Plastic Yield Surface Research Articles

Study on plastic yield surfaces of pyramidal lattice materials with circular and rectangular cross-sections

The yield surfaces and the evolution of yield surfaces of a bending-dominated pyramidal lattice material under complex stress states considering the coupling effect of axial force and bending moment of a strut are studied theoretically. The two-dimensional primary yield surfaces, secondary yield surfaces, and complete yield surfaces of pyramidal lattice materials of two kinds of cross-section struts are obtained, and the analytical expressions of yield surfaces are established. The effects of the inclination angle and diameter-to-length ratio of the struts on the yield surfaces of pyramidal lattice materials with circular cross-sections are discussed. The results provide a basis for analyzing homogenized plastic properties of complex lattice material structures.

Simulation of soil-structure interaction of integral abutment bridges using advanced constitutive relations

Integral bridges have been proposed as an attractive design concept that negate the requirements for expansion joints. However, owing to structural continuity, seasonal and diurnal thermal fluctuations have subjected the abutments to long-term cyclic interaction with approach backfills. This resulted in two notable geotechnical complications: ratcheting of lateral stresses and progressive soil deformation. To understand these phenomena, considerable research has been conducted using numerical methods. Here, either Winkler springs or simple constitutive relations, such as the Mohr-Coulomb model, are predominantly utilized. However, the former is based on monotonic displacement-dependent stiffness theories, and the latter is constrained by the limitations of the traditional yield surface plasticity. Hence, neither possesses the capability to simulate the hysteretic response, which is a characteristic of integral bridge approaches. Therefore, this study aims to present the advantages of using advanced constitutive relations to predict the long-term behavior of integral bridge approach backfills. First, using a lateral cyclic triaxial simulation, the significance of soil densification was highlighted by comparing the performance of the Mohr-Coulomb model with that of the bounding surface Dafalias-Manzari-2004 model. Subsequently, further comparisons were performed with the centrifuge model data of the integral abutment. At both the element and boundary value levels, the Mohr-Coulomb model could not capture the gradual accumulation of the lateral stresses. Conversely, the Dafalias-Manzari-2004 model can estimate the long-term passive pressure accumulation along the height of the abutment.

Anisotropic yielding model of lattice material and influencing factors

Experiments and numerical simulations were used to study and analyze the anisotropy of lattice materials, and the elastic-plastic properties of lattice materials were obtained. The equivalent continuum hypothesis was used to approximate the lattice material as a homogeneous continuous orthotropic material. The relevant equivalent yield parameters and plastic yield surfaces of the lattice materials were calculated using the Hill yield criterion. Comparisons of numerical simulations and experimental data show that they match very well. The effect of strut radius on anisotropic yield property and yield surface of lattice material was investigated.

A novel time-dependent damage constitutive model for hard rock materials

The mechanical behavior of deep hard rocks plays a crucial role in the long-term safety and stability of engineering structures. This work focuses on studying the instantaneous and time-dependent fracture propagation of hard rock materials from a theoretical perspective. A new unified constitutive model, based on a modified Mohr–Coulomb (M-C) criterion, is proposed to accurately represent the short and long-term mechanical properties of hard rocks. To capture the strain hardening and strain softening behaviors, damage is utilized as an internal mechanism-driven variable, controlling the expansion and contraction of the plastic yield surface. Additionally, a combination of time-dependent damage law and viscoplastic theory is employed to account for nonlinear creep deformation characteristics. By considering the time-dependent effects, the model can be applied to both instantaneous loading and creep conditions. The general algorithm format is derived in detail, and an explicit integration algorithm is utilized to update the time-dependent damage evolution. Finally, the proposed model is validated by comparing its predictions with the short and long-term mechanical responses of Beishan granite and Rumei dacite. This comprehensive constitutive model improves our understanding of continuum damage mechanics and provides a scientific basis for analyzing and evaluating the long-term safety and stability of deep hard rock engineering projects.

