliding is constituted by a rate-independent flow rule. The nonsmooth dissipation potential for each sliding system is defined as a product of the yield strength and the absolute value of the fiber sliding. The response of the cable segments is nonsmooth as well, since it shows asymmetry with respect to tension and compression. A principle of minimum incremental potential and a pure complementary energy principle are derived for the equilibrium incremental loading of the network at large deformations. They form a pair of primal and dual second-order cone programming problems with matching sets of displacement-based and force-based variables. These problems can be effectively solved by interior point methods that have many advantages compared to the gradient-based methods or dynamic relaxation. The model is extended by a simple mechanism of fiber pull-out resulting from fiber sliding at the free unconstrained ends. This can be used for the microstructural analysis of failure of needle-punched nonwoven materials.

Large deformation assessment of three-tower suspension bridges: An enhanced analytical model and solution algorithm

An enhanced nonlinear approach is proposed in this study to address the limitations of traditional deflection theory in analyzing large deformations of three-tower suspension bridges. In this approach, the three-tower suspension bridge is simplified as two quasi-independent bridges for analysis, and detailed considerations are given to the vertical and longitudinal deformations of the main cable, which produce the expression for the relationship between the cable's internal force and deformations. By employing the complementary energy principle and variational calculus, the nonlinear differential equations describing bridge deformations are then established. To ensure the displacement continuity at the middle tower, this paper proposes the use of Chebyshev polynomials to approximate the solutions of the differential equations, and the weighted residual method is then used to transform these differential equations into algebraic equations. On this basis, the developed nonlinear adjustment optimization algorithm enables efficient solving of the nonlinear equation and thus determination of bridge deformation under various load conditions. The accuracy and effectiveness of the proposed scheme are validated through the analysis of two suspension bridges with equal and unequal spans. Finally, the extended applications of the proposed scheme on the other type of multi-span suspension bridges are introduced for further applications.

Solution of planar elastic stress problems using stress basis functions

The use of global displacement basis functions to solve boundary-value problems in linear elasticity is well established. No prior work uses a global stress tensor basis for such solutions. We present two such methods for solving stress problems in linear elasticity. In both methods, we split the sought stress σ into two parts, where neither part is required to satisfy strain compatibility. The first part, σ p , is any stress in equilibrium with the loading. The second part, σ h is a self-equilibrated stress field on the unloaded body. In both methods, σ h is expanded using tensor-valued global stress basis functions developed elsewhere. In the first method, the coefficients in the expansion are found by minimizing the strain energy based on the well-known complementary energy principle. For the second method, which is restricted to planar homogeneous isotropic bodies, we show that we merely need to minimize the squared L2 norm of the trace of stress. For demonstration, we solve nine stress problems involving sharp corners, multiple-connectedness, non-zero net force and/or moment on an internal hole, body force, discontinuous surface traction, material inhomogeneity, and anisotropy. The first method presents a new application of a known principle. The second method presents a hitherto unreported principle, to the best of our knowledge.

Probabilistic relative entropy in elasto‐plasticity using the iterative generalized stochastic stress‐based Finite Element Method

AbstractThe generalized iterative stochastic perturbation approach to the stress‐based Finite Element Method has been proposed in this work. This approach is completed using the complementary energy principle, Taylor expansion of the general order applicable to all random functions and parameters as well as nodal polynomial response bases determined with the use of the Least Squares Method. The main aim of this elaboration is the usage of such a probabilistic approach to determine Bhattacharyya relative entropy for some nonlinear engineering stress analysis with uncertainty. Mathematical apparatus with its numerical implementation has been used to study elastoplastic torsion of some prismatic bar with Gaussian material uncertainty and the corresponding reliability measures. This problem has been solved using the Constant Stress Triangular (CST) plane finite elements, the modified Newton–Raphson algorithm, whereas the first four probabilistic characteristics resulting from the iterative generalized stochastic perturbation method have been contrasted with these obtained with the crude Monte‐Carlo sampling and the semi‐analytical probabilistic approach.

The Dual Approach to Optimal Control in the Coefficients of Nonlocal Nonlinear Diffusion

We derive the dual variational principle (principle of minimal complementary energy) for the nonlocal nonlinear scalar diffusion problem, which may be viewed as the nonlocal version of the p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p$$\\end{document}-Laplacian operator. We establish existence and uniqueness of solutions (two-point fluxes) as well as their quantitative stability, which holds uniformly with respect to the small parameter (nonlocal horizon) characterizing the nonlocality of the problem. We then focus on the nonlocal analogue of the classical optimal control in the coefficient problem associated with the dual variational principle, which may be interpreted as that of optimally distributing a limited amount of conductivity in order to minimize the complementary energy. We show that this nonlocal optimal control problem Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}-converges to its local counterpart, when the nonlocal horizon vanishes.

