Isotropic Linear Elastic Material Research Articles

Sub-surface stress analysis of slewing bearing ball based on plastic deformation

Slewing support is widely used in large slewing machinery such as rocket launching platform, wind turbine and port crane. The accurate analysis of the internal stress in the raceway of ball roller is an important basis for studying the cause of bearing fatigue failure and improving the reliability of slewing support. In this paper, the finite element model of thrust angular contact rotary support was established in ANSYS. The stress distribution of the raceway of the ball under different working conditions was calculated by using bilinear isotropic plastic materials and linear elastic materials, and the equivalent stress resulted of the contact surface were compared with the Hertz contact model. This paper studied the factors that affect the distribution of the equivalent stress on the contact surface, calculates the ratio of the equivalent shear stress and the normal stress on the contact surface by finite element method, namely the equivalent friction coefficient, and studied the effect of the equivalent friction coefficient on the distribution of the stress inside the rolling ball. Simulation results show that plasticity occurs in the rolling ball.

Numerical Study of Prosthetic Knee Replacement Using Finite Element Analysis

The knee at times undergoes a surgical process to substitute the weight-bearing surfaces of the knee joint. This procedure relieves the pain and disability around the knee joint. This research paper studied the knee arthroplasty, also referred to as knee replacement. This work was aided with computer vision for visual and accuracy. Autodesk fusion 360 and the stl files were used to generate cemented, posterior stabilised knee prosthesis and imported into the COMSOL Multiphysics software. Then, the three-dimensional models of the total knee arthroplasty (TKA) prosthetic structure are produced. The prosthetic components are modelled as linear isotropic elastic materials. Finite element (FE) simulations using COMSOL Multiphysics on a CAD model of a knee are effectuated to show the effect of several loads and strains on the knee. FE analysis of the model indicates that the orthotropic model depicts a more realistic stress distribution of the knee as it reveals the detailed anatomy of the entire knee structure. The computational results of this work displayed a fair agreement with experimental information from the literature.

A simplified numerical approach to the evaluation of residual shaft friction induced by concrete curing in drilled shafts on granular soils

ABSTRACT: Conventional interpretation procedures of load tests on instrumented piles rely upon measurements of strains that assume as zero for strains measured at the instant immediately before starting the test as reference configuration. However, some experimental evidence shows that concrete in drilled shafts undergoes strains induced by the curing process comparable in magnitude to the strains measured during the load tests. It is therefore expected that mobilization of shaft friction takes place before the load test. Several authors have performed experimental and numerical analyses aiming to quantify the influence of those pre-load test concrete volumetric strains on the measured bearing capacity using different approaches. The present work aimed to establish a reference framework for the existing and future works on this topic. In order to assess the influence of concrete strains induced by curing process on the shaft friction before the start of the load tests in drilled shafts, several finite element numerical simulations are performed, considering the thermal, autogenous and drying strains. The analyses consider concrete as an isotropic linear-elastic material and the soil as an elastic-plastic material using the Mohr-Coulomb constitutive model natively implemented in the software ABAQUS. The results are interpreted focusing on the relevancy on the bearing capacity and load distribution along drilled shafts considering or not the strains induced by concrete curing.

A Simple Theory of Asymmetric Linear Elasticity

Rotation is antisymmetric and therefore is not a coherent element of the classical elastic theory, which is characterized by symmetry. A new theory of linear elasticity is developed from the concept of asymmetric strain, which is defined as the transpose of the deformation gradient tensor to involve rotation as well as symmetric strain. The new theory basically differs from the prevailing micropolar theory or couple stress theory in that it maintains the same basis as the classical theory of linear elasticity and does not need extra concepts, such as “microrotation” and “couple stresses”. The constitutive relation of the new theory, the three-parameter Hooke’s law, comes from the theorem about isotropic asymmetric linear elastic materials. Concise differential equations of translational motion are derived consequently giving the same velocity formula for P-wave and a different one for S-wave. Differential equations of rotational motion are derived with the introduction of spin, which has an intrinsic connection with rotation. According to the new theory, S-wave essentially has rotation as large as deviatoric strain and should be referred to as “shear wave” in the context of asymmetric strain. There are nine partial differential equations for the deformation harmony condition in the new theory; these are given with the first spatial differentiations of asymmetric strain. Formulas for rotation energy, in addition to those for (symmetric) strain energy, are derived to form a complete set of formulas for the total mechanical energy.

