Strain Energy Equivalence Principle Research Articles

Folding stress analysis of a sandwich plate based on transversely isotropic and rate-dependent constitutive relationship of adhesive

ABSTRACT Flexible and foldable electronic products often contain adhesive layers, and failure generally initiates at the adhesive interfaces. To analyze the interfacial stress accurately, correct constitutive relationship of adhesive layers is necessary. However, soft adhesives used in foldable sandwich plates can show rate-dependent and anisotropic mechanical behaviors, which can hardly be implemented by traditional linear elastic and hyperelastic constitutive models in the folding process simulation. Through experiments in material’s principal axes and the strain energy equivalence principle, a transversely isotropic and rate-dependent constitutive relationship of adhesive has been developed in a three-dimensional form. By commercial finite element (FE) software ABAQUS and its user subroutine UMAT, the folding stress of a sandwich plate has been analyzed. The results of FE simulation show that folding time would affect stress distributions significantly due to the anisotropic and rate-dependent properties of adhesive.

Homogenized elastic response of random fiber networks based on strain gradient continuum models

The purpose of this work is to develop anisotropic strain gradient linear elastic continuum models for two-dimensional random fiber networks. The constitutive moduli of the strain gradient equivalent continuum are assessed based on the response of the explicit network representation in so-called windows of analysis, in which each fiber is modeled as a beam and the fibers are connected at crossing points with welded joints. The principle of strain energy equivalence based on the extension to the strain gradient of the Hill–Mandel macro homogeneity condition is employed to identify the classical and strain gradient moduli, based on the application of a sequential set of polynomial displacements on windows of analysis of different sizes. The scaling of the first- and second-order moduli with network parameters, such as network density and the ratio of fiber bending to axial stiffness, is determined. We observe a similar dependency of classical and strain gradient moduli on the same network parameters. The internal length scales associated with the gradient coefficients of the constitutive equation are also defined in terms of the network parameters. The strain gradient moduli prove to be size-independent in the affine regime, and they converge toward a size-independent value in the non-affine deformation regime after a rescaling of physical dimensions by the window size. The obtained results show that the strain gradient moduli scale uniformly with the square of the magnitude of the strain gradients applied to the window of analysis.

Unified elastoplastic model based on a strain energy equivalence principle

A unified elastoplastic model was proposed to describe the relation among load, displacement, and uniaxial constitutive parameters of ductile materials according to the von Mises energy equivalence principle at a special location or energy center in the deformed region of a structural component (SC). Two pairs of parameters were considered in the model: one was related to the volume of deformed region and the other to the Mises equivalent strain at the energy center. In addition, they are easily determined by finite element analysis (FEA). For eight kinds of SCs under proportional loading, the load–displacement behaviors of various materials predicted by the unified model were highly consistent with the results of FEA.

Coupling microplane-based damage and continuum plasticity models for analysis of damage-induced anisotropy in plain concrete

A novel plastic-damage constitutive model for plain concrete is developed in this paper. For this purpose, the microplane theory is proposed for overcoming the deficiency of available anisotropic continuum plastic-damage models in reproducing the true anisotropic nature of damage in multidimensional loadings. Based on the microplane theory, the degeneration process in concrete is considered along with plastic deformations. Using the principle of strain energy equivalence, a transformation between the nominal and effective states of material is achieved, that results in a decoupled formulation for damage and plasticity. A yield function with multiple internal variables and a non-associative flow function are employed to describe the plastic behavior of concrete. In order to discriminate between the behavior of concrete in tension and compression, stress tensor is decomposed into tensile and compressive parts, and two sets of microplanes are defined for each material point, leading to corresponding independent anisotropic damage tensors in tension and compression. Several numerical examples are analyzed using the proposed model and robustness and accuracy of the formulation are assessed using the available experimental tests. In addition, superiority of the proposed model over the available isotropic and anisotropic formulations is demonstrated by comparisons between the different approaches. The effects of three different cubature schemes on the response of concrete specimens are also analyzed.

