Over-deterministic Method Research Articles

Determination of notch stress intensity factors under mode I loading using the 3D-DIC and finite element over-deterministic methods

In this paper, the fracture behaviour of Single Edge Notched Tension Specimens (SENT) is investigated experimentally and numerically using the three-dimensional digital image correlation (3D-DIC) experimental method and the Finite Element Analysis (FEA) technique, respectively. The mode I notch stress intensity factors (NSIFs) for the SENT specimens are derived from the measured and simulated displacement fields using the Finite Element Over-Deterministic (FEOD) method. The step-by-step process of a 3D-DIC experimental system for the SENT specimens is described. The SENT specimens were tested with pinned and clamped boundary conditions (BCs) and different notch initial opening angles. There are three main contributions of the paper: (1) the derivation of the NSIFs under Mode I Loading for thin plates subjected not only to the common pinned BCs, but also to the rare, clamped BCs; (2) the application of the FEOD method to derive the NSIFs from the displacement fields measured using the 3D DIC technique and predicted using FEA; (3) the validation of the NSIFs derived numerically via experimentally-determined NSIFs for notched thin plates subjected to both pinned and clamped BCs. Thus, four sets of NSIF data are provided in the paper and enable confident application of the NSIF values for the conservative (pinned BCs) and light-weight (clamped BCs) design of notched thin plates.

Fast atomic crack kinking and branching in WS2

Abstract Dynamic nanocrack propagation in 1T- and 2H-WS2 strips is studied by molecular dynamics, and the T-stress and circumferential stress in linear elastic fracture mechanics are considered. As the crack propagates, the crack-tip speed (v) experiences a rapid acceleration, and then oscillates at ∼55% (1T) and ∼65% (2H) of the Rayleigh-wave speed followed by crack kinking/branching. The critical energy release rates of crack instability are estimated to be ∼1.5 J/m2 (1T) and ∼4.0 J/m2 (2H). The crack path in 1T-WS2 exhibits higher sensitivity of strain rates for atomic asymmetry around the crack tip. Examination of the dynamic crack-tip field shows that the T-stress obtained by the over-deterministic method always fluctuates in negative, and the theoretical circumferential stress curve does not accurately capture the v-dependent atomic stress distribution. Consequently, both T-stress and circumferential stress are limited in predicting the crack kinking/branching directions, which can be attributed to the discrete crystal lattice and local anisotropy of WS2, where a preferred crack path along the zigzag surface is observed. The fracture properties of WS2 might provide useful physics for its applications in nano-devices.

An Isogeometric Over-Deterministic Method (IG-ODM) to Determine Elastic Stress Intensity Factor (SIF) and T-Stress

In order to examine the significance of Stress Intensity Factor and T-stress (K-T parameters) in modeling pressure-cracked structures, we propose a novel method known as the Isogeometric Over-Deterministic Method IG-ODM. IG-ODM utilizes the computation of stress and displacement fields through Extended Isogeometric Analysis to improve the geometry and enhance the crack. Subsequently, these results are incorporated into the Williams expression, resulting in a set of deterministic equations that can be solved using a common solving method; this particular combination has never been attempted before. IG-ODM enables the computation of stress intensity factor SIF, T-stress, and higher-order parameters in the Williams expansion. To validate the effectiveness of this method, we conducted tests on a single-edge uniaxial-stress-cracked plate and a central uniaxial-stress-cracked plate. The results showed an error ranging from 0.06% to 2%. The obtained results demonstrate accuracy and satisfaction when compared to existing findings.

Cracking direction in graphene under mixed mode loading

Crack branching in graphene under complex stresses is investigated by molecular dynamics simulations and boundary-layer models. Using the maximum energy release rate (MERR) criterion, an analogy of Wulff curve is plotted considering the anisotropic fracture toughness of hexagonal graphene, which shows the direction of crack kinking (local bump on the curve) is consistent with that of the weak fracture toughness (along zigzag edge). The angle of crack kinking prefers 0°/30°/60°, rather than the angle predicted by maximum circumferential stress criterion. The T-stress along crack front, obtained by the over-deterministic method, can indicate the possibility of kink by its transformation between positive and negative, especially for zigzag cracks. This study incorporates Wulff-like curve and MERR criterion to provide an accurate prediction of nanocrack paths in graphene.