Computational simulation of damage accumulation processes in cracked bodies by the UMAT procedure of SIMULIA Abaqus

The paper presents the experience of using the user subroutine UMAT for finite element package SIMULIA Abaqus/CAE for damage accumulation processes in the vicinity of the crack. A continuum damage mechanics model based on the constitutive relations of linear elastic isotropic materials with the incorporated damage tensor components is used to describe the material behavior. The material nonlinearity arising from the deformation process is modeled by introducing an anisotropic damage tensor of the second rank into the constitutive equation. The material model is described by means of user procedure UMAT of SIMULIA Abaqus. The finite element (FE) mechanical constitutive model is implemented in Abaqus/Standard via a UMAT routine. Numerical experiments for a large series of cracked specimens have been performed. Computed stress and damage tensor components were found. It is shown that they are not dependent on the FE mesh refinement. Distributions of the damage tensor components in the vicinity of the crack tip in cracked specimens of different configurations under mixed mode loading in a wide range of mixed mode loadings are found. The configurations of active damage accumulation process zone in the cracked specimens are obtained. It is shown that the damage accumulation process has substantial influence on the stress-strain state in the vicinity of the crack tip and leads to decrease of the stress concentration in cracked specimens.

Modelling Cell Origami via a Tensegrity Model of the Cytoskeleton in Adherent Cells.

Cell origami has been widely used in the field of three-dimensional (3D) cell-populated microstructures due to their multiple advantages, including high biocompatibility, the lack of special requirements for substrate materials, and the lack of damage to cells. A 3D finite element method (FEM) model of an adherent cell based on the tensegrity structure is constructed to describe cell origami by using the principle of the origami folding technique and cell traction forces. Adherent cell models contain a cytoskeleton (CSK), which is primarily composed of microtubules (MTs), microfilaments (MFs), intermediate filaments (IFs), and a nucleoskeleton (NSK), which is mainly made up of the nuclear lamina and chromatin. The microplate is assumed to be an isotropic linear-elastic solid material with a flexible joint that is connected to the cell tensegrity structure model by spring elements representing focal adhesion complexes (FACs). To investigate the effects of the degree of complexity of the tensegrity structure and NSK on the folding angle of the microplate, four models are established in the study. The results demonstrate that the inclusion of the NSK can increase the folding angle of the microplate, indicating that the cell is closer to its physiological environment, while increased complexity can reduce the folding angle of the microplate since the folding angle is depended on the cell types. The proposed adherent cell FEM models are validated by comparisons with reported results. These findings can provide theoretical guidance for the application of biotechnology and the analysis of 3D structures of cells and have profound implications for the self-assembly of cell-based microscale medical devices.

Simulation of bubble growth during the foaming process and mechanics of the solid foam

Elastomeric foams are widely used in different types of applications where different material properties are of interest in each application. All of these properties are governed by the microstructure and the properties of the material matrix. Studying the evolution of the microstructure experimentally is extremely challenging. Thus, here we use direct numerical simulations to gain an insight into the changes that happen from the creation of the gas bubbles in the liquid state, until the solidification into a cellular morphology. Furthermore, the resulting microstructure is then used directly in simulations of solid mechanical testing to determine the mechanical properties of the foam. The matrix fluid is assumed to be Newtonian and incompressible. A linear elastic isotropic material model for the solidified polymer was used to obtain the solid foam properties. The foam was described by a representative volume element (RVE), where a small number of bubbles was randomly distributed. Using this approach, the RVE can describe the bulk behavior of the foam, while remaining computationally tractable. Microstructures with volumes fraction of over 90% (2D) and 45% (3D) are accurately captured. In addition, the influence that the bubble growth rate and the initial bubble distribution of the fluid simulations have on the solid foam properties was studied.

