Linear Rule Research Articles

The critical role of size effect on internal damage and mechanical properties of flax fiber reinforced composites

The effect of the size on the strength of laminated artificial fiber reinforced composites has been extensive discussed during the design of large composites structure. With the trial as the structures in aerospace, civil engineering, automobile industry, the scaling of the properties of plant fiber reinforced composite should be studied. In this paper, the size effect and failure mechanism of tensile and impact properties of flax fiber reinforced composites were valuated. The effects of different area, thickness and volume on the tensile properties of composites were explored. Additionally, the failure mechanism of size effect on tensile specimens was proposed through the damage morphologies of composites. It is found that the twist of fiber bundle plays an important role in the size effect of composite thickness. Besides, the relationship between impact properties and size effect of composites was conducted, including the size of hammer, different impact energy and sample size. The curves of different types of impact samples were normalized to verify the linear rule in response stage. The crack length after impact was measured and the size effect of crack length was discussed. The size effect of crack area was studied by calculating the crack area with ultrasonic C-scan. Different “size effects” between flax fibers and artificial fibers were explored. The results are expected to provide a theoretical basis for the structural design of plant fiber reinforced composites.

Speed of sound measurements of binary n-octane+ ethylcyclohexane mixture at liquid-gas phase transition curve

Bio-jet fuel is a key element in the aviation industry to reduce operating costs and environmental impacts. Bio-jet fuel is a complex mixture of four hydrocarbons (n-alkanes, isoalkanes, cycloalkanes and aromatics). In the present work, the speed of sound in pure n-octane, ethylcyclohexane, and their mixtures with six selected compositions of (0.3004, 0.4191, 0.4999, 0.5538, 0.6991, and 0.7852 mole fraction of ethylcyclohexane) has been measured along the l-G saturation curve in the temperature ranges from (286 to 443) K using the pulse method with a constant (acoustic) sounding base. The combined expanded absolute and relative uncertainties (0.95 level of confidence, k = 2) of the temperature, concentration, and speed of sound measurements are estimated to be 20 mK, 0.0006 mole fraction, and 0.2 %, respectively. The measured speed of sound data together with our previous reported density data for the pure component (ethylcyclohexane) and the mixture were used to calculate derived thermodynamic properties, such as isentropic compressibility kS and heat capacity ratios CPCV as a function of temperature along the l-G saturation curve for the pure components and the mixture for selected concentration of x = 0.8 mole fraction of ethylcyclohexane. The deviation of the measured speed of sound data for the mixture from the linear additive rule has been determined using the pure component data.

Damage evolution simulation and lifetime prediction for composite blades under continuous droplet impacts

Offshore wind power generation is a promising technology in renewable energy applications due to its high reserves of wind energy in sea areas. To improve the energy transformation efficiency, the blade length of the offshore wind turbines has become larger and larger, and it has made rain erosion be one of the most frequent failures during the turbine operation. As the natural rainfall is stochastic in spatial and time domains, it is difficult to depict the damage evolution process caused by rain impact exactly. Therefore, regular and continuous droplet impact simulation and experiments present an alternative methodology for this issue. With the composite structure of the blade and deformability of the liquid droplet in consideration, a fluid solid interaction model will be established to investigate the impact response and subsequent damage evolution. In which, the Smooth Particle Hydrodynamics (SPH) model is utilized to depict the constitutive relationship within the droplet, and Finite Element Method (FEM) is used to construct the Representative Volume Element (RVE) model of the blade leading edge. The impact process is simulated first to obtain the impact pressure distribution at the contact center and velocity field in the droplet. Furthermore, the stress wave propagation in the blade multilayer structure can be analyzed. Owing to the multiaxial fatigue feature of the continuous droplet impact, the continuum damage mechanics is integrated with the fatigue criterion and the Jump-in-Cycle procedure is used to simulate the high-cycle fatigue process. The damage factor distribution on the blade coating surface and its influence on mechanical properties are analyzed. Thereafter, the droplet impact fatigue life can be accumulated based on Miner’s linear rules. The theoretical achievements are validated by experimental data provided by Rain Erosion Testing (RET), which shows a good agreement between each other. As a result, V-N curves and D-N curves, i.e. quantitative relationship between droplet falling conditions and impact fatigue life, are established. The achievements in this study can provide an effective tool for rain erosion mechanism analysis and life prediction in industrial applications.