Fatigue crack propagation under corrosion of high-strength steel

Corrosion is nearly unavoidable phenomenon in most of the metal materials. In this paper, its effect on the sharp edge crack propagation under tensile loading is investigated. A rectangular plate with a perpendicular crack and an elliptical corrosion pit nearby is modelled via finite elements and fracture behavior of the crack is analyzed. The multi-parameter fracture mechanics concept is applied, i.e. the higher-order terms of the Williams expansion are calculated by means of the over-deterministic method and utilized for tangential stress approximation in the vicinity of the crack tip. Thus, the generalized MTS criterion could be used for estimation of the crack deflection angle. The calculations were performed for a selected corrosion pit size and location considering various crack lengths. The results are discussed and a crack length with the highest probability to deflect from its original perpendicular propagation direction is found.

An improved over-deterministic method for determining Young’s modulus and Poisson’s ratio of materials using specimens with cracks

Young’s modulus and Poisson’s ratio are the two most basic mechanical properties of materials that determine the linear elastic deformation behaviours of materials. In this paper, a new method was proposed for determining the Young’s modulus and Poisson’s ratio of materials using specimens with pure mode I, pure mode II or mixed mode I/II cracks, on the basis of the concept of the over-deterministic method (ODM). Using this method, the strains of the nodes around the crack tip obtained from numerical simulation or experiments were brought into the Williams’ strain series. Then, a set of over-deterministic matrix equations could be obtained and solved using the least square method. Several numerical and experimental examples of cracked bodies were conducted to verify the effectiveness and accuracy of the improved ODM. Finally, the effects of the number of terms in Williams’ series expansion, the number and location of the nodes selected around the crack tip and the specimen geometry on the results were discussed in detail. Results indicate that the improved ODM can be used to effectively and accurately determine the Young’s modulus and Poisson’s ratio of materials from specimens with pure mode I, pure mode II or mixed mode I/II cracks. Accurate Young’s modulus and Poisson’s ratio can be obtained if the strains of the nodes around the crack tip are sufficient and accurate.

Dynamic brittle fracture in sharp V-notched rock specimens using digital image correlation method

Notched semi-circular bend (NSCB) method suggested by the International Society for Rock Mechanics and Rock Engineering (ISRM) only studies the static or dynamic brittle fracture in rock specimens with a local crack defect. This study aims to provide further insights into dynamic fracture behaviors of rock specimens containing a sharp V-notch loaded by a modified split Hopkinson pressure bar (SHPB) system. Ultra-high speed photography is utilized for real-time observation of the dynamic fracture process, and the two-dimensional displacement fields on the specimen surface are calculated using digital image correlation (DIC) method. The fracture parameters including the dynamic notch stress intensity factor (NSIF) at the sharp V-notch tip, and the dynamic stress intensity factor (SIF) and crack velocity at the propagating crack tip are first extracted from the displacement fields using the over-deterministic method. The effects of notch angle and loading rate on the dynamic notch fracture toughness before macrocrack onset and the dynamic crack propagation toughness after macrocrack initiation are subsequently studied. The experimental results are eventually compared with the numerical data obtained by the finite element method (FEM) and the existing results in the previous works. These results validate that the DIC method is applicable to study the dynamic brittle fracture in sharp V-notched rock specimens and predict the fracture resistance of V-notched rock components in actual rock engineering.

Crack Deflection Under Mixed-Mode Loading Conditions in Fine-Grained Composites Based on Water Glass-Activated Slag

This paper is devoted to an analysis of the crack deflection angle in a semi-circular disc made of a fine-grained composite based on water glass-activated slag. Three-point bending together with an inclined crack ensures the crack propagation in I + II mixed mode. Generally, concrete material exhibits quasi-brittle fracture behavior, which is difficult to understand. Multi-parameter fracture mechanics (MPFMs) concept shall help us to describe the crack propagation and the influence of choice of a suitable distance for application of generalized fracture criteria is investigated. In this work, the well-known maximum tangential stress (MTS) criterion as well as strain energy density (SED) criterion are applied in their multi-parameter (MP) form to find the initial kink angle. Williams power expansion is used to approximate the crack-tip stress field. Its coefficients need to be calculated numerically via combination of the finite-element (FE) solution and over-deterministic method. Results are discussed and recommendations are stated regarding the critical length parameter as well as the number of the Williams expansion (WE) coefficients.

