Multi-surface Plasticity Research Articles

A comprehensive implicit substepping integration scheme for multisurface plasticity

AbstractA complex elastoplastic model requires a robust integration procedure of the evolution equations. The performance of the finite element solution is directly affected by the convergence characteristics of the state‐update procedure. Thereby, this study proposes a comprehensive numerical integration scheme to deal with generic multisurface plasticity models. This algorithm is based on the backward Euler method aiming at accuracy and stability, and on the Newton–Raphson method to solve the unconstrained optimization problem. In this scenario, a line search strategy is adopted to improve the convergence characteristics of the algorithm. The golden section method, an exact line search, is considered. Also, a substepping scheme is implemented to provide additional robustness to the state‐update procedure. Therefore, this work contributes to computational plasticity presenting an adaptive substep size scheme and a consistent tangent modulus according to the substepping technique. Finally, some numerical problems are evaluated using the proposed algorithm. Single‐surface and novel multisurface plasticity models are employed in these analyses. The results testify how the line search and substepping strategies can improve the robustness of the nonlinear analysis.

A sub-stepping iterative constitutive model for cyclic cracking-crushing-shearing in masonry interface elements

This paper proposes a sub-step based iterative constitutive model for line interface elements used to analyse masonry structures loaded in-plane. Based on a total deformation theory, the model adopts characteristics of multi-surface plasticity, including a Coulomb friction failure surface for shear, with tension and compression cut-off and softening for all three domains. The model is driven by two damage parameters, one for compression and one that couples tension and shear. The sub-stepping technique is demonstrated to be numerically stable and is used as an alternative to the traditional return-mapping algorithms, which are prone to convergence issues and instability. The proposed model has been validated against experimental tests performed on masonry walls subjected to cyclic, in-plane loading. The numerical simulations adequately identify the failure mode, the hysteretic behaviour and the crack pattern. When toe crushing is governing, the results appear to be sensitive to the assumed masonry compressive strength. It is shown that calibration of the lumped compressive strength makes possible to fully describe the damage evolution in walls that exhibit a mix of flexural crack-crush failure and shear failure. Overall, the model is demonstrated to be an efficient and robust tool for analysing the cyclic, in-plane behaviour of masonry walls.

Mechanical characterisation for numerical simulation of extrusion-based 3D concrete printing

Extrusion-based 3DCP is experiencing exponential advancement in process, control, material, and fresh-state analysis technologies, strategically poising 3DCP to become an industrial manufacturing process for infrastructure development. To elucidate the mechanical characterisation and numerical simulation of 3DCP elements synergy among experimental and computational mechanics is required. Such activities should characterise the unique material properties of printed components to permit limit state design procedures and the correct prescription of finite element (FE) model parameters. In this research, mechanical characterisation procedures that experimentally evaluate the anisotropic material characteristics of a fibre-reinforced printable concrete (FRPC) are presented. These procedures are comprised of direct tensile, uniaxial compression, Young's modulus, and four-point bending (FPB) crack mouth opening displacement (CMOD) tests. The results portrayed anisotropic non-linear and similar elastic behaviour (in terms of Young's modulus) in all experimental tests conducted. From the mechanical characterisation tests two elastic parameters, seven strength parameters and five inelastic parameters are ascertained. The experimental findings relating to the material characteristic parameters are validated via supplementary numerical evaluation, and suitable constitutive relations are selected. The mechanical parameters are implemented in an anisotropic Rankine-Hill continuum multi-surface plasticity FE model, and the FPB-CMOD fracture response of 3DCP specimens is simulate with respectable agreement.

Loss of ellipticity analysis in non-smooth plasticity

This work is concerned with the influence of non-smooth yield functions on material instabilities. The loss of ellipticity condition is examined in the case of a family of non-smooth yield functions in principal stress space. A procedure for the numerical detection of loss of ellipticity in multisurface plasticity is proposed. An explicit expression of the subdifferential of the non-smooth yield function is obtained, thereby extending existing results in the literature for the Tresca and Mohr–Coulomb criteria. Structural computations show that the use of non-smooth yield functions can lead to much earlier failure prediction than in the case of a commonly used von Mises criterion.

