Total 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.

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.

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.

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.

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.

A numerical three-dimensional contact model for rough, multilayered elastic/plastic solid surfaces

A numerical three-dimensional contact model is presented to investigate the contact behavior of rough, multilayered elastic-perfectly plastic solid surfaces. The model is based on a variational principle in which the contact pressure distributions are those which minimize the total complementary potential energy. The quasi-Newton method is used to find the minimum. The influence coefficients of the displacements and stresses for a multilayered contact model are determined using the Papkovich–Neuber potentials with a fast fourier transform (FFT) based scheme. Contact analyses are performed to study the effects of layer-to-substrate ratios of stiffness and surface roughness on the contact statistics of rough, two-layered elastic/plastic solids. Local contact pressure profiles, von Mises stresses, tensile and shear stress contours, and the maximum displacement as a function of material properties and applied load are generated. The results yield insight into the effects of stiffness of layers and substrates, surface roughness and applied load on the contact performance. Optimum layer parameters are identified to provide low friction/stiction and wear.

Variational principles for generalized plane strain problems and their applications

This paper is concerned with the variational principles for the generalized plane strain problem of elasticity, which do not seem to have been documented well in the literature, hitherto. Both the total potential energy and the total complementary potential energy principles have been formulated and presented. Their counterparts in the context of generalized variational principles have also been presented. As a result, of the introduction of a discrete degree of freedom, i.e. the uniform direct strain out of the plane, which characterizes the generalized plane strain problem, a fair bit of complications arises. The minimum nature of the stationary values of the energy functionals may not be taken for granted as expected in their counterparts in conventional plane stress or strain elasticity. The expression of the total complementary potential energy obtained here has not been found before in the literature to the best of the authors' knowledge. This might be responsible for the fact that the generalized plane strain problem has been avoided in published work employing the variational principle based on the total complementary potential energy. The presently formulated total complementary potential energy functional has been applied to some classic problems in composites materials, viz. the analysis of transversely cracked laminates and the micromechanics of unidirectionally fibre-reinforced composites. Some interesting and/or new results have been obtained.

A Numerical Three-Dimensional Model for the Contact of Layered Elastic/Plastic Solids With Rough Surfaces by a Variational Principle

A new numerical model for the three-dimensional contact analysis of a layered elastic–perfectly plastic half space with another rough surface is presented. The model is based on a variational principle in which the real area of contact and contact pressure distribution are those which minimize the total complementary potential energy. A quasi-Newton method is used to find the minimum. The influence coefficients matrix is determined using the Papkovich–Neuber potentials with fast Fourier transformation. The model is extended to elastic–perfectly plastic contacts in dry and wet conditions. Contact analyses have been performed to predict contact statistics of layered elastic/plastic solids with rough surfaces using this model. The effects of the stiffness of the layer and the substrate, layer thickness, as well as normal load are studied. Optimum layer parameters are identified to provide low friction/stiction and wear.

A Numerical Three-Dimensional Model for the Contact of Rough Surfaces by Variational Principle

Abstract A new numerical method for the analysis of elastic and elastic-plastic contacts of two rough surfaces has been developed. The method is based on a variational principle in which the real area of contact and contact pressure distribution are those which minimize the total complementary potential energy. The present variational approach guarantees the uniqueness of the solution of the contact problem and significantly reduces the computation time as compared with the conventional matrix inversion method, and thus, makes it feasible to solve 3-D contact problem with large number of contact points. The model is extended to elastic-perfectly plastic contacts. The model is used to predict contact statistics for computer generated surfaces.

Simplified analysis of wall-frame structures

A simplified method is presented for the analysis of planar, uniformly perforated walls under the action of lateral loads. The method is based on replacing the perforated wall by an elastically equivalent orthotropic plate. The principle of minimum total complementary potential energy is then used to determine stresses and displacements in the equivalent plate. Internal forces in the actual structure are obtained by direct integration of these stresses. An example is worked out to illustrate the application of the method and its accuracy. Design curves are presented for the rapid determination of internal forces.