A thermodynamic-based model for modeling thermo-elastoplastic behaviors of saturated clayey soils considering bound water dehydration

The non-isothermal deformation of soft mudrocks or clay soils is one of the most critical issues in energy and environmental related geotechnics. Clay-related geomaterials hold complex microstructure and mineral composition, which brings difficulty in investigating their thermo-mechanical behaviors. Previous studies pay little attention to the difference between a thermal plastic strain and the strain from clay dehydration. In this study, a new constitutive model is proposed for describing the thermo-elastoplastic behaviors of clayey soils under water-saturated condition. The effect of temperature variation and mechanical loading on elastoplastic strains and dehydration are investigated. The thermodynamics laws and the unconventional plasticity are applied to quantify the thermo-mechanical behavior. The irreversible strain is captured by using Cam-Clay plasticity and subloading yield surface concept. The dehydration strain is described by utilizing a novel method based on generalized thermodynamics approach and Helmholtz free energy function. The internal variables, and the first and second laws of thermodynamics are applied in the model. The hardening rule is established by implementing the laws of physical conservation, energy dissipation, and plastic flow. The proposed model is validated using specially designed thermal consolidation tests on laboratory prepared heavily consolidated clayey soils and some published data of clayey soils with different geological origins.

A thermodynamically consistent finite strain phase field approach to ductile fracture considering multi-axial stress states

Phase field models for ductile fracture have gained significant attention in the last two decades due to their ability in implicitly tracking the nucleation and propagation of cracks. However, most crack phase field formulations for elastoplastic solids focus only on the effects of plastic deformation, and do not consider the different multi-axial stress states that may arise in practical designs. In this work, a thermodynamically consistent phase field approach coupled with finite strain plasticity, considering multi-axial stress states is presented. In order to account for the coupling between plasticity and stress states, the Stress-Weighted Ductile Fracture Model (SWDFM) is utilized. The SWDFM represents a criterion for predicting ductile crack initiation under both monotonic and cyclic loadings based on histories of an internal plastic variable, stress triaxiality, and the Lode angle parameter. The excellent performance of the SWDFM for predicting ductile crack initiation motivates for its incorporation into a phase field approach for predicting both crack initiation and propagation through degradation of the fracture toughness. Moreover, based on the second law of thermodynamics, exact requirements are imposed on the rate at which the fracture toughness can evolve. A novel function for degrading the plastic yield surface during the evolution of damage is introduced. This function, in line with experimental observations, leads to an accumulation of plastic deformation in damaged regions of a solid, and avoids numerical instabilities arising from concentrations of large plastic deformations in severely damaged regions. For validating the proposed model, results of computational simulations are compared to data from selected tests considering different multi-axial stress states. Comparisons of the numerical results with data from laboratory experiments demonstrate the capabilities of the proposed framework.

Damage-based strength reduction factor spectra of structures subjected to bidirectional ground motions

The current research on the damage-based strength reduction factor has been completely based on the inelastic single-degree-of-freedom systems that are more suitable for two-dimensional analyses. Therefore, it is necessary to construct damage-based strength reduction factor spectra for systems subjected to bidirectional ground motions. In this study, an inelastic single-mass bi-degree-of-freedom (SMBDOF) system subjected to the orthogonal bidirectional ground motions with the hysteretic property of two-dimensional yield-surface plasticity function is presented, and the x-component of the system is used to construct the damage-based strength reduction factor spectra of the system ( R b spectra). The Park-Ang damage model is used to determine the constant damage index of the R b spectra. In addition, 178 ground motion records for site classes C, D, and E are considered as ground excitations of the SMBDOF system to construct the mean R b spectra with the format of R b- D- T for a given ductility capacity and R b- μ u- T for a given damage level. The R b spectra of the SMBDOF system is compared with the corresponding R spectra of the SDOF system. Statistical analysis considering different values of the natural period, ductility capacity, damage index, period ratio, and site condition is conducted. Analysis results show that the coupling of two-component responses of an inelastic system could increase the strength demand of the system in certain cases; the trend of the R b spectra presented in this study is approximately consistent for different site classes, but the spectral values have the characteristic of site conditions. Therefore, the influence of the period ratio should be considered because a large period difference between two components of the SMBDOF system could aggravate seismic damage in certain situations. Based on the statistical analysis, the expression of R b spectra curves with a period ratio γ = 1 is derived using the regression analysis, and the correction expression is obtained considering the influence of the period ratio on the R b spectra. Then, the construction procedure of the R b spectra is introduced. The predicted R b spectra obtained by the proposed procedure is compared with the actual mean spectra, and comparison results indicate a good match.

