This paper describes modern techniques used to solve contact problems within Computational Mechanics. On the basis of a continuum description of contact, the mathematical structure of the contact formulation for finite element methods is derived. Emphasis is also placed on the constitutive behavior at the contact interface for normal and tangential (frictional) contact. Furthermore, different discretization schemes currently applied to solve engineering problems are formulated for small and finite strain problems. These include isoparametric interpolations, node-to-segment discretizations and also mortar and Nitsche techniques. Furthermore, several algorithms related to spatial contact search and fulfillment of the inequality constraints at the contact interface are discussed. Here, especially the penalty and Lagrange multiplier schemes are considered and also SQP- and linear-programming methods are reviewed.


Keywords:

contact mechanics;
friction;
penalty method;
Lagrange multiplier method;
contact algorithms;
finite element method;
finite deformations;
discretization methods

Computational Contact Mechanics

Assessment of rolling resistance models in discrete element simulations

The past 20 years have witnessed ever-growing evidence that the mechanical properties of biological tissues, from nanoscale to macroscale dimensions, are fundamental for cellular behaviour and consequent tissue functionality. This knowledge, combined with previously known biochemical cues, has greatly advanced the field of biomaterial development, tissue engineering and regenerative medicine. It is now established that approaches to engineer biological tissues must integrate and approximate the mechanics, both static and dynamic, of native tissues. Nevertheless, the literature on the mechanical properties of biological tissues differs greatly in methodology, and the available data are widely dispersed. This Review gathers together the most important data on the stiffness of living tissues and discusses the intricacies of tissue stiffness from a materials perspective, highlighting the main challenges associated with engineering lifelike tissues and proposing a unified view of this as yet unreported topic. Emerging advances that might pave the way for the next decade’s take on bioengineered tissue stiffness are also presented, and differences and similarities between tissues in health and disease are discussed, along with various techniques for characterizing tissue stiffness at various dimensions from individual cells to organs. The complexity of biological tissue presents a challenge for engineering of mechanically compatible materials. In this Review, the stiffness of tissue components — from extracellular matrix and single cells to bulk tissue — is outlined, and how this understanding facilitates the engineering of materials with lifelike properties is discussed.

/pdf/the-stiffness-of-living-tissues-and-its-implications-for-49xikp5kk1.pdf

The stiffness of living tissues and its implications for tissue engineering

Phase-field modeling of brittle fracture in elastic solids is a well-established framework that overcomes the limitations of the classical Griffith theory in the prediction of crack nucleation and in the identification of complicated crack paths including branching and merging. We propose a novel phase-field model for ductile fracture of elasto-plastic solids in the quasi-static kinematically linear regime. The formulation is shown to capture the entire range of behavior of a ductile material exhibiting $$J_2$$J2-plasticity, encompassing plasticization, crack initiation, propagation and failure. Several examples demonstrate the ability of the model to reproduce some important phenomenological features of ductile fracture as reported in the experimental literature.

http://congress.cimne.com/coupled2015/admin/files/fileabstract/a482.pdf

Phase-field modeling of ductile fracture

This review article is concerned with the mathematical modelling of the mechanical properties of the soft biological tissues that constitute the walls of arteries. Many important aspects of the mechanical behaviour of arterial tissue can be treated on the basis of elasticity theory, and the focus of the article is therefore on the constitutive modelling of the anisotropic and highly nonlinear elastic properties of the artery wall. The discussion focuses primarily on developments over the last decade based on the theory of deformation invariants, in particular invariants that in part capture structural aspects of the tissue, specifically the orientation of collagen fibres, the dispersion in the orientation, and the associated anisotropy of the material properties. The main features of the relevant theory are summarized briefly and particular forms of the elastic strain-energy function are discussed and then applied to an artery considered as a thick-walled circular cylindrical tube in order to illustrate its extension–inflation behaviour. The wide range of applications of the constitutive modelling framework to artery walls in both health and disease and to the other fibrous soft tissues is discussed in detail. Since the main modelling effort in the literature has been on the passive response of arteries, this is also the concern of the major part of this article. A section is nevertheless devoted to reviewing the limited literature within the continuum mechanics framework on the active response of artery walls, i.e. the mechanical behaviour associated with the activation of smooth muscle, a very important but also very challenging topic that requires substantial further development. A final section provides a brief summary of the current state of arterial wall mechanical modelling and points to key areas that need further modelling effort in order to improve understanding of the biomechanics and mechanobiology of arteries and other soft tissues, from the molecular, to the cellular, tissue and organ levels.

