Hypothesis Of Energy Equivalence Research Articles

Simulation of ductile fracture propagation using the elastoplastic phase-field damage method calibrated by ultrasonic wave velocity measurement

The phase-field damage theory has become a powerful approach to estimate crack propagation via the finite element method (FEM). The sharp crack is smeared into a diffusive fracture propagation zone within a length scale parameter (ℓ). However, the calibration of the phase-field damage model is still difficult, especially for the length scale parameter, which is difficult to directly observe and accurately measure in laboratory testing. In most numerical studies, the length scale parameter is arbitrarily determined as the minimum element size in FEM. However, the selection of the length scale parameter may largely affect the accuracy of the phase-field model. Also, most current phase-field models only consider elasticity and few studies further consider the ductile (plastic) fracture propagation, since the plastic recoverable energy does not have a convincing form. In this study, we aim to calibrate the elastoplastic phase-field damage model based on the traditional continuum damage mechanics, with the assistance of ultrasonic wave velocity measurement. The traditional damage variable can be back-calculated from the drop of P wave velocity, following the energy equivalence hypothesis. This traditional damage variable is shown to be consistent with the phase-field damage variable as calculated in our proposed elastoplastic phase-field damage model. Therefore, the length scale parameter can be determined from fitting the rock damage evolution. The plastic flow and its corresponding plastic-free energy are considered, using a phenomenological function obeying preliminary thermodynamic assumptions. The proposed model shows good consistency with laboratory observations. Numerical implementation of the proposed model was achieved by inserting it into COMSOL Multiphysics using a weak form formulation. This paper introduces a pragmatic approach to calibrate the phase-field damage model based on laboratory testing results.

A Multiaxial Fatigue Damage Model Based on Constant Life Diagrams for Polymer Fiber-Reinforced Laminates.

In the last decade, fatigue damage models for fiber-reinforced polymer composites have been developed assuming the fracture energy equivalence hypothesis. These models are able to predict a fatigue life of composite laminates, but their identification requires a significant number of off-axial tests for various stress ratios. The present study proposes the stress ratio dependent model, which phenomenologically adopts a decrease in stiffness and residual strength of a unique ply according to experimental constant life diagrams. Hashin, Tsai-Hill, and the maximum stress failure criteria are utilized for damage initiation considering the residual strength of the ply. The obtained results indicate a sufficiency of using S-N curves for UD 0°, UD 45°, and UD 90° for identification of the model. The model was verified by S-N curves for UD 10°, UD 15°, and UD 30° and its applicability was demonstrated for prediction of a fatigue life of composite laminates with an arbitrary lay-up. The model is implemented into ABAQUS finite element software as a user subroutine.

Constitutive modelling of plastically deformable self-healing materials

A novel elasto-plastic constitutive model of self-healing materials is proposed coupling continuum damage–healing mechanics (CDHM) framework with kinematic and isotropic hardening cases. A novel secondary damage internal variable is presented, eliminating an assumption in the existing literature that the healed area does not undergo damage again, thus improving the existing CDHM framework. The necessity of the proposed secondary damage variable is successfully demonstrated by a four spring material model. The improved CDHM framework is coupled with the three-dimensional elasto-plastic formulation containing nonlinear kinematic hardening and isotropic strain hardening effects. The proposed formulation is finally numerically implemented by return mapping approach employing the strain energy equivalence hypothesis from CDHM, and the macro-scale stress update is performed applying several different strain histories. A continuum tangent modulus expression is also derived, and a boundary value problem is successfully solved employing an in-house developed plane strain nonlinear finite element code. It is finally demonstrated, by the numerical results, that the secondary damage variable is crucial while obtaining the physically realistic stress–strain response of self-healing materials leading to the complete failure upon exhaustion of the healing capabilities.

