Deformation Boundary Conditions Research Articles

Experimental and theoretical analysis of mechanical properties of FRP laminated rubber bearing

Fiber reinforced polymer (FRP) laminated rubber bearing is a kind of high-performance isolator with simple structure, low cost, light weight and excellent mechanical properties. The mechanical properties of FRP laminated rubber bearings are investigated by experimental and theoretical methods. Based on the pseudo-static vertical compression test and compression-shear test experiment, the vertical and horizontal mechanical properties of the bearings under different vertical pressures are investigated. According to the deformation compatibility and boundary conditions of elastic composites, the modified calculation equations for the vertical and horizontal stiffness of FRP laminated rubber bearings are derived, and the theoretical analysis is verified by the results of laboratory tests. The influence law of characteristic parameters such as shape factor on the mechanical properties of bearings has been investigated. Tests and theoretical analysis show that the increase of vertical pressure causes a certain degree of increase in the vertical stiffness and a significant reduction in the horizontal stiffness of the bearing. By studying the effect of horizontal and vertical stiffness, the manuscript provides a useful reference for the design of FRP laminated rubber bearings.

Closed-form solutions for clamped FGM plates via the unified formulation and boundary discontinuous method

There are very few precise and reliable solutions available in the existing literature for functionally graded plates with fully clamped ends under mechanical loads, indicating a significant research gap in this area. In this article, an analytical solution for clamped functionally graded plates subjected to mechanical load is introduced. The shear deformation theory, governing equations, and boundary conditions are derived based on the Carrera Unified Formulation (CUF) strategy, and the solution is obtained by using the boundary-discontinuous Fourier method. The mechanical properties of the plates are assumed to vary according to an exponential law and a power-law distribution along the thickness direction in terms of the volume fractions of the constituents. The results presented in this article cover a large spectrum of plate thicknesses, ranging from thick to thin, and encompass various values of the functionally graded parameter. Accurate results of shear deformation theories with various order of expansion are achieved. Finally, a gap in the literature is covered and benchmarks solution are provided.

Development of simplified analytical model to predict the roll separating force in bar rolling

ABSTRACT Prediction of roll separating force (RSF) is necessary to design new rolling mills, optimise the number of passes and for online control of dimensional accuracy. The intricacy of deformation and complex mixed boundary conditions of velocity and friction makes exact analytical prediction of RSF in a bar or shape rolling quite difficult. Hence, the formula for the prediction of RSF while bar rolling is developed in this research using sticking and slipping friction, assuming the process is equivalent to compression of the simple rectangular bar between two flat plates. The concept of the weak plane strain is also incorporated in the proposed formula. Sinel’nikove’s formula derives the contact area between the roll and material. Considering the direct significance of the contact area on the value of RSF, the factors in the said formula are corrected based on a graphical method. Shida’s formula is used to calculate the flow stress. The proposed formula is compared with more than fifty distinct experimental findings found in the literature. It is found that the formula is in good agreement with majority of the experimental data. The derived simplified analytical formula can be successfully applied to most types of passes in such industries.

Mechanical behavior of thermally-induced shape memory polymer composites thin-wall tube in non-uniform thermal field: Experiments and simulations

Thermally-induced shape memory polymer composites (SMPCs) have the capability to self-deploy when stimulated by a specific temperature. As a basic component of deployable truss structure, SMPCs thin-wall circular tubes will undergo complex non-linear deformation and changing thermal boundary conditions during application. This article aims to research the mechanical behavior of thermally-induced SMPCs thin-wall circular tubes in a non-uniform thermal field. Thermal properties of SMPCs including thermal conductivities and coefficients of thermal expansion are tested as fundamental temperature-dependent parameters. A 500 mm-long SMPCs thin-wall tube is conducted in the flattening-folding-cooling-reheating experiments through a specially manufactured loading apparatus. Combining phase transition theory and Fourier heat conduction, a coupling algorithm is proposed to describe the thermo-mechanical behavior of SMPCs, which is implemented by the user subroutines VUMAT by commercial finite element analysis packages ABAQUS. By comparing the results from the simulation of the non-uniform thermal field, the simulation of the uniform thermal field and experiments, the coupling algorithm is verified, and the effect on SMPCs from the non-uniform thermal field is given.

