Process zone in the Single Cantilever Beam under transverse loading - Part II: Experimental

This paper describes an experimental arrangement to evaluate stress/strain fields in the process zone of asymmetric adhesively bonded joints. A transparent polycarbonate flexible beam was bonded to an aluminium alloy rigid block with an epoxy adhesive in a Single Cantilever Beam (SCB) configuration. The flexible adherend was loaded in the direction parallel to the initial crack front at constant rate. To monitor strains induced by bending and shear along the beam, electric strain gauges were attached to the upper surface of the flexible adherend. Thus strain distribution was measured above the bonded surface, which could be used to monitor crack propagation and investigate stress redistribution in the process zone. A Timoshenko beam lying on a Pasternak elastic foundation model was used for the analysis of experimental findings. Subsequently, the Digital Image Correlation technique was used to measure the flexible substrate in-plane displacement field in the vicinity of the crack front and to assess the specimen kinematics. We found that strain gauge instrumentation of the fracture mechanics specimen was a very sensitive technique for experimental analysis of crack propagation under complex loading, offering fine investigation of stress distribution in the cohesive zone.     

Modern topics and challenges in dynamic fracture

The field of dynamic fracture has been enlivened over the last 5 years or so by a series of remarkable accomplishments in different fields—earthquake science, atomistic (classical and quantum) simulations, novel laboratory experiments, materials modeling, and continuum mechanics. Important concepts either discovered for the first time or elaborated in new ways reveal wider significance. Here the separate streams of the literature of this progress are reviewed comparatively to highlight commonality and contrasts in the mechanics and physics.Much of the value of the new work resides in the new questions it has raised, which suggests profitable areas for research in the next few years and beyond. From the viewpoint of fundamental science, excitement is greatest in the struggle to probe the character of dynamic fracture at the atomic scale, using Newtonian or quantum mechanics as appropriate (a qualifier to be debated!). But lively interest is also directed towards modeling and experimentation at macroscales, including the geological, where the science of fracture is pulled at once by fundamental issues, such as the curious effects of friction, and the structural, where dynamic effects are essential to proper design or certification and even in manufacture.     

A finite strain model of stress, diffusion, plastic flow, and electrochemical reactions in a lithium-ion half-cell

We formulate the continuum field equations and constitutive equations that govern deformation, stress, and electric current flow in a Li-ion half-cell. The model considers mass transport through the system, deformation and stress in the anode and cathode, electrostatic fields, as well as the electrochemical reactions at the electrode/electrolyte interfaces. It extends existing analyses by accounting for the effects of finite strains and plastic flow in the electrodes, and by exploring in detail the role of stress in the electrochemical reactions at the electrode-electrolyte interfaces. In particular, we find that that stress directly influences the rest potential at the interface, so that a term involving stress must be added to the Nernst equation if the stress in the solid is significant. The model is used to predict the variation of stress and electric potential in a model 1-D half-cell, consisting of a thin film of Si on a rigid substrate, a fluid electrolyte layer, and a solid Li cathode. The predicted cycles of stress and potential are shown to be in good agreement with experimental observations.     

Prediction of diffusion assisted hydrogen embrittlement failure in high strength martensitic steels

A stress assisted hydrogen diffusion transport model, a dislocation-density-based multiple-slip crystalline plasticity formulation, and an overlapping fracture method were used to investigate hydrogen diffusion and embrittlement in lath martensitic steels with distributions of M₂₃C₆ carbide precipitates. The formulation accounts for variant morphologies based on orientation relationships (ORs) that are uniquely inherent to lath martensitic microstructures. The interrelated effects of martensitic block and packet boundaries and carbide precipitates on hydrogen diffusion, hydrogen assisted crack nucleation and growth, are analyzed to characterize the competition between cleavage fracture and hydrogen diffusion assisted fracture along preferential microstructural fracture planes. Stresses along the three cleavage planes and the six hydrogen embrittlement fracture planes are monitored, such that crack nucleation and growth can nucleate along energetically favorable planes. High pressure gradients result in the accumulation of hydrogen, which embrittles martensite, and results in crack nucleation and growth along {110} planes. Cleavage fracture occurs along {100} planes when there is no significant hydrogen diffusion. The predictions indicate that hydrogen diffusion can suppress the emission and accumulation of dislocation density, and lead to fracture with low plastic strains.     

