Principle Of Complementary Energy Research Articles

Stochastic finite elements as a bridge between random material microstructure and global response

This paper presents a finite element method aimed at the introduction of microstructural material randomness below the level of a single finite element. A consideration of dependence of effective moduli on scale and on the defining boundary conditions leads to an identification of a finite element as a mesoscale window (or, a mesoscale finite element) in a stochastic finite element method (SFE). An estimation of the global response can be obtained through bounds stemming from minimum potential energy and complementary energy principles, which involve Dirichlet and Neumann boundary conditions on all the mesoscale finite elements, respectively. While in the classical case of a homogeneous material, these two bounds converge to each other as the finite element mesh becomes sufficiently fine, an optimal mesoscale with respect to the difference between both bounds may exist in the case of a heterogeneous material. The proposed SFE method is illustrated with a numerical example of a sample two-phase Voronoi composite (with some 26 000 grains), where a reference solution taking into account the entire microstructure without any smearing out, is shown to fall between both energy bounds. A generalization to an ensemble response is straightforward.

General Analytic Solutions and Complementary Variational Principles for Large Deformation Nonsmooth Mechanics

This paper presents a nonlinear dual transformation method and general complementary energy principle for solving large deformation theory of elastoplasticity governed by nonsmooth constitutive laws. It is shown that by using this method and principle, the nonconvex and nonsmooth total potential energy is dual to a smooth complementary energy functional, and fully nonlinear equilibrium equations in finite deformation problems can be converted into certain tensor equations. The algebraic relation between the first and the second Piola–Kirchhoff stresses are revealed. A closed form solution for general three-dimensional large deformation boundary value problems is obtained. The properties of this general solution are clarified by a triality extremum principle. This triality theory reveals an important phenomenon in nonconvex variational problems. Applications are illustrated by nonlinear, nonsmooth equilibrium problems in Hencky's plasticity, 3D cylindrical structures and post buckling analysis of elastoplastic bar with jumping and hardening effects. The idea and methods presented in this paper can be used and generalized to solve many nonlinear boundary value problems in finite deformation theory.

A novel axisymmetric variational analysis of stress transfer into fibreS through a partially debonded interface

Most existing models for the problem of fibre/matrix stress-transfer through a partially debonded interface roughly solve the stress distribution in the debonded zone, neglecting the presence of the perfectly bonded zone. However the stress interactions between two zones is what makes the problem essentially different from the stress-transfer problem for a perfectly bonded interface. This paper suggests a variational approach based on the principle of minimum complementary energy not only in a perfectly bonded zone but also in a zone with a discontinuous interface. The debonded interface is treated as an external boundary on which a presumed interfacial shear stress is specified. A new analytical model, including stress non-uniformity in the radial direction and crack interaction, is derived to describe the stress state around fibre breaks and debonding tips in a single fibre embedded in an infinite matrix. For the presumed shear stress at the debonded interface the minimisation procedure renders the most accurate closed-form solution (under used assumptions) for both interactive zones. Finally, the ‘best’ shear stress distribution at the debonded interface is found by using Coulomb’s friction law and simple numerical iterations. The stress profiles along both axial and radial directions are presented and compared with results from a numerical model [1] available in the literature and also from finite-element analysis. Good agreements are achieved. Extensive applications of this approach and the derived model are also discussed.

Duality, triality and complementary extremum principles in non-convex parametric variational problems with applications

