Williams Series Research Articles

A strain-based criterion for predicting size-dependent fracture resistance of quasi-brittle materials under mixed mode loading

In the present study, a strain-based approach called the modified maximum tangential strain criterion has been developed for evaluating size effect on the mixed-mode fracture resistance of quasi-brittle materials. This approach relies on the generalized maximum tangential strain criterion, which considers the singular (K) and the first non-singular (T) terms of Williams series expansion. Furthermore, as an important and size-dependent parameter in the proposed criterion, a new formulation is presented to calculate the critical distance (rc). The predictions of the new criterion are compared not only with the available experimental data, but also with the results estimated by another size effect criterion named the modified maximum tangential stress criterion. It is shown that the proposed strain-based criterion is highly capable of estimating the size-dependent mixed-mode fracture behavior of quasi-brittle materials without requiring to compute the other higher order terms.

A new strain-based approach to investigate the size and geometry effects on fracture resistance of rocks

In this paper, a new strain-based criterion is suggested for assessing the effects of size and geometry of specimen on the fracture resistance of rocks under mixed-mode (I/II) loading. The new approach named the modified maximum tangential strain (MMTSN) criterion is based on the classical maximum tangential strain (MTSN) criterion, in which the first non-singular term (T) of Williams series expansion is considered in addition to the singular terms (K). Furthermore, to provide more coherence, the critical distance (rc) from the crack tip is defined according to a new strain-based failure model. Unlike similar strain-based fracture models available in the literature, the critical distance rc in the MMTSN criterion is assumed to be size-dependent and a semi-empirical formulation is utilized for describing this size-dependency. To assess the ability of MMTSN for considering the size and geometry effects, the experimental data existing in the literature for a number of cracked Brazilian disk (CBD) and semi-circular bend (SCB) specimens manufactured from Guiting limestone are taken into account. It is demonstrated that the MMTSN criterion can predict the experimental data very well by taking into consideration the size and geometry effects without needing to calculate the other higher order terms.

Engineering Single‐Atom Sites with the Irving–Williams Series for the Simultaneous Co‐photocatalytic CO2 Reduction and CH3CHO Oxidation

AbstractThe bonding effects between 3d transition‐metal single sites and supports originate from crystal field stabilization energy (CFSE). The 3d transition‐metal atoms of the spontaneous geometrical distortions, that is the Jahn–Teller effect, can alter CFSE, thereby leading to the Irving–Williams series. However, engineering single‐atom sites (SASs) using the Irving–Williams series as an ideal guideline has not been reported to date. Herein, alkynyl‐linked covalent phenanthroline frameworks (CPFs) with phenanthroline units are developed to anchor the desired 3d single metal ions from d5 to d10 (Mn2+, Fe3+, Co2+, Ni2+, Cu2+, and Zn2+). The Irving–Williams series was employed to accurately predict the bonding effects between 3d transition‐metal atoms and phenanthroline units. To verify this, theoretical calculations and experimental results reveal that Cu‐SASs/CPFs exhibits higher stability and faster charge‐transfer efficiency, far surpassing other metal‐SASs/CPFs. As expected, Cu‐SASs/CPFs demonstrates a high photoreduction of CO2‐to‐CO activity (~30.3 μmol ⋅ g−1 ⋅ h−1) and an exceptional photooxidation of CH3CHO‐to‐CH3COOH activity (~24.7 μmol ⋅ g−1 ⋅ h−1). Interestingly, the generated *O2− is derived from the process of CO2 reduction, thereby triggering a CH3CHO oxidation reaction. This work provides a novel design concept for designing SASs by the Irving–Williams to regulate the catalytic performances.

