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

An incremental variational formulation of dissipative magnetostriction at the macroscopic continuum level

A variational formulation of quasi-static brittle fracture in elastic solids at small strains is proposed and an associated finite element implementation is presented. On the theoretical side, a consistent thermodynamic framework for brittle crack propagation is outlined. It is shown that both the elastic equilibrium response as well as the local crack evolution follow in a natural format by exploitation of a global Clausius–Planck inequality. Here, the canonical direction of the crack propagation associated with the classical Griffith criterion is the direction of the material configurational force which maximizes the local dissipation at the crack tip. On the numerical side, we first consider a standard finite element discretization in the two-dimensional space which yields a discrete formulation of the global dissipation in terms of configurational nodal forces. Next, consistent with the node-based setting, the discretization of the evolving crack discontinuity for two-dimensional problems is performed by the doubling of critical nodes and interface segments of the mesh. A crucial step for the success of this procedure is its embedding into a r-adaptive crack-segment re-orientation algorithm governed by configurational-force-based directional indicators. Here, successive crack propagation is performed by a staggered loading-release algorithm of energy minimization at frozen crack state followed by nodal releases at frozen deformation. We compare results obtained by the proposed formulation with other crack propagation criteria. The computational method proposed is extremely robust and shows an excellent performance for representative numerical simulations.

A computational framework of configurational-force-driven brittle fracture based on incremental energy minimization

The phase field modeling of brittle fracture was a topic of intense research in the last few years and is now well-established. We refer to the work [1-3], where a thermodynamically consistent framework was developed. The main advantage is that the phase-field-type diffusive crack approach is a smooth continuum formulation which avoids the modeling of discontinuities and can be implemented in a straightforward manner by multi-field finite element methods. Therefore complex crack patterns including branching can be resolved easily. In this paper, we extend the recently outlined phase field model of brittle crack propagation [1-3] towards the analysis of ductile fracture in elastic-plastic solids. In particular, we propose a formulation that is able to predict the brittle-to-ductile failure mode transition under dynamic loading that was first observed in experiments by Kalthoff and Winkler [4]. To this end, we outline a new thermodynamically consistent framework for phase field models of crack propagation in ductile elastic-plastic solids under dynamic loading, develop an incremental variational principle and consider its robust numerical implementation by a multi-field finite element method. The performance of the proposed phase field formulation of fracture is demonstrated by means of the numerical simulation of the classical Kalthoff-Winkler experiment that shows the dynamic failure mode transition. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

https://onlinelibrary.wiley.com/doi/pdf/10.1002/pamm.201310258

Phase Field Modeling of Brittle and Ductile Fracture

A formulation of isotropic thermoplasticity for arbitrary large elastic and plastic strains is presented. The underlying concept is the introduction of a metric transformation tensor which maps a locally defined six-dimensional plastic metric onto the metric of the current configuration. This mixed-variant tensor field provides a basis for the definition of a local isotropic hyperelastic stress response in the thermoplastic solid. Following this fundamental assumption, we derive a consistent internal variable formulation of thermoplasticity in a Lagrangian as well as a Eulerian geometric setting. On the numerical side, we discuss in detail an objective integration algorithm for the mixed-variant plastic flow rule. The special feature here is a new representation of the stress return and the algorithmic elastoplastic moduli in the eigenvalue space of the Eulerian plastic metric for plane problems. Furthermore, an algorithm for the solution of the coupled problem is formulated based on an operator split of the global field equations of thermoplasticity. The paper concludes with two representative numerical simulations of thermoplastic deformation processes.

A theory of large-strain isotropic thermoplasticity based on metric transformation tensors

We propose a fundamentally new approach to the treatment of shearband localizations in strain softening elastic–plastic solids at finite strains based on energy minimization principles associated with microstructure developments. The point of departure is a general internal variable formulation that determines the finite inelastic response as a standard dissipative medium. Consistent with this type of inelasticity we consider an incremental variational formulation of the local constitutive response where a quasi-hyperelastic stress potential is obtained from a local constitutive minimization problem with respect to the internal variables. The existence of this variational formulation allows the definition of the material stability of an inelastic solid based on weak convexity conditions of the incremental stress potential in analogy to treatments of finite elasticity. Furthermore, localization phenomena are interpreted as microstructure developments on multiple scales associated with non-convex incremental stress potentials in analogy to elastic phase decomposition problems. These microstructures can be resolved by the relaxation of non-convex energy functionals based on a convexification of the stress potential. The relaxed problem provides a well-posed formulation for a mesh-objective analysis of localizations as close as possible to the non-convex original problem. Based on an approximated rank-one convexification of the incremental stress potential we develop a computational two-scale procedure for a mesh-objective treatment of localization problems at finite strains. It constitutes a local minimization problem for a relaxed incremental stress potential with just one scalar variable representing the intensity of the microshearing of a rank-one laminate aligned to the shear band. This problem is sufficiently robust with regard to applications to large-scale inhomogeneous deformation processes of elastic–plastic solids. The performance of the proposed energy relaxation method is demonstrated for a representative set of numerical simulations of straight and curved shear bands which report on the mesh independence of the results. Copyright © 2003 John Wiley & Sons, Ltd.

Analysis of microstructure development in shearbands by energy relaxation of incremental stress potentials: Large-strain theory for standard dissipative solids

Christian Miehe

Papers

An incremental variational formulation of dissipative magnetostriction at the macroscopic continuum level

A computational framework of configurational-force-driven brittle fracture based on incremental energy minimization

Phase Field Modeling of Brittle and Ductile Fracture

A theory of large-strain isotropic thermoplasticity based on metric transformation tensors

Analysis of microstructure development in shearbands by energy relaxation of incremental stress potentials: Large-strain theory for standard dissipative solids