On the mathematical modelling of material behavior in continuum mechanics

Summary The classical theories of continuum mechanics — linear elasticity, viscoelasticity, plasticity and hydrodynamics — are defined by special constitutive equations. These can be understood to be asymptotic approximations of a quite general constitutive model, valid under restrictive assumptions for the stress functional or the input processes. The general theory of material behavior develops systematic methods to represent material properties in a context of physical evidence and mathematical consistency. According to experimental observations material behavior may be rate independent or rate dependent with or without equilibrium hysteresis. This motivates four different constitutive theories, namely elasticity, plasticity, viscoelasticity and viscoplasticity. Constitutive equations can be formulated explicitly as functionals. Then, the particular constitutive models correspond to continuity properties of these functionals, related to convenient function spaces. On the other hand, a system of differential equations may lead to an implicit definition of a stress functional. In this case additional variables are introduced, which are called internal variables. For these variables additional evolution equations must be formulated, specifying the rate of change of the internal variables in dependence on their present values and the strain (or stress) input. In the context of different models of inelastic material behavior the evolution equations have different mathematical characteristics. These concern the existence of equilibrium solutions and their stability properties. Rate independent material behavior is modelled by means of evolution equations, which are related to an arclength instead of the time as independent variable. It can be shown that the rate independent constitutive equations of elastoplasticity are the asymptotic limit of rate dependent viscoplasticity for slow deformation processes.This paper is an extended version of a lecture held at the First Conference of the GAMM working group on material theory in Stuttgart, Germany, February 28, 1992. The author thanks Prof. Dr. F. Ziegler for the opportunity to participate in this conference.     

Gradient damage modeling of brittle fracture in an explicit dynamics context

In this contribution, we propose a dynamic gradient damage model as a phase‐field approach for studying brutal fracture phenomena in quasi‐brittle materials under impact‐type loading conditions. Several existing approaches to account for the tension–compression asymmetry of fracture behavior of materials are reviewed. A better understanding of these models is provided through a uniaxial traction experiment. We then give an efficient numerical implementation of the model in an explicit dynamics context. Simulations results obtained with parallel computing are discussed both from a computational and physical point of view. Different damage constitutive laws and tension–compression asymmetry formulations are compared with respect to their aptitude to approximate brittle fracture. Copyright © 2016 John Wiley & Sons, Ltd.     

Algorithms for the model configuration problem

The model configuration problem is a combinatorial optimization problem that arises in the context of switching cabinet manufacturing in the telecommunication industry. We discuss the manufacturing environment and define the q-model problem in this context, for q ≥ 1. We then discuss the structural properties of the q-model problem, and propose an efficient procedure for solving the 1-model problem. We also propose several heuristic procedures for solving the 2-model problem, and present an evaluation of these procedures through an extensive computational experiment.     

Calibrating constitutive models with full-field data via physics informed neural networks

The calibration of solid constitutive models with full-field experimental data is a long-standing challenge, especially in materials that undergo large deformations. In this paper, we propose a physics-informed deep-learning framework for the discovery of hyperelastic constitutive model parameterizations given full-field surface displacement data and global force-displacement data. Contrary to the majority of recent literature in this field, we work with the weak form of the governing equations rather than the strong form to impose physical constraints upon the neural network predictions. The approach presented in this paper is computationally efficient, suitable for irregular geometric domains, and readily ingests displacement data without the need for interpolation onto a computational grid. A selection of canonical hyperelastic material models suitable for different material classes is considered including the Neo–Hookean, Gent, and Blatz–Ko constitutive models as exemplars for general non-linear elastic behaviour, elastomer behaviour with finite strain lock-up, and compressible foam behaviour, respectively. We demonstrate that physics informed machine learning is an enabling technology and may shift the paradigm of how full-field experimental data are utilized to calibrate constitutive models under finite deformations.     

