Traction Distribution Research Articles

Bayesian Inference for the Calibration of Progressive Damage Model of Dovetail Specimens from Laminated Composite Fan Blade

Abstract Composite fan blades are the preferred alternative for the fan stage of most advanced high bypass ratio turbofan engines. The ability of the dovetail to safely bear this load is one of the key points of the multi-level “test pyramid” approach of compliance demonstration for special airworthiness regulation. Debonding between adjacent layers is the main damage mode of laminated composite fan blades. However, there is difficulty in measuring the as manufactured interlaminar mechanical properties used in finite element models. In this study, tensile loading was applied to simulate the interacting centrifugal force and capture mixed mode damage evolution. Structural responses and material damages were calibrated with measured tensile loads through Bayesian inversion. Posterior probability distributions of maximum interface tractions and contact stresses were solved using Markov chain Monte Carlo sampler. Results indicated the two bilinear cohesive material models had a capacity of predicting empirical means of longitudinal reaction forces as that in test considering additional discrepancy term (0.035 kN and 0.96 kN respectively), while they made an significant impact on the prediction of tensile load history especially when two delamination cracks initiated and propagated. Interface elements provided a higher matching quality in predicting loading history and capturing damage mechanism in association with in-plane progressive damage analysis. This calibrated parameter set could be functioned as benchmark in numerically determining the ultimate tensile load of dovetail elements and reducing the necessary number of physical tests at elemental length level.

Numerical and experimental crack-tip cohesive zone laws with physics-informed neural networks

The cohesive zone law represents the constitutive traction versus separation response along the crack-tip process zone of a material, which bridges the microscopic fracture process to the macroscopic failure behavior. Elucidating the exact functional form of the cohesive zone law is a challenging inverse problem since it can only be inferred indirectly from the far-field in experiments. Here, we construct the full functional form of the cohesive traction and separation relationship along the fracture process zone from far-field stresses and displacements using a physics-informed neural network (PINN), which is constrained to satisfy the Maxwell-Betti's reciprocal theorem with a reciprocity gap to account for the plastically deforming background material. Our numerical studies simulating crack growth under small-scale yielding, mode I loading, show that the PINN is robust in inversely extracting the cohesive traction and separation distributions across a wide range of simulated cohesive zone shapes, even for those with sharp transitions in the traction-separation relationships. Using the far-field elastic strain and residual elastic strain measurements associated with a fatigue crack for a ZK60 magnesium alloy specimen from synchrotron X-ray diffraction experiments, we reconstruct the cohesive traction-separation relationship and observe distinct regimes corresponding to transitions in the micromechanical damage mechanisms.

Multiscale contact homogenisation: A novel perspective through the method of multiscale virtual power

The interaction between deformable bodies and rigid foundations undergoing finite strains is explored in this work with the Method of Multiscale Virtual Power, unlocking novel insights into contact homogenisation theories. The focus lies in establishing the foundational kinematical links across scales and achieving seamless homogenisation of the traction vector through rigorous duality arguments. Two distinct families of contact homogenisation models are formulated: surface–surface and surface–volume models. These families emerge from the virtual power balance established between the target macroscopic surface and the corresponding microscopic contact surface or volume. They not only represent fundamental kinematical entities at each scale but also enable the replacement of the heterogeneous contact traction distribution with a smooth, homogenised version. Microscale equilibrium problems and homogenisation relations, along with the relevant macro- and microscale quantities, are derived by means of straightforward variational arguments. Furthermore, potential finite size effects are addressed in the homogenised response, enhancing the reliability and applicability of the strategy across diverse scenarios. Promising submodels have been identified within both families, broadening the understanding of multiscale contact phenomena and including existing models as particular cases. The proposed continuum-based variational families of models naturally lead to generic finite element-based frameworks for contact homogenisation of heterogeneous solids, as described in a forthcoming paper.

