Principle Of Complementary Energy Research Articles

Special hybrid stress element for stress analyses around circular cutouts in laminated composites

A 3-dimensional hybrid stress element with a traction-free cylindrical surface based on a modified complementary energy principle has been derived for efficient and accurate analysis of stress concentration around circular cutouts in thin to thick laminated composites. New expressions of six stress components are developed by using three stress-functions in cylindrical co-ordinates, so that the homogeneous equilibrium equations, the interlayer surface transverse-stresses and the traction-free boundary condition on the cylindrical surface are satisfied exactly, while the interelement traction continuity has been relaxed via the Lagrange multiplier method. Transverse-shear deformation effects are incorporated in each layer with displacement continuity enforced along interlayer surface. Selected examples are used to demonstrate the efficiency and accuracy of the present special element.

Statics of the laterally loaded shear-flexure beam with hinged end and elastic interior supports

Applying the principle of the minimum complementary energy a simple method is developed for the determination of the response quantities of the shear-flexure beam with hinged laterally fix end supports and laterally elastic interior supports subjected to lateral loads at the nodes. For up to seven-span beams closed-form solutions are derived, whilst for beams with eight or more spans formulas are given for the coefficients and load-terms of the compatibility equations system, so that the response quantities can easily be obtained. The general method includes, as limit or special cases, the solutions for the shear beam with elastic supports, the flexure beam with elastic supports and the simple shear-flexure beam without interior supports. As practical applications of the developed theory, flat-roofed diaphragms in the form of either stressed-skin deckings or longitudinal trusses and grid structures are dealt with and the corresponding stiffnesses defined. A numerical example of a typical industrial building illustrates both the general and the special-case approaches and a discussion of the results points out the effect of the system's parameters onto its response. The paper intends to contribute to an easier design and preliminary and simple final analyses of some spatial building structures.

Stochastic second-order perturbation approach to the stress-based finite element method

The paper is devoted to the application of the second-order perturbation second probabilistic moment method to the stress-based finite element method (FEM). The approach is introduced for the linear elastic heterogeneous medium – up to the second-order, variational equations of the complementary energy principle are presented together with an additional stochastic finite element discretization based on Airy and Prandtl stress functions. The numerical examples shown in this paper illustrate the probabilistic stress and strain tensors in the cantilever beam under shear loading and torsioned square beam with randomly defined material and geometrical parameters. The results obtained in the tests can be applied in probabilistic analyses of the boundary value problems having any closed form mathematical solutions as well as in the stress-based stochastic FEM analysis of solids and structures.

The “free-corner effect” in thermally loaded laminates

The occurrence of localized high interlaminar stresses is a well-known phenomenon along the free edges of composite laminates, which is called “free-edge effect”. Beyond the generally considered situation of a straight free laminate edge, a more specific but also very important situation is that of a free rectangular laminate corner. Here the localized occurrence of interlaminar stresses can be named as “free-corner effect”. For this free-corner effect an approximate closed-form description is given for a class of relatively simple laminates subjected to thermal loading. The applied approach makes use of an equilibrium stress distribution and the principle of minimum complementary energy to describe the situation in the laminate corner region. The predicted interlaminar stress distribution at the ply interfaces is shown to be in good agreement with comparative 3-dimensional finite element analyses.

Finite element methods for variational problems based on nonconforming dual mixed discretisations

A novel hybrid mixed discretisation scheme for the dual formulation of elliptic boundary value problems is presented. It is designed to facilitate the application of reliable a posteriori error estimators based on complementary energy principles. The Lagrange multipliers for the continuity constraints to be imposed on the dual variables are introduced as traces of shape functions which are of Lagrange type. Thus a conforming primal approximation can be obtained by post-processing these multipliers, while the dual approximation need no longer be conforming. The scheme may be viewed as complementing the well known equivalence of conforming dual and nonconforming primal methods. Asymptotic rates of convergence can be established for both dual and primal variables that are of optimal order.

Hybrid Stress Finite Element Methods for Plate and Shell Structures

This paper is a survey of the use of assumed stress hybrid finite element methods for analyzing plate and shell structures. It describes the construction of Kirchhoff plate element by complementary energy principle, and of Reissner-Mindlin plate element by Hellinger-Reissner principle. Special hybrid stress elements are used to model joining of elements which are not coplanar. Such elements include the so-called semiLoof plate elements and 3D-solid elements tailored for analyzing thin plates and shells.