Literature review of the stress, strain and separating force evaluation during hot bar rolling of different types of steels

Designing new rolling mills, optimization of the number of passes and online control of dimensional accuracy are the processes in which the accurate prediction of roll separating force (RSF) for different types of material is necessary. The formula of RSF is mainly a function of flow stress of the material, strain, strain rate, mean effective pressure and area of contact which further depend on the complexity of deformation and complex diversified boundary conditions in velocity as well as friction. Researchers in the last decades have used finite element analysis (FEA), experimental methods, fluid-like consideration of the material and many more techniques to estimate these parameters for low carbon steel and alloy steel. Based on the reviewed literature, the available formulas are classified into two major categories i.e. (i) Analytical and Empirical solutions and (ii) Finite element-based solutions. The majority of the existing models in the literature rely on empirical adjustments of theories of flat rolling dating back to 1943, whereas FEA-based formulae are only applicable to a single circumstance. In conclusion, it is suggested that it is important to create a fully analytical and relatively simple formula that can more precisely predict the RSF for the bar rolling process.

Effect of shape on void growth: A coupled Extended Finite Element Method (XFEM) and Discrete Dislocation Plasticity (DDP) study

Voids are one of the many material defects present at the microscopic length scale. They are primarily responsible for the formation of cracks and hence contribute to ductile fracture. Circular voids tend to deform into elliptical voids just before their coalescence to form cracks. The principle aim of this study is to investigate the effect of void shape on the micro-mechanism of void growth by using Discrete Dislocation Plasticity simulations. For voided crystals, conventional DDP produces a continuous slip step throughout the material even if a dislocation escapes from a non-convex domain. To overcome this issue, the Extended Finite Element Method (XFEM) is used here to incorporate the displacement discontinuity. Different aspect ratios of elliptical voids are considered under uniaxial and biaxial deformation boundary conditions. The results suggest that voids having the largest surface area tend to have maximum growth rate as compared to void with lower surface area, i.e. “larger is faster”. Under biaxial loading, a higher magnitude of strain hardening, and void growth rate are observed as compared to uniaxial loading. The results also suggest that the orientation of slip planes as well as voids, affect the overall plastic behavior of the voided-ductile material. Furthermore, circular void tends to induce minimum growth rate but have the maximum strain hardening effect as compared to other void shapes under both loading conditions. The results of this study provide a deeper understanding of ductile fracture with applications in manufacturing industry, aerospace industry and in the design of nano/micro-electromechanical devices i.e. NEMS/MEMS.

Mathematical modeling of planar physically nonlinear inhomogeneous plates with rectangular cuts in the three-dimensional formulation

Mathematical models of planar physically nonlinear inhomogeneous plates with rectangular cuts are constructed based on the three-dimensional (3D) theory of elasticity, the Mises plasticity criterion, and Birger’s method of variable parameters. The theory is developed for arbitrary deformation diagrams, boundary conditions, transverse loads, and material inhomogeneities. Additionally, inhomogeneities in the form of holes of any size and shape are considered. The finite element method is employed to solve the problem, and the convergence of this method is examined. Finally, based on numerical experiments, the influence of various inhomogeneities in the plates on their stress–strain states under the action of static mechanical loads is presented and discussed. Results show that these imbalances existing with the plate’s structure lead to increased plastic deformation.