An iterative crack tip correction algorithm discovered by physical deep symbolic regression

Digital image correlation is a widely used technique in the field of experimental mechanics. In fracture mechanics, determining the precise location of the crack tip is crucial. In this paper, we introduce a novel crack tip detection algorithm based on displacement and strain fields obtained by digital image correlation. Iterative crack tip correction formulas are discovered by applying deep symbolic regression guided by physical unit constraints to a dataset of simulated cracks under mode I, II and mixed-mode conditions with variable T-stress. For the training dataset, we fit the Williams series expansion with super-singular terms to the simulated displacement fields at randomly chosen origins around the actual crack tip. We analyse the discovered formulas and apply the most promising one to digital image correlation data obtained from uniaxial and biaxial fatigue crack growth experiments of AA2024-T3 sheet material. Throughout the experiments, the crack tip positions are reliably detected leading to improved stability of the crack propagation curves.

Determining the stresses near crack tips in composite plates based on the Williams series and singular integral equations

The paper presents a numerical-analytical method developed to determine the stresses formed in composite plates near the tips of curved cracks by means of the Williams series. The coefficients of the terms of the Williams series and T-stresses have been calculated based on the derivatives of the displacement discontinuity of the crack edges. It has been shown that the calculations can be simplified implementing the Green's solutions by the Lekhnitskii method. The numerical algorithms based on Lobatto quadrature formulas have been used to solve the singular equations and determine both the coefficients the Williams power series and T-stresses. The proposed approach has been used to analyze rectilinear and curvilinear cracks in composite plates. In addition, the Green's solutions have been implemented to investigate the cracks located in a half-plane, near an elliptical hole, in rectangular plates, and in strip. One- and two-period problems have been considered for plates with cracks in the present calculations. The high accuracy of the performed calculations of the stresses near the crack tips in composite plates using elementary formulas contained by the Williams series has been confirmed on the test examples.

3D characterization of closure of surface and internal fatigue cracks in nodular cast iron using Digital Volume Correlation of laboratory X-ray tomography images

Digital volume correlation (DVC) of laboratory X-ray tomography images has been used to characterize the closure of artificial internal fatigue cracks in nodular graphite cast iron. DVC allows to produce crack opening displacement (COD) map from 3D tomography images at the sub-voxel scale thanks to the graphite nodules which can be used as internal natural markers. Fatigue samples containing controlled internal defects have been produced in a nodular cast iron by diffusion bonding plates containing laser-machined sharp notches. Specimens with internal cracks and surface cracks were respectively characterized by laboratory tomography at different loads during fatigue crack propagation tests. The local stress intensity factors (SIFs) have been extracted from the measured displacement fields by using Williams series, based on those results the closure levels of internal cracks and surface cracks were comparatively analyzed and discussed. Crack closure levels were found to be linearly related to the location of the surface crack front, and the level of internal crack closure was comparable to that of the surface crack in bulk.

Application of holography method for the restoration of the Williams series near the crack tip

The article describes the processing of the results of a series of experiments performed by the interference-optical method of holographic interferometry (holographic photoelasticity) aimed at computing the amplitude coefficients of the M. Williams series constituting the stress and displacement fields at the crack tip for several cracked configurations. The main objective of this study is the experimental and computational determination of the coefficients of the M. Williams series for the stress, strain and displacement fields associated with the crack tip in an isotropic linearly elastic medium taking into account regular (non-singular) terms in the multiparameter Williams series. These coefficients are named generalized stress intensity factors. The method of holographic interferometry is shown to be a convenient and efficacious tool for reconstructing the stress field near the tip of the crack, because during the experiment it is possible to obtain two families of interference fringe patterns: absolute retardation fringes (isodromics) for vertical and horizontal polarizations. Experimental outcomes were thoroughly processed using the developed digital application allowing us to accumulate the isodromics orders and coordinates of points belonging to absolute retardations. In this work, absolute retardation fringes in a plate with a central horizontal crack and a crack inclined at different angles are obtained. For each type of experimental sample, the coefficients of the Williams series were calculated taking into account non-singular terms (in the representation of M. Williams, fifteen terms were preserved). A procedure for linearization of nonlinear algebraic equations following from the relations of Favre’s law is proposed. By solving the obtained overdeterministic system of linear algebraic equations, the generalized stress intensity factors (coefficients of the M. Williams series) are estimated. Conjointly, the finite element analysis of the specimens with same geometry was effectuated. The experimentally determined values of the Williams series are compared with the results of the finite element calculation of the stress-strain state performed in the SIMULIA Abaqus software.The results of the numerical and experimental studies were found to be quite consistent. It is lucidly shown that it is imperative to keep the higher order terms in the Williams series expansions for the fields associated with the crack tip.

