## Abstract

Modern high strength and ductile steels are a key element of US Navy ship structural technology. The development of these alloys spurred the development of modern structural integrity analysis methods over the past 70 years. Strength and ductility provided the designers and builders of navy surface ships and submarines with the opportunity to reduce ship structural weight, increase hull stiffness, increase damage resistance, improve construction practices and reduce maintenance costs. This paper reviews how analytical and computational tools, driving simulation methods and experimental techniques, were developed to provide ongoing insights into the material, damage and fracture characteristics of these alloys. The need to understand alloy fracture mechanics provided unique motivations to measure and model performance from structural to microstructural scales. This was done while accounting for the highly nonlinear behaviours of both materials and underlying fracture processes. Theoretical methods, data acquisition strategies, computational simulation and scientific imaging were applied to increasingly smaller scales and complex materials phenomena under deformation. Knowledge gained about fracture resistance was used to meet minimum fracture initiation, crack growth and crack arrest characteristics as part of overall structural integrity considerations.

## 1. Introduction

Ship structural materials have been an important and enabling technology for US Navy ship and submarine design. This is particularly true for high strength and high ductility steels used in hulls and superstructures. Progression from HT steels in World War II to the HY steels in the 1950s and the HSLA steels in the 1980s was driven by the opportunities provided by enhanced properties and lower-cost fabrication practices [1,2]. Within these broad developments, the ‘modern ages’ of both structural integrity strategies and the mechanics of fracture have been closely and continually related.

Ductile fracture in naval steels is a very broad topic. Of particular interest here is to review the interplay of basic and applied research that addressed some of the important issues involving naval steels, fracture and structural prediction over the past 70 years. The evolution of theoretical, analytical and computational tools during this period provided new insights into deformation, material damage and the fracture characteristics of naval alloys.

Three primary themes emerge in approaching computational aspects of fracture in naval steels. First, the strong interplay between theoretical concepts and practical applications is noted. The second theme is the transition from standard material tests (to provide general and baseline qualification of materials) to customized analysis methodologies (to address specific and challenging structural performance issues). Third, the advances derived from computational integration of experiments and simulation methods to enable image-based modelling and quantitative analysis methods in a field where visual observation is so important.

The focus here will be primarily on fracture from quasi-static large strain deformations, microstructural damage leading to fracture initiation and elastic–plastic crack growth. The importance of dynamic and temperature effects will be noted for their importance in motivating a better understanding of microstructural effects. High cycle fatigue, welding residual stress and corrosion are also important to naval structure integrity. Their focus, however, is primarily on small strain deformation and they will not be included in this discussion.

It is important to note that there have been many approaches taken to elastic–plastic fracture parameters over many years. This has created a rich history for the field, for example global energy balances, crack tip singularity parameters, crack opening displacement and crack growth resistance curves [3]. The approaches discussed here highlight some uses of the stress intensity factor as a crack tip parameter and strain energy density as a local measure of fracture resistance. They were motivated by the scientific challenges of thin section ductile deformation and fracture for naval technologies addressed during the 1980s and 1990s. Organizations such as the Ship Structures Committee brought attention to specific problems and potential solutions in this area [4].

The 1970s and 1980s were also important eras for the nuclear power, pressure vessel and general industry sectors. Flaws in base metal and welds were key issues. Work in these areas culminated in the 1990s fracture mechanics principles and fracture acceptance criteria for nuclear power plants in the ASME Boiler and Pressure Vessel Code—Section XI in 1993 [5]. Structural integrity procedures for European industry were addressed by the 1999 SINTAP standard [6]. Welded structure flaw acceptance criteria and methods were also addressed in British Standards PD6493 [7] in 1991 and 7910 [8] in 1999. Common issues continue to be addressed and refined in these standards, including crack shape, specimen size effects on fracture parameters and loading history.

## 2. Nonlinear elastic–plastic fracture and crack tip parameters

The work of George Irwin at the US Naval Research Laboratory (NRL) will always be associated with the science of fracture mechanics, especially for naval steel and aluminium alloys. The start of NRL's fracture research efforts occurred when Irwin was hired in 1937 to head the Ballistics Branch in the Mechanics Division. He quickly became interested in the fracture process through his work on the ballistic response of steel armour plating. There was a distinct specimen size effect in armour materials, where penetration testing of large specimens generally exhibited brittle-like failures but small Charpy and Izod specimens machined directly from the large specimens were ductile. Irwin investigated several causes for this specimen size effect, which eventually led him to develop the theoretical and experimental basis for the new field of fracture mechanics. A few of Irwin's efforts in this field are highlighted here. Two significant compilations [9,10] explain the overall breadth and impact of his research.

In 1948, Irwin published his first seminal paper on the concept of linear elastic fracture mechanics (LEFM) [11]. In LEFM, he equated elastic energy release rate *G* to the areal rate of energy adsorption using the framework of the Griffith theory for the fracture of brittle materials. This allows for the development of the stress intensity factor *K*. In that paper, Irwin was able to derive *K* (denoted as *C* in this paper) for a centrally notched plate. His LEFM research was critical to understand the role of local stress concentrations on the fracture behaviour of materials and that the limiting velocity of fracture in a material can be used to explain crack branching in materials with limited ductility. In the following years, Irwin continued his efforts by extending LEFM to include different states of applied stress and transverse constraint (i.e. plane stress versus plane strain), specimen geometry, a plastic zone correction to the strain energy release rate and the basis for subcritical crack growth through various mechanisms, such as fatigue, stress corrosion cracking and corrosion fatigue [12–15].

Irwin was also integral to the development of fracture surface analysis techniques at NRL [16]. He used these methods to examine limiting crack velocities, crack growth direction and thus identify the fracture initiation sites in failed components. He was known to be interested in the development of ‘pin holes’ ahead of cracks in foil specimens and was able to explain discontinuous fracture through analysis of time under load and the fracture path in the plastic zone.

William Pellini's 1976 book titled *Principles of structural integrity technology* [17] captured the Navy's knowledge base and engineering analysis methods developed in NRL's Metallurgy Division during the post-World War II decades to design ships, aircraft and missiles resistant to fracture. Peter Palermo, who directed ship structural design at the US Naval Sea Systems Command, writes in the book's foreword that Pellini ‘… summariz[es] coherently more than 25 years of groundbreaking research on real problems …’ and that he has ‘… translat[ed] the physical phenomena of fracture and crack growth to a format that facilitates engineering analysis …’.