An Over-deterministic Method Based on Atomistic Stress Fields: Higher Order Terms of the Williams power expansion

The investigation of the atomistic near crack-tip stress fields under Mixed Mode loading via molecular dynamics method implemented in a flexible classical molecular dynamics method software LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) is presented. The molecular dynamics modelling is applied for evaluation of the coefficients of higher-order terms of the Williams’ expansion -based approximation for the near crack tip fields in isotropic linear elastic materials. Fracture mechanics parameters such as stress intensity factors, higher-order coefficients of the Williams power-series for a copper plate with a central crack under Mode I and Mixed Mode (Mode I+Mode II) loadings are obtained by atomistic simulations. All computational experiments are realized in LAMMPS with the Embedded Atom Method (EAM) potential. The atomistic values of higher-order coefficients of the Williams series expansion are computed by the over-deterministic method and liken to the values obtained from classical continuum fracture mechanics solutions. It is demonstrated that the conventional continuum fracture theory successfully describes fracture quantities at extremely limited singular stress field of only several nanometers.

Experimental study on the interaction between two cracks by digital photoelasticity method: construction of the Williams series expansion

The multi-point Williams series expansions for the near-crack-tip fields in the vicinity of two interacting cracks in an isotropic linear elastic material are experimentally and numerically determined. The experimental method used to study the interaction of two cracks is the digital photoelasticity method which is applied for determination of coefficients of the higher-order terms of the Williams series expansion for a plate of finite width with two interacting cracks. The digital image processing tool for handling of the entire set of experimental data isochromatic fringes and isoclinic phase maps obtained from the photoelasticity experiments is developed and exploited. The digital image processing tool is relied on the classical Ramesh technique but enables to scan the experimental image in any direction and to analyse the image after any number of logical operations. The realized digital image analysis allows us to extract photoelastic data from the entire field isochromatic and isoclinic fringe patterns. In the digital image processing fringe analysis, the optical data contained in the transmission photoelastic isochromatic fringe patterns in the vicinity of the crack tips are converted into text file and then the points of isochromatic fringes with minimum light intensity are used for estimating linear elastic fracture mechanics parameters. The multi-parameter stress field description with the first fifteen terms is used. The mixed mode fracture parameters, especially stress intensity factors, T-stresses and higher-order coefficients of the Williams series expansion are obtained for specimen configurations like plates with two interacting cracks using 1) the classical over-deterministic method in the form based on the stress field and 2) the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm for minimization of the objective function. It is demonstrated that the BFGS method has indisputable advantages over the classical over-deterministic method. The significance of higher-order terms in the Williams expansion is proved for the considered cracked specimen. It is shown that the higher order terms are needed for accurate characterization of the stress field in the vicinity of the crack tip. The experimental SIF values, T-stresses and higher-order coefficients estimated using the proposed method are compared with finite element analysis (FEA) results, and are found to be in good agreement. This verifies the experimental analysis

Computation of conventional fracture mechanics parameters via molecular dynamics simulations

The Williams series expansion is the fundamental solution of conventional continuum fracture mechanics for characterizing the stress fields around the crack tip in a homogeneous material in the linear elastic regime. Nowadays stress intensity factors, T-stresses and the higher-order coefficients of the Williams series expansion are proved to be important parameters describing the near crack-tip fields in isotropic linear elastic materials. The central aim of this study is to understand if one can obtain these fracture parameters of conventional fracture mechanics from atomistic simulations based on molecular dynamics simulations. The ability to describe fracture processes at atomic scale via stress intensity factors, T-stresses and higher-order coefficients will provide the opportunity to take into account many effects such as material microstructures, crystallographic orientation, chemical compositions and concentrations and others. In this research the values of stress intensity factors, T-stress and coefficients of the Williams series expansion are obtained using atomistic simulations based on molecular dynamics method with a classical molecular dynamics code Large-scale Atomic/Molecular Massively Parallel Simulator. The classical over-deterministic method is used to determine stress intensity factors, T-stresses and coefficients of higher-order terms of the Williams series expansion of the near crack-tip stress field from molecular dynamics modelling of a plate with a central crack. The accuracy of the proposed approach is tested for this rather simple cracked configuration. There is the theoretical analytical solution with all the coefficients of the higher-order terms in the Williams series expansion. The existing theoretical solution allows us to compare the angular distributions of the stress tensor components for a large plane with the central crack. It is shown that results obtained from molecular dynamics simulations and the theoretical analytical solutions are in good agreement.

