## DYNAMIC FRACTURE MECHANICS

## BY ARUN SHUKLA

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**CONTENTS**

- Modeling Dynamic Fracture Using Large-Scale Atomistic Simulations
- Dynamic Crack Initiation Toughness
- The Dynamics of Rapidly Moving Tensile Cracks in Brittle Amorphous Material
- Optical Methods for Dynamic Fracture Mechanics
- On the Use of Strain Gages in Dynamic Fracture
- Dynamic and Crack Arrest Fracture Toughness
- Dynamic Fracture in Graded Materials
- Dynamic Fracture Initiation Toughness at Elevated Temperatures With Application to the New Generation of Titanium Aluminides Alloys
- Dynamic Fracture of Nanocomposite Materials

**BRIEF INTRODUCTION TO "DYNAMIC FRACTURE MECHANICS BY ARUN SHUKLA"**

**Modeling Dynamic Fracture Using Large-Scale Atomistic Simulations**

Introduction

Why and how cracks spread in brittle materials is of essential interest to numerous scientific disciplines and technological applications. Large-scale molecular dynamics (MD) simulation is becoming an increasingly useful tool to investigate some of the most fundamental aspects of dynamic fracture . Studying rapidly propagating cracks using atomistic methods is particularly attractive, because cracks propagate at speeds of kilometers per second, corresponding to time-and

length scales of nano-meters per pico-seconds readily accessible within classical MD methods. This similarity in time and length scales partly explains the success of MD in describing the physics and mechanics of dynamic fracture.

In this chapter, we mainly focus on the work involved by the authors in using simplistic inter-atomic potentials to probe crack dynamics in model materials, with an aim to gain broad insights into fundamental, physical aspects of dynamic fracture. A particular focus of our studies is on understanding the effect of hyper-elasticity on dynamic fracture. Most existing theories of fracture assume a linear elastic stress-strain law. However, the relation between stress and strain in real solids is strongly nonlinear due to large deformations near a moving crack tip, a phenomenon referred to here as hyper-elasticity. Our studies strongly suggest that hyper-elasticity, in contrast to most of the classical linear theories of fracture, indeed has a major impact on crack dynamics.

The plan of this chapter is as follows. First, we review atomistic modeling techniques, in particular our approach of using simple model potentials to study dynamic fracture. We will then cover three topics: (i) confined crack dynamics along weak layers in homogeneous materials, focusing on the crack limiting speed (Section 3), (ii) instability dynamics of fracture, focusing on the critical speed for the onset of crack instability (Section 4), and (iii) dynamics of cracks at interfaces of dissimilar materials (Section 5). Whereas cracks are confined to propagate along a prescribed path in Section 3, they are completely unconstrained in Section 4. Section 5 contains studies on both constrained and unconstrained crack propagation. We conclude with a discussion and outlook to future research in this area.

**The Dynamics of Rapidly Moving Tensile Cracks in Brittle Amorphous Material**

The dynamics of fast fracture in brittle amorphous materials are reviewed. We first present a picture of fracture in which numerous effects commonly observed in

dynamic fracture may be understood as resulting from an intrinsic (micro-branching) instability of a rapidly moving crack. The instability, when a single crack state

undergoes frustrated microscopic crack branching, occurs at a critical propagation velocity. This micro-branching instability gives rise to large velocity oscillations, the formation of non-trivial fracture surface structure, a large increase in the overall fracture surface area, and a corresponding sharp increase of the fracture energy with the mean crack velocity. We present experimental evidence, obtained in a variety of different materials, in support of this picture. The dynamics of crack-front interactions with localized material inhomogeneities are then described. We demonstrate that the loss of translational invariance resulting from this interaction gives rise to both localized waves that propagate along the crack front and the acquisition of an effective inertia by the crack. Crack-front inertia, when coupled with the micro-branching instability, leads to an understanding of the chain-like form of the micro-branch induced patterns observed both on and beneath the fracture surface.

