Strain Energy Density Factor Research Articles

Analysis of high temperature and strain amplitude effects on low cycle fatigue behavior of pitting corroded killed E350 BR structural steel

High-tension structural steels are prone to accelerated fatigue damage from pitting corrosion and high-temperature. Despite adverse effects, research on their low cycle fatigue (LCF) behavior is limited. Specifically, studies analyzing temperature-dependent pit sensitivity effects, considering pit-related material’s susceptibility to surface topographic features variation and stress concentration are lacking. This study conducts LCF tests on pitting corroded killed E350 BR structural steel at multiple strain amplitudes and high temperatures. It develops temperature-dependent parameters, such as the cyclic softening pit sensitivity factor, suitable for integration into existing approaches like total cyclic plastic strain energy density (CPSED), power-law, average strain energy density (SED), Coffin-Manson, and pit stress intensity factor (pit-SIF). Further, it proposes multiple linear regression-based prediction models relating total CPSED and average SED with strain amplitude and temperature. Corroded specimens show higher plastic deformation and reduced peak stress, fatigue life, and total CPSED compared to uncorroded ones. The developed parameters, integrated with average SED approach, predicts LCF life within an error band of ±1.5, while power-law relationship reduces it to ±1.2. Moreover, pit-SIF approach estimates fatigue life within an error band of ±1.5. The findings provide critical knowledge for enhanced component design, leading to structural safety, performance, and fire resilience.

Theoretical optimization of micropillar arrays for structurally stable bioinspired dry adhesives

Inspired by the excellent adhesion performances of setae structure from organisms, micro/nano-pillar array has become one of the paradigms for adhesive surfaces. The micropillar arrays are composed of the resin pillars for adhesion and the substrate with different elastic modulus for supporting. The stress singularity at the bi-material corner between the pillars and the substrate can induce the failure of the micropillar-substrate corner and further hinder the fabrication and application of micropillar arrays, yet the design for the stability of the micropillar array lacks systematical and quantitative guides. In this work, we develop a semi-analytical method to provide the full expressions for the stress distribution within the bi-material corner combining analytical derivations and numerical calculations. The predictions for the stress within the singularity field can be obtained based on the full expressions of the stress. The good agreement between the predictions and the FEM results demonstrates the high reliability of our method. By adopting the strain energy density factor approach, the stability of the pillar-substrate corner is assessed by predicting the failure at the corner. For the elastic mismatch between the pillar and substrate given in this paper, the stability can be improved by increasing the ratio of the shear modulus of the substrate to that of the micropillar. Our study provides accurate predictions for the stress distribution at the bi-material corner and can guide the optimization of material combinations of the pillars and the substrate for more stable bioinspired dry adhesives.

Influence of distinct testing methods on the mode-I fracture toughness of Longmaxi shale

Abundant shale gas in shale reservoirs can be successfully extracted by the fracking technique. The formation of hydraulic fractures is related to the mode-I fracture toughness that represents the resistance to cracking propagation. Two typical Longmaxi shale samples with horizontal and vertical bedding planes are selected in this work. To adequately measure the mode-I fracture resistance of shale, this research adopts four mode-I test specimens with different geometries (cuboid, cylinder, disk, and semi-disk). The experimental results demonstrate that there is an acceptable linear correlation between mode-I fracture resistance and critical T-stress. Further, the discrepancies in mode-I fracture resistance are due to distinct T-stresses that exist in the mode-I test specimens. Consequently, the mode-I fracture toughness KIc for the shale with horizontal bedding is 1.57 MPa·m0.5, while that for the shale with vertical bedding is 1.36 MPa·m0.5. Considering the influences of singular and non-singular stress terms, the GMTSN (generalized maximum tangential strain) and GMTSEDF (generalized maximum tangential strain energy density factor) fracture criteria can provide theoretical explanations for the variation of mode-I fracture resistance measurements. Based on the AE (acoustic emission) technique, the classical RA-AF method can identify the fracture mechanisms of distinct mode-I test specimens. However, the differences in the high-density zones of the RA-AF distribution are conspicuous for distinct mode-I testing methods.

