Williams Series Research Articles

A Further Improved Maximum Tangential Stress Criterion for Assessing Mode I Fracture of Rocks Considering Non-singular Stress Terms of the Williams Expansion

Previous studies showed that the modified maximum tangential stress (MMTS) criterion considering two stress terms (containing r−1/2 and r1/2) of the well-known Williams series expansion can better assess the mode I fracture toughness (KIc) of rocks than the traditional maximum tangential stress (MTS) criterion. However, this study indicates that in some cases, only using the two stress terms cannot fully describe the tangential stress at the critical distance for rock specimens, and the higher order, non-singular stress terms can also play important roles in the tangential stress. The MMTS fracture criterion might still induce non-negligible errors for assessing mode I rock fracture. Thus, we propose a further improved MTS (FIMTS) criterion. The FIMTS criterion emphasizes that the number of non-singular stress terms used in a MTS-based criterion should be carefully chosen according to the following principle: the tangential stress at the critical distance can be accurately described using the selected stress terms. Mode I fracture tests on two kinds of rocks are conducted using a newly proposed V-notched short rod bend (VNSRB) specimen. The specimen- or loading-configuration-dependence of KIc indicated by the experimental data is theoretically assessed using the traditional MTS, the MMTS and the FIMTS criteria, and the last criterion is shown to be the best. Therefore, considering non-singular stress terms of the Williams expansion can better assess mode I fracture of rocks. Our study calls for more attention to the effects of non-singular stress terms on mode I fracture of rocks, especially for small-size specimens and for rocks with relatively large critical distances.

Derivation of complete crack-tip stress expansions from Westergaard–Sanford solutions

The description of mechanical fields at the vicinity of a bi-dimensional crack-tip can be performed using the classic Williams series expansion. While its general structure is well known, complete expressions are rarely available for specific problems. This article describes and applies a methodology to express complete expansions for four given fracture configurations. The procedure relates Williams series coefficients to those of expanded Westergaard–Sanford complex potentials for modes I and II. Actual expansions of complex solutions for mode I cases are derived using classical complex analysis techniques. Complete closed-form results, four power series and one Laurent series, have been determined with this approach. The correctness of analytical results and series convergence behavior have been conclusively investigated through numerical tests comparing reference complex solutions with truncated series representations. The methodology can be applied straightforwardly to new fracture configurations where complex solutions are known. Complete closed-form expressions can be used to derive, test and improve numerical and experimental techniques involving higher order terms in crack-tip expansions.

Coordination Behavior of Ni2+, Cu2+, and Zn2+ in Tetrahedral 1-Methylimidazole Complexes: A DFT/CSD Study.

The interaction between nickel (Ni2+), copper (Cu2+), and zinc (Zn2+) ions and 1-methylimidazole has been studied by exploring the geometries of eleven crystal structures in the Cambridge Structural Database (CSD). The coordination behavior of the respective ions was further investigated by means of density functional theory (DFT) methods. The gas-phase complexes were fully optimized using B3LYP/GENECP functionals with 6-31G∗ and LANL2DZ basis sets. The Ni2+ and Cu2+ complexes show distorted tetrahedral geometries around the central ions, with Zn2+ being a perfect tetrahedron. Natural bond orbital (NBO) analysis and natural population analysis (NPA) show substantial reduction in the formal charge on the respective ions. The interaction between metal d-orbitals (donor) and ligand orbitals (acceptor) was also explored using second-order perturbation of the Fock matrix. These interactions followed the order Ni2+ > Cu2+ > Zn2+ with Zn2+ having the least interaction with the ligand orbitals. Examination of the frontier orbitals shows the stability of the complexes in the order Ni2+ > Cu2+ < Zn2+ which is consistent with the Irving–Williams series.

Electrokinetic properties of metal phthalocyanite nanoparticles

The effect of the nature of the metal in metal phthalocyanites on the electrokinetic potential and рН of the isoelectric point (рН(I)) in suspensions of these substances in NaCl solution (0.1 mol/L) is elucidated. The рН(I) values of unmodified МРс increases in a sequence MnPc < CoPc < СuPc ≈ ZnPc on account of the decreasing effective charge of the coordinated metal ion that corresponds to the Irving–Williams series. The impact of the functional group upon heterogeneous modification of МРс nanoparticles is studied, as well. It is established that the modifying group (the oxygenic properties) exerts a stronger influence on (pH(I)) than the metal nature.

