Energy Equivalence Principle Research Articles

Linear free flexural vibration of cracked functionally graded plates in thermal environment

In this paper, the linear free flexural vibrations of functionally graded material plates with a through center crack is studied using an 8-noded shear flexible element. The material properties are assumed to be temperature dependent and graded in the thickness direction. The effective material properties are estimated using the Mori–Tanaka homogenization scheme. The formulation is developed based on first-order shear deformation theory. The shear correction factors are evaluated employing the energy equivalence principle. The variation of the plates natural frequency is studied considering various parameters such as the crack length, plate aspect ratio, skew angle, temperature, thickness and boundary conditions. The results obtained here reveal that the natural frequency of the plate decreases with increase in temperature gradient, crack length and gradient index.

A damage-based numerical analysis of brittle rocks failure mechanism

The dominant causes of irreversible rock deformations are damage process and plastic flow. Most of the existing elastic–plastic models employed in the analysis and design of rock structures only consider the plastic flow and ignore the full damage process. The common plastic models used to simulate the rock failure, does not model the rock realistically and often the important issues such as stiffness degradation, softening, and significant differences in rock response under tensile and compressive loadings are ignored. Therefore, the use of realistic damage models is essential in the design process of rock structures. In this paper, the pseudo-logarithmic damage tensor, its conjugate thermodynamic force, and energy equivalence principle were used. In the definition of rock damage yield function, many authors considered only the tensile microcracking (mode Ι). Since quasi-brittle materials such as rock degrade under tensile and shear microcracking (mode II), separate positive and negative damage yield functions were introduced. The proposed damage yield functions are formulated in the framework of a damage model which was coded in C++ environment and implemented into a commercial code. Accordingly, the proposed model was applied to the simulation of brittle rocks behavior. The uniaxial compressive and tensile testing of a brittle rock was simulated numerically and numerical findings were compared against experimental data and analytical solution. The analysis results show a very good match between numerical, experimental data, and analytical solution especially in the post-elastic region.

Beam-to-Beam Impact Behavior of Reinforced Concrete Frames under Severe Ground Motions

The beam-to-beam impact behavior during the progressive collapse progress of reinforced concrete frame structures under severe ground motions is studied in which three types of impact are included, i.e. 1) contact-impact between one-end failed active beam and intact passive beam; 2) contact-impact between two-end failed active beam and intact passive beam, and 3) contact-impact of one-end failed active beam and one-end failed passive beam. As the first step, the initial impact conditions are established based on the principles of the kinematics and the structural dynamics. Then, the velocities of the beams at the impact instant and the locations of the impact would occur are derived. A so-called impact mass factor is introduced in the paper and determined through the principle of equivalent energy. The history of contact force between the beams involved in the impact is evaluated by the Hertz-damp model. The responses of the passive beam are calculated by solving the equation of motion of a generalized single degree of freedom system.

Use of a Continuum Damage Model Based on Energy Equivalence to Predict the Response of a Single-Crystal Superalloy

Anisotropic viscoplasticity coupled with anisotropic damage has been modeled in previous works by using the energy equivalence principle appropriately adjusted. Isotropic and kinematic hardenings are present in the viscoplastic part of the model and the evolution equations for the hardening variables incorporate both static and dynamic recovery terms. The main difference to other approaches consists in the formulation of the energy equivalence principle for the plastic stress power and the rate of hardening energy stored in the material. As a practical consequence, a yield function has been established, which depends, besides effective stress variables, on specific functions of damage. The present paper addresses the capabilities of the model in predicting responses of deformation processes with complex specimen geometry. In particular, multiple notched circular specimens and plates with multiple holes under cyclic loading conditions are considered. Comparison of predicted responses with experimental results confirms the convenience of the proposed theory for describing anisotropic damage effects.

Finite element analysis of functionally graded plates under transverse load

The nonlinear behaviors of functionally graded material (FGM) plates under transverse distributed load are investigated here using a high precision plate bending finite element. Material properties of the plate are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of volume fractions of the constituents. The effective material properties are then evaluated based on the rule of mixture. The formulation is developed based on the first-order shear deformation theory considering the physical/exact neutral surface position. The shear correction factors are evaluated employing the energy equivalence principle. The transverse shear stresses and transverse normal stress components are obtained using the in-plane stresses evaluated from the constitutive equations and the three-dimensional equilibrium equations. The nonlinear governing equations are obtained following a standard finite element procedure and solved through Newton–Raphson iteration technique to predict the lateral pressure load versus central displacement relationship.