Analytical solutions for the plane thermoelastic problem of a nano-open crack in one-dimensional hexagonal quasicrystal non-periodic plane

As a stable material at high temperatures, the presence of cracks and other defects greatly reduces the performance of quasicrystals (QCs). The plane thermoelastic problem of one-dimensional hexagonal quasicrystals (1DHQ) non-periodic plane containing a nano-open crack is analyzed using surface elasticity theory. The closed form solutions are obtained to describe the thermoelastic field corresponding to the nano-crack by the Fourier integral transformation technique. The analytical solutions of the thermal stress intensity factors (TSIFs) and the strain energy density factor (SEDF) at the crack tip are derived explicitly. The influence of the applied loads, the surface effects and the multi-field coupling effects on the TSIFs and SEDF are discussed. The numerical results reveal that the surface effects have an obvious impact on TSIFs with small applied loads, and the surface effects will inhibit the propagation of nano-open cracks. The research results of this study provide some reliable theoretical references for the development of nano-QCs.

Controlled crack propagation of flexible Film/Substrate assembly by biaxial stress

Crack has long been regarded as a form of material failure. At the same time, it is also an effective processing method to create some patterns on the surface of objects. Currently, most crack-based processing methods are achieved by applying equai-biaxial thermal stress. However, the stress ratio of equi-biaxial stress state is fixed, and its flexibility is very limited. In this paper, a method is developed to control the crack propagation in thin film coated on substrate by using generalized biaxial stress. Under some reasonable assumptions on the shear stresses in the interface layer between the film and its substrate, an analytical stress distribution in film under biaxial stress state is obtained. Then, the strain energy density factor is used to predict the propagation direction of a prefabricated crack. By changing the biaxial stress state undergone by film, the crack propagation direction can be controlled. Some verification experiments are conducted, which are consistent with the theoretical prediction.

Dynamic failure behavior and damage evolution process of holed sandstone under impact loads

In this research, the damage process and dynamic failure behavior of some sandstone rocks, containing an elliptical hole, were evaluated. For this reason, a series of laboratory tests under both static and dynamic loads in 4 different orientations of the elliptical hole was performed. Modified Split Hopkinson Pressure Bar testing machine was employed to apply impact load. Dynamic fracture evolution was recorded by utilizing a high-speed camera to evaluate the macro damage evolution process. Scanning electron microscope analysis and thin section studies were also used to assess mineral's behavior regarding micro-crack evolution. Furthermore, the experimental tests were numerically simulated, and strain energy density and dynamic stress concentration factor were measured to evaluate the effect of elliptical hole inclination on dynamic response. The results indicated that when the largest diameter is perpendicular to core axis, the rock shows the lowest strength in both static and dynamic loading conditions. Also, with increasing hole inclination, both strain energy density and dynamic stress concentration factor decreased. Evaluation of fracture surface indicated that grain-boundary cracks are the dominant type of cracks and iron oxide cement distribution has a vital role in the development of the cracks.

Aging Stability Analysis of Slope Considering Cumulative Effect of Freeze–Thaw Damage—A Case Study

The change of physical and mechanical properties of slope rock mass in open-pit mines in seasonally frozen area under the action of freeze–thaw cycles is one of the main reasons for slope instability. In this paper, taking the mechanical parameters of coal seam and sandstone layer in the Beitashan Pasture Open-Pit Mine in Xinjiang as the research object, considering the combined effect of the frost-heave tensile stress in the crack perpendicular to the crack surface and the three-dimensional confining pressure in the crack, the criterion for cracking of fractured rock mass under freeze-thaw condition is determined by applying the principle of stress superposition and the theory of strain energy density factor, and the theoretical frost-heave stress required for cracking is deduced. On this basis, the sensitivity analysis of the fixed factors and variable factors to the theoretical frost-heave stress was performed, respectively. Finite element analysis was utilized to analyze the slope stability under the attenuation of five groups of different rock mass mechanical properties and to determine the slope angle required for the slope stability. Seven different slope angles of sidewall mining ranging from 36° to 51° are analyzed. The results of finite element analysis show that considering the timeliness difference of rock mass parameters with time, the safety factor of slope is reduced from the original 1.70 to 1.18, and 91,500 tons of coal resources can be recovered every year, with remarkable economic benefits.