A novel global algorithm for solving linear multiplicative problem by integrating linear combination rule and branch-and-bound framework

A novel global algorithm for solving linear multiplicative problem by integrating linear combination rule and branch-and-bound framework

Experimental study and life prediction for aero-engine turbine blade considering creep-fatigue interaction effect

As an important failure form of the turbine blade, creep-fatigue interaction damage affects the safe operation and maintenance strategy of aero-engine, and has been the focus of scientific research and the academic community. Firstly, based on the Kachanov-Rabotnov-Lemaitre continuum damage mechanics theory and the nonlinear symmetry of creep damage and fatigue damage, a creep-fatigue life prediction model is constructed considering the interaction effect in this paper. Then, based on the thermal-fluid–solid multi-physical field coupling numerical simulation of the turbine blade, the equivalent method of creep-fatigue load spectrum was explored according to the equal damage criterion and linear damage rule, and the creep-fatigue interaction test of smooth samples of the blade material was conducted to analyze the creep-fatigue fracture morphology. Finally, the stress term in the creep-fatigue life prediction model of blade material is modified by the correction factor α, and the modified creep-fatigue life prediction model of the turbine blade is constructed considering the interaction effect. The results show that the modified creep-fatigue life prediction model considering the interaction effect has a high life prediction ability with an error of 3.9%. The above research has important scientific research value for the life extension design of turbine blades and the improvement of aero-engine maintenance strategy.

Investigation of creep damage accumulation under variable loading conditions in nickel-based single crystal superalloys

Nickel-based single crystal (SX) turbine blades typically experience varying load conditions during operation. It is essential to consider the effects of changing stress and temperature when evaluating creep damage. The commonly used time-fraction method, based on the linear damage rule (LDR), has been demonstrated to be inconsistent with experimental results under certain loading conditions. There is a lack of research on damage accumulation under variable loading conditions specifically for SX superalloy. This paper conducted two-stage creep experiments for SX superalloy DD6 at 980 °C with stress levels of 230 MPa and 320 MPa, and at 1050 °C with stress levels of 135 MPa and 230 MPa. The load for the experiment was set based on the typical operating conditions of the SX turbine blades. The evolution of the microstructure under variable loading was observed using scanning electron microscope (SEM) and transmission electron microscope (TEM). The result of creep experiments revealed that the damage accumulation behavior of the SX superalloy does not conform to the LDR. Furthermore, in two-stage loading tests, the creep damage accumulation behavior differs from the classical sequence effect of fatigue loading. SEM and TEM analyses suggested that this phenomenon might be associated with the varying degradation rates of microstructures in SX superalloy.

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.

Time-domain fatigue reliability analysis for floating offshore wind turbine substructures using coupled nonlinear aero-hydro-servo-elastic simulations

The present study aims to develop a novel methodology for the fatigue reliability analysis and design for a floating offshore wind turbine substructure using its fully coupled nonlinear dynamic responses in the time domain. The developed methodology offers a computationally efficient yet robust assessment for designing fatigue-critical welded joints on floating substructures. The methodology starts with the sea state selection based on the fatigue damage contribution of all the sea states in the scatter diagram. To this end, the dynamic response is analysed by fully coupled aero-hydro-servo-elastic simulations using OpenFAST. The resulting short-term fatigue loading is then used to estimate the hotspot stress history using the analytical solution for global structural analysis and the empirical solution for the stress concentration factor. The long-term fatigue damage is calculated as the joint probability-weighted sum of the short-term fatigue damage using Palmgren-Miner's linear damage rule. Afterwards, the selected sea states are used to perform dynamic response simulations with multiple random seeds, ensuring no significant accuracy loss. To obtain the probabilistic fatigue life prediction, a novel time-domain fatigue reliability analysis algorithm is developed based on the bootstrapping method, which creates a pool of short-term fatigue responses with different random seeds and combines them within a Monte Carlo Simulation. Finally, the fatigue reliability analysis considering the S-N and damage-tolerant approaches for design is carried out.

