Principle Of Minimum Complementary Energy Research Articles

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.

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.

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.

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.

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.

Experimental observation and modeling of resin pocket cracking in unidirectional laminates with ply discontinuity

Experimental observations revealed that a resin pocket in a unidirectional laminate with a discontinuous ply may form several cracks prior to initiation of delamination cracks. The observed cracks typically occur within the resin pocket, rather than at the interfaces with the discontinuous ply, and are curved towards the center of the pocket. To study this unintuitive cracking behavior, a simple variational stress analysis based on the principle of minimum complementary energy is developed. It allowed to estimate the stress field in the laminate with a discontinuous ply, with or without cracks in the resin region. The observed cracking pattern is explained by assuming a higher level of residual stresses due to chemical shrinkage of the resin during curing in the confined conditions of the pocket. It is shown that the crack locations are best determined using the principle of maximum energy release rate, while the curved crack path can be governed by the distribution of the maximum principle stress. The predicted cracking sequence and the crack curvatures agree well with experimental observations.

Hybrid equilibrium element with high‐order stress fields for accurate elastic dynamic analysis

AbstractIn the present article the two‐dimensional hybrid equilibrium element formulation is initially developed, with quadratic, cubic, and quartic stress fields, for static analysis of compressible and quasi‐incompressible elastic solids in the variational framework of the minimum complementary energy principle. Thereafter, the high‐order hybrid equilibrium formulation is developed for dynamic analysis of elastic solids in the variational framework of the Toupin principle, which is the complementary form of the Hamilton principle. The Newmark time integration scheme is introduced for discretization of the stress fields in the time domain and dynamic analysis of both the compressible solid and quasi‐incompressible ones. The hybrid equilibrium element formulation provides very accurate solutions with a high‐order stress field and the results of the static and dynamic analyses are compared with the solution of the classic displacement‐based quadratic formulation, showing the convergence of the two formulations to the exact solution and the very satisfying performance of the proposed formulation, especially for analysis of quasi‐incompressible elastic solids.

Theoretical Solution of Elastic Foundation Beam based on the Principle of Minimum Complementary Energy

Theoretical Solution of Elastic Foundation Beam based on the Principle of Minimum Complementary Energy

Prebuckling and buckling analysis of moderately thick variable angle tow composite plates considering the extension-shear coupling

In this paper, an analytical model based on a generalized Rayleigh-Ritz method is developed to deal with both the prebuckling and buckling problems of moderately thick VAT composite plates under a more general in-plane boundary condition. The non-uniform in-plane stress fields of VAT composite plates under various in-plane boundary conditions is firstly determined based on the minimum complementary energy principle combined with the Airy’s stress function. Afterwards, the governing equation of buckling problem of VAT composite plates under non-uniform in-plane stress fields is derived based on third-order shear deformation theory of Reddy’s type. The innovation of this paper lies in that a linear fitting method combined with Chebyshev-Gauss-Labotto point distribution is introduced to retrieve the boundary stress coefficients from the stress constraint conditions generated by the Lagrangian multiplier method, and meanwhile an integral expression of work done by the unknown boundary force along the prescribed in-plane displacement is adopted to satisfy the displacement constraint condition along the boundary line. In so doing, three kinds of in-plane boundary conditions are taken into account, that is, pure stress boundary condition, pure displacement boundary condition and mixed stress and displacement boundary condition. In addition, no assumption is made on the extension-shear coupling terms and thus the proposed analytical model is suitable to the unbalanced layup. The accuracy and robustness of the proposed analytical model is validated by finite element analysis and with prior results. Effects of the in-plane boundary condition, extension-shear coupling and varying fibre orientation angles on both in-plane stress and buckling responses of VAT composite plates are studied by numerical examples. It is shown that for the balanced panels under uniform compression, the benign load redistribution mechanism offered by the VAT layups remain in effect even if there exists the non-uniform shear constraint along the loading edges. However, the buckling load of the panel can be reduced by the shear constraint along the loading edges to a certain degree. Furthermore, it is found that for some displacement-controlled shear loading cases, the extension-shear coupling plays a positive role in the load redistribution, which results in a sight improvement in the longitudinal load of the panel when critical buckling occurs.

Shape-free polygonal hybrid displacement-function element method for analyses of Mindlin–Reissner plates

A high-performance shape-free polygonal hybrid displacement-function finite-element method is proposed for analyses of Mindlin–Reissner plates. The analytical solutions of displacement functions are employed to construct element resultant fields, and the three-node Timoshenko’s beam formulae are adopted to simulate the boundary displacements. Then, the element stiffness matrix is obtained by the modified principle of minimum complementary energy. With a simple division, the integration of all the necessary matrices can be performed within polygonal element region. Five new polygonal plate elements containing a mid-side node on each element edge are developed, in which element HDF-PE is for general case, while the other four, HDF-PE-SS1, HDF-PE-Free, IHDF-PE-SS1, and IHDF-PE-Free, are for the edge effects at different boundary types. Furthermore, the shapes of these new elements are quite free, i.e., there is almost no limitation on the element shape and the number of element sides. Numerical examples show that the new elements are insensitive to mesh distortions, possess excellent and much better performance and flexibility in dealing with challenging problems with edge effects, complicated loading, and material distributions.

