Advances in Computational Coupling and Contact Mechanics

The following sections are included:

• About the Editors
• Preface
• List of Contributors
• Contents

A frictional mortar contact formulation for the simulation of solids undergoing finite strains with inelastic material behaviour is presented. The framework includes the modelling of finite strain inelastic deformation, the numerical treatment of frictional contact conditions and specific finite element technology. Several well-established and recent models are employed for each of these building blocks to capture the distinct physical aspects of the deformation behaviour. The approach is based on a mortar formulation, and the enforcement of contact constraints is realized with dual Lagrange multipliers. The introduction of nonlinear complementarity functions into the frictional contact conditions combined with the global equilibrium leads to a system of nonlinear equations, which is solved in terms of the semi-smooth Newton method. The resulting method can be interpreted as a primal–dual active set strategy (PDASS) which deals with contact nonlinearities, material and geometrical nonlinearities in one iterative scheme. The consistent linearization of all building blocks of the framework yields a robust and highly efficient approach for the analysis of metal forming problems. The effect of finite inelastic strains on the solution behaviour of the PDASS method is examined in detail based on the complementarity parameters. A comprehensive set of numerical examples is presented to demonstrate the accuracy and efficiency of the approach against the traditional node-to-segment (NTS) penalty contact formulation.

In this chapter, we present high-order mortar-based contact elements using Lagrange multiplier and penalty methods. We seek approximation solutions for 2D small and large deformation contact problems, through the standard and generalized high-order finite element methods (labelled as hp-FEM and GFEM, respectively). We validate the elements for small and large deformation with known solutions in the literature. The h- and p-refinements assert that the high-order interpolation has superior performance in contact problems with better accuracy of results for stresses with lower time/cost processing. In the case of the GFEM contact element, we observe that the enrichment benefits goes beyond the smoothing of the stress fields. Our findings show that the contact elements proposed here can provide more accurate results than the ones obtained by the conventional low-order FEM.

This chapter deals with numerical treatment of a large deformation frictionless contact problem by isogeometric analysis. The symmetry-preserving formulation, also known as two-half-pass formulation is presented together with the Gauss-point-to-segment discretization. The contact constraints regularization is performed by the penalty method. In the chapter, all key aspects of the numerical solution of contact problems are mentioned, including contact kinematics, spatial discretization by the finite element method, linearization, contact detection and solution of the nonlinear equation system. The performances of the resulting algorithm are showed by means of several contact benchmark problems including contact patch test, Hertz problem and ironing problem.

In this work, we consider the dynamic interaction of incompressible fluids with flexible structures. Due to the particular characteristics of fluid and solid systems, different formulations are usually applied to each medium producing non-matching interfaces. Also the movements of the fluid on the free surface due to sloshing motions are usually bigger than the elastic displacements of the structure and is also a source of interface mismatching. To investigate this problem, a variational formulation for the linear-vibration of bounded fluid-structure systems taking into account the combined effect of sloshing, acoustic waves and initial stress due to gravity, is introduced. The systems are modelled using finite elements and coupled with localized Lagrange multipliers to study the particular case of an inviscid homogeneous liquid in a closed container. A key approach of the present localized formulation is to treat the non-matching interfaces by the method of LLMs using the FEM with a Lagrangian description of motion for the fluid and the structure.

In this chapter, we propose an Eulerian-based thermo-flexible multi-body approach in order to simulate rig testing of disc brake systems accurately and efficiently. A multi-body model of the disc, an assembly of flywheels and the shaft connecting the disc and flywheels is coupled to an Eulerian-based thermo-mechanical finite element model of the disc-pad system. By utilizing the Eulerian framework of the disc, the contact interface is modelled most accurately with Signorini contact, Coulomb frictional heating with a new temperature-dependent friction model that includes fading at high temperatures and Archard wear with a temperature-dependent wear coefficient. The governing equations are treated with a sequential approach, where first the mechanical contact problem is solved using the augmented Lagrangian approach with a non-smooth Newton method. Then, the multi-body model is solved for the brake moment obtained from the mechanical contact analysis using the average acceleration method. Finally, heat balance for the system including the frictional power from the two previous steps is obtained with the trapezoidal rule and formulating the nonlinear equations as a system of linear equations. Here, the non-symmetric convection matrix is stabilized by adding artificial conduction according to the streamline-upwind approach. The proposed sequential approach is implemented in an in-house code and utilized to study a vented disc-pad system to a heavy truck. This is discussed at the end of the chapter, showing the development of heat bands, hot spots and corresponding residual stresses.