Behavior of short circular tubed-reinforced-concrete columns subjected to eccentric compression

The tubed-reinforced-concrete (TRC) column is a kind of confined reinforced-concrete (RC) columns using the outer encasing thin-walled steel tube which discontinues at the beam-column joints and thus does not carry any direct axial load. Although this composite column has been used in some practical applications in China, there is still a lack of sufficient experimental data and design methods for the TRC columns subjected to eccentric compression. In this paper, 18 short circular TRC columns are tested under axial and eccentric compression considering parameters of eccentricity and diameter-to-thickness ratio of the steel tube. Failure mode, axial load-carrying capacity, and stress in the steel tube of the specimens are discussed in detail. The test results indicate that the average confining stress in the columns with small eccentricity is close to that in the columns under axial load at the peak load. To validate the employed stress–strain relationship for the confined concrete, a fiber-based model is developed and the predicted results agree well with the test results. Based on the strain energy equivalence principle, a regression formula to estimate the ultimate compressive strain of concrete confined by the steel tube is proposed. The equivalent concrete stress block parameters of circular TRC members are theoretically studied and the modified values are purposely suggested for the calculation of capacity interaction diagram.

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.

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.

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.

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.

Thermodynamic framework for coupling of non-local viscoplasticity and non-local anisotropic viscodamage for dynamic localization problems using gradient theory

This study develops a general consistent and systematic framework for the analysis of heterogeneous media that assesses a strong coupling between rate-dependent plasticity and anisotropic rate-dependent damage for dynamic problems within the framework of thermodynamic laws and gradient theories. The proposed formulation includes thermo-elasto-viscoplasticity (rate-dependent plasticity) with anisotropic thermo-viscodamage (rate-dependent damage); a dynamic yield criterion of a von Mises type and a dynamic damage growth criterion; the associated flow rules; thermal softening; non-linear strain hardening; strain-rate hardening; strain hardening gradients; and strain-rate hardening gradients. Since the material macroscopic thermomechanical response under dynamic loading is governed by different physical mechanisms on the meso- and macroscale levels, the proposed three-dimensional kinematical model is introduced with manifold structure accounting for discontinuous fields of dislocation interactions (plastic flow) and crack and void interactions (damage growth). The gradient theory of rate-independent plasticity and rate-independent damage that incorporates macroscale interstate variables and their higher-order gradients is generalized here for rate-dependent plasticity and rate-dependent damage to properly describe the change in the internal structure and in order to investigate the size effect of statistical inhomogeneity of the evolution-related rate- and temperature dependent materials. The idea of bridging length-scales is made more general and complete by introducing spatial higher-order gradients in the temporal evolution equations of the internal state variables that describe hardening in coupled viscoplasticity and viscodamage models, which are considered here physically and mathematically related to their local counterparts. Furthermore, the constitutive equations for the damaged material are written according to the principle of strain energy equivalence between the virgin material and the damaged material; that is, the damaged material is modeled using the constitutive laws of the effective undamaged material in which the nominal stresses and strains are replaced by their effective ones. In addition, computational issues concerned with the current gradient-dependent formulation of initial-boundary value problems are introduced in a finite element context. A weak (virtual work) formulation of the non-local dynamic viscoplastic and viscodamage conditions is derived, which can serve as a basis for the numerical solution of initial boundary value problems in the sense of the finite element method. Explicit expressions for the generalized tangent stiffness matrix and the generalized nodal forces are given. The model presented in this paper can be considered as a feasible thermodynamic approach that enables one to derive various coupled gradient viscoplasticity and viscodamage theories by introducing simplifying assumptions.

On the coupling of anisotropic damage and plasticity models for ductile materials

In this contribution various aspects of an anisotropic damage model coupled to plasticity are considered. The model is formulated within the thermodynamic framework and implements a strong coupling between plasticity and damage. The constitutive equations for the damaged material are written according to the principle of strain energy equivalence between the virgin material and the damaged material. The damaged material is modeled using the constitutive laws of the effective undamaged material in which the nominal stresses are replaced by the effective stresses. The model considers different interaction mechanisms between damage and plasticity defects in such a way that two-isotropic and two-kinematic hardening evolution equations are derived, one of each for the plasticity and the other for the damage. An additive decomposition of the total strain into elastic and inelastic parts is adopted in this work. The elastic part is further decomposed into two portions, one is due to the elastic distortion of the material grains and the other is due to the crack closure and void contraction. The inelastic part is also decomposed into two portions, one is due to nucleation and propagation of dislocations and the other is due to the lack of crack closure and void contraction. Uniaxial tension tests with unloadings have been used to investigate the damage growth in high strength steel. A good agreement between the experimental results and the model is obtained.

