Complementary Potential Energy Research Articles

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.

A stress-driven computational homogenization method based on complementary potential energy variational principle for elastic composites

Based on the complementary potential energy variational principle, in this work, we proposed a stress-driven homogenization procedure to compute overall effective material properties for elastic composites with locally heterogenous micro-structures. We have developed a novel incremental variational formulation for homogenization problems of both infinitesimal and finite deformations where the macro-stress-based complementary potential energy is obtained for hyperelastic materials for a global minimization problem with respect to fine-scale displacement fluctuation field. The point of departure of our approach is a general complementary variational principle formulation that can determine material responses of elastic composites with heterogeneous micro structures. We have implemented the proposed stress-driven computational homogenization procedure with the finite element method. By comparing the numerical results with the analytical method and the strain-driven homogenization method, we find that the stress-driven homogenization offers the lower bound estimate of materials properties for elastic composites.

Micromechanical model for rapid prediction of plain weave fabric composite strengths under biaxial tension

The biaxial properties of plain weave fabric composites are important as they are more representative of the performance under complex loading conditions. Experimental determination of these properties is difficult and Finite Element Analysis provides accurate prediction but is computationally expensive and requires skilled users. To provide a simple and rapid prediction of the strength of plain weave fabric composites under biaxial tension a novel micromechanical model is proposed in this paper. To predict the biaxial tensile strengths the minimum total complementary potential energy principle is used on a micromechanical unit cell where the orthogonally interlaced yarns are idealised as curved beams. The new model is verified with a finite element method model on three warp/weft biaxial loading ratios: 1:1, 2:1 (1:2) and 3:1 (1:3) and uniaxial experimental data. The model is verified on four types of material, ranging in mechanical properties from carbon to glass fibres, and 11 yarn specifications, including five cases compared to experimental results and six cases compared to the FE model, giving a mean error of 9.85% and a maximum error of 16.74% compared to experimental results and a mean error of 10.71% and a maximum error of 14.67% compared to the FE model, which demonstrates the effectiveness of the model. The standard deviation of prediction errors among the 11 cases is 2.66%, which demonstrates the robustness of the model for a range of applications. The proposed model is able to predict the uniaxial and biaxial tensile strengths without experimental investigations at the fabric and laminate level and only requires the yarn mechanical properties and specifications.

Three-dimensional fracture analysis for a thin cracked hardening plate

A semi-analytical method is presented to investigate fracture behavior of a mode-I crack in a thin ductile plate. In the part of theoretical analysis, a total deformation theory of plasticity, in conjunction with a power law hardening stress-strain relation, is employed. Three-dimensional Maxwell stress functions, the minimum complementary potential energy principle and three dimensional J-integrals are used to obtain crack front stress fields. In the part of numerical analysis, three dimensional finite element simulations are carried out to obtain J-integrals, the out-of-plane constraint level and crack front stress fields. Comparison with numerical results of crack front stress fields shows the analytic solutions for crack front stress fields are valid. Numerical and analytic results show that the value of the opening stress ahead of the crack front deceases with increasing z (approaching the free surface) and lies between the corresponding plane strain and plane stress solutions.

On the effect of an out-of-plane constraint on the three-dimensional crack front fields in a thin elastic plate

Two-dimensional theories of fracture are still applied widely today and provide theoretical foundations for solutions to many practical problems. These two-dimensional theories are based on the plane strain or plane stress assumption. However, strictly speaking, for a thin elastic plate with a through-thickness crack under tension, plane strain conditions can be met only at the crack front (except the corn point) and plane stress conditions exist at a distance of about one half of the plate thickness from the crack front in the mid-plane. What are the stress fields in the region where both plane strain and plane stress conditions cannot be met? In this paper, further investigations into the problem are carried out. Three-dimensional Maxwell stress functions, the principle of minimum complementary potential energy and three-dimensional J-integrals are employed to obtain an analytical solution to depict the relationship among out-of-plane constraints, three-dimensional J-integrals and stress intensity factors. Three-dimensional finite element simulations with fine meshes are carried out to verify the analytical results. Compared with the corresponding plane strain solution, the solution proposed is valid in a larger region.

