Power-law Materials Research Articles

ON THE EFFECT OF GRADING ELASTIC MODULUS IN FRICTIONAL LINE CONTACTS

Abstract We study the effect of elastic material grading close to the surface in line contacts with friction on internal stresses, particularly to quantify the reduction of tensile principal stresses due to grading. We observe that for the classical indentation by a parabolic indenter and power-law material grading, tensile stresses are reduced, but both dilatational and deviatoric strain energy are increased, which points to a possible decrease of strength at the surface. We then move to the more realistic case of exponential grading, for a sinusoidal form of pressure and shear traction, which is taken as representative of a wide class of contact conditions. We observe that tensile stresses can be removed completely if friction is not too high and Poisson's ratio is not too low, but anyway the effect is likely to be sufficiently strong to remove surface cracks. Dilatational strain energy is also decreased for this type of functional elastic grading, while deviatoric strain energy is not increased much at the surface. We thus confirm in a more general way than previously known the possible advantages of using functional elastic grading to increase resistance to contact loading and provide a general set of results to quantify this for engineering purposes.

Analytical solution for the structural strain parameter of power-law hardening materials to evaluate low- and high-cycle fatigue life of welded components

Analytical solution for the structural strain parameter of power-law hardening materials to evaluate low- and high-cycle fatigue life of welded components

Magnetohydrodynamic double diffusion natural convection of power-law Non-Newtonian Nano-Encapsulated phase change materials in a trapezoidal enclosure

PurposeThe present numerical investigation examines the magnetohydrodynamic (MHD) double diffusion natural convection of power-law non-Newtonian nano-encapsulated phase change materials (NEPCMs) in a trapezoidal cavity.Design/methodology/approachThe governing Navier-Stokes, energy and concentration equations based on the Cartesian curvilinear coordinates are solved using the collocated grid arrangement’s finite volume method. The in-house FORTRAN code is validated with the different benchmark problems. The NEPCM nanoparticles consist of a core-shell structure with Phase Change Material (PCM) at the core. The enclosure, shaped as a trapezoidal hollow, features a warmed (Th) left wall and a cold (Tc) right wall. Various parameters are considered, including the power law index (0.6 ≤ n ≤ 1.4), Hartmann number (0 ≤ Ha ≤ 30), Rayleigh number (104 ≤ Ra ≤ 105) and fixed variables such as buoyancy ratio (Br = 0.8), Prandtl number (Pr = 6.2), Lewis number (Le = 5), fusion temperature (Θf = 0.5) and volume fraction (ϕ = 0.04).FindingsThe findings indicate a decrease in local Nusselt (Nu) and Sherwood (Sh) numbers with increasing Hartmann numbers (Ha). Additionally, for a shear-thinning fluid (n = 0.6) results in the maximum local Nu and Sh values. As the Rayleigh number (Ra) increases from 104 to 105, the structured vortex in the streamline pattern is disturbed. Furthermore, for different Ra values, an increase in n from 0.6 to 1.4 leads to a 67.43% to 76.88% decrease in average Nu and a 70% to 77% decrease in average Sh.Research limitations/implicationsThis research is for two-dimensioal laminar flow only.Practical implicationsPCMs represent a class of practical substances that behave as a function of temperature and have the innate ability to absorb, release and store heated energy in the form of hidden fusion enthalpy, or heat. They are valuable in these systems as they can store significant energy at a relatively constant temperature through their latent heat phase change.Originality/valueAs per the literature review and the authors’ understanding, an examination has never been conducted on MHD double diffusion natural convection of power-law non-Newtonian NEPCMs within a trapezoidal enclosure. The current work is innovative since it combines NEPCMs with the effect of magnetic field Double diffusion Natural Convection of power-law non-Newtonian NEPCMs in a Trapezoidal enclosure. This outcome can be used to improve thermal management in energy storage systems, increasing safety and effectiveness.

