Existence of solutions for the anti-plane stress for a new class of “strain-limiting” elastic bodies

The main purpose of this study is to establish the existence of a weak solution to the anti-plane stress problem on V-notch domains for a class of recently proposed new models that could describe elastic materials in which the stress can increase unboundedly while the strain yet remains small. We shall also investigate the qualitative properties of the solution that is established. Although the equations governing the deformation that are being considered share certain similarities with the minimal surface problem, the boundary conditions and the presence of an additional model parameter that appears in the equation and its specific range makes the problem, as well as the result, different from those associated with the minimal surface problem.     

The numerical analysis of the trapezoidal thread rolling process

The thread rolling is difficult technological process. Improve quality and contemporary reduce manufacture cost of the trapezoidal thread requires acquaintances of physical phenomena observed in the contact zone between rolls and deform work-pieces. Therefore, in this paper the physical and mathematical models of deformations (displacements and strains) and stress in the cold process of trapezoidal thread rolling, were developed. The process is considered as a geometrical and physical non-linear, initial as well as boundary value problem. The phenomena on a typical incremental step were described using a step-by-step incremental procedure, with an updated Lagrangian formulation. The state of strains was described by Green-Lagrange's tensor, while the state of stress by the second symmetrical Pioli-Kirchhoff's tensor. The object was treated as an elastic (in the reversible zone) and visco-plastic body (in non-reversible zone) with mixed hardening. The variational equation of motion in three dimensions for this case was proposed. Then, the finite elements methods (FEM) and dynamic explicit method (DEM) were used to obtain the solution. The application developed for in the ANSYS programme, which provides a complex time analysis for displacement, strains and stresses occurring in the object. The recommendations concern modeling the trapezoidal thread rolling process, where reduce degrees of freedom in numerical model is very important and provide convergence calculated results for maximum stress and strain values in the thread surface layer, were elaborated. The influence a various process conditions on the states deformation and stress for examples calculations, were presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Characterisation of filled rubber with a pronounced non-linear viscoelasticity

This contribution presents the characterisation of an incompressible carbon black-filled elastomer as one characteristical example for highly filled rubber. It shows a strongly pronounced non-linear viscoelastic behaviour and the most important characteristic is the extremely long relaxation time which has to be taken into account. The material model is developed with respect to uniaxial tension data. The basis in the development of a phenomenological model is given by the basic elasticity. For this evaluation the long term relaxation behaviour results in a complex experimental procedure. Therefore, special attention has to be paid according to an optimised experimental process in order to get the necessary reference data in an adequate and reproduceable way [1]. With this model basis further investigations are taken into account concerning the time-dependent viscoelasticity. Therefore, cyclic deformations from zero up to a maximum of deformation are considered for different strain rates. Furthermore, the relaxation behaviour is investigated for multiple strain levels. The phenomena which are observed in the experimental results yield in a purely viscoelastic model, based on a rheological analogous model consisting of an equilibrium spring and several Maxwell-elements which contain nonlinear relations for the relaxation times of the dashpot elements [1,2]. The material model's numerical realisation is accomplished in two ways. Because of its numerical simplicity especially according to the parameter identification the model is restricted only to the simple case of uniaxial tension. A second, alternative implementation is executed providing the benefit that more complex deformation conditions can also be taken into account. Therefore, the general, three-dimensional finite model is implemented in an open-source Finite Element library [3]. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Real-time simulation of thermal stresses and creep in plates subjected to transient heat input

This paper presents a novel numerical technique for solvingthe temperature and stress fields in a plate subjected to arbitrarilyvarying transient boundary conditions (transient temperatureand heat-flux variations) on a surface. The numerical methodis based on the control-volume finite-difference approach. Itapplies a general formulation which takes into account nonconstantmaterial properties (e.g. temperature, material, or time dependency),heat-transfer coefficients, and creep. The temperature calculationapplies a one-dimensional numerical model, whereas the stressanalysis is semi-two-dimensional. Both plane stress and planestrain conditions are considered as extreme cases. It is shownthat, by using the developed numerical technique, very fastreal-time simulations can be performed. The method has provedits applicability in e.g. high-pressure die-casting, and applicationsto this industrial process are considered.     