This paper presents a reasonably complete duality theory and a nonlinear dual transformation method for solving the fully nonlinear, non-convex parametric variational problem inf{W(Λu - μ) - F(u)}, and associated nonlinear boundary value problems, where Λ is a nonlinear operator, W is either convex or concave functional of p = Λu, and μ is a given parameter. Detailed mathematical proofs are provided for the complementary extremum principles proposed recently in finite deformation theory. A method for obtaining truly dual variational principles (without a dual gap and involving the dual variable p* of Λu only) in n-dimensional problems is proposed. It is proved that for convex W(p), the critical point of the associated Lagrangian Lμ(u, p*) is a saddle point if and only if the so-called complementary gap function is positive. In this case, the system has only one dual problem. However, if this gap function is negative, the critical point of the Lagrangian is a so-called super-critical point, which is equivalent to the Auchmuty's anomalous critical point in geometrically linear systems. We discover that, in this case, the system may have more than one primal-dual set of problems. The critical point of the Lagrangian either minimizes or maximizes both primal and dual problems. An interesting triality theorem in non-convex systems is proved, which contains a minimax complementary principle and a pair of minimum and maximum complementary principles. Applications in finite deformation theory are illustrated. An open problem left by Hellinger and Reissner is solved completely and a pure complementary energy principle is constructed. It is proved that the dual Euler-Lagrange equation is an algebraic equation, and hence, a general analytic solution for non-convex variational-boundary value problems is obtained. The connection between nonlinear differential equations and algebraic geometry is revealed. Keywords:duality theory; triality; complementary energy; nonlinear variational problem; non-convex optimization; fully nonlinear system; finite deformation theory; partial differential equations; analytic solution; phase transitions; variational inequality

A variational approach for straight edge effect stresses in laminated composites

An analytical technique is presented to calculate thermally and mechanically induced edge effect stresses at straight free boundaries of laminated composites. The involved three-dimensional stress functions, which satisfy equilibrium equations, all traction boundary conditions (including traction continuity conditions at interfaces between layers), are determined based on the variational principle of complementary energy. Numerical results of the present technique are in good agreement with those of the finite element analysis.

Edge Effects in Laminated Composites with Pin-Loaded Holes

An analytical technique is developed for determining the three-dimensional edge effect stresses around pin-loaded holes in symmetric laminated composites. Based on the superposition principle of linear elasticity theory, the stress state around pin-loaded holes is divided into two parts, one the in-plane stress state determined using the classical laminated plate theory and the other the boundary-layer stress state arising from the edge effects that only exist in the vicinity of the holes. The equilibrium equations for boundary-layer stress state with zeroth-order approximation are written based on a set of power series expansions. A closed-form solution for the boundary-layer stress state with zeroth-order approximation is then obtained by imposing stress functions on the variational principle of complementary energy. Numerical results obtained using the present technique for cross-ply laminates are in good agreement with those obtained using finite element methods.

Statistical theories for the elastic moduli of two-dimensional assemblies of granular materials

In this micro-mechanical study of the behaviour of granular materials, a minimum potential energy and a minimum complementary energy principle are derived for the elastic moduli of two-dimensional, bonded assemblies of particles. These extremum principles are used to obtain bounds on the elastic moduli. Statistical versions of these energy principles are developed. The theoretical results are verified by computer simulations using the discrete element method. Good agreement between theory and simulations is observed.

Adaptive selection of polynomial degrees on a finite element mesh

The problem of finding a nearly optimal distribution of polynomial degrees on a fixed finite element mesh is discussed. An a posteriori error estimator based on the minimum complementary energy principle is proposed which utilizes the displacement vector field computed from the finite element solution. This estimator, designed for p- and hp-extensions, is conceptually different from estimators based on residuals or patch recovery which are designed for h-extension procedures. The quality of the error estimator is demonstrated by examples. The results show that the effectivity index is reasonably close to unity and the sequences of p-distributions obtained with the error indicators closely follow the optimal trajectory. © 1998 John Wiley & Sons, Ltd.

On the analysis of cross-ply laminates with micro-cracks and inelastic deformation

