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

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. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

https://www.onlinelibrary.wiley.com/doi/pdf/10.1002/pamm.200700311

Rate‐dependent incremental variational formulation of ferroelectricity

Mehrgitterverfahren ermoglichen eine effiziente Losung groser Gleichungssysteme, die bei Finite Element-Diskretisierungen nichtlinearer Randwertprobleme auftreten konnen. Bei heterogenen Materialien bedingt der Einsatz solcher Loser die Konstruktion spezieller Transferoperatoren. Diese Aufgabe wird mit Hilfe von Homogenisierungstechniken gelost. Die neuen Strategien werden anhand reprasentativer Modellprobleme mit bekannten Verfahren verglichen.

https://onlinelibrary.wiley.com/doi/pdf/10.1002/pamm.200310396

Homogenisierungsbasierte Mehrgitter-Transferoperatoren für nichtlineare heterogene Materialien

We propose a variational framework for the description of rate–independent nonlocal materials with microstructure and outline details of its numerical implementation. On the theoretical side, we focus on a multifield approach to microstructural dissipative mechanisms. To this end, the current state of the evolving microstructure is described by global fields of microscopic order parameters and their gradients, which together with the macroscopic deformation field define the multifield character of the problem under consideration. Focussing on rate–independent standard dissipative materials, the constitutive response is governed by an energy storage and a non–smooth dissipation function. For this scenario, we outline an incremental variational framework whose Euler equations define the macroscopic equilibrium along with the non–smooth global evolution of the order parameters. The formulation features fully–coupled macroscopic and microscopic field equations and associated boundary conditions. The key difficulty on the numerical side is the implementation of the global non–smooth evolution of the order parameters. Here, we outline a new active set strategy for the global finite element discretization of the multifield problem, where inequality constraints on the microscopic fields are taken into account by active nodal sets. An important consequence of the proposed variational approach is the symmetry of the coupled algebraic system. The proposed setting is specified for model problems of isotropic damage mechanics and crystal plasticity. The applicability of the proposed approach to nonlocal solids is highlighted by means of representative numerical examples. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Variational Formulations and FE Active‐Set Strategies for Rate‐Independent Nonlocal Material Response

The paper discusses numerical formulations of the homogenization for solids with discrete crack development. We focus on multi–phase microstructures of heterogeneous materials, where fracture occurs in the form of debonding mechanisms as well as matrix cracking. The definition of overall properties critically depends on the developing discontinuities. To this end, we extend continuous formulations [1] to microstructures with discontinuities [2]. The basic underlying structure is a canonical variational formulation in the fully nonlinear range based on incremental energy minimization. We develop algorithms for numerical homogenization of fracturing solids in a deformation–driven context with non–trivial formulations of boundary conditions for (i) linear deformation and (ii) uniform tractions. The overall response of composite materials with fracturing microstructures are investigated. As a key result, we show the significance of the proposed non–trivial formulation of a traction–type boundary condition in the deformation–driven context. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Deformation Driven Homogenization of Fracturing Solids

This paper presents a constitutive model of finite strain elasto-visco-plasticity suitable for metals which is highly effective for numerical implementation. Noval aspects of the proposed formulation are (i) the derivation of power-type associative viscoplastic evolution equations based on a penalty augmentation of the viscoplastic dissipation and (ii) a particular logarithmic elastic model suitable for metal elasticity. The combination of (i) and (ii) allows a direct implementation of integration algorithms used in linear viscoplasticity.

Christian Miehe

Papers

Rate‐dependent incremental variational formulation of ferroelectricity

Homogenisierungsbasierte Mehrgitter-Transferoperatoren für nichtlineare heterogene Materialien

Variational Formulations and FE Active‐Set Strategies for Rate‐Independent Nonlocal Material Response

Deformation Driven Homogenization of Fracturing Solids

Theory and Finite Element Computation of Finite Elasto-Visco-Plastic Strains