The success of LEFM applied to relatively brittle materials also raised important questions about design with ductile materials, such as the steels, that are so important to naval ship structures. The interplay of ductile alloy stress–strain characteristics (quantified by stiffness, yield stress, strain hardening and failure strain) and fracture characteristics (associated with the crack stress intensity factors and concept of a local critical value for a material) was increasingly important, as was the recognition that material quality and service conditions were fundamental factors. Alloys and weld quality were governed by alloy and weld composition choices and processing practices.

These factors set the initial conditions for microstructural response to deformation. The subsequent service temperatures, strain states and dynamic loading rates controlled the relative brittle versus ductile responses of the naval steel microstructures. This, in turn, affected fracture initiation, crack growth and the potential for crack arrest and ship damage tolerance. Collectively, the problem was highly nonlinear and beyond a direct theoretical description.

The interplay between fracture toughness and material strength was recognized as one important factor and, in particular, the ratio of these two quantities. This was related to the concept of elastic constraint and the strain state enhancement or inhibition of plastic flow adjacent to the crack tip and in the far-field bulk material. This was increasingly important to the relatively thin section ship structures and any cracks, where many components of the structure were more likely to exhibit ‘low constraint’ affecting crack growth in the base metal, and ‘high constraint’ near welds and structural detail intersections. As a result, a direct method for laboratory determination of a unique fracture toughness parameter or behaviour was problematic for relevant structural dimensions.

The fracture analysis diagram (FAD) was developed to account for the effects of temperature on crack growth, microstructural mechanisms and large-scale plastic deformation. It addressed important and complex design issues above the nil ductility temperature by providing practical information on fracture initiation and the likelihood of sustained crack propagation versus crack arrest in ductile materials. The FAD combined what worked using LEFM with experimental methods and analytical insights to overcome what was not theoretically possible to describe based on theories of micromechanics and ductile fracture. It could also be constructed to account for higher loading rates, which also suppressed ductility.

The FAD was based on developing and applying careful, rigorous and relatively simple experimental test procedures to probe the microstructural effects on crack growth and gather sufficient data on which to base design decisions. The approach is important because it highlighted and accounted for the effects of microstructural fracture mechanisms without being able to model or predict them. Physical testing of small laboratory specimens at different temperatures, crack sizes and applied stress captured the changes in cleavage, quasi-cleavage and microvoid mechanisms and the subsequent effect on unstable brittle fracture and stable ductile fracture.

The FAD was constructed by combining the concepts of nil ductility temperature (NDT), crack arrest temperature (CAT), fracture transition to elastic (FTE) behaviour and fracture transition to plastic (FTP) behaviour, as shown in figure 1. These concepts were important for understanding fracture propagation and designing against it. If crack propagation initiated and the material failed predominantly by cleavage mode of fracture the prospects for crack arrest and limited damage to the ship structure were decreased. Yielding of the structural cross section ahead of the crack, on the other hand, could arrest the crack and contain the structural damage. The FAD generated a practical design space ensuring that structural components loaded to anticipated levels of applied stress could contain cracks up to a critical size without loss of structural integrity over a range of service temperatures. Since Pellini introduced the FAD, many researchers have continued to develop fracture mechanics methods to address specimen size, temperature and constraint effects on the ductile-to-brittle transition region of material response. These methods include the development of an ATSM standard (ASTM E-1921) [18] for determination of *T*_{o}, and the master curve and unified curve analysis techniques, designed for the pressure vessel and nuclear industries [19,20].

## 3. Large strain elastic–plastic material responses for ductile fracture

In the 1980s, finite-element analysis methods were being applied to the design of steel components and forming processes in the shipbuilding industry. As these techniques were developed, they would be applied to a mix of basic and applied research topics, often focused on specific concerns. In relation to Navy steels, a natural progression was towards better characterization of large strain behaviour now needed for the simulation techniques.

The elastic–plastic constitutive algorithms and numerical solution strategies of commercial simulation software were becoming powerful and reliable enough to address problems such as plate bending where large-scale strains of 100% would be produced. Ductile fracture of high-performance steels would also generate large strains near the crack tip.

In both cases accurate yield, hardening and flow parameters for basic elastic–plastic constitutive models were really not available. Analytical methods commonly used to estimate uniaxial strains and stresses from post-yield data of necking ductile uniaxial tension test specimens pre-dated the finite-element methods. Simulations of simple tension tests of HY and HSLA steels using conventionally derived stress–strain curves, however, began to diverge at modest average engineering strains of less than 10%.

NRL's Materials Science and Technology Division began to look carefully at improving the quality of elastic–plastic constitutive parameters. The material stress–strain characterization was approached as an inverse problem using the load–displacement response of multiple different tensile specimen geometries as input. The objective was to determine the stress–strain curve that generated the measured load–deformation responses of a parametric set of tensile specimens [21,22]. HY-100 steel parameters were determined through an interactive sequence of finite-element analyses that converged to the solution for the uniaxial stress–strain curve. The set of tensile specimens featured four different lengths and three different diameters, for a total of 12 length-to-diameter (L/D) aspect ratios that provided a range of deformation constraint during elongation and necking. Photographic and video imaging methods were developed and applied to the tension tests to automate the measurement of specimen deformations and profile shapes. The finite-element simulations replicated each experimental specimen load–deformation response using the HY-100 steel stress–strain solution parameters derived from this procedure.

The simulation results using the stress–strain parameters generated multiaxial stress state, strain state and energy density values from the tensile specimens at the locations where local fracture initiation occurred experimentally. Recognizing the influence of the multiaxial strain states and the observed microstructural features, the strain energy density values associated with higher elastic constraint specimens were tabulated for use in subsequent studies of fracture initiation and crack growth. As one of the first of many subsequent inverse parameter optimization strategies for nonlinear materials, the method was also used to obtain flow and local fracture parameters for HY-80, HSLA-80 and HSLA-100 steels [23,24] (figure 2).

A sensitive indicator of the method's capability was found in the unique and progressive bifurcation of the strain field and necking predicted in the set of HY-100 tensile specimens. Physically, the HY-100 data showed the neck moving from the symmetry plane, in low L/D specimens, to an asymmetric location closer to one of the specimen grip sections, in higher L/D specimens. These observations were guided by the simulations using the optimum parameters and then measured in the experimental data.