Directional independence of the Kaiser effect of sandstone with high diameter–thickness ratios

The Kaiser effect(KE) was investigated considering strain energy density factor theory in fracture mechanics and the formation mechanism of the independence of effect direction.The results were as follows: Felicity ratio (FR) increased with increasing rotation angle, indicating that the memory of the original rock stress was affected by the rotation angle. Thus, the directional independence of the KE was directly related to the rotation angle. In theoretical calculations, the directional independence of the KE was directly related to the relative value of critical stress. The change in the relative value of the critical stress increased the difficulty of cracking, affecting the accuracy of the KE point.Poisson’s ratio considerably affected the FR value, and the in situ stress values of the KE method on various types of rock samples differed. The results of the study can provide insight into the KE mechanism and development of effective methods.

The micromechanics‐based rate‐dependent constitutive model of flawed rocks at intermediate strain rate

AbstractIn the previous works conducted by Zhou and Yang, the deformation induced by wing crack coalescence was not investigated. In this paper, a novel micromechanics‐based rate‐dependent constitutive model is proposed to investigate the effect of wing crack coalescence on the mechanical properties of rocks at intermediate strain rate. The rate‐dependent damage mechanism is revealed using the frictional sliding crack model. Wing crack interaction is defined using pseudo‐traction method as well as the dynamic stress intensity factors. Strain energy density factor approach is applied to determine the critical condition of wing crack initiation, propagation, and coalescence. The strain softening behaviors of flawed rocks is studied by using thermodynamic theory.

Modeling and analysis of forced vibration of the thin-walled cylindrical shell with arbitrary multi-ring hard coating under elastic constraint

In this paper, a parameterized multi-partitioning method is presented for modeling arbitrary multi-ring hard coating treatment for the thin-walled cylindrical shell under different ring numbers and coating ratios, while the Rayleigh–Ritz method and the Chebyshev orthogonal polynomials of the second kind are employed to derive the governing equations of motion and the admissible displacement, respectively. Moreover, a unified formulation of external load based on the trigonometric and Chebyshev series is developed for forced vibration analysis of the shell with arbitrary axial half-wave number and circumferential wave number. Based on the experimental resonant frequencies, an improved inverse identification procedure is proposed in which the genetic algorithm and pattern search algorithm are sequentially employed to identify the stiffness coefficients of elastic constraint. Good agreement between the experimental and numerical resonant frequencies and responses exhibits the reliability and effectiveness of both the stiffness identification method and semi-analytical model. The identified results happen to find that the constraint stiffness may have the inherent property of frequency dependence due to the existence of micro sliding which would further result in soft nonlinearity in frequency response curve as shown in experimental results. Additionally, the influences of ring number and coating ratio on vibration and damping characteristics of the thin-walled cylindrical shell are investigated by introducing the average strain energy density and average modal loss factor.

Crack growth angle prediction of an internal crack under mixed mode load for unfilled elastomer using the strain energy density factor

ABSTRACTThis article investigates the prediction of the crack growth angle of an existing internal crack under mixed mode loading at the crack tip for an unfilled ethylene propylene diene terpolymer rubber (EPDM). For the realization of mixed mode loading, the cracks of the uniaxial loaded specimens were oriented with different angles to the loading direction. The energy density factor was used as a potential criterion for determining the crack growth angle. The determination of the strain energy density factor was carried out simulatively in Abaqus. The second‐order Ogden model was used to describe the rubber‐like material behavior. The relative local minimum of the strain energy density factor provides the possible growth angle. The experimental investigations show that the initial cracks grow orthogonally to the loading direction for the different crack orientation angles. For the crack orientation angle parallel to the load direction, the crack growth was observed because the strong stretching of the specimen caused strong necking in the crack region. The crack growth for the remaining crack orientation angles were induced due to shear loading at the crack tip. The predictive angle of different crack orientation angles shows very good accordance to the measured crack growth angles.

Analysis of Mixed-Mode I/II/III Fracture Toughness Based on a Three-Point Bending Sandstone Specimen with an Inclined Crack

The crack-propagation form may appear as an arbitrary mixed-mode fracture in an engineering structure due to an irregular internal crack. It is of great significance to research the mixed-mode fracture of materials with cracks. The coupling effect of multiple variables (crack height ratio, horizontal deflection angle and vertical deflection angle) on fracture parameters such as the stress intensity factors and the T-stress are the key points in this paper. A three-point bending specimen with an inclined crack was proposed and used to conduct mixed-mode fracture research. The fracture parameters were obtained by finite element analysis, and the computed results showed that the pure mode I fracture and mixed-mode fractures (mode I/II, mode I/III and mode I/II/III) can be realized by changing the deflection angles of the crack. The pure mode I and the mixed-mode fracture toughness of sandstone were obtained by a series of mixed-mode fracture experiments. The experimental results were analyzed with the generalized maximum tangential strain energy density factor criterion considering T-stress. The results showed that the non-singular term T-stress in the fracture parameters cannot be ignored in any mixed-mode fracture research, and the generalized maximum tangential strain energy density factor criterion considering T-stress can better predict the mixed-mode fracture toughness than other criteria.

