Direction Of Plastic Strain Increment Research Articles

Non-coaxial plasticity of similar weakly cemented soft rock under directional shear stress path

The plastic flow behavior of soft rock exhibits non-coaxial features under complex stress paths, while traditional plasticity theories are ill-equipped to adequately represent this, which leads to the mechanism of soft rock failure still unclear. To investigate the evolution law of strain increments and non-coaxial characteristics of weakly cemented soft rock, the directional shear tests are conducted using the hollow cylinder apparatus (HCA). The results show that non-coaxiality does not occur when α is distinct from 0° or 90°. The oscillation of the non-coaxial angle is significantly more variable in soft rock experiencing combined tension–torsion (45° < α < 90°), as opposed to those under the influence of combined compression-torsion (0° < α < 45°). The non-coaxiality swiftly dissipates when the sample is approaching the failure state. The stress rate is decomposed into stress magnitude and direction to describe non-coaxial features of plastic strain. And a new method for non-coaxial stress rate is proposed which can express the plastic strain increment directions. The spherical interpolation coefficient method is utilized to describe the continuous change in non-coaxial plastic flow direction between tangential and normal directions of the yield surface. The non-coaxial parameter (Δ) is introduced to quantify the non-coaxial characteristics of soft rock and its validity is confirmed through test results. This method effectively captures the principal stress direction influence on non-coaxial behavior of soft rock and have significance for rock mechanics.

A numerical method for calculating the long-term settlement of three-dimensional subgrade under numerous traffic loads

Settlement problems caused by various traffic loads significantly impact infrastructure sustainability. Accurately predicting the long-term settlement of subgrade is crucial for improving infrastructure resilience. This study determined the element strain of soil using an explicit strain model, based on the bounding surface model and associated flow rules. Using the method of generalized shear strain equivalent, a cyclic cumulative factor is proposed for determining the magnitude of cumulative strain increment. The direction of plastic strain increment was determined using the bounding surface model combined with associated flow rules. A cumulative strain calculation method integrating the bounding surface and explicit strain models under general stress states was established. This computational method, combined with equivalent finite elements, can be used for calculating the long-term settlement of three-dimensional subgrade under numerous cyclic loads at general stress states through a user-defined subroutine provided by the commercial finite element modeling (FEM) software. The reliability of the method was verified through unit tests.

A state-dependent non-orthogonal elastoplastic constitutive model for sand

The state-dependent deformation behavior of sand is reinterpreted from the perspective of strain increment magnitude and direction by analyzing the existing test data. A state-dependent non-orthogonal elastoplastic constitutive model for sand is established by incorporating the state variable into plastic strain increment magnitude and direction, respectively. For the magnitude of plastic strain increment, an S-type isotropic compression hardening rule applicable in large stress ranges is proposed, which is extended into a new hardening parameter for general stress states. Furthermore, the state variable is introduced into the newly proposed hardening parameter to consider the state dependence. On the other hand, the non-orthogonal plastic flow rule is employed to determine the plastic strain increment direction. To consider its state dependence, the relationship between the state variable and the parameter controlling the plastic flow direction is established. The comparison with the test results of four types of sand demonstrates that the proposed model can reasonably capture the state-dependent deformation behaviors of plastic strain increment magnitude and direction of sand.

Integrating multiple samples into full-field optimization of yield criteria

Describing the anisotropic yielding behavior of sheet metal requires the determination of a yield criterion along with its model parameters. A method often used for this task is inverse full-field optimization. The method is typically combined with either multiaxial experiments or complex-shaped specimens to cover various stress states, over which the parameters of the yield criterion are optimized. This study investigates the feasibility of replacing the complex experiments with multiple simpler ones simultaneously. As material, the aluminum alloy AA6014-T4 is used. The yield criterion considered is YLD2000 2D, assuming both the associated and non-associated flow rules. Firstly, it is demonstrated through identifiability analysis that the model parameters, although often attempted in the literature, cannot be uniquely identified using only conventional experiments if non-associated flow is assumed. Secondly, it is shown that parameter identifiability in the latter case is achieved by combining multiple specimen geometries and orientations to rolling direction simultaneously. Thirdly, it is quantitatively shown that the use of multiple specimens simultaneously generally enhances the identifiability of model parameters due to an increase in the aggregated stress heterogeneity. Furthermore, the result indicates that while the full-field optimization of yield criteria assuming the associated flow rule can rely on force–displacement or direction of plastic strain increment alone, yield criteria assuming the non-associated flow rule require both. All yield criteria are benchmarked based on the strain distribution of Nakazima experiments, demonstrating comparable or higher accuracy compared to conventionally derived models.