## 4. Elastic–plastic fracture of naval steels

Concern about fracture initiation and crack growth also drove nonlinear fracture mechanics in the 1980s and 1990s. The increasing ability of non-destructive evaluation to detect both larger voids and smaller cracks challenged predictive capabilities and structural design guidelines. Ship structural geometries and welded joints required new methods to account for complex geometries, material nonlinearity, voids and cracks.

The increasing use of finite-element methods further highlighted these issues. In ship structural design, the plastic deformation predicted by finite-element analysis was often found to be more prevalent than anticipated by the semi-empirical design guidelines for the same structural detail. This could include plasticity generated from the deformation forming of structural components, or operational loads on complex geometries and welded corners.

Initial investigations at NRL on modelling standard fracture specimens focused on the role of material nonlinearity and the non-uniform development of plasticity across the crack front. The strength and ductility of the HY and HSLA steels generated considerable plasticity and specimen surface deformation, more consistent with lower levels of elastic constraint mentioned previously. This complicated the interpretation of standard fracture toughness testing and its application to cracks in the structural section thicknesses comparable to those used in ships and submarines.

Meshing in the vicinity of the crack tip for all the analyses described below was a balance between the finite-element mesh refinement and the computational resources available. Crack tip meshing these elastic–plastic materials relied on eight-noded reduced integration, hybrid elements. Mesh refinement studies were performed resulting in element sizes less than 1% of the crack length. Non-uniform meshing was used to reduce element density away from the crack tip.

### (a) Predicting the onset of crack growth in compact fracture specimens

A thorough analysis was conducted of a standard compact fracture specimen of HY-100 steel using two-dimensional plane strain and three-dimensional finite-element simulations [25]. The specimen had a thickness of 22.9 mm (0.90 inch), width was 50.8 mm (2.0 inch), and the initial crack depth (i.e. the sum of the initial notch plus fatigue pre-cracking) was nominally 25.4 mm (1.0 inch), equal to one-half the specimen width. The predicted load and displacement values, along with the plastic zones and internal strain energy density values along the crack front, were generated from the simulation results. Physical testing of the compact specimen was conducted in the laboratory with automated data acquisition of load and displacement data, alternating current potential drop data for detecting crack growth, and visual observation of the specimen deformation.

The load versus displacement results from the two-dimensional simulations were in excellent agreement up to approximately 80% of the experimentally observed load at which crack growth began. Beyond this point, the two-dimensional simulation generated a higher load. This was consistent with a plane strain model generating too much constraint.

The three-dimensional simulation was in full agreement with the experimental load–displacement result and within 3.5% of the load at the predicted onset of crack growth. Onset was taken to occur when the strain energy density associated with the higher constraint tension specimens near the crack tip.

The simulation results showed the highest values of strain energy density at the centre of the crack front, in the mid-plane of the specimen. The energy density values along the crack front decreased for points away from the centre towards the free surfaces. This was consistent with the appearance of a curved crack front in the physical specimens once crack growth began.

In addition to the load–displacement response and the onset of local crack growth, the apparent stress intensity factor *K* fracture toughness values from the computational simulation were within 3.5% of the experimental result. Both values were determined in accordance with the ASTM-399 standard [26]. The computational and experimental *J* integral fracture toughness values also agreed within 2.5% of each other. Both these values were determined using ASTM E-813 [27]. Taken together, a more complete understanding of local material, crack tip and global specimen response associated with fracture was obtained using the compact specimen in a low constraint test.

### (b) Simulation of three-point bend fracture specimens

A study was also performed on two different geometries of HY-100 three-point bend fracture specimens [28]. The objective was to determine the effect of reduced specimen thickness with side grooves on enhancing the effective level of constraint along the crack front. One specimen was 0.98 mm (2.50 inch) thick without side grooves. The other specimen was 11.4 mm (0.45 inch) thick with 20% side grooves. Two-dimensional nonlinear finite-element analyses of the specimen were performed for the limiting conditions of plane stress and plane strain. The experimental and plane strain computational load versus crack-mouth-opening displacement (CMOD) curves for both specimens were in excellent agreement at lower loads. The divergence from plane strain occurred at a comparable fraction of load initiating crack growth. Both experimental results were between the bounds generated by the plane stress and plane strain predictions. The thinner specimen with side grooves was closer to plane strain, whereas the thicker specimen without side grooves was closer to plane stress. The side grooves as designed were able to more than compensate for the reduced thickness of the specimen. The computationally predicted CMOD values corresponding to crack growth initiation, using the local HY-100 fracture initiation energy density value, were about 25% less than either measured CMOD value. This again indicated the importance of the three-dimensional fracture characteristics of HY-100 steel in terms of both the continuum and crack at these specimen thicknesses.

### (c) Simulation of crack growth in compact tension fracture specimens

A computational parametric study was also conducted to simulate stable crack growth in a modified HY-100 compact tension specimen geometry using two-dimensional finite-element analyses [29]. The objectives were to numerically model stable crack growth and understand the relative effects of crack length, material strength and material toughness.

Two modified compact tension specimen models were developed, each with a different crack length-to-specimen width ratio of 0.136 and 0.250. The holes in the specimen for pin and clevis loading were deleted, so the specimen was solid in this area. Stress was applied at the top boundary of the specimen along the axis of the deleted holes to load the specimen. This modification avoided a larger model size and additional computational complexity to place pin contact on loading holes.

The HY-100 stress–strain curve and local fracture toughness was used as a baseline material behaviour. The parametric properties were constructed around the HY-100 properties as well as higher and lower values of both the yield stress and the critical strain energy density to produce fracture.

Node release, element removal and model remeshing were options considered to model crack growth. Node release was selected because it was suitable for predicting crack growth in ductile steels, with the necessary simulation practice to ensure solution convergence and stability. Element removal was not selected because it was better suited for materials sustaining more diffuse crack damage over element volumes. Model remeshing was not selected because, at that time, it was more difficult to implement owing to the computational demands of updating each new mesh with the prior material plastic deformation history.

Crack growth was implemented by nodal force release using debonding interface elements to specify the path, but not the time, of growth. This provided a prediction of crack growth based on the local conditions in the vicinity of the crack tip. Nodal release was triggered when the local energy density near the crack tip, averaged to account for the discretization of the model, reached a critical value for HY-100 steel. Nodal release was implemented by reducing the nodal forces at an essentially constant load.