Asymptotic Stress Field in a Degenerate Orthotropic Material Containing a Cohesive Zone ahead of a Crack Tip

Complex variable and eigenfunction expansion methods are used to derive elastic stress fields near the cohesive zone of a crack aligned with principal axes of a degenerate orthotropic material. Asymptotic field terms are obtained using coordinate rescaling and transformation techniques. The stress fields do not have stress singularity. Variations of stress components associated with the first-order or dominant stress terms are discussed in detail.      

A fracture criterion for finitely deforming crystalline solids—The dynamic fracture of single crystals

The major objective of this work has been to develop, within a continuum framework, a microstructurally-based computational theory to investigate dynamic failure in metals. To model the nucleation and propagation of failure surfaces at the microstructural scale, under large deformations and dynamic loading conditions, general finite-deformation theory, as relating to the decomposition of the deformation gradient, was tailored to monitor displacement incompatibilities and fracture in crystalline solids subjected to large deformations. Based on this proposed decomposition, a general fracture criterion for finitely deforming crystals, using the integral law of incompatibility, was developed. The analyses indicate that this newly proposed fracture formulation and criterion can be validated with experimental results, and can be used to accurately predict brittle and ductile failure modes for the large deformation of single crystals. As part of the newly proposed decomposition of the deformation gradient, sub-problems can also be solved for lattice distortions, such as twinning and geometrically necessary dislocation (GND) densities. Accordingly, the interactions of GND densities with cracks were investigated for single crystals. GND densities were shown to form as loops for stationary crack tips, but no loops formed for propagating cracks.     

Numerical simulations of stress generation and evolution in Volmer–Weber thin films

We present a detailed model of the stresses and shape changes that occur in polycrystalline thin films during Volmer–Weber growth. Our model tracks the shape of an array of islands as they grow and coalesce into a continuous film. The islands change shape as a result of the deposition flux, as well as surface and grain boundary diffusion. Stress is generated in the film as a result of forces exerted between neighboring islands as they meet to form a grain boundary. The internal stresses in the islands and the diffusive changes on their surfaces and grain boundaries are computed using a coupled finite element scheme. Interactions between neighboring islands are modeled using a cohesive zone law. Our model predicts stress-thickness vs. thickness behavior that is in excellent agreement with experiments. Specifically, we observe a three-stage growth process consisting of a stress-free pre-coalescence stage, a rapid tensile rise at coalescence, and an eventual transition to a steady-state. The steady-state stress may be tensile or compressive, depending on the deposition rate, the grain size, and the properties of the film. Detailed parametric studies are conducted to establish the influence of material properties and growth conditions on the stress history, and the results are compared with experimental observations and previous models.     

Interface damage and its effect on vibrations of slab track under temperature and vehicle dynamic loads

This paper presents a three-dimensional finite element model to investigate the interface damage occurred between prefabricated slab and CA (cement asphalt) mortar layer in the China Railway Track System (CRTS-II) slab track system. In the finite element model, a cohesive zone model with a non-linear constitutive law is introduced and utilized to model the damage, cracking and delamination at the interface. Combining with the temperature field database obtained from the three-dimensional transient heat transfer analysis, the interface damage evolution as a result of temperature change is analyzed. A three-dimensional coupled dynamic model of a vehicle and the slab track is then established to calculate the varying rail-supporting forces which are utilized as the inputs to the finite element model. The non-linearities of the wheel–rail contact geometry, the wheel–rail normal contact force and the wheel–rail tangential creep force are taken into account in the model. Setting the maximum interface damaged state calculated under temperature change as the initial condition, the interface damage evolution and its influence on the dynamic response of the slab track are investigated under the joint action of the temperature change and vehicle dynamic load. The analysis indicates that the proposed model is capable of predicting the initiation and propagation of cracks at the interface. The prefabricated slab presents lateral warping, resulting in severe interface damage on both the sides of the slab track along the longitudinal direction during temperature drop process, while the interface damage level does not change significantly under vehicle dynamic loads. The interface damage has great effects on the dynamic responses of the slab track.     

