Small Deformation Theory Research Articles

Influence of internal factor on crack resistance of shell mold by investment models

The process of evolution of the stress-strain state (SSS) of a multilayer shell mold (SM) is modeled at properties change between layers during cooling of poured steel casting. A mathematical model was constructed and a theoretical study of the stress state of the SM was carried out in absence of connection between the layers in a multilayer composite. The article describes a complex three-component system: liquid metal, solid metal, and ceramic SM. Solid metal and SM are considered to be isotropic. To solve this problem, the authors used the theory of small elastic-plastic deformations and equations of thermal conductivity, as well as proven numerical methods. Evolution of SSS in SM was traced by time steps. Thickness of the solidifying metal was determined through the equation of interphase transition. The article considers the process of heating an axisymmetric SM when pouring liquid metal into it. Stress state was estimated by stresses and displacements that occur in SM. At SM contact with support filler (SF), SM surface move away from the SF is possible during cooling of liquid metal. In this case, contact problem is solved. Taking into account the compiled algorithm for solving the problem, calculations were performed for the case of complete sliding of layers using developed numerical schemes and software complexes. Obtained results of numerical calculations are clearly displayed by graphic illustrations in form of plots and graphs. Detailed analysis of the obtained results is given. There is inconsistency of the previously expressed idea about the applicability of sliding between layers in a multilayer composite from the position of reducing its stress state. The research results can be useful in calculations of other functional multilayer shell systems.

Quantifying the subsurface damage and residual stress in ground silicon wafer using laser ultrasonic technology: A Bayesian approach

Dispersion nature of surface acoustic waves (SAWs) propagating along the machined surface contains key information about the elastic properties and stress state of the subsurface damage (SSD) layer. This enables the acoustical characterization of SSD. In this paper, we develop a mechanical properties and residual stress estimation technology for the machined surfaces, especially for ground silicon wafer, based on laser-excited surface acoustic waves (LSAWs). Bayesian inversion is applied to quantify nine orthotropic elastic constants and one residual stress parameter of the SSD layer together with the corresponding uncertainty estimates from dispersion data. The forward model of SAW propagation is derived based on Biot’s theory of small deformations influenced by initial stress. The solution of the inverse problem is given in terms of the posterior probability density (PPD) of the SSD model parameters. Markov chain Monte Carlo (MCMC) method is used to generate posterior samples by repeatedly solving the forward model to compute parameter uncertainties and inter-relationships. Laser ultrasonic experiments have been performed on two typical areas of the ground silicon wafer to determine the ability of the technology to characterize SSD and residual stress and the effect of grinding direction on the damage degree of SSD. The elastic constant values estimated by the proposed method can be reasonably explained by the fracture properties of single-crystal silicon. The reliability of the results is further verified via nanoindentation and micro-Raman spectroscopy.

Effects of electric and magnetic fields on dielectric and elastic properties of membranes composed of cotton fabric and carbonyl iron microparticles

This study investigates the influences of electric and magnetic fields applied both separately and superimposed on the dielectric and elastic properties of membranes (Ms) composed of cotton fabric and carbonyl iron microparticles (CI). To this end, three membranes were fabricated: one without CI microparticles and the remaining two with various proportions of CI microparticles. Additionally, scanning electron microscopy and energy emission X-ray spectroscopy were used to perform morphological and elemental analysis of the cotton fabric sample, CI microparticles, and the prepared Ms. Subsequently, flat capacitors (FCs) were manufactured using these Ms, encapsulated in a silicone rubber sheath. The FCs were introduced in static electric and magnetic fields superimposed over a sinusoidal electric field of frequency f = 1 kHz, and the electric capacity Cpand the tangent of dielectric loss angle Dpof the FCs were measured, depending on the E values of the static electric and magnetic fields superimposed over the medium-frequency electrical field. The values of the static electric field intensity and magnetic flux density were varied in steps of 4 kV/m within an interval 0 ≤ E (kV/m) ≤ 48 and in steps of 100 mT within an interval 0 ≤ B (mT) ≤ 300, respectively. Based on the obtained data, the components of the complex dielectric permittivity, those of the deformations, and the coefficients of elasticity of the Ms were determined to characterize the dependence of these values on the composition of Ms under the applied fields. Furthermore, the observed physical mechanisms were described based on the elements of physics of linear dielectrics and small deformation theory. Ultimately, the development of these materials can aid the design of various electrical devices for diverse applications such as sensors with electric field transducer, magnetic field, vibrations, and mechanical deformations.