Numerical analysis of flow anisotropy in rotated-square deterministic lateral displacement devices at moderate Reynolds number

Deterministic lateral displacement (DLD) is a microfluidic method for accurately separating particles by size or deformability. Recent efforts to operate DLD devices in the inertial, rather than in the Stokes flow regime, have been hindered by a loss of separation efficiency and difficulty predicting the separation behavior. One factor contributing to these problems is the onset of inertia-induced flow anisotropy where the average flow direction does not align with the direction of the pressure gradient in the device. We use the lattice-Boltzmann method to simulate two-dimensional flow through a rotated-square DLD geometry with circular pillars at a Reynolds number up to 100 for different gap sizes and rotation angles. We find that anisotropy in this geometry is a nonmonotonous function of Reynolds number and can be positive or negative. This finding is in contradiction to the naive expectation that inertia would always drive flow along the principal direction of the pillar array. Anisotropy tends to increase in magnitude with gap size and rotation angle. By analyzing the traction distribution along the pillar surface, we explain how the change of the flow field upon increasing inertia leads to the observed trends of anisotropy. Our work contributes to a better understanding of the inertial flow behavior in ordered cylindrical porous media, and it might contribute to improved DLD designs for operation in the inertial regime. Published by the American Physical Society 2024

Semi-Analytical Model for 3D Multilayered Rolling Contact Problems With Different Creepage Combinations

Abstract In this work, a semi-analytical model replicating 3D rolling contact problems with elastic coating layers has been proposed. The dimension reduced boundary element-based model rests on the conventional half-space assumptions, with influence coefficients describing the relationship between the stress and the displacement. In this model, the influence coefficient describes the stress–strain relationship for multilayered materials, which is derived from the corresponding frequency–response functions. The governing equations for 3D rolling contact problems are solved efficiently using the conjugate gradient method with the classical fast Fourier transformation (FFT)-based technique to accelerate the calculation of the integral equation relating the stresses to the strains, and all the possible creepages, i.e., the longitudinal, lateral, and spin creepages, are considered. The results obtained with the present numerical model are compared with existing results, and good agreement is found. Some selected results are presented to show the effect of the material elasticity and the thickness of the coating, and the layering, on the traction distribution and the traction coefficient under different creepage combinations. In this way, this work provides important information related to the composition of layered coatings in rolling contact applications.

Non-contact reconstitution of the traction distribution using incomplete deformation measurements: Methodology and experimental validation

Traction distribution is often unknown in solid mechanics. However, identifying traction distribution with both high accuracy and spatial resolution can sometimes be critical. In this paper, we consider the frequent situation where incomplete deformation measurements, such as partial surface measurements obtained by digital image correlation, are available. For such situations, we propose to achieve the complete identification of boundary tractions, using an optimization based inverse method. The feasibility and robustness of the proposed approach are firstly assessed with numerical simulations. We show that complete traction distributions can be identified with high accuracy even when there is noise added to the deformation measurements. More specifically, the relative error was less than 3% even with 10% noise. Moreover, the effect of noise depends significantly on the level of incompleteness of the deformation data. We show for instance that when the measurement area is 8.3% of the solid, the relative error can be 5 times larger than when the measurement area is 33.3%. However, even with 8.3% deformation measurement area of the solid, the relative error of the mapped traction distribution can be less than 6%. Besides, the accuracy of the mapped traction distribution can be further improved if the region of the deformation measurement is closer to the traction boundary. We eventually validate the proposed inverse approach using incomplete experimental datasets consisting in tensile tests carried out on two elastic samples. We obtain very promising results, with relative errors below 4% using the measured displacements of only 3 points, which is promising to open potential applications in different fields of engineering.