Introduction

The equation of motion for a rapidly propagating crack The detailed dynamics of a rapidly propagating crack have been the object of intensive study since the early scaling theories of Mott, which were initiated during the second World War. These theories generalized the concept of energy balance, which was first introduced by Griffith in 1920 to describe the onset of fracture. Griffith proposed that fracture occurs when the potential elastic energy per unit area released by a unit

extension of a crack is equal to the fracture energy, r , defined as the amount of energy necessary to create a new unit of fracture surface. This concept was generalized by Mott and, later Dulaney and Brace1' 2, to include a global accounting of the kinetic energy within the elastic medium that is released by a propagating crack. Utilizing dimensional analysis, these first theories predicted that, in a two-dimensional medium of infinite extent, a crack should continuously accelerate as a function of its instantaneous length to a limiting, but finite, asymptotic velocity. Stroh noted that this asymptotic limit could be reached by either asymptotically increasing the amount of energy stored in the material or, equivalently, by reducing r to zero. With r —> 0, a crack propagating at its asymptotic velocity is equivalent to a disturbance moving across two free surfaces. As the speed of a disturbance moving along a free surface is the Ralyeigh wave speed, VR, of the medium, Stroh concluded that this must be the asymptotic velocity of a dynamic crack. This intuitive idea was later shown to be rigorously correct, when a quantitative theory for dynamic fracture was developed.

**Dynamic Fracture in Graded Materials**

The dynamic crack propagation in materials with varying properties, i.e., functionally graded materials is presented. First, an elasto-dynamic solution for a propagating crack inclined to the direction of property variation is introduced. Crack tip stress, strain and displacement fields are obtained through an asymptotic analysis coupled with displacement potential approach. Next, a systematic theoretical analysis is provided to incorporate the effect of transient nature of growing crack-tip on the crack-tip stress, strain and displacement fields. The analysis revealed that crack tip stress fields retain the inverse square root singularity and only the higher order terms in the expansion are influenced by material inhomogeneity. Using these stress, strain and displacement fields, contours of constant maximum shear stress, constant first stress invariant and constant in- plane displacements are generated and the effect of non-homogeneity and transient nature of crack tip on these contours is discussed.

Introduction

The combination of several materials in one component offers, in many cases, significant improvements to its functional performance. Optimally, material properties throughout a component should be tailored to its specific application. This requires combinations of properties that are unattainable with a single homogeneous material. Functionally graded materials (FGMs) offer an advantageous means of combining materials, providing a spatial variation in composition and properties, as an

alternative to homogeneous materials and bi-material interface structures. Functionally graded materials (FGMs) are a new generation of engineered materials wherein the micro-structural details are spatially varied through nonuniform distribution of the reinforcement phase(s). The resulting micro-structure produces continuously or discretely changing thermal and mechanical properties at the macroscopic or continuum scale. FGMs may be used as points between different types of materials, graded surface coatings or simply as graded components. More recently, graded composites have become a focus for tailoring thermal, mechanical and other properties for enhanced performance in structural, high temperature and other specialized applications. Man made functionally graded materials (FGMs) copy natural bio-materials,

such as bamboo, bone and shell etc., with a gradient in chemical composition or micro-structure from one side to the other side in the material. The concept of functionally graded materials was proposed in 1984 by material scientists in Sendai (Japan). Since then studies to develop high-performance heat-resistant materials using functionally graded technology have continued. Various techniques have been employed to the fabrication of FGMs, including Chemical Vapor Deposition (CVD) / Physical Vapor Deposition (PVD), powder metallurgy, plasma spraying, electro-plating and combustion synthesis. The principal motivation for FGM development is that a spatial variation in composition and properties at a joint between two different materials has the potential to reduce the stresses at the joint as compared

to a bi-material interface. Additionally, there is an increased inter-facial strength at the joint and thus the likelihood of de-bonding is reduced. This is illustrated in Fig. 2 in the context of a potential application of FGMs, as thermal barrier coatings for high-speed space access vehicle applications. In this situation, a heat resistant ceramic must be strongly bonded to the metal structure to prevent spalling or surface cracking due to thermally induced stresses. A graded ceramic-metal composite interface has been shown to reduce thermal damage and delamination. Even though the initial research on FGMs was largely motivated by the practical applications of the concept on a wide variety of thermal shielding problems, materials with graded physical properties have almost unlimited potential in many other technological applications. Mechanics research on FGMs is needed to provide technical support for material scientists and for design and manufacturing engineers to take full advantage of their certain favorable properties in new product development.