Importance of the Higher Order Terms of the Williams series expansion: Experimental Aspects and Finite Element Simulations

The study is devoted to the multi-point Williams series expansion for the crack-tip fields in isotropic linear elastic materials and methods of estimating the coefficients of higher order terms in the power expansion. It is shown that higher order terms are very important in processing of the experimental data obtained from interference – optic methods. It is demonstrated that the accurate interpretation of the experimental data requires keeping, considering and evaluating the higher order terms of the Williams expansion. A large cohort of experiments on cracked specimens from optically sensitive materials using digital photoelasticity method was performed. The interference fringe patterns obtained show that the correct description of the stress fields in the vicinity can be given if we take into account higher order terms in the asymptotic presentation of the near crack tip fields. Two interference-optical methods were used. Digital photoelasticity method and digital holography interferometry technique are applied for determination of the coefficients of the Williams series expansion. It is shown that we can restore theoretically the stress field in the vicinity of the crack tip if at least fifteen terms in the power expansion are hold. The classical over-deterministic method is used for the evaluation of the coefficients.

Computational and experimental identification of coefficients of the Williams series expansion by considering higher order terms in the cracked specimens through digital image analysis

In the study, digital photoelasticity method is used for finding of the linear elastic fracture mechanics parameters for a plate of finite width with two inclined interacting cracks. The digital image processing tool for the whole set of experimental data namely isochromatic fringes and isoclinic phase maps obtained from the photoelasticity experiments is developed and exploited. The digital image processing tool is based on the well-known Ramesh technique but allows us to scan the image in any direction and to analyse the image after any number of logical operations. The performed digital image analysis allows us to extract photoelastic data from the whole field isochromatic and isoclinic fringe patterns. In the digital image processing fringe analysis, the optical data contained in the transmission photoelastic isochromatic fringe patterns near the crack tips were converted into text file and then the points of isochromatic fringes with minimum light intensity were used for evaluating fracture mechanics parameters. The multi-point stress field description is used. The mixed mode fracture parameters, especially stress intensity factors, T-stresses and higher-order coefficients of the Williams series expansion are obtained for specimen configurations like plates with two inclined interacting cracks using 1) the algorithm based on the classical over-deterministic method and 2) the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm for minimization of the objective function. It is shown that the latter method has undeniable advantages over the classical over-deterministic method. The effects of higher-order terms in the Williams expansion were analysed for different the considered cracked specimen. It is shown that the higher order terms are needed for accurate characterization of the stress field in the vicinity of the crack tip. The experimental SIF values, T-stresses and higher-order coefficients estimated using the proposed method are compared with finite element analysis (FEA) results, and are found to be in good agreement.

Experimental determination and finite element analysis of coefficients of the multi-parameter Williams series expansion in the vicinity of the crack tip in linear elastic materials. Part II

In this study coefficients of the multi-parameter Williams power series expansion for the stress field in the vicinity of the central crack in the rectangular plate and in the semi-circular notched disk under bending are obtained by the use of the finite element analysis. In SIMULIA Abaqus, the finite element analysis software, the numerical solutions for these two cracked geometries are found. The rectangular plate with the central crack has the geometry similar to the geometry used in the digital photoelasticity. Numerical simulations of the same cracked specimen as in the experimental photoelasticity method are performed. The numerical solutions obtained are utilized for the determination of the coefficients of the Williams series expansion. The higher-order coefficients are extracted from the finite element method calculations implemented in Simulia Abaqus software package and the outcomes are compared to experimental values. Determination of the coefficients of the terms of this series is performed using the least squares-based regression technique known as the over-deterministic method, for which stresses data obtained numerically in SIMULIA Abaqus software are taken as inputs. The plate with a small central crack has been considered either. This kind of the cracked specimen has been utilized for comparison of coefficients of the Williams series expansion obtained from the finite element analysis with the coefficients known from the theoretical solution based on the complex variable theory in plane elasticity. It is shown that the coefficients of the Williams series expansion match with good accuracy. The higher-order terms in the Williams series expansion for the semi-circular notch disk are found.