The Influence of Internal Factor on Crack Resistance of Shell Mold for Investment Models

The evolution of stress–strain state (SSS) of multilayer shell mold (SM) has been simulated upon variation of properties between the layers during cooling of casted steel. A mathematical model has been developed and the stress state of SM has been theoretically investigated in the case of absence of interrelation between the layers in multilayer composite material. The complex three-component system (liquid metal, solid metal, ceramic shell mold) has been analyzed. The solid metal and the SM are considered as isotropic. In order to solve the formulated problem, the theory of small elastic deformations and the equations of heat conductance have been applied as well as verified numerical methods. The SSS evolution in shell molds can be traced by time steps. The thickness of solidified metal is determined by the equation of interphase transition. Heating of axisymmetric SM during pouring of liquid metal is analyzed. The stress state has been estimated by stresses and displacements occurring in SM. During cooling of liquid metal at the contact between the shell mold and supporting filler (SF), the surface between them can be detached. In this case, the contact problem is solved. The calculations for the case of complete sliding of layers have been carried out with consideration of the compiled algorithm for problem solving by using the developed numerical schemes and software. The obtained numerical results are presented in the form of diagrams and plots. The obtained results have been analyzed in detail. The inconsistency of the previously proposed concept about applicability of sliding between the layers in multilayer composite material in terms of its strain state has been demonstrated. The experimental results can be useful for analysis of other functional multilayer shell systems.

A new method for estimating the bending stiffness curve of non-uniform Euler-Bernoulli beams using static deflection data

The static Euler-Bernoulli beam is a simplified model from linear elasticity theory for small deformations, with which the stresses and deflections of slender beams can be determined very well. The simplicity of Euler-Bernoulli beam theory makes it still an important tool in the sciences, particularly in construction and mechanical engineering. The behavior and the deflection of a static Euler-Bernoulli beam is completely given by the bending stiffness curve. The bending stiffness curve of a real beam structure can be determined using measured data from a bending test. However, the determination of the bending stiffness curve for non-uniform beams leads to a so-called inverse problem, which is ill-posed. In this work, a robust and computationally effective method for determining the bending stiffness curve of a non-uniform Euler-Bernoulli beam based on measured data from a static bending test and the calculus of variations is presented. With the calculus of variations, the complex optimization problem of the inverse problem is transformed into a boundary value problem, which is then solved by collocation with lower computational costs. In addition, the new method does not require the second derivative of the beam deflection and the problem of determining the bending stiffness at the boundaries does not arise, as with the traditional method. The new method to determine the bending stiffness curve of non-uniform beams is demonstrated in several test examples.

INVESTIGATION OF CONTACT INTERACTION WITH STRIPE WITH INITIAL STRESSES IF SHE REINFORCED BY THE ELASTIC THIN FINITE STRINGER

