This book is a tutorial written by researchers and developers behind the FEniCS Project and explores an advanced, expressive approach to the development of mathematical software. The presentation spans mathematical background, software design and the use of FEniCS in applications. Theoretical aspects are complemented with computer code which is available as free/open source software. The book begins with a special introductory tutorial for beginners. Followingare chapters in Part I addressing fundamental aspects of the approach to automating the creation of finite element solvers. Chapters in Part II address the design and implementation of the FEnicS software. Chapters in Part III present the application of FEniCS to a wide range of applications, including fluid flow, solid mechanics, electromagnetics and geophysics.

Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book

Interpretation of X-ray Powder Diffraction Patterns . By H. Lipson and H. Steeple. Pp. viii + 335 + 3 plates. (Mac-millan: London; St Martins Press: New York, May 1970.) £4.

X-Ray Diffraction

https://hal.archives-ouvertes.fr/hal-00773015/document

A review on the Mullins effect

A large quantity of small molecules may migrate into a network of long polymers, causing the network to swell, forming an aggregate known as a polymeric gel. This paper formulates a theory of the coupled mass transport and large deformation. The free energy of the gel results from two molecular processes: stretching the network and mixing the network with the small molecules. Both the small molecules and the long polymers are taken to be incompressible, a constraint that we enforce by using a Lagrange multiplier, which coincides with the osmosis pressure or the swelling stress. The gel can undergo large deformation of two modes. The first mode results from the fast process of local rearrangement of molecules, allowing the gel to change shape but not volume. The second mode results from the slow process of long-range migration of the small molecules, allowing the gel to change both shape and volume. We assume that the local rearrangement is instantaneous, and model the long-range migration by assuming that the small molecules diffuse inside the gel. The theory is illustrated with a layer of a gel constrained in its plane and subject to a weight in the normal direction. We also predict the scaling behavior of a gel under a conical indenter.

https://imechanica.org/files/Kinetics%202007%2010%2022%20correct.pdf

A theory of coupled diffusion and large deformation in polymeric gels

The extended and generalized finite element methods are reviewed with an emphasis on their applications to problems in material science: (1) fracture, (2) dislocations, (3) grain boundaries and (4) phases interfaces. These methods facilitate the modeling of complicated geometries and the evolution of such geometries, particularly when combined with level set methods, as for example in the simulation growing cracks or moving phase interfaces. The state of the art for these problems is described along with the history of developments.

/pdf/a-review-of-extended-generalized-finite-element-methods-for-1uqjazt5n2.pdf

A review of extended/generalized finite element methods for material modeling

The present paper proposes a thorough comparison of twenty hyperelastic models for rubber-like materials. The ability of these models to reproduce different types of loading conditions is analyzed thanks to two classical sets of experimental data. Both material parameters and the stretch range of validity of each model are determined by an efficient fitting procedure. Then, a ranking of these twenty models is established, highlighting new efficient constitutive equations that could advantageously replace well-known models, which are widely used by engineers for finite element simulation of rubber parts.

https://hal.archives-ouvertes.fr/hal-01004680/document

Comparison of Hyperelastic Models for Rubber-Like Materials

This paper reports on the development of a new network alteration theory to describe the Mullins effect. The stress-softening phenomenon that occurs in rubber-like materials during cyclic loading is analysed from a physical point of view. The Mullins effect is considered to be a consequence of the breakage of links inside the material. Both filler-matrix and chain interaction links are involved in the phenomenon. This new alteration theory is implemented by modifying the eight-chains constitutive equation of Arruda and Boyce (J. Mech. Phys. Solids 41 (2) (1993) 389). In the present method the parameters of the eight-chains model, denoted C-R and N in the bibliography, become functions of the maximum chain stretch ratio. The accuracy of the resulting constitutive equation is demonstrated on cyclic uniaxial experiments for both natural rubbers and synthetic elastomers.

https://hal.archives-ouvertes.fr/hal-01004954/document

A theory of network alteration for the Mullins effect

The present paper deals with the fatigue crack growth in a carbon black filled cis-1,4-polyisoprene rubber under relaxing loading conditions. The study focuses on the determination of the scenario of crack growth. For this purpose, an original “microcutting” method is employed to observe microscopic phenomena involved in the growth of the crack with a SEM. It reveals that the cavitation induced by the decohesion between zinc oxides and rubber matrix is the major fatigue damage and that the crack tip is composed of stretched elliptical areas surrounded by highly stretched and crystallized ligaments. Finally, the observations are considered to establish the fatigue crack growth mechanism.

https://hal.archives-ouvertes.fr/hal-01010941/document

Mechanism of Fatigue Crack Growth in Carbon Black Filled Natural Rubber

The present paper reports and rationalizes the use of Continuum Damage Mechanics (CDM) to describe the Mullins effect in elastomers. Thermodynamics of rubber-like hyperelastic materials with isotropic damage is considered. Since it is demonstrated that stress-softening is not strictly speaking a damage phenomenon, the limitations of the CDM approach are highlighted. Moreover, connections with two-network-based constitutive models proposed by other authors are exhibited through the choice of both the damage criterion and the measure of deformation. Experimental data are used to establish the evolution equation of the stress-softening variable, and the choice of the maximum deformation endured previously by the material as the damage criterion is revealed as questionable. Nevertheless, the present model agrees qualitatively well with experiments except to reproduce the strain-hardening phenomenon that takes place as reloading paths rejoin the primary loading path. Finally, the numerical implementation highlights the influence of loading paths on material response and thereby demonstrates the importance of considering the Mullins effect in industrial design.

https://hal.archives-ouvertes.fr/hal-01007152/document

On the relevance of Continuum Damage Mechanics as applied to the Mullins effect in elastomers

Fracture of rubber-like materials is still an open problem. Indeed, it deals with modelling issues (crack growth law, bulk behaviour) and computational issues (robust crack growth in 2D and 3D, incompressibility). The present study focuses on the application of the eXtended Finite Element Method (X-FEM) to large strain fracture mechanics for plane stress problems. Two important issues are investigated: the choice of the formulation used to solve the problem and the determination of suitable enrichment functions. It is demonstrated that the results obtained with the method are in good agreement with previously published works.

/pdf/stress-analysis-around-crack-tips-in-finite-strain-problems-3xyrzry38z.pdf

Erwan Verron

Papers

Comparison of Hyperelastic Models for Rubber-Like Materials

A theory of network alteration for the Mullins effect

Mechanism of Fatigue Crack Growth in Carbon Black Filled Natural Rubber

On the relevance of Continuum Damage Mechanics as applied to the Mullins effect in elastomers

Stress analysis around crack tips in finite strain problems using the eXtended Finite Element Method