Parameters of the Deformation Zone and Boundary Conditions of the Piercing Process

Parameters of the Deformation Zone and Boundary Conditions of the Piercing Process

ДО ПИТАННЯ ПРО ВИЗНАЧЕННЯ ГРАНИЧНИХ УМОВ ПРОЦЕСУ ПРОКАТКИ-ВОЛОЧІННЯ У ЗДВОЄНИХ РОЛИКОВИХ ВОЛОКАХ

There is a great demand in the world for steel strip profiles, which are widely used in mechanical engineering, instrument making and other sectors of the national economy. In Ukraine, the production of strip profiles is extremely limited. The production volumes of such products, if a wide range of sizes and brands are required, are relatively small. The creation of large specialized enterprises for the production of strip profiles in these conditions is economically unprofitable, the orientation towards the import of these profiles makes the economic and technological security of Ukraine dependent on foreign suppliers. For the production of high-precision profiles, various combinations of basic shaping and auxiliary processes are used, depending not only on the characteristics of the profile itself and the technical requirements of the supply, but also on the size of the ordered batch and the capabilities of the available equipment. This explains the high complexity of technological design. The aim of the work is to determine the boundary conditions of deformation in double roller dies, which ensure a stable flow of the drawing process according to the criterion of the safety factor of the deformed metal. It was found that when calculating the transitions of drawing strip profiles during deformation of the initial round billet with a diameter of 3.67 mm from steel St.70 with a width to thickness ratio of more than two, the maximum value of the relative reduction per transition should not exceed 40-50 % during deformation without counter tension, and 40 % – with tension. Analytical studies have shown that in the range of working reductions per transition (5-40 %), the safety factor that guarantees the process of metal deformation without breakage and does not allow local constrictions and distortion of the profile geometry is not exhausted. It was found that in the range of working reductions per transition (5-40 %), the deformation process of the initial round billet in a roller die with smooth rollers proceeds without exceeding the permissible values of the drawing forces. It has been determined that with an increase in the relative reduction in the first roller die, the range of working reductions in the second roller die, which affects the stability of the metal drawing process, decreases.

INVESTIGATION OF LONGITUDINAL STABILITY PROCESS AT ROLLING WITH STRIP TENSION

Earlier in the works, a qualitative construction of the diagram of changes in longitudinal normal stresses in the case of thin-sheet rolling with strip tension was proposed.. On the basis of the proposed stress distribution, the internal longitudinal forces were determined in a dimensionless form. Knowing these values, it is possible to determine the average resultant longitudinal force of the plastically deformed metal during rolling with the strip tension. It is shown that the value of the mean resulting internal forces is a more stringent boundary condition for the rolling process than the equality of the angle of the neutral section to zero.In this work, the influence of the strip tension mode on the ultimate stability of the metal in the rolls was determined, as well as the limiting condition of rolling, the change in the moment in the deformation zone and the angle of the neutral section. The theoretical analysis was carried out proceeding from the solution of the equilibrium equation with the specific friction forces specified by the Coulomb model. Using the technique proposed in previous works, we studied the effect of the magnitude and mode of strip tension on the longitudinal stability of the sheet during continuous rolling. It is shown that the rear tension has a significant effect on the average resultant longitudinal internal forces, and thereby predetermines the boundary conditions of the rolling process. The boundary conditions of deformation in a steady-state process presented in this work and generally accepted in theory differ significantly, which should be taken into account when developing a technology for continuous sheet rolling.

Thermo-Elastic Creep Analysis and Life Assessment of Thick Truncated Conical Shells with Variable Thickness

A semi-analytical method is presented to investigate time-dependent thermo-elastic creep behavior and life assessment of thick truncated conical shells with variable thickness subjected to internal pressure and thermal load. Based on the first-order shear deformation theory (FSDT), equilibrium equations and boundary conditions are derived using the minimum total potential energy principle. To the best of the researcher’s knowledge, in previous studies, thermo-elastic creep analysis of conical shell with variable thickness based on the FSDT has not been investigated. Norton’s law is assumed as the material creep constitutive model. The multilayered method is proposed to solve the resulting equations, which yields an accurate solution. Subsequently, the stresses at different creep times can be obtained by means of an iterative approach. Using Robinson’s linear life fraction damage rule, the creep damages of conical shells are determined and Larson–Miller parameter (LMP) is employed for assessing the remaining life. The results of the proposed approach are validated with those of the finite element method (FEM) and good agreement was found. The results indicate that the present analysis is accurate and computationally efficient.