A semi-analytical approach for two-dimensional frictional contact of anisotropic magneto-electro-elastic solids

In this work, we present a semi-analytical modeling approach for solving the frictional contact of two-dimensional anisotropic magneto-electro-elastic (MEE) solids. The method is established in a broad framework in which the contact body is made of generally anisotropic MEE materials, the punch head can be arbitrary, and the punch can be subjected to generalized loading conditions including mechanical forces and electric and magnetic charges. An analytical solution of generalized distributed load along the half-plane surface is derived and utilized to formulate the semi-analytical equations governing the relation between the generalized displacements and tractions. Based upon the semi-analytical equations, the semi-analytical model in which the influence matrices can be calculated explicitly and its effectiveness is established. An iterative procedure is then used for determining the contact region, contact status, generalized surface deformation and the distribution of contact tractions, as well as the punch’s indentation depth and electric and magnetic potentials. The present approach is verified by comparing its numerical results with those obtained by the existing analytical solutions and boundary element method. The influences of different factors, such as punch profile, mechanical load, electric and magnetic charges, friction coefficient and material properties, are also analyzed. The capability of this method to handle complicated geometry, such as rough surfaces, is finally discussed, showing the potential performance of the developed method for practical engineering applications.

Real area of contact and tractions on the patterned surfaces generated by spinodal decomposition and amplified instability

Past decades featured significant advancements in additive and micromanufacturing that facilitated the creation of functional patterned surfaces with impressive spatial resolutions. However, these techniques are expensive and require a considerable amount of time and energy, and hence lack scalability to practical surfaces. Recent techniques employing spinodal decomposition and instabilities amplified via centrifugal acceleration offer viable and cheaper alternatives. The patterns created by those techniques, however, vary randomly in geometry. When interfacing those patterned surfaces with other components and under self-contact scenarios, geometric variations lead to stress concentration and abrupt failure around the contact. In this study, we investigate numerically real contact areas, contact tractions, and stress concentration. We generate patterned surfaces in congruence with actual surfaces created by those techniques. Then, we conduct normal-contact analyses of those surfaces boundary element method (BEM) under nominal mean pressures ranging from 0.001E* to E*, where E* is the contact modulus. We record real contact areas and stress concentration as a function of nominal mean pressures. We compare these values with the analytical solutions from sinusoidally-patterned and randomly rough surfaces. Randomness in pattern geometry is primarily influenced by the processing parameters such as the degree of anisotropy in spinodal decomposition and acceleration in amplified instabilities. To understand the influence of the processing parameters, we perform a parametric study. We find isotropic spinodal decomposition creates patterns that deliver contact area and traction distributions similar to randomly rough surfaces, and lead to high-stress concentrations. Such high-stress concentrations are expected to occur under self-contact loading scenarios, and thus can explain the compromised resilience and strength in recently-proposed spinodal metamaterials. For patterned surfaces created by amplified instabilities, high-stress concentrations are obtained for the surfaces created at high accelerations. At high accelerations, increased elastic instabilities and stochastic growth result in a more skewed and broader distribution in heights. Therefore, high-stress concentrations are inevitable. To account for combined loading scenarios, we conduct additional simulations on the same surface patterns with frictional pre-sliding contacts. We find the frictional tractions play a secondary role in stress concentrations where the primary factor is the processing parameters determining the degree of randomness in pattern geometry.

The investigation of the contact and wear of railway axle bearings under various race conformities

The bearing race conformity is a design parameter that determines the degree of contact between rollers and races. This paper investigates the effect of race conformity on wear distribution in a railway spherical roller bearing under boundary lubricated condition in which contact behaves like a dry contact. The results showed that the traction/micro-slip distributions were substantially dependent on the level of conformity. The wear is found to be highly sensitive to the race conformity, with lower conformity resulting in lower wear. For example, 5.2% increase of conformity factor, a parameter that represents race conformity mathematically, reduces worn area by 52% in inner contact that is 60% in outer contact where conformity factor increase is 5.8%. This is mainly due to high value micro-slip being truncated at the edges as the traction distribution is compressed into the smaller contact patch. The bearings with ceramic rolling elements show lower wear than for steel rolling elements due to lower wear coefficient. The higher elastic modulus of ceramic causes the traction magnitude to be higher in the ceramic-steel case over the steel-steel case. Therefore, race conformity can be taken into consideration to optimize roller bearing wear life.