An Improved Support Domain Model of Smoothed Particle Hydrodynamics Method to Simulate Crack Propagation in Materials

Smoothed particle hydrodynamics (SPH) has its unique advantages in simulating large deformation. However, in its calculation, it is easy for some regional particles to break off. This paper puts forward the concept of virtual crack boundary to solve crack propagating problems in the framework of SPH. The parameters of virtual crack boundary are determined by the projection method of descending dimension, which improves the support domains of crack tip particles. Based on the principle of minimum strain energy density factor in fracture mechanics, the initiation and extended directions of cracks in materials are estimated. By simulating the generation and expansion of three modes of cracks in materials, the SPH calculation results of the three types of support domain forms are compared with the ABAQUS simulation results and experiment results. It indicates that the method of virtual crack boundary method is more stable than the one based on discontinuous medium. And it can generate more reasonable results than the method based on continuous medium. This paper proves the effectiveness of strain energy density factor based on fracture mechanics theory in searching the direction of crack expansion using SPH, and explores the application of new support domain model in SPH crack solution.

Thermoelastic analysis of a cracked orthotropic strip under heat conduction effect of crack interior

The problem of a Griffith crack embedded in an orthotropic strip is investigated under a combination of thermal and mechanical loadings. The extended thermal-medium crack model is used to address the effects of thermal conductivity of crack interior on the fracture parameters of concerns. Using the Fourier transform technique, the boundary-value problem is reduced to solving singular integral equations with Cauchy kernel. The closed-form solutions of temperature fields are determined. In solving the elastic fields, the Lobatto–Chebyshev integration formula is applied to discretize the obtained singular integral equations as a system of algebraic equations, which is solved by elaborating an algorithm. Numerical results are reported to illustrate the effects of the thermal conductivity inside crack and the thickness of orthotropic strip on the partial insulation coefficient, the thermal stress intensity factor, and the strain energy density factor. The crack initiation behavior is analyzed according to the normalized strain energy density factor. The observations reveal that the crack-tip thermoelastic fields in an orthotropic strip are dependent on the combined effects of the thermoelastic loadings, the physical and geometrical properties of the material and crack. Some comparisons with the existing works are offered to show the novelty and applicability of the obtained results.

Multi-parameter failure assessment of a bi-material V-notch – Crack initiation from a free-edge singularity

Multi-parameter fracture mechanics criteria are used to describe crack initiation conditions in general singular stress concentrators. In the cases of stress concentrators described by only one singular stress term, multi-parameter description provides a qualitative and quantitative increase of accuracy. In this paper multi-parameter criteria, based on an average tangential stress and average strain energy density factor, are described and used to determine a crack initiation direction and the critical load in the case of a free-edge singularity. The critical load expresses a failure/stability criterion for the studied general singular stress concentrator. Contrary to criteria employing singular stress terms only, a criterion of this kind is able to cover the influence of the direction of the external load.

Crack Growth Direction in Mixed Mode Loading: A Strain Energy Density Approach

The crack growth in metallic materials using fast and reliable simulations of 2-D and linear elastic finite element models is investigated. The effect of the stress intensity factor in mode I and II (KI, KII) on the fracture behavior of stainless steel and the associated strain energy density factor in mixed mode crack propagation were studied numerically to determine crack propagation angle θ in linear elastic fracture investigation. In order to implement the determination of the crack propagation direction using the strain energy density criterion S, the numerical finite element program ANSYS was used. ANSYS APDL macros were developed to generate the geometry, material properties, boundary conditions and mesh size of the model for the conducted analyses. To demonstrate the capability of crack propagation trajectories using the proposed method under mixed mode situation, an edge crack specimen was considered with initial crack having the same length but at different inclination angles under a uniaxial tension load. Results obtained from the developed models had a good agreement (average deviation of 4.63%) with the results available in the literatures.

