Endochronic Model Research Articles

Numerical investigation of the effect of improved endochronic constitutive model on rock material deformation under cyclic loading

Abstract Rock materials are often subjected to cyclic loads during construction. A full understanding of the deformation behavior of rock under cyclic loads is of great significance in engineering. In this study, by using the endochronic constitutive model proposed by Valanis, the reinforcement function form and constitutive parameters of the endochronic model are improved. An endochronic plasticity constitutive program for rock based on the finite element method is used to study the mechanical properties of rock under different cyclic loading conditions (of maximum loading stress, amplitude, and confining pressure), and through the use of red sandstone, marble, and basalt, experimental and simulation results are compared to verify the model. The results show that the theory can better simulate the hysteresis loop width and cumulative plastic strain of rock under cyclic loading and, under different loading conditions, the numerical simulation results are consistent with the actual rock experimental law. Therefore, the endochronic plasticity constitutive model in this study can be applied to investigate the influence of cyclic loading on the dynamic deformation behavior of rock.

A kinematic hardening model based on endochronic theory for complex stress histories

While the field of soil constitutive modeling is rich, the availability of models that capture the behavior of different soil types under complex loading is limited. This study presents the development and validation of the Multi-Mechanical Model (MMM). The MMM is an elasto-plastic kinematic hardening model, which extends an earlier endochronic model by introducing the third stress invariant, accounts for shear-volumetric coupling in granular materials, and displays ratcheting behavior during stress reversals. A series of triaxial and isotropic consolidation tests are used to calibrate the MMM to materials found in conventional unbound granular materials and subgrades, such as crushed limestone, Leighton Buzzard sand, and Vicksburg Buckshot clay. The MMM is tested against resilient modulus test data that present complex stress histories. The study shows the MMM is straightforward in its calibration, while also displaying a robust capability in capturing the complex behaviors of the different soil types under various loading conditions. While positioned for applications in other areas, the MMM’s capabilities in capturing the short-term pore pressure response of fine-grained materials, the shear-induced volume change and stress-hardening effects of granular materials, and ratcheting behavior during repeated loading conditions make it well-suited for addressing pavement problems.

An Endochronic-based approach for simulating pore water pressure variation during liquefaction of sand

Liquefaction is one of the consequences of earthquake in soil layers comprised of saturated loose sands. Various aspects of liquefaction have been investigated using different methods of field, laboratory, and numerical studies. Numerical simulation of liquefaction constitutes a major part of these researches. Application of a proper constitutive law in the numerical analysis is crucial for modeling thecomplicated undrained behavior of saturated sands during dynamic loading. Simplicity of formulation, limited number of parameters, and good performance are the main features of a proper constitutive law. In this study, an Endochronicbased model for simulating liquefaction of sand is implemented into the finite element program, PISA. Coupled dynamic field equations of Biot’s theory with u-p formulation are used to determine pore fluid and soil skeleton responses. Generalized Newmark method is employed for integration in time. The developed code is capable of predicting the magnitude of the generated pore water pressure, coupled with advanced constitutive laws such as critical state twosurface plasticity as well as simple models such as Mohr- Coulomb. Simulating cyclic simple shear and centrifuge tests using Endochronic model showed favorable performance of this model for predicting the variation of pore water pressure in saturated soil layers subjected to earthquake excitations.

Elastic-Plastic Endochronic Constitutive Model of 0Crl7Ni4Cu4Nb Stainless Steels

We presented an elastic-plastic endochronic constitutive model of 0Crl7Ni4Cu4Nb stainless steel based on the plastic endochronic theory (which does not need the yield surface) and experimental stress-strain curves. The key feature of the model is that it can precisely describe the relation of stress and strain under various loading histories, including uniaxial tension, cyclic loading-unloading, cyclic asymmetric-stress axial tension and compression, and cyclic asymmetric-stress axial tension and compression. The effects of both mean stress and amplitude of stress on hysteresis loop based on the elastic-plastic endochronic constitutive model were investigated. Compared with the experimental and calculated results, it is demonstrated that there was a good agreement between the model and the experiments. Therefore, the elastic-plastic endochronic constitutive model provides a method for the accurate prediction of mechanical behaviors of 0Crl7Ni4Cu4Nb stainless steel subjected to various loadings.