Nonlinear Decision Rules Made Scalable by Nonparametric Liftings

Sequential decision making often requires dynamic policies, which are computationally not tractable in general. Decision rules provide approximate solutions by restricting decisions to simple functions of uncertainties. In this paper, we consider a nonparametric lifting framework where the uncertainty space is lifted to higher dimensions to obtain nonlinear decision rules. Current lifting-based approaches require predetermined functions and are parametric. We propose two nonparametric liftings, which derive the nonlinear functions by leveraging the uncertainty set structure and problem coefficients. Both methods integrate the benefits from lifting and nonparametric approaches, and hence provide scalable decision rules with performance bounds. More specifically, the set-driven lifting is constructed by finding polyhedrons within uncertainty sets, inducing piecewise-linear decision rules with performance bounds. The dynamics-driven lifting, on the other hand, is constructed by extracting geometric information and accounting for problem coefficients. This is achieved by using linear decision rules of the original problem, also enabling one to quantify lower bounds of objective improvements over linear decision rules. Numerical comparisons with competing methods demonstrate superior computational scalability and comparable performance in objectives. These observations are magnified in multistage problems with extended time horizons, suggesting practical applicability of the proposed nonparametric liftings in large-scale dynamic robust optimization. This paper was accepted by David Simchi-Levi, optimization. Supplemental Material: The electronic companion and data files are available at https://doi.org/10.1287/mnsc.2024.4988 .

Development of a Flow Rule Based on a Unified Plasticity Model for 13Cr-4Ni Low-Carbon Martensitic Stainless Steel Subject to Post-Weld Heat Treatment

This study aims to develop a flow rule for evaluating the relaxation and redistribution of residual stresses during the post-weld heat treatment (PWHT) of hydroelectric runners made from low-carbon martensitic stainless steel (13Cr-4Ni composition). During the PWHT, austenite reforms in the filler metal and surrounding areas of the base metal near welded joints. The evolving inelastic strain rate with reformed austenite led to defining two distinct flow rules in the pure martensitic (α′) and austenitic (γ) phases. A linear rule of mixture was then applied to assess global effective stress based on the inelastic strain rate and current austenite fraction during the PWHT. A unified constitutive model incorporating drag stress and back stress, evolving with creep and plastic deformation mechanisms during the PWHT, described the stress–strain behavior. To validate this analysis, a third flow rule was determined in the 18% tempered austenitic microstructure, compared with the rule of mixture’s effective stress contribution from each phase on the inelastic strain rate. Isothermal constant strain rate tests in stabilized crystalline microstructures evaluated constants specific to their respective flow rules. This study demonstrates the stability of reformed austenite at elevated temperatures during slow cooling and its significant influence on the mechanical properties of 13Cr-4Ni steels. The effectiveness of estimating yield stress using the rule of mixture based on individual phase behaviors is also confirmed.

Decision algorithms for reversibility of 1D cellular automata under reflective boundary conditions

Reversibility is one of the most significant properties of cellular automata (CA). In this paper, we focus on the reversibility of one-dimensional finite CA under reflective boundary conditions (RBC). We present two algorithms for deciding the reversibility of one-dimensional CA under RBC. Both algorithms work for not only linear rules but also non-linear rules. The first algorithm is to determine what we call the “strict reversibility” of CA. The second algorithm is to compute what we call the “reversibility function” of CA. Reversibility functions are proved to be periodic. Based on the algorithms, we list some experiment results of one-dimensional CA under RBC and analyse some features of this family of CA.

A distributionally robust optimization approach for the potassium fertilizer product transportation considering transshipment through crossdocks

Potassium fertilizer is an essential input for agricultural productivity, and plays a critical role in various plant processes, influencing water uptake, enzyme activation, and photosynthesis. The efficient delivery of potassium fertilizer products to the end-users, typically farmers and agricultural enterprises, is of utmost importance. Due to the long traveling distance between the supplier and customers, decision makers of supplier normally set up multiple crossdocks to aid in transition and storage of potassium fertilizer products. In this situation, making an optimal transportation plan as well as the inventory level of crossdocks will directly enhance the efficiency and effectiveness of long-distance transportation. In this study, we model this problem as a dynamic transportation problem with transshipment considering the uncertain time-varying customers’ demand. To characterize the demand information, we first construct an ambiguity set of customers’ demand based in limited historical data and then develop a distributionally robust optimization-based (DRO) framework to optimize the transportation plan and related inventory level of crossdocks simultaneously. We also propose a general approach to overcome the computational challenges of DRO by transforming the original DRO into a second-order cone programming based on duality theory. Additionally, we introduce linear decision rules to adjust the optimization strategy based on the new observed demand, thus lead the model to handle the dynamic information flow in real time. In case study, all involved data are collected or derived from a real potassium fertilizer company located at Western China. The results show our model reduces the cost by 18.07% compared to stochastic model (sample average approximation), indicating a significant effectiveness of our model on the improvement of delivery efficiency and cost saving in real dynamic logistic systems. Also, we conclude the managerial insight that decision-makers should develop a comprehensive strategy, including improving communication to ensure order status updates, planning rationally to evenly distribute orders, and proactively allocating resources to meet operational demands.