https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2010.0058

Constitutive modelling of arteries

Part One Basic concepts 1 Introduction 1.1 Aims and scope 1.2 Layout 1.3 General scheme of notation 2 ELEMENTS OF TENSOR ANALYSIS 2.1 Vectors 2.2 Second-order tensors 2.3 Higher-order tensors 2.4 Isotropic tensors 2.5 Differentiation 2.6 Linearisation of nonlinear problems 3 THERMODYNAMICS 3.1 Kinematics of deformation 3.2 Infinitesimal deformations 3.3 Forces. Stress Measures 3.4 Fundamental laws of thermodynamics 3.5 Constitutive theory 3.6 Weak equilibrium. The principle of virtual work 3.7 The quasi-static initial boundary value problem 4 The finite element method in quasi-static nonlinear solid mechanics 4.1 Displacement-based finite elements 4.2 Path-dependent materials. The incremental finite element procedure 4.3 Large strain formulation 4.4 Unstable equilibrium. The arc-length method 5 Overview of the program structure 5.1 Introduction 5.2 The main program 5.3 Data input and initialisation 5.4 The load incrementation loop. Overview 5.5 Material and element modularity 5.6 Elements. Implementation and management 5.7 Material models: implementation and management Part Two Small strains 6 The mathematical theory of plasticity 6.1 Phenomenological aspects 6.2 One-dimensional constitutive model 6.3 General elastoplastic constitutive model 6.4 Classical yield criteria 6.5 Plastic flow rules 6.6 Hardening laws 7 Finite elements in small-strain plasticity problems 7.1 Preliminary implementation aspects 7.2 General numerical integration algorithm for elastoplastic constitutive equations 7.3 Application: integration algorithm for the isotropically hardening von Mises model 7.4 The consistent tangent modulus 7.5 Numerical examples with the von Mises model 7.6 Further application: the von Mises model with nonlinear mixed hardening 8 Computations with other basic plasticity models 8.1 The Tresca model 8.2 The Mohr-Coulomb model 8.3 The Drucker-Prager model 8.4 Examples 9 Plane stress plasticity 9.1 The basic plane stress plasticity problem 9.2 Plane stress constraint at the Gauss point level 9.3 Plane stress constraint at the structural level 9.4 Plane stress-projected plasticity models 9.5 Numerical examples 9.6 Other stress-constrained states 10 Advanced plasticity models 10.1 A modified Cam-Clay model for soils 10.2 A capped Drucker-Prager model for geomaterials 10.3 Anisotropic plasticity: the Hill, Hoffman and Barlat-Lian models 11 Viscoplasticity 11.1 Viscoplasticity: phenomenological aspects 11.2 One-dimensional viscoplasticity model 11.3 A von Mises-based multidimensional model 11.4 General viscoplastic constitutive model 11.5 General numerical framework 11.6 Application: computational implementation of a von Mises-based model 11.7 Examples 12 Damage mechanics 12.1 Physical aspects of internal damage in solids 12.2 Continuum damage mechanics 12.3 Lemaitre's elastoplastic damage theory 12.4 A simplified version of Lemaitre's model 12.5 Gurson's void growth model 12.6 Further issues in damage modelling Part Three Large strains 13 Finite strain hyperelasticity 13.1 Hyperelasticity: basic concepts 13.2 Some particular models 13.3 Isotropic finite hyperelasticity in plane stress 13.4 Tangent moduli: the elasticity tensors 13.5 Application: Ogden material implementation 13.6 Numerical examples 13.7 Hyperelasticity with damage: the Mullins effect 14 Finite strain elastoplasticity 14.1 Finite strain elastoplasticity: a brief review 14.2 One-dimensional finite plasticity model 14.3 General hyperelastic-based multiplicative plasticity model 14.4 The general elastic predictor/return-mapping algorithm 14.5 The consistent spatial tangent modulus 14.6 Principal stress space-based implementation 14.7 Finite plasticity in plane stress 14.8 Finite viscoplasticity 14.9 Examples 14.10 Rate forms: hypoelastic-based plasticity models 14.11 Finite plasticity with kinematic hardening 15 Finite elements for large-strain incompressibility 15.1 The F-bar methodology 15.2 Enhanced assumed strain methods 15.3 Mixed u/p formulations 16 Anisotropic finite plasticity: Single crystals 16.1 Physical aspects 16.2 Plastic slip and the Schmid resolved shear stress 16.3 Single crystal simulation: a brief review 16.4 A general continuum model of single crystals 16.5 A general integration algorithm 16.6 An algorithm for a planar double-slip model 16.7 The consistent spatial tangent modulus 16.8 Numerical examples 16.9 Viscoplastic single crystals Appendices A Isotropic functions of a symmetric tensor A.1 Isotropic scalar-valued functions A.1.1 Representation A.1.2 The derivative of anisotropic scalar function A.2 Isotropic tensor-valued functions A.2.1 Representation A.2.2 The derivative of anisotropic tensor function A.3 The two-dimensional case A.3.1 Tensor function derivative A.3.2 Plane strain and axisymmetric problems A.4 The three-dimensional case A.4.1 Function computation A.4.2 Computation of the function derivative A.5 A particular class of isotropic tensor functions A.5.1 Two dimensions A.5.2 Three dimensions A.6 Alternative procedures B The tensor exponential B.1 The tensor exponential function B.1.1 Some properties of the tensor exponential function B.1.2 Computation of the tensor exponential function B.2 The tensor exponential derivative B.2.1 Computer implementation B.3 Exponential map integrators B.3.1 The generalised exponential map midpoint rule C Linearisation of the virtual work C.1 Infinitesimal deformations C.2 Finite strains and deformations C.2.1 Material description C.2.2 Spatial description D Array notation for computations with tensors D.1 Second-order tensors D.2 Fourth-order tensors D.2.1 Operations with non-symmetric tensors References Index