Photoelastic study of a double edge notched plate for determination of the Williams series expansion

In this work, digital photoelasticity method is applied for assessment of the crack tip linear fracture mechanics parameters for a plate with double edge notches and different other crack configurations. The overarching objective of the study is to obtain the coefficients of the Williams series expansion for the stress and displacement fields in the vicinity of the crack tip by the digital photoelasticity technique for the double edge notched plate. The digital image processing tool for experimental data obtained from the photoelasticity experiments is developed and utilized. The digital image processing tool is based on the Ramesh approach but allows us to scan the image in any direction and to analyse the image after any number of logical operations. In the digital image processing isochromatic fringe analysis, the optical data contained in the transmission photoelastic isochromatics were converted into text file and then the points of isochromatic fringes with minimum light intensity were used for evaluating fracture mechanics parameters. The multi-parameter stress field approximation is used. The mixed mode fracture parameters, especially stress intensity factors (SIF) are estimated for specimen configurations like double edge notches and inclined center crack using the proposed algorithm based on the classical over-deterministic method. The effects of higher-order terms in the Williams expansion were analysed for different cracked specimens. It is shown that the higher order terms are needed for accurate characterization of the stress field in the vicinity of the crack tip. The experimental SIF values estimated using the proposed method are compared with analytical / finite element analysis (FEA) results, and are found to be in good agreement.

The digital photoelasticity method and finite element analysis for determination of the multi-point crack-tip field series expansions for notched semi-disks

The digital photoelastic method for the construction of the multi-point crack-tip series expansions in isotropic elastic semi-disks is employed. The series of cracked specimens in the form of the semi-disk with vertical and inclined notches from optically active materials (polycarbonates) were produced and tested under three-point bending. The principal intent of this experimental procedure of the phototelasticity technique was estimation of the amplitude coefficients in the Williams expansion in which the non-singular high-order terms are taken into account. The first ten coefficients in the multi-point series expansion have been ascertained. To retrieve the points belonging to the isochromatic fringe patterns and having the minimum light intensity the digital image processing scheme was utilized. The scheme of the over-deterministic method for evaluation of the amplitude coefficients (fracture mechanics parameters) is applied. Additionally, the problem was reduced to the problem of minimizing the objective function which is the quadratic form for the stress field in terms of unknown fracture mechanics parameters. The finite element analysis for the same series of the notched specimens was performed. The demeanour of the stress field in the neighbourhood of the notch tip for different types of mixed mode loading was elucidated from finite element analysis. The coefficients of the asymptotic presentation of the stress and displacement fields are recovered from finite element results. It is shown that the multi-point coefficients are in good agreement with experimental observations.

Atomistic Determination of Fracture Mechanics Parameters

Stress intensity factors, T-stresses and higher order coefficients in the Williams series expansion are the fundamental concepts of continuum fracture mechanics for characterizing the stress fields around the crack tip in a homogeneous material in the linear regime. It is well-known that critical values of stress intensity factors stipulate the of materials to growth a crack. Nowadays the modern multi-purpose computational tools such as Simulia Abaqus allow us to calculate T-stresses in cracked specimens and structures along with stress intensity factors. The aim of this study is to understand if one can obtain this fracture parameters of conventional fracture mechanics from atomistic simulations based on molecular dynamics method. The ability to describe fracture processes at atomic scale by means of stress intensity factors, T-stresses and higher-order coefficients will provide the opportunity to take into account many effects such as material microstructures, chemical compositions and concentrations and others. In this study the values of stress intensity factors, T-stress and coefficients of the Williams series expansion are determined using atomistic simulations based on molecular dynamics method with a classical molecular dynamics code Large-scale Atomic/Molecular Massively Parallel Simulator. The over-deterministic method is used to determine SIFs, T-stresses and coefficients of higher-order terms from molecular dynamics modelling of a plate with a central crack. The accuracy of the proposed approach is tested for this rather simple cracked configuration. There is the theoretical analytical solution with all the coefficients of the higher-order terms in the Williams series expansion. The existing theoretical solution allows us to compare the angular distributions of the stress tensor components for a large plane with the central crack. It is shown that results obtained from molecular dynamics simulations and the theoretical analytical solutions coincide qualitatively.

