Numerical Methods for Strong Nonlinearities in Mechanics (eBook)

Contact and Fracture

Jacques Besson, Frederic Lebon, Eric Lorentz (Herausgeber)

eBook Download: EPUB

2024
597 Seiten
Wiley-Iste (Verlag)
978-1-394-34049-1 (ISBN)

Infos

Numerical Methods for Strong Nonlinearities in Mechanics deals with recent advances in the numerical treatment of contact/friction and damage phenomena. Although physically distinct, these phenomena both lead to a strong nonlinearity in the mechanical problem, therefore limiting the regularity of the problem, which is now non-differentiable.

This has two direct consequences: on the one hand, the mathematical characteristics of the problem deviate from wellestablished forms, requiring innovative discretization schemes; on the other hand, the low regularity makes it particularly difficult to solve the corresponding large-scale algebraic systems robustly and efficiently. In addition, neither the uniqueness, nor the existence of solutions, remain assured, resulting in bifurcation points, limit loads and structural instabilities, which are always tricky to overcome numerically.

Jacques Besson is Research Director at the CNRS, France, where he conducts research into damage and fracture modeling of metallic materials.

Frédéric Lebon is Professor of Solid Mechanics at Aix-Marseille University and the Mechanics and Acoustics Laboratory (LMA), France.

Eric Lorentz is a senior expert at EDF R&D, France, where he conducts studies on damage modeling, applied to the performance of power generation structures.

Textauszug

1
Lagrangian and Nitsche Methods for Frictional Contact

Franz CHOULY1,2, Patrick HILD3 and Yves RENARD4,5

1 Université Bourgogne-Franche-Comté, Dijon, France

2 University of Chile, Santiago, Chile

3 CNRS, Université de Toulouse, France

4 Institut Camille Jordan, Université de Lyon, France

5 CNRS, INSA Lyon, France

1.1. Introduction

Augmented Lagrangians and Lagrangians are constrained optimization tools that very early have naturally been applied to contact problems with deformable solids (see, for example, Rockafellar 1974, 1976). The augmented Lagrangian has since quite widely become established for the approximation and resolution of contact problems in small and large strains, mainly following the research of Curnier and Alart (1988); Alart and Curnier (1991); Simo and Laursen (1992). The method by Nitsche (1971) was originally proposed to allow a Dirichlet-type boundary condition to be weakly taken into account, precisely avoiding the use of Lagrange multipliers. Only recently has it been extended to contact conditions with or without friction in Chouly and Hild (2013a); Annavarapu et al. (2014); Chouly (2014); Chouly et al. (2015). The close connection between Nitsche and Lagrangian methods is however quite clear and it is the objective of this chapter to shed some light on this relationship. This is achieved namely by looking into the mechanisms underlying these methods, and also by way of presenting some recent developments within the framework of small and large elastic strains.

Section 1.2 first presents the continuous problem of frictional contact between two elastic solids, within the framework of small strains. Section 1.3 is dedicated to finite element approximation within the framework of small strains, where mathematical analysis of numerical methods is possible. Section 1.4 finally presents the extension of the methods described in previous sections to the regime of large elastic transformations, as well as numerical results related to this context.

1.2. Small-strains frictional contact between two elastic bodies

The problem of frictional contact between two elastic solids is first described in section 1.2.1, and then in section 1.2.2, this problem is reformulated as a quasi variational inequality. Then, section 1.2.3 introduces the weak multiplier form, and section 1.2.4 introduces the proximal augmented Lagrangian formulation. These reformulations are the basis of the numerical approximations presented in sections 1.3 and 1.4.

1.2.1. Contact between two elastic bodies

We consider two elastic solids whose respective reference configurations are denoted by Ω1 and Ω2 corresponding to two domains of ℝd (d = 2 or 3) of regular boundaries (piecewise of class ), as shown in Figure 1.1. At the boundaries ∂Ω1 and ∂Ω2 of Ω1 and Ω2, we can identify the boundaries and (with non-empty interiors) on which the elastic bodies are clamped, the boundaries and with an imposed force density and and which are the potential contact boundaries, slave and master, respectively. We assume that these boundaries form a partition without boundaries overlapping of ∂Ω1 and ∂Ω2.

The two elastic bodies are subjected to force densities (volumic forces if d = 3) denoted as f1 and f2 and on and to force densities (surface forces if d = 3) denoted as ℓ1 and ℓ2. The focus is now on expressing the contact condition with Coulomb friction. To this end, we consider the slave surface . For a point the point that potentially comes into contact therewith must be determined. This is called contact pairing. In the contact condition of small–strain approximation, this correspondence is determined on the reference configuration and is not questioned during deformation. In general, a projection is used, but it is not the only possible choice. Let us consider this correspondence:

Figure 1.1. Two bodies with their respective potential contact boundaries.

