Constitutive relations are fundamental to the solution of problems in continuum mechanics, and are required in the study of, for example, mechanically dominated clinical interventions involving soft biological tissues. Structural continuum constitutive models of arterial layers integrate information about the tissue morphology and therefore allow investigation of the interrelation between structure and function in response to mechanical loading. Collagen fibres are key ingredients in the structure of arteries. In the media (the middle layer of the artery wall) they are arranged in two helically distributed families with a small pitch and very little dispersion in their orientation (i.e. they are aligned quite close to the circumferential direction). By contrast, in the adventitial and intimal layers, the orientation of the collagen fibres is dispersed, as shown by polarized light microscopy of stained arterial tissue. As a result, continuum models that do not account for the dispersion are not able to capture accurately the stress–strain response of these layers. The purpose of this paper, therefore, is to develop a structural continuum framework that is able to represent the dispersion of the collagen fibre orientation. This then allows the development of a new hyperelastic free-energy function that is particularly suited for representing the anisotropic elastic properties of adventitial and intimal layers of arterial walls, and is a generalization of the fibre-reinforced structural model introduced by Holzapfel & Gasser (Holzapfel & Gasser 2001 Comput. Meth. Appl. Mech. Eng. 190, 4379–4403) and Holzapfel et al. (Holzapfel et al. 2000 J. Elast. 61, 1–48). The model incorporates an additional scalar structure parameter that characterizes the dispersed collagen orientation. An efficient finite element implementation of the model is then presented and numerical examples show that the dispersion of the orientation of collagen fibres in the adventitia of human iliac arteries has a significant effect on their mechanical response.

https://royalsocietypublishing.org/doi/pdf/10.1098/rsif.2005.0073

Hyperelastic modelling of arterial layers with distributed collagen fibre orientations

Designing engineering components that make optimal use of materials requires consideration of the nonlinear characteristics associated with both manufacturing and working environments. The modeling of these characteristics can only be done through numerical formulation and simulation, and this requires an understanding of both the theoretical background and associated computer solution techniques. By presenting both nonlinear continuum analysis and associated finite element techniques under one roof, Bonet and Wood provide, in this edition of this successful text, a complete, clear, and unified treatment of these important subjects. New chapters dealing with hyperelastic plastic behavior are included, and the authors have thoroughly updated the FLagSHyP program, freely accessible at www.flagshyp.com. Worked examples and exercises complete each chapter, making the text an essential resource for postgraduates studying nonlinear continuum mechanics. It is also ideal for those in industry requiring an appreciation of the way in which their computer simulation programs work.

https://cdn.preterhuman.net/texts/science_and_technology/physics/Mechanics/Nonlinear%20Continuum%20Mechanics%20For%20Finite%20Element%20Analysis%20-%20Bonet,%20Wood.pdf

Nonlinear Continuum Mechanics for Finite Element Analysis

https://www.crm.sns.it/media/course/3060/miehe+hofacker+welschinger10.pdf

A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits

Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: Theory, experiments, applications

The computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies. This can be overcome by a diffusive crack modeling based on the introduction of a crack phase-field. In this paper, we outline a thermodynamically consistent framework for phase-field models of crack propagation in elastic solids, develop incremental variational principles and consider their numerical implementations by multi-field finite element methods. We start our investigation with an intuitive and descriptive derivation of a regularized crack surface functional that Γ-converges for vanishing length-scale parameter to a sharp crack topology functional. This functional provides the basis for the definition of suitable convex dissipation functions that govern the evolution of the crack phase-field. Here, we propose alternative rate-independent and viscous over-force models that ensure the local growth of the phase-field. Next, we define an energy storage function whose positive tensile part degrades with increasing phase-field. With these constitutive functionals at hand, we derive the coupled balances of quasi-static stress equilibrium and gradient-type phase-field evolution in the solid from the argument of virtual power. Here, we consider a canonical two-field setting for rate-independent response and a time-regularized three-field formulation with viscous over-force response. It is then shown that these balances follow as the Euler equations of incremental variational principles that govern the multi-field problems. These principles make the proposed formulation extremely compact and provide a perfect base for the finite element implementation, including features such as the symmetry of the monolithic tangent matrices. We demonstrate the performance of the proposed phase-field formulations of fracture by means of representative numerical examples. Copyright © 2010 John Wiley & Sons, Ltd.