Photoelastic evaluation of stress fields and coefficients of multi-parameter asymptotic expansion of the crack-tip stress field

In this work, digital photoelasticity method is appliedfor estimation of the crack tip fracture mechanics parameters for a plate with double edge notches and different crack configurations.The digital image processing tool for experimental data obtained from the photoelasticity experiments is developedand utilized. The digital image processing tool is based on the Ramesh approach but allows us to scan the image in any direction and to analyse the image after any number of logical operations. In the digital image processing fringe analysis, the optical data contained in the transmission photoelasticisochromatics were converted into text file and then the points of isochromatic fringes with minimum light intensity were used for evaluating fracture mechanics parameters. The multi-parameter stress field approximation is used. The mixed mode fracture parameters, especially stress intensity factors (SIF) are estimated for specimen configurations like double edge notches and inclined center crack using the proposed algorithm based on the classical over-deterministic method. The effects of higher-order terms in the Williams expansion were analysed for different cracked specimens. It is shown that the higher order terms are needed for accurate characterization of the stress field in the vicinity of the crack tip. The experimental SIF values estimated using the proposed method are compared with analytical/finite element analysis (FEA) results, and are found to be in good agreement.

Experimental determination and finite element analysis of coefficients of the multi-parameter Williams series expansion in the vicinity of the crack tip in linear elastic materials. Part I

This study aims at obtaining coefficients of the multi-parameter Williams series expansion for the stress field in the vicinity of the central crack in the rectangular plate and in the semi-circular notched disk under bending by the use of the digital photoelasticity method. The higher-order terms in the Williams asymptotic expansion are retained. It allows us to give a more accurate estimation of the near-crack-tip stress, strain and displacement fields and extend the domain of validity for the Williams power series expansion. The program is specially developed for the interpretation and processing of experimental data from the phototelasticity experiments. By means of the developed tool, the fringe patterns that contain the whole field stress information in terms of the difference in principal stresses (isochromatics) are captured as a digital image, which is processed for quantitative evaluations. The developed tool allows us to find points that belong to isochromatic fringes with the minimal light intensity. The digital image processing with the aid of the developed tool is performed. The points determined with the adopted tool are used further for the calculations of the stress intensity factor, T-stresses and coefficients of higher-order terms in the Williams series expansion. The iterative procedure of the over-deterministic method is utilized to find the higher order terms of the Williams series expansion. The procedure is based on the consistent correction of the coefficients of the Williams series expansion. The first fifteen coefficients are obtained. The experimentally obtained coefficients are used for the reconstruction of the isochromatic fringe pattern in the vicinity of the crack tip. The comparison of the theoretically reconstructed and experimental isochromatic fringe patterns shows that the coefficients of the Williams series expansion have a good match.

Determination of SIFs and T-stress using an over-deterministic method based on stress fields: Static and dynamic

An over-deterministic method based on crack-tip stress fields is proposed for evaluating stress intensity factors (SIFs) and T-stress at crack tip for mode I, mode II or mixed-mode I/II cracks under static or dynamic loading. In this method, the coordinates and stresses of the nodes around the crack tip, obtained from finite element method (FEM) or extended finite element method (XFEM), are brought into the expansion of Williams’ stress series when the cracked body is subjected to static or dynamic loading. Then, a set of over-deterministic equations can be obtained, and the optimal solution of SIFs as well as T-stress can be achieved by solving these equations using the least square method. Various cracked bodies are employed to verify the accuracy of the proposed method, and the effects of the number of terms in Williams’ series expansion, the number and location of the nodes selected around the crack tip, and the mesh around the crack tip on SIFs and T-stress are detailed discussed. Results show that the proposed method is quite simple and convenient to obtain SIFs and T-stress under static or dynamic loading, with being in good agreement with the solutions from previous works. Satisfactory results can be achieved only if the coordinates and stresses of the nodes around the crack tip are accurate enough. In addition, it can be found that the proposed method can also be used to calculate higher-order coefficients of Williams’ series expansion.