Four-Point Bending of Basic Rails: Theory and Experimental Verification

Mathematical models of prediction provide theoretical support for basic rail automation. The three-point bending method for basic rails is characterized by its simplicity and flexibility, and, as such, it is widely used in bending processes. However, due to the significant curvature changes that occur after bending, it is not suitable for scenarios requiring large arc bending, and its range of achievable deflections is limited. This study focuses on four-point bending, dividing the bending process into three stages and using a power-law material hardening model to establish different bending moment expressions for each stage. We derived the relationships between curvature, elastic zone ratio, load, and deflection, ultimately creating a load–deflection model. Based on the simple springback law, we developed the final bending prediction model. Finite element simulations were conducted to simulate the bending process under various conditions, using top punch distances ranging from 200 mm to 400 mm and die distances ranging from 600 mm to 1000 mm. These simulations validated the advantages and accuracy of the four-point bending prediction model in large arc bending. Additionally, a four-point bending experimental setup was established under specified conditions. The experimental results were compared with the theoretical model calculations, showing errors within 0.2 mm and thus verifying the accuracy of the four-point bending prediction model. The mathematical model developed in this study provides theoretical support for the automation of basic rail bending.

The Effect of Hyperelasticity and Nonlinearity on the Dynamic Behaviors of Hyperelastic Functionally Graded Beams on Nonlinear Elastic Foundation

Hyperelastic functionally graded materials have a wide range of application prospects in soft robotics and biomedical fields. This paper investigates the nonlinear free and forced vibrations of a hyperelastic functionally graded beam (HFGB) based on higher-order shear deformation beam theory. The geometrical nonlinearity is considered by using the von-Kármán’s nonlinear theory. The three-material-parameter free energy function named as Ishihara model is employed to characterize the hyperelastic material. The power-law gradient form along the thickness direction is adopted. The HFGB is resting on the elastic foundation. The Winkler, Pasternak and nonlinear stiffness coefficients are considered. The time-harmonic external force is applied to the HFGB. The nonlinear governing equations for the vibration of the HFGB are derived by using Hamilton’s principle, and are subsequently transformed into ordinary differential equations via Galerkin’s method. The nonlinear free vibration and primary resonance of the HFGB are investigated analytically by employing the extended Hamiltonian method and multiple scales method, respectively. The results indicate that the power-law index, slenderness ratio, material properties, and elastic foundation parameters have significant influences on the nonlinear frequency of free vibration as well as the frequency–response and force–response curves of forced vibration. The phase plane method is employed to analyze the system’s stability states under various excitation amplitudes. The relative error between the results of the current computational model and the published literature is less than 0.1 percent.

Measuring the viscoelastic relaxation function of cells with a time-dependent interpretation of the Hertz-Sneddon indentation model

The Hertz-Sneddon elastic indentation model is widely adopted in the biomechanical investigation of living cells and other soft materials using atomic force microscopy despite the explicit viscoelastic nature of these materials. In this work, we demonstrate that an exact analytical viscoelastic force model for power-law materials, can be interpreted as a time-dependent Hertz-Sneddon-like model. Characterizing fibroblasts (L929) and osteoblasts (OFCOLII) demonstrates the model's accuracy. Our results show that the difference between Young's modulus EY obtained by fitting force curves with the Hertz-Sneddon model and the effective Young's modulus derived from the viscoelastic force model is less than 3%, even when cells are probed at large forces where nonlinear deformation effects become significant. We also propose a measurement protocol that involves probing samples at different indentation speeds and forces, enabling the construction of the average viscoelastic relaxation function of samples by conveniently fitting the force curves with the Hertz-Sneddon model.

