Complementary Energy Formulation Research Articles

Efficient dynamic topology optimization of 2D metamaterials based on a complementary energy formulation

The advent of additive manufacturing has revolutionized the design and development of hierarchical structures, with potential applications in compliant, auxetic, and band-gap structures. This paper presents an innovative approach to developing a dynamic Topology Optimization (TO) framework for designing printable lattice structures that exhibit specific dynamic properties. Utilizing parametrically defined filament-based unit cell structures for topology optimization, we achieve desired natural frequency bandgaps in the structures composed of these unit cells. To enhance computational efficiency, we employ a complementary energy-based formulation to (semi)analytically derive the flexibility and stiffness matrices of the unit cell structure, thus, eliminating extensive finite element discretization. Consequently, a wide variety of parametrically defined filament-based meso-structures can be mathematically explored. We apply this innovative framework specifically for band-gap maximization of 2D lattice structures. By tuning the geometry within each cell using TO, we maximize the band gap. Our results show the potential of this approach to create more efficient and effective hierarchical structures with desired band-gap properties.

Variational Convergences of Dual Energy Functionals for Elastic Materials with a ε-Thin Strong Inclusion

We give a new derivation, based on the complementary energy formulation, of a simplified model for a multi-structure made up of two anisotropic hyper-elastic bodies connected by a thin strong material layer. The model is obtained by identifying the Mosco-limit of the stored complementary energy functional when the thickness is of order e and the stiffness of order 1/e where e is a positive real adimensional parameter. In order to prove the existence of the displacement associated with the stress we use a suitable weak version of the Saint-Venant compatibility condition also known as Donati’s theorem.

A complementary energy formulation of no tension masonry-like solids

In the present work a complementary energy formulation for the solution of the static equilibrium problem of no-tension solids is presented. The main feature of the work is systematic use of the augmented Lagrangian approach for the regularisation of the nonsmooth and nondifferentiable anelastic deformation potential. The proposed formulation is applied to derive a finite element formulation using the subspace of the self-equilibrated stresses as primal variables. The code implements the augmented Lagrangian algorithm for the determination of the inelastic deformations. Some numerical examples are included to illustrate the performance of the proposed approach.

A new approach to mechanical testing and modeling of biological tissues, with application to blood vessels.

The collection and processing of data from mechanical tests of biological tissues usually follow classical principles appropriate for studying engineering materials. However, difficulties specific to biological tissues have generally kept such methods from producing quantitative results for statistically-oriented studies. This paper demonstrates a different approach linking testing and data reduction with modern statistical tools. Experimental design theory is used to minimize the detrimental effects of collinearity on the stability of the parameters in constitutive equations. The numerical effects of time-dependent biasing factors such as viscoelasticity are reduced by randomizing the order of collection of data points. Some of the parameters of the model are allowed to vary from specimen to specimen while the others are computed once from a database of designed experiments on several specimens. Finally, a new self-modeling algorithm based on principal component analysis is used to generate uncorrelated parameters for a model that is linear in its specimen-dependent parameters. The method, associated with a recently published complementary energy formulation for vascular mechanics, is illustrated with biaxial canine saphenous vein data. Results show that three specimen-dependent linear parameters are enough to characterize the experimental data and that they can be repeatedly estimated from different data sets. Independently collected biaxial inflation data can also be predicted reasonably well with this model.

An alternate formulation of blood vessel mechanics and the meaning of the in vivo property

In an effort to bridge the gap between theoretically based vascular mechanics and simpler, clinically relevant compliance models, an alternative complementary energy formulation as well as an improved experimental method are proposed. This formulation generalizes the uniaxial compliance models to multiaxial stretch and twist and clarifies the role of the total in vivo force (measured in biaxial vascular testing) in vessel stability. The measurement of wall thickness when incompressibility is assumed, and the assumption of the existence of an unloaded state are unnecessary. Scattered results from both clinical and mechanical literature are reinterpreted in view of this new understanding of vascular mechanics.Canine saphenous veins (4–5 mm in diameter) conform to the behavior expected from the analysis. Their in vivo property is shown to be a passive property, independent of smooth muscle tone. Redundant particle tracking, control of force and pressure values and the use of papaverine, result, respectively, in strain measurements accurate to within 0.002, experimental control accurate to 0.5% of full range, and repeatable experiments over a few days. The validated system and the new formulation show great potential for exploiting the benefits of innovative experimental design and statistically reliable data analysis.