Numerical Investigation of Reinforced Concrete Beam Due to Shear Failure

This paper investigates the possibility of using a multi-surface plasticity model to predict shear failure in reinforced concrete beams. The analysis is carried out using the in-house software called 3D-NLFEA. The constitutive model for the concrete material is based on the plasticity-fracture model, which had previously developed to simulate the behavior of concrete cover spalling in reinforced concrete columns. To obtain the asymmetric shear failure pattern, random material properties imperfection for each meshed element is used. Two beams available in the literature are investigated and compared with the analysis results using 3D-NLFEA. From the comparisons, excellent agreement between the analysis and the test result was obtained. 3D-NLFEA can predict the peak load accurately. The peak load prediction only varies 2.19% for beam OA1 and 3.28 % for beam OA2, and it was lower than the test results. The failure crack patterns also show a typical diagonal crack extension from the support to the loading steel plate.

An inertial macroelement for bridge abutments

In recent years, the designers of long girder bridges in seismic areas have frequently opted for a continuous deck. One implication of this choice is that in many instances bridge abutments are called upon to carry large seismic forces, engaging the dynamic response of the soil–abutment system. To deal with this problem, this paper describes the formulation of a novel one-dimensional, inertial macroelement for simulating the dynamic behaviour of bridge abutments. The non-linear force–displacement relationship is characterised by a multi-surface plasticity model using a rigorous thermodynamic approach. The plastic response of the model is bounded by the ultimate capacity of the soil–abutment system that includes dissymmetry of the soil response in active and passive loading directions, while inertial effects transferred by the near-field approach embankment are simulated through appropriate participating masses in the macroelement formulation. The paper describes a straightforward calibration procedure of the proposed macroelement for horizontal, longitudinal loading of the abutment. The macroelement has been incorporated into a simplified, global, finite-element model of a multi-span girder bridge and validated through comparisons with results from a full three-dimensional (3D) dynamic time domain analysis under seismic loading. The inertial macroelement predictions of abutment deformations, axial deck loads and pier reaction forces are in very good agreement with the 3D soil–structure interaction model, and are achieved at much lower computational costs. The proposed inertial macroelement represents a significant improvement over existing simplified models based on linear response of the soil–abutment system.

A multisurface kinematic hardening model for the behavior of clays under combined static and undrained cyclic loading

AbstractIn dynamic geotechnical problems, soils are often subjected to a combination of sustained static and fast cyclic loading. Under such loading conditions, saturated and normally consolidated clays generally experience a build‐up of excess pore water pressure along with a degradation of stiffness and strength. If the strength of the soil falls below the static stress demand, a self‐driven failure is triggered. In this paper, a constitutive model is presented for the analysis of such problems, based on a general multisurface plasticity framework. The hardening behavior, the initial arrangement of the surfaces, and the nonassociated volumetric flow rule are defined to capture important aspects of cyclic clay behavior. This includes nonlinear hysteretic stress‐strain behavior, the effect of anisotropic consolidation, and the generation of excess pore water pressure during undrained cyclic loading along with a degradation of stiffness and strength. The model requires nine independent parameters, which can be derived from standard laboratory tests. A customized experimental program has been performed to validate the model performance. The model predictions show a good agreement with test results from monotonic and cyclic undrained triaxial tests, in particular with respect to the strain‐softening response and the number of loading cycles to failure. A procedure for a general stress‐space implicit numerical implementation for undrained, total stress‐based finite element analyses is presented, including the derivation of the consistent tangent operator. Finally, a simulation of the seismic response of a submarine slope is shown to illustrate a possible application of the presented model.