In this paper, within the framework of the linearized theory of elasticity, the contact problem of reinforcement of a prestressed elastic band by a finite elastic stringer is considered. Elastic strip, free from reinforcement and load face, rigidly clamped to the base. The problem is solved for the cases of equal and unequal roots of the defining equation in the case of elastic potentials of an arbitrary structure. The research is carried out in general for the theory of large initial deformations and two variants of the theory of small initial deformations within the linearized theory of elasticity at arbitrary structure of elastic potential. It is assumed that the initial stress state of the elastic stringer and the band in the contact area is homogeneous. The research is carried out in the coordinates of the initial deformed state, which are related to the Lagrangian coordinates (natural state). In addition, the external concentrated load causes small perturbations of the corresponding values of the ground stress-strain state. The general solutions of the basic differential equations of the linearized theory of elasticity are given. Assuming that the stringer is loaded with both vertical and horizontal forces, it should be noted that the elastic stringer bends in the vertical direction like a normal beam, and in the horizontal direction is compressed or stretched like a normal rod with finite stiffness, which is in uniaxial stress. deformed state, the problem is mathematically formulated as a system of integro-differential equations with respect to unknown contact stresses. Using the Fourier transform, the system is solved in a closed form. Ultimately, the expressions for the contact voltages are represented as Fourier integrals. In the case of uneven roots for chemically active rubber SKU-6 and Treloir potential (neogukov type material), the results of numerical analysis are presented, presented in the form of a graph illustrating a fairly significant effect of initial stresses. Therefore, the effect of initial stresses on the stressstrain state of the elastic band is that the initial stresses in the strip lead to a decrease in stress in the contact area in the case of compression, and in the case of tension to increase them. As for the movements in the area of contact of the stringer with the strip, the opposite is true.

PERIODIC REINFORCEMENT OF TWO PRE-STRESSED STRIPS BY FINISHED SUPPORTING ELEMENTS

The qualitative and quantitative influence of initial (residual) stresses on the law of distribution of contact characteristics at interaction of elastic finished overlays (stringers), at their placement, with two prestressed strips which are clamped by one edge. In general, the research was carried out for the theory of great initial (ultimate) and two variants of the theory of small initial deformations within the framework of linearized theory of elasticity with the elastic potential having arbitrary structure for compressible and incompressible bodies. Fourier integral methods, methods for solving harmonic differential equations, singular integral-differential equations and numerical methods were used too. It is assumed that in the contact area the initial stress state, with a certain degree of accuracy, is considered homogeneous. We make the assumption that the elastic band with initial (residual) stresses is in the conditions of plane deformation. Also, for an elastic lining loaded with both vertical and horizontal forces, the generally accepted model of beam bending in combination with the model of uniaxial stress-strain state of the elastic lining is valid. A singular integral-differential equation with the Gilbert kernel was derived, which allows us to solve this problem. We find the analytical solution of the equation in the form of series from the Jacobi function. Numerical studies were performed for materials with elastic potentials of harmonic type (compressible bodies) and elastic potentials of Bartenev-Khazanovich and Treloar (incompressible bodies). Numerical studies have been performed for materials with elastic potentials of harmonic type (compressible bodies) and elastic potentials of Bartenev-Khazanovich and Treloir (incompressible bodies). We have considered the case when all periodically placed pads that support elastic bands with the initial stresses were loaded with tangential force. Analysis of numerical results shows a significant effect of initial (residual) stresses on the distribution of contact characteristics of periodically reinforced strips with thin reinforcing elements. The results obtained can be used for engineering calculations for the strength and durability of structures, taking into account the initial stresses for a wide range of structural materials.

Bending and undamped free vibration analysis of laminated bimodular composite Material thin curved beam

This paper presents the numerical results and its analysis of transverse deflection and first mode free vibration of bimodular curved beam. The analysis has been carried out using classical beam theory, and the problem is formulated based on small deformation, slight rotation, and small strain theory. In this analysis, aramid rubber and polyester rubber bimodular material composite laminated curved beams are considered. The [ABD] stiffness parameters are determined by using Vinson-Sierakowski (VS) and Rios-Chan’s (RC) equivalent stiffness methods. For bending analysis, uniformly distributed transverse load is considered. The numerical results are presented for neutral surface location, non-dimensional transverse deflection, strain distributions, the first mode of natural frequencies for different lengths to the radius of curvature of the beam, and lamination schemes. The numerical results obtained for non-dimensional transverse deflections and the first mode of natural frequencies using VS and RC are almost identical for cross-ply laminates but differ significantly for angle-ply laminated beams.