Thermodynamically consistent phase‐field models of fracture: Variational principles and multi‐field FE implementations

In the recent years phase-field modeling of fracture has become a promising tool to describe complex crack patterns in all kinds of solid materials. Many of the models assume an isotropic material behavior, which of course is not a meaningful assumption for e.g. biological tissues such as arterial walls. Since the phase-field approach introduces an additional (smeared) phase describing the evolution of the crack, this method is well suited to be extended to anisotropic materials without thinking about an adaption of the discretization technique. Anisotropy can be incorporated in several ways, like by an extension of the surface energy, i.e. by making the energy release rate orientation dependent, as considered in [1]. Our ansatz is based on a pure geometrical approach, namely on an anisotropic formulation of the crack surface itself. Here, we will focus on transversely isotropic and cubically anisotropic solids, where the latter one makes the incorporation of the second gradient of the crack phase field necessary. At the end one numerical example is shown, which conceptually shows the influence of the anisotropy on the crack path. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

Phase‐Field Modeling of Fracture in Anisotropic Media

The aim of this work is to present new variational-based modeling and homogenization approaches in dissipative magneto-mechanics. On the modeling side, the presented constitutive models are motivated by the underlying physical phenomena at the micro- and nano-scale that are embedded in appropriate variational-based finite element frameworks. We discuss aspects that are characteristic to magneto-mechanical problems such as the unity constraint on the magnetization, the inclusion of the surrounding free-space, the micro-mechanics of the coupled response etc. Following a hierarchical structure of scales, we present new variational principles for three model scenarios: (i) The modeling of magnetic domains at a micro-level, (ii) the modeling of magnetorheological elastomers at a macro-level and (iii) a homogenization-based scale bridging scenario. Numerical simulations are outlined in each focus area in order to demonstrate the salient features of the variational-based formulations. From the theoretical and computational standpoint, this work aims to contribute a building block to the ultimate goal of construction of a compatible hierarchy of computational models for magneto-mechanically coupled materials. (© 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Variational modeling and homogenization in dissipative magneto-mechanics

The paper presents continuous and discrete variational formulations for the treatment of the non-linear response
of piezoceramics under electrical loading. The point of departure is a general internal variable formulation that
determines the hysteretic response of the material as a generalized standard medium in terms of an energy
storage and a rate-dependent dissipation function. Consistent with this type of standard dissipative continua,
we develop an incremental variational formulation of the coupled electromechanical boundary value problem.
We specify the variational formulation for a setting based on a smooth rate-dependent dissipation function
which governs the hysteretic response. Such a formulation allows us to reproduce the dielectric and butterfly
hysteresis responses characteristic of the ferroelectric materials together with their rate-dependency and to
account for macroscopically non-uniform distribution of the polarization in the specimen. An important aspect
is the numerical implementation of the coupled problem. The discretization of the two-field problem appears,
as a consequence of the proposed incremental variational principle, in a symmetric format. The performance of the proposed methods is demonstrated by means of benchmark problems.

A rate-dependent incremental variational formulation of ferroelectricity

Parameter identification processes concern the determination of parameters in a material model in order to fit experimental data. We provide a distinct, unified algorithmic setting of a generic class of material models and discuss the associated gradient–based optimization problem. Gradient–based optimization algorithms need derivatives of the objective function with respect to the material parameter vector κ . In order to obtain the necessary derivatives, an analytical sensitivity analysis is pointed out for the unified class of algorithmic material models. The quality of the parameter identification is demonstrated for a representative example.

An Unified Computational Framework for Parameter Identification of Material Models in Finite Inelasticity

We discuss different concepts for the construction of transfer operators with regard to the application of multigrid methods to an efficient solving of strongly heterogeneous, linear elastic materials. The micro-structure is assumed to be resolved sufficiently well. The key contribution is the formulation of new physically motivated approaches for linear-elastic composites with arbitrary micro-structures. This task is achieved by means of a numerical homogenization concept based on a variational principle. The efficiency of the new approaches will be compared with alternative techniques in the scope of a representative model problem.

Christian Miehe

Papers

Phase‐Field Modeling of Fracture in Anisotropic Media

Variational modeling and homogenization in dissipative magneto-mechanics

A rate-dependent incremental variational formulation of ferroelectricity

An Unified Computational Framework for Parameter Identification of Material Models in Finite Inelasticity

Coupling of Homogenization Techniques with Multigrid Solvers for Unstructured Meshes