FEA of displacements and stresses of aortic heart valve leaflets during the opening phase

Purpose: Modelling of biomechanical behaviour of heart valve materials aids improvement of biofunctional feature. The aim of the work was assessment of influence of material thickness of leaflets of artificial aortic valve on displacements and stresses during opening phase using finite element analysis (FEA). Design/methodology/approach: The model of aortic valve was developed on the basis of average anatomical valve shapes and dimensions. Nonlinear dynamic large displacements analysis with assumption of isotropic linear elastic material behaviour was used in simulation (Solidworks). The modulus of elasticity of 5.0 MPa was assumed and Poisson ratio set to 0.45. The rigidly supported leaflets was loaded by pressure increasing in the range 0-55 mmHg in time 0.1 s. Leaflets with material thickness 0.13 and 0.15 and 0.17 mm were analysed. The thickness was simulated with shell finite elements. Findings: The highest stresses were observed in the areas of fixation of the leaflets near the scaffold and were lower than dangerous value of fatigue of polyurethanes. Increasing the thickness of valve leaflet material in the range of 40 micrometres resulted in reduction of the valve outlet by almost 10 percent. Research limitations/implications: The FEA was limited to the isotropic linear-elastic behaviour of the material albeit can be used to assess leaflet deformation during dynamic load. Practical implications: Leaflets design may be start from efficient FEA which helps estimation of material impact on stress and fold formation which can affect local blood flow. Originality/value: Aortic heart valve leaflet material can be initially tested in dynamic conditions during opening phase with using FEA.

Estimation of crack propagation direction angles under mixed mode loading in linear elastic isotropic materials by generalized fracture mechanics criteria and by molecular dynamics method

The crack growth directional angles in the isotropic linear elastic plane with the central crack under mixed-mode loading conditions for the full range of the mixity parameter are found. The traditional strain energy density, maximum tangential stress and maximum tangential strain criteria are introduced and analyzed. The accuracy of each model is investigated and discussed by comparing its results to the molecular dynamics modelling. Three fracture criteria of traditional linear fracture mechanics (maximum tangential stress, and minimum strain energy density criteria) are generalized by mean the multi-parameter stress expansion in the vicinity of the crack tip. Atomistic simulations of the central crack growth process in an infinite plane medium under mixed-mode loading using Large-scale Molecular Massively Parallel Simulator (LAMMPS), a classical molecular dynamics code, are performed. The inter-atomic potential used in this investigation is Embedded Atom Method (EAM) potential. The plane specimens with initial central crack were subjected to Mixed-Mode loadings. The simulation cell contains 400000 atoms. The crack propagation direction angles under different values of the mixity parameter in a wide range of values from pure tensile loading to pure shear loading in a wide diapason of temperatures (from 0.1 К to 800 К) are obtained and analyzed. It is shown that the crack propagation direction angles obtained by molecular dynamics method coincide with the crack propagation direction angles given by the multi-parameter fracture criteria based on the strain energy density and the multi-parameter description of the crack-tip fields.

Analysis of Cone Wave Reflection in Finite-Size Elastic Membranes and Extension of the Ballistic Impact Problem From Elastic to Viscoelastic Membranes

The transverse ballistic impact on a two-dimensional (2D) membrane causes a truncated deformation cone to develop in the wake of tensile implosion waves. Here, the cone wave reflected from the finite boundaries of the elastic membrane has been studied analytically. A first-order linear nonhomogeneous differential equation for the ratio of the reflected cone wave front velocity to the speed of tensile waves is derived, which is further used to calculate the traveling time taken by the reflected cone wave to reach to the projectile surface. Since the reflected wave starts when the membrane is already in a deformed configuration, the speed of the reflected cone wave is a function of radius r in the cylindrical coordinates as opposed to almost constant speed of the incoming cone wave studied in the literature. The analytical results are validated with molecular dynamics (MD) simulations of the ballistic impact of projectiles onto a single layer of coarse-grained (CG) graphene. In the second part of the paper, we analyze the membrane impact problem for linear isotropic viscoelastic materials and find that the tensile wave speed for stresses and displacements is the same as that obtained in the case of a linear isotropic elastic material. We also show that only under special conditions, self-similar solutions for the cone wave are possible in viscoelastic materials modeled by Maxwell, Kelvin–Voigt, or a combination of similar models. Our findings lay some grounds on which further studies on the ballistic response of viscoelastic materials can be performed.

