Nonlinear Criterion Research Articles

Stability analysis for viscoelastic fluid model with thermorheological effects: Linear and nonlinear approaches

This study investigates the stability analysis of Rayleigh-Bénard configuration for a viscoelastic fluid subject to thermorheological effects, using the D2-Chebyshev-τ method. The fluid is modeled as a third-order viscoelastic fluid. This study accentuates how salting the fluid layer affects the thresholds for the onset of instability in a fluid of third order encompassing physically realistic rigid boundaries. The dynamic model incorporates advection-diffusion of temperature and solute concentration and a modified Navier–Stokes equation. We determine instability thresholds for the complex non-Newtonian fluid by analyzing the linear stability of the steady-state conduction solution. Our analysis proves the strong form of the principle of exchange of stabilities, demonstrating that convective motions can only occur through stationary motion. Additionally, a nonlinear stability analysis using the energy method is performed, deriving an unconditional nonlinear stability criterion. The results provide a comprehensive understanding of how variable viscosity and viscoelasticity impact system stability. Both the viscosity parameter and the third-grade fluid parameter exhibit stabilizing effects. Notably, we observe a discrepancy between the linear and global nonlinear stability results, indicating the presence of a subcritical instability region. This study contributes to the understanding of complex fluid dynamics in non-linear mechanical systems, with potential applications in various industrial and natural processes.

Batch-sizing and machinability data systems for milling operations: An optimal sustainable cost of quality approach

Nowadays, manufacturers make every effort to achieve a higher quality of their products at an attractive cost. With all the introduced legislation and incentives in the developed world to address global warming, machining shops in the West also strive to cut greenhouse emissions. This article offers an optimal approach to the micro Computer-Aided Process Planning (CAPP) problem to optimize the internal quality cost and buffering effect while keeping the environmental impact low. To optimize the machining parameters, the mathematical model is developed for different milling operations, face, side, and peripheral. cutting speed, feed rate, axial depth of cut, radial depth of cut, nose radius, and batch sizing while maximizing profit and meeting customer demand. A Mixed-Integer Nonlinear Programming (MINLP) model is formulated and solved using Classical Constrained Nonlinear Optimization (CCNO) and Genetic Algorithms (GAs). Surface roughness, used as a metric to evaluate the desired quality level of a finished machined part type, is modeled as a Gaussian random variable to model the surface roughness of the machined part utilizing a cumulative normal distribution. The ratio of rework and scrap is calculated in terms of the surface roughness of the machined part shifting away from the target and exceeding upper and lower specification limits. The internal failure cost model, addressing both scrap and rework, is developed based on Taguchi’s quadratic loss function. CCNO is employed to validate the results obtained by GAs, relaxing the lot-sizing integrality constraint and, thus, the convexity of the produced relaxed model. An iterative method employing a developed multi-regression model is used to solve for the expended power consumption (an inherent highly nonlinear environmental criterion of the developed model) within both GAs and CCNO. This study reveals that the machining parameters substantially impact the cost components of the objective function as well as the scrap and rework quantities. A stringent quality cost target can force the model to optimize the feed rate and nose radius to minimize the internal failure quality cost while improving the environmental impact, including direct and indirect power consumption and CO2 emissions considerations.

Assessing the Reliability of Rock Slopes Based on the Nonlinear Strength Criterion

Assessing the Reliability of Rock Slopes Based on the Nonlinear Strength Criterion

Analyzing the Stability of a Connected Moving Cart on an Inclined Surface with a Damped Nonlinear Spring