Cohesive fracture simulation and failure modes of FRP–concrete bonded interfaces

A work-of-fracture method using three-point bend beam (3PBB) specimen, commonly employed to determine the fracture energy of concrete, is adapted to evaluate the mode-I cohesive fracture of fiber reinforced plastic (FRP) composite–concrete adhesively bonded interfaces. In this study, a bilinear damage cohesive zone model (CZM) is used to simulate cohesive fracture of FRP–concrete bonded interfaces. The interface cohesive process damage model is proposed to simulate the adhesive–concrete interface debonding; while a tensile plastic damage model is used to account for the cohesive cracking of concrete near the bond line. The influences of the important interface parameters, such as the interface cohesive strength, concrete tensile strength, critical interface energy, and concrete fracture energy, on the interface failure modes and load-carrying capacity are discussed in detail through a numerical finite element parametric study. The results of numerical simulations indicate that there is a transition of the failure modes controlling the interface fracture process. Three failure modes in the mode-I fracture of FRP–concrete interface bond are identified: (1) complete adhesive–concrete interface debonding (a weak bond), (2) complete concrete cohesive cracking near the bond line (a strong bond), and (3) a combined failure of interface debonding and concrete cohesive cracking. With the change of interface parameters, the transition of failure modes from interface debonding to concrete cohesive cracking is captured, and such a transition cannot be revealed by using a conventional fracture mechanics-based approach, in which only an energy criterion for fracture is employed. The proposed cohesive damage models for the interface and concrete combined with the numerical finite element simulation can be used to analyze the interface fracture process, predict the load-carrying capacity and ductility, and optimize the interface design, and they can further shed new light on the interface failure modes and transition mechanism which emulate the practical application.     

Cohesive zone laws for void growth — II. Numerical field projection of elasto-plastic fracture processes with vapor pressure

Modeling ductile fracture processes using Gurson-type cell elements has achieved considerable success in recent years. However, incorporating the full mechanisms of void growth and coalescence in cohesive zone laws for ductile fracture still remains an open challenge. In this work, a planar field projection method, combined with equilibrium field regularization, is used to extract crack-tip cohesive zone laws of void growth in an elastic-plastic solid. To this end, a single row of void-containing cell elements is deployed directly ahead of a crack in an elastic-plastic medium subjected to a remote K-field loading; the macroscopic behavior of each cell element is governed by the Gurson porous material relation, extended to incorporate vapor pressure effects. A thin elastic strip surrounding this fracture process zone is introduced, from which the cohesive zone variables can be extracted via the planar field projection method. We show that the material's initial porosity induces a highly convex traction-separation relationship — the cohesive traction reaches the peak almost instantaneously and decreases gradually with void growth, before succumbing to rapid softening during coalescence. The profile of this numerically extracted cohesive zone law is consistent with experimentally determined cohesive zone law in Part I for multiple micro-crazing in HIPS. In the presence of vapor pressure, both the cohesive traction and energy are dramatically lowered; the shape of the cohesive zone law, however, remains highly convex, which suggests that diffusive damage is still the governing failure mechanism.     

Molecular-dynamics simulation-based cohesive zone representation of intergranular fracture processes in aluminum

A traction-displacement relationship that may be embedded into a cohesive zone model for microscale problems of intergranular fracture is extracted from atomistic molecular-dynamics (MD) simulations. An MD model for crack propagation under steady-state conditions is developed to analyze intergranular fracture along a flat Σ99 [1 1 0] symmetric tilt grain boundary in aluminum. Under hydrostatic tensile load, the simulation reveals asymmetric crack propagation in the two opposite directions along the grain boundary. In one direction, the crack propagates in a brittle manner by cleavage with very little or no dislocation emission, and in the other direction, the propagation is ductile through the mechanism of deformation twinning. This behavior is consistent with the Rice criterion for cleavage vs. dislocation blunting transition at the crack tip. The preference for twinning to dislocation slip is in agreement with the predictions of the Tadmor and Hai criterion. A comparison with finite element calculations shows that while the stress field around the brittle crack tip follows the expected elastic solution for the given boundary conditions of the model, the stress field around the twinning crack tip has a strong plastic contribution. Through the definition of a Cohesive-Zone-Volume-Element—an atomistic analog to a continuum cohesive zone model element—the results from the MD simulation are recast to obtain an average continuum traction-displacement relationship to represent cohesive zone interaction along a characteristic length of the grain boundary interface for the cases of ductile and brittle decohesion.     