Enhanced balancing Neumann–Neumann preconditioning in computational fluid and solid mechanics

In this work, we propose an enhanced implementation of balancing Neumann–Neumann (BNN) preconditioning together with a detailed numerical comparison against the balancing domain decomposition by constraints (BDDC) preconditioner. As model problems, we consider the Poisson and linear elasticity problems. On one hand, we propose a novel way to deal with singular matrices and pseudo‐inverses appearing in local solvers. It is based on a kernel identification strategy that allows us to efficiently compute the action of the pseudo‐inverse via local indefinite solvers. We further show how, identifying a minimum set of degrees of freedom to be fixed, an equivalent definite system can be solved instead, even in the elastic case. On the other hand, we propose a simple implementation of the algorithm that reduces the number of Dirichlet solvers to only one per iteration, leading to similar computational cost as additive methods. After these improvements of the BNN preconditioned conjugate gradient algorithm, we compare its performance against that of the BDDC preconditioners on a pair of large‐scale distributed‐memory platforms. The enhanced BNN method is a competitive preconditioner for three‐dimensional Poisson and elasticity problems and outperforms the BDDC method in many cases. Copyright © 2013 John Wiley & Sons, Ltd.     

Materiomics: An ‐omics Approach to Biomaterials Research

The past fifty years have seen a surge in the use of materials for clinical application, but in order to understand and exploit their full potential, the scientific complexity at both sides of the interface—the material on the one hand and the living organism on the other hand—needs to be considered. Technologies such as combinatorial chemistry, recombinant DNA as well as computational multi‐scale methods can generate libraries with a very large number of material properties whereas on the other side, the body will respond to them depending on the biological context. Typically, biological systems are investigated using both holistic and reductionist approaches such as whole genome expression profiling, systems biology and high throughput genetic or compound screening, as already seen, for example, in pharmacology and genetics. The field of biomaterials research is only beginning to develop and adopt these approaches, an effort which we refer to as “materiomics”. In this review, we describe the current status of the field, and its past and future impact on the biomedical sciences. We outline how materiomics sets the stage for a transformative change in the approach to biomaterials research to enable the design of tailored and functional materials for a variety of properties in fields as diverse as tissue engineering, disease diagnosis and de novo materials design, by combining powerful computational modelling and screening with advanced experimental techniques.     

Nonlinear Multi-Scale Modelling,Simulation and Validation of 3D Knitted Textiles

Three-dimensionally (3D) knitted technical textiles are spreading into industrial applications, since their geometric, structural and functional performance can be tailored and optimized on fibre-, yarn- and fabric levels by customizing yarn materials, knit patterns and geometric shapes. The ability to simulate their complex mechanical behaviour is thus an essential ingredient in the development of a digital workflow for optimal design and manufacture of 3D knitted textiles. Here, we present a multi-scale modelling and simulation framework for the prediction of the nonlinear orthotropic mechanical behaviour of single jersey knitted textiles and its experimental validation. On the meso-scale, representative volume elements (RVEs) of the fabric are modelled as single, interlocked yarn loops and their mechanical deformation behaviour is homogenized using periodic boundary conditions. Yarns are modelled as nonlinear 3D beam elements and numerically discretized using an isogeometric collocation method, where a frictional contact formulation is used to model inter-yarn interactions. On the macro-scale, fabrics are modelled as membrane elements with nonlinear orthotropic material behaviour, which is parameterized by a response surface constitutive model obtained from the meso-scale homogenization. The input parameters of the yarn-level simulation, i.e., mechanical properties of yarns and geometric dimensions of yarn loops in the fabrics, are determined experimentally and subsequent meso- and macro-scale simulation results are evaluated against reference results and mechanical tests of knitted fabric samples. Good agreement between computational predictions and experimental results is achieved for samples with varying stitch values, thus validating our novel computational approach combining efficient meso-scale simulation using 3D beam modelling of yarns with numerical homogenization and nonlinear orthotropic response surface constitutive modelling on the macro-scale.     