The application of strain energy density criterion to fatigue crack growth behavior of cracked components

The aim of this paper is to study the effects of specimen geometry on fatigue crack growth rate and fatigue life in cracked components by using a modified form of averaged strain energy density criterion. The modified criterion takes into account the first non-singular term of the Williams series expansion in addition to the classical stress intensity factors. Different samples are used to conduct high cycle fatigue experiments on Al 7075-T6. The fatigue behavior of cracked Al 7075-T6 samples is investigated experimentally and theoretically for the compact tension (CT), double cantilever beam (DCB) and Brazilian disc (BD) specimens. It is shown that the fatigue crack growth in these specimens is significantly geometry-dependent. The modified criterion can predict the fatigue crack growth rate and the fatigue life of the pre-cracked specimens well.

Stress intensity factor calculation through thermoelastic stress analysis, finite element and RPIM meshless method

This work aims at determining the stress intensity factor (SIF) for a compact tension specimen (CT) during a fatigue crack growth test with Thermoelastic Stress Analysis (TSA). Additionally, the problem is addressed using advanced discretization techniques, Radial Point Interpolation Meshless method (RPIM) and Finite Element Method (FEM) to obtain a stress distribution and calculate SIF range. Based on an optimization procedure, a functional stress relation was designated where the TSA analysis relies upon. The stress fields obtained with the FEM and the RPIM, are graphically represented and both methodologies are compared with the experimental TSA analysis in the presence of mode I fatigue loading test. The stress field in front of the crack tip is obtained with TSA and used in Williams series expansion, together with an overdeterministic algorithm to compute the SIF under mode I loading. This proposed procedure possesses a hybrid experimental-computational feature in which the SIF determination hinges on a stress field obtained with an optical technique, TSA. The robustness of the numerical FE and RPIM computations were thereby experimentally validated with the achieved SIF solution.

Evaluation of size effect on mixed-mode fracture behavior of epoxy/silica nanocomposites

The aim of this study was to examine the effect of size on the mixed-mode fracture toughness of quasi-brittle nanocomposites with the help of modified maximum tangential stress criterion. The literature reveals that the effect of size on mixed-mode fracture behavior of brittle nanocomposites has not been well investigated previously using modified maximum tangential stress criterion. The studied nanocomposites were made of epoxy resin reinforced with 7 wt%, 20–30 nm nanosilica. The accuracy of the method was assessed by taking into account the high-order terms of Williams series expansion along with finite element over-deterministic method. To investigate the effect of size on fracture toughness, a number of three-point semi-circular bending tests with different radii and four angles of edge–crack orientation were conducted and subjected to mixed-mode loading. The size of fracture process zone and apparent fracture toughness ( Kc) were also evaluated as a function of sample size. Experimental results showed that the proposed approach can accurately predict the fracture behavior of studied nanocomposites.

A bifunctional 2,2′:6′,2″-terpyridine-based ligand for ratiometric Cu(II) sensing

Synthesis and characterization of the bifunctional sensor receptor ligand N-([2,2′:6′,2″-terpyridine]-4′-yl)methyl)-N-propylacrylamide (1) and the model ligand N-([2,2′:6′,2″-terpyridine]-4′-yl)methyl)-N-propylisobutyramide (2) are described. Ligand 1 is a receptor for Cu(II) that is copolymerizable with N-isopropylacrylamide giving a ratiometric sensor of weakly bound Cu(II) in environmental waters. Ligand 2 is a model for copolymerized 1 whereby the reactive acrylamide group is replaced by isobutyramide. Solution speciation of complexes of Cu(II) and Zn(II) with 2 were investigated spectroscopically and their solid-state structures were studied through single-crystal X-ray diffraction. Solution UV–vis and fluorescence studies show a preference of 2 toward Cu(II) over Na(I), Zn(II), Cu(II), Co(II), Mn(II), Ni(II), Cd(II), and Pb(II) in accord with the Irving–Williams series and other coordination principles. Solution speciation determined in a weakly coordinating aqueous-organic (60 : 40 DMF/H2O) medium indicates 1 : 1 Cu(II):2 binding as desired in that formation of [Cu(2)2]2+ would crosslink the polymer sensor. The crystal structures of [Cu(2)(NO3)2] and [Zn(2)(NO3)2]·MeOH·1/2Et2O display distorted octahedral geometries where 2 coordinates meridionally and two nitrate groups occupy the remaining sites around the metal center.

