Equivalent Strain Energy Research Articles

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.

A multiscale eigenelement method and its application to periodical composite structures

This paper develops a multiscale eigenelement method based on multilevel substructure technology for the multiscale analysis of periodical composite structures, and provides a comparison study with user point of view for the multiscale methods including the mathematical homogenization method, the heterogeneous multiscale method, the multiscale finite element method and the generalized finite element method, laying emphasis on their ideas, implementations, adaptabilities and similarities. It is shown that the newly developed eigenelement method satisfies the two essential homogenization conditions, one is the strain energy equivalence which is crucial to the lower order frequencies, and the other is the deformation similarity which is crucial to the local or micro stresses. Numerical experiments validate the present method.

An eigenelement method and two homogenization conditions

Under inspiration from the structure-preserving property of symplectic difference schemes for Hamiltonian systems, two homogenization conditions for a representative unit cell of the periodical composites are proposed, one condition is the equivalence of strain energy, and the other is the deformation similarity. Based on these two homogenization conditions, an eigenelement method is presented, which is characteristic of structure-preserving property. It follows from the frequency comparisons that the eigenelement method is more accurate than the stiffness average method and the compliance average method.

138 POSTER Design, synthesis, biochemical and biological evaluations of novel and potent small-molecule inhibitors of STAT3

The equivalent strain energy density (ESED) method was originally suggested and is still widely used to calculate the stresses and strains at notch root. It is well recognized, however, that the ESED method tends to underestimate the notch-tip stresses and strains. In order to obtain better predictions, in the present paper, a factor ((1 + νe)/(1 + νeff)) was introduced to modify the dissipated heat energy contained in the ESED method. A computational modeling technique coupling with Jiang-Sehitoglu plasticity model was also developed to calculate the multiaxial elastic-plastic notch-tip stress-strain responses of notched components. Extensive validations of the proposed method show that the calculated stresses and strains are in accord with experimental data and show reasonable accuracy.

Theoretical Formulation of a Coupled Elastic—Plastic Anisotropic Damage Model for Concrete using the Strain Energy Equivalence Concept

An anisotropic damage constitutive model for concrete is developed within the framework of elastoplasticity and continuum damage mechanics. The transformation from the effective (undamaged) to the damaged configuration in the elastic regime is obtained by using the hypothesis of elastic strain energy equivalence. Damage in plasticity is accounted for by developing a new formulation relating the plastic strains rate tensors in the effective and damaged configurations. Two anisotropic damage criteria are introduced to account for the different concrete behavior effects under tensile and compressive loadings. The total stress is decomposed into tensile and compressive components in order to satisfy these damage criteria. The plasticity yield criterion presented in this work accounts for the spectral decomposition of the stress tensor and will be used simultaneously with the damage criteria. The transformation of stresses from the effective to the damaged configuration is achieved by using a fourth order transformation tensor that is based on second order tensile and compressive damage tensors. Expressions are derived for the elastoplastic tangent operator in the effective and damaged configurations. The formulations are derived consistently based on sound thermodynamic principles.

Hybrid lattice particle modeling: Theoretical considerations for a 2D elastic spring network for dynamic fracture simulations

Abstract A new hybrid lattice particle modeling (HLPM) scheme is proposed. The particle–particle interaction is derived from lattice modeling (LM) theory, whereas the computational scheme follows particle modeling (PM) technique. The newly proposed HLPM considers different particle interaction schemes, involving not only particles in the nearest neighborhood, but also the second nearest neighborhood. Different mesh structures with triangular or rectangular unit cells can be used. The current paper is concerned with the mathematical derivations of elastic interaction between contiguous particles in 2D lattice networks, accounting for different types of linkage mechanism and different shapes of lattice. Axial (α) and combined axial-angular (α − β) models are considered. Derivations are based on the equivalence of strain energy stored in a unit cell with its associated continuum structure in the case of in-plane elasticity. Conventional PM technique was restricted to a fixed Poisson’s ratio and had a strong bias in crack propagation direction, as a result of the geometry of the adopted lattice network. The current HLPM is free from the above-mentioned deficiencies and can be applied to a wide range of impact and dynamic fracture failure problems. Although the current analysis is based on the linear elastic spring model, inelastic considerations can be easily implemented, as HLPM has the same force interaction scheme as PM, based on the Lennard–Jones potential.

