Equivalent Strain Energy Research Articles

Equivalence and dynamic analysis for jointed trusses based on improved finite elements

Deployable trusses that are widely applied in space missions utilize many hinged members to adapt to different geometrical shapes and work conditions. This paper presents two improved finite elements for a link and a bending beam to calculate dynamic characteristics of non-jointed and jointed trusses. First, the axial and transverse wave motion equations of a beam are used to get the shape functions of the link and the beading beam in their axial and transverse directions. Second, the dynamic matrices of stiffness and mass for a link and a bending beam are established on the basis of virtual work principle. The dynamic matrix and a theoretical equation are used to calculate the natural frequency of a cantilever bar. The comparison of the two results shows the effectiveness of the dynamic stiffness matrix. These improved elements are proved to be more accurate in calculating the responses of structures at high frequencies than common elements. Equivalent models of non-jointed and jointed structures are obtained on the basis of the displacement and strain energy equivalence of a truss unit. The result of the period truss confirms the accuracy of the equivalent model of the non-jointed truss expressed by dynamic matrices. The dynamic influences of cubic joints on a jointed structure are evaluated by giving the simulations of the equivalent jointed structure based on improved matrices. The results indicate that the natural frequencies of the jointed structure increase with the excitation force and the stiffness of joints. A relational function among the natural frequencies, joint stiffness, and excitation force is also presented.

Modelling matrix damage and fibre–matrix interfacial decohesion in composite laminates via a multi-fibre multi-layer representative volume element (M2RVE)

A three-dimensional multi-fibre multi-layer micromechanical finite element model was developed for the prediction of mechanical behaviour and damage response of composite laminates. Material response and micro-scale damage mechanism of cross-ply, [0/90]ns, and angle-ply, [±45]ns, glass-fibre/epoxy laminates were captured using multi-scale modelling via computational micromechanics. The framework of the homogenization theory for periodic media was used for the analysis of the proposed ‘multi-fibre multi-layer representative volume element’ (M2RVE). Each layer in M2RVE was represented by a unit cube with multiple randomly distributed, but longitudinally aligned, fibres of equal diameter and with a volume fraction corresponding to that of each lamina (equal in the present case). Periodic boundary conditions were applied to all the faces of the M2RVE. The non-homogeneous stress–strain fields within the M2RVE were related to the average stresses and strains by using Gauss’ theorem in conjunction with the Hill–Mandal strain energy equivalence principle. The global material response predicted by the M2RVE was found to be in good agreement with experimental results for both laminates. The model was used to study effect of matrix friction angle and cohesive strength of the fibre–matrix interface on the global material response. In addition, the M2RVE was also used to predict initiation and propagation of fibre–matrix interfacial decohesion and propagation at every point in the laminae.

Thermal vibration and transient response of magnetostrictive functionally graded material plates

Abstract The results of functionally graded material (FGM) plate with mounted magnetostrictive layer are investigated in thermal vibration and transient response by using the generalized differential quadrature (GDQ) method. The modified shear correction coefficient can be obtained based on the total strain energy equivalence principle. The computational real value solutions of Terfenol-D FGM plate with four edges in simply supported boundary conditions are obtained for the center displacement. Some parametric effects on the Terfenol-D FGM plates are analyzed, there are: shear correction coefficient values, thickness of mounted magnetostrictive layer, control gain values, temperature of environment and power law index of FGM. The effect of different mechanical boundary conditions on the results of numerical GDQ method is also investigated.

Analysis of damage in 5083 aluminum alloy deformed at different strainrates

This paper presents a study on damage evolution in 5083 marine-grade Aluminum alloy while deformed under different strain rates. The concept of continuum damage mechanics (CDM) was utilized to evaluate the evolution of damage throughout the deformation process of 5083 Aluminum alloy. Tensile tests with several unloading and reloading stages were conducted at room temperature to generate true stress–true strain curves at three different strain rates of 0.001s−1, 0.01s−1, and 0.1s−1. A scanning electron microscope (SEM) was utilized to characterize the damage at several strain levels for each strain rate. SEM results showed that void area fraction increases with strain rate and with accumulation of plastic strain. Reduction in stiffness with accumulation of plastic strain was also evaluated. Isotropic damage values were predicted using the hypotheses of strain equivalence and strain energy equivalence, as well as a recently developed energy-based model. The predicted damage values were verified via comparisons with the present SEM results. The energy-based model showed good comparisons, while the other two hypotheses overestimated the damage, especially at higher strainrates.