A variational method for analysis of laminates with parallel arrays of intralaminar cracks

SummaryThis paper presents a robust variational method for prediction of the effective thermoelastic properties of laminates with parallel, but not necessarily coplanar intralaminar cracks. Arbitrary laminate layup, crack patterns, and any in‐plane forces and bending moments, and temperature change can be treated simultaneously. The method is based on the principle of minimum complementary energy. The admissible in‐plane stress components are assumed to be linear through the ply thickness. Simple matrix expressions are derived for the effective compliance matrix, thermal expansions and curvatures, and specific heat of a cracked laminate. Predictions for the Young's modulus, Poisson's ratio, shear modulus, flexural rigidity, and thermal expansion as functions of crack density for various laminates of symmetric and nonsymmetric layups demonstrated excellent agreement with experimental results of seven independently published studies. Accuracy and robustness of the method are complemented by the fact that no experimentally fitted parameters are required. It is shown that the same formalism can be applied to analysis of closed intralaminar cracks, as well as nonuniform and nonsymmetrical distributions of cracks.

Principle of cluster minimum complementary energy of FEM-cluster-based reduced order method: fast updating the interaction matrix and predicting effective nonlinear properties of heterogeneous material

The present paper studies the efficient prediction of effective mechanical properties of heterogeneous material by the FCA (FEM-Cluster based reduced order model Analysis) method proposed in Cheng et al. (Comput Methods Appl Mech Eng 348:157–168, 2019). The principle of minimum complementary energy and its cluster form for the RUC subjected to applied uniform eigenstrains and the PHBCs (Periodic Homogeneous Boundary Conditions) are developed. By using the known interaction matrix, an alternative form of the principle of cluster minimum complementary energy is constructed and proved very efficient for updating the interaction matrix and the effective elastic modulus when the material properties of clusters change. Moreover, the proposed principle of cluster minimum complementary energy is applied for the incremental nonlinear analysis of the cluster reduced order model, and thus greatly improves the prediction of nonlinear effective properties of the RUC in online stage computed in Cheng et al. (Comput Methods Appl Mech Eng 348:157–168, 2019). A number of numerical examples illustrate the effectiveness and efficiency of the FCA approach with the proposed principle of cluster minimum complementary energy.

Three stress-based triangular elements

To obtain proper stresses , three new triangular elements are formulated in this study. First, a complementary energy functional is used within an element for the analysis of plane problems. In this energy manifestation, the Airy stress function will be applied as a functional variable. Then, some basic analytical solutions are assigned for the stress functions. These trial functions are matched with each element number of degrees of freedom. The result is a number of equations with anonymous constants. Subsequently, according to the principle of minimum complementary energy, the unknown constants can be expressed in terms of displacements. Finally, this system can be rewritten in terms of the nodal displacement. In this way, three new triangular elements are formulated. To validate the results, extensive numerical studies are performed. The findings clearly demonstrate accuracies of structural displacements as well as stresses.

Parameter analysis of damaged region for laminates with matrix defects

Considering the influence of bridge fibers, a representative volume element (RVE) with damaged regions of vanishing thickness is developed to study for estimation of laminates with transverse matrix cracks. For the open and closed cracks, the tensile force, the compressive force and the shear force are mainly studied by the stress-based variational approach with the pending partition loads at the statically indeterminate boundary, with considering not only three cases of the ideal transverse cracks, the smooth plane-surface open crack, the smooth plane-surface closed crack and the perfect meshing open crack, but also two cases of the non-ideal transverse cracks, the open cracks and closed cracks with the resistance resisting the relative sliding of crack surfaces. The expressions of the stress components in the RVE are obtained by means of assumption of the function of inter-laminar shear stresses. The governing equations are derived from the variational analysis based on the principle of minimum complementary energy. Both the effective mechanical properties and the distributions stresses of the damaged laminates with the ideal transverse cracks and those of the laminates with the non-ideal transverse cracks are estimated on the basis of the analysis of deformation on surface of the RVE.

Enhanced variational approach for damage analysis of laminated composite

The main objective of this research is to introduce a new coupled analytical–numerical procedure to calculate stiffness reduction due to matrix cracks in composite laminates subjected to general loading conditions. A novel micromodel is developed for transverse cracking of general symmetric laminates based on a variational approach. The proposed method overcomes the restrictions of previous similar approaches regarding damage in laminates with general symmetric stacking sequences. A polynomial series is used for calculating the stress perturbation due to intralaminar cracks in conjunction with the principle of minimum complementary energy. The numerical part deals with well-known Newton–Raphson procedure to solve the governing equations for the unknown parameters for the stress functions. Furthermore, the analytical part can be solved just one time by a symbolic calculation for unknown geometry parameters, material properties, and loading conditions, which has the benefit to be ready for Finite Element software implementations. Comparison between the existing experimental and analytical results and the result of the present study are in good agreement.