The effect of out-of-plane constraint on the stress fields near the front of a crack in a thin ductile plate

Classical plane solutions based on the elastic-plastic fracture mechanics are applied widely in practical engineering. These plane solutions follow the plane stress or plane strain assumption. However, in the case of a thin ductile plate with a through-thickness crack under tension, plane stress conditions exist at a distance of about one half of the plate thickness ahead of the crack front cross the thickness of the plate. What is the stress state in the region where both plane strain and plane stress conditions cannot be met? In the current paper, a semi-analytical method is presented to investigate the problem. Three dimensional Maxwell stress functions, the minimum complementary potential energy principle and three dimensional J-integrals are employed to obtain solutions for crack front fields in a thin ductile plate. Three-dimensional finite element (FE) analyses are carried out to verify the current solutions. FE results reveal that the out-of-plane constraint level Tz increases with increasing remote loading near the crack front. FE results also show that the in-plane stress fields near the crack front can be characterized by the current J-Tz solutions. Both FE and theoretical results illustrate that opening stresses decrease gradually to the corresponding plane stress HRR-field solutions with increasing radial distance r from the crack front in the mid-plane when remote loading is large enough.

Optimal design of triaxial weave fabric composites under tension

Triaxial weave fabrics are increasingly used in ultralight structures, such as the wings of unmanned aerial vehicles (UAVs) and deployable antenna on spacecraft. The tensile strength to stiffness ratio for these applications is important, requiring an optimal weave pattern; in this paper Genetic Algorithms are used to improve these designs. The mechanical response is obtained using the minimum total complementary potential energy principle where the yarns are approximated as curved beams in a micromechanical unit cell. Leading Genetic Algorithms are benchmarked to determine which perform best. The results form a disconnected Pareto front where the left hand part can be used for flexible structures but is difficult to find. An overall improvement in strength to stiffness ratio of 1191% is made with 643 designs found better than a current example. The selection of the Genetic Algorithm is shown to be crucial with only MLSGA-NSGAII regularly finding the entire Pareto front.

A micromechanical model for predicting biaxial tensile moduli of plain weave fabric composites

This article presents a new micromechanical model to predict biaxial tensile moduli of plain weave fabric composites by considering the interaction between the orthogonal interlacing strands. The two orthogonal yarns in micromechanical unit cell were idealized as curved beams with a path depicted using sinusoidal shape functions. The biaxial tensile moduli of plain weave fabric composites were derived by means of the minimum total complementary potential energy principle founded on micromechanics. Biaxial tensile tests were conducted on the resin transfer molding–made EW220/5284 plain weave fabric composites at five biaxial loading ratios of 0, 1, 2, 3 and ∞ to validate the new model. Predictions from the new model were compared with experimental data. Good correlation was achieved between the predictions and actual experiments, demonstrating the practical and effective use of the proposed model. Using the new model, the biaxial tensile moduli of plain weave fabric composites can be predicted based only on the properties of basic woven fabric.

Analytical solutions for predicting tensile and in-plane shear strengths of triaxial weave fabric composites

This paper deals with new analytical solutions to predict tensile and in-plane shear strengths of triaxial weave fabric (TWF) composites accounting for the interaction between angularly interlacing yarns. The triaxial yarns in three directions of 0° and ±60° in micromechanical unit cell (UC) are idealized as the curved beams with a path depicted by using sinusoidal shape functions. The tensile and in-plane shear strengths of TWF composites are derived by means of the minimum total complementary potential energy principle founded on micromechanics. In order to validate the new model, the predictions are compared with experimental data in prior literatures. It is shown that the predictions from the new model agree well with experimental results.

Semi-analytical model based on generalized plane strain assumption for unidirectional composites with matrix micro cracks

ABSTRACTGeneralized plane strain state is assumed and stress-based finite strip method is formulated for analysis of unidirectional laminates with matrix microcracks. Total complementary potential energy is minimized and fourth-order Euler Lagrange governing equations are presented. This stress-based generalized plane strain approach analyzes general layup and loading conditions. It provides flexibility to control the number of finite strip nodal lines within each lamina; hence, stress behavior can be predicted across each lamina at the desired location of the structure. Along with all of the capabilities which are common with finite strip methodology based on plane strain assumption, this current work has extended the analysis of the cracked laminate. For example, by incorporating behavior of the out of the plane shear stress, boundary conditions including natural boundary conditions are imposed appropriately to solve governing Euler's equations. Results are compared with previously developed displacement-based formulation in the literature for cracked laminates. It has already been shown that a stress-based plane strain approach enhances variation-based cracked laminate analysis where only the case of cross-ply laminate under tension is considered. This current work applies generalized plane strain-based finite strip methodology to carry out analysis under different loading conditions.

