Structural models based on 3D constitutive laws: Variational structure and numerical solution

https://doi.org/10.1016/j.cma.2020.112872Get rights and content

Highlights

  • A description of the role of 3D constitutive laws in abstract structural models.

  • A new incremental potential that reframes structural problems as optimization ones.

  • Local and global problems are shown to be solvable with descend-type solvers.

  • Error indicators are introduced based on the incremental potential.

Abstract

In all structural models, the section or fiber response is a relation between the strain measures and the stress resultants. This relation can only be expressed in a simple analytical form when the material response is linear elastic. For other, more complex and interesting situations, kinematic and kinetic hypotheses need to be invoked, and a constrained three-dimensional constitutive relation has to be employed at every point of the section in order to implement non-linear and dissipative constitutive laws into dimensionally reduced structural models. In this article we explain in which sense reduced constitutive models can be expressed as minimization problems, helping to formulate the global equilibrium as a single optimization problem. Casting the problem this way has implications from the mathematical and numerical points of view, naturally defining error indicators. General purpose solution algorithms for constrained material response, with and without optimization character, are discussed and provided in an open-source library.

Introduction

In linear and nonlinear structural theories (including bars, beams, plates, membranes, shells, etc.), the kinematic definitions and the equilibrium equations need to be closed with a relation between the strain measures and the stress resultants, possibly with time-dependent and dissipative effects. In the simplest situations, often restricted to the linear elastic case, there exists an analytical expression that relates stresses and strain resultants, leading to well-known equations that have been employed for decades in structural analyses (see [1], [2], [3], for example, among many books fully describing this process).

Structural analyses involving nonlinear elastic or inelastic materials can rarely make use of section response laws and must, therefore, rely directly on three-dimensional constitutive laws evaluated at every point of the cross section. Such an evaluation is complicated due to the fact that every structural theory constrains, from its outset, the kind of strains and stresses that are allowed on the sections. The hypotheses imply that the constitutive response of these material points is constrained and its evaluation is rarely possible by analytical means.

These constrained constitutive laws must provide, given a proper subset of the strain components and knowing that some stress components vanish, the remaining parts of the strain and stress tensors. Numerical methods designed to solve this problem are iterative by nature, since, except for the linear elastic case, the strain–stress relations are often implicit and nonlinear. Standard Newton–Raphson iterative methods — or related ones — have been proposed for this purpose in the past [4], [5], [6], [7].

Newton-like methods solve the constrained constitutive problem in a fast and robust fashion. By focusing on the nonlinear (and constrained) strain–stress relations, however, they fail to notice that there is an underlying variational structure behind the problem. In this article we reveal the precise meaning of this framework, and identify the form of the energy functional involved. Such a characterization allows to show that the equilibrium problem of structures, even when the section constitutive law is evaluated pointwise from constrained three-dimensional models, has a global variational character.

This assertion has at least three consequences. First, from the theoretical point of view, theorems for the existence of solutions can now be derived based on minimizing sequences and properties of functionals, even in the constrained case. Second, for more pragmatic reasons, numerical optimization methods can be used to solve the global problem and local constitutive relations, for example of nonlinear conjugate gradient type. And third, since a single energy functional is shown to be the quantity to minimize for a given solution, it provides a natural error indicator for Galerkin-type approximations. The first of these three implications falls outside the scope of the current work, but the second and the third point are discussed and exemplified in Sections 4 Solution of the reduced constitutive equations, 5 Numerical examples.

The setting employed for the derivations in this article is that of linearized kinematics and nonlinear materials with internal variables. This framework is general enough to accommodate many problems of interest, and can be extended, in a fairly straightforward manner to include geometrical nonlinearities. Due to the similarities with the small strain case, the details of such generalization are not worked out in this article, although both small and finite strain simulation examples will be provided.

The way that we choose to deal with material nonlinearities, rate and history effects is by means of incremental potentials [8]. With them, a variational structure is preserved even for problems involving plasticity, viscoelasticity, etc., as long as this is interpreted in an incremental fashion. Such ideas, widespread in Computational Mechanics, are extended by our results to structural models of inelastic materials.

The remainder of this article is organized in the following way. In Section 2 we describe structural models in an abstract way that can encompass all types. In this completely general setting, we introduce the connection between the section response and the constrained material constitutive law. The underlying variational setting is discussed in Section 3 by revealing that there is a compatible minimization structure at the level of material point, section, and structural member. In addition to the theoretical advantages of the formulation, as already mentioned, some numerical implications are studied in Section 4 and illustrated by means of examples in Section 5. Section 6 closes the article summarizing the main findings.

As will be detailed in Section 4, we classify two types of methods that can be employed to solve numerically the constrained or reduced constitutive model. The material open-source library MUESLI [9] implements both, and readers can access it to verify the details of the numerical implementation.

Section snippets

From structural models to 3D, and back

We explore in this section the relationship between the mechanics of three-dimensional solids and structural theories. There are many interesting points of view for this analysis, including asymptotic behavior, model reduction, numerical discretization, etc., but we restrict our exposition to those features that help to explain the central point of this article: how three-dimensional material constitutive laws can be employed to formulate section response relations. We address fully inelastic

Variational structure

We show in this section that the constitutive relation of a structural section has a variational structure that encompasses the kinematic and kinetic hypotheses. Such a property allows the formulation of the whole quasistatic problem of equilibrium as a structural problem in the form of a minimization problem, including the section response derived from three-dimensional constitutive theory. When the material response is inelastic, however, such variational statement can only be made

Solution of the reduced constitutive equations

The structural models described in this article can employ arbitrary (small strain) material models making use of incremental updates and setting the problem statement under a single variational framework. Key to this unified formulation is the identification of the reduced effective stored energy function W̄n that accounts for the kinetic constraints.

As explained in Section 2, the reduced constitutive model consists of finding part of the stress and the strain tensors at a point, given the

Numerical examples

We examine next several numerical examples involving bars, beams, and shells combined with various three-dimensional constitutive laws. The purpose of these simulations is, first, to demonstrate that the two solution strategies outlined in Section 4 are both valid routes for the integration of the section response for complex material models. Secondly, we will show that the variational structure behind the constitutive response of reduced models opens the door to error indicators.

In all

Summary and outlook

Structural models employing complex material models (inelastic, rate-dependent, etc.) require the solution of constrained three-dimensional material constitutive laws for the evaluation of their section response. This article starts by describing in an abstract way how these two problems are linked in an arbitrary structural model.

The main result of this work is the identification of a variational setting behind the problem of general structural models, encompassing three-dimensional laws, both

Acknowledgment

Partial support for I.R. has been provided by project DPI2017-92526-EXP from the Spanish Ministry of Science, Innovation, and Universities .

References (28)

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