An Economic Hybrid Shunt Compensator for Power Factor Correction and Mitigation of Large Unbalance as in HV Traction Distribution System

This paper presents a delta-connected hybrid shunt FACTS compensator along with its control architecture for unbalance mitigation and power factor correction in high voltage (HV) distribution systems consisting of single-phase traction loads. The compensator comprises of static VAr compensator (SVC) in series with static synchronous compensator (STATCOM) in each of its delta branch. Design and control strategy of the SVC and the STATCOM is formulated such that STATCOM rating reduces substantially. The proposed compensator can be an economical alternative to STATCOM only solution when applied in HV systems. Simulations are carried out in PSCAD/EMTDC platform to validate the unbalance mitigation and power factor correction capability of the compensator under different single-phase traction loading conditions. Further, experiments are carried out on a scaled down laboratory prototype and results are presented.

A fast version of ‘CONTACT’ for normal problem calculations

In its different versions, the CONTACT method developed by Prof. Kalker is the primary reference in wheel-rail contact mechanics. Despite adopting simplifications associated with the elastic behaviour of the solids and being a non-conformal contact theory, CONTACT provides precise solutions for most wheel-rail contact conditions, with lower computational and modelling costs than other numerical methods such as Finite Elements. Nevertheless, the computational cost of CONTACT is still too high for its implementation in dynamic simulation.The present work proposes a fast and accurate wheel-rail contact method for normal problems based on Kalker's CONTACT algorithm. Dissimilarly to CONTACT, the new method formulates the normal traction distribution through a suitable basis, which reduces the dimension of the problem. This method is able to faithfully reproduce the contact patch and the normal traction distribution, even when the yaw angle of the wheelset is non-zero. Results obtained with this method are compared with the ones calculated with CONTACT, and errors about 0.05% are obtained in normal contact forces, with a reduction on the computation cost between 30 and 60 times.

Computational Technique for Crack Propagation Simulation in Viscoelastic Solid Propellant

To further investigate the fracture response in propellant grain, numerical methodology is proposed to cope with crack propagation simulation especially for the mixed mode condition. The numerical discrete scheme of the propellant linear viscoelastic constitutive model is proposed, which provides a key means for the simulation of crack propagation. In order to simulate the cohesive traction distribution on the new crack surface, the extrinsic Park-Paulino-Roesler (PPR) cohesive zone model (CZM) is introduced. To let the crack propagate along any direction determined, element splitting technique and its corresponding topological operations are proposed step by step. Then, computational simulation implementation process is explained in greater detail. Typical fracture problem, single edge-notched tension test (SENT) is solved to demonstrate the efficiency and accuracy of the proposed method. In addition, double edge-notched tension test (DENT) as well as plate tension test with a slant crack is conducted to show the special fracture characters in viscoelastic solid propellant, like time dependence. Computational results reveal that the method proposed can be utilized in further fracture investigation in solid propellant combined with the experimental findings.

Steady-State Crack Growth in Nanostructured Quasi-Brittle Materials Governed by Second Gradient Elastodynamics

The elastodynamic stress field near a crack tip propagating at a constant speed in isotropic quasi-brittle material was investigated, taking into account the strain gradient and inertia gradient effects. An asymptotic solution for a steady-state Mode-I crack was developed within the simplified strain gradient elasticity by using a representation of the general solution in terms of Lamé potentials in the moving framework. It was shown that the derived solution predicts the nonsingular stress state and smooth opening profile for the growing cracks that can be related to the presence of the fracture process zone in the micro-/nanostructured quasi-brittle materials. Note that similar asymptotic solutions have been derived previously only for Mode-III cracks (under antiplane shear loading). Thus, the aim of this study is to show the possibility of analytical assessments on the elastodynamic crack tip fields for in-plane loading within gradient theories. By using the derived solution, we also performed analysis of the angular distribution of stresses and tractions for the moderate speed of cracks. It was shown that the usage of the maximum principal stress criterion within second gradient elastodynamics allows us to describe a directional stability of Mode-I crack growth and an increase in the dynamic fracture toughness with the crack propagation speed that were observed in the experiments with quasi-brittle materials. Therefore, the possibility of the effective application of regularized solutions of strain gradient elasticity for the refined analysis of dynamic fracture processes in the quasi-brittle materials with phenomenological assessments on the cohesive zone effects is shown.