There are then two outward vectors of interest at point x (see Figure 1.1): the outward unit normal vector of Ω1, which we will denote by n1, and the outward unit vector in the direction of y = Π(x), which we will denote by n and which can be defined by:

since the last two cases (i.e. for (y − x) · n1 ≤ 0) are expected, either when contact is established in the reference configuration, or if both domains are overlapping, which is a priori not prohibited. There is no reason that these two vectors n1 and n should be equal, in general, except by using the “ray tracing” strategy exposed in section 1.4.1. The vector n is usually called the contact normal. To express the contact condition, we need to determine what is the normal component of the stress. Let u1 : Ω1 → ℝd be the displacement of the first body and σ(u1)its Cauchy stress tensor. So, we will denote:

the normal and tangential component decomposition of the stress on the slave contact boundary. We will also denote:

the initial gap between the two potential contact surfaces as well as:

the jumps of the displacements and of the normal displacements. Therefrom, the non-interpenetration condition, or Signorini condition, can be written on as the following complementarity relation:

[1.1]

To write the friction condition, a coefficient of friction is of course needed, which will be denoted by ℱ ≥ 0 and rigorously a notion of sliding velocity. Here, in a supposedly quasi-static evolution, we will not use a sliding velocity but a tangential displacement increment that we will denote by dt. Duvaut and Lions (1972) and Kikuchi and Oden (1988) use the expression , leading to a problem which, although artificial, exhibits the same characteristics as that obtained for an expression of dt that would derive from a time discretization that can be written as:

where is the displacement jump at the previous time step. The friction condition is then written as:

[1.2]

The second Newton law, or action-reaction principle, imposes that:

[1.3]

where n2 is the outward unit normal to at point y = Π(x)and JΠ is the Jacobian of the transformation Π between the two surfaces and .

The description of the linearized elasticity law is carried out by the intermediate of the small strain tensor ε(u) = (∇u + ∇uT)/2. The Cauchy stress tensor is then connected to the strain tensor by the fourth-order elasticity tensor A with the usual symmetry and coerciveness properties. This relationship is written as σ(u) = A ε(u). The displacements u1,u2 of the two elastic bodies are then subjected to the following equations on Ωi, i = 1, 2 in addition to the contact and friction equations [1.1], [1.2] and [1.3]:

[1.4]

1.2.2. The classical weak inequality form

The weak formulation in the form of inequality that can be found in Duvaut and Lions (1972) and Kikuchi and Oden (1988) can be constructed by introducing the following spaces:

and the set of admissible displacements:

The spaces of normal and tangential traces on are also introduced as:

as well as their respective topological duals and . Let (f1,f2) be in , (ℓ1,ℓ2) in , the following bilinear and linear forms on V are defined:

as well as the functional corresponding to the virtual work of the frictional force:

where s = −ℱσn(u) is the friction threshold. The notation denotes the product of duality between the spaces and XN. When s is regular, this product is reduced to the integral . Therefore, the classical weak form associated with [1.1]–[1.4] is written as:

[1.5]

When there is no friction (ℱ = 0), the weak problem [1.5] is a variational inequality of the first kind. Then, the Stampacchia theorem allows us to conclude that it admits a unique solution, which moreover is the only minimizer of the functional on the convex set K. In the presence of friction, results of existence could be shown, for example, in Eck et al. (2005), given that the coefficient of friction ℱ is small. A recent result in Ballard and Iurlano (2023) claims that existence holds for any friction coefficient. Regarding the uniqueness of the solution, counterexamples were presented for large friction coefficients in Hild (2003, 2004) and a criterion for characterizing the uniqueness of the solution was presented in Renard (2006). The uniqueness of the solution for a sufficiently small coefficient is still an open problem.

It should be noted that when the friction threshold is known...

Produkt-Details

• Erscheint lt. Verlag: 3.12.2024
• Reihe/Serie: ISTE Consignment
• Sprache: englisch
• Themenwelt: Technik ► Maschinenbau
• Schlagworte: Algebraic Systems • bifurcation points • contact/friction • damage phenomena • limit loads • nonlinearity • Numerical Methods • Structural Instabilities
• ISBN-10: 1-394-34049-4 / 1394340494
• ISBN-13: 978-1-394-34049-1 / 9781394340491