Understanding the influence of structural hierarchy and its coupling with chemical environment on the strength of idealized tropocollagen–hydroxyapatite biomaterials

Hard biomaterials such as bone, dentin, and nacre have primarily an organic phase (e.g. tropocollagen (TC)) and a mineral phase (e.g. hydroxyapatite (HAP) or aragonite) arranged in a staggered arrangement at the nanoscopic length scale. Interfacial interactions between the organic phase and the mineral phase as well as the structural effects arising due to the staggered arrangement significantly affect the strength of such biomaterials. The effect of such factors is intricately intertwined with the chemical environment of such materials. In the present investigation, an idealized TC–HAP composite system under tensile loading is analyzed using explicit three-dimensional (3-D) molecular dynamics (MD) simulations to develop an understanding of these factors. The material system is analyzed in three different environments: (1) in the absence of water molecules (non-hydrated), (2) in the presence of water molecules (hydrated), and (3) in the presence of water molecules with calcium ions (ionized water). The analyses focus on understanding the correlations among factors such as the structural arrangement, the peak stress during deformation, Young's modulus, the peak interfacial strength, and the length scale of the localization of peak stress during deformation. Analyses show that maximizing the contact area between the TC and HAP phases results in higher interfacial strength as well as higher fracture strength. Due to the staggered arrangement, the orientation of HAP crystals has insignificant effect on the biomaterial strength. Analyses based on strength scaling as a function of structural hierarchy level reveal that while peak strength follows a multiscaling relation, the fracture strength does not. The peak strain for failure was found to be independent of the changes in levels of structural hierarchy. Overall, the analyses, being limited in size due to the computational time constraint, point out important correlations between the mechanical strength and chemically influenced structural hierarchy of biomaterials.     

Theoretical development and experimental validation of a thermally dissipative cohesive zone model for dynamic fracture of amorphous polymers

A thermally dissipative cohesive zone model is developed for predicting the temperature increase at the tip of a crack propagating dynamically in a nominally brittle material exhibiting a cohesive-type failure such as crazing. The model assumes that fracture energy supplied to the crack tip region that is in excess of that needed for the creation of new free surfaces during crack advance is converted to heat within the cohesive zone. Bulk dissipation mechanisms, such as plasticity, are not accounted for. Several cohesive traction laws are examined, and the model is then used to make predictions of crack tip heating at various crack propagation speeds in the nominally brittle amorphous polymer PMMA, observed to fail by a crazing-type mechanism. The heating predictions are compared to experimental data where the temperature field surrounding a high speed crack in PMMA was measured. Measurements are made in real time using a multi-point high speed HgCdTe infrared radiation detector array. At the same time as temperature, simultaneous measurement of fracture energy is made by a strain gauge technique, and crack tip speed is monitored through a resistance ladder method. Material strength can be estimated through uniaxial tension tests, thus minimizing the need for parameter fitting in the stress-opening traction law. Excellent agreement between experiments and theory is found for two of the cohesive traction law temperature predictions, but only for the case where a single craze is active during the dynamic fracture of PMMA, i.e. crack tip speed up to approximately 0.2c_R. For higher speed fracture where subsurface damage becomes prominent, the line dissipation model of a cohesive zone is inadequate, and a distributed damage model is needed.     

Steady growth of a crack with a rate and temperature sensitive cohesive zone

The steady-state dynamic propagation of a crack in a heat conducting elastic body is numerically simulated. Specifically, a mode III semi-infinite crack with a nonlinear temperature dependent cohesive zone is assumed to be moving in an unbounded homogeneous linear thermoelastic continuum. The numerical results are obtained via a semi-analytical technique based on complex variables and integral transforms. The relation between the thermo-mechanical properties of the failure zone and the resulting crack growth regime are investigated. The results show that temperature dependent solutions are substantially different from purely mechanical ones in that their existence and stability strongly depends on the cohesive zone thermal properties.     