Numerical Analysis of Fracture Behavior of Functionally Graded Materials using 3D-XFEM

Abstract This paper presents the numerical evaluation of mixed stress intensity factors (SIFs) and non-singular terms of William's series (T-stress) of functionally graded materials (FGMs) using three-dimensional extended finite element method (3D-XFEM). Four-point bending specimen with crack perpendicular to material gradation have been used in this investigation in order to study the effect of some parameters (crack position, crack size, specimen thickness) on the failure of FGMs materials. The fracture parameters (KI KII, phase angle ψ and T-stress) obtained by the present simulation are compared with available experimental and numerical results. An excellent correlation was found of the 3D-XFEM simulations with those available in the literature. From the numerical results, a fitting procedure is performed in order to propose an analytical formulation and subsequently are validated against the 3D-XFEM results.

Mixed‐metal Ionothermal Synthesis of Metallophthalocyanine Covalent Organic Frameworks for CO2 Capture and Conversion

AbstractPhthalocyanines (PCs) are intriguing building blocks owing to their stability, physicochemical and catalytic properties. Although PC‐based polymers have been reported before, many suffer from relatively low stability, crystallinity, and low surface areas. Utilizing a mixed‐metal salt ionothermal approach, we report the synthesis of a series of metallophthalocyanine‐based covalent organic frameworks (COFs) starting from 1,2,4,5‐tetracyanobenzene and 2,3,6,7‐tetracyanoanthracene to form the corresponding COFs named M‐pPPCs and M‐anPPCs, respectively. The obtained COFs followed the Irving–Williams series in their metal contents, surface areas, and pore volume and featured excellent CO2 uptake capacities up to 7.6 mmol g−1 at 273 K, 1.1 bar. We also investigated the growth of the Co‐pPPC and Co‐anPPC on a highly conductive carbon nanofiber and demonstrated their high catalytic activity in the electrochemical CO2 reduction, which showed Faradaic efficiencies towards CO up to 74 % at −0.64 V vs. RHE.

