Advances in Computational Plasticity

In the last few years, several authors have proposed different phase-field models aimed at describing ductile fracture phenomena. Most of these models fall within the class of variational approaches to fracture proposed by Francfort and Marigo [13]. For the case of brittle materials, the key concept due to Griffith consists in viewing crack growth as the result of a competition between bulk elastic energy and surface energy. For ductile materials, however, an additional contribution to the energy dissipation is present, related to plastic deformations. Of crucial importance, for the performance of the modeling approaches, is the way the coupling is realized between plasticity and phase field evolution. Our aim is a critical revision of the main constitutive choices underlying the available models and a comparative study of the resulting predictive capabilities.

The title of this work represents a figurative counterpart of bridging the topographical gap between Hongkong and Macao. Presently under construction, the bridge that connects these two cities at opposite sides of the mouth of the Pearl River is interrupted by a submersed tunnel, which is the actual research object of this paper. By means of four different topics in the framework of the general subject of this contribution, reflected by its title, the scientific progress resulting from multiscale structural analyses of tunnel linings is documented. The four topics are: (a) microstructural analysis of impact and blast loading in tunnel linings, (b) multiscale analysis of thermal stresses in concrete linings due to sudden temperature changes, (c) experiments and finite element modeling of concrete hinges in Mechanized Tunneling, and (d) multiscale structural analysis of a segmented tunnel ring.

Phase-field models have been a topic of much research in recent years. Results have shown that these models are able to produce complex crack patterns in both two and three dimensions. A number of extensions from brittle to ductile materials have been proposed and results are promising. To date, however, these extensions have not accurately represented strains after crack initiation or included important aspects of ductile fracture such as stress triaxiality. This work describes a number of contributions to further develop phase-field models for fracture in ductile materials.

An overview of various numerical methods developed for speeding-up computations is presented in the field of the bulk material forming under solid state, which is characterized by complex and evolving geometries requiring frequent remeshings and numerous time increments. These methods are oriented around the axis that constitutes the meshing problem. The multi-mesh method allows to optimally solve several physics involved on the same domain, according to its finite element discretization with several different meshes, for example in the cogging or cold pilgering processes. For quasi steady-state problems and problems with quite pronounced localization of deformation, such as Friction Stir Welding (FSW) or High Speed Machining, an Arbitrary Lagrangian or Eulerian formulation (ALE) with mesh adaptation shows to be imperative. When the problem is perfectly steady, as for the rolling of long products, the direct search for the stationary state allows huge accelerations. In the general case, where no process specificity can be used to solve the implicit equations, the multigrid method makes it possible to construct a much more efficient iterative solver, which is especially characterized by an almost linear asymptotic cost.

The simulation of penetration problems into granular materials is a challenging problem as it involves large deformations and displacements as well as strong non-linearities affecting material behaviour, geometry and contact surfaces. In this contribution, the Discrete Element Method (DEM) has been adopted as the modelling formulation. Attention is focused on the simulation of cone penetration, a basic reconnaissance tool in geotechnical engineering, although the approach can be readily extended to other penetration problems. It is shown that DEM analysis results in a very close quantitative representation of the cone resistance obtained in calibration chambers under a wide range of conditions. DEM analyses also provides, using appropriate averaging techniques, relevant information concerning mesoscale continuum variables (stresses and strains) that appear to be in agreement with physical calibration chamber observations. The examination of microstructural variables contributes to a better understanding of the mechanisms underlying the observed effects of a number of experimental and analysis features of the cone penetration test.