A gradient micromorphic modeling for plasticity softening

Localization of internal quantities, which induce material softening, potentially leads to an ill-posed global partial differential equation. Along with further developments of singularity issues in the simulations, an unwanted divergent numerical solution may occur. The work at hand attempts to formulate a ductile softening model by considering a size reduction of the plastic yield surface. An internal variable to reduce the size of the plastic yield surface is constituted by a gradient micromorphic approach. In detail, a degradation function induced by an internal softening quantity is employed to multiply the thermodynamic force of plastic hardening in the yield function. In this regard, a gradually shrinking plastic yield surface is constituted to model the overall softening response. The evolution of the internal softening variable is governed by a Kuhn–Tucker condition together with its non-local extension. The present constitutive model of the coupled softening problem is derived based on a thermodynamically consistent algorithm from a well-defined Helmholtz free energy potential, which is implemented into the context of a conventional Finite Element Method. A representative and meaningful numerical example is studied to demonstrate the capability of the present model.

Discovering plasticity models without stress data

We propose an approach for data-driven automated discovery of material laws, which we call EUCLID (Efficient Unsupervised Constitutive Law Identification and Discovery), and we apply it here to the discovery of plasticity models, including arbitrarily shaped yield surfaces and isotropic and/or kinematic hardening laws. The approach is unsupervised, i.e., it requires no stress data but only full-field displacement and global force data; it delivers interpretable models, i.e., models that are embodied by parsimonious mathematical expressions discovered through sparse regression of a potentially large catalog of candidate functions; it is one-shot, i.e., discovery only needs one experiment. The material model library is constructed by expanding the yield function with a Fourier series, whereas isotropic and kinematic hardening is introduced by assuming a yield function dependency on internal history variables that evolve with the plastic deformation. For selecting the most relevant Fourier modes and identifying the hardening behavior, EUCLID employs physics knowledge, i.e., the optimization problem that governs the discovery enforces the equilibrium constraints in the bulk and at the loaded boundary of the domain. Sparsity promoting regularization is deployed to generate a set of solutions out of which a solution with low cost and high parsimony is automatically selected. Through virtual experiments, we demonstrate the ability of EUCLID to accurately discover several plastic yield surfaces and hardening mechanisms of different complexity.

Pure cross-anisotropy for geotechnical elastic potentials

The pure cross-anisotropy is understood as a special scaling of strain (or stress). The scaled tensor is used as an argument in the elastic stiffness (or compliance). Such anisotropy can be overlaid on the top of any elastic stiffness, in particular on one obtained from an elastic potential with its own stress-induced anisotropy. This superposition does not violate the Second Law. The method can be also applied to other functions like plastic potentials or yield surfaces, wherever some cross-anisotropy is desired. The pure cross-anisotropy is described by the sedimentation vector and at most two constants. Scaling with more than two purely anisotropic constants is shown impossible. The formulation was compared with experiments and alternative approaches. Static and dynamic calibration of the pure anisotropy is also discussed. Graphic representation of stiffness with the popular response envelopes requires some enhancement for anisotropy. Several examples are presented. All derivations and examples were accomplished using the algebra program Mathematica.

Equivalent continuum method of plane-stress dominated plate-lattice materials

An equivalent continuum method based on micro–macro transformations of strains and stresses is developed to analyze the effective mechanical performances of three-dimensional plane-stress dominated plate-lattice materials. The equivalent stiffness matrix of the plate-lattice material is deduced by the equivalent continuum method. Based on the two-dimensional Mises yield criterion of the micro-plate, the yield criterion, yield strength and three-dimensional plastic yield surfaces of the plate-lattice materials are deduced. Anisotropy of the stiffness and strength is discussed for six typical plate-lattice materials. The predicted stiffness and strength are in good agreement with the experimental data, validating the present model in the prediction of the mechanical properties of the plate-lattice materials. Through the research, it is found that the plane-stress dominated topology makes the plate-lattice material stiffer and stronger than stretching-dominated truss-lattice material when there is no elastic buckling in both structures.