Influence of Curing Agents Molecular Structures on Interfacial Characteristics of Graphene/Epoxy Nanocomposites: A Molecular Dynamics Framework

AbstractThis paper presents a comprehensive molecular dynamics study on the effects of the stoichiometric ratio of epoxy:hardener, hardener's linear and cyclic structure, and number of aromatic rings on the interfacial characteristics of graphene/epoxy nanocomposite. The van der Waals gap and polymer peak density as a function of the type of the hardener is calculated by analyzing the local mass density profile. Additionally, steered molecular dynamics are used to conduct normal pull‐out of graphene to study the effect of the mentioned features of hardeners on the interfacial mechanical properties of nanocomposites, including traction force, separation distance, and distribution quality of reacted epoxide rings in the epoxy. Influence of the hardeners on the damage mechanism and its initiation point are also studied by analyzing the evolution of local mass density profile during the normal pull‐out simulation. It is seen that stoichiometric ratio and geometrical structure of the hardeners affect the interfacial strength. It is also revealed that the hardener type can change the epoxy damage initiation point. The damage occurs in the interphase region for a higher stoichiometric ratio or cyclic structure of hardener. In comparison, for hardener's lower stoichiometric ratio and non‐cyclic structure, failure begins in the epoxy near graphene layers.

The use of modelling to determine the limiting conditions for resuming soil loading by tractor in an area of sugar cane under reduced tillage

The marked growth of the sugar and alcohol sector has promoted an increase in the fleet of heavy agricultural vehicles to meet the demand for commodities; however, the intense traffic of machines in areas of insufficient soil moisture has resulted in compaction, compromising the productivity of the sugar cane. In order to determine the optimal conditions for resuming the use of tractors under a reduced tillage system, the bearing capacity was evaluated and stress distribution was modelled in a dystroferric Red Latosol cultivated with sugar cane. Soil sampling was carried out during the first ratoon crop at 120 days, using metallic cylinders installed at a depth of 0.05 m in the crop rows and 0.15 m between rows. The samples were submitted to the uniaxial compression test to determine the preconsolidation pressure of the soil. The reduced tillage gave the lowest bearing capacity at the soil surface (0.05 m), while the highest values ​​were found at a depth of 0.15 m. The bearing capacity proved to be an efficient indicator for analysing the structural degradation of the soil by demonstrating the compressibility of the various layers and sampling positions resulting from reduced tillage and the impact of agricultural machines in the field. The prediction model for traction and soil stress distribution that was employed was able to predict the optimal performance of the tractor, and provide stress data that would allow a study of the limiting water content for the traffic of agricultural tyre tractors.

Elastic Solutions to 2D Plane Strain Problems: Nonlinear Contact and Settlement Analysis for Shallow Foundations

The classical Neumann boundary value problem of an isotropic, homogeneous elastic half-plane under plane strain conditions is readdressed as the limiting case of the fully three-dimensional problem. Analytical solutions of the stress and strain tensors are obtained by taking the limit from known three-dimensional solutions. It is shown that the displacement fields for the plane strain problem are not well defined. A small number of simple expressions are developed, which provide a general solution for linearly-varying traction over arbitrary regions on the boundary. A simple, efficient, and rapidly convergent algorithm is developed which uses these solutions as analytic elements and provides a solution approach to the general boundary value problem. The method is verified against known solutions for Hertzian contact between parallel cylinders. Two numerical examples are presented for the analysis of shallow foundation systems. In the first, the boundary conditions are informed by analytical elastoplastic calculations and a strain influence analysis is performed and compared with the Schmertmann method. Subsequently, empirical laboratory contact traction distributions measured by Bauer et al., in both the normal and tangential directions are employed as boundary conditions for an analysis of the underlying stress field.