Two basic approaches in a search of the crack propagation angle

AbstractTwo basic approaches in predicting crack paths have been analysed, namely, incremental method and integral (global) method. Incremental methods are associated with the local criterion of fracture mechanics and pointed out on the estimation of an angle between directions of the initial crack propagation and succeeding crack increment. The local fracture criterion of the maximum average tangential stress in the vicinity of fracture process zone including two terms of the Williams series solution for the mixed mode I/II loading has been used for predicting the angle of surface crack growth under conditions of rolling–sliding. The integral method is associated with a variation principle of fracture mechanics. This principle postulates the minimum of the certain functional. This functional contains the weight function depending on the stress and strain state in the uncracked solid and search functions which are described the crack propagation angle on the surface. To validate the variation principle in the search of crack trajectory, this trajectory has been estimated for plates under different loading conditions.

Цифровая обработка результатов оптоэлектронных измерений. Метод фотоупругости и его применение для определения коэффициентов многопараметрического асимптотического разложения М. Уильямса поля напряжений

Приведены результаты числовой обработки результатов оптоэлектронных измерений, проведенных с помощью метода фотоупругости. Целью экспериментов являлось исследование смешанного деформирования пластины с двумя горизонтальными и наклонными трещинами, проведенных с помощью метода фотоупругости, и построение многопараметрических асимптотических разложений М. Уильямса поля напряжений в окрестности каждой из вершин трещин. Приведен эффективный алгоритм вычисления коэффициентов высших приближений в асимптотических разложениях компонент тензора напряжений. Проведено сравнение коэффициентов с имеющимся точным решением задачи о смешанном нагружении бесконечной изотропной линейно-упругой пластины с двумя коллинеарными трещинами. Показано, что имеющееся точное решение для бесконечной плоскости с разрезами может служить начальным приближением для разработанного алгоритма. В работе приводятся вычисленные значения коэффициентов высших приближений как для пластины с двумя трещинами одинаковой длины, так и для пластины с разрезами различной длины.

Analysis of accuracy of Williams series approximation of stress field in cracked body – influence of area of interest around crack-tip on multi-parameter regression performance

A study on the accuracy of an approximation of the stress field in a cracked body is presented. Crack-tip stress tensor is expressed using the linear elastic fracture mechanics (LEFM) theory in this work, more precisely via its multi-parameter formulation, i.e. by Williams power series (WPS). Determination of coefficients of terms of this series is performed using a least squares-based regression technique known as over-deterministic method (ODM) for which results from finite element (FE) method computation are usually taken as inputs. Main attention is paid to a detailed analysis of a suitable selection of FE nodes whose results serve as the inputs to the employed method. Two different ways of FE nodal selection are compared – nodes selected from the crack tip vicinity lying at a ring of a certain radius versus nodes selected more or less uniformly from a specified part of the test specimen body. Comparison of these approaches is made with the help of procedures developed by the authors which enable both the determination of the coefficients of terms of the analytical WPS approximation of the stress field based on the FE results and the backward reconstruction of the field (again using WPS) from those determined terms’ coefficients/functions. The wedge-splitting test (WST) specimen with a crack is taken as example for the study.

Complete Williams Asymptotic Expansion near the Crack Tips of Collinear Cracks of Equal Lengths in an Infinite Plane

The present study is aimed at analytical determination of coefficients in crack tip expansion for two collinear finite cracks of equal lengths in an infinite plane medium. The study is based on the solutions of the complex variable theory in plane elasticity theory. The analytical dependence of the coefficients on the geometrical parameters and the applied loads for two finite cracks in an infinite plane medium is given. It is shown that the effect of the higher order terms of the Williams series expansion becomes more considerable at large distances from the crack tips. The knowledge of more terms of the stress asymptotic expansions allows us to approximate the stress field near the crack tips with high accuracy.