Design and Analysis of Pressure Vessel Assembly for Testing of Missile Canister Sections Under Differential Pressures

In this work, a pressure vessel is designed to simulate differential pressure conditions on components of missiles and canisters. This design mainly concerned with two pressure chamber mounted concentrically, out of which outer chamber experiences internal pressure and the other experiences external pressure. The operating pressure conditions are 45×105 Pa external and 10×105 Pa internal. The chamber is designed for 100×105 Pa external and 50×105 Pa internal pressure with consideration of safety issues involved in the operation. Primarily, the design is based on IS 2825 unfired pressure vessel code and materials are chosen as per the standard ASTM A516 Gr. 70 pressure vessel steels. The other mounting fixtures, supporting channels and beams are as per the standard and all the other materials are of IS 2062 structural steel. The design is iterated many times to satisfy the desired requirements. The equivalent stresses and strain energy stored in the critical location of both pressure vessels are calculated. The fatigue life of the entire system are also estimated based on the stress and strain based designs with consideration of the fully reversed and zero based loading conditions.

MICROMORPHIC TWO-SCALE MODELLING OF PERIODIC GRID STRUCTURES

The present contribution focuses on the numerical homogenization of periodic grid structures. In order to investigate the micro-to-macroscale transition, a consistent numerical homogenization scheme will be presented, replacing a heterogeneous Cauchy microcontinuum by a homogeneous micromorphic substitute continuum on the macroscale. The extended degrees of freedom, namely, the microdeformation and its gradient, are to be interpreted in terms of geometrical deformation modes and the related loading conditions of the underlying unit cells. Assuming strain energy equivalence of the macro- and the microscale, the effective constitutive properties of a square and a honeycomb grid structure are identified and quantitatively validated in comparison to reference calculations with microscopic resolution.

Calculation of effective section stiffness properties for wind turbine blades using homogenization

The structural behavior of wind turbine blades can be accurately modeled using beam models. The accuracy in the predictions using beam models depends on the degree of accuracy in the prediction of the effective section properties. A simple and efficient method compared with full 3D analysis is developed for the prediction of effective section properties for prismatic slender members of any arbitrary cross-section shape. The technique, which is based on the superposition principle, is developed to include the shear deformation modes in predicting the effective stiffness properties. A homogenization scheme based on strain energy equivalence is employed for calculating the effective properties. The method can accurately predict the full stiffness matrix including the off-diagonal terms. Also a procedure for finding the locations of the weighted centroid and the shear center is presented. The results for four different cross-section shapes including a wind turbine blade section are validated using an existing analysis technique, Variational Asymptotic Beam Sectional Analysis. Copyright © 2012 John Wiley & Sons, Ltd.

Modal analysis of the FGM beams with effect of the shear correction function

In the proposed contribution the effect of the shear correction function is originally studied and evaluated in modal analysis of the functionally graded material (FGM) beams. Spatially continuous variations of the material properties are considered. The shear correction function is calculated from the shear strain energy equation including spatial Poisson′s ratio variations. The equations of the homogenized FGM beam free vibration and their solution is presented including the shear correction function. Four coupled differential equations are derived and used in the modal analysis of beams with polynomial continuous longitudinal and transversal variations of material properties. Further, 2nd order beam effects and longitudinal varying elastic beam foundations are considered. The influence of using an average shear correction factor is evaluated through numerical experiments. Additionally, the longitudinal eigenfrequencies are calculated. Continuum solutions using solid finite elements are taken as a reference for comparison purposes.

Variation of Etch Pit Size by Screw Dislocation Tilt in 4H-SiC Wafer

The wide size distribution of the hexagonal etch pit of screw dislocations (SD) in 4H-SiC wafer was found in spite of the narrow size distribution of the SD pit in epitaxial film. Calculation on the basis of the strain energy equation indicated that etch pit size depends on the Burgers vector and dislocation tilt. Size variation of SD etch pits in 4H-SiC wafer fabricated by sublimation method is explained to be caused by the dislocation tilt by observing the sizes and the positions of etch pits from the surface of the epitaxial film to the inside of 4H-SiC wafer. The SDs in 4H-SiC wafer fabricated by sublimation method propagate to c-axis direction in macroscopic but changing tilt in microscopic.

New Constitutive Model for Two-Dimensional Jointed Rock Mass

Mechanical analysis for jointed rock mass is very significant in geotechnical engineering. In particular, the mechanical characteristics of jointed rock mass can result in nonlinear response of geotechnical engineering and in consequence mechanical analysis for the engineering is difficult. In order to simulate jointed rock mass, a new equivalent model is proposed for two-dimensional jointed rock mass based on strain energy equivalence principle. Firstly, the equivalent elastic modulus and Poisson's ratio for a horizontal joint in unit square is deduced. Then, the equivalent flexibility matrix for a joint of specified angle in unit square is given. Moreover, the equivalent flexibility matrix for multiple joints, which have different angles in unit square, is obtained. Finally, a numerical simulation is given to verify the model. Compared with other conventional algorithm, the proposed approach has some strong points: it is easy to implement in the finite element program. Furthermore, it provides a new model to compute the deformation of joints and rock and the precision is high.