A nano-scale frictional contact problem incorporating the size dependency and the surface effects

In this paper, the nano-scale sliding contact problem of an isotropic semi-infinite medium is considered. The small scale effects are taken into account by means of the couple stresses and the surface energy theories. Employing the Fourier integral method, the plane elasticity problem is formulated in terms of the integral equations. The numerical results are provided for the plane strain state. The parametric study reveals that the surface material constants remarkably alter the surface stresses and the contact length. The results indicate that the stress singularity substantially decreases due to size effect compared with the classical theory predictions. For example, the in-plane stress tensile peak decreases by a %80 with respect to the Hertzian contact.     

The longitudinal deformation of a rod in friction interaction with a compressing body

The deformation of a rod, confined in a fixed external housing, is considered. The friction forces in the contact surface are related to the deformation of the rod by a power relation. A wide range of variation of the friction parameter and the preliminary clearance parameter with which the rod is inserted into the housing is investigated and the characteristic features of the stress and strain distributions are revealed. The dissipation of energy due to friction and the formation of a hysteresis loop in the dependence of the stresses in the loaded end face on its displacement are considered. The problem is solved in a quasistatic formulation. Analytical relations are found for a number of important cases. Other results are obtained by numerical integration of the initial differential problem.     

Numerical investigation of the deformation characteristics and heat generation in pneumatic aircraft tires Part II. Thermal modeling

A numerical procedure to determine the temperature rise in aircraft tires under free rolling conditions is presented in this article. Energy dissipation from cyclic inelastic deformation is considered the main heat generation source. This modeling considers the deformation process of the tire to be a steady-state problem, where all concurrent cycles are assumed to be the same as the first. The inelastic energy is determined by imposing a phase lag between the strain and the stress fields. The phase lag is assumed to be frequency independent in the range of interest, in keeping with the experimental observations in aircraft tire materials. It is further assumed that the inelastic energy is completely converted into volumetric heat input for a transient thermal conduction analysis. A conduction model is described and results are compared against thermocouple data recorded by Clark and Dodge [1].     

Modelling of the deformation processes and the localization of plastic deformations in the torsion–tension of solids of revolution

Generalized two-dimensional problems of the torsion of elastoplastic solids of revolution of arbitrary shape for large deformations under non-uniform stress-strain conditions are formulated and a method for their numerical solution is proposed. The use of this method to construct strain diagrams of materials based on experiments on the torsion of axisysmmetric samples of variable thickness until fracture occurs is described. Experimental and numerical investigations of processes of elastoplastic deformation, loss of stability and supercritical behaviour of solid cylindrical steel samples of variable thickness under conditions of monotonic kinematic loading with a torque, a tension and a combined load are presented. The mutual influence of torsion and tension on the deformation process and the limit states is estimated, and the universality (the independence of the form of the stress-strain state) of the “stress intensity – Odqvist parameter” diagram for steel for large deformations is proved.     

Numerical analysis of chip formation during machining for different value of failure strain

Grinding is a very complicated processing. To increase quality of product and minimize the cost of abrasive machining, we should know physical phenomena which exist during the process. The first step to solution of this problem is analysis of machining process with a single abrasive grain. In the papers [1, 2] the thermo-mechanical models of this process are presented, but in this work attention is concentrated on chip formation and his separation from object. The influence of failure strain ε_f on states of strain and stress in surface layer during machining is explained. The phenomena on a typical incremental step were described using step-by-step incremental procedure, with updated Lagrangian formulation. Then, the Finite Element Method (FEM) and Dynamic Explicit Method (DEM) were used to obtain the solution. Application was developed in the ANSYS system, which makes possible a complex time analysis of the physical phenomena: states of displacements, strains and stress. Numerical computations of the strain have been conducted with the use of two methodologies. The first one requires an introduction of boundary conditions for displacements in the contact area determined in modeling investigation, while the second – a proper definition of the contact zone through the introduction of finite elements of TARGET and CONTACT types, without the necessity to introduce boundary conditions. This model includes variational equations of the object's motion and deformation. Examples of calculations for the displacement, strain and stress field in the surface layer zones were presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Antiplane strain of an elliptical cylinder with a crack