Stress analysis of double-strap bonded joints using a variational method

The design of bonded composite repairs to cracked structure requires not only accurate stress analysis but also the confidence that the design formulae used are truly conservative. This paper describes a stress analysis of the double-strap joint based on a modification of Hashin's variational method for orthogonally cracked composite laminates in which the principle of minimum complementary energy has been employed. The modification is essentially to introduce an adhesive layer and to include isotropic as well as orthotropic material as is typically found in bonded repairs. A simple analytical expression is found for the equivalent spring constant for the joint, which is a parameter that is useful in calculating the stress intensity factor for a patched crack. The expression is shown to yield a truly conservative estimate, in this case a lower bound, for the equivalent spring constant.

Variational solution for a cracked mosaic model of woven fabric composites

A variational solution for a cracked mosaic laminate model of woven fabric composites is presented using the principle of minimum complementary energy. The solution is derived for the woven laminate in either the plane strain or the plane stress state, with the warp/fill yarn materials being either orthotropic or transversely isotropic, unlike other existing solutions in the literature of laminate elasticity. The stress components are given in closed-form expressions in terms of a perturbation function, which is governed by two (uncoupled) fourth-order inhomogeneous ordinary differential equations (i.e., Euler–Lagrange equations) when the thermal effects are included. All possible expressions of this perturbation function are obtained in closed forms. Young’s modulus of the cracked laminate is calculated using the determined minimum complementary energy. The present closed-form solution can account for different yarn materials, applied loads (crack densities), geometrical dimensions, or their combinations. To demonstrate the solution, a total of 60 sample cases are analyzed using three different composite systems (i.e., glass fiber/epoxy, graphite fiber/epoxy and ceramic fiber/ceramic) and ten different crack densities. The obtained numerical results are also compared to two existing elasticity solutions for cross-ply laminates.

Hamilton Principle and Generalized Variational Principles of Linear Thermopiezoelectricity

Via the semi-inverse method, a family of various variational principles is established for thermopiezoelectricity, including a Hamilton principle and a minimum complementary energy principle.

Uncertainty limits for the macroscopic elastic moduli of random polycrystalline aggregates

Practical polycrystalline aggregates are expected to have macroscopic properties that depend upon the properties of constituent crystals and the aggregate geometry. Since that microgeometry is usually random, there will be some uncertainty in the observed macroscopic behavior of the aggregates. The general shape-independent upper and lower estimates for those uncertainty intervals for the elastic moduli of completely random polycrystals are constructed from the minimum energy and complementary energy principles. Applications to aggregates of cubic crystals are presented.

Effects of Friction on Three-Dimensional Contact Stresses in Pin-Loaded Laminated Composites

An investigation is performed to study the effects of friction on the three-dimensional (3D) stress state around pin-loaded holes in laminated composites. The composite pinned joints are modeled as a linear elastic contact problem with friction; therefore, the interactions between the pin with the laminate follow the law of Coulomb friction. An analytical solution for the 3D stress state involved in this problem is developed based upon the variational principle of complementary energy. Compared to the previous research based on a two-dimensional stress assumption, the present solution is innovative, since the out-of-plane effects (i.e., edge effects) near the pin-loaded holes are considered. The present formulations are restricted to the laminated composites with bi-directional fiber orientations, because all the traction and displacement boundary conditions can be satisfied in such type of laminates. Numerical results for pin-loaded cross-ply laminates with different coefficients of friction are presented. It is shown that the effects of friction on the magnitude and distribution of the 3D stress state around pin-loaded holes in laminated composites are significant.

Prediction of energy release rate due to the growth of an interface crack by variational analysis

In our previous studies (Wu W, Verpoest I, Varna J. A novel axisymmetric variational analysis of the stress transfer into fibre through a partially debonded interface. Composites Science and Technology 1998;58:1863–77 and Wu W, Jacobs E, Verpoest I, Varna J. Variational approach to the stress transfer problem through partially debonded interfaces in a three phase composite. Composites Science and Technology 1999;59:519–35) on the stress transfer problem for a single fragment of a fibre or a fibre with an additional interphase, embedded in an infinite matrix with a partially debonded interface, we presented two axisymmetric models, based on the principle of minimum complementary energy. In this paper, some parts of models, useful in the application of fracture mechanics, are first summarised. An expression is then given for the strain energy release rate due to a crack extension in an arbitrary composite system subjected to mixed traction-displacement boundary conditions and with thermal residual stresses included. This expression is suitable for a stress-based variational model. An important advantage is that it can include the friction work at the crack surfaces, in a rather simple but exact way, without using both the displacement and stress distributions at the crack surfaces as inputs. As an application of the expression, we illustrate how to calculate the energy release rate due to the growth of an interface crack in a single fibre fragment with a partially debonded interface under thermomechanical load, by using models in the above references. It is found that the energy release rate can be calculated by using only the rate of change of the strain energy related to the perturbation stress components. A second application is the energy change which is due to fibre breakage. The numerical results indicate that the energy release rate due to the growth of an interface debond or a similar problem can now be determined in a reliable way for both 2- and 3-phase composites.