Numerical Investigation of the Multiscale Characteristics of Elastoplasticity in Microheterogeneous Continuum Media Using a Simple Hollow Sphere Model

In this paper, we investigate the fundamental characteristics of plasticity in microheterogeneous continuum materials. Plasticity in such materials is a multiscale phenomenon with contrasting characteristics at different scales. A hollow sphere model, including a matrix embedding an oblate spheroidal pore, is considered as a sample representative elementary volume (REV) of a microheterogeneous material where, at the local level (microscale), a simple Drucker–Prager plasticity model is assumed for the matrix obeying the classical continuum theory of plasticity. The plasticity model is Lode angle independent with associated plastic flow and no hardening. We perform finite element numerical simulations on the REV to investigate the characteristics of the homogenized plasticity of the REV. It is seen that the homogenized plasticity shows complex characteristics such as Lode angle dependency, presence of cap, hardening, and nonassociativity. Well-established as these aspects are in the literature, they are often presumed in the plasticity models of heterogeneous materials in an ad hoc form. In contrast, we present here a systematic and lucid way to capture these aspects as emergent phenomena of heterogeneous microstructures. This has crucial implications in guiding the theoretical constitutive developments for elastoplastic materials toward adopting a multiscale approach that properly considers the effect of microstructure and naturally captures these aspects without the need for ad hoc assumptions. A fundamental assumption in the classical theory of plasticity is the plastic flow rule postulate, which states that the direction of plastic strain increment is governed only by the current state variables, independent of the loading increment direction. We show, through a stress probing exercise in 3D principal stress space, that the flow rule, assumed for the microscale plasticity, is not valid for the macroscopic plasticity. Indeed, the homogenized plastic flow direction in heterogeneous materials is governed by the current state variables and loading increment direction. Finally, we show that the plastic response is also a function of the stress path taken prior to loading. As such, nonclassical characteristics are observed for the plasticity of microheterogeneous materials. These nonclassical characteristics for the simple REV considered in this work (the hollow sphere model) align with some of the results observed in experiments or only reported for microdiscrete materials; herein, we show they are also valid for a broader range of materials endowed with microstructure, i.e., microcontinuous materials, and, as such, the fundamental assumptions in the classical theory of plasticity require revision before being applicable to the plastic behavior of heterogeneous materials.

A cohesion-friction combined hardening plastic model of concrete with the nonorthogonal flow rule: Theory and numerical implementation

A cohesion-friction combined hardening model of concrete was developed. Combined with the Mohr-Coulomb strength criterion, the yield curve is determined directly by the test data of concrete. Then, a method to determine the hardening law from the evolutionary process of the yield curve is elaborated. To consider the influence of the multiaxial stress state on the ductility of concrete, a stress dependent plastic internal variable is proposed. Based on the nonorthogonal flow rule implemented by fractional order derivative, the direction of plastic strain increment is directly determined by the nonorthogonal gradient of the yield function without the need for plastic potential function. The comparison with the test data shows that the proposed model can capture the strength and deformation characteristics of concrete reasonably. Finally, the model is implemented into ABAQUS via an open-source user defined material subroutine UMAT (https://github.com/zhouxin615/CFCH_model.git) based on the implicit return mapping algorithm. In particular, the integral-type nonlocal approach was adopted to regularise FE solution. Moreover, through the structural analysis of reinforced concrete columns, the operation process of the friction and cohesion mechanism of concrete materials is well reproduced.

Deformation simulations of sand under Bi-directional shear loading

A well-developed sand plasticity model is further revised and validated based on bi-directional simple shear test data on medium to high relative density sands, which includes circular, oval and figure-8 type stress paths at different cyclic stress ratio (CSR) levels. The sand properties revealed from non-proportional loading tests, such as extended contractive behavior for medium to dense sands, flow rule dependency on plastic strain increment direction, stress increment direction dependency of plastic moduli, as well as the so-called non-coaxial property are incorporated into the new model development. Specifically, the recently proposed strain increment direction dependent flow rule is formulated and validated using bi-directional simple shear test results; and the relatively simple plastic moduli revised for the non-proportional loading are validated.After the model parameters are calibrated using a circular stress path, predictions are made for oval and figure-8 type stress paths at two cyclic stress levels. The modeling results show that the computed deformations up to a large strain state (i.e., 6% double amplitude strain) match fairly well with the test results; and the predicted pore-water pressure generation (or the effective stress change) followed the trend of test measurements including the repeating contractive and dilative cycles near the failure surface. A fabric dilatancy concept-based stiffness reduction approach is adopted that is active for the simulations of oval and figure-8 type stress paths.By comparing with test results, the model predictions demonstrate the capability of the present model in simulating the complex deformation magnitudes and patterns for sands under bi-directional loadings.

