6 lectures on: Toughening mechanisms in biological materials: Experiments, modeling and bio-inspiration

I. Experimental tools for fracture mechanics and application to biological materials This first lecture will provide a brief overview of the experimental techniques relevant for characterizing the fracture properties of materials and their application to biological materials. Linear and nonlinear fracture mechanics of key biological materials will be subsequently introduced. II. Why biological materials are so tough We will present the specificity of biological materials in relationship with their extraordinary fracture performance. First, the compositions and microstructures of key biological structural materials will be presented and compare with engineering materials. The relationship between their specific architecture (based on mineralized tissues, fibrous tissues, natural elastomers and gels) and their toughness will then be largely discussed. III. Toughness of bio-inspired materials We will give an overview of recent bio-inspired materials and introduced basic rules for the design of tough composites. The fabrication and testing of these solids, their fracture performance as well as their applications will be presented.

6 lectures on: Non-linear fracture mechanics and friction

I. Fast fracture in slow motion: Dynamic fracture in brittle gels and the structure of the near-tip region - experiment and theory This first lecture will provide an introduction to dynamic fracture mechanics through the comparison of theory and experiments performed in brittle gels. We will then show how linear elastic fracture mechanics can be generalized to include nonlinear elasticity. II. Ever more singular: Instability in Dynamic Fracture The second lecture will address open problems that are currently not included in usual fracture mechanics. We will focus on the experimental study of waves propagating along the crack front and the recent investigation of crack branching taking place during fast fracture. These studies suggest a new view of crack instabilities that will finally be presented. III. Friction is fracture This third lecture will introduce experimental results showing how friction can be seen as a "classical" shear fracture problem. The relevance of this new paradigm for addressing friction problems will be presented.

6 lectures on: Perturbations of cracks.

This course will be devoted to perturbations of cracks in linear elastic media, in both 2D and 3D, with applications to the prediction of crack paths. It will encompass three topics: I. Crack perturbations under mixed-mode I+II This first lecture will present the tools required to understand crack path in 2D solids. The concepts introduced during this lecture will be applied to the prediction of crack kinking. II. Coplanar crack perturbations under pure mode I The second lecture will focus on in-plane fracture and the effect of crack front deformation on the stress distribution along the front. As an application, we will analyze the problem of the pinning of a crack front by an array of tough obstacles. III. Non-coplanar crack perturbations under mixed mode I+III The tools required to calculate the stress distribution along crack fronts with both in-plane and out-of-plane perturbations will be introduced. They will then be applied to the prediction of the formation and development of tilted facets in mode I+III. The theoretical concepts introduced pedagogically during this course will be subsequently applied for the analysis of crack path instabilities (Pr. Ravi-Chandar’s course) and crack branching (Pr. Fineberg’s course) and the prediction of the failure properties of heterogeneous materials (Dr. Ponson’s course).

6 lectures on: Ductile crack growth

I. Mechanisms of ductile failure and theoretical modeling We present an overview of the mechanism of ductile fracture (void nucleation, growth and coalescence) and introduce the basic theoretical concepts used to model this process. As a consequence of progressive cavitation, ductile fracture typically involves softening, which can lead to material instabilities. The analysis and modeling of such instabilities will be presented. II. Computational fracture mechanics and application to ductile failure The second lecture is an introduction to computational fracture mechanics. We will introduce the concept of cohesive zone models for fracture and their numerical implementation. The rest of the lecture will be devoted to the numerical framework for the analysis of both quasi-static and dynamic ductile crack growth, including phenomenological damage type constitutive relations and cohesive zone type formulations. Numerical issues associated with localization of deformation and with the creation of new free surface will be discussed. A material length scale needs to be incorporated into any fracture prediction, if only from dimensional considerations, and this provides a modeling and a computational challenge. III. From microstructure to ductile crack growth resistance and vice & versa The discussion in this lecture will center on the analysis of the ductile fracture mechanism in the fracture process zone and relating analyses of the micromechanics of ductile fracture to the prediction of values of component/structure level measures of crack growth resistance. A question in this regard is whether modeling of ductile fracture mechanisms can serve as a tool for developing materials with improved ductile crack growth resistance.