Fully optimized second-order estimates for the macroscopic behavior and field statistics of particle-reinforced viscoplastic composites

This paper is concerned with the characterization of the macroscopic behavior and statistics for the distribution of the stress and strain-rate fields in composites consisting of random and isotropic suspensions of rigid spherical particles in power-law viscoplastic materials. For this purpose, use is made of the Fully Optimized Second-Order (FOSO) homogenization method (Ponte Castañeda, 2016) in combination with recently developed estimates (Kammer and Ponte Castañeda, 2022) for the macroscopic properties of the associated ‘linear comparison composite’ (LCC). Special attention is devoted to the method’s ability to account for the dependence of the homogenized properties of the nonlinear composite on the Lode angle (third invariant) of the applied loading. It is found that, while, for large particle volume fractions c, the effective flow stress is only weakly dependent on the Lode angle, for dilute volume fractions, the dependence on the Lode angle becomes more pronounced. In the ideally plastic limit, as c tends to zero, the effective yield stress is shown to depend linearly on c for axisymmetric shear, while this dependence becomes weaker with a non-analytic leading-order correction of O(c/lnc) for pure shear loading. This strong dependence on the Lode angle at dilute concentrations is shown to be due to significant differences in the local deformation patterns, which become strongly anisotropic and localize for pure shear conditions, but do not for axisymmetric shear. In turn, the FOSO homogenization method is able to capture the statistical features of these different deformation patterns by providing consistent estimates for the covariance tensor of the strain-rate field fluctuations in the matrix phase, which tend to become more strongly anisotropic for the pure shear case. As c increases, the shear bands are deflected by the randomly dispersed spheres leading to a more isotropic distribution of the stress and strain-rate fields, which is consistent with a weaker Lode angle effect. The estimates can also capture the effect of strong particle interactions, including the existence of a rigidity threshold where the macroscopic flow stress and field fluctuations blow up.

Unified theoretical solutions for describing the crack-tip stress fields of mode-II cracked specimens under the fully plastic condition

Potential novel singular solutions to the crack-tip stress fields beyond the asymptotic solution frame are discussed in this study. It is believed that one of them is not only suitable for the crack-tip stress fields of mode-I cracked specimens but also for those of mode-II cracked specimens. The crack-tip stress fields of mode-II cracked specimens are investigated based on the singular solution and the stress factor of mode-II cracked specimen derived from energy density equivalence principle. Further, special functions for describing the coefficients of normalized stress distributions of mode-II cracked specimens are proposed. The method to determine the parameters in the theoretical solutions are introduced. To give an example, one set of parameters for compact shear specimen under plane-strain conditions are determined by the finite element analysis of a few typical materials and geometries conditions. Finally, comparisons of radial and circumferential stress distributions and contour lines of equivalent stress are completed for the compact shear specimens and Arcan-shaped shear specimens of power-law plastic materials under plane-strain and plane-stress conditions. And the results show that the applicability and reliability of the proposed unified theoretical solutions are verified.

Lubricated stagnation point flow of Walters-B nanofluid with Cattaneo-Christov heat flux model: Hybrid homotopy analysis simulations

Thermal mechanism of non-Newtonian nanofluid coated by lubricated power law material has been theoretically worked out. The slip interaction constraints has been proposed for inspecting the thermal transport. The modelling of heat transfer is processed via updated Cattaneo-Christov (CC) expressions. The rheological constraints of nonlinear model are observed by Walter B fluid. The flow phenomenon is governed by stagnation point flow. The differential system is associated to interaction of similarity variables. The solution technique is based on new developed hybrid homotopy analysis scheme. The hybrid homotopy algorithm is developed by combining the shooting technique and traditional HAM. The accuracy regarding the developed scheme is carefully inspected after making comparison of results with earlier claimed studies. The different physical aspect of transport phenomenon is attributed in view of parameters.

Axisymmetric stagnation-point flow of non-Newtonian nanomaterial and heat transport over a lubricated surface: Hybrid homotopy analysis method simulations

Abstract The heat transfer phenomenon associated with the lubricated surfaces offers applications in the manufacturing processes, thermal systems, industrial systems, and engineering phenomenon. It is a well-established fact that improvement in heat transfer is recently successfully claimed with the interaction of nanoparticles. Following such motivation in mind, the prime objective of current continuation is to perform the prediction of heat transfer in second-grade material subject to the lubricated surface. The lubricants are filled with non-Newtonian power law material. The varying thickness of the thin lubricating layer permits an imperfect slip surface. The second-grade fluid interfaces with the boundary condition. The modified semi-analytical tool termed as hybrid homotopy scheme is used to perform the simulations. Shooting and homotopy methods are combined in this new approach. Relevant effects of parameters on physical phenomenon are explained. The importance of influencing parameters in relation to the velocity field, temperature, and concentration profiles is investigated graphically. It is claimed that analytical computations existed for shear thinning case. It is observed that there is a noticeable drop in concentration and thermal profiles due to the variation of viscoelastic parameter. The control of free stream velocity is claimed due to the interaction of slip parameters.