Spatial Rigid-Flexible-Liquid Coupling Dynamics of Towed System Analyzed by a Hamiltonian Finite Element Method

An effective Hamiltonian finite element method is presented in this paper to investigate the three-dimensional dynamic responses of a towed cable-payload system with large deformation. The dynamics of a flexible towed system moving in a medium is a classical and complex rigid-flexible-liquid coupling problem. The dynamic governing equation is derived from the Hamiltonian system and built-in canonical form. A Symplectic algorithm is built to analyze the canonical equations numerically. Logarithmic strain is applied to estimate the large deformation effect and the system stiffness matrix will be updated for each calculation time step. A direct integral solution of the medium drag effect is derived in which the traditional coordinate transformation is avoided. A conical pendulum system and a 180° U-turn towed cable system are conducted and the results are compared with those retraced from the existing Hamiltonian method based on small deformation theory and the dynamic software of Livermore software technology corp. (LS-DYNA). Furthermore, a circularly towed system is analyzed and compared with experimental data. The comparisons show that the presented method is more accurate than the existing Hamiltonian method when large deformation occurred in the towed cable due to the application of logarithmic strain. Furthermore, it is more effective than LS-DYNA to treat the rigid-flexible-liquid coupling problems in the costs of CPU time.

Numerical simulation of laser-generated Rayleigh wave pulses propagation in the machined surface with residual stress using finite-difference method

Laser-excited broadband Rayleigh wave propagation behaviors can be used for characterizing material surface characteristics non-destructively. The present paper is concerned with the Rayleigh wave propagation in the machined surface with a damaged layer with residual stress. Biot’s theory of small deformations affected by initial stress forms the basis for this study. Taking ground silicon wafer as an example, we adopted a staggered finite-difference (FD) scheme for numerical simulation. By comparing the dispersion images with the theoretical dispersion curves of Rayleigh wave, the accuracy of the FD algorithm is verified. Then, some two-layer models without or with residual compressive stress are utilized to further analyze the dispersion energy characteristics of Rayleigh waves in the machined surface. The results indicate that damage layer thickness and damage degree determine the main Rayleigh wave high-order mode and the energy of the high-order mode, respectively. In addition, residual compressive stress can reduce the phase velocity of the Rayleigh wave to a small extent. Meanwhile it can significantly increase the total energy of the Rayleigh wave which excited by the same excitation source. This paper will provide a valuable reference for laser ultrasonic evaluation of residual stress on the machined surface.

Analytical solutions for process-induced spring-in of U-shaped composite parts

Analytical solutions for predicting spring-in angles of U-shaped composite parts generating in rubbery and glassy states are proposed. In the newly developed analytical solutions, the process-induced distortions of web and flange are mathematically converted to spring-in angles within the scope of small deformation theory, and then corresponding analytical expressions of web, corner, flange and total spring-in are separately derived. These new solutions can quantitatively give the correlation of web, corner, flange and total spring-in with material properties-related and geometrical configuration-dependent parameters, and evaluate the spring-in angles of multi-angle ply composite parts just using lamina properties. The reliability of newly proposed analytical solutions is checked by the comparison between the analytically calculated and experimentally measured spring-in angles for unidirectional and quasi-isotropic U-shaped composite parts. Meanwhile, numerical simulations on the U-shaped composite parts with various geometries and layups also provide a favourable agreement with analytical results.

Elastic Stability of Steel Structural Systems Restrained by Initially Curved Members.(Dept.C)

The axial stiffness of elastic, initially Curved, members is considered using small-deformation theory. Simple relations are obtained for the tangent axial stiffness. The use of this equivalent spring stiffness is described in an attempt to provide a unified description of the restraint given to a main column member by the provision of initially curved members at mid-height. The buckling of a column element in the Plane of the nonlinear restraints is investigated. Further the purpose of the analysis to describe precise values to the spring stiffness in the light of the analysis referred to the above. Further, the matrix technique illustrated in the present paper is capable of generalization, especially using automatic Computational procedures. Finally the critical load of the structural system is investigated with respect to the presence of imperfection in the lateral restraints.

On the tension of a cylindrical rod of variable cross-section

The paper considers the solution of linearized equations of the theory of small elastic-plastic deformations in the case of an axisymmetric problem. It is assumed that in the initial state there is a simple stretching. In the first approximation, the relations for the components of displacements, deformations and stresses are obtained. Solutions are expressed in terms of zero-and first-order Bessel functions. The paper deals with the stretching of an infinitely long cylindrical rod of variable cross-section.