Physiological and pathological aspects of soft biological tissues in terms of, e.g., aortic dissection, aneurysmatic and atherosclerotic rupture, tears in tendons and ligaments are of significant concern in medical science. The past few decades have witnessed noticeable advances in the fundamental understanding of the mechanics of soft biological tissues. Furthermore, computational biomechanics, with an ever-increasing number of publications, has now become a third pillar of investigation, next to theory and experiment. In the present chapter we provide a brief review of some constitutive frameworks and related computational models with the potential to predict the clinically relevant phenomena of rupture of soft biological tissues. Accordingly, Euler-Lagrange equations are presented in regard to a recently developed crack phase-field method (CPFM) for soft tissues. The theoretical framework is supplemented by some recently documented numerical results, with a focus on evolving failure surfaces that are predicted by a range of different failure criteria. A peel test of arterial tissue is analyzed using the crack phase-field approach. Subsequently, discontinuous models of tissue rupture are described, namely the cohesive zone model (CZM) and the extended finite element method (XFEM). Traction-separation laws used to determine the crack growth are described, together with the kinematic and numerical foundations. Simulation of a peel test of arterial tissue is then presented for both the CZM and the XFEM. Finally we provide a critical discussion and overview of some open problems and possible improvements of the computational modeling concepts for soft tissue rupture.

For a truly meshfree technique, Galerkin meshfree methods rely chiefly on nodal integration of the weak form. In the case of Strong Form Collocation meshfree methods, direct collocation at the nodes can be employed. In this paper, performance of these node-based Galerkin and collocation meshfree methods is compared in terms of accuracy, efficiency, and stability. Considering both accuracy and efficiency, the overall effectiveness in terms of CPU time versus error is also assessed. Based on the numerical experiments, nodally integrated Galerkin meshfree methods with smoothed gradients and variationally consistent integration yield the most effective solution technique, while direct collocation of the strong form at nodal locations has comparable effectiveness.

Data-Driven Computing is a new field of computational analysis which uses provided data to directly produce predictive outcomes. Recent works in this developing field have established important properties of Data-Driven solvers, accommodated noisy data sets and demonstrated both quasi-static and dynamic solutions within mechanics. This work reviews this initial progress and advances some of the many possible improvements and applications that might best advance the field. Possible method improvements discuss incorporation of data quality metrics, and adaptive data additions while new applications focus on multi-scale analysis and the need for public databases to support constitutive data collaboration.

We present an application of the third-order plate theory for investigation of the elasto-plastic response of thick plates made of functionally graded material. The theory was originally developed by Reddy and Kim [1]. In their formulation they expanded the in-plane displacements up to the cubic term and the transverse displacement up to the quadratic term with respect to the coordinate perpendicular to the plate surface, obtaining a quadratic variation of the transverse shear strains through the plate thickness. FGM properties are modelled following the power law distribution of constituent ratio across the thickness. The plates are modelled using a 16-noded lagrangian elements using Lobatto integration rules. The problem is solved using Newton-Raphson method applying modified Crisfield constant arc-length procedure. Numerical examples are provided to illustrate the advantages of the method proposed.

Reduced models and especially those based on Proper Generalized Decomposition (PGD) are decision-making tools which are about to revolutionize many domains. Unfortunately, their calculation remains problematic for problems involving many parameters, for which one can invoke the “curse of dimensionality”. The paper starts with the state-of-the-art for nonlinear problems involving stochastic parameters. Then, an answer to the challenge of many parameters is given in solid mechanics with the so-called “parameter-multiscale PGD”, which is based on the Saint-Venant principle.

To analyze complex, heterogeneous materials, a fast and accurate method is needed. This means going beyond the classical finite element method, in a search for the ability to compute, with modest computational resources, solutions previously infeasible even with large cluster computers. In particular, this advance is motivated by composites design. Here, we apply similar principle to another complex, heterogeneous system: additively manufactured metals.

A viscoelastic-viscoplastic combined constitutive model is presented to represent large deformations of amorphous thermoplastic resins. The model is endowed with viscoelastic and viscoplastic rheology elements connected in series. The standard generalized Maxwell model is used to determine the stress and characterize the viscoelastic material behavior at small or moderate strain regime. To realize the transient creep deformations along with kinematic hardening due to frictional resistance and orientation of molecular chains, a proven finite strain viscoplastic model is employed. After identifying the material parameters with reference to experimental data, we verify and demonstrate the fundamental performances of the proposed model in reproducing typical material behavior of resin.