This approach was different from some common practices at the time. These practices modelled crack growth by specifying *a priori* the crack length and the applied forces (or displacements) as boundary conditions based on the experimentally observed values of both. Prescribing the crack length by incrementally releasing nodes ahead of the crack tip was computationally simple to implement. The method cast the crack length as an independent variable rather than as a dependent variable, and therefore did not require a crack growth criterion to determine crack tip position as a dependent variable of the analysis. Defining crack growth from experimental data did facilitate the study of plastic deformation fields, crack tip parameters and global energy balances as the load and crack length increased in the specified manner.

The results of the simulations for HY-100 provided insights into the numerical aspects of implementing crack growth predictive algorithms and the physical influences of crack size, material strength and local fracture toughness. Crack growth of over 60% of the initial crack size was achieved for some parameter combinations, and the discrete crack growth of the node release algorithm was a practical approximation of stable crack growth. The predicted crack growth rates were consistent with the general trends observed for HY-100, which sustained significant crack growth in the compact tension specimen geometry prior to gross plasticity and plastic hinge formation in the area between the crack tip and specimen surface. A direct comparison would have required a three-dimensional crack growth simulation, not practical at that time, to account for the higher crack growth rates near the centre of the specimen and the lagging rates near the surface.

Each node release event produced a partial unloading of the specimen as force equilibrium was re-established for the new crack length. As the crack grew, the magnitude of the local crack tip unloading decreased, as quantified by the local-to-critical strain energy density ratio. This produced conditions moving progressively closer to crack instability with each crack growth increment. For the cases of longer crack and higher material strength, the rate of growth increased as more elastic energy was available in the specimen to drive plastic deformations. The influence of local fracture toughness was more closely related to the interplay between it and the strain hardening of the material, which again influenced the relative elastic energy available in the system and the relative ability of the material to sustain greater plastic deformation prior to fracture.

The results of these three studies demonstrated a range of computational approaches for fracture initiation, crack growth and material property influences on plastic zone size and fracture for relevant specimen thicknesses. Given that the HY steels featured very high strength and ductility, the methods applied and insights obtained from these studies were also directly applicable to other materials which would be easier to model.

## 5. Naval structural integrity analyses

A series of structural design concepts and ship components were analysed to understand new crack formation or existing crack growth. These analyses, conducted during the late 1980s and well into the 1990s, included submarine bulkhead test panel material influences on deformation, ship structure weld flaw criticality, weld metal strength influences on fracture initiation and the grounding characteristics of a double hull ship concept.

Common themes were the complex geometries; analyses enabled by nonlinear finite-element simulation with the potential to reduce design or maintenance costs; utilization of the large strain elastic–plastic local fracture material parameters discussed earlier; and careful decisions in model development to balance physical detail against model size. The results from these analyses demonstrated practical application of finite-element simulation to the structural design and structural integrity processes for naval structures.

It is also noted that the effects of structural fatigue are not explicitly addressed in terms of N-cycle effects on material fracture. Fatigue was captured implicitly when material specimens taken from ship structures were tested to obtain local fracture parameters. A separate line of inquiry in the naval structures community was focused on fatigue loading characterizations, waveforms, cumulative damage and probabilistic analysis methods at that time.

### (a) Submarine bulkhead hydrostatic test panel

A parametric computational study was conducted for a submarine bulkhead design using candidate steels. Bulkhead test panels were used to evaluate design and material resistance to deformation and fracture when subjected to hydrostatic pressure from one side. The test panel was a circular plate with T-stiffeners welded in the pattern of an inscribed square. The panel was welded at its periphery into a hydrostatic loading chamber. Pressure was applied from the reinforced side, deforming the plate away from the T-stiffeners.

Physical panel tests conducted by the Naval Surface Warfare Center-Carderock Division (NSWC-CD) had shown that HSLA-80 steel had produced a desirable and somewhat unexpected result of higher panel deflections at a given pressure load than the HY-80, HY-100 or HSLA-100 steels under consideration. The results for HSLA-80 also exhibited corresponding less deformation near the plate and stiffener intersections. The result meant that more energy was being dissipated by HSLA-80. The objectives of bulkhead test panel simulations conducted by NRL [24] were to understand how the properties of fabricating it with either HY-80, HY-100, HSLA-80 or HSLA-100 steel affected panel deformations; map how plastic flow fields develop in the test panel; identify locations where cracks may initiate; and predict the structural integrity with and without small cracks at the panel and T-stiffener welded intersection.

A three-dimensional finite-element shell model of the panel captured the large-scale panel response, whereas a two-dimensional continuum model was used for the intersection. The simulations were able to understand the role of the more uniform work hardening in the stress–strain response of the HSLA-80 in the panel response. The higher yield stress in the 80 and 100 psi yield steels did produce a stiffer panel response. The more uniform and generally higher strain-hardening slope in the HSLA-80 distributed high plastic strains across a larger volume of the plate and was able to dissipate more plastic energy in resisting the pressure load. Conversely, the rapid decrease in strain hardening beyond 0.20 uniaxial strain for either HY-80, HY-100 or HSLA-100 steel produced more localized plasticity just inside the stiffener boundary. These influences, along with a sufficiently high strain energy density to fracture, also indicated the HSLA-80 bulkhead panel would be comparable to HSLA-100, and exceed that of HY-80 and HY-100, in resisting crack formation or existing crack growth (figure 3).

### (b) Lack-of-penetration weld flaw integrity in ship structures

In the same time period of the late 1980s, opportunities developed to apply these methods to assess the integrity of HT steel ship structures with lack of penetration weld flaws. Two complementary studies were conducted using finite-element simulations on similar flaw geometries found in surface ship hull T-stiffeners. These flaws were internal, three-dimensional, macroscopic, high aspect ratio voids in the butt welds joining the flanges of the T-stiffeners.

The flaws had not been detected during fabrication when revision of the welds would have been more practical. Operational experience of the ships had not produced widespread incidents of crack formation or crack propagation. So, the discovery of the flaws was considered somewhat of a surprise considering their size.

Initial estimates of flaw criticality were made using simpler analytical approaches based on application of fracture mechanics to conservative idealizations of the flaws. The projected area of the three-dimensional flaw was used to create an ‘equivalent crack’. Values of *K*_{IC} and *J*_{IC} for the base and weld materials were used in the fracture mechanics calculations. They predicted that crack growth would initiate at much lower than expected stress levels, corresponding to the service loads, which was not observed.

The cost of repairing existing flaws in an operational ship structure was very high. The ship had to be removed from service and the interior spaces cleared of equipment in order to gain access to the welds. It was an important factor motivating the use of finite-element methods and the accurate stress–strain curves for an elastic–plastic analysis.