Nonlinear Analysis of Masonry Buildings Under Seismic Actions with a Multifan Finite Element

This paper presented an innovative method for nonlinear simulation of masonry structures under seismic loading conditions. The method is based on a macro-modeling formulation of a masonry panel. A new finite element (FE), called a “multi-fan” (MF) element, is developed on the macro scale, whereby the stress field within the panel is determined at each load step by minimizing the current complementary potential energy. Because only one element is required to model a single masonry panel in the structure, this approach enables a reduction in the total number of degrees of freedom (DOFs) of the system and the total computational effort. The ability of the proposed approach to estimate the ultimate capacity of masonry walls was analyzed both by comparing its predictions with available experimental results and through FE analyses based on a micro-modeling approach. The comparisons show satisfactory accuracy at both the global and local levels. Therefore, the new macro-element can be effectively used in safety assessments of masonry buildings.

ENERGY-BASED ANALYSIS OF TRANSVERSE CRACKING AND INDUCED DELAMINATION IN [S'/90n/S]s LAMINATES: A VARIATIONAL APPROACH

Analysis of transverse-ply cracking and small delaminations emanating from the tips of transverse cracks is extended to symmetric laminates having more general layup configurations. Almost all studies on induced delamination due to transverse-ply cracking are limited to [S/90n]s laminates, where (S) is any orthotropic sublaminate that is much stiffer than [90n]. In this paper, the evolution of transverse matrix cracking and the induced delamination from it is investigated for laminates with more general layups [S'/90n/S]s. In this configuration, the cracked layer is offset to the midplane of the laminate and is constrained by the two orthotropic sublayers. The micromechanics-based model is developed using a variational approach to find the stress field of a unit-cell in a generalized plane strain condition. A combined analytical/numerical methodology is used to solve the boundary value problem. The total complementary potential energy of the unit-cell is calculated, as is the energy release rate for transverse cracking and induced delamination damage. The calculated energy release rate as a function of crack density and delamination length is obtained for different layup configurations, and in some cases the results are compared with results from the literature. Because the distribution of cracks in a structure is uniform, the results for the unit cell scale can be used for the structural scale.

A new three-dimensional analytical model to simulate microresidual stresses in polymer matrix composites

Based on the energy method and the principle of minimum total complementary potential energy, a new three-dimensional analytical model is developed. A three-dimensional analysis is performed for a unit-cell representative volume element (RVE) consisting of a monofiber, a matrix, and an interphase region. The model is evaluated using the finite-element technique. In order to study the effects of neighboring fibers on the residual stresses, several microstructural finite-element models are considered. It is proved that the RVE used in the analytical model is suitable for predicting the thermal microresidual stresses in polymer composites. In comparison with the analytical model, the finite-element analysis presents a little higher residual stresses. According to the results of the analytical model, the interfacial shear stress reaches a maximum in the vicinity of fiber ends, while the finite-element analysis gives a maximum shear stress at composite ends. This is due to the edge singularity, which is not considered in the finite-element model. Although the interphase region affects the distribution of interfacial radial stresses only slightly, it has a significant effect on the maximum interfacial radial stresses at fiber ends.

Mixed Lagrangian Formulation for Linear Thermoelastic Response of Structures

AbstractAlthough a complete unified theory for elasticity, plasticity, and damage does not yet exist, an approach on the basis of thermomechanical principles may be able to serve as the foundation for such a theory. With this in mind, as a first step, a mixed formulation is developed for fully coupled, spatially discretized linear thermoelasticity under the Lagrangian formalism by using Hamilton’s principle. A variational integration scheme is then proposed for the temporal discretization of the resulting Euler-Lagrange equations. With this discrete numerical time-step solution, it becomes possible, for proper choices of state variables, to restate the problem in the form of an optimization. Ultimately, this allows the formulation of a principle of minimum generalized complementary potential energy for the discrete-time thermoelastic system.

El concepto de hiperviga como generalización de los modelos unidimensionales en la estática de piezas alargadas

This paper is concerned with the general formulation of linear problems in rod elastostatics, and with the identification of their common formal and structural features, valid for every kinematical hypothesis. The generalized variables (section forces and generalized deformations) defining the 1-dimensional model are introduced in a consistent and natural way, through a convenient factorization of the density of complementary potential energy, for every kinematical constraint which can be expressed as a linear combination of the generalized displacements. The identification of this complementary energy function with the Hamiltonian functional of analytical mechanics allows a systematic procedure to construct the equations which rule this class of problems. In this frame, the main target is to establish the required conditions to write the rod equilibrium equations in a purely statical form (with no interplay of kinematical variables). We primarily conclude that this is possible when the kinematics of the cross-sections is constrained to a rigid body movement. As a consequence, the concept of hyperbeam is proposed in order to define those models with deformation modes beyond rigid body movements of the cross section: in them, section forces and generalized displacements are coupled in the equilibrium equations. This is related to the idea of local static indeterminacy ( hyperstaticity) and justifies the new name.