This paper examines the stability behavior of the nonlinear dynamical motion of a vibrating cart with two degrees of freedom (DOFs). Lagrange’s equations are employed to establish the mechanical regulating system of the examined motion. The proposed approximate solutions (ASs) of this system are estimated through the use of the multiple-scales method (MSM). These solutions are considered novel as the MSM is being applied to a new dynamical model. Secular terms have been eliminated to meet the solvability criteria, and every instance of resonance that arises is categorized, where two of them are examined concurrently. Therefore, the modulation equations are developed based on the representations of the unknown complex function in polar form. The solutions for the steady state are calculated using the corresponding fixed points. The achieved solutions are displayed graphically to illustrate the impact of manipulating the system’s parameters and are compared to the numerical solutions (NSs) of the system’s original equations. This comparison shows a great deal of consistency with the numerical solution, which indicates the accuracy of the applied method. The nonlinear stability criteria of Routh–Hurwitz are employed to assess the stability and instability zones. The value of the proposed model is exhibited by its wide range of applications involving ship motion, swaying architecture, transportation infrastructure, and rotor dynamics.

A novel three dimensional failure criterion for quasi-brittle materials based on multi-scale damage approach

In this paper, we propose a novel three-dimensional micromechanics-based failure criterion to assess the load-bearing capacity of quasi-brittle materials under complex multiaxial stress conditions. This criterion not only inherits benefits of the multi-scale friction-damage coupling modeling approach but also accounts for the effect of the intermediate principal stress. Physically, the initiation and propagation of microcracks contribute to the damage, and the failure of the material ultimately occurs due to the unstable growth of microcracks. Simultaneously, plastic deformation, which results from frictional sliding along microcracks, is intimately coupled with the damage process. Employing friction-damage coupling up-scale analyses and introducing a novel parabolic local frictional law, we derive a new nonlinear compression meridian criterion within the upscaling framework. Moreover, by incorporating a Lode dependence function, this criterion effectively addresses variations in strength induced by the intermediate principal stress. To validate this criterion, we utilize data from triaxial compression, triaxial extension, and true triaxial tests conducted on various rock materials and concrete, all of which demonstrate excellent agreement.

Upper Bound Limit Analysis of Deep Tunnel Face Support Pressure with Nonlinear Failure Criterion under Pore Water Conditions

Based on the nonlinear failure criterion and modified tangential technique, the upper bound solutions of the critical supporting pressure on the deep tunnel face were obtained under pore water pressure conditions. The influence of parameters on the critical supporting pressure and collapse range was investigated according to the unlimited block failure mechanism. It was found that the upper bound solutions of the critical supporting pressure increase with the growth of the nonlinear coefficient and pore water pressure coefficient. The collapse range of the tunnel face scales out with the increase in the nonlinear coefficient and shrinks with an increasing pore water pressure coefficient. Moreover, with the increase in the nonlinear coefficient, the impact strength on critical supporting pressure and collapse range declines gradually. According to the calculated results, both the pore water pressure and nonlinear criterion factors have negative impacts on the stability of the tunnel face. Thus, more attention should be paid to these parameters to ensure face stability in deep tunnel construction.

Fatigue damage accumulation in a bolted joint subjected to the random loading

Fatigue damage in structures, which escalates with cyclic loading, is traditionally analyzed using blocks of constant amplitude. This study aims to derive the fatigue loading spectrum for the Cirrus SR20 multipurpose airplane and predict the fatigue life of its main spar connection. Using available reference loading data from acrobatic and training aircraft, the loading spectrum was established, and constant amplitude loading blocks were determined through the Rainflow cycle counting and statistical methods. Fatigue damage in the bolted joint was assessed using both linear and nonlinear Miner rules, revealing that results from nonlinear criteria tend to converge towards those from linear predictions as loading blocks increase. This approach provides a robust predictive framework for assessing the fatigue life of airplane components under variable amplitude loadings, potentially enhancing maintenance protocols and structural design durability.