Thermodynamic based model for coupled elastoplastic damage-healing behaviour of unsaturated geomaterials

The capacity for self-healing of fractures in geomaterials, especially clay, is a well-documented phenomenon that is important in many problems encountered in geotechnical and geo-environmental engineering (e.g., foundation design, slope stability, waste barrier systems, hydraulic fracturing). Several constitutive models that can account for damage and fracture of geomaterials have been proposed, many of which are formulated at microstructural scale with high computational costs that often preclude them from usage in engineering practice. In addition, mechanical constitutive laws describing healing (i.e., damage reversal) have been mainly developed for concrete and rock but not for soil.A coupled elastoplastic-damage-healing constitutive model is presented in this paper to investigate the behaviour of geomaterials subjected to hydro-mechanical loadings. The mathematical formalism is presented within a continuum mechanics framework, with damaged-healed configuration defined through an energy equivalence hypothesis. Damage and healing evolution laws are also established within the framework of thermodynamics, taking into account the effect of plastic hardening, strain rate, stress ratio, suction hardening as well as confining pressure. The elastoplastic response of geomaterials is captured using a bounding surface plasticity model from critical state soil mechanics. Effects of damage, healing and suction are also accounted for by defining a proper hardening rule based on the consistency condition. The presented model's ability to simulate experimental results taken from the literature is assessed. Several examples are solved under different loading conditions in order to study the effect of damage-healing on the behaviour of unsaturated soils.

Effective properties of composite material based on total strain energy equivalence

The main problem referred in the present research is how to correctly and efficiently predict the averaged effective properties of a composite on the example of an irregular particle-reinforced composite in which the distribution, shape and orientation of particles are fully disordered (chaotic). In the present analysis a new method is proposed in which such composite is considered as a multi-phase material, the constitutive description of which is derived on the basis of the total strain energy equivalence hypothesis. The method is verified both theoretically (by comparison with the results of other methods) and experimentally. A good agreement between theoretical and experimental results is observed.

A thermodynamically consistent viscoelastic–viscoplastic constitutive model for self-healing materials

Self-healing materials, as a class of intelligent materials, are capable to recover a part of the weakened mechanical properties induced by damage. In this article, based on the thermodynamics of irreversible processes, employing the effective configuration in the continuum damage-healing mechanics, a viscoelastic–viscoplastic constitutive model is presented. In the constitutive model development, we adopt an additive decomposition of the total strain into elastic, viscoelastic, and viscoplastic parts. In this regard, defining the damage and healing variables and employing the strain energy equivalence hypothesis, stress and strain tensors in the effective configuration are obtained. Satisfying the Clausius–Duhem inequality, the evolution equations are introduced for viscoelastic and viscoplastic strains. The damage and healing variables also evolve according to two different evolutions. To employ the proposed model in different loading conditions, the time-discrete form of the model in the semi-implicit form is presented. Material parameters of the model are identified with creep, creep-recovery, and repeated creep-recovery tests on asphalt concrete materials in the literature. Finally, the capability of the proposed model is demonstrated applying the model prediction for creep-recovery and repeated creep-recovery case studies and comparing the results with the experimental data available in the literature.

A Nonlinear Damage Model of Hardening-Softening Materials

In this paper, an elastoplastic-damage constitutive model is presented. The formulation is cast within the framework of continuum damage mechanics (CDM) by means of the internal variable theory of thermodynamics. The damage is assumed as a tensor type variable and its evolution is developed based on the energy equivalence hypothesis. In order to discriminate the plastic and damage deformation, two surfaces named as plastic and damage are introduced. The damage surface has been developed so that it can model the nonlinear variation of damage. The details of the model besides its implicit integration algorithm are presented. The model is implemented as a user-defined subroutine user-defined material (UMAT) in the abaqus/standard finite element program for numerical simulation purposes. In the regard of investigating the capability of model, the shear and tensile tests are experimentally conducted, and corresponding results are compared with those predicted numerically. These comparisons are also accomplished for several experiments available in the literature. Satisfactory agreement between experiments and numerical predictions provided by the model implies the capability of the model to predict the plastic deformation as well as damage evolution in the materials.

Total energy equivalence in constitutive modeling of multidissipative materials

In the present work, the total energy equivalence hypothesis was applied in constitutive modeling of engineering materials. The approach originally developed for damaged materials, was extended to modeling not only damage but also other dissipative phenomena, like phase transformation, in a consistent manner. The proposed model was examined by means of parametric studies to show its ability to reflect different experimentally observed features of real materials.