Closed-Form Analysis of the Stress Field at Rectangular Corners of Reinforcement Patches

Laminate reinforcement patch corners are locations of stress localizations which may cause a premature failure of the structure. A closed-form approach for the calculation of the occurring interlaminar stress concentrations in the surroundings of rectangular reinforcement patch corners of thermally loaded composite cross-ply laminates is presented. The interlaminar stresses are obtained by the application of a perturbing stress field which comprises separable stress functions. The separation is of the kind that the dependency of the stress functions in transverse direction is assumed a priori to be known and that the stress functions' in-plane part is determined with the help of the principle of minimum complementary potential energy. Afterwards boundary conditions are applied using an expanded Ritz-approach with Lagrange multipliers in order to obtain the perturbing stress field. This stress field is then superposed to the results of the classical laminate plate theory. The method is in good agreement with results from a comparative finite element analysis, is easily applicable and requires very low computational effort.

Dual integrals in small strain elasticity with body forces

Dual integrals of small strain elasticity are derived and related to the energy release rates associated with a defect motion in the presence of body forces. A modified energy momentum tensor is used, which includes a work term due to body forces, and which yields simple expressions for the configurational forces in terms of the J k , L k , and M integrals. Since the complementary potential energy does not include body forces explicitly, the complementary energy momentum tensor has the same structure as in the elasticity without body forces. The expressions for the nonconserved J k , L k , and M integrals, and their duals, are related to the volume integrals whose integrands depend on body forces and their gradients. If body forces are spatially uniform, the conservation laws J k = J ˆ k = 0 hold for both 2D and 3D problems, and L 3 = L ˆ 3 = 0 for the antiplane strain problems. The conservation law M = M ˆ = 0 holds if body forces are absent, or if they are homogeneous functions of particular degree in spatial coordinates. To cite this article: V.A. Lubarda, X. Markenscoff, C. R. Mecanique 336 (2008).

Complementary energy release rates and dual conservation integrals in micropolar elasticity

The complementary energy momentum tensor, expressed in terms of the spatial gradients of stress and couple-stress, is used to construct the J ^ k and L ^ k conservation integrals of infinitesimal micropolar elasticity. The derived integrals are related to the release rates of the complementary potential energy associated with a defect translation or rotation. A nonconserved M ^ integral is also derived and related to the energy release rate that is associated with a self-similar cavity expansion. The results are compared to those obtained on the basis of the classical energy momentum tensor, expressed in terms of the spatial gradients of displacement and rotation, and the release rates of the potential energy. It is shown that the evaluation of the complementary conservation integrals is of similar complexity to that of the classical conservation integrals, so that either can be effectively used in the energetic analysis of the mechanics of defects. The two-dimensional versions of the dual conservation integrals are then derived and applied to an out-of-plane shearing of a long cracked slab.

Dual conservation integrals and energy release rates

Simple derivation of the dual conservation integrals in small strain elasticity is presented, without the aid of Noether’s theorem on invariant variational principles. The derived integrals are related to the release rates of the potential and complementary potential energy associated with the defect motion. The analysis corrects the errors in earlier derivation of the relationship between the dual integrals and the release rates of the complementary potential energy. Selected examples in plane and anti-plane strain illustrate the calculation of dual integrals and their application. It is shown that the evaluation of dual integrals is of similar complexity to that of classical integrals, so that either can be used to determine the stress intensity factors or the forces between defects, without solving the corresponding boundary value problems. An advantage of combining the two calculations is discussed.

Three-Dimensional Sliding Contact Analysis of Multilayered Solids With Rough Surfaces

Friction/stiction and wear are among the main issues in magnetic storage devices and microelectromechanical systems/nanoelectromechanical systems having contact interfaces. A numerical model which simulates the actual contact situations of those devices is needed to obtain optimum design parameters including materials with desired mechanical properties, layers thickness, and to predict and analyze the contact behavior of devices in operation. This study presents a first attempt to develop a numerical three-dimensional multilayered elastic–perfectly plastic rough solids model to investigate the contact behavior under combined normal loading and tangential traction. Energy method is used to formulate the problem, and variational principle in which the contact pressure distributions are those which minimize the total complementary potential energy is applied. A quasi-Newton method is used to find the minimum, and fast Fourier transform is applied to enhance the computation efficiency. In-depth analyses of the effects of friction force, layers properties, and layers thickness to contact statistics and stresses are performed. The optimum layer parameters which decrease friction/stiction and wear are investigated and identified.