Comparison of Inelastic Behaviors of Lead-Free and Sn—Pb Solder Joints

In this study, finite element analysis is used to investigate the inelastic behaviors of lead-free solders, 96.5Sn—3.5Ag and 95.5Sn—3.9Ag—0.6Cu, and classical 63Sn—37Pb solder-bumped wafer level chip scale package (WLCSP) on printed circuit board (PCB) assemblies subjected to four different temperature cycle tests. The behaviors of equivalent strain range and strain energy density at the outmost corner solder joint adjoining to the upper pad are examined. It is found that: (1) the 95.5Sn—3.9Ag—0.6Cu solder has the best performance on the effect of plastic behavior among these three solder alloys, but it accumulates more plastic strain while the number of temperature cycling increased; (2) the 96.5Sn—3.5Ag solder alloy can sustain more creep than others; (3) the difference exists between using the equivalent total strain range and the strain energy density, respectively, to assess the thermo-mechanical behaviors of the solder-bumped WLCSP on PCB assemblies.

A unified expression of elastic–plastic notch stress–strain calculation in bodies subjected to multiaxial cyclic loading

In this paper, a simplified thermodynamics analysis of cyclic plastic deformation is performed in order to establish an energy transition relation for describing the elastic–plastic stress and strain behavior of the notch-tip material element in bodies subjected to multiaxial cyclic loads. Based on the thermodynamics analysis, it is deduced that in the case of elastic–plastic deformation, Neuber’s rule inevitably overestimates the actual stress and strain at the notch tip, while the equivalent strain energy density (ESED) method tends to underestimate the actual notch-tip stress and strain. According to the actual energy conversion occurring in the notch-tip material element during cyclic plastic deformation, a unified expression for estimating the elastic–plastic notch stress–strain responses in bodies subjected to multiaxial cyclic loads is developed, of which Neuber’s rule and the ESED method become two particular cases, i.e. upper and lower bound limits of the notch stress and strain estimations. This expression is verified experimentally under both proportional and non-proportional multiaxial cyclic loads and a good agreement between the calculated and the measured notch strains has been achieved. It is also shown that in the case of multiaxial cyclic loading, the unified expression distinctly improves the accuracy of the notch-tip stress–strain estimations in comparison with Neuber’s rule and the ESED method. The unified expression of the notch stress–strain calculation developed in this paper can thus provide a more logical approximate approach for estimating the elastic–plastic notch-tip stress and strain responses of components subjected to lengthy multiaxial cyclic loading histories for local strain approach-based fatigue-crack-initiation life prediction.

Nonlinear active control of damaged piezoelectric smart laminated plates and damage detection

Considering mass and stiffness of piezoelectric layers and damage effects of composite layers, nonlinear dynamic equations of damaged piezoelectric smart laminated plates are derived. The derivation is based on the Hamilton’s principle, the higher-order shear deformation plate theory, von Karman type geometrically nonlinear strain-displacement relations, and the strain energy equivalence theory. A negative velocity feedback control algorithm coupling the direct and converse piezoelectric effects is used to realize the active control and damage detection with a closed control loop. Simply supported rectangular laminated plates with immovable edges are used in numerical computation. Influence of the piezoelectric layers’ location on the vibration control is investigated. In addition, effects of the degree and location of damage on the sensor output voltage are discussed. A method for damage detection is introduced.

Free flexural vibration analysis of symmetric rectangular honeycomb panels using the improved Reddy’s third-order plate theory

The free flexural vibration of symmetric rectangular honeycomb panels is investigated in this paper using the improved Reddy’s third-order theory. Reddy’s third-order theory provides a parabolic distribution of the transverse shear stresses in the thickness direction and represents a close approximation to the true shear strain distribution for a shallow single-layer plate. However, for the case of multilayered laminates, the continuity condition on the inter-laminar shear stresses implies a piecewise continuous shear strain distribution in the thickness direction. To accommodate the effect of the continuity condition of inter-laminar transverse shear stresses, the shear correction factors are introduced to modify the shear strains in the Reddy’s third-order theory. The shear correction factors are calculated using an iterative formulation based on the shear strain energy equivalence. The improved Reddy’s third-order theory coupled with shear correction factors in predicting free flexural vibration of symmetric honeycomb panels is examined by being compared with the experimental value and the finite element analyses based on three-dimensional models.