Endochronic model of plasticity generalizing Sanders theory

The Sanders’s theory of plasticity and quasi-statistical variant of incremental plastic theory with isotropic and kinematic hardening in Novozhilov’s version are generalized in the frameworks of the endochronic approach. The constitutive equations of endochronic theory of inelasticity including the ideas of Sanders, Novozhilov and Valanis are formulated. The relations for the calculation of stresses and strains in uniaxial active and reversible material loadings are proposed. The formulas are obtained by using the elementary average principle of local values and by the simplest set of material constants and functions. Two types of initial conditions are considered in the calculations. The results of numerical modeling of inelastic material behavior under uniaxial active and cyclic loadings are presented. The results are compared with each other and with original Sanders’s theory. The similarity and differences between the generalized endochronic theory and the Sanders’s version are demonstrated. Several unusual manifestations of inelastic material behavior that require further theoretical analysis, calculations on the complex loading paths and the experimental verification are noted.

The Application of Endochronic Damage Constitutive Model in Cracking Analysis of Mass Concrete

This paper provides a brief introduction of endochronic damage constitutive model and studies the application of endochronic damage constitutive model in cracking analysis of mass concrete. This paper proved the model is feasible in cracking analysis of mass concrete and it is worth deep study.

Pseudo-potentials and bipotential: A constructive procedure for non-associated plasticity and unilateral contact

Pseudo-potentials are very useful tools to define thermodynamically admissible constitutive rules. Bipotentials are convenient for numerical purposes, in particular for non-associative rules. Unfortunately, these functionals are not always easy to construct starting from a given constitutive law. This work proposes a procedure to find the pseudo-potentials and the bipotential starting from the usual description of a non-associative constitutive law. This method is applied to different non-associative plasticity models such as the Drucker-Prager model and the non-linear kinematic hardening model. The same procedure allows one to obtain the pseudo-potentials of an endochronic plasticity model. The pseudo-potentials for the contact problem with dissipation are constructed using the same ideas. For all these non-associative constitutive laws a bipotential is then automatically deduced.

THEORETICAL SIMULATION OF STRESS-STRAIN RELATIONS FOR SOME IRAQI CLAYS USING THE ENDOCHRONIC MODEL

A constitutive law can be defined as a mathematical functional relation between physical quantities such as stress and strain and may take other factors like time ,temperature and additional material properties into account. In this paper , the endochronic model is used to predict the stress-strain relations of two Iraqi clays. This model is a viscoplastic one but without introducing a yield surface. It encompasses material behaviour such that the current stress state is a function of strain history through a time scale called “intrinsic time” which is not the absolute time but a material property. The simulation showed that the model overestimates the strains for all cases studied. This may be attributed to the material parameters which require a parametric study to determine their actual values for Iraqi clays.

Mroz' Local Response Moduli as a Means for Comparing Constitutive Equations in Plasticity

Linear and nonlinear relations of time independent plasticity are compared and their local response for varying stress or strain increment orientation is studied. The applicability of incremental relations to stability and post-critical response analysis can be assessed from such comparative study. In particular, the endochronic model providing nonlinear incremental relations is considered.

A combined viscoelastic–viscoplastic behavior of particle reinforced composites

This study formulates a micromechanical model for predicting effective viscoelastic–viscoplastic responses of composites. The studied composites consist of solid spherical particle reinforcements dispersed in a homogeneous matrix. The particle constituent is assumed linear elastic, while the matrix exhibits combined viscoelastic–viscoplastic responses. The Schapery integral model is used for the 3D isotropic non-linear viscoelastic responses. Two viscoplastic models are considered: the Perzyna model, having a rate-independent yield surface and an overstress function, and the Valanis endochronic model based on an irreversible thermodynamics without a yield surface. The Valanis model is suitable for materials when viscoplastic responses occur at early loadings (small stress levels). A unit-cell model with four particle and polymer sub-cells is generated to obtain homogenized responses of the particle-reinforced composites. Available micromechanical models and experimental data in the literature are used to verify the proposed micromechanical model in predicting effective time-dependent and inelastic responses of composites. Field variables in the homogenized composites are compared to the ones in heterogeneous composites. The heterogeneous composites, having detailed particle geometries, are modeled using finite element (FE) method.