Experimental Determination of Crack Driving Forces for Small Cracks

The crack tip field of small fatigue cracks was measured using digital image correlation DIC. For this purpose, smooth specimen were fatigued until a certain amount of damage was visible on the surface. This specimens were then placed in a micro-tensile device in the SEM. The grey value patterns obtained at two different load levels were analyzed with DIC. Fitting the Williams series for the crack tip field to the corresponding DIC-displacement field in the crack tip area yielded the mode-I, and the mode-II stress intensity factors together with the T-stresses. It was found that the experimental values of the stress intensity factor were comparatively high with pronounced mode-II contributions.

Digital image correlation method for calculating coefficients of Williams expansion in compact tension specimen

The digital image correlation (DIC) method is used to obtain the coefficients of higher-order terms in the Williams expansion in a compact tension (CT) specimens made of polymethyl methacrylate (PMMA). The displacement field is determined by the correlation between reference image (i.e., before deformation) and deformed image. The part of displacements resulting from rigid body motion and rotation is eliminated from the displacement field. For a large number of points in the vicinity of the crack tip, an over-determined set of simultaneous linear equations is collected, and by using the fundamental concepts of the least-squares method, the coefficients of the Williams expansion are calculated for pure mode I conditions. The experimental results are then compared with the numerical results calculated by finite element method (FEM). Very good agreement is shown to exist between the DIC and FE results confirming the effectiveness of the DIC technique in obtaining the coefficients of higher order terms of Williams series expansion from the displacement field around the crack tip.

Scaling effect on the mixed-mode fracture path of rock materials

In this paper, a new approach is presented to predict the crack growth path in the rock materials by taking into account the size effect. The proposed approach is an incremental method in which the crack initiation angle for each step is determined from the modified forms of the maximum tangential stress criterion. These modified maximum tangential stress criteria take into account the influence of the higher order terms of the stress series at the crack tip in addition to the singular terms. As an important parameter in the proposed method, the critical distance rc is also assumed to be size dependent. Finally the incremental method is evaluated by experimental results obtained from Guiting limestone and CJhorveh marble specimens reported in the previous studies. It is shown that the proposed approach can predict the fracture trajectory of cracked specimens with different sizes in good agreement with the experimental results when three terms of Williams series expansion are considered for characterizing the stress field around the crack tip.

Mixed mode brittle fracture analysis of high strength cement mortar using strain-based criteria

Strain based fracture criteria are employed to predict the brittle fracture of a cracked high strength cement mortar subjected to combined tensile/shear loading. Theoretical background of a traditional and extended versions of maximum tangential strain criteria are introduced, and the mixed mode fracture toughness and crack propagation angle are calculated for centrally cracked Brazilian disk specimens with various dimensions. In addition to the strain based criteria, a traditional stress based criterion and strain energy based criteria are used to predict the crack propagation conditions of the Brazilian disk specimens. The validity of the fracture criteria are evaluated by comparing the predicted fracture conditions to the previously published experimental data. The comparisons show that the extended version of the maximum tangential strain criterion, which takes into account the effect of both the first nonsingular strain term and the singular strain terms in Williams series expansion for strain field near crack tip, provides highly accurate predictions for the crack propagation conditions. The results indicate that the T-strain, the first non-singular term, plays an important role in the crack propagation of the cement mortar.