Static Analysis of Sandwich Panels with Square Honeycomb Core

Static analysis of sandwich panels with a square honeycomb core is performed using the finite element approach applied to the plate theories. The constitutive behavior of an equivalent continuum to the core is obtained using the strain energy approach. The displacement field of the sandwich panel modeled using the equivalent continuum is then compared with results obtained from a highly detailed finite element method created in ABAQUS. The comparison shows the results of the higher-order shear deformation theory in error of 7.6% compared with the detailed model results. An equivalent single layer finite element method for the sandwich panel was then created in ABAQUS and the displacement results of this equivalent single layer model are compared with those obtained from the detailed finite element method. This comparison reveals an error of 6.1% in the results between the two models. A comparison between the equivalent single layer model and the highly detailed model is presented for different core relative densities at two locations of the sandwich panel. A detailed finite element method analysis of the unit cell of the square honeycomb was next performed using ABAQUS and the flexibility approach. Comparison between the equivalent strain energy approach and flexibility approach applied to detailed ABAQUS models of the unit cell proved the equivalent strain energy approach to be efficient.

Analytical study of the elastic–plastic stress fields ahead of parabolic notches under antiplane shear loading

An analytical study is carried out on the elastic–plastic stress and strain distributions and on the shape of the plastic zone ahead of parabolic notches under antiplane shear loading and small scale yielding. The material is thought of as obeying an elastic-perfectly-plastic or a strain hardening law. When the notch root radius becomes zero, the analytical frame matches the solutions for the crack case due to Hult–McClintock (elastic-perfectly-plastic material) and Rice (strain hardening material). The analytical frame provides an explicit link between the plastic stress and the elastic stress at the notch tip. Neuber’solution for blunt notches under antiplane shear is also obtained and the conditions under which such a solution is valid are discussed in detail by using elastic and plastic notch stress intensity factors. Finally, revisiting Glinka and Molski’s equivalent strain energy density (ESED), these factors are used also to give, under antiplane shear loading, the increment of the strain energy at the notch tip with respect to the linear elastic case.

Research on Fracture Damage Coupled with Crack Arrest by Anchorage Model of Discontinuous Jointed Rock Mass under the State of Complex Stress

Based on the hypothesis of equivalent strain energy and the theories of fracture mechanics and damage mechanics，the constitutive model and fracture damage mechanism of bolted discontinuous jointed rockmass are systematically studied under the state of complex stresses. Initially ， considering the equivalent strain energy ， the constitutive relation of anchored discontinuous jointed rockmass is derived under the state of compression-shear stresses. The constitutive relation under the state of tension-shear stresses is also developed according to the theory of self-consistence. Next，the damage evolution equations of discontinuous multi-crack rockmass under compression-shear and tension-shear are put forward according to the wing crack-initiating criterion. Finally，based on the above constitutive models and the damage evolution equations three-dimensional finite element procedures have been developed to evaluate the stability and deformability of the surrounding rock mass during excavation and supporting. The calculated results indicate that above-mentioned constitutive relation and the damage evolution equations are available.

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.

Fracture-Damage Model for Anchored Discontinuous Jointed Rockmass and its Application

According to the theories of fracture mechanics and damage mechanics, the constructive model and fracture damage mechanism of brittle discontinuous jointed rockmass are systematically studied under the state of complex stress in this paper. By the aid of the method of equivalent strain energy, the constitutive relation of anchored brittle discontinuous jointed rockmass is derived under the state of compression-shearing. The constitutive relation under the state of tension-shearing is also developed according to the theory of self-consistence. Finally, based on the above constructive models, the three-dimensional finite element procedure has been developed to model the ground movements that occur when underground power-houses of pumped-storage power station are installed in discontinuous jointed rockmass. The anchor supporting is an important component of this underground power-houses excavation work. Besides the displacement field and the secondary state of stress induced by the excavation disturbance, the effect of anchoring and the damage evolution around the power-houses have been particularly described during the process of installation. The numerical results obtained by numerical simulation were compared with that of field monitoring in order to verify the validity of the proposed models.

