Modelling of Non-Newtonian Starved Thermal-elastohydrodynamic Lubrication of Heterogeneous Materials in Impact Motion

This study presents a numerical model for the thermal-elastohydrodynamic lubrication of heterogeneous materials in impact motion, in which a rigid ball bounces on a starved non-Newtonian oil-covered plane surface of an elastic semi-infinite heterogeneous solid with inhomogeneous inclusions. The impact–rebound process and the microscopic response of the subsurface inhomogeneous inclusions are investigated. The inclusions are homogenized according to Eshelby’s equivalent inclusion method. The Elrod algorithm is adopted to determine the lubrication starvation based on the solutions of pressure and film thickness, while the lubricant velocity and shear rate of the non-Newtonian lubricant are derived by using the separation flow method. The dynamic response of the cases subjected to constant impact mass, momentum, and energy is discussed to reveal the influence of the initial drop height on the impact–rebound process. The results imply that the inclusion disturbs the subsurface stress field and affects the dynamic response of the contact system when the surface pressure is high. The impact energy is the decisive factor for the stress peak, maximum hydrodynamic force, and restitution coefficient, while the dynamic response during the early approaching process is controlled by the drop height.

A Computational Scheme for Evaluating the Stress Field of Thermally and Pressure Induced Unconventional Reservoir

The fluid flow connecting the hydraulic fracture and associated unconventional gas or oil reservoir is of great importance to explore such unconventional resource. The deformation of unconventional reservoir caused by heat transport and pore pressure fluctuation may change the stress field of surrounding layer. In this paper, the stress distribution around a penny-shaped reservoir, whose shape is more versatile to cover a wide variety of special case, is investigated via the numerical equivalent inclusion method. Fluid production or hydraulic injection in a subsurface resource caused by the change of pore pressure and temperature within the reservoir may be simulated with the help of the Eshelby inclusion model. By employing the approach of classical eigenstrain, a computational scheme for solving the disturbance produced by the thermally and pressure induced unconventional reservoir is coded to study the effect of Biot coefficient and some other important factors. Moreover, thermo-poro transformation strain and arbitrarily orientated reservoir existing within the surrounding layer are also considered.

Numerical study on butterfly wings around inclusion based on damage evolution and semi-analytical method

Butterfly wings are closely related to the premature failure of rolling element bearings. In this study, butterfly formation is investigated using the developed semi-analytical three-dimensional (3D) contact model incorporating inclusion and material property degradation. The 3D elastic field introduced by inhomogeneous inclusion is solved by using numerical approaches, which include the equivalent inclusion method (EIM) and the conjugate gradient method (CGM). The accumulation of fatigue damage surrounding inclusions is described using continuum damage mechanics. The coupling between the development of the damaged zone and the stress field is considered. The effects of the inclusion properties on the contact status and butterfly formation are discussed in detail. The model provides a potential method for quantifying material defects and fatigue behavior in terms of the deterioration of material properties.

Effects of Near-surface Composites on Frictional Rolling Contact Solved by a Semi-analytical Model

A semi-analytical model (SAM) to tackle the steady-state elastic frictional rolling contact problem involving composites is presented. Specifically, the frictional rolling contact is categorized into two subtypes, namely normal and tangential problems, and the conjugate gradient method (CGM) is used to figure out the normal pressure and tangential traction. In SAM, the equivalent inclusion method (EIM) is applied to analyze the influence of composites on the matrix, and the displacement disturbance resulting from such composites is added to the total surface displacement, which implements the coupling between surface contact and composites. The accuracy of the proposed model is verified by the finite element model. The effects of composites on the frictional rolling contact behavior are investigated. The results indicate that Young's modulus, as well as the size and location of the composites, are correlated with the distributions of tangential traction, subsurface stresses and the sizes of stick and sliding zones.

Elastic solution of a polyhedral particle with a polynomial eigenstrain and particle discretization

The paper extends the recent work (JAM, 88, 061002, 2021) of the Eshelby's tensors for polynomial eigenstrains from a two dimensional (2D) to three dimensional (3D) domain, which provides the solution to the elastic field with continuously distributed eigenstrain on a polyhedral inclusion approximated by the Taylor series of polynomials. Similarly, the polynomial eigenstrain is expanded at the centroid of the polyhedral inclusion with uniform, linear and quadratic order terms, which provides tailorable accuracy of the elastic solutions of polyhedral inhomogeneity by using Eshelby's equivalent inclusion method. However, for both 2D and 3D cases, the stress distribution in the inhomogeneity exhibits a certain discrepancy from the finite element results at the neighborhood of the vertices due to the singularity of Eshelby's tensors, which makes it inaccurate to use the Taylor series of polynomials at the centroid to catch the eigenstrain at the vertices. This paper formulates the domain discretization with tetrahedral elements to accurately solve for eigenstrain distribution and predict the stress field. With the eigenstrain determined at each node, the elastic field can be predicted with the closed-form domain integral of Green's function. The parametric analysis shows the performance difference between the polynomial eigenstrain by the Taylor expansion at the centroid and the 𝐶0 continuous eigenstrain by particle discretization. Because the stress singularity is evaluated by the analytical form of the Eshelby's tensor, the elastic analysis is robust, stable and efficient.