Coupled multi‐scale cohesive modeling of failure in heterogeneous adhesives

AbstractA multi‐scale cohesive numerical framework is proposed to simulate the failure of heterogeneous adhesively bonded systems. This multi‐scale scheme is based on Hill's variational principle of energy equivalence between the higher and lower level scales. It provides an easy way to obtain accurate homogenized macroscopic properties while capturing the physics of failure processes at the micro‐scale in sufficient detail. We use an isotropic rate‐dependent damage model to mimic the failure response of the constituents of heterogeneous adhesives. The finite element method is used to solve the equilibrium equation at each scale. A nested iterative scheme inspired by the return mapping algorithm used in computational inelasticity is implemented. We propose a computationally attractive technique to couple the macro‐ and micro‐scales for rate‐dependent constitutive laws. We introduce an adhesive patch test to study the numerical performance, including spatial and temporal convergence of the multi‐scale scheme. We compare the solution of the multi‐scale cohesive scheme with a direct numerical simulation. Finally, we solve mode I and mode II fracture problems to demonstrate failure at the macro‐scale. Copyright © 2010 John Wiley & Sons, Ltd.

Damage Theoretical Analysis of 2.5D C/SiC Composites under Tensile and Shear Loading

An anisotropic damage constitutive model is developed to describe the damage behavior of C/SiC composites. Different kinematic and isotropic hardening functions were employed in damage yield function to describe accurately the damage nonlinear hardening. The damage variable is defined by the principle of energy equivalence. The degradation of stiffness and the unrecoverable deformation induced by micro-crack propagation were considered in this model. The constants of constitutive model are identified and the damage evolution processes under tensile and shear loading. Uniaxial tension and shear tests have been used to valid the constitutive model to C/SiC composites.

Energy based time equivalent approach for evaluating fire resistance of reinforced concrete beams

The current prescriptive approaches for evaluating fire resistance of reinforced concrete beams under standard fire exposure have a number of drawbacks and do not provide realistic performance assessment. As an alternative, time equivalent concept can relate the severity of design fire exposure to that of standard fire exposure. However, the current empirical formulae for evaluating such time equivalency are mainly derived for protected steel members and may not be applicable for reinforced concrete members. This paper presents an energy-based time equivalency method for evaluating the fire resistance of reinforced concrete beams under design fire scenarios. The proposed method is based on the principle of equivalent energy, and estimates fire resistance based on the equivalency between standard and design fire exposures. The validity of the method is established by comparing the predictions from the proposed approach with those from existing methods (equivalent area method and empirical formulae) and with nonlinear finite element analysis. The applicability of the proposed approach to design situations is illustrated through a numerical example. It is shown that the proposed energy based method is capable of predicting equivalent fire resistance under design fire scenarios with an accuracy that is sufficient for practical purposes.

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.

Boundary element method for calculation of effective elastic moduli in 3D linear elasticity

AbstractA boundary integral formulation of the linear elasticity problem for a multi‐component composite is given. The fast BEM solver based on the adaptive cross approximation is then obtained by the data‐sparse representation of the resulting Galerkin matrices. The solver is used to obtain effective elastic moduli of fibre and particle reinforced composites in three dimensions by means of the strain energy equivalence principle. Copyright © 2009 John Wiley & Sons, Ltd.

New discrete element models for elastoplastic problems

The discrete element method (DEM) has attractive features for problems with severe damages, but lack of theoretical basis for continua behavior especially for nonlinear behavior has seriously restricted its application. The present study proposes a new approach to developing the DEM as a general and robust technique for modeling the elastoplastic behavior of solid materials. New types of connective links between elements are proposed, the inter-element parameters are theoretically determined based on the principle of energy equivalence and a yield criterion and a flow rule for DEM are given for describing nonlinear behavior of materials. Moreover, a numerical scheme, which can be applied to modeling the behavior of a continuum as well as the transformation from a continuum to a discontinuum, is obtained by introducing a fracture criterion and a contact model into the DEM. The elastoplastic stress wave propagations and the tensile failure process of a steel plate are simulated, and the numerical results agree well with those obtained from the finite element method (FEM) and corresponding experiment, and thus the accuracy and efficiency of the DEM scheme are demonstrated.

Parametric Identification of Dynamic Model of Assembled Flexible Manipulator

In order to restrain the vibration of flexible manipulator,friction damping of assembled flexible link can be used to restrain the vibration of flexible manipulator. The contrast experiment with solid flexible link indicates that assembled flexible link has effective vibration-restraining capability. The dynamic model of assembled flexible link is built with an equivalent mass,a nonlinear spring,a damper and a spring-slider. According to the data obtained from FEA simulation of forced vibration of assembled flexible link,its shape function and reaction force hysteresis curve are determined. The dynamic model parameter of assembled flexible link is determined according to kinetic energy equivalence principle,analysis of reaction force hysteresis curve and nonlinear least-squares solution. In addition,S-function block is written in M-file to compute spring-slider. Finally,the simulink model is created according to the dynamic model. The response of free vibration from simulation is consistent with experiment. This method provides an effective way for dynamics modeling and vibration-damping design of assembled flexible link.

Continuum Damage Models based on Energy Equivalence: Part II — Anisotropic Material Response

Anisotropic viscoplasticity coupled with anisotropic damage is modeled in a thermodynamically consistent way. Isotropic and kinematic hardening are present in the viscoplasticity part of the model and the evolution equations for the hardening variables incorporate both, static and dynamic recovery terms. Damage effects are captured in the framework of the concept of effective stress and effective strain combined with the principle of energy equivalence as adopted in Part I. The theory is employed to determine stress distributions for a single-crystal superalloy under complex loading histories. The results are compared with experimental measurements in order to examine the capabilities of the proposed theory.