Fast construction of cluster interaction matrix for data-driven cluster-based reduced-order model and prediction of elastoplastic stress-strain curves and yield surface

The data-driven cluster-based reduced-order model (CROM) methods[47–50] are very effective for predicting the nonlinear effective properties of heterogeneous materials. The Cluster Interaction Matrix (CIM) in these methods can be regarded as a clustered discrete Green's function and is the critical part of CROM methods, which more and more scholars have realized. However, the explicit expression of Green's function is available only for homogeneous material. The CIM for heterogeneous materials needs to be calculated through numerical approaches, which face the challenge of computational efficiency. Therefore, this paper proposes an efficient method to construct the CIM of heterogeneous materials. By deriving the Green's function in Fourier space under the governing equation of FCA, the CIM of homogeneous materials that can provide the self-equilibrium stress field is calculated efficiently. Then, to avoid the effects of stress averaging on clusters, we augment the proposed cluster-element interaction matrix with the self-equilibrium stress basis vector of heterogeneous material without introducing additional computational cost. Based on the augmented cluster-element interaction matrix, the self-equilibrium stress space is improved, and the CIM of highly heterogeneous materials is calculated using the minimum complementary energy principle. The proposed algorithm significantly improves the offline computational accuracy of FCA, enabling efficient prediction of nonlinear effective properties and yield surface under complex load paths. Numerical examples demonstrate the efficiency and accuracy of the proposed method. In addition, we provide the relationship between the CIMs of a homogeneous material of FCA and SCA. Based on the proposed algorithm in this paper, it can be helpful for other CROM methods.

Mesoscopic Analysis of Rounded and Hybrid Aggregates in Recycled Rubber Concrete.

Recycled rubber concrete (RRC), a sustainable building material, provides a solution to the environmental issues posed by rubber waste. This research introduces a sophisticated hybrid random aggregate model for RRC. The model is established by combining convex polygon aggregates and rounded rubber co-casting schemes with supplemental tools developed in MATLAB and Fortran for processing. Numerical analyses, based on the base force element method (BFEM) of the complementary energy principle, are performed on RRC's uniaxial tensile and compressive behaviors using the proposed aggregate models. This study identified the interfacial transition zone (ITZ) around the rubber as RRC's weakest area. Here, cracks originate and progress to the aggregate, leading to widespread cracking. Primary cracks form perpendicular to the load under tension, whereas bifurcated cracks result from compression, echoing conventional concrete's failure mechanisms. Additionally, the hybrid aggregate model outperformed the rounded aggregate model, exhibiting closer peak strengths and more accurate aggregate shapes. The method's validity is supported by experimental findings, resulting In detailed stress-strain curves and damage contour diagrams.

A unified variational framework of no-tension and no-compression solids and its application to finite element analysis

No-tension and no-compression constitutive models have important applications in solid mechanics, such as modelling of masonry, wrinkled membrane, unilateral contact interface, etc. Although lots of studies on no-tension and no-compression solids have been found, the variational principle constructing the cornerstone of elasticity is not studied thoroughly. The paper presents two concise variational formulations, a principle of minimum potential energy and a principle of minimum complementary energy, which are available both for no-tension and no-compression solids. Linearization of the conic yield surfaces leads to a series of linear complementary constitutive equations that are embedded into the proposed variational framework. Differing from other variational formulations, an approximate total solution rather than the Newton iteration is achieved in finite element analysis. It makes the algorithm stable. The applications include a no-tension panel benchmark test, two masonry structures and a wrinkled membrane. Compared with our previous study on bi-modulus materials, the newly developed variational formulation is capable of capturing the evolution of wrinkles in membranes, and can be used for the analysis and design of wrinkle-free structures.

Study on the Control Parameter of the Complex Underground Space Engineering Based on the Complementary Energy Principle with Mixed Variables

The control parameter of the complex underground space structure with the Track Panel Tunnel and wall columns was studied. An analysis method of the control parameter was established based on the complementary energy principle with mixed variables. The general analytical solution of the structure under the trapezoidal load was obtained. Then, the correctness of the solution was verified by two-dimensional finite element simulation. The three-dimensional global model is built to analyze models with different story heights, the distance of two adjacent wall columns, and the thickness of the earth covering; the consequences prove the assumptions’ rationality and the engineering applicability of the analytical solution. The sensitive region of the control parameter is found through the analytical solution, which is meaningful in determining the reasonable stiffness ratio of column and beam for structural design optimization with cost savings. It can be the reference for complex underground space engineering designs.