Nonlinear free vibration analysis of functionally graded beams by using different shear deformation theories

This study investigates the nonlinear free vibration of functionally graded material (FGM) beams by different shear deformation theories. The volume fractions of the material constituents and effective material properties are assumed to be changing in the thickness direction according to the power-law form. The von Kármán geometric nonlinearity has been considered in the formulation. The Ritz method and Lagrange equation are adopted to yield the discrete formulations. A direct numerical integration method for the motion equation in matrix form is developed to solve the nonlinear frequencies of FGM beams. Comparing with the global concordant deformation assumption (GCDA), a new deformation assumption named as local concordant deformation assumption (LCDA) is proposed in this study. The LCDA fits with the real deformation of the vibrating beam better, thus more accurate results of the nonlinear frequency can be expected. In numerical results, the comparison study of the GCDA and LCDA is carried out. In addition, the effects of power-law index, slenderness ratio and maximum deflection for different shear deformation theories and boundary conditions on the nonlinear frequency of the beam are discussed.

Research on deformation law and mechanism for milling micro thin wall with mixed boundaries of titanium alloy in mesoscale

At mesoscale, the blade of micro impeller with Ti-6Al-4V titanium alloy (micro thin wall with mixed boundaries) is a thin-walled structure, characterized by small size, low stiffness, high surface precision, complex boundaries, and difficult-to-control machining deformation. To control and reduce the deformation of micro thin wall, it is necessary to study the micro-deformation mechanisms during milling micro thin-wall parts. In side milling of micro thin wall with mixed boundaries, the micro thin wall suffers from dynamic alternating forces, and the deformation modes are complex, so the direct and accurate theoretical modelling is difficult to make. Since the feed speed in milling micro thin wall is less, the small deformation of micro thin wall can be described based on the flexure deformation model of the thin plate subjected to elastic loadings. Firstly, accompanied by the reciprocal theorem of work for a curved thin plate, Kirchhoff-Love small deformation model of micro thin wall is used to establish the deformation equation and boundary conditions of micro thin wall with mixed boundaries under concentrated milling forces. Thus, the obtained boundary conditions are mainly applied to the subsequent finite element simulation of three-dimensional milling deformation of micro thin wall. Then a three-dimensional deformation simulation of the micro thin wall under different milling parameters is carried out by considering the micro-walled structure, material elasto-plastic constitutive model, the stiffness and geometric structure of micro milling tools in finite element method. In the workpiece constitutive modelling, considering the strain gradient plastic model in micro milling of Ti-6Al-4V titanium alloy, the workpiece plastic constitutive model is modified by introducing the intrinsic characteristic length of the material to describe the size effects in mesoscale micro milling. Finally, a series of the corresponding experiments on the milling of titanium alloy micro thin walls are carried out. By comparing and analyzing the experimental values and finite element numerical results of the micro thin-walled deformation, the micro deformation mechanisms in milling of thin wall are revealed. It also verifies the accuracy and effectiveness of this established milling finite element model of thin wall, and provides theoretical basis and technical support for controlling the milling deformation of micro thin-walled parts.

A viscoelastic constitutive relation for the rate-dependent mechanical behavior of rubber-like elastomers based on thermodynamic theory

Because of large deformation and viscoelastic properties of rubber-like elastomers, it is very challenging to establish its rate-dependent constitutive relation. Based on the basic theory of thermodynamics, the energy dissipation rate of the materials is derived in this paper. Then, a viscoelastic constitutive relation suitable for describing the rate-dependent behavior of rubber-like elastomers is proposed based on the viscoelastic theory and energy dissipation rate. In order to explain the application of constitutive relation, the constitutive equation is simplified according to the deformation characteristics and boundary conditions of rubber-like elastomers under working conditions, and the simplified one- or two-dimensional constitutive forms are obtained. The existing classical data is fitted and analyzed through the simplified form of the model, and found that the new model has the ability to describe the stress-strain relationship of the materials under common deformation modes. Meanwhile, through comparative analysis found that the overall prediction accuracy of the new model is better than that of some commonly used models. Finally, the newly proposed constitutive relation is used to predict the nonlinear mechanical behavior of three kinds of rubber-like elastomers at different strain rates, and the validity and universality of the constitutive relation are verified through comparative analyses.

