Symposium 4: Brittle Fracture: 100 years After Publication of Griffith’s Theory

This paper provides an overview of recent progress in probabilistic modeling of cleavage fracture phrased in terms of a local approach to fracture (LAF) and the Weibull stress concept. Emphasis is placed on the incorporation of plastic strain effects into the probabilistic framework by approaching the strong influence of constraint variations on (macroscopic) cleavage fracture toughness in terms of the number of eligible Griffith-like microcracks which effectively control unstable crack propagation by cleavage. Some recent results based on a modified Weibull stress model to predict specimen geometry effects on Jc-values for pressure vessel grade steels are summarized in connection with an engineering procedure to calibrate the Weibull stress parameters. These results are compared against corresponding fracture toughness predictions derived from application of the standard Beremin model. Finally, the robustness of LAF methodologies, including specifically the Weibull stress approach, is critically examined along with a discussion of key issues and challenges related to engineering applications in fracture assessments of structural components.
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The presentation will briefly review the history of the development of fracture mechanics from 1921 to the present, including the evolution of its basic concept. Arguments will be made that Griffith's basic concept, properly implemented in the context of modern non-equilibrium thermodynamics, remains valid.
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The scale- or the size-dependence of mechanical strength in many brittle materials appears to follow a ‘universal law,’ of the form: strength proportional as:L^-n or V^-n, where n is a number, L is the length and V is the volume of the specimen or structure. Broadly known as the “size-effect” in geology, civil engineering, mining and materials science, this behavior determines the strength of large structures such as ice sheets, rock formations, coal pillars in mines and concrete beams and columns in civil infrastructure. As of now, there is no reliable scientific basis or theory to explain the size effect or for determining a reliable value of ‘n’. This has been the missing link in strength of materials for nearly a century since the Griffith’s crack theory Here, we show that the change in net-section strain energy, due an initial crack in a structure, and its dissipation within a crack layer of finite thickness, leads to the necessary and sufficient physical basis to explain the size-dependence of strength as L^-0.5. Further, size-independence of strength is explained simultaneously when the crack layer volume approaches the specimen volume.
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This study proposes a new fracture model to correlate cleavage fracture toughness with microstructure of steel having bainitic structure with/without M-A constituent based on the Local Approach. In this model, a new fracture parameter to predict fracture toughness is derived through the proposal of microstructural characteristic of the material to control fracture toughness and on the basis of weakest link theory assumed Griffith crack. The material properties required for applying the fracture model are microstructural properties, those are 1) representative volume and 2) maximum size distribution of micro-crack nuclei, 3) mechanical properties and 4) effective energy release rate of matrix material. The applicability of the theoretical fracture model is demonstrated by experiments for upper bainitic steel with different microstructural morphology. This model can correlate materials properties which are microstructural and mechanical properties with fracture toughness.
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Fracture of brittle solids is ultimately executed by atomistic-scale, discrete, and ultrafast bond-breaking mechanisms along the crack path. Here, we show new fracture behavior and properties of brittle materials, based on macroscopic fracture cleavage experiments of silicon crystal specimens and atomistic-scale semi-empirical model for bond-breaking mechanisms along the curved crack front, to relate micro to macro in fracture.
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In this talk, we will describe damage accumulation and failure of free-standing micro-cantilevers made of 7YSZ coatings during cyclic bending in a nanoindentation system in both, the as-sprayed condition as well as after low-temperature thermal cycling up to 700 oC while attached to the substrate. The technique has been established as a means of tracking elastic modulus, hysteresis/creep, and fracture behavior as a function of coating densification during isothermal treatment at high temperatures. In contrast, low-temperature thermal cycling is designed to simulate operating conditions during which crack healing and sintering, which are known to lead to stiffening, are minimal. The load-displacement curves typically display hysteretic behavior with an increasing permanent residual displacement (ratcheting) after each cycle which increases with an increase in load, accompanied by a reduction in stiffness that is characteristic of damage accumulation. Failure appears to result from the formation of macrocracks after a critical amount of ratcheting. The number of mechanical cycles to failure reduces with the number of prior thermal cycles and with increasing maximum load/stress. Thus, mechanical cycling can act as a proxy for thermal cycling in evaluating progressive damage accumulation in TBCs.
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This study proposes the new fracture model to assess the fracture toughness under complex loading mode subjected to cracked component on the brittle fracture toughness assuming combined stress state in plastic zone near crack-tip. This model newly considers non-linear energy release rate named Local-J as the elastic-plastic local fracture driving force for micro-crack nucleus in plastic zone. The effect of 3-dimentioinal combined stress state on local-J, which is different from the effect on the linear elastic energy release rate for Griffith crack, is formulated as the Local-J equivalent stress by conducting numerical analysis of unit-cell including a penny-shaped crack. Based on weakest link theory assuming this new model under combined stress field, Extended Weibull stress is derived as a new fracture parameter for cracked component. The characteristics of the proposal model is examined by predicting the critical load for pure mode II or III from fracture toughness assumed under pure mode I load. Fracture toughness assessed by this new model under mode II or III load is smaller than that assessed by conventional model. This result of numerical analysis implies the possibility of rational assessment of the effect of loading mode by applying the new model.
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Brittle failure by transgranular and intergranular mechanisms is commonly addressed by probabilistic methods based on the weakest-link concept. For homogeneous materials this approach is straightforward and well established. Different methods have been proposed in the past to incorporate the presence of heterogeneities, e.g. due to welding or segregated zones. A key issue in this context is the length that characterizes variations in the heterogeneous microstructure in relation to a representative size of the zone where brittle fracture typically has been observed to occure, i.e., fracture process zone (FPZ). Here, a new approach for weakest-link modelling of heterogeneous materials is proposed that accounts for the interplay between the different scales.