Analysis of second grade nanofluid flow confined by lubricated disk with bioconvection

Thermal mechanism of non-Newtonian nanofluid coated by lubricated power law material has been theoretically worked out. The slip interaction constraints have been proposed for inspecting the thermal transport. The concept of microorganism is utilized to achieve the stabilization of the suspension of nanoparticles. The rheological constraints of nonlinear model are observed by second grade fluid. The flow phenomenon is governed by stagnation point flow. The second-grade (SG) axisymmetric bio-convective nanofluid flow across a moving surface is considered. The flow equations are simplified by the implementation of compatible similarity transformations. The resulting system is solved numerically and analytically by using a hybrid homotopy technique. With the use of a graph and tabular data, the importance of influencing parameters in relation to the velocity field, motile density microorganism’s, temperature and concentration profiles is investigated. The thermal profile is observed to be lowering with higher Prandtl number values. There is a noticeable drop in concentration and thermal profiles against higher viscoelastic parameter values. The microorganism profile is lower in the presence of bio-convected Lewis number. Moving from Newtonian fluid to non-Newtonian fluid, a decrease in motile organism and Nusselt number.

Plane and axisymmetric problems of an interfacial crack in a power-law graded material

Plane and axisymmetric problems of interfacial cracks in a power-law graded infinite medium are examined. It is shown that the models are governed by integral equations of Mellin’s convolution type whose kernels are expressed through the hypergeometric function. Exact solutions are derived by the method of orthogonal Jacobi polynomials in a series form and by the Wiener–Hopf method by quadratures. Mode-I stress intensity factors and the associated weight functions for power-law graded materials in the plane and axisymmetric cases are introduced and evaluated. It is shown that, although the displacement jump and the normal traction component have power singularities at the crack tip different from 1/2, the strain energy variation is proportional to the crack length change as in the case of homogeneous materials. A Griffith-type criterion of crack propagation in power-law graded materials is proposed and results of numerical tests are reported.

Data-driven nonparametric identification of material behavior based on physics-informed neural network with full-field data

A Physics-Informed Neural Network (PINN) model is developed to extract material behavior from full-field displacement data. The PINN model consists of independent networks describing mechanical fields such as displacement, deformation gradient, stress, and strain energy density function. The displacement and deformation gradient networks learn from the deformation data of a specimen, while the stress and material networks predict the stress distribution within the specimen and learn the behavior of the target material, respectively. Each mechanical field should satisfy the physics laws of momentum balance, compatibility, constitutive relations, and boundary conditions. The constraints are wrapped up in the loss function, and the weights and biases of each hidden layer are determined by minimizing the loss function. This study introduces a two-stage learning strategy to deal with the minimization process, which is a multi-objective optimization problem. The accuracy of the proposed method is verified for linear elastic, power-law, and incompressible hyperelastic materials. The data-driven identification method successfully identifies the response of target materials in uniaxial tension, simple shear, and tension-shear-coupled loading cases. The robustness of the proposed method is verified on full-field data with noise and missing points. Data-driven identification can handle full-field data with missing points and maintain a reasonable accuracy level as the noise amplitude increases. Furthermore, data-driven identification results of an incompressible hyperelastic material are compared with conventional parametric identification results. The proposed method has advantages over the conventional method as it requires no predefined model. The results indicate that the data-driven identification method can be applied to new materials without a developed constitutive model or materials with manufacturing uncertainty.