A thermodynamic consistent rate-dependent elastoplastic-damage model

In this article, a rate-dependent elastoplastic-damage constitutive model considering the effect of strain rate has been developed. The derivation of this model has been established based on the irreversible thermodynamics with internal variables within the fundamentals of Continuum damage mechanics (CDM). For investigating the rate effect, an additional power function dependent on the effective strain rate has been involved in the plastic dissipation function (dynamic plastic yield surface). The damage has been assumed as a tensor-type variable and based on the energy equivalence hypothesis, the damage evolution has been developed. The proposed constitutive model has been implemented into user-defined subroutines UMAT and VUMAT in the finite-element program ABAQUS/(Standard and Explicit). For this purpose, the implicit and explicit stress integration algorithms of the model have been explained. The model has been validated by comparing the predicted results with experimental data conducted on Al2024-T3. These experiments are including the tensile and double-notched tests. Furthermore, the numerical results have been compared with some data available in the literature. The numerical examples show the excellent correlation between experiments and simulations for stress (or force) and damage results.

A damage model incorporating dynamic plastic yield surface

In this paper, a general elastoplastic-damage constitutive model considering the effect of strain rate has been developed. The derivation of this model has been cast into the irreversible thermodynamics with internal variables within the fundamentals of Continuum Damage Mechanics (CDM). The rate effect has been involved as an additional term into the plastic yield surface (dynamic plastic yield surface). Therefore, the plastic surface has been presented in the category of Consistency–type model in which the rate of state variables is considered as independent state variables. The damage has been assumed as a tensor type variable and based on the energy equivalence hypothesis the damage evolution has been developed. The proposed model has been validated for both rate-independent and rate-dependent deformation. For this manner, the generalized trapezoidal stress integration algorithm of the model has been explained and the model has been implemented into user-defined subroutines (UMAT and VUMAT) in the finite element program ABAQUS. The results of numerical simulation, statically and dynamically, have been compared to the experimental results of three aluminum and two steel alloys. Also, the results of simulation for shear and double-notched tests have been compared to their experiments. By comparing the predicted results with experimental data, the capability and validity of the model have been verified.

Complex Damage Variables in Continuum Damage Mechanics

The concept of complex damage variables is introduced in this work. These damage variables have both real and imaginary parts. They are introduced not to use them in practical applications but to try to derive a direct relationship between the damage due to cross-sectional area reduction and the damage due to elastic stiffness degradation. In addition this concept can provide an insight in addressing the concept of healing that the authors have extensively published as well as the concept of undamageable materials. Toward this goal some success is achieved in the sense that some complicated relationships between the two damage processes are formulated. These relationships and the complex variable approach are novel ideas in Continuum Damage Mechanics. Throughout the formulation the hypothesis of strain equivalence is used in order to simplify the mathematical equations. This work can be extended and generalized by substituting the hypothesis of energy equivalence but this will complicate the equations unnecessarily.

Temporal patterns of energy equivalence in temperate soil invertebrates.

The question whether total population energy use is invariant to species body size (the energy equivalence hypothesis) is central to metabolic ecology and continues to be controversial. While recent comparative field work and meta-analyses pointed to systematic deviations of the underlying allometric scaling laws from predictions of metabolic theory none of these studies included the variability of metabolic scaling in ecological time. Here we used extensive data on the invertebrate soil fauna of Kampinos National Park (Poland) obtained from six consecutive quantitative sampling seasons to show that phylogenetically corrected species density—body weight and population energy use—body weight relationships across all soil fauna species and within trophic groups and body weight classes were highly variable in time. On average, population energy use tended to increase with species body weight in decomposers and phytophages, but not in predators. Despite these trends, our data do not exclude the possibility that energy equivalence marks the central tendency of energy use in the edaphon. Our results highlight the need for long-term studies on energy use to unequivocally assess predictions of metabolic theory.Electronic supplementary materialThe online version of this article (doi:10.1007/s00442-015-3317-3) contains supplementary material, which is available to authorized users.