Hyper‐reduction of mechanical models involving internal variables

We propose to improve the efficiency of the computation of the reduced‐state variables related to a given reduced basis. This basis is supposed to be built by using the snapshot proper orthogonal decomposition (POD) model reduction method. In the framework of non‐linear mechanical problems involving internal variables, the local integration of the constitutive laws can dramatically limit the computational savings provided by the reduction of the order of the model. This drawback is due to the fact that, using a Galerkin formulation, the size of the reduced basis has no effect on the complexity of the constitutive equations. In this paper we show how a reduced‐basis approximation and a Petrov–Galerkin formulation enable to reduce the computational effort related to the internal variables. The key concept is a reduced integration domain where the integration of the constitutive equations is performed. The local computations being not made over the entire domain, we extrapolate the computed internal variable over the full domain using POD vectors related to the internal variables. This paper shows the improvement of the computational saving obtained by the hyper‐reduction of the elasto‐plastic model of a simple structure. Copyright © 2008 John Wiley & Sons, Ltd.     

Towards simplified and optimized a posteriori error estimation using PGD reduced models

The paper deals with the use of model order reduction within a posteriori error estimation procedures in the context of the finite element method. More specifically, it focuses on the constitutive relation error concept, which has been widely used over the last 40 years for FEM verification of computational mechanics models. A technical key‐point when using constitutive relation error is the construction of admissible fields, and we propose here to use the proper generalized decomposition to facilitate this task. In addition to making the implementation into commercial FE software easier, it is shown that the use of proper generalized decomposition enables to optimize the verification procedure and to get both accurate and reasonably expensive upper bounds on the discretization error. Numerical illustrations are presented to assess the performance of the proposed approach.     

Systematic study of thermal properties of CNT composites by the fast multipole hybrid boundary node method

Carbon nanotubes (CNTs) are predicted to possess superior heat conductivity, which makes the CNTs promising in development of fundamentally new composite material. With the current advancement in nanotechnology, it is possible to design materials with desired properties for specific applications. On the other hand, the overall properties of CNT composites are usually evaluated using a representative volume element (RVE) with a number of CNTs embedded. For realistic modeling, an RVE including a large number of CNTs, for example, tens or hundreds, is necessary. However, analysis of such an RVE using standard numerical methods faces two severe difficulties: discretization of the geometry into elements and the very large computational scale. In this paper, the first difficulty is alleviated by developing the hybrid boundary node method (HdBNM), which is a boundary-type meshless method. To overcome the second difficulty, a simplified mathematical model for thermal analysis of CNT analysis is first proposed, by which the size of the linear system can be reduced by nearly half. Then, the HdBNM is combined with the Fast Multipole Method (FMM) based on the model to further reduce the computational scale. A variety of RVEs containing different numbers of CNTs, from small to large scales, have been studied in an attempt to investigate the influence of CNT length, distribution, orientation and volume fraction on the overall thermal properties of the composites. Insights have been gained into the thermal behavior of the CNT composite material.     

On Quasi-Newton methods in fast Fourier transform-based micromechanics

This work is devoted to investigating the computational power of Quasi-Newton methods in the context of fast Fourier transform (FFT)-based computational micromechanics. We revisit FFT-based Newton-Krylov solvers as well as modern Quasi-Newton approaches such as the recently introduced Anderson accelerated basic scheme. In this context, we propose two algorithms based on the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method, one of the most powerful Quasi-Newton schemes. To be specific, we use the BFGS update formula to approximate the global Hessian or, alternatively, the local material tangent stiffness. Both for Newton and Quasi-Newton methods, a globalization technique is necessary to ensure global convergence. Specific to the FFT-based context, we promote a Dong-type line search, avoiding function evaluations altogether. Furthermore, we investigate the influence of the forcing term, that is, the accuracy for solving the linear system, on the overall performance of inexact (Quasi-)Newton methods. This work concludes with numerical experiments, comparing the convergence characteristics and runtime of the proposed techniques for complex microstructures with nonlinear material behavior and finite as well as infinite material contrast.     