Pseudopotentials and Loading Surfaces for an Endochronic Plasticity Theory with Isotropic Damage

The endochronic theory, developed in the early 70s, allows the plastic behavior of materials to be represented by introducing the notion of intrinsic time. With different viewpoints, several authors discussed the relationship between this theory and the classical theory of plasticity. Two major differences are the presence of plastic strains during unloading phases and the absence of an elastic domain. Later, the endochronic plasticity theory was modified in order to introduce the effect of damage. In the present paper, a basic endochronic model with isotropic damage is formulated starting from the postulate of strain equivalence. Unlike the previous similar analyses, in this presentation the formal tools chosen to formulate the model are those of convex analysis, often used in classical plasticity: namely pseudopotentials, indicator functions, subdifferentials, etc. As a result, the notion of loading surface for an endochronic model of plasticity with damage is investigated and an insightful comparison w...

A constitutive model for casting magnesium alloy ZL101 based on the analysis of spherical void evolution

Since casting magnesium alloys contain numerous spherical microvoids, the aggregate of microvoids and matrix can be analyzed using a representative volume element. The representative element can be idealized as a cell containing a spherical void. Through the analysis on the velocity field of the spherical void-cell model, the strain field of the spherical void-cell model was obtained. Defining an intrinsic time that involves the hardening due to plastic deformation and the softening due to voids, a new endochronic model was derived for the elastoplastic and damage behavior of casting magnesium alloys. The corresponding numerical algorithm and finite element procedure were developed and applied to the analysis of the elastoplastic response and the porosity of casting magnesium alloy ZL101. The computed results show satisfactory agreement with experiments.

Endochronic-Based Approach to the Failure of the Lower San Fernando Dam in 1971

The Lower San Fernando dam failure, which took place in 1971, is one of the most reported cases of seismic liquefaction damage in the geotechnical literature. For this reason, it has been analyzed by almost all of the numerical models developed since that year. In this paper, a comparison between numerical simulations, using a new endochronic model, with measured response of this dam during the earthquake of 1971, as well as numerical results previously obtained by other researchers, is presented. The main particularity of this new constitutive law is a nonassociative flow rule, related to a parameter quantifying degradation with shaking duration, in terms of stiffness reduction. It is incorporated in the model to represent soil dilation. Furthermore, contractive, dilative, and collapse trends of soil behavior are jointly embodied into the new developed constitutive law, which has been implemented in a two-dimensional coupled finite-element model. By so doing, the failure mechanism and the critical locations of the dam are identified and compared with field observations, and the approximate time for the beginning of the upstream sand fill slide is determined.

Endochronic damage constitutive model for fully-graded aggregate mass concrete

The behavior of deformation and strength of a fully-graded aggregate concrete under complex stress state is the basis of the nonlinear analysis and design for mass concrete structures such as concrete dams. In this paper, based on the combined endochronic theory with damage mechanics and on the quantities of experimental data, an endochronic damage constitutive model suitable for various aggregate grade concrete is proposed. This model takes into consideration the scale effect and the effect of wet screen sieve of aggregate in which the concept of yield surface is not needed and the difference of damage evolution rule of various graded aggregate concrete is a concern. The proposed model is used to analyze the deformation and strength of fully-graded aggregate mass concrete and the wet screened aggregate concrete specimens. The calculated results are in good agreement with experimental data, which can be used as a guide for the design of arch dams and other mass concrete structures.

Validation of a new endochronic liquefaction model for granular soil by using centrifuge test data

The application of a new liquefaction constitutive model, based on the endochronic theory applied to densification of sandy soil, to a set of centrifuge tests from the University of British Columbia, is presented in this paper. The model employed herein takes into account, in a unified formulation, contractive and dilative behaviours, and considers the soil collapse as well. First of all, the model is calibrated by means of undrained cyclic simple shear stress test data. The constitutive law of the soil is implemented in the bidimensional coupled finite element code CMLIQ (Cyclic Mobility and LIQuefaction), developed by the authors. Three centrifuge tests are analysed, the seismic loading and the geometry being the same for all of them, namely an improved slope with drain devices or denser soil at some locations. Comparisons between the data provided by the numerical model and the experimental measurements are shown, and, as a result, the accuracy of the model is explored and evaluated.