A revisiting of the elasticity solution for a transversely isotropic functionally graded thick-walled tube based on the Mori–Tanaka method

In this paper, we present the elastic solutions for the problem of an internal pressurized functionally graded thick-walled tube based on the Voigt method in Xin et al. (Int J Mech Sci 89:344–349, 2014); a transversely isotropic functionally graded thick-walled tube subjected to internal pressure is studied. It is assumed that the functionally graded tube is made up of two linear isotropic elastic materials; the matrix is reinforced by fibers with circular cross section all aligned in the circumferential direction. The volume fraction of the reinforced material is identical with our previous work (i.e., Xin et al. in Int J Mech Sci 89:344–349, 2014). By using the Mori–Tanaka method, this paper obtains the differential equation of the radial displacement and then the numerical results of the radial displacement and the stresses are deduced. The approximate analytical solutions are also derived which agree well with the numerical results on the basis of the Mori–Tanaka method. Further, both based on the Mori–Tanaka method, the results received by the present model are compared with those by a particle model for solving an isotropic inner-pressurized FGM tube problem. Finally, in the numerical part the influences of the volume fraction and the elastic moduli’s ratio on radial displacement and the stresses are discussed.

Mathematical modeling of the crack growth in linear elastic isotropic materials by conventional fracture mechanics approaches and by molecular dynamics method: crack propagation direction angle under mixed mode loading

The crack growth directional angles in the isotropic linear elastic plane with the central crack under mixed-mode loading conditions for the full range of the mixity parameter are found. Two fracture criteria of traditional linear fracture mechanics (maximum tangential stress and minimum strain energy density criteria) are used. Atomistic simulations of the central crack growth process in an infinite plane medium under mixed-mode loading using Large-scale Molecular Massively Parallel Simulator (LAMMPS), a classical molecular dynamics code, are performed. The inter-atomic potential used in this investigation is Embedded Atom Method (EAM) potential. The plane specimens with initial central crack were subjected to Mixed-Mode loadings. The simulation cell contains 400000 atoms. The crack propagation direction angles under different values of the mixity parameter in a wide range of values from pure tensile loading to pure shear loading in a wide diapason of temperatures (from 0.1 К to 800 К) are obtained and analyzed. It is shown that the crack propagation direction angles obtained by molecular dynamics method coincide with the crack propagation direction angles given by the multi-parameter fracture criteria based on the strain energy density and the multi-parameter description of the crack-tip fields.

The two-dimensional elasticity of a chiral hinge lattice metamaterial

We present a lattice structure defined by patterns of slits that follow a rotational symmetry (chiral) configuration. The chiral pattern of the slits creates a series of hinges that produce deformation mechanisms for the lattice due to bending of the ribs, leading to a marginal negative Poisson's ratio. The engineering constants are modeled using theoretical and numerical Finite Element simulations. The results are benchmarked with experimental data obtained from uniaxial and off-axis tensile tests, with an overall excellent agreement. The chiral hinge lattice is almost one order of magnitude more compliant than other configurations with patterned slits and – in contrast to other chiral micropolar media – exhibits an in-plane shear modulus that closely obeys the relation between Young's modulus and Poisson's ratio in homogeneous isotropic linear elastic materials.