Analysis of Factors Affecting the Direction of Plastic Strain Increment in Sand

Taking the standard sand of Fujian as the test material, this paper concentrates on studying the influence of different stress increment directions on the direction of plastic strain increment of sand materials under different stress states and the underlying mechanism. The test results show that the plastic strain increment angle rotates counterclockwise with the increase of stress increment direction angle, but the two angles do not coincide; the higher the stress state parameters is, the larger the stress increment direction angle range corresponding to sand dilatation is, the smaller the plastic strain increment direction angle range is, and the plastic volumetric strain still increases in critical state.The plastic flow mechanism of sand is explained from the average stress increment, generalized shear stress increment and stress state, which may provide theoretical reference and numerical support for the related research of plastic strain increment direction of sand.

A 3D non-orthogonal plastic damage model for concrete

A plastic damage model with the non-orthogonal flow rule is developed, which is consisted of the damage part driven by plastic strain and the plastic part based on effective stress. In the damage characterization, the degeneration law of elastic stiffness is described by the damage variable in the form of exponential function. In the plastic characterization, the evolution for plastic strain increment (PSI) is characterized from the magnitude and direction perspective. The direction of PSI is obtained directly by the fractional gradient of yield surface and the need for the construction of plastic potential function is bypassed. The magnitude of PSI is calculated based on the effective hardening function and the yield function, where the effective hardening function corresponding to the undamaged material is derived by both the normal hardening/softening function corresponding to the damaged material and the damage variable. The Next Increment Corrects Error (NICE) algorithm integrating the computational accuracy and efficiency is used for the numerical implementation of model. The ability of model to capture the complex mechanical behavior of concrete is assessed by the monotonous and cyclic test data from the literature.

Implicit integration of incrementally non-linear, zero- elastic range, bounding surface plasticity

The zero elastic range, bounding surface plasticity framework is a suitable choice for modeling materials which exhibit zero purely elastic response during shearing. As a consequence of zero elastic range the plastic strain increment direction, and consequently the elastic-plastic moduli fourth order tensor depends on the direction of the stress increment, rendering the model incrementally non-linear. The system of ordinary differential equations of the elastic-plastic formulation is intrinsically implicit. Thus, an iterative algorithm based on the Backward Euler numerical integration method and the damped Newton-Raphson method with an adaptive trial step is proposed in this work. The proposed methodology is applied to the zero elastic range SANISAND-Z model for sands and a thorough verification is done for various applied strain histories. The proposed integration scheme allows the use of SANISAND-Z or any other bounding surface zero elastic range model in displacement-driven finite element analysis.

A 3D fractional elastoplastic constitutive model for concrete material

The core mission to establish elastoplastic constitutive model is how to determine the magnitude and direction of plastic strain increment for material under the given load. Fractional derivative provides a powerful tool for determining the nonorthogonal direction of a certain differentiable function. A three-dimensional (3D) fractional plastic flow rule suitable for concrete material is presented for the first time utilizing fractional derivative. The direction of plastic strain increment is determined by the proposed fractional plastic flow rule. A closed form yield function is developed in the transformed stress (TS) space based on the nonlinear unified strength criterion, and further used to describe the plastic strain induced by hydrostatic loading. A united hardening/softening function proposed by the authors in previous works is adopted to describe the pre-peak hardening and post-peak softening behaviour of concrete. Furthermore, a 3D fractional elastoplastic constitutive model for concrete material is presented. Only nine material parameters with the clear physical meaning are need for the proposed model, and the determination method of each material parameter is given. A series of test results of concrete with different strength under biaxial and triaxial stress conditions are used to assess the performance of proposed model.