A simple model of elastoplasticity for nonproportional cyclic loading

On the basis of the analysis of experimental data, we formulate requirements to the constitutive relations of plasticity under the conditions of complex cyclic loading. We propose a version of constitutive relations obtained by a simple generalization of the Mazing model to the three-dimensional case and introduction of a function of cyclic hardening. We also suggest a procedure for the identification of this function. According to the results of numerical analysis, this model adequately describes the main effects of cyclic plasticity for austenitic stainless steels. Perm State Technical University, Perm, Russia. Translated from Problemy Prochnosti, No. 1, pp. 15 – 24, January – February, 1998.     

On tension–compression asymmetry in ultrafine-grained and nanocrystalline metals

We present a physically motivated computational study explaining the tension/compression (T/C) asymmetry phenomenon in nanocrystalline (nc) and ultrafine-grained (ufg) face centered cubic (fcc) metals utilizing a variational constitutive model where the nc-metal is modeled as a two-phase material consisting of a grain interior phase and a grain boundary affected zone (GBAZ). We show that the existence of voids and their growth in GBAZ renders the material pressure sensitivity due to porous plasticity and that the utilized model provides a physically sound mechanism to capture the experimentally observed T/C asymmetry in nc- and ufg-metals.     

A two scale damage concept applied to fatigue

The ductile type of damage is a phenomenon now well understood. Once the fully coupled set of constitutive equations is identified, Damage Mechanics is a powerful tool to predict failure. Brittle materials do not exhibit such a damageable macroscopic behavior. Nevertheless, they still fail. On the idea that damage is localized at the microscopic scale, a scale smaller than the mesoscopic one of the Representative Volume Element (RVE), we propose a three-dimensional failure modeling for monotonic as well as for fatigue loading. We develop a two scale model of what we shall call brittle damage: at the microscopic scale, micro-cracks or micro-voids exhibit a damageable plastic-like behavior with no effect on the global (mesoscopic) elastic behavior. Microscopic failure is assumed to coincide with the RVE failure. This model turns out to represent quite well physical phenomena related to high cycle fatigue such as the mean stress effect, the nonlinear accumulation of damage, initial strain hardening or damage effect and the nonproportional loading effect for bi-axial fatigue. The model has been implemented as a post-processor computer code. A simplified identification procedure for the determination of the material properties is given. This revised version was published online in July 2006 with corrections to the Cover Date.     

Prediction of the corrosion cracking of structures under the conditions of high-temperature creep

On the basis of the continual model of corrosion crack growth proposed earlier and the well-known incremental-type creep theory, we make an attempt to predict the corrosion cracking of structures under the conditions of high-temperature creep. We propose the mathematical statement of the problem taking into account the influence on corrosion cracking of the properties of corrosive media and the redistribution of stresses in time caused by creep. The Bubnov–Galerkin method is applied for the solution of this problem. An example of prediction of the phenomenon of corrosion cracking in the case of creep of a pipe under the action of internal pressure is analyzed.     

Bounds for element size in a variable stiffness cohesive finite element model

The cohesive finite element method (CFEM) allows explicit modelling of fracture processes. One form of CFEM models integrates cohesive surfaces along all finite element boundaries, facilitating the explicit resolution of arbitrary fracture paths and fracture patterns. This framework also permits explicit account of arbitrary microstructures with multiple length scales, allowing the effects of material heterogeneity, phase morphology, phase size and phase distribution to be quantified. However, use of this form of CFEM with cohesive traction–separation laws with finite initial stiffness imposes two competing requirements on the finite element size. On one hand, an upper bound is needed to ensure that fields within crack‐tip cohesive zones are accurately described. On the other hand, a lower bound is also required to ensure that the discrete model closely approximates the physical problem at hand. Both issues are analysed in this paper within the context of fracture in multi‐phase composite microstructures and a variable stiffness bilinear cohesive model. The resulting criterion for solution convergence is given for meshes with uniform, cross‐triangle elements. A series of calculations is carried out to illustrate the issues discussed and to verify the criterion given. These simulations concern dynamic crack growth in an Al₂O₃ ceramic and in an Al₂O₃/TiB₂ ceramic composite whose phases are modelled as being hyperelastic in constitutive behaviour. Copyright © 2004 John Wiley & Sons, Ltd.