Torsion of a Cosserat elastic bar with square cross section: theory and experiment

An approximate analytical solution for the displacement and microrotation vector fields is derived for pure torsion of a prismatic bar with square cross section comprised of homogeneous, isotropic linear Cosserat elastic material. This is accomplished by analytical simplification coupled with use of the principle of minimum potential energy together with polynomial representations for the desired field components. Explicit approximate expressions are derived for cross section warp and for applied torque versus angle of twist of the bar. These show that torsional rigidity exceeds the classical elasticity value, the difference being larger for slender bars, and that cross section warp is less than the classical amount. Experimental measurements on two sets of 3D printed square cross section polymeric bars, each set having a different microstructure and four different cross section sizes, revealed size effects not captured by classical elasticity but consistent with the present analysis for physically sensible values of the Cosserat moduli. The warp can allow inference of Cosserat elastic constants independently of any sensitivity the material may have to dilatation gradients; warp also facilitates inference of Cosserat constants that are difficult to obtain via size effects.

КОМПЬЮТЕРНОЕ МОДЕЛИРОВАНИЕ ПРОЦЕССОВ НАКОПЛЕНИЯ ПОВРЕЖДЕНИЙ В ТВЕРДЫХ ТЕЛАХ С ТРЕЩИНАМИ С ПОМОЩЬЮ ПОЛЬЗОВАТЕЛЬСКОЙ ПРОЦЕДУРЫ UMAT ВЫЧИСЛИТЕЛЬНОГО КОМПЛЕКСА SIMULIA ABAQUS

The paper presents the experience of using the user subroutine UMAT for FEM package SIMULIA Abaqus/CAE for damage accumulation processes in the vicinity of the crack. A continuum damage mechanics model based on the constitutive relations of linear elastic isotropic materials with the incorporated damage tensor components is used to describe the material behavior. The material nonlinearity arising from the deformation process is modelled by introducing an anisotropic damage tensor of the second rank into the constitutive equation. The material model is described by means of user procedure UMAT of SIMULIA Abaqus. The finite element (FE) mechanical constitutive model is implemented in Abaqus/Standard via a UMAT routine. Numerical experiments for a large series of cracked specimens have been performed. Computed stress and damage tensor components were found. It is shown that they are not dependent on the FE mesh refinement. Distributions of the damage tensor components in the vicinity of the crack tip in cracked specimens of different configurations under mixed mode loading in a wide range of mixed mode loadings are found. The configurations of active damage accumulation process zone in the cracked specimens are obtained. It is shown that the damage accumulation process has a substantial influence on the stress-strain state in the vicinity of the crack tip and leads to a decrease of the stress concentration in cracked specimens.

Numerical study of temperature effects on the poro-viscoelastic behavior of articular cartilage.

This paper presents a new approach to study the effects of temperature on the poro- elastic and viscoelastic behavior of articular cartilage. Biphasic solid-fluid mixture theory is applied to study the poro-mechanical behavior of articular cartilage in a fully saturated state. The balance of linear momentum, mass, and energy are considered to describe deformation of the solid skeleton, pore fluid pressure, and temperature distribution in the mixture. The mechanical model assumes both linear elastic and viscoelastic isotropic materials, infinitesimal strain theory, and a time-dependent response. The influence of temperature on the mixture behavior is modeled through temperature dependent mass density and volumetric thermal strain. The fluid flow through the porous medium is described by the Darcy's law. The stress-strain relation for time-dependent viscoelastic deformation in the solid skeleton is described using the generalized Maxwell model. A verification example is presented to illustrate accuracy and efficiency of the developed finite element model. The influence of temperature is studied through examining the behavior of articular cartilage for confined and unconfined boundary conditions. Furthermore, articular cartilage under partial loading condition is modeled to investigate the deformation, pore fluid pressure, and temperature dissipation processes. The results suggest significant impacts of temperature on both poro- elastic and viscoelastic behavior of articular cartilage.

Relative identifiability of anisotropic properties from magnetic resonance elastography.