Numerical study on ratcheting performance of heavy haul rail flash-butt welds in curved tracks

Ratcheting at the rail heads due to cyclic rolling contact stresses is often observed in heavy haul railways, especially when there are large steering forces developed in low-radius curves. Additionally, ratcheting behaviour can be exacerbated when associated with material strength loss in the heat affected zones of rail welds. However, the numerical evaluation of this failure mode requires a sophisticated representation of elastic–plastic contact pressures and traction distributions, which, to the best of the authors’ knowledge, cannot be directly measured by any existing tools. In this paper, in order to accurately evaluate the ratcheting performance at rail welds in curved tracks, attempts were conducted to modify the FaStrip algorithm to analytically estimate the traction distributions based on contact pressures and creepages obtained from static finite element analysis and multi-body dynamic simulations, respectively. Cyclic rolling contact was then simulated by repeatedly applying contact pressure and traction on the rail surface consisting of both the rail weld and the parent rail regions. The weld region was divided into 23 sub-zones in the rolling direction to represent the material inhomogeneity within the heat affected zones. Cases on a 2000 m radius curved track and a tangent track were studied for comparison. It was found that the ratcheting strain rate in curved tracks can be significantly elevated due to the higher magnitude of traction, and the location of the elements with the highest ratcheting strain rate was dependent on the traction distribution patterns. The proposed method provides a general tool for predicting rolling contact fatigue initiation life and location, which will benefit the maintenance efficiency of heavy haul rail welds in curved track.

Wrinkle force microscopy: a machine learning based approach to predict cell mechanics from images

Combining experiments with artificial intelligence algorithms, we propose a machine learning based approach called wrinkle force microscopy (WFM) to extract the cellular force distributions from the microscope images. The full process can be divided into three steps. First, we culture the cells on a special substrate allowing to measure both the cellular traction force on the substrate and the corresponding substrate wrinkles simultaneously. The cellular forces are obtained using the traction force microscopy (TFM), at the same time that cell-generated contractile forces wrinkle their underlying substrate. Second, the wrinkle positions are extracted from the microscope images. Third, we train the machine learning system with GAN (generative adversarial network) by using sets of corresponding two images, the traction field and the input images (raw microscope images or extracted wrinkle images), as the training data. The network understands the way to convert the input images of the substrate wrinkles to the traction distribution from the training. After sufficient training, the network is utilized to predict the cellular forces just from the input images. Our system provides a powerful tool to evaluate the cellular forces efficiently because the forces can be predicted just by observing the cells under the microscope, which is much simpler method compared to the TFM experiment. Additionally, the machine learning based approach presented here has the profound potential for being applied to diverse cellular assays for studying mechanobiology of cells.

Analysing the accuracy of asymptotic approximations in incomplete contact problems

The error incurred in the representation of the contact pressure at the edges of incomplete contacts by first order asymptotes is treated, and the maximum value of the relative error found for a range of geometries, both symmetric and non-symmetric. For a symmetric power-law geometry, we identify when the first-order asymptote achieves maximum fidelity. Shear tractions are excited by both the application of a shear force and the application of bulk tension in one body. An asymptotic representation of the shear traction distribution under conditions of full stick is presented.

Elastic disk with isoperimetric Cosserat coating

A circular elastic disk is coated with an elastic beam, absorbing shear and normal forces without deformation and linearly reacting to a bending moment with a change in curvature. The inextensibility of the elastic beam introduces an isoperimetric constraint, so that the length of the initial circumference of the disk is constrained to remain fixed during the loading of the disk/coating system. The mechanical model for this system is formulated, solved for general loading, and particularized to the case of two equal and opposite traction distributions, each applied on a small boundary segment (thus modeling indentation of a coated fiber). The stress fields, obtained via complex potentials, are shown to evidence a nice correspondence with photoelastic experiments, ad hoc designed and performed. The presented results are useful for the design of coated fibers at the micro and nano scales.