Further investigation of two-parameter J-A estimation for clamped SET specimens

Based on the J-A two-parameter characterization, this paper presents a systematic study of the crack-tip constraint for clamped single-edge notched tension (SET) specimens. Extensive finite element analyses (FEA) are conducted to obtain the A solutions for the modified boundary layer (MBL) model and the SET specimen. The material used in the analysis is the Ramberg-Osgood power-law material, with varying yield stress and strain hardening exponents. Based on previous studies, the T-stress-related A solutions for MBL problems under small-scale yielding conditions are obtained from FEA results, and the A solutions for the SET specimen are found to scale with normalized applied load under large-scale yielding conditions. By summing up these two solutions, the total A for the SET specimen can be estimated, covering the entire range from small-scale to large-scale yielding conditions. A key geometry factor is introduced as a function of material properties, crack depths and applied load. Some fitting equations are proposed in this estimation method, and a Python program is developed to quickly and accurately obtain the total A for the SET specimen. The results of the present study show good agreement for all the simulated cases. The proposed solutions of constraint parameter A can be used to establish the constraint-corrected J-R curve for SET specimens.

Investigation of composed charged particles with suspension of ternary hybrid nanoparticles in 3D-power law model computed by Galerkin algorithm

Transport of heat visualizes a vital role in many industrial developments. Current study is discussing the role of Joule heating, solar thermal radiation, heat generation/absorption, reactions (homogeneous and heterogeneous) with variable thermal conductivity on partially ionized power law material past over a three-dimensional heated stretched surface. The power law model is assumed to have the thermal characteristics of ethylene glycol material. The phenomenon of momentum and energy balance is derived in Cartesian coordinates and developed PD (partial differential)-equations. Swimming pools, solar collectors, food processing, electronic gadgets, cooling systems, magnetic field measurement, computer chips, thermal enhancement, semiconductor characterization, nuclear fusion research and other physical applications are examples of ongoing research. The principle of boundary layer simplified the governing problem. The complex coupled PD (partial differential)-equations have been converted into ordinary differential equations OD (ordinary differential)-equations by using appropriate similarity transformation. The converted boundary value problem is complex and highly nonlinear which does not have the exact solution. The approximate solution is computed numerically via finite element scheme (FES) which is coded in MAPLE 18.0 symbolic package. The convergence of the scheme is established through grid independent survey and the solution is plotted against numerous involved parameters. Thermal performance produced by Si{O}_{2}-Ti{O}_{2}-{Al}_{2}{O}_{3}/EG is higher thermal performance produced by Si{O}_{2}-Ti{O}_{2}/EG. Ion slip and Hall forces are responsible for generating Joule heating mechanism that is responsible for reduction of velocity curve and generating shear stresses. Hence, tangential stresses are declined against increasing {beta }_{i} and {beta }_{e}.

Theoretical solutions for 2D mode-I crack-tip stress fields in power-law plastic materials based on the stress factor derived from the developed median-energy–density equivalence method

This study presents the energy density equivalence equation, demonstrating that the strain energy density of an arbitrary representative volume element (RVE) under a complex stress state equals to that under a uniaxial stress state when volume change is neglected. Based on the developed median-energy–density equivalence method, the analytical solution for equivalent stress of the RVE at the energy density equivalence point in power-law plastic materials is derived in a new method and the concept of stress factor is proposed. Additionally, novel special functions are introduced to accurately describe the angular distributions of stresses in this study. By combining the stress factor, integral theoretical solutions for mode-I crack tip stress fields in power-law plastic materials under finite plane conditions are proposed. Finally, parameterized studies are conducted on bending and tension specimens under plane-strain and plane-stress conditions. The results of stress distributions and contour lines predicted by the integral theoretical solutions are compared with the finite element analysis and HRR solutions for five mode-I plane specimens, providing verification of the solutions’ applicability.

