Geometric nonlinearity: potential energy, complementary energy, and the gap function

In this paper by the use of “convex analysis” the general nonlinear unilateral static problem of structural mechanics is formulated. By means of the “subdifferentiability” concept the unilateral nonlinear boundary conditions and the unilateral nonlinear constitutive laws are formulated as “subdifferential relations”, permitting the derivation of the variational inequalities of virtual and complementary virtual work. Thus the minimum potential and complementary energy theorems are proved. These minimum theorems are used to formulate the analysis as a nonlinear programming problem. Numerical examples on structures with nonlinear unilateral constitutive laws and nonlinear unilateral boundary conditions illustrate the theory.

The aim of this note is to analyse the results in the paper by D.Y. Gao and G. Strang [Geometric nonlinearity: Potential energy, complementary energy, and the gap function, Quart. Appl. Math. 47 (1989), pp. 487–504] where, according to D.Y. Gao, the complementary gap function was first introduced, and several flaws, mistakes, and inconsistencies are present, and to discuss the mathematical manner in which the mentioned paper presents itself, especially from the point of view of convex analysis.

In the sintering of advanced ceramics, a digital twin consisted of a finite element model can predict the final shape and microstructure of a sintered ceramic. However, to minimise uncertainties, it is essential to continually update the mechanical properties in the digital twin using data collected from the manufacturing process. One promising approach for achieving this is through artificial neural networks (ANN). This study introduces a machine learning strategy to update the constitutive behaviour of advanced ceramics in finite element analysis of sintering deformation. A major challenge in implementing machine learning in material processing is the huge amount of data required by the training and validation of an ANN, which are often unavailable or incomplete in a real manufacturing process. This study demonstrates that the data requirement can be reduced by employing a two-step training technique. Firstly, the ANN is trained using a nonlinear constitutive law, which describes a general relationship between the strain rates and stresses. Subsequently, the weights and biases of the ANN are transferred for the retraining using limited experimental data for an actual ceramic. It is shown that such approach can successfully capture the nonlinear constitutive behaviour of fine-grained alumina without demanding for large amount of experimental data. A case study is provided, highlighting the feasibility of implementing the ANN in a commercial finite element package, to replace the constitutive law and predict the shrinkage and distortion of a sintering part. In particular, the sintered dumb-bell shape part, simulated using the retrained ANN, showed grain size and relative density that markedly different from those using the nonlinear constitutive law. It is important to note that the proposed methodology is generic and can be used to create ANNs to replace constitutive laws in finite element analysis in digital twins for a wide range of other engineering processes.

Variability response functions for beams with nonlinear constitutive laws

VAM applied to dimensional reduction of non-linear hyperelastic plates

A minimum principle of complementary energy is established for cable networks involving only the stress components as variables with geometrical nonlinearities and nonlinear elastic materials. The minimization problem of total potential energy is reformulated as a variational problem with a convex objective functional and an infinite number of second-order cone constraints; its Fenchel dual problem is shown to coincide with the minimization problem of the complementary energy. It is of interest to note that the obtained complementary energy attains always its minimum value at the equilibrium state irrespective of the stability of the cable networks, contrary to the fact that only stationary principles have been presented for elastic trusses and continua, even in the case of a stable equilibrium state.

The famous bifurcation analysis performed by Flügge on compressed thin-walled cylinders is based on a series of simplifying assumptions, which allow to obtain the bifurcation landscape, together with explicit expressions for limit behaviours: surface instability, wrinkling, and Euler rod buckling. The most severe assumption introduced by Flügge is the use of an incremental constitutive equation, which does not follow from any nonlinear hyperelastic constitutive law. This is a strong limitation for the applicability of the theory, which becomes questionable when is utilized for a material characterized by a different constitutive equation, such as for instance a Mooney-Rivlin material. We re-derive the entire Flügge's formulation, thus obtaining a framework where any constitutive equation fits. The use of two different nonlinear hyperelastic constitutive equations, referred to compressible materials, leads to incremental equations, which reduce to those derived by Flügge under suitable simplifications. His results are confirmed, together with all the limit equations, now rigorously obtained, and his theory is extended. This extension of the theory of buckling of thin shells allows for computationally efficient determination of bifurcation landscapes for nonlinear constitutive laws, which may for instance be used to model biomechanics of arteries, or soft pneumatic robot arms.