Some notes on the early history of hybrid stress finite element method

The first remark of this paper is to disclose a historical fact that the assumed stress hybrid finite element method pioneered by Pian [1] in 1964 was actually developed originally based on the Hellinger–Reissner principle and not on the complementary energy principle as indicated in the published paper. The other remark is that Dick Gallagher was the first person to prove the fact that the rectangular plane stress elements by Pian is identical to the Turner element presented in 1956. Finally, through a formulation by Hellinger–Reissner principle a formal proof is given of the fact that the Pian element is also identical to Wilson's incompatible element. Copyright © 2000 John Willey & Sons, Ltd.

Adaptive multigrid method using duality in plane elasticity

This work presents an adaptive multigrid method for the mixed formulation of plane elasticity problems. First, a mixed-hybrid formulation is introduced where the continuity of the normal components of the stress tensor is indirectly imposed using a Lagrange multiplier. Two different numerical approximations, naturally associated with the primal problem and the dual problem, are then proposed. The Complementary Energy Principle provides an a posteriori error estimate. For the effective solving of both systems of equations, a non-standard multigrid algorithm has been designed that allows us to solve the two problems, dual and primal, with reasonable cost and in an integrated way. Finally, a significant numerical application is presented to check the efficiency of the error estimator and the good performance of the algorithm. Copyright © 2001 John Wiley & Sons, Ltd.

Application of variational principles to the axial extension of a circular cylindical nonlinearly elastic membrane

In this paper stationary potential-energy and complementary-energy principles are formulated for boundary-value problems for compressible or incompressible nonlinearly elastic membranes, and full justification for adoption of the complementary principle is provided. The stationary principles are then extended to extremum principles, which provide upper and lower bounds on the energy functional associated with the solution of a given problem. The principles are then illustrated by their application to the nonlinear problem of the axially symmetric static deformation of an isotropic elastic membrane. In its undeformed natural configuration the membrane has the form of a circular cylindrical surface. The cylinder is subject to a prescribed (tensile) axial force with the ends of the cylinder constrained so that their radii remain constant. The alternative boundary condition in which the axial displacement of the ends is prescribed instead of the axial force is also considered. The extremum principles are applied first without restriction on the form of strain-energy function in order to obtain primitive bounds on the energy of Voigt and Reuss type commonly used in composite-material mechanics. Then, for particular forms of strain-energy function, specific bounds are obtained by selecting suitable trial deformation and stress fields and the bounds are optimized using a numerical procedure (which is readily adapted for other forms of strain-energy function). It is found that these bounds are very close and hence give a good estimate of the actual energy. The associated deformed geometry of the membrane is described together with the resulting principal stresses.

Finite deformation beam models and triality theory in dynamical post-buckling analysis

Two new finitely deformed dynamical beam models are established for serious study on non-linear vibrations of thick beams subjected to arbitrarily given external loads. The total potentials of these beam models are non-convex with double-well structures, which can be used in post-buckling analysis and frictional contact problems. Dual extremum principles in unstable dynamic systems are developed. A pure complementary energy principle (in terms of the second Piola–Kirchhoff’s type stress only) in finite deformation mechanics is actually constructed. An interesting triality theory in post-buckling analysis is proved. This theory shows that if the gap function introduced by Gao and Strang in 1989 in positive, the generalized pure complementary energy has only one saddle point, which gives a global stable buckling state. However, if the gap function is negative, the generalized complementary energy may have two so-called super-critical points: the one which minimizes the pure complementary energy gives another relatively stable buckling state; and the other one which maximizes the complementary energy is a unstable buckling state. Application in unilateral buckling problem is illustrated, and an analytic solution for non-linear complementarity problem is obtained. Moreover, the general duality theory proposed recently is generalized into the non-linear dynamical systems. A pair of dual Duffing equations are obtained.