3D architected isotropic materials with tunable stiffness and buckling strength

This paper presents a class of 3D single-scale isotropic materials with tunable stiffness and buckling strength obtained via topology optimization and subsequent shape optimization. Compared to stiffness-optimal closed-cell plate material, the material class reduces the Young’s modulus to a range from 79% to 58%, but improves the uniaxial buckling strength to a range from 180% to 767%. Based on small deformation theory, material stiffness is evaluated using the homogenization method. Buckling strength under a given macroscopic stress state is estimated using linear buckling analysis with Block–Floquet boundary conditions to capture both short and long wavelength buckling modes. The 3D isotropic single-scale materials with tunable properties are designed using topology optimization, and are then further simplified using shape optimization. Both topology and shape optimized results demonstrate that material buckling strength can be significantly enhanced by hybrids between truss and variable thickness plate structures.

Reliability-based resistance of RC element subjected to membrane action and their sensitivity to uncertainties

Traditional design of reinforced concrete structures is usually based on small deformation theories whereas in order to assess the ultimate load bearing capacity of RC structures against progressive collapse significant deformations should be taken into account. The alternate load paths facilitated by membrane action which develop under large deformation can activate a significant strength reserve, redistribute the loads and hence limit the damage progression in the structure. Up to date, no uniform approach is available on how to incorporate these membrane effects in the structural design or verification procedures against progressive collapse. In this article, sensitivity analyses were performed to investigate the effect of the stochastic nature of membrane action on the load bearing capacity for three distinct RC elements, i.e. a beam, one-way slab and two-way slab. It was found that variables which are usually not explicitly taken into account by traditional design methods, such as the axial restraint stiffness and ultimate reinforcement strain have a significant influence on the resistance of the considered elements. Subsequently, the design resistance of the elements was assessed using a probabilistic method and the ECOV method. It was found that the ECOV method and the probabilistic method considering a lognormal distribution provide similar results in case one observes a single failure mode for which the variability of the resistance is limited. In case of multiple failure modes and a large variability of the resistance, a mixed lognormal distribution proved to enable a more accurate design resistance estimation.

Thick-walled spherical shell problem

Introduction. Cylindrical and spherical shells are extensively used in engineering. They face internal and/or external pressure and heat. Stresses and strains distribution in elastoplastic shells has been studied by many scientists. Numerous works involve the use of the von Mises yield conditions, maximum shear stress, maximum reduced stress. These condi- tions do not include the dependence on the first invariant of the stress tensor and the sign of the third invariant of the stress deviator. In some cases, it is possible to obtain numerical-analytical solutions for stresses, displacements and de- formations for bodies with spherical and cylindrical symmetry under axisymmetric thermal and force action. Materials and Methods. The problem on the state of a thick-walled elastoplastic shell is solved within the framework of the theory of small deformations. A plasticity condition is proposed, which takes into account the dependence of the stress tensor on three independent invariants, and also considers the sign of the third invariant of the stress deviator and translational hardening of the material. A disconnected thermoelastoplastic problem is being solved. To estimate the stresses in the region of the elastic state of a spherical shell, an equivalent stress is introduced, which is similar to the selected plasticity function. The construction of the stress vector hodograph is used as a method for verification of the stress state. Results. The problem has an analytical solution for linear plasticity functions. A solution is obtained when the strength- ening of the material is taken into account. Analytical and graphical relationships between the parameters of external action for the elastic or elastoplastic states of the sphere are determined. For a combined load, variants are possible when the plastic region is generated at the inner and outer boundaries of the sphere or between these boundaries. Discussion and Conclusions. The calculation results have shown that taking into account the plastic compressibility and the dependence of the plastic limit on temperature can have a significant impact on the stress and strain state of a hollow sphere. In this case, taking into account the first invariant of the stress tensor under the plasticity condition leads to the fact that not only the pressure drop between the outer and inner boundaries of the spherical shell, but the pressure values at these boundaries, can vary within a limited range. In this formulation of the problem, when there is only thermal action, the hollow sphere does not completely pass into the plastic state. The research results provide predicting the behavior of an object (a hollow sphere) that experiences centrally symmetric distributed power and thermal external influences.