Why the stress trajectories in the Dean-Hutchinson plastic sector of the growing mode III crack are an unfocused fan

The plastic zone of the growing mode III crack in an elastic perfectly plastic solid consists of two sectors in contact with each other. The sector closer to the crack plane, first studied analytically by Chitaley and McClintock (CM), consists of a fan of straight maximum shear stress trajectories that are focused on the crack tip. The other sector, first analyzed numerically by Dean and Hutchinson (DH), is a ‘radial’ fan of straight lines that are not focused at the crack tip or at another common point. In this paper it is shown with use of the dislocation density field that the need that the stress magnitude in the plastic wake be below the yield stress requires the existence of an unfocused fan in the DH sector. It appears unlikely that this result can be obtained without explicit use of dislocations.     

The effect of indentation sequence on rock breakages: A study based on laboratory and numerical tests

Rock may response differently to external loads applied in different sequences. Thus, we conducted indentation tests to investigate the effect of the indentation sequence on rock breakages. Sequential indentations, consuming less indentation energy, usually resulted in larger and deeper grooves and then led to lower specific energies. Thus, we conclude that sequential indentations occur instead of simultaneous indentations form larger grooves with the same indentation energy. To further validate this conclusion, we performed a series of numerical tests. The numerical analysis of stress evolution shows that, for simultaneous indentations, the propagation of an internal crack from an inner rim restrained the propagation of the other internal crack from the other inner rim. However, the chipping pattern varied for sequential indentations. In the first indentation process, an internal crack, initiating from an inner rim, is usually connected with an internal crack caused by the second indentation. The deflection angles of the internal cracks for the sequential indentations were smaller because of the lower compressive stress in the horizontal direction. Then, these smaller deflection angles led to larger chips.     

A continuum treatment of growth in biological tissue: the coupling of mass transport and mechanics

Growth (and resorption) of biological tissue is formulated in the continuum setting. The treatment is macroscopic, rather than cellular or sub-cellular. Certain assumptions that are central to classical continuum mechanics are revisited, the theory is reformulated, and consequences for balance laws and constitutive relations are deduced. The treatment incorporates multiple species. Sources and fluxes of mass, and terms for momentum and energy transfer between species are introduced to enhance the classical balance laws. The transported species include: (i) a fluid phase, and (ii) the precursors and byproducts of the reactions that create and break down tissue. A notable feature is that the full extent of coupling between mass transport and mechanics emerges from the thermodynamics. Contributions to fluxes from the concentration gradient, chemical potential gradient, stress gradient, body force and inertia have not emerged in a unified fashion from previous formulations of the problem. The present work demonstrates these effects via a physically consistent treatment. The presence of multiple, interacting species requires that the formulation be consistent with mixture theory. This requirement has far-reaching consequences. A preliminary numerical example is included to demonstrate some aspects of the coupled formulation.     

Electrically driven cracks in piezoelectric ceramics: experiments and fracture mechanics analysis

Piezoelectric systems like multilayer actuators are susceptible to damage by crack propagation induced by strain incompatibilities. These can arise under electric fields for example between the electroded and external regions. Such incompatibilities have been realised in thin rectangular model specimens from PZT-piezoelectric ceramics with top and bottom electrodes only close to one edge. Under an electric field, controlled crack propagation has been observed in situ in an optical microscope. The crack paths are reproducible with very high accuracy. Small electrode widths lead to straight cracks with two transitions between stable and unstable crack growth regions, while large electrode widths result in curved cracks with four transitions. Fracture mechanics analysis is able to explain the different crack paths. An iteration method is developed to simulate the curved crack propagation also for strong curvature of the crack paths using the finite element method. The computed crack contours exhibit excellent quantitative agreement with the experiment with respect to their shape, the stages of stable and unstable crack propagation and the transitions between them. Finally, also the crack length as a function of the electric field can be predicted.