An Elastic Contact Model for the Prediction of Workpiece-Fixture Contact Forces in Clamping

The design and evaluation of machining fixture performance requires accurate knowledge of workpiece-fixture contact forces since they strongly impact workpiece accuracy during clamping and machining. This paper presents an elastic contact model for the prediction of workpiece-fixture contact forces due to clamping. The fixture and workpiece are considered to be elastic bodies in the vicinity of the contact region. The model is formulated as a constrained quadratic program by applying the principle of minimum total complementary energy. The model predicts the normal force, and the magnitude and direction of the tangential (friction) force at each workpiece-fixture contact due to clamping forces. Experimental verification of the model under different clamping loads shows good agreement between predicted and measured normal and tangential contact forces. The model can be used to analyze fixture performance in terms of the contact forces and contact region deformation.

Variational approach to the stress-transfer problem through partially debonded interfaces in a three-phase composite

In our previous study (Wu W, Verpoest I, Varna J. Compos Sci Technol 1998;58(12):1863–77) of the stress-transfer problem around a single fibre, we presented a variational approach based on the principle of minimum complementary energy, not only in the perfectly bonded zone but also in the zone with a discontinuous interface of a two-phase composite. This approach is adapted in this paper to derive an accurate axisymmetric analytical model for the description of the stress state around the fibre breaks and partially debonded interfaces of a three-phase composite with a fibre, coating and matrix. The debonded fibre/coating interface is treated as a special external boundary on which a presumed interfacial shear stress with some free parameters is specified. Once the parameters are given, minimisation of complementary energy can be applied for both debonded- and bonded-interface zones together, to extract the most accurate closed-form solution. The shear stress at the debonded interface (the right values of free parameters) is finally found by substituting the calculated radial stresses in the Coulomb friction law and minimising the discrepancy by a simple numerical iteration. As the minimisation procedure is applied for the both debonded and bonded zones simultaneously, the strong interaction of the two zones is correctly described. This model also includes the matrix axial stress non-uniformity in the radial direction. The stress profiles along both axial and radial directions are presented and closely compared with the results from a finite element model and both agree quite well. A number of applications of this model are also discussed.

A Variational Approach to Analyzing Laminates with Cracks and Viscoplasticity—The Model and its Validation

For cross-ply laminates, matrix cracks and viscoplasticity of the 90 degree ply are two major forms of damage which affects the performance of the composite laminates, especially at high temperatures and under complicated loading patterns. To predict the durability of such kinds of materials, one of the tasks is to analyze stress distribution in each ply of the laminates taking into the interaction between the cracks and viscoplastic deformation. A unified viscoplasticity model is utilized together with the principle of minimum complementary energy to establish a general framework for cross-ply laminates containing cracks and undergoing general viscoplastic deformation. The problem is reduced to an ordinary differential equation for a stress function with an inhomogeneous "force" term related to the viscoplastic strain accumulation. An algorithm to solve the equation for the stress function is presented. The results are compared with those from the finite element method to validate the proposed approach. It is found that although the variational approach, like most of the analytical solutions, does not represent the near crack tip stress singularity field, it does provide fairly accurate through-thickness average stress within the 0 and 90 degree plies. In reality, local inelastic deformation occurs near the crack tip so that the stress is finite. If the local inelastic deformation near the crack tip is taken into consideration, the results from the variational approach match more closely with the finite element solution. Furthermore, the variational approach yields rather accurate in-plane shear stress along the interface between the 90 and 0 degree plies.

Pure complementary energy principle and triality theory in finite elasticity

It is well-known that the complementary energy principle for large deformation elasticity was first proposed by Hellinger in 1914. Since Reissner clarified the boundary conditions in 1953, the complementary energy principles and methods in finite deformation mechanics have been studied extensively during the last forty years (cf. e.g. Koiter, 1976; Nemat-Nasser, 1977; Atluri, 1980; Lee & Shield, 1980; Buffer, 1983; Oden & Reddy, 1983; Ogden, 1984; Tabarrok, 1984 and much more). But the Hellinger-Reissner principle involves both the second Piola-Kirchhoff stress and the displacement, it is not considered as a pure complementary energy principle. For more than 80 years, this principle was considered only as a stationary principle. Its extremum property has been an open problem, which yielded many arguments. The Levinson-Zubov principle involves only the first Piola-Kirchhoff stress r . Unfortunately in finite deformation problems with compressive external loads, the stored energy function W(F) is usually nonconvex in the deformation gradient F (cf. e.g. Ogden, 1984). In this case, even for one-dimensional problems, the complementary energy W e obtained by the Legendre transformation