In the first of two studies conducted by NRL for the Naval Sea Systems Command [30], two T-stiffener specimens, one with a weld flaw and one without a weld flaw, were removed from a ship for physical tension testing. They also provided the dimensions for a three-dimensional finite-element model of the T-stiffener with the flaw. Some smaller material samples for tensile specimens were also removed.

The weld flaw in the T-section was modelled as a long internal ‘slot’ 0.100 inch high, 0.025 inch wide and running through the 2.0 inch long transverse (LT) weld joining the two T-frame flange sections. The net section thickness around the flaw was made comparable to the base metal plate thickness, owing to local weld crown asymmetry diagonally across the flaw, and justified a decision to omit the weld crown. The model geometry was then symmetric across both transverse and longitudinal symmetry planes, which reduced the model size by a factor of four.

HT base metal and associated weld metal stress–strain curves and local fracture toughness values were developed using the flat tensile specimens from higher-quality welds. The specimens featured a serial composition of base–weld–base metal along its length, without any large weld flaws. Tensile testing produced a range of results in specimen load–displacement and fracture locations, probably owing to smaller scattered voids, but the dominant trend was that the specimens that failed in the base metal had markedly higher reduction in area than the specimens that failed in the weld metal.

The stress–strain curves and local fracture toughness values for both the base metal and weld metal were determined by adapting the procedures discussed earlier. The weld material overmatched the base metal in yield stress, by about 20%, and was significantly less ductile compared with the base metal, by a factor of 10. Local critical strain energy density at fracture in the weld metal was about 7% of the base metal value. No account for a heat-affected zone in the base metal was made for these analyses.

The T-stiffener finite-element model was subjected to a longitudinal tension load applied at the boundary. The maximum energy density, relative to the critical material value for fracture, occurred in the weld material. The location of this point was on the periphery of the flaw at approximately 80% of the distance from the flange centre to the flange edge. The value increased monotonically to the predicted load of fracture initiation.

The physical testing of the two T-stiffener sections, one with and one without the weld flaws, was done after the finite-element analysis of the section with the weld flaw was performed. The tests were used to compare the load–time plots and identify when the flaw began to fracture and exhibit crack growth based on the onset of load differences between the two plots. Fractography was used to identify the location of fracture initiation. The predicted fracture initiation load was within 5% of the measured load, and the predicted location agreed favourably with the fractographic assessment.

A supplementary two-dimensional finite-element analysis was performed to understand how the presence of a weld crown would affect the T-stiffener tension results [31]. A butt-welded HT plate was modelled featuring a weld crown on both sides of the weld material and an interior weld flaw. For the early stages of tensile loading, the strain energy density relative to the weld metal critical value was initially higher at the flaw tip compared with the corresponding relative maximum in the base metal near the base and weld metal boundary.

As the tensile loading progressed, the base metal cross section yielded, which resulted in unloading of the of the stronger weld material with the wider net cross section. Continued loading increased the maximum relative strain energy to the point where the fracture initiation location shifted from the weld metal to an area near the weld metal and base metal interface. This change in location was a direct result of the weld and base metal nonlinear material responses and the presence of the weld crown. This analysis also demonstrated that the defect was a less critical feature of this system and the base metal would in fact govern fracture initiation.

The analyses provided excellent insights into the role of these weld flaws in fracture initiation and quantitative input into the ship maintenance decision process. They were based on the material behaviours and fracture properties, and predicted weld flaw fracture initiation without an *a priori* specification of location.

In the second study of weld flaws, a three-dimensional finite-element analysis was conducted for similar weld-defect geometries in a T-stiffener flange reinforcing a section of hull plate [32]. The hull plate was subjected to transverse pressure loading. The stiffener flange was 4.0 inches wide, the web was 10.0 inches tall and both sections were 0.2 inches thick. The weld with symmetric weld crown geometry was a maximum of 0.4 inches thick. The weld defect was a slot shape 0.05 inches in width, 0.10 inches in height and ran the width of the flange. The hull was 0.375 inches thick.

To analyse this component, a preliminary analysis was performed using a more economical shell model of the hull plate and stiffener without the weld and defect. The hull was loaded by the pressure, and the results of this model used to establish local displacement loading conditions for the three-dimensional T-stiffener model.

The analysis showed that for the bending generated by the pressure loading, constraint at the intersection of the flange and the web produced the maximum stain energy density prediction of fracture initiation at the outer edge of the weld flaw just above the web–flange intersection. This location was different from that in the first study of the T-stiffener under tension, where fracture initiation occurred at the flaw edge in the flange, at some distance away from the web. Again, the analysis provided data for ship maintenance decisions.

### (c) Parametric simulations of over-, even- and under-matched weld designs

The standard practice in weld design is to use a weld material that features even- or over-matched yield stress when compared with the base metal. The technical challenge and cost of developing higher strength weld materials to keep pace with the increases in base metal yield strength were considerable. The occurrence of fabrication-related cracking with even- or over-matched base metal was a major consideration. Cold cracking could be avoided by pre-heating the base metal. In the late 1980s, the option of using undermatched yield strength weld material was under consideration for use with the higher strength Navy steels to avoid the cost of pre-heating.

Elastic–plastic finite-element analysis was again an excellent tool to evaluate the influence of weld geometry, weld material yield stress and applied load on a weld system. General trends were established in lieu of an experimental test programme for the Naval Sea Systems Command. Parametric two-dimensional computational studies were conducted for double-V welds joining two plates of HY-100 base metal [33]. A four region heat-affected zone, with progressively higher yield stress and lower critical strain energy value for local fracture initiation, was included in the base metal on each side of the weld. The heat-affected zone parameters were maintained for all the analyses. The weld geometry parameter was the presence or absence of a weld crown on each side of the weld. The welded plate was subjected to two loading conditions, one tensile and one bending load.

Representative parametric material parameter sets were developed for the weld metal analyses, constructed from representative standard and candidate weld designs. The weld metal yield stress was even with the base metal, overmatched by 18% or undermatched by 18%. All three weld materials featured similar strain-hardening elastic–plastic flow curves beyond the yield point. The strain energy density to initiate fracture was assumed to have a constant value, generating the lower ductilities observed at higher strengths.

The computational simulations were conducted to determine the load–displacement response of the weld, the location of maximum local strain energy density in each of the different material parameter regions of the model, and the predicted fracture initiation point of the weld where the local maximum strain energy reached the critical value for that material.