Finite element vibration analysis of multi-scale hybrid nanocomposite beams via a refined beam theory

Present manuscript is mainly arranged to take into consider the influences of nanofillers' aggregation, beam's shear deformation and various boundary conditions on the vibration frequency of multi-scale hybrid nanocomposites in the framework of finite element based Rayleigh-Ritz method. The constituent material is made from three phases, namely polymer matrix, macro-scale carbon fibers and nano-scale carbon nanotubes. Homogenization procedure is procured based on Eshelby-Mori-Tanaka approach incorporated with a micromechanical scheme to obtain the effective material properties via a two-step method. In addition, a new refined higher-order beam theory is introduced to govern shear stress and strain through the thickness direction. Furthermore, influences of different boundary conditions are included, too. The accuracy of the presented finite element formulations is examined by setting a comparison between the dimensionless frequency of multi-scale hybrid nanocomposite beams via both analytical and finite element solutions. Afterwards, parametric studies are adopted to put emphasize on the influence of various terms on the vibrational behaviors of nanocomposite beams. It is reported that influence of different parameters deeply depends on the magnitude of volume fraction of nanofillers inside the inclusions.

Application of Chebyshev–Ritz method for static stability and vibration analysis of nonlocal microstructure-dependent nanostructures

This research develops a nonlocal couple stress theory to investigate static stability and free vibration characteristics of functionally graded (FG) nanobeams. The theory introduces two parameters based on nonlocal elasticity theory and modified couple stress theory to capture the size effects much accurately. Therefore, a nonlocal stress field parameter and a material length scale parameter are used to involve both stiffness-softening and stiffness-hardening effects on responses of FG nanobeams. The FG nanobeam is modeled via a higher order refined beam theory in which shear deformation effect is verified needless of shear correction factor. A power-law distribution is used to describe the graded material properties. The governing equations and the related boundary conditions are derived by Hamilton’s principle and they are solved applying Chebyshev–Ritz method which satisfies various boundary conditions. A comparison study is performed to verify the present formulation with the provided data in the literature and a good agreement is observed. The parametric study covered in this paper includes several parameters such as nonlocal and length scale parameters, power-law exponent, slenderness ratio, shear deformation and various boundary conditions on natural frequencies and buckling loads of FG nanobeams in detail.

On the stressed state of the joint upon bending beam with an elastic coating

Joint deformation of two elastic layers is considered, one of them being a coating applied to the surface of another thicker layer. It is assumed that slippage is absent in the plane of layer contact. External load is normal to the coating surface and unchanging in width. In the absence of lateral loading, the junction can be considered a two-layer beam and described by the equations of the plane elasticity using the asymptotic method. The required functions, normal and tangential stresses, and also the components of the displacement of an arbitrary point of the beam and coating are expanded in power series of the small parameter (accepted as a half thickness of the corresponding layer). In contrast to the known asymptotic expansions, an alternative asymptotic method is proposed in which all the required functions are asymptotically equivalent, i.e., expanded in asymptotic series of the same structure. When all positive powers of the small parameter are present in the series, the asymptotic algorithm leads to the appearance of two independent recurrent systems of linear equations, which greatly simplifies their solution. In each approach for one elastic layer the algorithm generates five indefinite functions of the longitudinal coordinate. Due to a rapid convergence of the asymptotic series, the first asymptotic approximation is used to construct the mathematical model of the deformation of a beam-to-coating coupling. Ten indefinite functions of the coordinate x provide fulfilling of eight conditions of continuity of the stress-strain state in the linkage depth, two functions remaining indefinite. Equilibrium equations (a system of two linear differential equations of the sixth order) are derived for them proceeding from the principle of minimum potential energy of deformation and corresponding boundary conditions (that can be realized both in static and kinematic variants). Numerical results are obtained in static approximation for two types of surface loading — steady and sinusoidally varying — along the length of the coating. An emphasis is made on the necessity of allowing for tangential stresses when studying the stress state of the coating as they can be rather significant upon bending.