Research on indirect measurement and estimation method of shearer drum cutting load

For the difficult problem that the shearer drum load is difficult to be detected and perceived directly, a shearer drum load indirect estimation measurement method and technique is proposed and invented. Based on the structural characteristics of the rocker arm, the basic rod system skeleton of the rocker arm is extracted. Based on the transfer route of the drum load on the rocker arm, the feasibility of using the rocker arm pin shaft load for drum load prediction is analyzed, and the correlation mechanics equation of load from the drum to the rocker arm pin shaft is constructed. To address the statically indeterminate problem in the equation, the Lagrange multiplier is introduced to construct the energy function of the rocker rod system skeleton. Based on the principle of minimum complementary energy and solving the extreme value of energy function, the statically indeterminate equation is transformed into the statically determinate equation, and a rough prediction model of drum load based on the force of the rocker arm hinged ear pin shaft is derived, and the maximum relative error of the model is 25.15 %. To address the low prediction accuracy of the model due to the dynamic characteristics of the drum rocker arm, the drum load is derived from the rocker arm hinged ear pin shaft load and the rough prediction model as input, and the experimentally measured drum load as output. The bp neural network error correction model is constructed, and the accurate estimation and perception of the drum load are realized by the prediction model and the bp neural network correction model. Finally, the accuracy of the prediction algorithm is verified by experiments. The results show that after correction by bp neural network, the average error of the drum three directional force is 0.14 × 104N ∼ 0.55 × 104N, the mean square error is 0.18 × 104N ∼ 0.65 × 104N, and the relative error is 3.99 %∼10.12 %. The average error of the drum three directional moment is 0.31 × 104N·m ∼ 0.40 × 104N·m, the mean square error is 0.16 × 104N·m ∼ 0.50 × 104N·m, and the relative error is 2.04 %∼11.44 %. The accuracy of drum load prediction is higher than 11.44 % in all cases. The research results can provide theoretical and technical support for intelligent shearer design, development, and healthy operation and maintenance.

Simulation study on the effect of the initial defect of recycled aggregate concrete based on BFEM

To study the effect of the initial defect on the mechanical properties and failure pattern of recycled aggregate concrete (RAC), several numerical models containing the initial defects were established and simulated under uniaxial compression. The recycled aggregate was simulated as an ellipse geometry, which can be constructed by compressing the circular geometry. The base force element method on the complementary energy principle was applied for the loading condition analysis. In addition, the stress–strain curves, deformation, and crack extension patterns of RAC were explored. Moreover, the maximum principal stress distribution and stress concentration were also investigated. The results demonstrate that the angle and length of the initial defect have a significant effect on the mechanical properties of the RAC. The initial defect interrupts the loading transfer path, forcing it into other continuous regions and forming a new transfer path. In addition, the longer the projection length of the initial defect in the loading direction, the most substantial influence on the mechanical properties, stress transfer, and damage mode.

A theoretical model of amplitude-dependent dynamical stiffness for cord-rubber air springs

Based on the principle of minimum complementary energy, a theoretical model considering the additional dynamic stiffness caused by the structural deformation of the cord-rubber diaphragm is proposed. The capability of modeling the dynamic stiffness change of the rolling lobe air spring provides guidance for material selection and the design of the diaphragm structure. The directly measured geometric parameters in the proposed model have physical meanings, while the relevant material parameters can be identified by combining the mechanics of composite materials and the Payne effect theory of rubber diaphragm structures. Three air springs with different designs and structures were selected for the indicator test and compared with the theoretical value. The comparison results show that the relative error was less than 3%, which verifies the universality and accuracy of the proposed dynamic stiffness theory of the rubber diaphragm. The results demonstrate that the amplitude-dependent additional stiffness generated by the rubber diaphragm is significant, and the lobe bending moment has the most considerable stiffness contribution to the rolling lobe air spring at low amplitudes. In the end, the changing trend of the entire dynamic stiffness with different design factors is given. The rubber diaphragm stiffness theory of rolling lobe air springs proposed in this paper provides powerful assistance for material selection and shape design for car air springs.

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.