The results from the complete set of 12 analyses showed that system nonlinearities would make the *a priori* prediction of the fracture initiation location in each case difficult. General trends showed that initiation shifted from interior to exterior surface locations and from the weld material to the heat-affected zones when (i) the weld metal strength increased, (ii) the load changed from tension to bending, and (iii) the crown was added to the weld profile. The effect of weld strength was most significant for tensile loads, whereas the effect of weld crown was most significant for bending loads.

### (d) Simulation of a double hull ship grounding

In the early 1990s, interest in double hull designs returned owing to the Exxon Valdez oil spill and subsequent legislation, with the damage resistance of the inner hull of paramount importance. The advanced double hull (ADH) concept for naval and commercial ships, developed at the NSWC-CD, featured intersecting transverse and longitudinal webs between inner and outer hulls spaced 79 inches apart. A quarter-scale structural test model to support the overall ADH programme was designed to evaluate the double hull design performance in stranding and grounding events [34]. The distance between the inner and outer hull in the quarter-scale model was 18.5 inches. A conical penetrator was used to simulate an object at the bottom of a harbour or shallow waterway. The quarter-scale model simulated the stranding event by using the penetrator to deform the hull structure.

A computational model was developed and analysed by NRL, whereas NSWC-CD fabricated the physical model for testing at the National Institute of Standards and Technology [35,36]. The primary objectives were to understand the deformation of the double hull structure, the ability of the web components to collapse and absorb energy, and the likelihood of inner hull fracture. It was assumed that the outer hull would probably be breached during such an event. The A36 steel was characterized by tensile specimen tests and computational simulations to optimize the elastic–plastic parameters for large strains and fracture. The combination of large strain plastic flow, penetrator-to-hull contact, structure surface-to-surface deforming contact, and fracture of thin section components required accurate consideration of material behaviour, complex structural response and fracture in one large analysis.

The analysis and test showed that the penetrator deformed the hull section globally and breached the outer hull locally. Complex collapse of the longitudinal and transverse stiffeners followed, effectively protecting the inner hull. The predicted and measured load and displacement at outer hull fracture were in good agreement. The subsequent loads were predicted to be higher in the analysis than in the test, but these differences were traced to boundary conditions and welds. Deformations of the test model were greater owing to the higher compliance of its bolted attachments to the test fixtures, compared with the smaller deformations generated by ideal boundary conditions in the model. Localized weld failures at the longitudinal webs developed in the test model, whereas the finite-element shell model did not have the resolution to account for welds.

Both these differences were valuable in highlighting the importance of ADH design parameters and fabrication quality. In the computational analysis, the inner hull was not predicted to fracture, meeting the design objective. The test inner hull did fracture owing to the partial loss of the longitudinal webs, which prematurely initiated penetrator contact with the inner hull and produced localized deformation. The ability of advanced simulation techniques and structural integrity methods to add this type of insight and value to the design and testing process was clear (figure 4).

These analyses of structural components, structural concepts and weld details demonstrated practical application of finite-element simulations to high strength and high ductility naval steels in the structural integrity and structural design processes. The simulations were used by the Navy ship structures community in combination with existing standards, engineering expertise and cost–benefit considerations to enhance their decision process with new tools.

## 6. Image-based modelling of material damage and microstructures

By the late 1980s, a major challenge to modelling material damage and predicting fracture in naval steels was the geometric complexity of the microstructural features associated with microvoid initiation, propagation and coalescence. Complex arrays of microvoids could be seen in cross sections of ductile tensile specimens loaded to within a few percentage of fracture surface initiation. Cleavage, quasi-cleavage and microvoids were relatively easy to identify on a two-dimensional fracture surface. Predicting the evolution of actual microdamage to understand how it modulated the evolution of local fracture would ultimately require comprehensive three-dimensional visual imaging of material volumes, accurate geometrical models and quantitative simulation methods able to predict and track damage evolution and coalescence leading to fracture initiation and crack growth.

### (a) Microdamage evolution in ductile fracture

Many naval steels fail by the accumulation of damage through the process of ductile fracture. It is well known that ductile fracture consists of three stages that occur simultaneously: (i) void nucleation, (ii) void growth, and (iii) void coalescence. By inhibiting any of these three stages, the ductility and strength of these alloys can be improved. The process of void nucleation is fairly well understood by both experiment and theory [37–41]. Void nucleation usually occurs at second phase precipitates or inclusions. In HY-80 and HY-100 naval steels, voids are nucleated with little or no loading at the large MnS particles that occur in large stringers that form during processing. Thus, for these steels, most of the ductility occurs under the void growth and coalescence stages. The process of strain-induced void growth has been studied both experimentally and computationally for isolated voids [39,42–45].

As detailed below, NRL has studied the role of void interactions effects when the multiple voids are closely spaced to better understand the ductile fracture process. Of the three stages of the ductile fracture process, void coalescence is the least understood. Many studies of the void linking process use regular arrays of holes (in two dimensions) or voids (in three dimensions) [43,46]. It has been shown that these void geometries overestimate the strains needed for voids to coalesce [47]. In order to examine the role of void spacing and clustering, NRL has developed experimental and computational tools to measure these parameters and use them to generate accurate finite-element meshes of actual microstructures for predictive models in two and three dimensions.

NRL performed initial two-dimensional experiments and computational modelling to determine the role of geometrical perturbations, in the form of pairs of through-thickness holes, in the deformation localization behaviour of sheet specimens [48]. The effect of loading path was examined considering both uniaxial and equal biaxial tension to initiate localized necking in the ligament between the hole pairs. While the experiments were performed on sheet specimens of 3003 aluminium consisting of either 1, 2 or 3 hole diameter spacings, the results provided insights applicable to ductile steel alloys. Additionally, finite-element models of the hole pairs were used to examine the influence of matrix strain hardening on ligament localization. It is noted that, while the hole interactions in two dimensions are more pronounced than void interactions in three dimensions, the results relate to forms of void linking which occur owing to plastic flow localization. In the two-dimensional study, the strain at the onset of localized necking in the ligament between the holes was shown to increase with increasing hole spacing and strain-hardening exponent, as well as when the biaxiality of the loading path increases. The ligament localization event was also found to follow the Hill criterion, if the local strain states are properly accounted for in the analysis.