For cross-ply laminates micro-matrix cracks in the 90° plies and inelastic deformation in the 0° and 90° plies are two major forms of damage which affect the long-term durability of the materials. It is crucial to develop a mechanics based approach to incorporate both micro-cracks and inelastic deformation of the composite materials based on the response of the 0° and 90° plies in order to accurately predict the response of the materials in service environments and to assist in optimizing the materials system for best performance. However, such a task leads to severe mathematical difficulties, primarily due to the zero traction conditions on the crack surfaces, the complicated constitutive relations governing the inelastic deformation and the interaction between the cracks and the inelastic deformation. In this paper, a general framework for the analysis of crossply laminates with micro-matric cracks and inelastic deformation is proposed. For this purpose admissible stress fields are constructed which satisfy equilibrium and all boundary and interface conditions. The principle of minimum complementary energy is utilized to derive a differential equation for the stress function from which the stress field of the composite can be derived. The inhomogeneous terms of the differential equation involve the inelastic strains which are loading history dependent. The Green's function of the differential equation is then obtained. Using the Green's function and a constitutive equation, two-dimensional stress and strain states in the composite at any time are represented by an integral of the Green's function and the inelastic strains accumulated up to that time. This new analysis takes into consideration the microcrack-microcrack interaction, as well as the interaction between the microcracks and the inelastic deformation, and provides a point-wise variation of the stress field instead of average stress field as most of the analytical approaches yield.The interactions of matrix cracks and creep deformation of an eight-harness satin weave (SHSW) Nextel 610/Aluminosilicate ceramic matrix composite is studied using the proposed model. The predicted creep strain of the composite shows good correlation with experimental data at different levels of temperature and stress conditions. The distribution of stresses and strains provides important information on the response of the composite.

An improved analysis of the stresses in a single-fibre fragmentation test—II. 3-phase model

A new model has been developed for the analyses of the stress state in a single-fibre specimen containing a coated fibre. This model is a generalization of the 2-phase model developed in Part I[1] for the case of 3 phases: fibre/coating/matrix. The stress distributions in radial and axial directions are obtained by using the principle of minimum complementary energy. Assumptions made to simplify the stress analysis are: (a) the axial stress distribution is uniform across the fibre cross-section; (b) the axial stress distribution is uniform across the coating thickness; (c) the non-uniform axial stress distribution in the matrix is given by a decreasing function which is obtained by a minimization procedure. The model developed has been used in a parametric analysis to determine the effect of the coating properties and geometry on the rate of stress transfer and shear stress concentration at the fibre end.

The Effect of Temperature Gradient on the Free-Edge Interlaminar Stresses in Multi-Layered Structures

For a multilayered anisotropic laminate, the intense and localized interlaminar stresses in the vicinity of a free edge may be significantly affected by non-uniformity of the temperature load. An efficient analysis method, based on polynomial expansions of the stress functions with respect to the thickness coordinate and the principle of complementary energy, is further developed in the present work to evaluate localized free-edge stresses under non-uniform temperature loads. The variational method yields closed-form solutions of the interlaminar stresses in terms of eigenfunctions. Interlaminar stresses are obtained for symmetric, four-layer, cross-ply and angle ply laminates subjected to a uniform temperature load and to linearly varying non-uniform temperature loads. It is shown that a temperature gradient may contribute significantly to the free-edge interlaminar stresses even if the local value of the temperature increment vanishes at the intersection of the interface and the free edge.

Transverse cracks in cross-ply laminates. 1. Stress analysis

During service loading of cross-ply laminates, transverse cracks occur in plies. The cracks parallel to the fiber direction are extended over the full thickness of transverse plies and often cross the entire test specimen width. It is widely recognized that the changes of laminate thermomechanical constants, caused by the transverse cracking of composite laminates, can be significant. Theoretical stress analysis in the cross-ply laminates in the vicinity of cracks is performed using numerical (FE) and analytical methods. The effect of transverse cracks on the degradation of elastic properties will be discussed in Part 2 [1]. Approximate analytical micromechanical models based on shear lag predictions, variational analysis, and numerical 2D finite element calculations were verified in their predictive abilities. The three variational models used are based on the principle of minimum complementary energy and have different degrees of accuracy with respect to the stress assumptions used (Hashin's, 2D 0° and 2D 0°/90° models). Using FEM, the plane stress and strain state were analyzed. The effect of material properties and layer thickness on the stress distribution in a 90° layer was evaluated by varying the crack spacing. The crack opening displacement (COD), normalized with respect to the far field strain, is proposed as a measure of reduction of the mechanical properties. Since the CODs are rather insensitive to the crack spacing (crack density) in a wide region, they will be used in modeling the stiffness reduction in these laminates [1].

On the variational equations of thermoelasticity for inhomogeneous anisotropic shells

On the basis of the complete system of equations that describes the thermoelastic behavior of an inhomogeneous anisotropic shell under the action of force and thermal loads we derive variational equations that are analogs of the principle of virtual work, the principle of complementary energy, and the Biot equation in three-dimensional thermoelasticity.