Elastic Strain Energy of a Nanowire in Three-Point Bending Test with the Consideration of Surface Effects

Three-point bending tests of nanowires with Contact atomic force microscopy reveal that the Young’s modulus of a nanowire is size-dependent. The modulus changes with the diameter of a nanowire. This size dependency can be explained within the framework of classical continuum mechanics by including the effects of surface stress. In this study, an analytical solution has been derived for the elastic strain energy of a nanowire with both ends clamped and contacted by an AFM tip at its midpoint. Different from previous theoretical models, the present model can handle the case of large deflection, where the displacement of the nanowire is in the same order of the diameter. Based on the equivalence of elastic strain energy, the apparent Young’s modulus of a nanowire is expressed as a function of the elastic modulus of the bulk and that of the surface, and the dimensions of a nanowire.

Bond failure of steel beams strengthened with FRP laminates – Part 2: Verification

The application of the new bond failure model developed in Part 1 to predict the bond failure load of fiber reinforced polymer (FRP) strengthened steel beam is presented in this paper. Full-scale experiments on FRP strengthened steel beams under four-point bending were first carried out. The effects of different strengthening parameters including laminate thickness, bond length and adhesive thickness on the bond strength were investigated experimentally. Finite element analyses (FEA) were then conducted on the FRP strengthened steel beams. By incorporating the proposed bond failure model into the FEA, the equivalent strain energy density in the bondline was calculated for bond strength prediction. The validity of the model is assessed by comparing the bond strengths obtained from numerical analyses against the experimental results. The advantages of this model are proven by comparing its results against those predicted by the maximum value based model. Finally, a parametric study was conducted to investigate the effects of the thickness of FRP laminate, the tensile modulus of FRP laminate, the thickness of epoxy adhesive and the bond length on bond strength.

Quasi-linear viscoelastic modeling of arterial wall for surgical simulation

Realistic soft tissue deformation modeling and haptic rendering for surgical simulation require accurate knowledge of tissue material characteristics. Biomechanical experiments on porcine tissue were performed, and a reduced quasi-linear viscoelastic model was developed to describe the strain-dependent relaxation behavior of the arterial wall. This information is used in surgical simulation to provide a realistic sensation of reduction in strength when the user holds a virtual blood vessel strained at different levels. Twelve pieces of porcine abdominal artery were tested with uniaxial elongation and relaxation test in both circumferential and longitudinal directions. The mechanical property testing system consists of automated environment control, testing, and data collection mechanism. A combined logarithm and polynomial strain energy equation was applied to model the elastic response of the specimens. The reduced relaxation function was modified by integrating a rational equation as a corrective factor to precisely describe the strain-dependent relaxation effects. The experiments revealed that (1) stress is insensitive to strain rate in arterial tissue when the loading rate is low, and (2) the rate of stress relaxation of arterial wall is highly strain dependent. The proposed model can accurately represent the experimental data. Stress-strain function derived from the combined strain energy function is able to fit the tensile experimental data with R(2) equals to 0.9995 in circumferential direction and 0.999 in longitudinal direction. Modified reduced relaxation function is able to model the strain-dependent relaxation with R(2) equals to 0.9686 in circumferential direction and 0.988 in longitudinal direction. The proposed model, based on extensive biomechanical experiments, can be used for accurate simulation of arterial deformation and haptic rendering in surgical simulation. The resultant model enables stress relaxation status to be determined when subjected to different strain levels.

Bond failure of steel beams strengthened with FRP laminates – Part 1: Model development

To strengthen deteriorated steel structures, bonding fiber reinforced polymer (FRP) laminate applied externally to the steel surface is a promising method. For FRP strengthened steel structures, the bond performance between the FRP laminate and the steel structure is a crucial consideration which will directly influence strengthening effect and determine the final capacity of the strengthened structures. To investigate the bond failure mechanism of FRP bonded steel structures, experiments on three types of FRP-steel joints were conducted. The bond strengths of different configuration joints were found out. Besides the experimental investigation, finite element analyses were also carried out to study in detail the stress and strain distributions along the bondline. It was found that the most important factors that influence the final bond failure is the stress concentration at the end of the bondline. After analyzing the mechanism of bond failure, a bond failure model based on the failure criterion of equivalent strain energy density was proposed.