Monitoring trace metals in seawater using a diffusive gradient in thin film probe in Ulsan Bay, East Sea, Korea: Comparison with transplanted mussels

To evaluate the applicability of a diffusive gradient in thin film (DGT) probe for monitoring dissolved metals in coastal seawater, DGT-labile metal concentrations were compared with total dissolved metal concentrations using spiked and natural seawater samples in the laboratory and transplanted mussels (Mytilus galloprovincialis). This was achieved through the simultaneous deployment of DGT probes and transplanted mussels in Ulsan Bay during winter and summer. DGT-labile metal concentrations were 45% (Cu) ~ 90% (Zn) of total dissolved concentrations, and the order of non-labile concentrations was Cu > Pb > Co ~ Ni > Cd ~ Zn in both metal-contaminated and non-contaminated seawater samples, which was similar to the order of stability of metal complexes in the Irving–Williams series. The overall variability of the DGT probe results within and between tanks was less than 10% (relative standard deviation: RSD) for all the metals tested during a 48-h deployment. The accumulation of metals, as determined by DGT probes, represented the spatial gradients better than the transplanted mussels did for all of the metals tested, and the extent of metal accumulation in mussels differed depending on the metal. The comparison of results for the DGT probe and the transplanted mussels in two seasons (winter and summer) suggested that metal accumulation in mussels was controlled by the physiological factors of mussels and partly by their diet (particulate metal loadings). The DGT probe could be used as a monitoring tool for dissolved metals in coastal seawater because its results explained only labile species. When using the DGT probe, slightly more than half of the total dissolved concentration in seawater samples for all the metals investigated displayed timeintegrated properties and distinct spatial gradients from pristine to metal-contaminated seawater.

Fracture behaviors of concrete CSTBD specimens by the modified MTS criterion under the condition of compression loading

A group of concrete cracked straight-through Brazilian discs are investigated under the condition of quasi-static compression loading, and the fracture initiation angles and the fracture resistances of pure mode II and I-II mixed mode on concrete are obtained by changing the directions of initial cracks and concrete mix proportion. The predictive fracture resistances of different proportions and crack modes (pure mode II and I-II mixed mode) on concrete can be obtained by the modified maximum tangential stress criterion (MMTS), which is proposed by Ayatollahi et al. The predictive results from MMTS criterion are compared and analysed with those from experiments. Especially, the MMTS criterion takes into account the effects of non-singular stress terms (i.e. the T-stress, A 3 and B 3 from the expansion of Williams series and the higher order terms are ignored) in addition to the singular stress terms around the crack tip. The dimensionless parameters in the expansion of Williams series are calculated from “finite element over-deterministic” (FEOD) method. The size of the fracture process zone is calculated by using Schmidt model containing A 3 term. It is found that the results from MMTS criterion are more reliable compared to those from traditional MTS criterion or GMTS criterion when used for predicting the fracture initiation angles and the fracture resistances of pure mode II and I-II mixed mode on concrete, particularly for mode II dominant loading condition.

Measuring benzohydroxamate complexation with Mg2+, Mn2+, Co2+, and Ni2+ using isothermal titration calorimetry

Isothermal titration calorimetry has been used to determine the stoichiometry and first (K1) and second (K2) conditional stability constants, where applicable, for the formation of complexes of benzohydroxamate (BzH) with Mg2+, Mn2+, Co2+, and Ni2+ ions, and benzoylphosphonate with Ni2+ ions. Binding determinations were conducted in HEPES buffer (0.1 mol L−1, pH 7.5) at 25.0 °C by titration of the ligand with metal ion. BzH and Mg2+, and benzoylphosphonate and Ni2+, formed complexes with a 1:1 stoichiometry; however, BzH formed complexes with Mn2+, Co2+, and Ni2+ ions in a step-wise binding model with a stoichiometry of 2:1 (BzH:M2+). In addition, the Gibbs energy change accompanying the binding of BzH to each of the divalent metal ions was found to follow the order of the Irving–Williams series (Mg2+ < Mn2+ < Co2+ < Ni2+) and to correlate with the electron affinities of the metal ions. Conditional stability constants, determined in this manner, may be useful for calculating the concentrations of free inhibitor when inhibition studies are conducted with metalloenzymes and enzyme inhibitors containing a hydroxamic acid group.

Cysteine containing dipeptides show a metal specificity that matches the composition of seawater

Model prebiotic dipeptide sequences were identified by bioinformatics and DFT and molecular dynamics calculations. The peptides were then synthesized and evaluated for metal affinity and specificity. Cysteine containing dipeptides were not associated with metal affinities that followed the Irving-Williams series but did follow the concentration trends found in seawater.