A bi-dissipative damage model for concrete

This work deals with an improvement of an anisotropic damage model in order to analyze reinforced concrete structures submitted to reversal loading. The original constitutive model is based on the fundamental hypothesis of energy equivalence between real and continuous media following the concepts of the Continuum Damage Mechanics. The concrete is assumed as an initial elastic isotropic medium presenting anisotropy, permanent strains and bimodularity induced by damage evolution. In order to take into account the bimodularity, two damage tensors governing the rigidity in tension or compression regimes are introduced. However, the original model is not capable to simulate the influence of the previous damage processes in compression regimes. In order to avoid this problem, some conditions are introduced to simulate the damage unilateral effect. It has noted that the damage model is agreement with to micromechanical theory conditions when dealing to unilateral effect in concrete material. Finally, the proposed model is applied in the analyses of reinforced concrete framed structures submitted to reversal loading. These numerical applications show the good performance of the model and its potentialities to simulate practical problems in structural engineering.

Interactions between multiple large macrofauna species and nematode communities — Mechanisms for indirect impacts of trawling disturbance

Diverse assemblages of infauna in sediments provide important physical and biogeochemical services, but are under increasing pressure by anthropogenic activities, such as benthic trawling. It is known that trawling disturbance has a substantial effect on the larger benthic fauna, with reductions in density and diversity, and changes in community structure, benthic biomass, production, and bioturbation and biogeochemical processes. Largely unknown, however, are the mechanisms by which the trawling impacts on the large benthic macro- and megafauna may influence the smaller meiofauna. To investigate this, a mesocosm experiment was conducted whereby benthic nematode communities from a non-trawled area were exposed to three different densities (absent, low, normal) of 7 large (>10mm) naturally co-occurring, bioturbating species which are potentially vulnerable to trawling disturbance. The results showed that total abundances of nematodes were lower if these large macrofauna species were present, but no clear nematode abundance effects could be assigned to the macrofauna density differences. Nematode community structure changed in response to macrofauna presence and density, mainly as a result of the reduced abundance of a few dominant nematode species. Any detectable effects seemed similar for nearly all macrofauna species treatments, supporting the idea that there may be a general indirect, macrofauna-mediated trawling impact on nematode communities. Explanations for these results may be, firstly, competition for food resources, resulting in spatial segregation of the meio- and macrobenthic components. Secondly, different densities of large macrofauna organisms may affect the nematode community structure through different intensities of bioturbatory disturbance or resource competition. These results suggest that removal or reduced densities of larger macrofauna species as a result of trawling disturbance may lead to increased nematode abundance and hints at the validity of interference competition between large macrofauna organisms and the smaller meiofauna, and the energy equivalence hypothesis, where a trade-off is observed between groups of organisms that are dependent on a common source of energy.

Thermodynamic Significance and Basis of Damage Variables and Equivalences

In this article, thermodynamic significance and basis of damage variables and equivalences are addressed within the irreversible thermodynamic framework with internal variables by Rice [Rice, J.R. (1971). Inelastic Constitutive Relations for Solids: An Internal Variable Theory and its Application to Metal Plasticity. Journal of the Mechanics and Physics of Solids, 19:433—455]. It is revealed that, as soon as the multiscale kinematic relation is prescribed, the condition of free or complementary energy equivalence leads to the full equivalence of the microscale and macroscale thermodynamic formulations. In order to achieve the fully thermodynamic equivalence, the damage equivalence hypothesis should be based on free energy or (thermodynamic) complementary energy from a thermodynamic viewpoint. In this sense, the complementary energy equivalence hypothesis by Sidoroff [Sidoroff, F. (1981) Description of Anisotropic Damage Application to Elasticity, IUTAM Colloquium on Physical Nonlinearities in Structural Analysis, pp. 237—244, Springer, Berlin] is much better than the strain equivalence hypothesis by Lemaitre and Chaboche [Lemaitre, J. and Chaboche, J. L. (1978). Aspect Phenomnologique de la Rupture par Endommagement (in French). J. de Mecanique Appliquee, 2:317—365]. One single scalar damage variable plays a role of fictitious or intrinsic time. The introduction of the scalar damage variable implies that the microscale thermodynamic fluxes measured by the intrinsic time should be potential or irrotational and the corresponding flow potential function is just thermodynamic force conjugate to the damage variable.