Model experiment on dynamic behavior of jointed rock mass under blasting at high-stress conditions

The blasting response of jointed rock mass under high-stress conditions is studied in a laboratory experiment using model specimens. The crack initiation mode and initiation angle at the joint are analyzed, as well as the stress intensity factor and crack velocity. The blasting stress wave increases the stress concentration at the joint which promotes crack initiation and propagation. The dynamic behavior of the crack can be divided into two stages – the initiation stage and the propagation stage. The theoretical criterion based on the minimum strain energy density factor can well predict and explain the crack initiation behavior under blasting loading. The high-stress condition enhances the degree of shear failure of the crack initiation at the joint, which has significant effects on the dynamic behavior of subsequent crack propagation and leads to larger crack initiation angles.

Calculation of stress intensity factor in two-dimensional cracks by strain energy density factor procedure

In order to calculate the stress intensity factor (SIF) of crack tips in two-dimensional cracks from the viewpoint of strain energy density, a procedure to use the strain energy density factor to calculate the SIF is proposed. In this paper, the procedure is presented to calculate the SIF of crack tips in mode I cracks, mode II cracks and I+II mixed mode cracks. Meanwhile, the results are compared to those calculated by traditional approaches or other approaches based on strain energy density and verified by theoretical solutions. Furthermore, the effect of mesh density near the crack tip is discussed, and the proper location where the strain energy density factor is calculated is also studied. The results show that the SIF calculated by this procedure is close to not only those calculated by other approaches but also the theoretical solutions, thus it is capable of achieving accurate results. Besides, the mesh density around the crack tip should meet such requirements that, in the circular area created, the first layer of singular elements should have a radius about 0.05 mm and each element has a circumferential directional meshing angle to be 15°–20°. Furthermore, for a single element around the crack tip, the strain energy density factor is suggested to be calculated in the location where half of the sector element’s radius from the crack tip.

Multi-parameter average strain energy density factor criterion applied on the bi-material notch problem

The stress field near a bi-material notch tip has the singular character. The power of singularity is lower in comparison to the case of a crack. The stress field can be described by the asymptotic stress series, in which each term contains generalized stress intensity factor and stress term exponent. The terms can be either singular or non-singular depending on the value of stress term exponent. The exponents are determined as solution of eigenvalue problem and depend on material properties and local geometry. The factors are calculated by analytical-numerical approaches and depend on global geometry and applied loading. The singular terms become unbounded when approaching the bi-material notch tip while the non-singular terms vanish. The non-singular terms increase precision in stress description on larger distances from the notch tip. The crack initiation conditions from the notch tip can be assessed by the average strain energy density factor (SEDF) criterion. In the case of bi-material notch problem, the crack can initiate in the direction of global minimum, the local minimum or the interface. The actual crack initiation load is calculated for each of abovementioned directions considering the individual fracture toughnesses. Contrary to the case of a crack, the direction of minimum of SEDF changes in dependence on the distance from the tip, therefore the average value over specific distance is used. The multi-parameter criterion considers singular and non-singular stress terms. Therefore in cases, when the specific distance is large, the non-singular terms provide means to improve precision of prediction of crack initiation parameters.

Multi-parameter average strain energy density factor criterion applied on the sharp material inclusion problem

The tip of a sharp material inclusion (SMI) embedded in a parent material is considered as the singular stress concentrator. The SMI tip is modelled as a special case of a multi-material junction. The singular stress behaviour is caused by the material mismatch and geometrical discontinuity. The power of singularity is lower in comparison to the case of a crack. The stress field can be described by the asymptotic stress series, in which each term contains generalized stress intensity factor and stress terms exponent. The exponents are determined as a solution of eigenvalue problem. The factors are calculated by combination of analytical and numerical approaches. The terms can be either singular or non-singular depending on the stress term exponent. When approaching the concentrator, the singular terms become unbounded while the non-singular terms vanish. The non-singular terms increase precision of stress description on larger distances from the point of singularity. In some cases they provide the only means to describe the stress field well. Because of the singular stress behaviour near SMI tip, this location is prone to crack initiation. The crack initiation conditions are calculated by the average strain energy density factor (SEDF) criterion. Contrary to the case of a crack, the direction of minimum of SEDF changes with distance from the singular point. Therefore, an averaged value over specific distance is used. If the specific distance is relatively large, the employment of non-singular terms in multi-parameter criterion can significantly improve the critical parameters prediction. Thanks to that e.g. the particle composite design can be optimized.