A three-dimensional small-deformation theory for electrohydrodynamics of dielectric drops

Abstract

Повторное знакопеременное нагружение упругопластической трехслойной пластины в температурном поле

Axisymmetric deformation of a three-layer circular plate under repeated alternating loading from the plastic region by a local load is considered. To describe kinematics of asymmetrical on the thickness of the plate pack is adopted the hypothesis of a broken line. In a thin elastic-plastic load-bearing layers are used the hypothesis of Kirchhoff. A non-linearly elastic relatively thick filler is incompressible in thickness. It is taken to be a hypothesis of Tymoshenko regarding the straightness and the incompressibility of the deformed normals with linear approximation of the displacements through the thickness layer. The work of the filler in the tangential direction is taken into account. The physical relations of stress-strain relations correspond to the theory of small elastic-plastic deformations. The effect of heat flow is taken into account. The temperature field in the plate was calculated by the formula obtained by averaging the thermophysical parameters over the thickness of the package. The system of differential equations of equilibrium under loading of the plate from the natural state is obtained by the Lagrange variational method. Boundary conditions on the plate contour are formulated. The solution of the corresponding boundary value problem is reduced to finding the three desired functions: deflection, shear and radial displacement of the shear surface of the filler. A non-uniform system of ordinary nonlinear differential equations is written for these functions. Its analytical iterative solution is obtained in Bessel functions by the method of elastic solutions of Ilyushin. In case of repeated alternating loading of the plate, the solution of the boundary value problem is constructed using the theory of variable loading of Moskvitin. In this case, the hypothesis of similarity of plasticity functions at each loading step is used. Their analytical form is taken independent of the point of unloading. However, the material constants included in the approximation formulas will be different. The cyclic hardening of the material of the bearing layers is taken into account. The parametric analysis of the obtained solutions under different boundary conditions in the case of a local load distributed in a circle is carried out. The influence of temperature and nonlinearity of layer materials on the displacements in the plate is numerically investigated.

On the quasi-static approximation in the initial boundary value problem of linearised elastodynamics

Continuous data dependence estimates are employed to rigorously derive conditions that validate the quasi-static approximation for the initial homogeneous boundary value problem in the theory of small elastic deformations superposed upon large elastic deformations. This theory imposes no sign-definite assumptions on the linearised elastic moduli and in consequence the requisite estimates are established using methods principally motivated by known Lagrange identity arguments.

Inversion of surface damage and residual stress in ground silicon wafers by laser surface acoustic wave technology

The paper presents a study of surface acoustic waves propagation in a damage layer with finite thickness and residual stress on an orthotropic semi-infinite medium to reveal the application of laser ultrasound in the surface inspection of ground silicon wafers. Biot’s theory of small deformations influenced by initial stress forms the basis for this study. Considering the case that the displacement and boundary forces are continuous at the interface and the forces vanish on the free surface, the required dispersion relation is obtained. We consider a sample of (100) silicon wafer by grinding with fine abrasive grains, which has a machined face with a micrometer-level thickness of surface damage and residual stress. In order to discuss the impact of propagation directions, degree of surface damage, residual compressive stress on the velocity characteristics of surface waves, the numerical computation of the dispersion equations is performed. It has been found that surface damage has a significant effect on the dispersion curve, while the residual compressive stress can only cause a small decrease Δc in the surface wave velocity. The velocity decrease Δc becomes obvious at high frequencies. For a fixed residual compressive stress and frequency, Δc hardly changes with the degree of surface damage and propagation directions. Based on the above characteristics, we study the inverse problem on detecting both surface damage and residual stress simultaneously by SAW velocities and give a corresponding iterative algorithm. This study may provide theoretical guidance for non-destructive testing of residual stress.