A numerical strain-softening solution of a circular opening in nonlinear yield rock masses

Strain-softening behavior occurs when the surrounding rock reaches its peak load, complicating the prediction of stress and displacement. The linear Mohr-Coulomb yield criterion and semi-linear Hoek-Brown yield criterion are widely used in elastoplastic analysis. However, the established criteria struggle to universally describe the nonlinear behavior of various rock types. This study aims to overcome the difficulty and propose the solution of surrounding rock in a circular tunnel following any nonlinear yield criterion and plastic potential function. The derivation of the equivalent cohesive strength and frictional angle for an arbitrary nonlinear yield criterion in the form of σ1=fσ3 using the secant method, along with the determination of the equivalent dilatancy angle for any nonlinear plastic potential functions, was conducted. Concerning the elastoplastic solution of a thick-cylinder, a numerical strain-softening solution for a circular tunnel in nonlinear yield rock masses was derived. Verification of the proposed solution was performed against the established analytical and/or numerical elasto-(brittle-)plastic and strain-softening solutions in rock masses following the Mohr-Coulomb, (generalized) Hoek-Brown, and unified strength criteria. The impact of critical plastic shear strain on displacement, plastic radius, and residual radius was further analyzed. The predictions of displacement and residual region for the Jinping II project also meet the field test results.

Multiscale modeling of diffuse damage and localized cracking in quasi-brittle materials under compression with a quadratic friction law

The diffuse damage and localized cracking of quasi-brittle materials (i.e., rocks and concretes) under compression can be delineated by a matrix-microcrack system, wherein a solid matrix phase is weakened by a large number of randomly oriented and distributed microcracks, and the macroscopic cracking is formed by a progressive evolution of microcracks. Several homogenization-based multiscale models have been proposed to describe this matrix-microcrack system, but most of them are based on a linear friction law on the microcrack surface, rendering a linear strength criterion. In this paper, we propose a new quadratic friction law within the local multiscale friction-damage (LMFD) model to capture the plastic distortion due to frictional sliding along the rough microcrack surface. Following that, a macroscopic Ottosen-type nonlinear strength criterion is rationally derived with up-scaling friction-damage coupling analysis. An enhanced semi-implicit return mapping (ESRM) algorithm with a substepping scheme is then developed to integrate the complex nonlinear constitutive model. The performance of LMFD model is evaluated compared to a wide range of experimental data on plain concretes, and the robustness of ESRM algorithm is assessed through a series of numerical tests. Subsequently, to effectively describe the localized cracking process, a regularization scheme is proposed by combining the phase-field model with the established LMFD model, and the discretization independent crack localization is numerically verified.

Inspection of the nonlinear instability of electrified Casson fluids: a novel approach

The nonlinear electrohydrodynamic (EHD) instability of a circular cylindrical interface, separating two Casson prototypes is the aim of the present inspection. The motivation for the issue is owing to the increasing significance of several engineering and practicle physics. Because of the complicated mathematical processes, the contribution is performed through the viscous potential theory (VPT). The methodology resulted in a nonlinear characteristic equation of the interface displacement. This equation is transformed to simulate a hybrid Rayleigh-Helmholtz Duffing oscillator (HRDO) with complex coefficients. Guaranteeing the stability of fluid interfaces is crucial in industrial coating operations to provide consistent and even coatings. Gaining comprehension of the nonlinear stability criterion helps to enhance the management of coating thickness and uniformity. A novel methodology via the Non-perturbative approach (NPA) is employed. Through the case of the real coefficients, the later form is validated using a numerical solution (NS). It is concluded that the Ohnesorge contribution has a destabilizing effect on the stability zone in contrast with permeability factor. The electrical coefficients are found to be decaying factors in the stability configuration. It is also established that the viscosity decays the nonlinear steadiness requirement of the structure. The approach is a simple, promising, effective, and interesting.

A unified nonlinear yield criterion for fracture and fault slip in quasi-brittle rocks

The micromechanical damage model has been developed to better describe multiscale fracture behavior in quasi-brittle rocks. While the deduced yield criterion under tension and compression has been carefully investigated in different stress spaces, it fails to capture a smooth transition from extension to shear fracture and overestimates the tensile strength of the rock. This paper introduces a unified nonlinear yield criterion based on a stress-dependent fracture energy parameter. In particular, the effect of the intermediate principal stress is encapsulated. The proposed yield criterion is then validated in the principal stress space based on a detailed parameter calibration procedure. Subsequently, four verification examples corresponding to Beishan granite, Lac du Bonnect granite, Carrara marble and Berea sandstone are carried out to show a desirable predictive capacity of the proposed yield criterion on progressive failure, extension to compressive-shear fracture transition and fault slip in quasi-brittle rocks.