Generalized potential formulations for elastostatics of constitutively non-linear structures under general loading

Extremum problem formulations are presented that are applicable for the equilibrium analysis of continuum structures made up of a non-specific, aeleotropic and inhomogeneous, constitutively non-linear material. The formulations are described for linear deformation kinematics, and they amount to generalizations of the classical characterization in terms of potentials for elastostatics with non-linear materials. In one case, a generalized version of the classical minimum potential energy formulation is obtained using a representation of total strain expressed as a linear composition of independent fields. Each such constituent field is to have a separate potential, and the contribution of each constituent is limited by specified constraint. The resulting generalized potential energy formulation for non-linear elasticity provides for the simulation of overall properties for an arbitrary material, in terms of the set of individually specified potential functions. A second formulation provides a generalized model of the classical complementary energy principle. Here total stress is expressed as a composition of independent fields, and there is a separate potential associated with each of these constituent fields. These two problem statements are complementary in concept and in their structure. The net constitutive properites, which are implicit in the models, generally may be non-smooth. In both formulations, load path is represented in terms of a load path (or process) parameter, and this provides for interpretation of the dependence of evolution of response on the form of loading.

FINITE ELEMENT STRESS EXTRACTION BY THE COMPLEMENTARY ENERGY PRINCIPLE

This paper presents a new method for accurate pointwise stress extraction from finite element solutions, applied to two-dimensional linear elastostatic problem shaving bounded value stresses. The method, denoted by SEC(Stress Extraction by Complementary principle), is based on the complementary energy principle applied overalocal domain in thepost-processing phase. Detailed formulation of the SEC method is provided, and numerical experiments with the h- and p-versions of the finite element method are presented for a family of exact solutions characterized by varying degree of smoothness. It is shown that on the boundaries of the domain, as well as in the interior, accurate pointwise stresses are obtained, and the relative error in the pointwise stresses converges with a rate which is as fast as the relative error measured in energy norm or faster. Importantly, the SEC method in conjunction with the p-version of the finite element method is virtually independent of the Poisson's ratio, and is equally applicable to nearly in compressible materials. © 1997 by John Wiley & Sons, Ltd.

An improved analysis of the stresses in a single-fibre fragmentation test: I. Two-phase model

In order to improve the analysis of the single-fibre fragmentation test, a modified variational analysis, based on the principle of minimum complementary energy, has been derived to describe the stress states around breaks in a single fibre embedded in an infinite matrix. The variational analysis includes the non-uniform stress distribution in the radial direction of the matrix cylinder, and allows for crack interaction. The stress profiles along the interface between the fibre and matrix are presented and compared with the prediction by the variational approach presented by Nairn. The predictions are also compared with the results from finite element analysis and good agreement is obtained. While the present analysis is more accurate than existing analytical models, it leads to very simple final expressions. Thus, it is convenient to be used for data reduction of the single-fibre fragmentation test results. In particular, it can be easily generalized to a three-phase model, including an interphase between fibre and matrix. The extensive applications of this model are also presented.