Complementary energy principle associated with modified couple stress theory for Euler micro-beams considering support movements and negative Poisson's ratio

Complementary energy principle associated with modified couple stress theory for Euler micro-beams considering support movements and negative Poisson's ratio

Shear behavior of UHPC beams with small shear span to depth ratios based on MSTM

Ultra-high performance concrete (UHPC) has been gradually used in structural engineering due to its excellent mechanical performance, however, predicting the shear behavior of the UHPC beams at service load conditions is still a big challenge, especially for the beams with small shear span to depth ratio. To address this issue, this paper devotes to develop a modified strut and tie model (MSTM) to predict the shear behavior of the UHPC beams with stirrups throughout the loading process. The principle of stationary complementary energy has been introduced into the MSTM to decouple the contribution of arch action and truss action on shear forces. Based on the MSTM, the shear force resistance mechanism has been revealed, and simplified models at each load step have been developed to estimate the shear behavior of UHPC beams. 12 UHPC beams were designed and tested to validate the proposed model. The proposed model was verified to be sufficiently accurate. Moreover, it can be noted that the current design codes, including SETRA-AFGC and SIA, give overly conservative values for UHPC beams when the shear span to depth ratio is less than 2.5.

Variational Stress Analysis of Laminates Containing Regions with Different Material Properties in Loading Direction

In this study, a variational stress analysis was performed on a laminate that contained regions with varying material properties in the loading direction. Investigation of material properties and damage behavior of laminates with fiber discontinuity (resin-rich region) has been the main motivation for this study. The analysis considered an arbitrary number of regions with different material properties. For simplicity, we assumed a plane stress condition or stress distribution, we assumed that the normal stress in the loading direction was constant along the direction of thickness in each ply. The admissible stress state that satisfied both equilibrium and boundary/continuity conditions was constructed with unknown functions that were determined using the principle of minimum complementary energy. The analysis was applied to laminates with single fiber discontinuity and with distributed fiber discontinuities. These results were compared using finite element analyses, which obtained good agreement. The results show the validity of this analysis, which can be used to investigate the effects of fiber discontinuity on the laminate mechanical properties and damage behavior in laminates with fiber discontinuities.

Partitioned parametrized variational procedure for the generation of theorems of structural analysis

Structural field modeling of displacements, stresses, and strains is conceptually linked with the generation of functionals. In particular, this work introduces for the first time the Partitioned Parametrized Variational Procedure (PPVP). In-plane stresses and strains are mathematically treated differently than the transverse counterparts. This allows one to generate all the classical variational theorems of elasticity (such as the Principles of Virtual Displacements and Complementary Energy, Hellinger-Reissner, and the Veubeke-Hu-Washizu theorems) and additional new types of variational theorems which can be exploited for the solution of structural problems in various areas, such as reinforced panels and variable angle tow composites. With the proposed variational procedure energetically consistent constraints naturally arise directly from the formulation. PPVP could also be used to formulate new enhancements of the finite element or meshless methods and to quantify functional errors not related to the numerical discretization.

Hellinger–Reissner variational principle for a class of specified stress problems

Abstract Aiming at the problem of the specified stress condition in a partial region of the structure, the Hellinger–Reissner (H–R) variational principle is studied to provide a theoretical basis for the finite element numerical analysis. By introducing the unknown non-elastic strain as an additional unknown quantity to fulfill the specified stress condition, the elastic mechanics governing equations for the specified stress problem are given. Both stress and unknown non-elastic strain are taken as independent variables to establish the complementary energy principle and virtual work equation which are equivalent to the elastic mechanical control equation of the specified stress problem. Based on the conventional H–R variational principle, using displacement, stress and unknown non-elastic strain as independent variables, a H–R variational functional that satisfies the specified stress conditions is established by using Lagrange multiplier method. Also the variational functional with displacement, elastic strain and unknown non-elastic strain as independent variables is deduced by transforming the stress into the elastic strain. The corresponding finite formulae are derived based on an intra-element stress hybridization method. The H–R variational principle for the specified stress problem takes non-elastic strain as an independent variable, so that the stress explicitly appears in the equilibrium equation of the element and structure, which expands the application range and capabilities of the existing variational principle and finite element method. The correctness and accuracy of the theory and algorithm are verified by numerical examples.

Analysis of Tensile Strength and Failure Mechanism Based on Parallel Homogenization Model for Recycled Concrete.

In this paper, a parallel homogenization model for recycled concrete was proposed. A new type of finite element method, the base force element method, based on the complementary energy principle and the parallel homogenization model, is used to conduct meso-level damage research on recycled concrete. The stress–strain softening curve and failure mechanism of the recycled concrete under uniaxial tensile load are analyzed using the nonlinear damage analysis program of the base force element method based on the parallel homogenization model. The tensile strength and destructive mechanisms of recycled concrete materials are studied using this parallel homogenization model. The calculation results are compared with the results of the experiments and meso-level random aggregate model analysis methods. The research results show that this parallel homogenization analysis method can be used to analyze the nonlinear damage analysis of recycled concrete materials. The tensile strength, stress–strain softening curve, and crack propagation process of recycled concrete materials can be obtained using the present method.