Principle Of Complementary Virtual Work Research Articles

Series Solution-Based Approach for the Interlaminar Stress Analysis of Smart Composites under Thermo-Electro-Mechanical Loading

This paper introduces a new loading condition considering the combined thermo-electro-mechanical coupling effect in a series solution-based approach to analyze the free-edge interlaminar stresses in smart composite laminates. The governing equations are developed using the principle of complementary virtual work. The assumed stress fields satisfy the traction-free and free-edge boundary conditions. The accurate stress states of the composite structures are acquired through the procedure of generalized eigenvalue problems. The uniform temperature is employed throughout the laminate, and the electric field loading is applied to the symmetric piezo-bonded actuators to examine the combined effect of thermal and electrical stresses on the overall deformation of smart composite laminates. It was observed that the magnitude of the peeling stresses generated by mechanical loading was reduced by the combined thermal and electric excitation loading (up to 25.3%), which in turn resulted in expanding the service life of the smart composite structures. The proposed approach is implemented on three different layup configurations. The efficiency of the current methodology is confirmed by comparing the results with the 3D finite element (FEM) solution computed by ABAQUS.

3D Thermal Stresses in Composite Laminates Under Steady-State Through-Thickness Thermal Conduction

In this study, 3D thermal stresses in composite laminates under steady-state through thickness thermal conduction are investigated by means of a stress function-based approach. One-dimensional thermal conduction is solved for composite laminate and the layerwise temperature distribution is calculated first. The principle of complementary virtual work is employed to develop the governing equations. Their solutions are obtained by using the stress function-based approach, where the stress functions are taken from the Lekhnitskii stress functions in terms of in-plane stress functions and out-of-plane stress functions. With the Rayleigh–Ritz method, the stress fields can be solved by first solving a standard eigenvalue problem. The proposed method is not merely computationally efficient and accurate. The stress fields also strictly satisfy the prescribed boundary conditions validated by the results of finite element method (FEM) results. Finally, some of the results will be given for discussion considering different layup stacking sequences, thermal conductivities and overall temperature differences. From the results, we find that the thermal conductivity greatly affects the stress distributions and peak values of stresses increase linearly for the present model. The proposed method can be used for predicting 3D thermal stresses in composite laminates when subjected to thermal loading.

A stress function based model for transient thermal stresses of composite laminates in various time-variant thermal environments

We present a study of transient thermal stresses of composite laminate in this work by means of stress function based approach considering various time-variant thermal environments. Two-dimensional temperature distribution is considered for both linear and quadratic temperature distributions. For each type of distribution, we consider three different cases to examine the effect off thermal transfer point on the distribution of thermal stresses. The proposed methodology uses the principle of complementary virtual work to develop the governing equations and consider the thermomechanical coupling in its constitutive equation. The Lekhnitskii stress functions are adopted as stress fields and the mode shape functions of clamped beam are adopted as stress trial functions. By means of the Rayleigh–Ritz method, we obtain a standard eigenvalue problem and further solve the eigen problem to obtain the solutions. We find that different temperature distributions affect the inhomogeneous equations, but they don't affect the homogeneous equations in the present model. The results will demonstrate the distributions of transient thermal stresses in angle-ply and cross-ply laminates under linear and quadratic temperature distributions. The present methodology is computationally efficient and accurate in calculating the transient thermal stresses, and it can be used in thermal stress analysis of laminated structures.

Bending stress analysis of a piezoelectric nanoplate with flexoelectricity under inhomogeneous electric fields

Flexoelectricity has been found as an unneglectable electro-mechanical coupling effect in micro/nanoscale dielectric materials. We present a stress function based bending stress analysis method for piezoelectric nanoplate under inhomogeneous electric fields considering both piezoelectric effect and flexoelectric effect in this work. A Ritz type solution procedure is developed by means of the quasi-three dimensional stress functions with the initial assumption of out-of-plane stress functions. A standard eigenvalue problem is constructed to obtain the general solutions of governing equations which are obtained by the principle of complementary virtual work. For the numerical analysis, we investigate the bending stresses in laminated piezoelectric nanoplate with or without flexoelectricity and the size-dependent effect on the bending stress distributions. Two kinds of inhomogeneous electric fields are considered for discussion. The present work is expected to help understanding the influence of flexoelectric effect on the bending stress distributions for the future design and application of piezoelectric nanostructures.