Prediction of buckling characteristics of carbon nanotubes

In this paper, to investigate the buckling characteristics of carbon nanotubes, an equivalent beam model is first constructed. The molecular mechanics potentials in a C–C covalent bond are transformed into the form of equivalent strain energy stored in a three dimensional (3D) virtual beam element connecting two carbon atoms. Then, the equivalent stiffness parameters of the beam element can be estimated from the force field constants of the molecular mechanics theory. To evaluate the buckling loads of multi-walled carbon nanotubes, the effects of van-der Waals forces are further modeled using a newly proposed rod element. Then, the buckling characteristics of nanotubes can be easily obtained using a 3D beam and rod model of the traditional finite element method (FEM). The results of this numerical model are in good agreement with some previous results, such as those obtained from molecular dynamics computations. This method, designated as molecular structural mechanics approach, is thus proved to be an efficient means to predict the buckling characteristics of carbon nanotubes. Moreover, in the case of nanotubes with large length/diameter, the validity of Euler’s beam buckling theory and a shell model with the proper material properties defined from the results of present 3D FEM beam model is investigated to reduce the computational cost. The results of these simple theoretical models are found to agree well with the existing experimental results.

Swing-up Control of Mass Body Interlinked Tether

The tethered system, which composed with a mother ship and satellite connected with tether, is an example of recent satellites. In the analysis and modeling of tether, large deformation and large displacements must be considered. The Absolute Nodal Coordinate Formulation can describe the motion of tether. In the modeling method, internal damping and fluid resistance are not considered, and the identification for such parameters is still problem to be solved. In this study, the effect of damping and fluid resistance based on strain energy and drag equation are formulated while considering a parameter identification method. These parameter and number of elements are identified by the experiment at the same time. In order to verify the modeling method, an example of swing-up control using a genetic algorithm is performed in simulation and experiment. Validity of model and availability of motion control based on multibody dynamics are shown comparison between experiment and simulation.

Nonlinear finite element analysis of anular lesions in the L4/5 intervertebral disc

Degenerate intervertebral discs exhibit both material and structural changes. Structural defects (lesions) develop in the anulus fibrosus with age. While degeneration has been simulated in numerous previous studies, the effects of structural lesions on disc mechanics are not well known. In this study, a finite element model (FEM) of the L4/5 intervertebral disc was developed in order to study the effects of anular lesions and loss of hydrostatic pressure in the nucleus pulposus on the disc mechanics. Models were developed to simulate both healthy and degenerate discs. Degeneration was simulated with either rim, radial or circumferential anular lesions and by equating nucleus pressure to zero. The anulus fibrosus ground substance was represented as a nonlinear incompressible material using a second-order polynomial, hyperelastic strain energy equation. Hyperelastic material parameters were derived from experimentation on sheep discs. Endplates were assumed to be rigid, and annulus lamellae were assumed to be vertical in the unloaded state. Loading conditions corresponding to physiological ranges of rotational motion were applied to the models and peak rotation moments compared between models. Loss of nucleus pulposus pressure had a much greater effect on the disc mechanics than the presence of anular lesions. This indicated that the development of anular lesions alone (prior to degeneration of the nucleus) has minimal effect on disc mechanics, but that disc stiffness is significantly reduced by the loss of hydrostatic pressure in the nucleus. With the degeneration of the nucleus, the outer innervated anulus or surrounding osseo-ligamentous anatomy may therefore experience increased strains.

Influence of uniaxial and biaxial tension on meso-scale geometry and strain fields in a woven composite

Representative volume element (RVE) of a woven composite is often idealised to have identical geometry in the warp and weft directions. While this may be true in the case of consolidating a completely relaxed preform with equal warp and weft densities, majority of woven composites have non-ideal RVE geometry. The present paper investigates the influence of membrane stresses – applied during the moulding process – on the mechanical properties of the finished composite. Crimp interchange due to uniaxial stress and tow flattening due to biaxial stress have been computed by a fabric compliance model developed by the authors. Compliance model is based on the principle of stationary potential energy by taking into consideration tow bending, compression and extensional energies, and the external work done by the tensile forces. It has been shown that a uniaxial tensile stress applied to the preform results in an increase in tensile modulus and a corresponding reduction in the transverse modulus, as a result of reduced crimp in the loading direction with a corresponding increase in the transverse direction. Tensile strength and stiffness increase in the loading direction, and the mode of failure initiation changes from ‘knee phenomena’ to tow failure. A biaxial stress applied during the forming stage results in crimp reduction in both axial and transverse directions; this leads to an improved tensile modulus and strength of a composite laminate. RVE of the deformed geometry has been modelled using ABAQUS CAE pre-processor. The elastic moduli were computed using the concept of equivalent strain energy proposed by Zhang and Harding. The resulting strain fields were compared for various cases of uniaxial and biaxial extension.

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.