It is well known that the void linking process usually initiates in regions of local high void content and clustering. However, many continuum localization models are based on unit cells, which assume some form of periodic void arrays, and over predict the applied strains for void coalescence. In order to better understand the role of void clustering, NRL continued the two-dimensional analysis of hole linking using randomly spaced arrays of multiple holes in sheet materials tested under uniaxial or equal biaxial tension [49]. The results indicate that hole linking owing to flow localization is dependent on stress state in a number of ways. As mentioned above, the localization strain to initiate hole linking increases as the biaxiality of the applied stress state is increased. However, when the specimens contain multiple holes, the linking process is accelerated under stress biaxiality, because the flow localization between the holes is dependent solely on hole spacing. Under uniaxial tension, the flow localization between holes is dependent on both the hole spacing and the directionality of the linking path. As a consequence, it was determined that the flow localization process is more gradual with considerable damage accumulation prior to specimen fracture.

As mentioned above, voids are thought to nucleate at MnS inclusions at little to no strain in high strength, low alloy steels such as HY-100. Thus, NRL measured local MnS inclusion densities on polished sections in three orthogonal directions, the rolling direction (RD), the LT and the short transverse (ST) [50]. While the RD and LT sections were found to contain uniformly distributed inclusion arrays, the ST data exhibited banding of the inclusions. Thus, it was determined that actual inclusion/void distributions with different levels of clustering were needed for accurate predictive models of void linking in these types of ductile steels.

An examination of subcritical and pre-critical voids in HY-100 using X-ray computed microtomography (XCMT) was performed [51]. D-notch tensile specimens have been tested by Penn State University in the LT orientation to fracture and to two levels of strain prior to fracture. The levels of strain were approximately 80% and 85% of the strain to failure. At the lower strain, banded clusters of voids elongated in the RD are present. This agrees with the two-dimensional observations of banded MnS inclusions by metallography. The higher strain specimen shows a remarkable growth in the level of damage. The growth initially occurs along the RD followed by void sheeting perpendicular to the RD, as seen in figure 5*a*. Voids inside the intense strain localization band grow faster than the voids outside the localization. Using the XCMT data with image-based computational models of actual void geometries in three dimensions, a new void linking model for HY-100 was determined. First, the primary long stringer MnS inclusions nucleate cigar-shaped voids with little or no applied strain. Void growth occurs in the elongated direction through a field of smaller inclusions. Critical groups of elongated voids coalesce into bands through flow localization ahead of the crack tip, as shown in figure 5*b*. Large bands then combine into large cracks using a void sheeting mechanism and small MnS and carbide inclusions. Figure 5*c* shows the results from a finite-element simulation of two three-dimensional voids under high triaxial conditions, similar to the above experiments. The surface contour represents the elements that have a plastic equivalent strain of 0.50. The similarity of the contour to the linking voids in figure 5*b* suggests that the void linking criterion may have the form of a strain to failure.

### (b) Image-based simulation of microvoid arrays

An initial foray into image-based modelling began with videography of a soft elastomer with submillimetre-scale voids [52,53]. A cross section of the elastomer featured a complex array of voids that intersected with the free surface. The surface of the bulk sample was imaged as it was subjected to increasing uniaxial compressive deformation. The resulting video images clearly showed deformation localization and collapse of the voids at the cross section. Two-dimensional finite-element models of the cross section were constructed from an image of the undeformed cross section. Nonlinear elastic simulations of compressive loading produced void deformation and collapse very similar to that observed in the video, even while recognizing the difference between the three-dimensional physical material and the two-dimensional model.

Because it was shown that the ductile fracture of HY-100 steel occurs by a void sheet mechanism influenced by the clustering of MnS inclusions, NRL employed image-based models to examine the deformation localization behaviour within void arrays based on experimentally measured inclusion microstructures [54]. Treating the elongated voids as through thickness holes in arrays of over 100 holes determined from standard micrographs in two-dimensional models of flow localization, critical features such as void size, shape, spacing and clustering have been evaluated. Examples of the computer models are shown in figure 6. As seen in figure 6, deformation localization across the specimen was especially sensitive to the presence of a few large voids that were spaced within the bands of inclusions noted above. The void linking process was enhanced by voids pairs oriented at 45°±15°. In cases where multiple large voids were not present near this orientation, smaller voids coalesced in order to generate the required stress conditions for void sheeting. The results also showed that deformation localized more readily in specimens tested under multiaxial applied strain. Within microstructures consisting solely of small voids, high density clusters can cause intense strain localization between closely spaced voids, but it is normally confined to occur within the scale of the cluster itself. The models with actual void microstructures were also compared with regular, periodic arrays of voids. As expected, the periodic arrays greatly increased the applied strain required for flow localization across the specimen.

NRL also developed novel cellular automaton methods to rapidly quantify initial microstructural information and simulate the evolution of material damage [55,56]. These image-quality simulations were focused on mesoscale systems of microvoids and microcracks that span a wide range of engineering materials. The use of cellular automaton methods produced rapid simulations that cover the full range of damage morphologies. The model is based on an expanding representative volume element (RVE) in a cellular domain. The state of a cell, which can be either solid material or void, is determined by mapping existing voids through plastic convection and creating new damage through the cellular automaton. The cellular automaton formulation addresses the effect of damage initiation, propagation and coalescence through the iterative evaluation of individual cells based on that cell's current state and the current state of the eight neighbouring cells. The amount of damage in each time step is controlled by conservation of mass in the expanding RVE. By modifying the ratio of damage to plastic convection and the probabilities in the cellular automaton rule table, the model can generate microcrack or microvoid damage morphologies exhibiting different spatial scales. Very rapid simulations are achieved using this formulation. Thus, a wide range of material damage morphologies can be rapidly modelled in a single simulation architecture based solely on local cell geometry.

### (c) Macromodel analogues for insight into three-dimensional microvoid interactions

In an initial study on the effect of void–void interactions on the mechanisms of intervoid ligament failure in three dimensions, NRL developed simple cylindrical specimens with blind macroscale holes to model the complex void interactions [57]. The goal of the study was to determine the co-planar cavity arrangements that reproduce the observed ligament failure mechanisms in ductile materials. In this study, cold-rolled 1018 steel bar stock was used. The specimens contained blind radial spherical holes with the spacing between the holes equal to the spherical cavity radius. Two of the specimen geometries produced the typical ligament necking found during the ductile void coalescence process. The first specimen contained three deep blind holes spaced 120° apart with three shallow holes placed in the middle ligament between the deep hole pairs. This specimen was effective in generating the Y-shaped intervoid ligaments observed between microvoids in ductile materials. The second specimen consisted of six radial holes with a central longitudinal hole. This specimen produced similar void–void interactions with annular symmetry. The global material response was similar for both specimens. It was noted that the amount of ligament material plasticity was extremely dependent on the number of interacting voids and that three voids were required to set up flow instabilities within the intervoid ligaments in three dimensions.