An endochronic model of material function and its application to plastic behavior of metals under asymmetric cyclic loadings

By using the incremental form of the endochronic theory of plasticity, a model of material function is proposed in this paper to investigate plastic behavior. By comparing the stress-strain hysteresis loop, the theory is shown to agree well with the experimental results, especially in the evolution of peak stress values of SAE 4340 steel loaded by cyclic loading with various amplitudes. Depending on the choice of material parameters, the present model can substantially result in six categories of material function, each of which can behave differently with respect to an identical deformation history. In addition, the present model of material function is shown to be capable of describing the behavior of erasure of memory of materials, as experimentally observed by Lamba and Sidebottom (1978).

THEORETICAL SIMULATION OF STRESS-STRAIN RELATIONS FOR SOME IRAQICLAYS USING THE ENDOCHRONIC MODEL

A constitutive law can be defined as a mathematical functional relation between physical quantitiessuch as stress and strain and may take other factors like time, temperature and additional materialproperties into account. In this paper, the endochronic model is used to predict the stress-strain relations of two Iraqi clays. This model is a viscoplastic one but without introducing a yield surface. It encompassesmaterial behaviour such that the current stress state is a function of strain history through a time scale called “intrinsic time” which is not the absolute time but a material property. The simulation showed that the model overestimates the strains for all cases studied. This may be attributed to the material parameters which require a parametric study to determine their actual values for Iraqi clays.

Creep damage modelling for quasi-brittle materials

Softening and time-dependence of fracture are two complex and coupled phenomena that have to be taken into account in order to simulate realistic concrete or rock behaviour. Understanding the interaction between these two phenomena is important to design reliable civil engineering structures subjected to high level loading during a long time. The aim of this paper is to develop a simple time-dependent softening model applied to quasi-brittle materials such as rock or concrete. A three-dimensional constitutive visco-damage model describes phenomena like relaxation, creep and rate-dependent loading using a unified framework. The model could be viewed as a generalisation of a time-independent damage model and is based on strong thermodynamical arguments. A general material stability analysis is proposed in case of uniaxial monotonic loading. Phenomena as creep failure under high-sustained load are explained quite simply within stability theory. Creep failure appears as the manifestation of a bifurcation phenomenon. Consequently, this model is able to predict creep failure for various stress level. Conversely, for low-sustained load, the motion asymptotically converges towards an equilibrium configuration, also called equilibrium curve. As for endochronic models, no initial threshold is assumed. Nevertheless, an apparent elastic domain is identified for constant strain rate tests, which reveals the competition between the internal kinetics of damage and the strain rate imposed on the material.

A hypoelasto-viscoplastic endochronic model for numerical simulation of shear band localization

In this paper, a hypoelasto-viscoplastic endochronic model is developed to capture the strain localization phenomena. The elastic response is stated in terms of hypoelastic model and endochronic constitutive equations are stated in unrotated frame of reference. The infinitesimal theory of endochronic plasticity is extended to large strain range on the basis of the additive decomposition of the strain rate tensor and hypoelasticity. Constitutive equations are stated in unrotated frame of reference that greatly simplifies endochronic constitutive relations in finite plasticity and yields the efficiency of the presented algorithm by total uncoupling material and geometrical nonlinearities. An implicit scheme is employed, which is the efficient method for the type of nonlinear problems considered here. Finally, the efficiency of the proposed constitutive model and computational algorithms is demonstrated by evolution of the shear band in a plane strain strip and vertical slope with strain softening endochronic plasticity.

An endochronic model of yield surface accounting for deformation induced anisotropy

Abstract In this article, an endochronic model of yield surface is proposed. Based on this model, the yield surface is simulated such that the forward and rear parts of the yield surface are described by different ellipses which are characterized by corresponding aspect ratio functions, respectively. Verification of the endochronic theory used the experimental results of yield surfaces obtained by Wu and Yeh for 304 stainless steel (Wu, H.C., Yeh, W.C., 1991. On the experimental determination of yield surfaces and some results of annealed 304 stainless steel. Int. J. Plasticity 7, 803–826). The experiments were performed cyclically under uniaxial, torsional, and combined axial–torsional loading conditions. The result has shown that the agreement between the prediction and experiments is quite satisfactory. In addition to the distortion of the yield surface plastically behaving a sharp front accompanied by a blunt rear, the anisotropic kinematic hardening effect has been addressed in this investigation. Although the experimental results of yield surfaces subjected to non-proportional loading conditions can be found in the literature, lack of information about the plastic strain history makes it impossible to verify the theory under such complicated loading conditions. The domain of applicability and validity of the theory, which is defined in terms of plastic strain increments, need be further investigated with the aim to set up related experiments.