Influence of the intermediate principal stress and principal stress direction on the mechanical behavior of cohesionless soils using the discrete element method

In this paper, the Discrete Element Method (DEM) is employed to numerically explore the response of hollow cylinder specimens of granular soils under complex stress paths. Two series of numerical tests are conducted to clarify the effects of the principal stress direction α and the intermediate principal stress through the b-value on the mechanical response of granular materials. The effects of α and b-value on the non-coaxiality of the principal stress and the principal plastic strain increment directions are investigated. It is observed that b-value and α significantly affect the non-coaxial behavior of granular materials. Finally, the results are discussed and compared with those obtained from physical laboratory tests.

Noncoaxial Behavior of a Highly Angular Granular Material Subjected to Stress Variations in Simple Vertical Excavation

AbstractExperiments were carried out using a dynamic hollow-cylinder apparatus (HCA) to investigate the effects of stress paths and principal stress rotations experienced during a simple vertical excavation on the element-scale deformation and noncoaxial behavior of a highly angular granular material. For this purpose, an improved specimen preparation technique was used to generate homogeneous HCA specimens using a multilayer undercompaction method. The stress paths used in the HCA experiments were extracted from the numerical results obtained in a discrete element simulation of a simple vertical excavation, where complex stress paths and principal stress rotations have been observed. The experimental results show that different stress paths and rotations in excavation significantly affect the element-scale deformation behavior. Noncoaxial behavior (i.e., the deviation of plastic strain increment direction from the principal stress direction) was observed in the HCA tests, which was a result of the combin...

Undrained anisotropy and non-coaxial behavior of clayey soil under principal stress rotation

In this study, a series of undrained tests were conducted on both intact and reconstituted clay using an automatic hollow cylinder apparatus. Monotonic shearing tests with fixed principal stress directions were carried out, pure and cyclic principal stress rotation tests were also performed. The non-coaxiality, defined as the non-coincidence of the principal plastic strain increment direction and the corresponding principal stress direction, of clayey soil was studied experimentally. The effects of the intermediate principal stress, shear stress level, and inherent anisotropy were highlighted. Clear non-coaxiality was observed during pure principal stress rotation, in both intact and reconstituted clay. The influence of the intermediate principal stress parameter, shear stress level, and inherent anisotropy on the non-coaxial behavior of the clayey soil was found to be insignificant when compared with the sand. The non-coaxial behavior of the clayey soil depended more on the stress paths. Under undrained conditions, the contribution of elastic strain to the direction of the total principal strain increment cannot be ignored.

On the validity of the flow rule postulate for geomaterials

SUMMARYThis paper is concerned with a fundamental assumption in the theory of plasticity: the direction of plastic strain increments is independent of the loading (stress) increment direction. This assumption, also known as plastic flow rule postulate, works quite well for metal‐like materials. However, geomaterials such as sand present deformational mechanisms that are distinctive from those of metals when they are loaded. As such, we hereby examine the validity of this postulate for granular media accounting for their discrete nature. This is accomplished by analysing the mechanical behaviour of a cubic assembly of polydispersed spherical articles using a particle flow code. An extension to Gudehus' response envelope to three‐dimensional conditions is used to study the incremental character and influence of loading direction on the behaviour of these materials. It is found that plastic flow in granular media is governed by both current state variables and incremental loading direction and magnitude, especially under non‐axisymmetric stress conditions. The flow rule postulate of plasticity remains valid only in axisymmetric and biaxial conditions. We also verified that the plastic response might be significantly influenced by the stress path (or history) taken prior to loading. These findings raise the question of whether or not classic elastoplastic models based on the above postulate will have serious shortcomings, especially in true‐triaxial conditions. Copyright © 2013 John Wiley &amp; Sons, Ltd.

Non-Coaxial Plasticity Constitutive Modeling of Sands

Classical coaxial plasticity constitutive models implicate an inevitable limitation that directions for principal stress and that for principal plastic strain increment are always coaxial. They are not capable of simulating non-coaxial phenomena during the rotation of principal stress axis. In this paper, a three-dimensional, non-coaxial plasticity constitutive model for sands with a modification of Lade angle dependent shape function is introduced to describe the non-coaxial behavior under principal axes rotation. A series of numerical simulations of hollow cylindrical torsional shear tests are performed. The results show that the proposed constitutive model can predict the variations of principal plastic strain increment directions with principal stress directions reasonably.