Based on the 1D differential matrix derived from the Lagrange series expansion, the Finite Block Method (FBM) was recently developed to solve both the elasticity and transient heat conduction problems of anisotropic and functionally graded materials. In this chapter, the formulation of the Lagrange FBM with boundary type in the strong form is presented and applied to nonlinear analysis including non-conforming contact and large deformation of Reissner plate problems with functionally graded materials subjected to either static or dynamic loads. The first-order partial differential matrices are only needed both in the governing equations and in the Neumann boundary condition. By introducing the mapping technique, a block of quadratic type is transformed from the Cartesian coordinate of global system to the normalized coordinate with 8 seeds. Time-dependent partial differential equations are analyzed in the Laplace transformed domain, and the Durbin’s inversion method is applied to determine all the physical values in the time domain. Conforming and nonconforming contacts are investigated by using the iterative algorithm with full load technique. However, the load incremental technique and displacement control scheme are developed for the large deformation and post-buckling investigations of plates under pressure and compressive loads. Illustrative numerical examples are given and comparisons have been made with analytical solutions and numerical solutions using the finite element method (ABAQUS).

In the dynamic analysis of mechanics, some of the large deformation problems such as contact and impact problems involve not only extreme deformations with material damage and separation but also finite and small deformations without material damage and separation. Consequently, it brings challenges to traditional finite element method in terms of element distortion and mesh entanglement due to extreme deformation and to advanced mesh-free methods in terms of low computational efficiency, especially when the size of finite/small deformation region is considerable. This chapter gives a state-of-the-art review of the coupled finite element material point (CFEMP) method, which takes both advantages of traditional finite element method (FEM) and a type of mesh-free methods, material point method (MPM), and avoids their disadvantages with the aim to overcome the challenges, and examples of its applications to the large deformation problems are also presented. The CFEMP method includes two variants, static CEMP based on a contact method and adaptive CFEMP further based on an element-to-particle conversion algorithm and alternative coupling method. In addition, MPM is also introduced in detail for completeness.

A recently developed approach for using the material point method to model complex phenomena involving fracture, contact and large deformation is presented. Motivations, applications and implementation considerations are discussed, along with the benefits and limitations of the method. With this approach, we demonstrate that it is possible to model complex behaviours such as fragmentation and comminution while avoiding the numerical artifacts and limitations of other approaches.

The development of a 3D microstructural model for the analysis of degradation and failure in polycrystalline materials is reviewed in the present chapter. The material is explicitly modelled at the grain level, using integral equations in conjunction with a phenomenological crystal plasticity framework for the bulk grains, and with cohesive-frictional laws to represent inter-granular micro-cracking processes. The method allows to capture the initiation, development and coalescence of damage or plasticity at the aggregate scale. The formulation’s key feature is the representation of the mechanical problem in terms of inter-granular variables only, which allows to reduce the computational cost of the analysis. In the present chapter, the building blocks of the technique are described and emphasis is given to the description of the adopted cohesive-frictional laws, which are the fundamental ingredient for the study of inter-granular degradation processes. The potential of the framework is illustrated through micro-cracking and crystal plasticity simulations. The effect of inter-granular friction on the macroscopic material properties is highlighted through micro-cracking simulations under compressive loading.

This chapter deals with the frictional contact problem for coupled piezoelectric and magneto-electro-elastic materials under indentation conditions. Due to the relevant technological applications, this topic of research has been treated in some analytical works. However, analytical solutions lack the generality of numerical methodologies, being restricted typically to simple geometries, loading conditions, idealized contact conditions and mostly taking into account transversely isotropic material symmetry with the symmetry axis normal to the contact surface. In this chapter, the numerical procedure proposed by the authors is presented. This numerical scheme makes it possible to model the 3D frictional contact of anisotropic piezoelectric or magneto-electro-elastic materials in the presence of both electric and magnetic fields. The methodology uses the boundary element method with explicit evaluation of the fundamental solutions in order to compute the magneto-electro-elastic influence coefficients. The contact model is based on an augmented Lagrangian formulation and it uses an iterative Uzawa scheme of resolution. Conducting, semi-conducting and insulated electric and/or magnetic indentation conditions, as well as orthotropic frictional contact conditions are considered (so anisotropy is present both in the bulk and on the surface). After the validation by comparison with benchmark analytical solutions, additional exploration examples are presented and discussed in detail. Results reveal the influence of the conductivity on the indentation force and contact pressure distributions. The influence of friction in electric and magnetic potential responses has been also proved to be very significant. Moreover, tangential loads exhibit an important influence both on the maximum values of the electric and magnetic potentials as well as on their distributions.

The following section is included:

• Index