Out-of-Plane Elastic Deformation of Triangle Honeycomb

In this research, the equivalent rigidities for the out-of-plane deformation of honeycombs, consisting of triangle cells, were studied theoretically based on the equivalence of strain energy. Honeycombs composed of unsymmetrical cells are anisotropic and the out-of-plane deformation of honeycombs should be expressed by considering six equivalent rigidities. The validity of the analytical results was confirmed by comparing with the results of numerical analysis of FEM. Also, in this research, two kinds of honeycombs composed of the same shape triangle cells are treated. One is such honeycomb where each triangle cell joins mutually at a vertex. And the other is such that a vertex of a triangle is on another triangular base. The former is called type A and latter is called type B. The mean density for two types of honeycombs consisting of same triangles is the same. However, one of the bending rigidities of type B is very low compared with type A, because the load conditions of the cell walls differ.

Variability response functions for effective material properties

The variability response function ( VRF) is a well-established concept for efficient evaluation of the variance and sensitivity of the response of stochastic systems where properties are modeled by random fields that circumvents the need for computationally expensive Monte Carlo (MC) simulations. Homogenization of material properties is an important procedure in the analysis of structural mechanics problems in which the material properties fluctuate randomly, yet no method other than MC simulation exists for evaluating the variability of the effective material properties. The concept of a VRF for effective material properties is introduced in this paper based on the equivalence of elastic strain energy in the heterogeneous and equivalent homogeneous bodies. It is shown that such a VRF exists for the effective material properties of statically determinate structures. The VRF for effective material properties can be calculated exactly or by Fast MC simulation and depends on extending the classical displacement VRF to consider the covariance of the response displacement at two points in a statically determinate beam with randomly fluctuating material properties modeled using random fields. Two numerical examples are presented that demonstrate the character of the VRF for effective material properties, the method of calculation, and results that can be obtained from it.

An eigenelement method of periodical composite structures

Eigenelement method is an eigenvector expansion based finite element method , which was proposed by the authors to solve the macro behaviors of composites with less computational cost. To improve the macroscopic accuracy of the classical eigenelement method (CEEM), a serendipity eigenelement method (SEEM) is proposed, which takes the geometry and elastic properties of different phases of composites into account to some extent. Moreover, the shape function and its construction method of a multiscale eigenelement method (MEM) are presented, and the results of SEEM and MEM are compared with that of CEEM and the mathematical homogenization method (MHM) whose physical interpretation is revealed for the first time. It is shown that MEM is the most accurate eigenelement, SEEM is more accurate than CEEM, and MEM satisfies the two essential homogenization conditions: the strain energy equivalence and the deformation similarity. The extensive numerical comparison is given for stresses, displacements and frequencies.

Size-dependent optimal microstructure design based on couple-stress theory

The purpose of this paper is to propose a size-dependent topology optimization formulation of periodic cellular material microstructures, based on the effective couple-stress continuum model. The present formulation consists of finding the optimal layout of material that minimizes the mean compliance of the macrostructure subject to the constraint of permitted material volume fraction. We determine the effective macroscopic couple-stress constitutive constants by analyzing a unit cell with specified boundary conditions with the representative volume element (RVE) method, based on equivalence of strain energy. The computational model is established by the finite element (FE) method, and the design density and FE stiffness of the RVE are related by the solid isotropic material with penalization power (SIMP) law. The required sensitivity formulation for gradient-based optimization algorithm is also derived. Numerical examples demonstrate that this present formulation can express the size effect during the optimization procedure and provide precise topologies without increase in computational cost.

ハニカムコーア材の面外弾性変形 : 第1報,ひずみエネルギーの等価性に基づく弾性定数の解析

ハニカムコーア材の面外弾性変形 : 第1報,ひずみエネルギーの等価性に基づく弾性定数の解析

Application of a hybrid of particle swarm and genetic algorithm for structural damage detection

This study presents a novel optimization algorithm which is a hybrid of particle swarm optimization (PSO) method and genetic algorithm (GA). Using the Ackley and Schwefel multimodal benchmark functions incorporating up to 25 variables, the performance of the hybrid is compared with pure PSO and GA and found to be far superior in convergence and accuracy. The hybrid algorithm is then used to identify multiple crack damages in a thin plate using an inverse time-domain formulation. The damage is detected using an orthotropic finite element (FE) model based on the strain energy equivalence principle. The identification is carried out using time-domain acceleration responses. The principle is to minimize the difference between the measured and theoretically predicted accelerations. Since the computational effort of identifying the use of global FE model proved prohibitive, a quarter substructure was identified which contains 72 damage variables. Using numerically simulated experiments, three cracks in a plate were reliably detected using this method in the presence of noise. While the pure particle swarm algorithm proved to be fast, the hybrid algorithm proved to be more accurate in damage prediction. GA performed worst in speed and accuracy.