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 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 -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 which govern the evolution of the crack phase field. Here, we propose alternative rate-independent and viscous over-force models which 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.

Thermodynamically-Consistent Phase Field Models of Fracture: Variational Principles and Multi-Field

Computational homogenization analysis in finite plasticity Simulation of texture development in polycrystalline materials

The paper investigates algorithms for the computation of homogenized stresses and overall tangent moduli of microstructures undergoing small strains. Typically, these microstructures define representative volumes of nonlinear heterogeneous materials such as inelastic composites, polycrystalline aggregates or particle assemblies. We consider a priori given discretized microstructures, without focusing on details of specific discretization techniques in space and time. The key contribution of the paper is the construction of a family of algorithms and matrix representations of the overall properties of discretized microstructures. It is shown that the overall stresses and tangent moduli of a typical microstructure may exclusively be defined in terms of discrete forces and stiffness properties on the boundary. We focus on deformation-driven microstructures, where the overall macroscopic deformation is controlled. In this context, three classical types of boundary conditions are investigated: (i) linear displacements, (ii) constant tractions and (iii) periodic displacements and antiperiodic tractions. Incorporated by the Lagrangian multiplier method, these constraints generate three classes of algorithms for the computation of equilibrium states and the overall properties of microstructures. The proposed algorithms and matrix representations of the overall properties are formally independent of the interior spatial structure and the local constitutive response of the microstructure and are therefore applicable to a broad class of model problems. We demonstrate their performance for some representative model problems including elastic–plastic deformations of composite materials.

Christian Miehe

Papers

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

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

Thermodynamically-Consistent Phase Field Models of Fracture: Variational Principles and Multi-Field

Computational homogenization analysis in finite plasticity Simulation of texture development in polycrystalline materials

Computational micro-to-macro transitions of discretized microstructures undergoing small strains