### (d) Three-dimensional materials science for naval steels

As part of the Office of Naval Research's ‘Naval Steels-by-Design’ programme during 2000–2005, researchers at NRL aimed to develop a computationally based methodology to assist in the design of advanced naval steels [58]. An additional focus of this research was to integrate these computational methods with advanced experimental measurement techniques. Both the computational and experimental tools ranged across length and time scales. Relevant information about material microstructures was incorporated into simulations, from first-principles models at the atomistic length scale to continuum response at the mesoscale. The first-principles electronic structure calculations employing density functional theory were combined with Monte Carlo and molecular dynamic simulations to query the magnetic and atomistic structure of iron and nickel alloys. Accurate angular-dependent potentials were developed for calculations of grain boundary structure, energy and element segregation.

During this programme, NRL also developed three-dimensional reconstruction techniques to characterize the microstructure, including grain and grain boundary statistics, three-dimensional boundary plane analysis (i.e. frequency of special grain boundary character), compositional analysis of second-phase precipitates and matrix phase, and three-dimensional morphology, connectivity and crystallography of grains. Figure 7*a* shows the application of these techniques to the three-dimensional reconstruction of an AL6XN austenitic stainless steel microstructure using serial sectioning [59]. With the three-dimensional reconstructions, image-based finite-element modelling was performed to identify critical microstructural features in the mechanical response of the alloy. NRL developed mesh generation techniques to use the experimental data reconstructions as direct input into models [60]. Particular emphasis was placed on the identification of microstructural features that initiate localized plasticity under a range of applied loading conditions from uniaxial tension to biaxial tension and simple shearing, as shown in figure 7*b*. Unique visualization techniques were developed to highlight the regions of the microstructure where failure and fracture will initiate.

## 7. Future challenges and trends

The progress in ductile fracture mechanics, as discussed here with a focus on naval relevant alloys and demonstrative applications, highlighted important changes in experimental data acquisition strategies, computational methods, material parameter characterization and structural analysis complexity. Current materials science and technology efforts should impact future understanding of ductile fracture and the improvement of ductile naval steels. These efforts include

— Recent advances in three-dimensional materials science, such as faster imaging, faster data acquisition and more efficient image segmentation algorithms, enabling larger and more accurate three-dimensional microstructural reconstructions.

— Advanced application of destructive (e.g. mechanical polishing and focused ion beam serial sectioning) and non-destructive (e.g. conventional and synchrotron tomography) imaging techniques will aid in

*ex situ*and*in situ*measurements of fracture initiation processes and crack growth evolution.— High resolution flash X-ray imaging of dynamic material damage and fracture in extreme environments will help to ensure ballistic and shock resistance.

— Progress in the computational modelling of fracture and fatigue over the past 25 years, first with the advent of cohesive zone theory [52] and more recently with the extended finite-element method [53], along with other finite-element techniques will improve the basic understanding of fracture processes.

— Computational modelling of fracture and fatigue using statistical techniques will address outlier phenomena, operating from atomistic to bulk scales and governed by fundamental fracture and fatigue mechanisms. This will build on the recent statistical methods used to design materials for stiffness and strength based on processing–microstructure–property linkages in material knowledge systems [54].

Progress in these areas will enable new understanding of persistent problems such as the evolution of stress corrosion cracks from corrosion pits, another important and costly failure mechanism for naval ship structures. Scientific challenges include determining the underlying causes with advanced detection, measurement and prediction capabilities. Integrating new experimental and computational methods, such as those described above, should produce a more comprehensive understanding of localized corrosion, pit formation, the transition of growing pits to short stress corrosion cracks and the fracture processes of naval materials in the near future.

## 8. Summary

The evolution of fracture mechanics for naval steels was driven by the need to use high strength, high ductility and high fracture toughness alloys (e.g. HT, HY and HSLA steels) in naval ships designed using sound principles of structural integrity. The operational requirements for successive classes of ships and submarines required better understanding of material behaviour and continuous advancement of fracture mechanics to meet this goal.

The challenge of exploiting small-scale or large-scale plastic strains in relatively thin hull, stiffener and bulkhead structural components (from the fracture toughness versus yield stress perspective) was met only through the developments made in nonlinear fracture mechanics. Methods were needed to predict and avoid new crack formation, minimize existing crack growth and instability, and rely on large-scale plastic flow to create damage tolerant structures.

Over the past 70 years, progress was made through successive advancements in experimental observations, theoretical approaches and the application of computational methods to both. At the more macrocrack level, the theoretical constructs of elastic stress intensity factors preceded the construction of the FAD to conceptually account for how plasticity, temperature and loading rate can influence design. Later, computational simulations integrated elastic–plastic analysis to predict deformation, fracture initiation locations and crack growth in complex structural geometries for combinations of materials. This was particularly important for welding, where the weld metal, heat-affected zones and base metal combined materials of different properties and shifted the location of maximum stress, strain and energy density points in each material.

At the more microlevel, methods were needed to account for the inherent complexity of the microstructural material deformation and damage mechanisms underlying fracture initiation and crack growth. The experimental observations of cleavage, quasi-cleavage and microvoids associated with crack growth preceded the experimental quantification and imaging of void arrays to support three-dimensional micromechanics. This was followed by translation of these images into computational models for simulation of deformation localization and better understanding of how arrays modulate damage initiation, propagation and coalescence into fracture.

The rate of progress was generally not as uniform or as synchronized as can be imagined looking back from today's perspective. The interplay between micro- and macroscales, and the sensitivity to loading rates and temperature, still pose challenges. The recent rapid acquisition of detailed microstructural data on polycrystalline material (e.g. grain composition, geometry, connectivity and crystallographic orientation) will provide the opportunity to simulate the underlying phenomena responsible for fracture leading back to its atomic origins.

## Acknowledgements

The authors thank Drs Richard Everett, Virginia DeGiorgi and Siddiq Qidwai of the Naval Research Laboratory for their helpful discussions.

## Footnotes

One contribution of 19 to a theme issue ‘Fracturing across the multi-scales of diverse materials’.

- © 2015 The Author(s) Published by the Royal Society. All rights reserved.