A thermodynamic framework to predict ductile damage in thermoviscoplastic porous metals

In this work, we develop and study a ductile damage model for thermoviscoplastic metallic materials taking into account void growth, thermal softening and heat conduction. The novel feature of the study is the extension of a Gurson-type dilatational plasticity yield surface, to include the combined effect of void shearing, thermoviscoplasticity and thermal conductivity. Several complex phenomena, such as plastically-induced void growth and nucleation, coalescence, shearing void growth, strain hardening, strain-rate dependence, heat generation by plastic work and diffusive thermal softening, are considered. Numerical examples considering 2D plates are presented to validate the proposed formulation. The influence of the strain rate on the material response is investigated in detail and the results are shown to be consistent with data reported in the scientific literature. The present formulation can reproduce shear banding coupled with damage for a range of strain rates, including the stress collapse due to porosity growth and temperature rise. The importance of the proposed model is also highlighted to other well known simplified models.

Prediction of plastic yield surface for porous materials by a machine learning approach

The present paper focuses on the prediction of effective plastic yield surface of porous materials having a von Mises type solid matrix. Some typical explicit yield criteria obtained by different analytical homogenization methods are briefly reviewed and evaluated by using numerical results obtained from direct finite element simulations with different values of porosity. Each criterion has its own advantage and weakness. In order to get a better prediction, the Artificial Neuron Network (ANN) algorithm is adopted specially for the prediction of macroscopic yield stress of porous materials, seen as a regression problem with two input parameters and one output value. For the training purpose which is a key step in the ANN approach, new numerical results are presented in the present work with a wide range of porosity and of macroscopic stress triaxiality. Based on these data, the ANN approach is trained and it converges quickly. Then the ANN predictions are compared with numerical test data, a good agreement is found for all loading cases. Comparing with the existing yield criteria, the prediction given by the ANN approach is much more accuracy and easy to apply.

Granular viscosity from plastic yield surfaces: The role of the deformation type in granular flows

Numerical simulations of granular flows with Navier–Stokes type models emerged in the last decade, challenging the well established depth-averaged models. The structure of these equations allows for extension to general rheologies based on complex and realistic constitutive models. Substantial effort has been put into describing the effect of the shear rate, i.e. the magnitude of the velocity gradient, on the shear stress. Here we analyse the effect of the deformation type. We apply the theories of Mohr–Coulomb and Matsuoka–Nakai to calculate the stresses under different deformation types and compare results to the theory of Drucker–Prager, which is formulated independently of the deformation type. This model is particularly relevant because it is the basis for many granular rheologies, such as the μ ( I ) –rheology. All models have been implemented into the open-source toolkit OpenFOAM® for a practical application. We found that, within the context of these models, the deformation type has a large influence on the stress. However, for the geometries considered here, these differences are limited to specific zones which have little influence on the landslide kinematics. Finally we are able to give indicators on when the deformation type should be considered in modelling of landslides and when it can be neglected.

Collapse response of two-dimensional cellular solids by plasticity and cracking: application to wood

The competition between fracture and plasticity in periodic hexagonal honeycomb structures subjected to (i) intercell cracking, (ii) intrawall cracking and (iii) transwall cracking is examined, and their effect upon the macroscopic collapse response is explored using dedicated FEM analyses of unit cell configurations. These three cracking mechanisms are regularly observed in wood microstructures, and insight into their influence on the macroscopic collapse behavior is necessary for adequately designing timber structures against failure. The numerical results are presented by means of collapse contours in the hydrostatic-deviatoric stress space, illustrating the effects of wall slenderness, relative fracture (versus yield) strength, and the relative size of the plastic zone at the crack tip. Both the hydrostatic and deviatoric collapse strengths of the honeycomb strongly increase in the transition from brittle cell walls with low relative fracture strength to ductile cell walls with high relative fracture strength. This strength increase typically changes the shape of the collapse contour, and is the largest for transwall cracking, followed by intercell cracking and finally intrawall cracking. The ultimate collapse strength of the honeycomb is significantly more sensitive to the fracture strength than to the fracture toughness of the cell walls, and correctly approaches the plastic yield surface under increasing relative fracture strength. The numerical results may serve as a useful guideline in the experimental calibration of the local fracture and yield strengths of cell walls in wood.

Increment Dynamic Analysis for One-Story Asymmetric Plan Systems under Hard Soil Site

Considering the multi-dimensional feature of earthquake motions and the torsion response can aggravate destroying of structure, the one-storey asymmetric plan system with three degree of freedom is established. The yield rule of this asymmetric system is determined by two-dimensional yield-surface plasticity function. Bi-directional Earthquake motion records for hard soil site are selected as the excitations of this asymmetric system. The increment dynamic analysis for this system is conducted, and increment dynamic analysis capacity curves are given in which peak ground acceleration is used as earthquake intensity parameter, the ductility factor and the maximum rotation angle are used as performance parameters, respectively. The effect of eccentricity, frequency ratio on responses of the one-storey asymmetric plan system, are analyzed based on the two types of capacity curves.