The stability condition of cantilevered masonry piers subjected to their own weight and to a concentrated compressive top load is investigated, considering no-tension material with nonlinear stress-strain law in compression, of an experimental nature, and including softening behavior. The analysis is carried out by improving a numerical approach adopted in previous works where stability problems were solved assuming an infinitely elastic linear constitutive law in compression. Before the geometrical nonlinearity is considered, the limit equilibrium of a typical pier rectangular cross section is detected assuming unlimited available compressive strain. This preliminary analysis allows one to determine analytically the limit value that has to be imposed on the eccentricity of the resultant compressive force and to derive the moment-curvature relationships on which the second-order effects depend. Then the stability domains are derived in dimensionless form and their boundaries are modeled by analytical approximate expressions of practical use. Some numerical examples show that, depending on the average normal stress level, the assumption of an approximate linear constitutive law in compression, affected by the same elasticity modulus as that at the origin of the actual stress-strain law, can provide an unacceptable overestimate of the critical load.

The notion of material stability is examined in the context of active muscle tissue modeling, where the nonlinear constitutive law is dependent both on the physiologically driven muscle contraction and finite mechanical deformation. First, the governing equations and constitutive laws for a general active-elastic material with a single preferred direction are linearized about a homogeneous underlying configuration with respect to both the active contraction and deformation gradient. In order to obtain mathematical restrictions analogous to those found in elastic materials, stability conditions are derived based on the propagation of homogeneous plane waves with real wave speeds, and the generalized acoustic tensor is obtained. Focusing on 2D motions, and considering a simplified, decoupled transversely isotropic energy function, the restriction on the active acoustic tensor is recast in terms of a generally applicable constitutive law, with specific attention paid to the fiber contribution. The implication of the material stability conditions on material parameters, active contraction, and elastic stretch is investigated for prototype material models of muscle tissue.

Artificial neural network (ANN) models are used to learn the nonlinear constitutive laws based on indirectly measurable data. The real input and output of the ANN model are derived from indirect data using a mechanical system, which is composed of several subsystems including the ANN model. As the ANN model is coupled with other subsystems, the input of the ANN model needs to be determined during the training. This approach integrates measurable data, mechanics, and ANN models so that the ANN models can be trained without direct data which is usually not available from experiments. Two examples are provided as an illustration of the proposed approach. The first example uses two-dimensional (2D) finite element (FE) analysis to train an ANN model to learn the nonlinear in-plane shear constitutive law. The second example applies a continuum damage model to train an ANN model to learn the damage accumulation law. The results show that the trained ANN models achieve great accuracy based on the proposed approach.

We propose a numerical procedure for nonlinear electromechanical problems in smart ceramics. Two essential features of the present procedure are the use of a Landau's phenomenological potential energy as nonlinear constitutive law and the finite element discretization of an alternative governing equation with vector potential. For solving the resulting semi-discrete nonlinear system, the Newmark-β method is employed. Numerical examples in two dimensions are shown to validate the present procedure.

Introduction of (1) material frame indifference, (2) indifference with respect to rigid-body rotations of the reference configuration, and (3) material symmetry classification guides the development of non-linear (both Cauchy and hyper-) elastic constitutive laws for isotropic and anisotropic behaviour (for materials with a micro-structure). Incremental elasticity is introduced with reference to the relative Lagrangean description, and the Biot framework for incompressible elasticity is derived. Hypo-elasticity is also briefly included. Broadly speaking, constitutive laws set a bridge between strain and stress, so they have to keep into account the behaviour of the specific material under consideration. This behaviour can be fully reversible and described by elasticity (e.g., in the case of rubber) or can be partially irreversible and described by elastoplasticity (e.g., in the case of mild steel), or it may be time-dependent and described by viscous laws (e.g., in the case of a Newtonian fluid). We exclude for the moment the elastoplastic behaviour, and we present constitutive laws in this chapter for anisotropic elastic materials and for incrementally deformed solids (elastoplasticity is deferred to Chapter 8). The introduction of constitutive laws for materials subject to large strains requires preliminary statements of general principles, such as the so-called material frame indifference and indifference with respect to rigid-body rotations of the reference configuration. When these concepts are specified, the development of constitutive equations to model different materials is greatly simplified. After introduction of these general principles, we conclude the present chapter with the formulation of elastic constitutive laws, generalising concepts proposed in Chapter 4, now to includemicro-structural anisotropy.