Liturgical Song and Practice in Dante's 'Commedia' by Helena Phillips-Robins

Reviewed by: Liturgical Song and Practice in Dante's 'Commedia' by Helena Phillips-Robins Alejandro Cuadrado Liturgical Song and Practice in Dante's 'Commedia'. By Helena Phillips-Robins. (The William and Katherine Devers Series in Dante and Medieval Italian Literature, 19) Notre Dame: University of Notre Dame Press. 2021. xvi+307 pp. $60. ISBN 978–0–268–20068–8. The impressive slate of Dante-related publications by Notre Dame's Devers Series has its most recent instalment in Helena Phillips-Robins's Liturgical Song and Practice in Dante's 'Commedia', a well-researched and interdisciplinary reading of the [End Page 249] role of liturgy in Dante's poem. In this book Phillips-Robins not only articulates the presence of liturgical song in Purgatorio and Paradiso, but argues that these songs invite readers of the Commedia to participate in the same performance as the souls depicted in the poem. The first chapter, 'Liturgy and Community', examines two liturgical moments that bookend a soul's time in Purgatory: the 'In exitu Israel de Aegypto' sung aboard the angelic ferry to Purgatory in Purgatorio ii and the Gloria sung in celebration of Statius's completed purgation in Purgatorio xx. Using evidence from liturgical practices attested in medieval Italian ordinals, Phillips-Robins hypothesizes a reader response in which Dante's reader would pause to complete the psalm or Gloria, and in doing so participate in a transformation similar to that experienced by the penitent souls in Purgatorio. The second chapter, 'Liturgy and Participation in Christ', continues the argument that Dante invites readers to participate in the psalms of the Commedia, first by examining the two psalms of Purgatorio xxiii ('Labïa mea, Domine' and 'Elì'). Leaning on patristic sources, Phillips-Robins argues that these psalms invite the reader to adopt the principle of imitatio Christi. The chapter then broadens its scope to include the liturgical songs in Paradiso x and Purgatorio xi as 'explicit invitations to the reader to engage in devotional activity' (p. 108). A section in the middle of the chapter catalogues at breakneck pace the passages in the Commedia in which Dante indicates that he is only partially citing a liturgical song. Given Phillips-Robins's convincing close reading of Purgatorio xxiii, one wishes she had given more attention to each passage listed. In 'The Shared Voice of Liturgical Prayer' (Chapter 3) Phillips-Robins turns her attention to the liturgical songs in the final cantos of Purgatorio: the 'In te, Domine, speravi' of Purgatorio xxx and the 'Asperges me' of Purgatorio xxxi. She also treats the Prayer to the Virgin of Paradiso xxxiii, arguing that intercessory prayer in Dante's Commedia involves both those praying and those being prayed for. The second half of this chapter turns to Giovanni di Paolo's celebrated manuscript illuminations in Yates Thompson 36, especially those that depict Marian devotion (three of these are reproduced in black and white). Phillips-Robins argues that 'Giovanni di Paolo shows himself to be particularly attuned to Dante's invitations to his readers to engage, in the first person, with the divine' (p. 163). The final chapter, 'Liturgy and Love', is at once the most successful and the one that strays furthest from the book's central theme. In this chapter Phillips-Robins traces a historical itinerary of the idea of 'believing in' from Augustine through to thirteenth-century theologians, including the prominent preacher Giordano da Pisa. Against this historical and theological context, Phillips-Robins offers a careful close reading of Dante's Credo in the Heaven of the Fixed Stars, arguing that his creed engages in a 'movement in love toward' God (p. 191). One limitation is the extent to which Phillips-Robins insists that certain materials such as the Florentine ordinals are 'all but unknown in Dante studies' (p. 13) or 'as yet all but untapped in Dante studies' (pp. 14 and 31). A visit to her extensive and well-documented footnotes (especially nn. 50 and 57 in the Introduction) shows [End Page 250] that Dante scholars have tapped some of these sources, most notably in Francesco Ciabattoni's Dante's Journey to Polyphony (Toronto: University of Toronto Press, 2010). Similarly, Alessandro Vettori's recent...

To the scrutiny of the experiential and computational elicitation of factors of the Williams multi-parameter expansion

The article defines stress fields near the tips of mathematical cracks in an isotropic linearly elastic plate with two horizontal collinear cracks lying on a straight line of different lengths under the uniaxial tensile condition, using two approaches - experimental, based on the method of digital photomechanics, and numerical, based on finite element calculations. To represent the stress field at the tip of the section, the Williams polynomial series is used - the canonical representation of the field at the top of the mathematical section of a two-dimensional problem of elasticity theory for isotropic media. The main idea of the current study is to take into consideration the regular (non-singular) terms of the series and analyze their impact on the holistic description of the stress field in the immediate vicinity of the top of the section. The first fifteen coefficients of the Max Williams series were preserved and determined in accordance with experimental patterns of isochromatic bands and finite element modeling. To extract the coefficients of the Williams series used a redefined method designed to solve systems of algebraic equations, the number of which is significantly greater than the unknown - amplitude multipliers. The influence of the non-singular terms of the Williams series on the processing of the experimental pattern of interference fringes is demonstrated. It is validated that the preservation of the terms of a high order of smallness makes it possible to expand the area adjacent to the tip of the crack, from which experimental points can be selected. The finite element study was carried out in the SIMULIA Abaqus engineering analysis system, in which experimental samples tested in a full-scale experiment were reproduced. It is revealed that the results obtained by the two methods are in good agreement with each other.