Phase field fracture in elasto-plastic solids: Variational formulation for multi-surface plasticity and effects of plastic yield surfaces and hardening

The phase field modelling has been extended from brittle fracture to ductile fracture by incorporating plasticity. However, the effects of plastic yield functions and hardening on the fracture behaviour have not been examined systematically to date. The phase field fracture coupled with multi-surface plasticity is formulated in the variational framework for the unified yield criterion, which is able to facilitate the study on different yield surfaces. First, the homogeneous solutions of fracture in elasto-plastic solids are derived analytically for 1D and 2D cases. The results show that a greater hardening modulus would lead to an ascending branch of the stress versus strain curve; and the yield function may significantly affect the stress state and phase field damage. Second, the finite element (FE) technique is implemented for modelling the phase field fracture in elasto-plastic solids, in which the stress update and consistent tangent modular matrix are derived for the unified yield criterion. Finally, three numerical examples are presented to explore the effects of the yield function and material hardening. It is found that the yield function and material hardening could significantly affect the crack propagation and the final fracture pattern. In particular, the Tresca yield function tends to create a straight crack path orthogonal to the first principal stress, while the other yield functions show no sizeable difference in their crack paths.

A 3D mesoscale damage-plasticity approach for masonry structures under cyclic loading

This paper deals with the accurate modelling of unreinforced masonry (URM) behaviour using a 3D mesoscale description consisting of quadratic solid elements for masonry units combined with zero-thickness interface elements, the latter representing in a unified way the mortar and brick–mortar interfaces. A new constitutive model for the unified joint interfaces under cyclic loading is proposed. The model is based upon the combination of plasticity and damage. A multi-surface yield criterion in the stress domain governs the development of permanent plastic strains. Both strength and stiffness degradation are captured through the evolution of an anisotropic damage tensor, which is coupled to the plastic work produced. The restitution of normal stiffness in compression is taken into account by employing two separate damage variables for tension and compression in the normal direction. A simplified plastic yield surface is considered and the coupling of plasticity and damage is implemented in an efficient step by step approach for increased robustness. The computational cost of simulations performed using the mesoscale masonry description is reduced by employing a partitioning framework for parallel computation, which enables the application of the model at structural scale. Numerical results are compared against experimental data on realistic masonry components and structures subjected to monotonic and cyclic loading to show the ability of the proposed strategy to accurately capture the behaviour of URM under different types of loading.

A three-dimensional plasticity-damage constitutive model for timber under cyclic loads

The performance of timber structures is governed by the nonlinear response at their connections, where high deformation levels and stress concentrations are developed, particularly when subjected to load reversals. To date, no constitutive model for wood under cyclic load exists which is able to incorporate its most important failure modes while considering plastic deformations and cyclic stiffness and strength degradation simultaneously. This paper presents the formulation and implementation of a plasticity-damage model with these characteristics within a continuum mechanics approach. The theoretical framework of both plasticity and damage models is described, and a detailed derivation of the constitutive equations required for their computational implementation and coupling as well as the return mapping and iterative algorithms for their integration are presented. The damage evolution process is handled by two independent scalar variables for tension and compression. A general orthotropic plasticity yield surface with isotropic hardening is employed to incorporate timber plastic flow in compression. A closed-form expression for the plasticity-damage consistent tangent operator is derived. It is demonstrated that the proposed constitutive model captures all the key characteristics required for an accurate modelling of timber under large deformation levels until failure.

Investigation of the Optimum Height of Railway Embankments during Earthquake Based on Their Stability in Liquefaction

ABSTRACTOne of the most reported problems in the railway embankments is loss of their overall stability and major settlements due to liquefaction. Although several studies have been made on the stabilization of embankments, the effects of the influencing parameters on the stability of embankments during liquefaction have not been sufficiently investigated. The series of two dimensional plane strain finite element (FE) models were developed using multi yield surface plasticity model namely Prevost. The results were represented and discussed in terms of foundation excess pore pressure, embankment settlements and their failure modes. By analyzing the results, the optimum height of embankment which minimizes the settlement and failure cracks was derived.