Although magnetic resonance elastography (MRE) has been used to estimate isotropic stiffness in the heart, myocardium is known to have anisotropic properties. This study investigated the determinability of global transversely isotropic material parameters using MRE and finite-element modeling (FEM). A FEM-based material parameter identification method, using a displacement-matching objective function, was evaluated in a gel phantom and simulations of a left ventricular (LV) geometry with a histology-derived fiber field. Material parameter estimation was performed in the presence of Gaussian noise. Parameter sweeps were analyzed and characteristics of the Hessian matrix at the optimal solution were used to evaluate the determinability of each constitutive parameter. Four out of five material stiffness parameters (Young's modulii E1 and E3 , shear modulus G13 and damping coefficient s), which describe a transversely isotropic linear elastic material, were well determined from the MRE displacement field using an iterative FEM inversion method. However, the remaining parameter, Poisson's ratio, was less identifiable. In conclusion, Young's modulii, shear modulii and damping can theoretically be well determined from MRE data, but Poisson's ratio is not as well determined and could be set to a reasonable value for biological tissue (close to 0.5).

Method to analyse multiple site damage fatigue before and after crack coalescence

The Multiple Site Damage is a phenomenon that appears in e.g. aircraft fuselages and weld geometries. This article introduces a method for solving the Multiple Site Damage fatigue in a 2D specimen. The cases studied are for Linear Elastic Fracture Mechanics situations, isotropic materials and crack growth governed by the Paris Law regime. Already known fracture mechanics concepts are merged together in an innovative algorithm that increases computational efficiency by optimizing the number of Finite Element Analyses necessary for fatigue calculations. A new approach to crack coalescence is presented by using the application limit of Linear Elastic Fracture Mechanics. Comparisons with analytical, experimental and other software results have shown the reliability of this method for several cases. A final explanatory example of a plate with multiple cracks and a hole shows the capabilities of the method proposed.

Projector representation of isotropic linear elastic material laws for directed surfaces

AbstractIn the framework of linear elasticity, it is possible to use eigenspace projectors to describe the elasticity tensor, at least for special cases of material symmetries. A similar procedure is also advantageous in the context of directed surfaces. It is thus possible to introduce this representation to stiffness measures of thin‐walled members. Hereby, we reduce our considerations to an elastic mid‐surface, where the in‐plane, out‐of‐plane, and transverse shear states are uncoupled, but superposed eventually. Thereby, we introduce a unique decomposition of the stiffness measures for the considered deformation states, while we limit our considerations to homogeneous materials without pronounced orientation dependency. For each of these three states, eigenvalues of the stiffness tensors are evaluated based on engineering material parameters. Finally, the whole procedure allows for the clear distinction of dilatoric and deviatoric portions in the constitutive equations.After all, a compact and mathematically easy‐to‐handle representation for the stiffness tensors with respect to in‐plane, out‐of‐plane, and transverse shear state has been found. Thereby, we show correlations to classical representations as well as advantages due to the clarity of present scheme.

Uncertainties in the estimation of in situ stresses: effects of heterogeneity and thermal perturbation

In situ stress directions and magnitudes in subsurface are commonly estimated using two different methods: (1) borehole break-out—drilling-induced fracture interpretation complemented by extended leak-off tests; (2) stress estimation from acoustic measurements using advanced wireline tools. Both methods use stress perturbations around boreholes to estimate far field stresses employing Kirsch’s equations, which are only valid for boreholes aligned with one of the principal stresses, linear elastic deformation and homogeneous isotropic materials. Furthermore, the original stress state may have been altered by drilling-mud-circulation induced temperature changes. Using heuristic models including heterogeneity, this study investigates potential errors in the estimation of far-field stress due to (a) the generalisation of Kirsch’s equations to heterogeneous media and (b) plausible temperature perturbations. First, errors due to uncertainties in measuring wave slowness are analysed for an idealised homogeneous material. Second, errors due to application of Kirsch’s equations are investigated considering potential effects of frictional interfaces between layers, pore pressure in the rock matrix and thermal perturbations induced by drilling for example cases. To analyse stress in these complex scenarios finite element analysis was used, revealing strong effects of lithological and thermal variations. Stress magnitude was amplified by stiff layers and was attenuated by soft ones. At layer interfaces, substantial changes in stress orientation occurred. Kirsch’s equations for the considered cases resulted in errors in far field stresses as large as 44% in the magnitude and 90° in orientations. An uncertainty propagation analysis indicated a high accuracy of acoustic estimates for homogenous materials. However, a dramatic impact of small-scale heterogeneity may not be resolved by the logging process.