Medium-entropy state quinary keplerate clusters as a remarkable electrocatalyst for small molecule electrooxidation

We fabricate a library of binary to quinary element Keplerate clusters stabilized in a medium entropy state. Different ions follow the Irving–Williams series to be incorporated in the clusters.

Application of digital image correlation and Williams series approximation to characterize mode I stress intensity factor

Application of digital image correlation and Williams series approximation to characterize mode I stress intensity factor

COMPARATIVE ANALYSIS OF STRESS STATE IN THE VICINITY OF THE CRACK-TIPS AND NOTCHES USING TRUNCATED WILLIAMS SERIES EXPANSION

In the present study the asymptotic analysis of the multi-parameter Williams series expansion of the stress-strain state in the vicinity of seven different configurations is performed. The aim of the asymptotic analysis is to educate the sensitivity of different cracked and notched configurations to the number of kept terms in Williams series expansion. It is shown that all configurations with the concentrated forces possess the more sensitivity to the number of terms in the truncated Williams series expansions and, therefore, require the more terms in the series expansion. The cracked configurations with distributed loadings are less sensitive to the number of terms in Williams series expansion. As a whole, it is demonstrated that high order terms have principal role in the description of the stress-strain in the vicinity of the crack tip.

Variation of Fracture Toughness with Biaxial Load and T-Stress under Mode I Condition

The effect of a biaxial load on brittle fractures of solids under predominantly elastic deformation and mode I loading conditions is studied in this article. A cross-shaped specimen is used in this work, and the variation of T-stress is achieved by changing the load applied to the arm parallel to the crack. In the tests, under a series of different loads parallel to the crack, a series of load values perpendicular to the crack are obtained, and the stress intensity factor is calculated by FEM (Finite Element Analysis). The test data demonstrate that the apparent fracture toughness of the material varies with the load parallel to the crack and the T-stress, since the T-stress is directly related to the load parallel to the crack. If only the first term of the Williams series solution is used to describe the stress field near the crack tip, the variation of KC is unexplained; therefore, more than one term in the Williams series solutions is used to develop the theory of biaxial effect on solid fracture behavior. The fracture criterion based on this consideration is also used to predict crack instability or crack curving under mode I load conditions. Experimental data are presented and compared with the theory.

Directing the Morphology, Packing, and Properties of Chiral Metal–Organic Frameworks by Cation Exchange**

AbstractWe show that metal–organic frameworks, based on tetrahedral pyridyl ligands, can be used as a morphological and structural template to form a series of isostructural crystals having different metal ions and properties. An iterative crystal‐to‐crystal conversion has been demonstrated by consecutive cation exchanges. The primary manganese‐based crystals are characterized by an uncommon space group (P622). The packing includes chiral channels that can mediate the cation exchange, as indicated by energy‐dispersive X‐ray spectroscopy on microtome‐sectioned crystals. The observed cation exchange is in excellent agreement with the Irving–Williams series (Mn<Fe<Co<Ni< Cu>Zn) associated with the relative stability of the resulting coordination nodes. Furthermore, we demonstrate how the metal cation controls the optical and magnetic properties. The crystals maintain their morphology, allowing a quantitative comparison of their properties at both the ensemble and single‐crystal level.

Helena Phillips-Robins, Liturgical Song and Practice in Dante’s “Commedia”. (William and Katherine Devers Series in Dante and Medieval Italian Literature 19.) Notre Dame: University of Notre Dame Press, 2021. Pp. xvi, 307; black-and-white figures. $60. ISBN: 978-0-2682-0068-8.

Previous articleNext article No AccessReviewsHelena Phillips-Robins, Liturgical Song and Practice in Dante’s “Commedia”. (William and Katherine Devers Series in Dante and Medieval Italian Literature 19.) Notre Dame: University of Notre Dame Press, 2021. Pp. xvi, 307; black-and-white figures. $60. ISBN: 978-0-2682-0068-8.Glenn A. SteinbergGlenn A. SteinbergThe College of New Jersey Search for more articles by this author PDFPDF PLUSFull Text Add to favoritesDownload CitationTrack CitationsPermissionsReprints Share onFacebookTwitterLinkedInRedditEmail SectionsMoreDetailsFiguresReferencesCited by Speculum Volume 97, Number 3July 2022Emerging Issues in Medieval Iberian Studies The journal of the Medieval Academy of America Article DOIhttps://doi.org/10.1086/720534 Views: 24Total views on this site For permission to reuse, please contact [email protected]PDF download Crossref reports no articles citing this article.