Forming Limit of AZ31 Magnesium Alloy Sheet under Non-Proportional Deformation at Elevated Temperature

In the present work, FLDs of AZ31 magnesium alloy sheet for non-proportional strain paths were investigated by performing two-step stretch forming experiments at various forming speeds (3, 30 and 300 mm.min-1) at elevated temperatures of 150, 200 and 250°C. The forming limit strainSuperscript texts, both for proportional and non-proportional deformations, increased with temperature rise and with decreasing forming speed. A FLC after a uniaxial pre-strain lies outside of the proportional FLC for a given condition of temperature and forming speed, whereas a biaxially pre-strained FLC lies inside of the proportional FLC. It was found that the accumulated effective plastic strain and the direction of plastic strain increment at the final stage of forming are two major factors that influence the forming limits for non-proportional deformations.

Yielding Characteristic and Non-Coaxiality of Toyoura Sand on p′-Constant Shear Stress Plane

ABSTRACT In order to investigate the deformation characteristics under a wide selection of stress history, ten series of stress probing tests on p′-constant shear stress plane on dense Toyoura sand are conducted by using hollow cylinder apparatus. This paper presents yielding behavior and non-coaxiality of Toyoura sand obtained from the tests. The stress probing tests consist of ten series of shearing tests each of which starts from systematically chosen individual initial stress point. Each initial stress point is subjected to load-unload stress history. The yielding characteristic is interpreted by means of a concept of multiple yield surfaces model which has three yield surfaces representing linearly elastic behavior limit, commencement point of rapid development in plastic strain and elasto-plastic range. As a result, an experimental evidence of the isotropic hardening, i.e., isotropic expansion of yield surface, induced by anisotropic loadings is obtained. This is what was tacitly assumed in the application of plastic flow rules. However, misalignment of directions between principal stress and principal plastic strain increment is observed. Moreover, plastic strain increment direction is found to be influenced by the given stress increment direction. These facts suggest the non-coaxiality between stress and plastic strain increment due to shear stress increment, even during monotonic loading with isotropic hardening. The plastic strain increment direction diverges from the normal direction to a surface which is a circle with its center nearly at the origin on p′-constant plane and which passes through the current stress point.

Interpretation of strains in torsion shear tests

The normal and shear strains obtained in torsion shear tests may be interpreted in two different ways to gain insight into (1) the type of plastic potential to be employed in hardening plasticity stress–strain models, and (2) the coincidence in physical space of the plastic strain increment direction with the stress direction during principal stress rotation. Thirty-four drained torsion shear tests were performed on hollow cylinder specimens of Santa Monica Beach sand deposited by dry pluviation. Twenty-six tests were performed on tall specimens with height of 40 cm, and eight tests were performed on short specimens with height of 25 cm to investigate the effect of the specimen height on the soil behavior in hollow cylinder specimens. Each test was conducted with the same, constant inside and outside confining pressure, σ r , thus tying the value of b = ( σ 2– σ 3)/( σ 1– σ 3) to the inclination, β, of the major principal stress. The directions of strain increment vectors at failure are compared with the directions corresponding to associated and non-associated flow. The relation between the directions of major principal strain increment and major principal stress during rotation of principal stress axes in physical space are investigated.

Non-Coaxiality and Energy Dissipation in Granular Materials

The paper presents a theoretical and experimental study of the effects of non-coaxiality or non-coincidence of the principal stress and the principal plastic strain increment directions on the behaviour of granular materials. Experimental results from hollow cylindrical tests on sand involving principal stress rotation which support previously published results on non-coaxiality are presented. These results imply that constitutive relations cannot be sufficiently formulated in the principal stress space unless the deviations between the principal stress and plastic strain increment directions are taken into consideration. It is shown that plasticity formulations with plastic potentials that are scalar functions of the stress invariants alone implicitly assume coaxiality and cannot be used for loading involving principal stress rotation. The paper presents a comprehensive analysis of the effects of non-coaxiality on the energy dissipation of sand. The paper shows that energy dissipation calculated from the principal stresses and the principal plastic strain increments or from the stress and plastic strain increment invariants, would be erroneous and would over-estimate the amount of dissipated energy during loading in the case of non-coaxial flow. A non-coaxiality factor is introduced in order to account for the effects of non-coaxiality on the energy dissipation equation and in a stress-dilatancy relation for granular materials. Explicit expressions of the non-coaxiality factor for two-and three-dimensional loading conditions are given at the end of the paper. Experimental results are presented to show the validity of the proposed energy dissipation and stress-dilatancy equations.