In this chapter we illustrate the use of the abstract results obtained in Chap. 4, in the study of three representative static frictional contact problems for deformable bodies. In the first two problems we model the material’s behavior with a nonlinear elastic constitutive law and with a viscoelastic constitutive law with long memory, respectively, and we describe the frictional contact with subdifferential boundary conditions. In the third problem the deformable body is assumed to be piezoelectric and, therefore, we model its behavior with an electro-elastic constitutive law. And, again, the contact conditions, including the electrical conditions on the contact surface, are of subdifferential type. For each problem we provide a variational formulation. For the first two problems it is in a form of a hemivariational inequality for the displacement field and, for the third one, it is in a form of a system of hemivariational inequalities in which the unknowns are the displacement and electric potential fields. Then, we use the abstract existence and uniqueness results presented in Chap. 4 to prove the weak solvability of the corresponding contact problems and, under additional assumptions, their unique weak solvability. Finally, we present concrete examples of constitutive laws and frictional contact conditions for which our results work and provide the related mechanical interpretation. Everywhere in this chapter we use the notation introduced in Chap. 6.

A new reduced formulation for modeling elastic structures or structural components with geometric nonlinear- ities is presented. This formulation is obtained as a generalization of the concept of foreshortening, which has been successfully used by other authors to model beams. Some examples are provided that illustrate the generality and validity of this formulation. The proposed theory is compared to previous work related to foreshortening, variable reduction in nonlinear elastic problems, and the use of a geometric stiffness matrix. Some guidelines are also given to implement the proposed method and to interpret the concepts that it involves. The reduction method presented is general, but a corotational approach is followed to simplify the exposition. The use of the foreshortening 9 is another method that has been proposed to solve this problem. This interesting concept provides certain importantcalculation advantagesdespitethat,unfortunately, it has been impossible to generalize structural members other than cantileverbeams.Thepresentpaperdevelopsa generalizationofthe concept of foreshortening that can be applied to any type of struc- ture or structural component with elastic behavior and geometric nonlinearities. The result is a method of structural analysis with variable reduc- tion that provides considerable savings in calculation times, as well as an interesting theoretical model for the purpose of studying these types of problems analytically. In addition, the presented deriva- tion allows a better understanding of the relationships between the different approaches that are used to solve the geometric stiffening problem. 9 Thispaperisorganizedasfollows.First,abriefintroductiontothe foreshortening concept is presented (Sec. II). This is followed by a second-order corotational formulation of elastic structural analysis includinggeometric nonlinearities thatprovideaneasy introduction to the ideas presented (Sec. III). Next, foreshortening is redee ned and generalized in the framework of the structural formulation just mentioned, up to a new reduced formulation of the nonlinear prob- lem (Sec. IV). This derivation is subsequently ree ned, and relevant terms are identie ed and interpreted (Sec. V). A brief description of the particularization of the proposed formalism in the case of a planar cantilever beam (Sec. VI) and the key steps of the numerical algorithmofthestaticproblem (Sec.VII)aresubsequentlyincluded. InSec.VII,itis also explained howto usethepresented formulation togeneratereducedcomponentmodelsvalidformultibodyanalysis. In the succeeding section, the most relevant connections with other derivations where foreshortening or variable reduction are applied are discussed. Finally, some examples are provided that illustrate the potential of this theory (Sec. IX).

The classical linear stability equations for columns take the form of equilibrium equations in terms of displacements. From a variational point of view these equations emerge as conditions of extremum from an energy or virtual work functionals. In principle it should be possible to arrive at the governing equations for stability of columns from a complementary energy or complementary virtual work functional involving force quantities alone. This is rarely done. Both the energy and the complementary energy approaches depend upon the relevant constitutive equations. In this study a formulation is employed wherein focus is maintained on the constitutive equations. It is shown that under certain admissibility conditions a least squares functional, forcing the satisfaction of the constitutive equations, yields the energy and the so-called pure complementary energy functionals as subsets.