Continuum Damage Models based on Energy Equivalence: Part I — Isotropic Material Response

An energy equivalence method for modeling damage effects in material response is proposed. In the present article, the main issues of the method are discussed for the less complicated case of isotropic constitutive functions. Otherwise, the material response addressed is supposed to be (rate-independent) elasto-plastic exhibiting isotropic and kinematic hardening. In order to make clear the difference to other continuum damage models, it suffices to deal here with isotropic damage expressed in terms of a scalar state variable. Our approach is based on the concept of effective stress and effective strain combined with a principle of energy equivalence as explained in the article. As a result, both the yield function and the evolution equations governing the hardening response of the damaged material are obtained from a given undamaged model material. Characteristic properties of the damage theory proposed are illustrated by comparing predicted responses with those according to damage models based on the principle of strain equivalence.

Computational efficient inverse dynamics of 6-DOF fully parallel manipulators by using the Lagrangian formalism

Abstract This paper readdress the issue of deriving explicit equations of motion for parallel robots by using the Lagrangian formalism. Reputed of being inefficient for closed-loop mechanisms, the formalism became unpopular for parallel robots. This paper argue the converse by giving an alternative, though intuitive and simple approach. The idea is to consider open-loop subchains of the manipulator and to derive their dynamics by the Lagrangian formalism and with respect to an own set of generalized coordinates and velocities. The paper discusses and proves the importance of the formal choice of these coordinates. The principle of energy equivalence is derived to allow for joining the different equations of motion. To obtain the final form with respect to the robot’s active coordinates, attention has to be paid to the transformation of the subchains dynamics, which is introduced in this paper. Finally, a systematic study of the resulting computational effort is presented and discussed in relation to other methods and approaches given in the literature.

Evaluation of the Structural Properties of Single-Walled Carbon Nanotubes Using a Dynamic Continuum Modeling Method

The structural properties of single-walled carbon nanotubes (SWCNTs) are evaluated by a new continuum modeling method, called the dynamic continuum modeling method (DCMM). In the DCMM, a repeating cell unit isolated from a SWCNT is represented as an equivalent continuum beam element by using the energy equivalence principle. The method may allow us to accomplish the static or dynamic analysis for SWCNTs without needing to assume their wall thickness. The structural rigidities of armchair SWCNTs evaluated by using the DCMM are within the range predicted by using the Young's moduli reported in existing references. It is also found that SWCNTs have quite high transverse shear rigidity, but with negligible coupling rigidities.

Torsional structural properties of single-walled carbon nanotubes

In this paper, a dynamic continuum modelling method is developed based on the molecular mechanics as well as on the energy equivalence principle for the single-walled carbon nanotubes subject to torsional vibration. The dynamic continuum modelling method enables one to evaluate torsional structural properties without assuming any fictitious wall thickness of single-walled carbon nanotubes. The present dynamic continuum modelling method is then applied to armchair single-walled carbon nanotubes to evaluate torsional rigidities and mass inertia properties. By using the evaluated torsional structural properties, the natural frequencies of carbon nanotubes are also investigated.

Evaluation of a Strain Energy Equivalence Principle (SEEP)-Based Continuum Damage Model

A new continuum damage theory (CDT) has been proposed by Lee et al. (1997) based on the SEEP. The CDT has the apparent advantage over the other related theories because the complete constitutive law can be readily derived by simply replacing the virgin elastic stiffness with the effective orthotropic elastic stiffness obtained by using the proposed continuum damage theory. In this paper, the CDT is evaluated by comparing the mode shapes and natural frequencies of a square plate containing a small line-through crack with those of the same plate with a damaged site replaced with the effective orthotropic elastic stiffness computed by using the CDT.

Effective Material Properties of Damaged Elastic Solids

By using a continuum modeling approach based on the equivalent elliptical crack representation of a local damage and the strain energy equivalence principle, the effective elastic compliances and the effective engineering constants are derived in closed forms in terms of the virgin (undamaged) elastic properties and a scalar damage variable for damaged two- and threedimensional isotropic solids. It is shown that the effective Young’s modulus in the direction normal to the crack surfaces is always smaller than its intact value.

Analysis of nonlinear vibration for symmetric angle-ply laminated viscoelastic plates with damage

The behavior of nonlinear vibration for symmetric angle-ply laminated plates including the material viscoelasticity and damage evolution is investigated. By employing the von Karman's nonlinear theory, strain energy equivalence principle and Boltzmann superposition principle, a set of governing equations of nonlinear integro-differential type are derived. By applying the finite difference method, Newmark method and iterative procedure, the governing equations are solved. The effects of loading amplitudes, exciting frequencies and different ply orientations on the critical time to failure initiation and nonlinear vibration amplitudes of the structures are discussed. Numerical results are presented for the different parameters and compared with the available data.