LOCALIZATION ANALYSIS OF AN ORTHOTROPIC MULTI-SURFACE PLASTICITY MODEL UNDER UNIAXIAL STRESS

Numerical simulations of masonry structures are often based on continuum macromodelling approaches that need constitutive laws able to phenomenologically reproduce the behavior of the material. To capture the deformation process up to failure, appropriate softening laws are needed to take into account the contraction of the yield stress domain caused by cracking and crushing. It is well known that softening may lead to localization of inelastic strain. This paper focuses on localization analysis of an orthotropic macro-scale model in the framework of multi-surface plasticity, which describes the in-plane behavior of masonry structures. Preliminary results reported in this short paper are limited to uniaxial stress states. Analytical localization conditions are first derived for uniaxial stress states with principal axes aligned with the material axes of orthotropy. Then, localization analysis is extended to an arbitrary angle between the principal stress axes and the axes of orthotropy.

Modeling refractory concrete lining of fluid catalytic cracking units of oil refineries

This work presents a numerical modeling procedure to simulate the refractory concrete lining in fluid catalytic cracking units of oil refineries. The model includes the simulation of the anchors that reinforce the contact between the refractory concrete and the steel casing. For this purpose, the constitutive relations of an interface finite element are set to values that represent the homogenized behavior of the anchored interface. The parameters of this constitutive relation can be obtained by experimental tests. The model includes also multi-surface plasticity, in order to represent the behavior of the refractory concrete lining. Since the complexity of real case applications leads to high computational costs, the models presented here were implemented in a high-performance parallelized finite element platform. A case study representing a riser similar to the ones used by the refinery industry demonstrates the potential of the model.

Rigid block and finite element analysis of settlement-induced failure mechanisms in historic masonry wall panels

The paper is related to the assessment of collapse mechanisms of historic masonry structures suffering settlements induced by ground movements. Two numerical strategies are adopted in order to study the influence of the settled zone on the cracking of masonry buildings: a discrete rigid block model and a continuous homogenized model. The first approach provides an estimate of the collapse load and failure pattern of masonry based on the lower bound theorem of limit analysis. The second approach is formulated in the framework of multi-surface plasticity and implemented in a FE code for the path-following non-linear analysis of masonry wall described as continuous anisotropic plate. Several settlement configurations, of masonry walls under moving ground support are investigated and the corresponding failure patterns resulting from the analysis are obtained resulting in local or global failure modes. The results of the two modeling formulations are compared and discussed in order to highlight the features of the two different approaches in the prediction of settlement-induced damage.

A plastic model based on the bounding surface formulation for cyclic behavior of soil

Cyclic behavior of saturated soil under cyclic loading, such as earthquake, is one significant cause of the strain accumulation which can lead to the foundation destruction. This paper presents the relationship between hardening rules in the multi-surface plasticity and hardening function in the bounding surface plasticity, because both of them need to calculate the plastic modulus at every current stress state, applying the newly developed hardening rule into the hardening function to calculate the variable plastic modulus, this new idea adds the adversity of plasticity model, and a new bounding surface is established. The simulation result of undrained triaxial test verified the reliability of the new bounding surface model.

A novel algorithm for rate independent small strain crystal plasticity based on the infeasible primal-dual interior point method

Single crystal plasticity plays a major role in the analysis of material anisotropy and texture evolution, treats each crystalline grain individually. The polycrystalline material response is obtained upon considering a structure consisting of various individual grains, often also considering interface effects at the grain boundaries. On the individual grain level, single crystal plasticity can be treated in the mathematical framework of multi-surface plasticity, leading to a constrained optimization problem wherein multiple constraints are defined as yield criteria on the different slip systems. In this work, we present a new algorithm for the solution of the constrained optimization problem based on the Infeasible Primal Dual Interior Point method (IPDIPM). The main motivation herein is the handling of the ill-posed problem without the use of simple perturbation technique, see e.g. Miehe and Schröder [2001]. The proposed algorithm, involving slack variables, is developed for the framework of small strain single crystal plasticity. The use of slack variables therein stabilizes the conventional method and allows for a temporary violation of the constraint condition during the optimization. Moreover, all slip systems are considered simultaneously, omitting an iterative active set search. Several numerical examples are simulated to show the performance of the developed algorithm.