It is now generally recognized that mode I fracturing in saturated geomaterials is a stepwise process. This is true both for mechanical loading and for pressure induced fracturing. Evidence comes from geophysics, from unconventional hydrocarbon extraction, and from experiments. Despite the evidence only very few numerical models capture this behavior. From our numerical experiments, both with a model based on Standard Galerkin Finite Elements in conjunction with a cohesive fracture model, and with a truss lattice model in combination with Monte Carlo simulations, it appears that already in dry geomaterials under mechanical loading the fracturing process is time discontinuous. In a two-phase fracture context, in case of mechanical loading, the fluid not only follows the fate of the solid phase material and gives rise to pressure peaks at the fracturing event, but it also influences this event. In case of pressure induced fracture clearly pressure peaks appear too but are of opposite sign: we observe pressure drops at fracturing. In mode II fracturing, the behavior is brittle while in mixed mode there appears a combination of pressure rises and drops.

This work deals on the optimization and computational material design using the topological derivative concept. The necessary details to obtain the anisotropic topological derivative are first presented. In the context of multi-scale topology optimization, it is crucial since the homogenization of the constitutive tensor of a micro-structure confers in general an anisotropic response. In addition, this work addresses the multi-scale material design problem in which the goal is then to minimize the structural (macro-scale) compliance by appropriately designing the material distribution (micro-structure) at a lower scale (micro-scale). To overcome the exorbitant computational cost, a consultation during the iterative process of a discrete material catalog (computed off-line) of micro-scale optimized topologies (Computational Vademecum) is proposed in this work. This results into a large diminution of the resulting computational costs, which make affordable the proposed methodology for multi-scale computational material design. Some representative examples assess the performance of the considered approach.

In this chapter we present recent advances on the Discrete Element Method (DEM) and on the coupling of the DEM with the Finite Element Method (FEM) for solving a variety of problems in non linear solid mechanics involving damage, plasticity and multifracture situations.

The hierarchical architecture of the stromal collagen is strictly related to the optical function of the human cornea. The basic features of the corneal collagen organization have been known for a while, but recently the advance of optical imaging has revealed changes across the thickness that might be related to particular aspects of the corneal behavior. It is worth to investigate whether the actual structure possesses some relevance in the overall mechanical behavior of the cornea and whether it should not be disregarded in computational models with predictive purposes. In this study, finite element analyses of the human cornea considering four different architectures of the collagen are presented. Results of the numerical simulations of quasistatic and dynamic tests are compared and discussed.

This contribution was presented by the author at COMPLAS XII in 2013 as a plenary lecture but not published so far. The historical presentation of physical und mathematical modeling together with the computational foundation of related FEMs seems to be of current importance, also regarding the algorithms and applications of elasto-plastic deformations based on \(C^1\)-continuous kinematics for engineering applications, including a posteriori error analysis and adaptivity.

The virtual element method (VEM) is a generalization of the finite element method recently introduced.

In a variety of engineering applications knowledge of accurate contact stress is of great importance.

Virtual elements were introduced in the last decade and applied to problems in solid mechanics. The success of this methodology when applied to linear problems asks for an extension to the nonlinear regime. This work is concerned with the numerical simulation of structures made of anisotropic material undergoing large deformations. Especially problems with hyperelastic matrix materials and transversly isotropic behaviour will be investigated. The virtual element formulation is based on a low-order formulations for problems in two dimensions. The elements can be arbitrary polygons. The formulation considered relies on minimization of energy, with a novel construction of the stabilization energy and a mixed approximation for the fibers describing the anisotropic behaviour. The formulation is investigated through a several numerical examples, which demonstrate their efficiency, robustness, convergence properties, and locking-free behaviour.