René de Borst

Non-Linear Finite Element Analysis of Solids and Structures: de Borst/Non-Linear Finite Element Analysis of Solids and Structures

Built upon the two original books by Mike Crisfield and their own lecture notes, renowned scientist Rene de Borst and his team offer a thoroughly updated yet condensed edition that retains and builds upon the excellent reputation and appeal amongst students and engineers alike for which Crisfields first edition is acclaimed. Together with numerous additions and updates, the new authors have retained the core content of the original publication, while bringing an improved focus on new developments and ideas. This edition offers the latest insights in non-linear finite element technology, including non-linear solution strategies, computational plasticity, damage mechanics, time-dependent effects, hyperelasticity and large-strain elasto-plasticity. The authors integrated and consistent style and unrivalled engineering approach assures this books unique position within the computational mechanics literature.

Numerical Modelling of Self Healing Mechanisms

A number of self healing mechanisms for composite materials have been presented in the previous chapters of this book. These methods vary from the classical concept of micro-encapsulating of healing agents in polymer systems to the autonomous healing of concrete. The key feature of these self healing mechanisms is the transport of material to the damaged zone in order to establish the healing process. Generally, this material is a fluid and its motion is driven by capillary action which enables transportation over relatively large distances requiring little or no work. In the microencapsulated polymers as developed by White et al. [1], this liquid material is a healing agent, which is enclosed in the material by micro-encapsulation. When the capsule is ruptured by a crack, the healing agent will flow into the crack, driven by capillary action. Polymerisation of this healing agent is triggered by contact with catalysts which are inserted in the material and whose position is fixed. The new polymerised material will rebond the crack surfaces.

Isogeometric Failure Analysis

Isogeometric analysis is a versatile tool for failure analysis. On the one hand, the excellent control over the inter-element continuity conditions enables a natural incorporation of continuum constitutive relations that incorporate higher-order strain gradients, as in gradient plasticity or damage. On the other hand, the possibility of enhancing a basis with discontinuities by means of knot insertion makes isogeometric finite elements a suitable candidate for modeling discrete cracks. Both possibilities are described and will be illustrated by examples.

Computational Methods for Debonding in Composites

This contribution starts with a discussion of various phenomena in laminated composite structures that can lead to failure: matrix cracking, delamination between plies, and debonding and subsequent pull-out between fibres and the matrix material. The different scales are discussed at which the effect of these nonlinearities can be analysed. From these scales – the macro, meso and micro-levels – the meso-level is normally used for the analysis of delamination, which is the focus of this contribution. At this level, the plies are modelled as continua and interface elements between them conventionally serve as the framework to model delamination and debonding. After a a derivation of interface elements and a brief discussion of the cohesive–zone concept and its importance for the analysis of delamination, a particular finite element model for the plies is elaborated: the solid–like shell. Next, a more recent method to numerically model delamination is discussed, which exploits the partition–of–unity property of finite element shape functions. This approach offers advantages over interface elements, as will be discussed in detail.

A Discussion on Gradient Damage and Phase-Field Models for Brittle Fracture

Gradient-enhanced damage models find their roots in damage mechanics, which is a smeared approach from the onset, and gradients were added to restore well-posedness beyond a critical strain level. The phase-field approach to brittle fracture departs from a discontinuous description of failure, where the distribution function is regularised, which also leads to the inclusion of spatial gradients. Herein, we will consider both approaches, and discuss their similarities and differences.

Application of the Discontinuous Solid-Like Shell Element to Delamination

A possible failure mechanism in layered composite materials is delamination. This mechanism can be analysed numerically using the discontinuous solid-like shell element. In this element, the delamination crack can occur at arbitrary locations and is incorporated as a jump in the displacement field by using the partition-of-unity property of finite element shape functions. The performance of the element is demonstrated by a small example.

Numerical Modelling of Fibre Metal Laminates Under Thermomechanical Loading

1 General Introduction A thermo-mechanical finite element model, based on a solid-like shell element, has been developed. The use of standard continuum elements to model thinwalled structures, such as a fuselage skin, may lead to problems as they tend to show Poisson-thickness locking for high aspect ratios. Therefore a solid-like shell element has been extended to include the temperature field and thermal expansion. The coupled system of equations is solved simultaneously. This numerical model is used to characterise the behaviour of fibre metal laminates under thermo-mechanical loadings. A bi-material strip subjected to a heat source is presented as a benchmark test to demonstrate the performance of the thermo-mechanical solid-like shell element. With a minimum amount of elements and a high aspect ratio the results are accurate and in agreement with the analytical solution.

Variational Germano Approach for Multiscale Formulations

In this chapter the recently introduced Variational Germano procedure is revisited. The procedure is explained using commutativity diagrams. A general Germano identity for all types of discretizations is derived. This relation is similar to the Variational Germano identity, but is not restricted to variational numerical methods. Based on the general Germano identity an alternative algorithm, in the context of stabilized methods, is proposed. This partitioned algorithm consists of distinct building blocks. Several options for these building blocks are presented and analyzed and their performance is tested using a stabilized finite element formulation for the convectionU? diffusion equation. Non-homogenous boundary conditions are shown to pose a serious problem for the dissipation method. This is not the case for the leastsquares method although here the issue of basis dependence occurs. The latter can be circumvented by minimizing a dual-norm of the weak relation instead of the Euclidean norm of the discrete residual.

Computational Multi-Scale Methods and Evolving Discontinuities

This contribution discusses modern concepts in multi-scale analysis. Emphasis is placed in the discussion on so-called concurrent approaches, in which computations are carried out simultaneously at two or more scales. Since analyses at a lower level typically involve more discontinuities to be considered, attention is also paid to the proper modelling of evolving discontinuities. Another related problem is the treatment of discontinuities for problems that involve the modelling of diffusion phenomena in addition to a stress analysis, since this also requires the application of multi-scale concepts. As a further step the coupling of dissimilar media is considered like continuum to discrete models.

On the monolithic and staggered solution of cell contractility and focal adhesion growth: On the monolithic and staggered solution of cell contractility and focal adhesion growth

The mechanical response of cells to stimuli tightly couples biochemical and biomechanical processes, which describe fundamental properties such as cell growth and reorientation. Cells interact continuously with their external surroundings, the extracellular matrix (ECM), by establishing a bond between cell and ECM through the formation of focal adhesions. Focal adhesions are made up of integrins, which are mechanosensitive proteins and responsible for the communication between the cell and the ECM. The governing biochemomechanical processes can be modeled by means of a continuum approach considering mechanical and thermodynamic equilibrium to describe cell contractility and focal adhesion growth. The immanent multiphysical character of cell mechanics involves important aspects such as the coupling of fields of different scales and corresponding interface conditions that are sensitive to the solution of the governing numerical problem. These aspects become even more relevant when considering a feedback loop among the multiphysical solutions fields. In this contribution, we consider solution properties and sensitivity aspects of a nonlinear mechanical continuum model for the prognosis of stress fiber growth and reorientation incorporating a mechanosensitive feedback loop. We provide the governing equations of a Hill model-based stress fiber growth, which is coupled to a thermodynamical approach modeling the focal adhesions. Furthermore, a variational formulation including the algebraic equations is derived for staggered and monolithic solution approaches and the reaction-diffusion equation that models the feedback mechanism. We test both schemes with regard to reliability, accuracy, and numerical efficiency for different model parameters and loading scenarios. We present algorithmic aspects of the considered solution schemes and reveal their robustness with regard to model refinement in space and time and finally provide an assessment of their overall solution performance for multiphysics problems in the context of cell mechanics.