The Improved Irwin’s Model for Crack Problems in Power-Law Hardening Materials

In this paper, an improved Irwin’s model for power-law hardening materials is established through the static equivalence between the linear elastic stress solution and the stress redistribution in the plasticity-affected zone (PAZ) under small-scale yielding (SSY) conditions. HRR solution is adopted to describe the stress field within the crack-tip plastic zone (CTPZ). Analytical expressions of the effective stress intensity factor (SIF) and the size of PAZ are derived to illustrate the effect of CTPZ on the fracture behavior for power-law hardening materials. Influences of external load and hardening exponent on fracture behavior are explored. The proposed model can assess the toughening effect of CTPZ accurately using the SIF ratio, which is always smaller than 1. This model can be applied to various materials obeying power-law hardening constitutive equations. It will reduce to the extended Irwin’s model for elastic-perfectly plastic materials in the limiting case.

Asymmetrical cutting-edge design of broaching tool based on FEM simulation

Finite element method (FEM) based model of broaching Inconel 718 was established to optimize an asymmetrical cutting-edge by the commercial software AdvantEdge in this paper. The asymmetrical cutting-edge form by two circles was proposed. The power-law (P-L) material constitutive model, including strain hardening, strain rate hardening and thermal softening was developed by split Hopkinson pressure bar (SHPB) test and high-temperature Vickers hardness test. The parameters of thermal properties were measured by laser flash diffusivity apparatus. The friction model between TiN (The outer layer coating material of the tool) and Inconel 718 was obtained by ball on disc friction test. The simulated and experimental results have been compared comprehensively in the items of chip formation and tool wear width. The results showed that the simulated results were consistent with the experimental results. The average errors of chip curl radius, chip thickness, wear width of rake face and flank face were 7.68%, 3.56%, 13.64% and 12.24%, respectively, which proved the accuracy of the simulation model. Then the optimal asymmetrical cutting-edge of broach were obtained through orthogonal test based on the accurate FEM model. The rise per tooth of the optimal broach tool is 0.04 mm/tooth, the rake angle is 13°, the circle radius R1 is 0.13 mm, and the circle radius R2 is 0.02 mm.

Rigorous plane-stress solution and buckling analysis of rectangular functionally graded plates subjected to non-uniform edge loads

In this article, analytical solutions for the buckling analysis of simply supported rectangular functionally graded (FG) plates subjected to various forms of non-uniform in-plane loads are presented. Both perfect and porous FG plates with power-law material property variation are considered. The analytical solution is obtained by adopting a two-step approach: First, a rigorous plane-stress solution is developed based on the superposition of Airy’s stress functions, and later, these stress solutions are used in the buckling load computation of the FG plate based on Galerkin’s technique. One key aspect of the proposed solution is that it can consider different non-uniform waveforms of the in-plane loads at two opposite sides of the plate. Results show that the nature of the edge load has a profound effect on the stress solution of the FG plates. Several tables and graphs are presented to obtain a qualitative and quantitative understanding of the buckling behavior of FG plates influenced by various plate design parameters which include plate aspect ratios, power-law exponent, and porosity index. Comparative studies are also presented to highlight some erroneous results published in the open literature. It can be emphasized that the analytical results reported here can be used as a reference to judge the accuracy of other numerical solutions such as the finite element method.

Peristaltic flow of chemically reactive Ellis fluid through an asymmetric channel: Heat and mass transfer analysis

The importance of the peristaltic phenomena in biological and biomedical engineering has recently sparked a significant interest. In the current study, the peristaltic pattern of Ellis fluid in a channel has been inspected following the propagation of wave train with infinite magnitude. The confined flow regime is assumed due to asymmetric channel. The chemical reaction features are also considered in the modelling. The flow modelling is followed with the small Reynolds number hypothesis while the long wavelength is premised. The influence of various emerging physical parameters in the obtained solutions is observed. The computed result is presented in graphical form and discussed. The streamlines for flow parameters are also presented and discussed under different flow situations. The significant results of the current studies are that, the fluid velocity declined with higher values of Ellis fluid parameters. The pressure gradient improves with the material parameters. The higher fluctuation in power-law index α improves the temperature profile. The concentration reduces with the Soret constant and Schmidt number. The more significant change in power-law index and material perimeter result in a progressive concentration profile.