A new microplane model for non-proportionally multiaxial deformation of shape memory alloys addressing both the martensite transformation and reorientation

A new microplane model is established to predict the non-proportionally multiaxial deformation of pseudoelastic shape memory alloys (SMAs) by considering martensite transformation and reorientation. In each microplane, various martensite variants with certain shear directions are introduced. Adopting the static constraint formulation, the shear and normal stresses in each microplane are obtained through a macroscopic stress tensor. Thermodynamically-consistent kinetic equations of internal variables are derived based on the Clausius's dissipative inequality. Employing the principle of complementary virtual work, a macroscopic strain tensor is obtained by integrating the strains on the microplanes with all orientations. Comparisons between the simulated results and corresponding experimental ones show that the non-proportionally multiaxial deformations of pseudoelastic SMAs can be reasonably predicted. Moreover, the effects of the martensite reorientation and the number of martensite variants on the macroscopic stress-strain responses of pseudoelastic SMAs are discussed.

Mechanical model of suture joints with fibrous connective layer

A composite model for suture joints with a connective layer of aligned fibers embedded in soft matrix is proposed. Based on the principle of complementary virtual work, composite cylinder assemblage (CCA) approach and generalized self-consistent micro-mechanical models, a hierarchical homogenization methodology is developed to systematically quantify the synergistic effects of suture morphology and fiber orientation on the overall mechanical properties of sutures. Suture joints with regular triangular wave-form serve as an example material system to apply this methodology. Both theoretical and finite element mechanical models are developed and compared to evaluate the overall normal stiffness of sutures as a function of wavy morphology of sutures, fiber orientation, fiber volume fraction, and the mechanical properties of fibers and matrix in the interfacial layer. It is found that generally due to the anisotropy-induced coupling effects between tensile and shear deformation, the effective normal stiffness of sutures is highly dependent on the fiber orientation in the connective layer. Also, the effective shear modulus of the connective layer and the stiffness ratio between the fiber and matrix significantly influence the effects of fiber orientation. In addition, optimal fiber orientations are found to maximize the stiffness of suture joints.

Obtenção de um elemento de viga unificado utilizando a formulação corrotacional e o princípio dos trabalhos virtuais complementares

In this work is described a shear flexible beam element model without shear locking and integrating Euler-Bernoulli’s and Timoshenko’s beam theories. The unified stiffness matrix of this beam is obtained by using the co-rotational formulation that is based on the separation of displacement fields into rigid body modes and deformation modes. The principle of complementary virtual work is also used for finding the unified stiffness matrix. The stiffness matrix is then implemented in a finite element software for linear analysis of plane frames. Some examples are presented and numerical results demonstrate that a shear flexible beam element do not show shear locking effects. Keywords: Co-rotational approach, Complementary virtual work, Shear flexible beam element.

Extended Kantorovich method for local stresses in composite laminates upon polynomial stress functions

The extended Kantorovich method is employed to study the local stress concentrations at the vicinity of free edges in symmetrically layered composite laminates subjected to uniaxial tensile load upon polynomial stress functions. The stress fields are initially assumed by means of the Lekhnitskii stress functions under the plane strain state. Applying the principle of complementary virtual work, the coupled ordinary differential equations are obtained in which the solutions can be obtained by solving a generalized eigenvalue problem. Then an iterative procedure is established to achieve convergent stress distributions. It should be noted that the stress function based extended Kantorovich method can satisfy both the traction-free and free edge stress boundary conditions during the iterative processes. The stress components near the free edges and in the interior regions are calculated and compared with those obtained results by finite element method (FEM). The convergent stresses have good agreements with those results obtained by three dimensional (3D) FEM. For generality, various layup configurations are considered for the numerical analysis. The results show that the proposed polynomial stress function based extended Kantorovich method is accurate and efficient in predicting the local stresses in composite laminates and computationally much more efficient than the 3D FEM.