A hybrid formulation for determining torsion and warping constants

Theoretically, the torsion constant of a bar subjected to a torque can be calculated irrespective of its cross-sectional shape, either with a potential energy (displacement) or with a complementary energy (stress function) formulation. The potential energy formulation, however, also provides the so-called warping constant, which may be important in the case of thin-walled sections, but the complementary energy formulation does not. On the other hand, the complementary energy formulation leads to simple analytical expressions for the torsion constant of thin-walled sections, while a finite element calculation based on potential energy requires a relatively large number of elements to obtain comparable results. Since hybrid formulations combine some potential energy features with complementary ones, a hybrid formulation was derived for the torsion problem, with the purpose of combining the advantages of both—i.e. an ultimate displacement formulation and possibly greater accuracy. Both the torsion and warping constants can be calculated consistently with this formulation. The hybrid variational formulation that was derived, together with the general discretization procedure, can be used to formulate finite elements for calculating the torsional constants for all kinds of cross-section, whether solid or thin-walled, open or closed, symmetrical or unsymmetrical. The initial application, to rectangular solid cross-sections, resulted in an improved warping constant.

On the use of the complementary energy in the solution of buckling problems

A systematic derivation of the expression for the complementary energy in elastic buckling problems is presented. Compatibility is identified with variation with respect to the stress components, and the resulting eigenvalue problem is shown to be equivalent to, and sometimes more convenient than, the corresponding formulation in terms of the potential energy. Similarly, approximate techniques may lead to better as well as simpler estimates, whose upper bound property can, however, be assured only through appropriate safeguards.The method is applied in some detail to buckling of columns of arbitrary boundary conditions and axial force distribution. Another example is the problem of lateral beam buckling, with the effect of warping restraint included. In both cases (and presumably in many others) the complementary energy formulation is of lower order than the conventional potential energy formulation, and it is clear that the same simplification should also apply to finite elements or other discrete formats. The method is restricted to the (technically significant) case of a linear prebuckling state.

An iterative penalty function method in structural analysis

AbstractAn iterative solution process combined with the penalty function method employed in constrained variational priciples is described. As an example of the use of the method an axi‐symmetric plate with the complementary energy formulation is considered.

A new variational principle for finite elastic displacements

Variational principles for finite elastic displacements have been formulated in terms of Green strain and Kirchhoff-Trefftz stress tensors. The first is a functional of the displacement field only and implies stationarity of the total potential. The second is a canonical principle, in the sense of Friedrichs [1], involving both stresses and displacements and generalizing Reissner's principle [5]. In contrast with the geometrically linearized elasticity theory, it cannot be reduced to a complementary energy principle involving equilibrium stresses only [9,10]. The paper discusses the Levinson[9] and Zubov[11] formulation in terms of displacement gradients and the Piola stress tensor, which, however interesting from a theoretical viewpoint, does not appear suitable for practical applications. A new set of variational principles, of displacement, canonical or complementary energy types, is found to derive from the use of the polar decomposition theorem of the jacobian. It involves the engineering strain tensor and its conjugate stress tensor is to be regarded as a function of the Piola tensor and the material rotation. The complementary energy formulation is discussed in terms of first order stress functions. The presence of the rotational degrees of freedom opens the possibility of discretizing the rotational equilibrium equations in approximate solutions.

Approximate Three-Dimensional Solutions for Symmetric Laminates Under Inplane Loading *

A three-dimensional equilibrium finite element analysis is used to obtain approximate stress solutions for some finite symmetric laminates under inplane loading. The analysis is based on a comple mentary energy formulation. Obtained stress solutions at the center section of the laminate are compared with solutions for uniform axial strain that are available in the literature. A comparison of the stress distributions obtained at the center and end sections of the laminate shows that a large τyz stress can occur at the end section and a large τxz can occur at the center section. The influence of laminate stacking sequence on the predicted σz is also presented.

An incremental complementary energy method of nonlinear stress analysis

A stress analysis capability to treat orthotropic physically nonlinear materials that behave differently in tension and compression is presented. The analysis is an energy search method based on an incremental complementary energy formulation. Load incrementation is used. Stress solutions are obtained by formulating and minimizing the incremental complementary energy associated with each load increment. Material behavior is conservative during each load increment. Structures are idealized as an assemblage of discrete elements. An Airy stress function is approximated by a sum of unknown coefficients times interpolation polynomials for each element. Equilibrium between adjacent elements is satisfied identically by imposing equality constraints on the unknown coefficients. Applications to structural elements loaded in plane stress or plane strain are presented. Correlations with some experimental results available in the literature are made.

Calculations of plate frequencies from complementary energy formulation

AbstractThe complementary variational principle has been used to derive the differential equations and the associated boundary conditions of the vibrating plate in terms of bending moments. It is shown that in this formulation, the plate possesses an infinite number of zero frequency modes in which the plate remains in a state of constant strain under a set of self‐equilibrating bending moments. In applying the Rayleigh Ritz procedure for the non‐zero frequency modes of the plate, it is shown that it the assumed functions are orthogonal to only a finite number of zero frequency modes, then one may obtain frequencies which are lower than the true frequencies of the plate. An iliustrative example is given in the paper.