Rolling simulation system for non-symmetric groove types

An extension of a previously developed rolling simulation system is presented. Complicated groove dimensions, which have no longer symmetry on any of the coordinate axis, are successfully implemented. The thermo-mechanical numerical solution, considering ideal plastic deformation and obeying von-Mises flow rule, is based on meshless local radial basis function collocation method. During the solution procedure, 2D discrete cross sectional slice of a groove and rolled material become a computational domain at a given time, parallel to each other and aligned perpendicular to the rolling direction. Groove slice, or groove surface lines, are created with the help of imaginary groove surface points for each slice position. The implementation of roll contact at the boundaries is done by considering possible contacts of a slice and groove surface line at multiple sections with different deformation rates and boundary conditions. Coulomb model of friction is considered at the contact boundaries. The design of non-symmetric (in vertical or/and horizontal direction) type of rolling schedule is quite complicated since keeping the rolled product at the center may become problematic and unexpected deformation in one direction could ruin the desired final shape. The computational model, capable of coping with non-symmetric groove geometries provides an essential simulation tool for these type of roll pass designs. To increase the stability and the accuracy, the number of collocation nodes over the slices as well as groove surface points may be suitably increased or redistributed. The simulation results are shown in terms of temperature, displacement, strain and stress fields as well as roll forces and torques. A user friendly computer application is created for industrial use based on C# and .NET.

A modified nonlocal couple stress-based beam model for vibration analysis of higher-order FG nanobeams

ABSTRACTIn the present paper, nonlocal couple stress theory is developed to investigate free vibration characteristics of functionally graded (FG) nanobeams considering exact position of neutral axis. The theory introduces two parameters based on nonlocal elasticity theory and modified couple stress theory to capture the size effects much accurately. Therefore, a nonlocal stress field parameter and a material length scale parameter are used to involve both stiffness-softening and stiffness-hardening effects on responses of FG nanobeams. The FG nanobeam is modeled via a higher-order refined beam theory in which shear deformation effect is verified needless of shear correction factor. A power-law distribution is used to describe the graded material properties. The governing equations and the related boundary conditions are derived by Hamilton's principle and they are solved applying Galerkin's method, which satisfies various boundary conditions. A comparison study is performed to verify the present formulation with the provided data in the literature and a good agreement is observed. The parametric study covered in this paper includes several parameters, such as nonlocal and length scale parameters, power-law exponent, slenderness ratio, shear deformation, and various boundary conditions on natural frequencies of FG nanobeams in detail.

Size-dependant behaviour of functionally graded microplates based on the modified strain gradient elasticity theory and isogeometric analysis

This paper presents a robust numerical model, which takes into account both size-dependent and shear deformation effects, for the bending, buckling and free vibration analyses of functionally graded microplates. The size-dependent effect is captured by using the modified strain gradient elasticity theory with three length scale parameters, whilst the shear deformation effect is accounted by using the third-order shear deformation theory. The rule of mixture is employed to describe the distributions of material phrases through the plate thickness. By using Hamilton’s principle, the governing equations are derived and then discretized by employing an Isogeometric Analysis (IGA) approach, where the Non-Uniform Rational B-Splines (NURBS) basis functions are adopted to meet the C2-continuity requirement. Physical mesh convergence and verification studies are performed to prove the accuracy and reliability of the present model. In addition, parametric studies are also carried out to investigate the size effect in conjunction with the influences of gradient index, shear deformation effect and boundary conditions on the responses of microplates.