On extremal factors of de Bruijn-like graphs

In 1972, Mykkeltveit confirmed Golomb’s conjecture on the de Bruijn graphs: he proved that the pure cycling register rule yields the maximum number of vertex-disjoint cycles in de Bruijn graphs. We show that this result encompasses the tensor product of the de Bruijn graph for strings of length n with a simple cycle of size k, when n divides k or vice versa. Furthermore, we give counting formulas for an array of cycling register rules, which includes Golomb’s well-studied linear register rules.

Dielectric Properties of PEEK/PEI Blends as Substrate Material in High-Frequency Circuit Board Applications.

Substrate materials for printed circuit boards must meet ever-increasing requirements to keep up with electronics technology development. Especially in the field of high-frequency applications such as radar and cellular broadcasting, low permittivity and the dielectric loss factor are key material parameters. In this work, the dielectric properties of a high-temperature, thermoplastic PEEK/PEI blend system are investigated at frequencies of 5 and 10 GHz under dried and ambient conditions. This material blend, modified with a suitable filler system, is capable of being used in the laser direct structuring (LDS) process. It is revealed that the degree of crystallinity of neat PEEK has a notable influence on the dielectric properties, as well as the PEEK phase structure in the blend system developed through annealing. This phenomenon can in turn be exploited to minimize permittivity values at 30 to 40 wt.-% PEI in the blend, even taking into account the water uptake present in thermoplastics. The dielectric loss follows a linear mixing rule over the blend range, which proved to be true also for PEEK/PEI LDS compounds.

Nonlinear fatigue damage constrained topology optimization

In engineering applications, plenty of components are subjected to variable-amplitude cyclic loading, resulting in fatigue damage, which is one of the main forms of structural damage. While the linear damage rule has long served as a fundamental approach, its limitations necessitate advancements for more accurate fatigue life predictions. Hence, this paper introduces a pioneering method termed nonlinear fatigue damage constrained topology optimization (NFDCTO). This method integrates several key components: the rainflow counting method to evaluate the non-proportional cyclic load levels, Basquin's equation to describe the S-N curve, and Morrow's plastic work interaction rule to calculate the nonlinear cumulative damage of the structure. Consequently, we establish a mathematical model for the NFDCTO method based on these components. The incorporation of penalized damage aggregation with the P-Normal function is used to address significant constraints and singularity challenges. Furthermore, by employing the adjoint method, we derive sensitivity equations for both the objective function and fatigue constraint function concerning the design variables. Subsequently, the superiority of the NFDCTO method over traditional linear fatigue damage method were verified applying 2D l-shaped beam and simply supported structures as examples. Concurrently, bridge structures were employed to investigate the effect of the sensitivity index of the material to the history of variable amplitude stresses on the optimization results. In addition, the influence of the fatigue penalty factor on the topology optimization results was assessed using a 2D cantilever beam. Finaly, we verify the effectiveness and feasibility of the NFDCTO method in addressing fatigue optimization challenges for 3D structures utilizing two 3D examples.

Study on fatigue-healing performance and life prediction of hot recycled asphalt mixture

The extensive application of recycled asphalt mixtures (RAM) requires a better understanding of its fatigue performance and there is need to establish an accurate prediction model for fatigue life. The fatigue-healing performance of RAM containing different recycled asphalt pavement (RAP) contents was investigated by using four-point bending test with virgin mixtures as a reference. The linear portion during the steady decline of the flexural stiffness was utilized to establish a linear method for fatigue life prediction. The effects of RAP content, aging, and immersion factors on the fatigue-healing performance of RAM were studied. A fatigue-healing life prediction model considering RAP content, initial flexural stiffness (E0), strain level, long-term aging at 85 °C and moisture immersion at 60 °C was developed through regression analysis of data obtained from various fatigue tests. The test results indicated that E0 at the 500th loading cycle exhibited higher accuracy for life prediction based on linear rule when compared with the traditional E0 at the 50th loading cycle. With an increase in RAP content, the fatigue life of RAM decreased, while the E0 showed a linear increase and the healing index gradually decreased. The decrease in fatigue life of RAM due to moisture effect was greater than that due to long-term aging effect. Aging effect for 5 d and moisture immersion for 7 d reduced the fatigue life of RAM by 17.9 % and 47.4 % respectively. The fatigue-healing life prediction model can accurately predict the fatigue life of RAM within the range of 0–45 % RAP content. Aging was found to have the greatest negative effect on the healing performance of RAM, followed by moisture effect, while the effect of RAP content was minimal. This study contributes to the prediction of remaining fatigue life for RAM after long-term service and improving its service levels.