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Charpy impact tests are used in the nuclear industry to certify forging processes. However, the results of these tests may exhibit a strong variability in the context of large metal parts manufactured by Framatome. Preliminary studies have shown that the steel is highly heterogeneous at the millimeter scale in certain areas of forged parts. These heterogeneities are surmised to be the main cause of the variability observed in the results of impact tests. The aim of this study is to qualify and numerically quantify the effect of these heterogeneities on the distribution of fracture energies thanks to an innovative computational approach featuring deep learning to generate 3D realizations of the mechanical properties from sparse experimental results, and high-fidelity modeling of brittle fracture in heterogeneous Charpy specimens.
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A computational strategy to evaluate the Weibull stress for the Beremin model
is proposed to simultaneously solve problems caused by volumic locking and extreme element
distortion at the crack tip. It is based on the use of mixed elements and remeshing. It is shown that
a single simulation can be used to evaluate the Weibull stress for any range for the CTOD at failure.
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Ductile-To-Brittle transition modeling for ferritic steels used in the nuclear industry has been studied for years. This paper proposes a two-step coupled modeling representing ductile crack growth thanks to a gradient-enhanced GTN model and applying a modified Beremin model to evaluate the probability of failure of CT specimens at -50°C.
Both models have been calibrated separately at temperatures where there is no coupling. Beremin model is first studied at low temperature. It is shown that the model could be applied to different geometries. GTN model is then calibtrated at -20°C. The identified GTN model could be transferred to lower temperatures. Finally the coupling is studied.This work emphasizes the necessity of a special treatment of the stress field resulting from the GTN model to compute the Weibull stress and the use of a modified Beremin model accounting for the void volume fraction.
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The dynamic fracture of a brittle ceramic, B4C, is investigated using our in-house split-Hopkinson pressure bar (SHPB), and Sandia’s peridynamics simulation code, PERIDIGM. In order to study the dynamics of this particular SHPB, the initial boundary value problem (IBVP) is solved for a 1-D impact in which a finite striker bar collides with the front face of a stationary incident bar bonded to a specimen of finite thickness, with the back face of the specimen bonded to another finite transmission bar; this is the classic SHPB experiment. Laplace transform domain solutions are numerically inverted to the time domain using a modified Dubner-Abate-Crump algorithm. The new IBVP solutions for particle velocity in the SHPB composed of maraging steel bars, and B4C specimen, are used to verify the commercial FE codes COMSOL and ABAQUS, and PERIDIGM. Subscale SHPB simulations are conducted using PERIDIGM on jacketed/unjacketed B4C specimens with a critical stretch failure condition proportional to the ceramic’s critical energy release rate, Gc; also investigated is the effect of initial defect populations governed by, Weibull, uniform random, and Bobaru critical stretch distributions, on the ceramic failure behavior. Computationally expensive full-scale PERIDIGM simulations are also currently underway to compare with the subscale simulation results.
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The new generation of advanced high strength steels for oil & gas transportation exhibit better mechanical response and fracture toughness not only in corrosive media, but also in arctic environments. In particular, under these latter conditions, X65 Q&T pipeline steels do not reveal a clear ductile-to-brittle transition (DBT) temperature and, in some cases, inverse fracture. It is still unclear the actual causes of this phenomenon typically observed in impact tests such as Charpy and drop-weight tear experiments. This study aims at the understanding of the underlying mechanisms controlling this abnormal behavior, which leads mostly to disqualifying a particular material for a certain engineering application. In general, thorough mechanical and material characterizations are intended to be conducted in order to unveil the relationship between microstructure characteristics and structural configurations. By means of a phenomenological fracture model, the statistical nature of the brittle fracture and the size effects will be deemed into a more general computational damage framework incorporating also ductile fracture from the upper shelf energy region.
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The oil and gas industry favors to use less conservative fracture toughness measured from a single edge notched tension (SENT) specimen in the end-clamped conditions in terms of J-integral or crack-tip opening displacement (CTOD) or their resistance curves, where the elastic unloading compliance technique is usually utilized to monitor the incremental crack growth during the single specimen test. Several numerical solutions of crack mouth opening displacement (CMOD) compliance obtained from the finite element analysis (FEA) are available for the end-clamped SENT specimens. However, they have different accuracies and different applicable ranges of crack length ratio a/W, and they may be inconsistent with the existing solutions of their stress intensity factor (K) solutions for the same end-clamped SENT specimen because both the compliance and the K factor were determined separately by FEA. Based on a full-range analytical K solution, this work develops a more accurate, analytical solution of CMOD compliance equation for the end-clamped SENT specimens. Comparisons with various existing FEA results confirm the higher accuracy of the proposed analytical compliance solution. As a result, the proposed CMOD compliance solution can be used to determine more accurate crack length for the SENT testing.
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Stress intensity factors are viewed as specializations of a family of drivers of crack propagation, defined on three-dimensional stress fields, to two-dimensional stress fields. The question of which driver is best suited for the prediction of crack propagation in three dimensions will have to be decided on the basis of evidence developed through the application of a model development process. The procedure for rational choice of a predictor of crack propagation in metals, caused by cyclic loading, is addressed.
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A novel modified cantilever beam method and modified clamped beam method with DIC (Digital Image Correlation) is developed to measure the interface fracture energy of ceramic/metal interface. The experimental execution for these geometries is demonstrated on an Air Plasma Sprayed (APS) YSZ coating on a steel substrate. For modified cantilever method, a pre-crack is first made along the interface, followed by the interface test. These methods use the same geometry for both pre-cracking and testing. The value of interface fracture energy is obtained as the critical energy release rate, Gc, using numerically computed values of J-integral. The results of both the geometries are compared.
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