Time-dependent stress variations in symmetrically viscoelastic composite laminates under uniaxial tensile load

In this paper, we report an analytical approach for time-dependent stress variations in symmetrically viscoelastic composite laminates under plane deformation state based on a stress function based equivalent single layer theory. The theory initially adopts the stress function separated by the in-plane stress function and out-of-plane stress function as field assumption, instead of using the displacement field. The viscoelastic properties are simply expressed by the Maxwell model for the analysis of relaxation effect on the free edge stresses in viscoelastic laminates. The constitutive equation in the integral form for linear viscoelastic materials under constant uniaxial strain load can be simplified. By taking the principle of complementary virtual work, the governing equation can be obtained and further solved by solving a general eigenproblem. Convergent stress distributions are obtained and validated by the 3-D finite element method using commercial package. The free edge stresses are function of time and loading conditions in viscoelastic composite laminates and the relaxation effect on the free edge stresses is clearly shown in the numerical results of viscoelastic composite laminates with various layup stacking sequences.

Interlaminar stress analysis of magneto-electro-elastic composite layered laminates using a stress function based iterative approach

In this paper, we propose a stress function based iterative approach for predicting the interlaminar stresses in magneto-electro-elastic (MEE) composite layered laminates induced by both mechanical and piezomagnetic excitations. The proposed iterative methodology adopts the Lekhnitskii stress functions as stress fields which satisfy the pointwise equilibrium condition under plane strain state and the stress trial functions are separated into the in-plane stress function and out-of-plane stress function. The iteration initiates from an assumption of the out-of-plane stress function as a combination of harmonic functions. By taking the principle of complementary virtual work, the governing equations are derived and the in-plane stress function can be obtained by solving a general eigenproblem. The obtained in-plane stress function is further used as the known function and the out-of-plane stress function is solved in the next iteration. Through the iterations, the interlaminar stress distributions are calculated from the iterative process and the accuracy is improved. The results are validated by the 3-D FEM results under the mechanical load and the convergence study is also conducted. The proposed approach is accurate and efficient in predicting the interlaminar stresses of MEE composite layered laminates.

Free edge stress prediction for magneto-electro-elastic laminates using a stress function based equivalent single layer theory

A stress function based equivalent single layer theory upon the Lekhnitskii stress functions is proposed for predicting the localized free edge stresses of magneto-electro-elastic composite layered laminates under extensional strain load and magnetic load. The stress trial functions are separated into the in-plane stress functions and out-of-plane stress functions, in which the out-of-plane stress functions are assumed by the combination of harmonic and hyperbolic functions. The in-plane unknown functions can be calculated from the governing equations obtained by taking the principle of complementary virtual work. A standard eigenvalue problem is formed to solve the ordinary differential equations. The obtained free edge stress components satisfy the prescribed traction free boundary condition. To verify the proposed method, both cross-ply and angle-ply cases are considered as the substrate laminate for both the mechanical and piezomagnetic loads. The results are also compared with those obtained by the finite element method to validate the efficiency of the proposed approach.

Interlaminar stress analysis of piezo-bonded composite laminates using the extended Kantorovich method

An iterative method has been applied to analyze the free edge interlaminar stresses of piezo-bonded composite laminates. Electric fields are applied to two symmetrically bonded piezo actuators that can generate induced strain, resulting in pure extension of the whole structure. The stresses, which satisfy the traction-free and free edge boundary conditions, are obtained by taking the principle of complementary virtual work, and conducting the extended Kantorovich method. In order to obtain accurate interlaminar stress distributions, static and kinematic continuity conditions are applied at the interfaces between plies through iterations. The stress components were obtained under the plane strain assumption as well as antiplane shear assumption. The results were compared with those obtained by the finite element method, to demonstrate the validity of the proposed method. Through the iteration processes, the interlaminar stresses converged well, and predicted maximum peak stress at the interface of piezo and composite layers. The present method provides accurate stresses distribution near the free edges, of piezo-bonded composite laminates.