Understanding the interface fracture mechanism by composite BFPMPC mortar and cement concrete bending beam

This study aims to utilize basalt fiber-reinforced polymer-modified magnesium phosphate cement (BFPMPC) mortar as an overlay for rapidly repairing surface defects such as raveling and delamination on Portland cement concrete (PCC) pavements, thereby swiftly restoring traffic flow and enhancing pavement service durability. Initially, double layered three-point bending fracture (DLTPB) tests were conducted on composite BFPMPC and PCC specimen with different predetermined crack lengths. The flexural strength was measured, fracture parameters were obtained based on the double-K fracture criterion, and crack propagation paths were determined using digital image correlation (DIC) methods. Subsequently, the fracture behavior of the composite beam was predicted by boundary effect model, and the interface microstructure between BFPMPC and PCC was characterized using nanoindentation, nano-scratch, and SEM/EDS techniques. Finally, by analyzing the fracture characteristics of the composite specimens, the crack propagation model was established and validated by finite element analysis (FEA). The results indicated that BFPMPC mortar overlay can enhance the flexural strength of composite structure. The unstable fracture toughness, cohesive fracture toughness, and fracture energy decreased with increase of crack length-to-height ratio (a0/D), while the initiation fracture toughness increased with increase of a0/D. DIC measurements provided insights into the morphology of the crack tip and the length of the fracture process zone (FPZ). The boundary effect model predictions aligned with the fracture characteristics of the specimens, which were controlled by nonlinear fracture criteria. Nanoindentation tests revealed that the mechanical properties of the interface are influenced by multiple sub-interfaces, with the elastic modulus relationship being ITZ-4 > ITZ-2 > ITZ-3 > ITZ-1. Further analysis of scratch hardness and scratch fracture toughness confirmed that the BFPMPC-PCC interface exhibited high fracture toughness, which was crucial for improving structural resistance to fracture. SEM/EDS observations revealed significant cracks at the interface between BFPMPC and ordinary Portland cement (OPC), while cracks between BFPMPC and coarse aggregates were smaller. Subsequently, the fracture analysis model for composite specimens was proposed to finely reveal interface fracture failure, and the FEA results exhibited a high degree of consistency with experimental results.

Estimation of fracture behavior of CFRP/CFRP adhesively bonded joints under mixed-mode conditions using a cohesive zone model

Finite element (FE) simulations of adhesively bonded composite joints were conducted using the cohesive zone model (CZM). To clarify the effect of fracture criteria on the accuracy of the CZM simulation, two types of mixed-mode criteria including the ­linear summation model of Modes I and II energy release rates, G I / G IC + G II / G IIC = 1 , and the nonlinear power-law, ( G I / G IC ) n + ( G II / G IIC ) n = 1 , were employed. A CZM formulation using the power-law fracture criterion was developed in this study. The load–displacement curves calculated by the two models were compared with the experimental results obtained from mixed-mode Fernlund–Spelt-type double cantilever beam tests, and the Mode II end-notched flexure (ENF) tests. Calculations using the nonlinear fracture criterion were closer to the experimental results. Consequently, it is concluded that nonlinear fracture modeling is of vital importance for accurate strength analysis of mixed-mode adhesive joints.