Stress fields in general composite laminates

A direct approach is employed to obtain a general boundary integral formulation for the analysis of composite laminates subjected to uniform axial strain. The integral equations governing the problem are directly deduced from the reciprocity theorem, employing the generalized orthotropic elasticity fundamental solutions expressly inferred. The solution is achieved by the boundary element method, which gives, once the traction-free boundary conditions and the interfacial continuity conditions are enforced, a linear system of algebraic equations. The formulation does not present restrictions with regard to the laminate stacking sequence and it does not require any aprioristic assumption. The interlaminar stress field near the free edge of generally stacked composite laminates subjected to uniaxial extension is investigated through this boundary integral equation formulation. The numerical applications show good agreement with those already available in literature, and they demonstrate the accuracy and the efficiency of the proposed method. approach where a complete three-dimensional analysis is performed was presented by Pipes and Pagano,1 who employed the finite dif- ference technique to obtain the solution of the governing elastic- ity equations. Many solutions obtained by using the finite element method are available.21() These differ from each other in the formu- lation, in the kind of employed elements, and in the discretization schemes. In the literature analytical solutions of approximate the- ories are also present. The techniques employed to achieve these solutions include the perturbation method,11 series solution,12'13 Lekhintskii's complex stress potentials coupled with an eigenfunc- tion expansion14'15 or a polynomial expansion,16'll the extension of Reissner's variational principle,18'19 and the force balance method coupled with the minimum complementary energy principle.20'21 The stress distributions obtained with these approaches show good agreement between them for points away from the free edge. Con- siderable disagreement exists instead for points near the free edge location among the various numerical and analytical solutions. This is to be expected as a result of a priori assumptions or because the boundary conditions of the continuum problem have been trans- formed into conditions on the generalized data. Actually the numer- ical solution techniques are not able to predict the singular trend in the stress field at the free edge, and hence, to achieve satisfactory solutions a lengthy extrapolation procedure is required. On the other hand, the analytical approximate solutions often rest on assumptions about the problem unknowns that enforce the free edge stress field structure. In the present paper, the stress field in multilayered composite laminates under uniform axial extension is analyzed on the basis of the integral equation theory.22'23 The laminate is considered as composed by prismatic elements having different elastic proper- ties. The generic element or ply is treated as homogeneous, and in terms of constitutive equations it is described by a generalized orthotropic law. The integral equations governing the exact elastic- ity solution of the problem are directly obtained by applying the reciprocity theorem with the fundamental solutions of the general- ized orthotropic elasticity.2427 The

Accurate stress extraction for nearly incompressible materials by the displacement formulation of the p-version FEM

Direct methods for computing the pointwise stresses for nearly incompressible elastic materials fail to provide meaningful results when applied to the displacement formulation of the finite element method (FEM). A new extraction method for accurate computation of pointwise stresses for nearly incompressible elastic materials is presented. It is based on the complementary energy principle applied over a local domain in the postprocessing phase in conjunction with the p-version finite element solution. It is shown that accurate pointwise stresses are obtained, that the relative error in the pointwise stresses converges at a rate which is as fast as the relative error measured in the energy norm or faster, and importantly, the extracted stresses are virtually independent of Poisson's ratio. Numerical results for two problems, one having a smooth solution and the other containing a singular point, are provided.

Micromechanically based stochastic finite elements: length scales and anisotropy

The present stochastic finite element (SFE) study amplifies a recently developed micromechanically based approach in which two estimates (upper and lower) of the finite element stiffness matrix and of the global response need first to be calculated. These two estimates correspond, respectively, to the principles of stationary potential and complementary energy on which the SFE is based. Both estimates of the stiffness matrix are anisotropic and tend to converge towards one another only in the finite scale limit; this points to the fact that an approximating meso-scale continuum random field is neither unique nor isotropic. The SFE methodology based on this approach is implemented in a Monte Carlo sense for a conductivity (equivalently, out-of-plane elasticity) problem of a matrix-inclusion composite under mixed boundary conditions. Two versions are developed: in one an exact calculation of all the elements' stiffness matrices from the microstructures over the entire finite element mesh is carried out, while in the second one a second-order statistical characterization of the mesoscale continuum random field is used to generate these matrices.

Mechanics of integral armor: discontinuous ceramic-cored sandwich structure under tension and shear

An integral armor sandwich structure consisting of S2-glass-reinforced face sheets with adhesively bonded discontinuous ceramic tiles provides potential optimal ballistic protection and structural performance at minimum weight for composite armored vehicle applications. The structural response is investigated by defining an idealized unit cell subjected to tensile and in-plane shear membrane loading. In the model, an admissible stress system that satisfies equilibrium and all boundary and interface conditions is constructed, and the principle of minimum complementary energy is employed to find an optimal approximation. Typical stress variations are presented, and their implications for progressive damage and failure of the structure are discussed. Based on the stress and deformation fields in the core and face sheet, the overall effective stiffness of the structure can be evaluated. By selecting some critical parameters, such as the core length and the adhesive stiffness and thickness, an optimized design for the structure can be achieved.