Free-edge stress analysis of functionally graded material layered biocomposite laminates.

A stress function based theory is proposed to obtain free-edge stress distributions for three-dimensional, orthotropic, linearly elastic rectangular biocomposite laminates with surface-bonded functionally graded materials (FGM). The assumed stress fields automatically satisfy the pointwise equilibrium equation, as well as traction-free and free edge boundary conditions. The complementary virtual work principle, followed by the general eigenvalue solution procedure, is used to obtain 3-D free edge stress states. A typical stacking sequence of composite laminate is used as numerical investigation with surface bonded FGMs. It is shown that with proper exponential factor of FGMs, the interlaminar stresses at the FGM layer interface can be reduced significantly, in return to prevent debonding of FGM layers. This approach can be useful in the design of functionally graded material layered biocomposite structures.

Control of free-edge interlaminar stresses in composite laminates using piezoelectric actuators

The control of free-edge interlaminar stresses in laminated composite structures using a stress function-based approach is proposed. The assumed stress fields satisfy pointwise traction and free boundary conditions at surfaces. Governing equations are derived using the principle of complementary virtual work. A general eigenvalue solution procedure was adopted to obtain accurate stress states of the laminated composite structure. The results obtained from the proposed method were compared with those obtained by three-dimensional finite element analyses. It was found that interlaminar stresses generated by mechanical loadings could be significantly reduced by applying proper electric fields to piezoelectric actuators, which were surface bonded or embedded in composite laminates. Locations of piezoelectric actuators also influenced the distributions of interlaminar stresses. The results provided that piezoelectric actuators have potential in the application to actively control interlaminar stresses in composite laminates.

A thermodynamically-consistent microplane model for shape memory alloys

In microplane theory, it is assumed that a macroscopic stress tensor is projected to the microplane stresses. It is also assumed that 1D constitutive laws are defined for associated stress and strain components on all microplanes passing through a material point. The macroscopic strain tensor is obtained by strain integration on microplanes of all orientations at a point by using a homogenization process. Traditionally, microplane formulation has been based on the Volumetric–Deviatoric–Tangential split and macroscopic strain tensor was derived using the principle of complementary virtual work. It has been shown that this formulation could violate the second law of thermodynamics in some loading conditions. The present paper focuses on modeling of shape memory alloys using microplane formulation in a thermodynamically-consistent framework. To this end, a free energy potential is defined at the microplane level. Integrating this potential over all orientations provides the macroscopic free energy. Based on this free energy, a new formulation based on Volumetric–Deviatoric split is proposed. This formulation in a thermodynamic-consistent framework captures the behavior of shape memory alloys. Using experimental results for various loading conditions, the validity of the model has been verified.

Reduction of free edge peeling stress of laminated composites using active piezoelectric layers.

An analytical approach is proposed in the reduction of free edge peeling stresses of laminated composites using active piezoelectric layers. The approach is the extended Kantorovich method which is an iterative method. Multiterms of trial function are employed and governing equations are derived by taking the principle of complementary virtual work. The solutions are obtained by solving a generalized eigenvalue problem. By this approach, the stresses automatically satisfy not only the traction-free boundary conditions, but also the free edge boundary conditions. Through the iteration processes, the free edge stresses converge very quickly. It is found that the peeling stresses generated by mechanical loadings are significantly reduced by applying a proper electric field to the piezoelectric actuators.