Micromechanics of stress transfer in shape memory alloy reinforced three-phase composites

Due to the weak interface in shape memory alloy wire–reinforced composites, the influence of interphase on the mechanical properties and stress distribution of hybrid composites is of considerable importance. In this article, a three-cylinder axisymmetric model using a pull-out test is developed to predict stress transfer and interfacial behavior between shape memory alloy wire, interphase, and matrix. In this article, only superelasticity behavior of the shape memory alloy wire is considered. Based on the stress function method and the principle of minimum complementary energy, stress distribution is derived for three different cases in terms of loading and boundary conditions (thermal loading model, intact model, and partially debonded model). Inhomogeneous interphase and different radial and hoop stress components in each phase are considered to achieve deeper physical understanding. Finite element analysis also performed to simulate stress transfer from the wire to the matrix through the interphase. To evaluate the accuracy of this model, the results of the work are compared with the results of the two-cylinder model proposed by Wang et al. and finite element results.

A micromorphic approach to stress gradient elasticity theory with an assessment of the boundary conditions and size effects

AbstractBasing on thermodynamics principles, a stress gradient elasticity theory is presented whereby the material is modeled as a centro‐symmetric micromorphic anisotropic material. A principle of minimum complementary energy and a stationarity principle of Hellinger–Reissner type are provided, by which the boundary conditions are uniquely determined in a form consistent with continuum mechanics. For 3‐D solids, these conditions include, beside the three ordinary boundary conditions for the assigned boundary data (displacements and/or tractions), six extra boundary conditions by which a microstructure/continuum compatibility condition is enforced, generally through the vanishing of the normal derivative of the stress, , on the whole boundary surface.A typical boundary‐value problem for isotropic materials with a single length scale parameter under static loads is discussed, showing that it is governed by the same Navier PDEs of classical elasticity, along with a set of Helmholtz PDEs as constitutive equations. Size effects are shown to arise from two distinct sources, that is, the spatial fluctuating of the body force and the double curl of the Hookean stress field. It is proved that a formulation of stress gradient elasticity theory based on the pure notion of nonsimple material (like the ones existing in the literature) leads to anomalies in the ordinary and extra boundary conditions. A comparison of the present theory with that by Eringen is also discussed.Two examples of stress gradient structural models are analytically worked out, namely, a thick‐walled cylinder and a Kirchhoff–Love circular plate. For comparison, the solutions for strain gradient material are also presented. Graphic illustrations show that in both examples stiffening size effects like with the ancillary strain gradient models are found.

Experimental and variational-based analytical investigation of multiple cracked angle-ply laminates

A unit cell based method is developed based on variational principles to derive the stress state and effective mechanical properties of angle-ply laminates of type [θm(o)/θn(i)]s containing whether mid-ply or outer-ply matrix cracks positioning in a symmetric or staggered pattern through the laminate. All plausible matrix crack distribution patterns are considered and implementing the principle of minimum complementary energy, an admissible stress field is derived. For the first time effects of ply refinement technique, layup configuration and crack distribution pattern on the stress state and degradation of mechanical properties of angle-ply laminates are investigated and compared to finite element results. CFRP test specimens are prepared and optical microscopy is utilized to detect matrix crack densities during tensile tests. Obtained analytical stiffness reductions for different crack densities are confirmed to be in agreement with experimental observations.

Solution of a Spherically Symmetric Static Problem of General Relativity for an Elastic Solid Sphere

The paper is concerned with the spherically symmetric static problem of the General Relativity Theory (GRT). The classical interior solution of this problem found in 1916 by K. Schwarzschild for a fluid sphere is generalized for a linear elastic isotropic solid sphere. The GRT equations are supplemented with the equation for the stresses which is similar to the compatibility equation of the theory of elasticity and is derived using the principle of minimum complementary energy for an elastic solid. Numerical analysis of the obtained solution is undertaken.

A variational approach for predicting initiation of matrix cracking and induced delamination in symmetric composite laminates under in-plane loading

Abstract This paper discusses the formation of matrix cracking and induced delaminations in a [ ϕ m ( 2 ) / ψ n ( 1 ) ] s $[\phi _m^{(2)}/\psi _n^{(1)}]s$ laminate subject to arbitrary in-plane loading using a variational approach. To this end, the effects of delaminations coming from the tips of transverse cracks in the middle sublaminates are considered to specify the critical crack density points and finally understand which modes of damage are prospects for various lay-ups in the presence of transverse cracks as well as induced delamination. After deriving a very good approximation of the principle of minimum complementary energy by considering complex equations, the stress fields are provided. In this analysis, a unit cell in the ply level of a composite laminate containing both matrix cracking and delamination is considered. Then, the values of the admissible stresses and compliance of cracked laminate are employed to evaluate the energy release rate of each mentioned damage mode. Eventually, the graphs for different lay-ups in such symmetric laminate are drawn, which indicate important points for designers in practical applications. Afterwards, the effects of thickness on the dislocation of critical crack density points are checked. It can be emphasized that the current approach opens a new insight for perceiving the composites’ structural behavior in vulnerable positions.