The antiplane strain of an anisotropic elliptical cylinder with a crack is examined. An infinite system of linear algebraic equations is obtained from the boundary conditions on the cylinder surface to determine the constants in the complex potential of the problem. Detailed numerical investigations are performed of the influence of the geometric and elastic characteristics of the cylinder on the magnitude of the stress intensity coefficients near the outer edge.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 18, pp. 28–34, 1987.     

An adaptive approach for modeling a fiber-matrix composite with the FE2 method

In materials with a complicated microstructre [1], the macroscopic material behaviour is unknown. In this work a Fiber-Matrix composite is considered with elasto-plastic fibers. A homogenization of the microscale leads to the macroscopic material properties. In the present work, this is realized in the frame of a FE² formulation. It combines two nested finite element simulations. On the macroscale, the boundary value problem is modelled by finite elements, at each integration point a second finite element simulation on the microscale is employed to calculate the stress response and the material tangent modulus. One huge disadvantage of the approach is the high computational effort. Certainly, an accompanying homogenization is not necessary if the material behaves linear elastic. This motivates the present approach to deal with an adaptive scheme. An indicator, which makes use of the different boundary conditions (BC) of the BVP on microscale, is suggested. The homogenization with the Dirichlet BC overestimates the material tangent modulus whereas the Neumann BC underestimates the modulus [2]. The idea for an adaptive modeling is to use both of the BCs during the loading process of the macrostructure. Starting initially with the Neumann BC leads to an overestimation of the displacement response and thus the strain state of the boundary value problem on the macroscale. An accompanying homogenization is performed after the strain reaches a limit strain. Dirichlet BCs are employed for the accompanying homogenization. Some numerical examples demonstrate the capability of the presented method. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Homogenisation of elasto-plastic open-celled foams

The effective yield behaviour of open-celled metal foams is studied by a micro-mechanical model. As a surrogate model for the sponge-like microstructure the simplified Kelvin foam is used. The yield loci in strain space and stress space are constructed by conducting numerical experiments. For the determination of the effective stresses a strain-energy based homogenisation procedure is adopted. The numerical examples show that the initial yield surface in the normal strain space is similar to a polyhedron with sharp corners. The further evolution of the yield surface is characterized by kinematic and isotropic hardening effects. In addition, the stress yield surface may rotate under certain loading conditions. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Elastoplastic deformation during projectile–wall collision

The Smoothed Particle Hydrodynamics method for elastic solid deformation is modified to include von Mises plasticity with linear isotropic hardening and is then used to investigate high speed collisions of elastic and elastoplastic bodies. The Lagrangian mesh-free nature of SPH makes is very well suited to these extreme deformation problems eliminating issues relating to poor element quality at high strains that limits finite element usage for these types of problems. It demonstrates excellent numerical stability at very high strains (of more than 200%). SPH can naturally track history dependent material properties such as the cumulative plastic strain and the degree of work hardening produced by its strain history. The high speed collisions modelled here demonstrate that the method can cope easily with collisions of multiple bodies and can also naturally resolve self-collisions of bodies undergoing high levels of plastic strain. The nature and the extent of the elastic and plastic deformation of a rectangular body impacting on an elastic wall and of an elastic projectile impacting on a thin elastic wall are investigated. The final plastically deformed shapes of the projectile and wall are compared for a range of material properties and the evolution of the maximum plastic strain throughout each collision and the coefficient of restitution are used to make quantitative comparisons. Both the elastoplastic projectile–elastic wall and the elastic projectile–elastoplastic wall type collisions have two distinct plastic flow regimes that create complex relationships between the yield stress and the responses of the solid bodies.     