Computational methods for plasticity : theory and applications

Computational methods for plasticity

Design of simple low order finite elements for large strain analysis of nearly incompressible solids

This paper proposes a new technique which allows the use of simplex finite elements (linear triangles in 2D and linear tetrahedra in 3D) in the large strain analysis of nearly incompressible solids. The new technique extends the F-bar method proposed by de Souza Neto et al. (Int. J. Solids and Struct. 1996; 33:3277-3296) and is conceptually very simple: It relies on the enforcement of (near-)incompressibility over a patch of simplex elements (rather than the point-wise enforcement of conventional displacement-based finite elements). Within the framework of the F-bar method, this is achieved by assuming, for each element of a mesh, a modified (F-bar) deformation gradient whose volumetric component is defined as the volume change ratio of a pre-defined patch of elements. The resulting constraint relaxation effectively overcomes volumetric locking and allows the successful use of simplex elements under finite strain near-incompressibility. As the original F-bar procedure, the present methodology preserves the displacement-based structure of the finite element equations as well as the strain-driven format of standard algorithms for numerical integration of path-dependent constitutive equations and can be used regardless of the constitutive model adopted. The new elements are implemented within an implicit quasi-static environment. In this context, a closed form expression for the exact tangent stiffness of the new elements is derived. This allows the use of the full Newton-Raphson scheme for equilibrium iterations. The performance of the proposed elements is assessed by means of a comprehensive set of benchmarking two- and three-dimensional numerical examples.

F‐bar‐based linear triangles and tetrahedra for finite strain analysis of nearly incompressible solids. Part I: formulation and benchmarking

On micro‐to‐macro transitions for multi‐scale analysis of non‐linear heterogeneous materials: unified variational basis and finite element implementation

E.A. de Souza Neto

Papers

Computational methods for plasticity : theory and applications

Computational methods for plasticity

Design of simple low order finite elements for large strain analysis of nearly incompressible solids

F‐bar‐based linear triangles and tetrahedra for finite strain analysis of nearly incompressible solids. Part I: formulation and benchmarking

On micro‐to‐macro transitions for multi‐scale analysis of non‐linear heterogeneous materials: unified variational basis and finite element implementation