Non-local coupling of viscoplasticity and anisotropic viscodamage for impact problems using the gradient theory

A general framework for the analysis of heterogeneous media that assesses a strong coupling between viscoplasticity and anisotropic viscodamage evolution is formulated for-impact related problems within the framework of thermodynamic laws and nonlinear continuum mechanics. The proposed formulations include thermo-elasto-viscoplastici- ty with anisotropic thermo-elasto-viscodamage, a dynamic yield criterion of a von Mises type and a dynamic viscodamage criterion, the associated flow rules, non-linear strain hardening, strain-rate hardening, and temperature softening. The constitutive equations for the damaged material are written according to the principle of strain energy equivalence between the virgin material and the damaged material. That is, the damaged material is modeled using the constitutive laws of the effective undamaged material in which the nominal stresses are replaced by the effective stresses. The evolution laws are impeded in a finite deformation framework based on the multiplicative decomposition of the deformation gradient into elastic, viscoplastic, and viscodamage parts. Since the material macroscopic thermomechanical response under high-impact loading is governed by different physical mechanisms on the macroscale level, the proposed three-dimensional kinematical model is introduced with manifold structure accounting for discontinuous fields of dislocation interactions (plastic hardening), and crack and void interactions (damage hardening). The non-local theory of viscoplasticity and viscodamage that incorporates macroscale interstate variables and their higher-order gradients is used here to describe the change in the internal structure and in order to investigate the size effect of statistical inhomogeneity of the evolution-related viscoplasticity and viscodamage hardening variables. The gradients are introduced here in the hardening internal state variables and are considered to be independent of their local counterparts. It also incorporates the thermomechanical coupling effects as well as the internal dissipative effects through the rate-type covariance constitutive structure with a finite set of internal state variables. The model presented in this paper can be considered as a framework, which enables one to derive various non-local and gradient viscoplasticity and viscodamage theories by introducing simplifying assumptions.

A theoretical and computational framework for anisotropic continuum damage mechanics at large strains

Abstract The main objective of this work is the formulation and algorithmic treatment of anisotropic continuum damage mechanics at large strains. Based on the concept of a fictitious, isotropic, undamaged configuration an additional linear tangent map is introduced which allows the interpretation as a damage deformation gradient. Then, the corresponding Finger tensor – denoted as damage metric – constructs a second order, internal variable. Due to the principle of strain energy equivalence with respect to the fictitious, effective space and the standard reference configuration, the free energy function can be computed via push-forward operations within the nominal setting. Referring to the framework of standard dissipative materials, associated evolution equations are constructed which substantially affect the anisotropic nature of the damage formulation. The numerical integration of these ordinary differential equations is highlighted whereby two different schemes and higher order methods are taken into account. Finally, some numerical examples demonstrate the applicability of the proposed framework.

On the theory and computation of anisotropic damage at large strains

The goal of this contribution is the formulation and algorithmic treatment of anisotropic continuum damage mechanics at large strains. Based on the concept of an isotropic, fictitious and undamaged configuration a Finger—type damage metric tensor in terms of the fictitious linear tangent map is introduced. Referring to the principle of strain energy equivalence with respect to the fictitious, effective space and the standard reference configuration the free Helmholtz energy can be formulated within the standard reference configuration and treated in the spirit of standard dissipative materials. Thereby, the introduced damage potential substantially affects the anisotropic nature of the damage formulation. Finally, some numerical examples demonstrate the applicability of the proposed framework.

Effective Elastic Compliances and Engineering Constants for Damaged Isotropic Solids

In continuum damage mechanics, damaged solids have been represented by the effective elastic stiffness into which local damages are smoothly smeared. In parallel to the effective elastic stiffness representation, damaged solids may be represented in terms of effective elastic compliance. It then becomes easier to derive the effective engineering constants (i.e., effective elastic moduli and Poisson's ratios) for damaged solids, all in closed forms, from the effective elastic compliance rather than from the effective elastic stiffness. Thus, in this paper, by using a continuum modeling approach previously developed by the author based on both the principle of strain energy equivalence and the equivalent elliptical micro-crack representation of a local damage, the effective elastic compliance and effective engineering constants are derived in closed forms for both damaged two- and three-dimensional isotropic solids. They are derived in terms of the undamaged (virgin) elastic properties and a scalar damage variable.

A Theory of Continuum Damage Mechanics for Anisotropic Solids

This paper develops a fracture mechanics based continuum damage theory for initially anisotropic solids by extending the author’s previous damage theory for isotropic solids. The concepts of strain energy equivalence principle (SEEP) and equivalent line-crack modeling are used to develop the effective continuum elastic properties of a damaged solid in terms of the undamaged anisotropic elastic properties and a scalar damage variable.