Directing the Morphology, Packing, and Properties of Chiral Metal-Organic Frameworks by Cation Exchange.

We show that metal–organic frameworks, based on tetrahedral pyridyl ligands, can be used as a morphological and structural template to form a series of isostructural crystals having different metal ions and properties. An iterative crystal‐to‐crystal conversion has been demonstrated by consecutive cation exchanges. The primary manganese‐based crystals are characterized by an uncommon space group (P622). The packing includes chiral channels that can mediate the cation exchange, as indicated by energy‐dispersive X‐ray spectroscopy on microtome‐sectioned crystals. The observed cation exchange is in excellent agreement with the Irving–Williams series (Mn<Fe<Co<Ni< Cu>Zn) associated with the relative stability of the resulting coordination nodes. Furthermore, we demonstrate how the metal cation controls the optical and magnetic properties. The crystals maintain their morphology, allowing a quantitative comparison of their properties at both the ensemble and single‐crystal level.

Stress intensity factors, T-stresses and higher order coefficients of the Williams series expansion and their evaluation through molecular dynamics simulations

Molecular dynamics method and finite element analysis are applied for the analysis of the stress field in the neighborhood of the crack tip in a copper plate with a single edge notch. The molecular dynamics simulations implemented in a classical molecular dynamics code LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) are aimed at evaluating classical continuum linear elastic fracture mechanics parameters such as stress intensity factors, T-stresses and higher order coefficients of the Williams power series expansion of the near crack stress field for Mode I, Mode II and Mixed Mode (Mode I + Mode II) loadings of the cracked specimen in isotropic linear elastic materials. The key objective of the study is the comparison of continuum and atomistic approaches for the estimation of the near crack tip fields using the example of one of the most common cracked configurations. Stress intensity factors, T-stresses and higher order coefficients of the Williams series expansion for a copper plate with the single edge notch under Mode I and Mixed Mode loadings are evaluated by atomistic modeling and by finite element method. The wide class of the computations in LAMMPS is performed. The atomistic values of stress intensity factors and higher order terms of the Williams series expansion are compared with the values obtained from the numerical solutions given by finite element method. It is shown that the continuum fracture theory successfully describes fracture and the near crack tip fields even at extremely confined singular stress field of only several nanometers. The angular distributions of the stress components from atomistic modeling are retrieved and compared with the angular distributions of the stresses from continuum linear elastic fracture mechanics. The comparison shows good agreement between two approaches.

T-stress for the double-edge cracked Brazilian disc under compression

As the nonsingular term in the Williams series solution of the stress field at the crack tip, T-stress has a great influence on the crack propagation behavior. Under the loading condition for pure mode II fracture, negative T-stress (compressive T-stress) reduces the deflection angle for crack initiation and promotes the self-similar propagation of crack; while positive T-stress increases the crack deflection angle, and the crack kinks into tensile wing crack. The closed-form solution of the T-stress for the double-edge cracked Brazilian disc (DCBD) under compression is obtained by the weight function method. The closed-form solution in series form has a good convergence. The normalized T-stresses are calculated for different conditions, then the effects of distribution form, loading angle, contact angle, and relative crack length on T-stresses are analyzed. The results show that the effect of contact angle γ on T-stress should be considered when the loading angle θ and the relative crack length α are both small or great; while the effect of distribution form on T-stress cannot be ignored if and only if θ and α are small. The T-stress in the DCBD specimen is always negative when θ ≤ 20° and α ≥ 0.4. This is advantageous for the realization of true mode II fracture.