Free-edge interlaminar stress analysis of piezo-bonded composite laminates under symmetric electric excitation

A stress function-based approach is proposed to analyze the free-edge interlaminar stresses of piezo-bonded symmetric laminates. The proposed method satisfies the traction free boundary conditions, as well as surface free conditions. The symmetric laminated structure was excited under electric fields that can generate induced strain, resulting in pure extension in the laminated plate. The governing equations were obtained by taking the principle of complementary virtual work. To verify the proposed method, cross-ply, angle-ply and quasi-isotropic laminates were analyzed. The stress concentrations predicted by the present method were compared with those analyzed by the finite element method. The results show that the stress function-based analysis of piezo-bonded laminated composite structures is an efficient and accurate method for the initial design stage of piezo-bonded composite structures.

A generalized mechanical model for suture interfaces of arbitrary geometry

Suture interfaces with a triangular wave form commonly found in nature have recently been shown to exhibit exceptional mechanical behavior, where geometric parameters such as amplitude, frequency, and hierarchy can be used to nonlinearly tailor and amplify mechanical properties. In this study, using the principle of complementary virtual work, we formulate a generalized, composite mechanical model for arbitrarily-shaped interdigitating suture interfaces in order to more broadly investigate the influence of wave-form geometry on load transmission, deformation mechanisms, anisotropy, and stiffness, strength, and toughness of the suture interface for tensile and shear loading conditions. The application of this suture interface model is exemplified for the case of the general trapezoidal wave-form. Expressions for the in-plane stiffness, strength and fracture toughness and failure mechanisms are derived as nonlinear functions of shape factor β (which characterizes the general trapezoidal shape as triangular, trapezoidal, rectangular or anti-trapezoidal), the wavelength/amplitude ratio, the interface width/wavelength ratio, and the stiffness and strength ratios of the skeletal/interfacial phases. These results provide guidelines for choosing and tailoring interface geometry to optimize the mechanical performance in resisting different loads. The presented model provides insights into the relation between the mechanical function and the morphological diversity of suture interface geometries observed in natural systems.

Free-Edge Interlaminar Stress Analysis of Composite Laminates Using Interface Modeling

AbstractInterlaminar stresses near free edges in composite laminates were analyzed by considering interface modeling. Previous regular analytical solutions for free edge stress are not rigorous in their linear elastic modeling with dissimilar interfaces between homogeneous plies of laminated composites. Singular stress is not practical in reality at the free edge because it nearly never occurs in real situations. Until now, the analytical nonsingular approximate solutions have been accepted even though they cannot show the rigorous singular behavior at the free edges; this shortcoming is corrected within linear elasticity dissimilar interface modeling. In this study, a regular analytical solution is provided, which is rigorous within the linear elastic model featuring smoothly varying material properties through the thickness of the laminates. This interface modeling provides not only nonsingular stresses but concentrated finite interlaminar stresses using the principle of complementary virtual work and t...

Bending analysis of a laminated composite patch considering the free-edge effect using a stress-based equivalent single-layer composite model

The interlaminar stresses of a laminated composite patch, which is made up of reinforcing fibers (carbon/graphite) and epoxy matrix are analyzed using a stress-based equivalent single-layer model under a bending load. The composite patch is frequently used as reinforcement for a metallic adherend of mechanical/aerospace structures (i.e., aluminum alloy, etc) by attaching the film- or paste-type adhesive (i.e., epoxy, BMI, etc). To calculate the adhesive stresses transferred from the substrate, an interlayer model is introduced. The adhesive stresses are obtained by solving the equilibrium equations. The stress fields of the patch are determined by assuming certain stress functions. To satisfy the equilibrium state of the patch, the stress functions are divided into homogeneous and particular parts. The adhesive stresses act as prescribed stress boundary conditions of the laminated composite patch. The stress functions are substituted into a complementary virtual work principle, and from this, two coupled ordinary differential equations are obtained. General eigenvalue problems are derived to solve the coupled governing equations. To demonstrate the validity and efficiency of the proposed method, cross-ply, angle-ply and quasi-isotropic laminated composite patches are studied. From the observations made, the authors found that the stress function-based approach is suitable for solving the stress prescribed boundary value problem with accuracy and efficiency compared to a displacement-based approach such as the finite element method. The proposed method can be used as an efficient tool in the initial design stage of structural components when it is necessary to consider the free-edge effect.