Full inference for the anisotropic fractional Brownian field

The anisotropic fractional Brownian field (AFBF) is a non-stationary Gaussian random field which has been used for the modeling of textured images. In this paper, we address the open issue of estimating the functional parameters of this field, namely the topothesy and Hurst functions. We propose an original method which fits the empirical semi-variogram of an image to the semi-variogram of a turning-band field that approximates the AFBF. Expressing the fitting criterion in terms of a separable non-linear least square criterion, we design a minimization algorithm inspired by the variable projection approach. This algorithm also includes a coarse-to-fine multigrid strategy based on approximations of functional parameters. Compared to existing methods, the new method enables to estimate both functional parameters on their whole definition domain. On simulated textures, we show that it has a low estimation error, even when the parameters are approximated with a high precision. We also apply the method to characterize mammograms and sample images with synthetic parenchymal patterns.

Numerical Analysis of the Stress Shadow Effects in Multistage Hydrofracturing Considering Natural Fracture and Leak-Off Effect

As a critical technological approach, multistage fracturing is frequently used to boost gas recovery in compact hydrocarbon reservoirs. Determining an ideal cluster distance that effectively integrates pre-existing natural fractures in the deposit creates a fracture network conducive to gas movement. Fracturing fluid leak-off also impacts water resources. In our study, we use a versatile finite element–discrete element method that improves the auto-refinement of the grid and the detection of multiple fracture movements to model staged fracturing in naturally fractured reservoirs. This computational model illustrates the interaction between hydraulic fractures and pre-existing fractures and employs the nonlinear Carter leak-off criterion to portray fluid leakage and the impacts of hydromechanical coupling during multistage fracturing. Numerical results show that sequential fracturing exhibits the maximum length in unfractured and naturally fractured models, and the leak-off volume of parallel fracturing is the smallest. Our study proposes an innovative technique for identifying and optimizing the spacing of fracturing clusters in unconventional reservoirs.

Mode I fracture propagation and post-peak behavior of sandstone: Insight from AE and DIC observation

Microcracks generated at the crack tip, particularly in quasi-brittle materials such as sandstone, give rise to a non-linear region known as the fracture process zone (FPZ), which cannot be analyzed using linear elastic fracture mechanics (LEFM). To gain insights into the nonlinear fracturing behavior of rock, this study conducted three-point bending experiments on sandstone beams with prefabricated notch. Stable mode-Ⅰ fracture propagation during the post-peak stage was achieved by the monotonic increment of crack mouth opening displacement (CMOD) measured by a clip gauge. The acoustic emission (AE) and digital image correlation (DIC) techniques were utilized to visually track the detailed crack propagation process from both micro and macro perspectives. Nonlinear fracture parameters, including FPZ length, were identified by AE and DIC measurements and compared with the theoretical calculation. A nonlinear fracture criterion relating applied load, equivalent crack length, and stress intensity factor were established, from which the theoretical CMOD-P curve at the post-peak stage were obtained and validated with experimental measurement from MTS machine. The results reveal that microcracks formulating the FPZ materialize in the pre-peak stage, with the majority emerging post-peak to facilitate the propagation of the traction-free crack and FPZ in the loading direction, as evidenced by AE locations and DIC displacement gradient. Throughout the crack propagation process, over 65 % of microcracks are estimated to be tensile mode based on both AF/RA and polarity analysis, aligning with anticipated outcomes from macroscopic three-point bending fracture tests. Energy dissipation primarily occurs within the FPZ to facilitate separation of crack faces, with the dissipated energy region advancing towards the beam boundary. By integrating AE source locations and DIC displacement gradient, the FPZ length of the 3-point-bending sandstone specimens is determined to be lp = 8.1 mm, slightly exceeding the theoretical FPZ lengths calculated by the Irwin model (lp = 7.0 mm) and the Dugdale-Barenblatt model (lp = 7.6 mm). Using a non-linear fracture criterion that incorporates the FPZ and employing the in-time equivalent crack length measured by AE and DIC as an intermediate variable, the theoretical applied load-crack length curve during the post-peak stage is calculated, showing close agreement with load cell measurements with a negligible error of approximately 2 %. Furthermore, the theoretical post-peak CMOD vs. load curve aligns with experimentally obtained results, demonstrating a non-linear crack propagation speed at a constant CMOD rate, wherein crack propagation speed decreases as crack length increases.