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.     

An exact finite deformation elasto-plastic solution for the outside-in free eversion problem of a tube of elastic linear-hardening material

An exact solution is obtained in this paper for the elasto-plasticoutside-in free eversion problem of a tube of elastic linear-hardeningmaterial using a tensorial formulation. The solution is basedon a finite-strain version of Hencky's deformation theory, thevon Mises yield criterion, and the assumptions of volume incompressibilityand axial length constancy. All expressions for the stress,strain distributions and the eversion load are derived in anexplicit form. In addition, with both the linear-elastic andstrain-hardening-plastic responses of the material being includedand with the thickness effect of the tube being incorporated,this solution provides a rigorous and complete theoretical analysisof the elasto-plastic eversion problem, unlike existing solutions.Two specific solutions are also presented as limiting casesof the solution. Also provided are some numerical results andthe related observations to show quantitatively applicationsof the solution.     

The problem of the stability of quasilinear flows with respect to perturbations of the hardening function

A formulation of the linearized boundary-value problem of the stability of a deformation process with respect to small perturbations of the hardening function (of the scalar constitutive relation of the material) is presented. The characteristic vector relations of the medium are assumed to be linear. The occurrence of rigid zones in the domain of the solid and the change in their boundaries in the perturbed motion are taken into account. A perfect rigid plastic deformation and the flow of a Newtonian fluid are considered explicitly as the basic flow. In the latter case, the equation of the asymptotic boundary of the rigid zone, which appears when there is a small variation in the yield stress and a transition to a viscoplastic material, is derived.     

A geometrically nonlinear (3,2) reduced degree-of-freedom unified zigzag laminated beam element for large deformation analysis

A geometrically nonlinear (3,2) unified zigzag beam element is developed with a reduced number of degree-of-freedom for the large deformation analysis. The main merit of the beam element model is the Kirchhoff and Cauchy shear stress solution for large deformation and large strain analysis is more accurate. The geometrically nonlinearity is considered in the calculation of the zigzag coefficients. Thus, the results of shear Cauchy stress are matching well with solid element analysis in case of the beam with aspect ratio greater than 20 under large deformation. The zigzag coefficients are derived explicitly. The Green strain and the second Piola Kirchhoff stress are used. The second Piola Kirchhoff shear stress is continuous at the interface between adjacent layers priori. The bottom surface second Piola Kirchhoff shear stress condition is used to determine the zigzag coefficient and the top surface second Piola Kirchhoff shear stress condition is used to reduce one degree-of-freedom. The nonlinear finite element equations are derived. In the numerical tests, several benchmark problems with large deformation are solved to verify the accuracy. It is observed that the proposed beam has accurate solution for beam with aspect ratio greater than 20. The second Piola Kirchhoff and Cauchy shear stress accuracy is also good. A convergence study is also presented.     

Analysis of the coupling effects of the longitudinal and transverse displacements on the deformation and internal forces of functionally graded beams

Functionally graded beams (FGBs) with an arbitrary gradation of the material properties along the thickness of the beams are analyzed. Such FGBs are of special interest in civil and mechanical engineering to improve both the thermal and the mechanical behaviour of the beams. In [1] and [3] free vibrations of functionally graded Timoshenko and Euler-Bernoulli beams have been considered. The obtained analytical solutions are based on the work of Li [2], where closed-form solutions of stress distributions, eigenfrequencies and eigenfunctions have been derived by means of a single differential equation of motion for the deflection. However, these previous works did not take into account the coupling between the longitudinal and the transverse displacements and its effects on the deformation and internal forces of the FGBs. This approach is appropriate only for a symmetrical material gradation but it may not be valid for general cases with an arbitrary material gradation. In this paper, the coupling effects of the longitudinal and transverse displacements on the deformation and internal forces of FGBs are investigated for different beam support conditions. Analytical solutions of the corresponding boundary value problem are derived. A comparison is also made with the numerical results obtained by the finite element method (FEM). (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)