6 lectures on: Fracture mechanics of heterogeneous materials

This course aims at presenting the basic mechanisms underlying crack propagation in heterogeneous materials and introducing tools that allow the prediction of their effective failure response from their microscale features. Contrary to effective elastic properties that emerge from a weighted average over the whole material volume, failure results from the propagation of a singularity so that the overall material response is dominated by the properties of a limited region of the fracturing solid. Therefore, crack propagation problems involve dedicated homogeneization techniques that will be presented through experimental and numerical examples. I. Effective toughness of heterogeneous brittle solids The first lecture deals with the calculation of the effective toughness of heterogeneous linear elastic solids. We will present first pedagogical examples of crack propagation in 2D materials with heterogeneous failure and elastic properties that illustrate how the crack path controls their failure behavior by selecting local properties. We will then address this homogeneization problem in 3D by considering the propagation of a planar crack through an interface with heterogeneous failure properties. Introducing the concepts of weak and strong pinning of the crack front by obstacles, we will show that the effective toughness can be either representative of an average over the properties of the interface or dominated by the strongest regions of the interface. Some numerical tools useful for the rational design of solids with tailored failure properties will finally be presented. II. Failure of disordered materials: Intermittent crack dynamics and depinning transition The second lecture focuses on the failure of solids with random microstructures. The presence of disorder and its competition with the crack front elasticity gives rise to specific failure response, characterized by a strong intermittency of the fracture process. The concept of depinning transition required to describe and understand this highly fluctuating crack dynamics will be presented through the experimental study of crack growth through interfaces with random properties. We will show how, in some regime, the largest fluctuations survive at the macroscopic scale and dominate the material failure response. III. Roughening mechanisms and application in quantitative fractography The third lecture is devoted to out-of-plane excursions of cracks in heterogeneous solids. We will introduce the statistical tools relevant to describe the roughness of fracture surfaces and will present some theoretical approaches that allow deciphering their geometry in terms of microscopic failure mechanisms. The application of this concept to quantitative fractography that aims at characterizing the fracture properties of materials from the statistical analysis of fracture surfaces will also be presented.

6 lectures on: Introduction to fracture mechanics

This course will present a unified continuum treatment of modern day fracture mechanics. Particular attention is directed to instabilities taking place during the slow and fact propagation of cracks. The concepts introduced during the three lectures will serve as a basis for the more advanced topics addressed in the subsequent courses. I. Introduction to quasi-static fracture mechanics This first lecture develops fracture mechanics from a traditional continuum perspective by emphasizing basic principles that govern the behavior of cracks in materials. The underlying theme is the fundamental Griffith energy-balance concept of crack propagation. We describe the crack tip fields and introduce the concepts of J-integral that is used to relate the singularity of these mechanical fields with the fracture driving force. These concepts are illustrated through the prediction of fracture for some simple crack geometries and loading conditions. II. Dynamic fracture and instabilities This third lecture deals with fast fracture. The elasto-dynamics analysis of the crack tip fields provides a dynamic fracture criterion and insights on the behavior of fast cracks. New fracture phenomena like crack branching and dynamic roughening emerging at large crack speed are presented and interpreted from the concepts introduced during the lecture. III. Fracture path and instabilities during slow fracture This second lecture is devoted to the prediction of the fracture path in homogeneous brittle solids. Failure criteria are introduced for in-plane loading geometries, and subsequently extended to mixed mode loading conditions that result in out-of-plane crack excursion. Two archetypes of fracture paths instabilities are investigated: Wavy crack trajectory during thermal fracture and crack front fragmentation under mixed-mode loading. The theoretical tools introduced by Pr. Leblond for dealing with perturbed crack paths will be practically implemented for the interpretation of experimental fracture patterns.