Modified Mohr–Coulomb criterion for nonlinear strength characteristics of rocks

AbstractThe classical Mohr–Coulomb criterion has an extremely wide range of applications in the field of engineering geology, while it cannot accurately characterize rock failure under high confining stresses. Therefore, a nonlinear modification is implemented based on the Mohr–Coulomb criterion for achieving wider usage in all stress environments. The cohesion and internal friction angle are firstly calculated by regression based on the triaxial experiments of diorite and other typical rocks in different confining stress ranges, and the variation modes of cohesion and internal friction angle are then formalized and integrated into the classical Mohr–Coulomb criterion. Using the experimental results of 435 triaxial compression tests in existing publications, the predictions of the modified criterion are compared with those of the previous nonlinear criteria, which show that the predicted values of the modified criterion are in good agreement with the measured strength. Compared with the previous nonlinear criteria, the modified criterion has a wider application range and always maintains a higher accuracy.

A Three-Dimensional Nonlinear Strength Criterion for Rocks Considering Both Brittle and Ductile Domains

A Three-Dimensional Nonlinear Strength Criterion for Rocks Considering Both Brittle and Ductile Domains

A generalized nonlinear three-dimensional Hoek‒Brown failure criterion

To understand the strengths of rocks under complex stress states, a generalized nonlinear three-dimensional (3D) Hoek‒Brown failure (NGHB) criterion was proposed in this study. This criterion shares the same parameters with the generalized HB (GHB) criterion and inherits the parameter advantages of GHB. Two new parameters, β, and n, were introduced into the NGHB criterion that primarily controls the deviatoric plane shape of the NGHB criterion under triaxial tension and compression, respectively. The NGHB criterion can consider the influence of intermediate principal stress (IPS), where the deviatoric plane shape satisfies the smoothness requirements, while the HB criterion not. This criterion can degenerate into the two modified 3D HB criteria, the Priest criterion under triaxial compression condition and the HB criterion under triaxial compression and tension condition. This criterion was verified using true triaxial test data for different parameters, six types of rocks, and two kinds of in situ rock masses. For comparison, three existing 3D HB criteria were selected for performance comparison research. The result showed that the NGHB criterion gave better prediction performance than other criteria. The prediction errors of the strength of six types of rocks and two kinds of in situ rock masses were in the range of 2.0724%–3.5091% and 1.0144%–3.2321%, respectively. The proposed criterion lays a preliminary theoretical foundation for prediction of engineering rock mass strength under complex in situ stress conditions.

Effects of different tillage and fertilizer on soil quality under wheat‐maize rotation in the North China Plain

AbstractThe North China Plain plays an important role in China's crop cultivation regions, with the wheat‐maize rotation serving as the predominant cropping system. Nonetheless, a dearth of research refers to the impact of tillage and fertilizer employment on soil quality (SQ) in this region. In this study, the primary objective centered on utilizing the soil quality index (SQI) derived from both minimum and total datasets (MDS, TDS) by the linear and nonlinear scoring criteria to assess SQ. The results indicated that there was a significant correlation between wheat‐maize yield and SQI, revealing a significant association between yield and key soil parameters such as bulk density, soil porosity, total phosphorus, alkali‐hydrolyzed nitrogen, available phosphorus and cation exchange capacity at 0–60 cm soil layer. The study identifies that substantially elevated SQI mean values were completely under T8 whose values were 0.980 (linear), 0.672 (nonlinear) under MDS and 0.908 (linear), 0.701 (nonlinear) under TDS at 20–40 cm. Importantly, the annual yields were the highest when the SQI values were the maximum. Furthermore, SQ levels within this locale demonstrate a moderate to highly favorable status, ultimately underscoring the area's propitious conditions for crop cultivation. This research provides indispensable theoretical guidance for agricultural practitioners and offers a foundational framework supporting enhanced yields in local grain crops.