High dimensional discriminant rules with shrinkage estimators of the covariance matrix and mean vector

Linear discriminant analysis (LDA) is a typical method for classification problems with large dimensions and small samples. There are various types of LDA methods that are based on the different types of estimators for the covariance matrices and mean vectors. In this paper, we consider shrinkage methods based on a non-parametric approach. For the precision matrix, methods based on the sparsity structure or data splitting are examined. Regarding the estimation of mean vectors, Non-parametric Empirical Bayes (NPEB) methods and Non-parametric Maximum Likelihood Estimation (NPMLE) methods, also known as f-modeling and g-modeling, respectively, are adopted. The performance of linear discriminant rules based on combined estimation strategies of the covariance matrix and mean vectors are analyzed in this study. Particularly, the study presents a theoretical result on the performance of the NPEB method and compares it with previous studies. Simulation studies with various covariance matrices and mean vector structures are conducted to evaluate the methods discussed in this paper. Furthermore, real data examples such as gene expressions and EEG data are also presented.

A peer‐to‐peer joint energy and reserve market considering renewable generation uncertainty: A generalized Nash equilibrium approach

AbstractPeer‐to‐peer (P2P) energy trading enhances distribution network resilience by reducing energy demand from central power plants and enabling distributed energy resources to support critical loads after extreme events. However, adequate reserves from main grids are still required to ensure real‐time energy balance in distribution networks due to the uncertainty in renewable generation. This paper introduces a novel two‐stage joint energy and reserve market for prosumers, wherein local flexible resources are fully utilized to manage renewable generation uncertainty. In contrast to cooperative optimization methods, the interactions between prosumers are modelled as a generalized Nash game (GNG), considering that prosumers are self‐interested and should follow distribution network constraints. Then, linear decision rules are employed to ensure a feasible market equilibrium and develop a privacy‐preserving algorithm to guide prosumers toward the market equilibrium with a proven convergence. Finally, the numerical study on a modified IEEE 33‐power system demonstrates that the designed market effectively manages renewable generation uncertainty, and that the algorithm converges to the market equilibrium.

Bi-level distributionally robust optimization model for low-carbon planning of integrated electricity and heat systems

Interdependence among diverse energy forms within integrated electricity and heat systems (IEHSs) holds the potential to harness cooperation effects. This paper proposes a bi-level optimization model for the low-carbon planning of IEHSs, which realizes the coordination between IEHSs and their upstream networks. The precise carbon emissions of IEHSs are traced through carbon emission flows of components, power distribution networks, and district heating networks. Moreover, the distributionally robust optimization approach is employed to account for the inherent uncertainty in renewable power outputs faced by IEHS planning, by incorporating the moment information into an ambiguity set. The bi-level distributionally robust planning model of IEHSs is initially reconfigured into a second-order cone structure, employing the strong duality theory, second-order cone duality theory, and linear decision rules. Subsequently, an iterative approach is introduced to attain convergence of the reformulated bi-level model. Numerical results demonstrate the effectiveness and superiority of the proposed approach for reducing carbon emissions and economic costs of IEHSs.

Fatigue life of RC bridge decks affected by speed and load weight of wheel-type moving loads

Over the service life of a bridge, vehicles vary in terms of speed and load weight, leading to varying degrees of fatigue damage. Particularly under wet condition, the fatigue life of reinforced concrete (RC) bridge decks experiences a significant decline. In this study, the results of fatigue life of RC bridge decks are systematically calculated for under wheel-type moving loads with various speed and load weight according to practical situation of bridges in both dry and wet condition by an acceptable approach involves a multi-scale and multi-chemo–physics analysis system (DuCOM-COM3) that incorporates the water-crack interaction. The results indicate that the effect of speed on fatigue life is not significant and follows a complex pattern, and the effect of load weight on fatigue life is dominant, with a heightened sensitivity to load weight variations under wet condition. Based on the simulation results of fatigue life, calculation formulations for fatigue life are presented accurately for dry and wet condition respectively, considering both load weight and speed as parameters, with the determination coefficients R2 are 0.977 and 0.952, respectively. In addition, a predictive method for the service life under constant environmental condition is proposed based on the Palmgren-Miner’s linear rule, and a case analysis is conducted by utilizing practical traffic flow data monitored by a piezoelectric-based dynamic weigh-in-motion system set on Jiashao Bridge, showing that the service life under dry condition is 55.41 years, while that under wet condition reduces to 1.03 years. Finally, a comprehensive service life predictive method considering actual and alternating environmental conditions is proposed.