Isotropic Elastoplasticity Fully Coupled with Non-Local Damage

This paper presents a simple damage-gradient based elastoplastic model with non linear isotropic hardening in order to regularize the associated initial and boundary value problem (IBVP). Using the total energy equivalence hypothesis, fully coupled constitutive equations are used to describe the non local damage induced softening leading to a mesh independent solution. An additional partial differential equation governing the evolution of the non local isotropic damage is added to the classical equilibrium equations and associated weak forms derived. This leads to discretized IBVP governed by two algebric systems. The first one, associated with equilibrium equations, is highly non linear and can be solved by an iterative Newton Raphson method. The second one, related to the non local damage, is a linear algebric system and can be solved directly to compute the non local damage variable at each load increment. Two fields, linear interpolation triangular element with additional degree of freedom is terms of the non local damage variable is constructed. The non local damage variable is then transferred from mesh nodes to the quadrature (or Gauss) points to affect strongly the elastoplastic behavior. Two simple 2D examples are worked out in order to investigate the ability of proposed approach to deliver a mesh independent solution in the softening stage.

Equivalence principles in continuum damage mechanics

In this paper, as usual in continuum damage mechanics, an effective continuum is introduced, but this continuum is here considered as an auxiliary body. The main purpose is then to connect together these two materials for a possible comparison, i.e. to propose geometrical and mechanical constraints between these materials. In this paper, we recall briefly the three-terms multiplicative decomposition of the deformation gradient by using a natural geometrical constraint, and we propose a new theoretical method available for obtaining mechanical constraints between the two materials. The proposed approach is then applied to generalize the hypothesis of strain equivalence and the hypothesis of energy equivalence. In this approach, new equivalence principles are obtained, and a new mechanical constraint based on reciprocity is analysed. This paper is restricted to mechanical processes and time-independant plasticity, but large strains are considered.

Kinematic Description of Damage

In this paper the kinematics of damage for finite elastic deformations is introduced using the fourth-order damage effect tensor through the concept of the effective stress within the framework of continuum damage mechanics. However, the absence of the kinematic description of damage deformation leads one to adopt one of the following two different hypotheses. One uses either the hypothesis of strain equivalence or the hypothesis of energy equivalence in order to characterize the damage of the material. The proposed approach in this work provides a relation between the effective strain and the damage elastic strain that is also applicable to finite strains. This is accomplished in this work by directly considering the kinematics of the deformation field and furthermore it is not confined to small strains as in the case of the strain equivalence or the strain energy equivalence approaches. The proposed approach shows that it is equivalent to the hypothesis of energy equivalence for finite strains. In this work, the damage is described kinematically in the elastic domain using the fourth-order damage effect tensor which is a function of the second-order damage tensor. The damage effect tensor is explicitly characterized in terms of a kinematic measure of damage through a second-order damage tensor. The constitutive equations of the elastic-damage behavior are derived through the kinematics of damage using the simple mapping instead of the other two hypotheses.

Analysis of Failure Properties and Strength of Structural Adhesive Joints with Damage Mechanics

The present paper deals with the strength and failure properties of adhe sive joints weakened by initial imperfections such as flaws or debonding of the interface. The problem is analyzed by applying the principles of the newly developed damage mechanics with suitable modifications made to accommodate the inherent physical and mechanical properties of structural adhesives. The reduction in effective load-carrying area due to damage accumulation is represented by a symmetric damage tensor of second rank that is related to the decrease of elastic modulus via a hypothesis of energy equiva lence. Damage evolution is assumed to be controlled by the dilatational part of total en ergy. The model is then implemented onto a finite element program by using the updated Lagrangian formulation. A butt adhesive joint weakened by a central crack in the adhesive is subsequently analyzed to demonstrate the potential capability of the damage model. Detailed stress and damage distributions in the near and far fields are given as functions of adherend/adhesive modulus ratio. The issue of fracture initiation angle and fracture load is also studied by applying two novel damage fracture criteria.

A fully coupled elastoplastic damage modeling and fracture criteria in metalforming processes

In this paper a fully coupled elastoplastic damage theory at finite strain is presented. The energy-based Von Mises yield criterion and the damage evolution criterion with two damage variables are postulated through the hypothesis of energy equivalence. The local distributions of variables connected with the history of stress, strain and damage are determined by numerical simulation. These values are then used with six previously published fracture criteria to predict fracture initiation sites and to simulate the propagation of fracture in metalforming processes.