Analytical Model for Studying the Influence of Thickness on the Protective Effect

This paper presents an analytical model to study the influence of the thickness of the built-up layer (BUL) / built-up edge (BUE) on its protective effect during cutting. A new elastic-plastic contact model at the tool-chip interface is proposed to analyze the sliding contact problem with a layer of adhesion (including the BUL and BUE). The equivalent inclusion method (EIM) is utilized to analyze the stress disturbance caused by the adhesion and to evaluate the protective effect of the adhesion. In this method, the adhesion is considered as an equivalent elliptical inclusion at the tool-chip interface. The protective effect of the adhesion and the influence of the adhesion thickness on its protective effect can be evaluated. The proposed analytical model was verified based on experimental data obtained from dry cutting of SUS304 stainless steel. From the results, it can be confirmed that BUL/BUE can protect the cutting tool by affecting the stress distributions in the tool, the positions of yield initiation, and the tangential force acting on the tool. It can also be concluded that a greater thickness improves the protective effect of the BUL/BUE. Furthermore, the proposed model can also provide a clear understanding of the BUL/BUE formation phenomenon.

Factors of Stress Concentration around Spherical Cavity Embedded in Cylinder Subjected to Internal Pressure

In this paper, an accurate distribution of stress as well as corresponding factors of stress concentration determination around a spherical cavity, which is considered as embedded in a cylinder exposed to the internal pressure only, is presented. This approach was applied at three main meridians of the porosity by combining the Eshelby’s equivalent inclusion method with Mura and Chang’s methodology employing the jump condition across the interface of the cavity and matrix, respectively. The distribution of stresses around the spherical flaw and their concentration factors were formulated in the form of newly formulated analytical relations involving the geometric ratio of the cylinder, such as external radius and thickness, the angle around the cavity, depth of the porosity, as well as the material Poisson ratio. Subsequently, a comparison of the analytical results and the numerical simulation results is applied to validate obtained results. The results show that the stress concentration factors (SCFs) are not constant for an incorporated flaw and vary with both the porosity depth and the Poisson ratio, regardless of whether the cylinder geometric ratio is thin or thick.

Multiple ellipsoidal/elliptical inhomogeneities embedded in infinite matrix by equivalent inhomogeneous inclusion method

Traditional equivalent inclusion method provides unreliable predictions of the stress concentrations of two spherical inhomogeneities with small separation distance. This paper determines the stress and strain fields of multiple ellipsoidal/elliptical inhomogeneities by equivalent inhomogeneous inclusion method. Equivalent inhomogeneous inclusion method is an inverse of equivalent inclusion method and substitutes the subdomains of matrix with known strains by equivalent inhomogeneous inclusions. The stress and strain fields of multiple inhomogeneities are decomposed into the superposition of matrix under applied load and each solitary inhomogeneous inclusion with polynomial eigenstrains by the iteration of equivalent inhomogeneous inclusion method. Multiple circular and spherical inhomogeneities are respectively used as examples and examined by the finite element method. The stress concentrations of multiple inhomogeneities with small separation distances are well predicted by equivalent inhomogeneous inclusion method and the accuracies improve with the increase of eigenstrain orders. Equivalent inhomogeneous inclusion method gives more accurate stress predictions than equivalent inclusion method in the problem of two spherical inhomogeneities.

Numerical methods for solving the equivalent inclusion equation in semi-analytical models

Equivalent inclusion method is the basis for semi-analytical models in tackling inhomogeneity problems. Equivalent eigenstrains are obtained by solving the consistency equation system of the equivalent inclusion method and then stress disturbances caused by inhomogeneities are determined. The equivalent inclusion method equation system can only be solved numerically, but the current fixed-point iteration method may not be able to achieve deep convergence when the Young's modulus of inhomogeneity is lower than that of the matrix material. The most significant innovation of this paper is to reveal the non-convergence mechanism of the current method. Considering the limitation, the Jacobian-free Newton Krylov algorithm is selected to solve the equivalent inclusion method equation. Results indicate that the new algorithm has significant advantages of computing accuracy and efficiency compared with the classic method.

Elastic Solution of a Polygon-Shaped Inclusion With a Polynomial Eigenstrain

This paper presents the Eshelby’s tensor of a polygonal inclusion with a polynomial eigenstrain, which can provide an elastic solution to an arbitrary, convex inclusion with a continuously distributed eigenstrain by the Taylor series approximation. The Eshelby’s tensor for plane strain problem is derived from the fundamental solution of isotropic Green’s function with the Hadmard regularization, which is composed of the integrals of the derivatives of the harmonic and biharmonic potentials over the source domain. Using the Green’s theorem, they are converted to two line (contour) integrals over the polygonal cross section. This paper evaluates them by direct analytical integrals. Following Mura’s work, this paper formulates the method to derive linear, quadratic, and higher order of the Eshelby’s tensor in the polynomial form for arbitrary, convex polygonal shapes of inclusions. Numerical case studies were performed to verify the analytic results with the original Eshelby’s solution for a uniform eigenstrain in an ellipsoidal domain. It is of significance to consider higher order terms of eigenstrain for the polygon-shape inclusion problem because the eigenstrain distribution is generally non-uniform when Eshelby’s equivalent inclusion method is used. The stress disturbance due to a triangle particle in an infinite domain is demonstrated by comparison with the results of the finite element method (FEM). The present solution paves the way to accurately simulate the particle-particle, partial-boundary interactions of polygon-shape particles.