A three-invariant cap-viscoplastic rate-dependent-damage model for cementitious materials with return mapping integration in Haigh-Westergaard coordinate space

We present a new strain-rate dependent elasto-viscoplastic damage constitutive model for cementitious materials by incorporating Duvaut-Lions viscoplasticity generalized to multi-surface plasticity (c.f. Simo et al., 1988b; Simo and Hughes, 1998) followed by strain-rate-dependent damage initiation and evolution under multiaxial loading, to our previous inviscid elastoplastic-damage model (Paliwal et al., 2017). We also present a unique numerical scheme in which the integration algorithm for modeling elastoplastic response utilizes fully implicit, backward Euler method in which the governing system of non-linear equations are solved in the space of invariants based on Haigh-Westergaard coordinate system, hardening parameter and discretized plastic multiplier. It uses the framework by (Matzenmiller and Taylor, 1994), which is modified in this work. This is particularly useful as the yield criteria, flow rule, and the non-linear hardening utilize functions that are dependent on the pressure, effective shear stress, and Lode angle components of the stress tensor. To the best of authors’ knowledge, such integration algorithm in Haigh-Westergaard coordinate space has never been developed for Lode angle dependent yield criteria and the flow rule, and non-linear hardening plasticity. Furthermore, we also develop an accurate form for the algorithmic elastoplastic-damage tangent modulus consistent with the proposed return mapping technique. Numerical simulations of uniaxial tension and compression cases demonstrate that both the local integration point algorithm as well as the global algorithm using the algorithmic elastoplastic-damage tangent modulus exhibit quadratic convergence. Finally, the predictive capability of the proposed model is demonstrated by virtue of several numerical examples under various loading conditions covering different stress paths and different dynamic strain-rates. These predictions are also compared against experimental results and experimentally observed features from tests on various concrete specimens, and demonstrate excellent agreement.

Coupled Horizontal and Vertical Component Analysis of Strong Ground Motions for Soil–Pile–Superstructure Systems: Application to a Bridge Pier with Soil–Structure Interaction

ABSTRACT This study investigates the influence of vertical components (VCs) of seismic excitations on the response of short and tall (slender) superstructures, and compares them with the system response under only horizontal components (HCs). The first part of the study examines the capability of the bounding multi-surface plasticity model to identify dynamic soil properties and soil–pile–superstructure system response to the HCs of a strong ground motion using existing results from dynamic centrifuge tests. The influence of VCs on short and tall superstructures is then investigated, and compared with system response under only HCs. A procedure is proposed to select strong vertical ground motion to complete a nonlinear time-history analysis with a three-dimensional finite element model. Step-by-step modelling techniques are described with a direct-method framework to simulate complex soil–pile–superstructure systems. This study emphasises the role of VCs in the general response of the system in regard to selected damage parameters for pile foundations. The results indicate that VCs may have increasing effects on axial loads and maximum pile-cap displacements for short superstructures. In addition, contrary to the vertical vibration period of the superstructure, the variation of the superstructure’s horizontal vibration period with the peak ground acceleration ratio (V/H) may be significant.

Study on axial resonance magneto-electric (ME) effects of layered magneto-electric composites

An experimental investigation has been carried out to capture the magneto-mechanical behavior and material parameters of magnetostrictive material. Further, the experimental study is extended to capture the magnetoelectric (ME) coupling response of trilayered composites by applying both alternating and static magnetic fields. From the experiments, it is observed that the magnetostrictive material has a nonlinear hysteretic behavior. Hence, a three dimensional (3-D) nonlinear constitutive model has been proposed based on a thermodynamic framework and multisurface plasticity. In ME composites, it is also observed that a large ME coupling response can be obtained while operating near resonance frequencies. To capture the resonant ME effects, a theoretical formulation has been proposed based on axial vibrations theory and the proposed constitutive model has been incorporated in the formulation. The simulated resonant ME coupling response based on the proposed formulation is compared with the experimental data and found to be in agreement with each other. Further, a parametric study has been performed to address the variation of the resonant ME response as a function of various parameters such as axial compliance ratio, volume fraction of constituents, temperature, direction of magnetic loading and aspect ratio. This study reveals that the coupling response can be optimized by choosing the suitable parameters based on design requirement.

Computational investigation into the role of localisation on yield of a porous ductile solid

In ductile materials containing micro-voids, diffuse plasticity or localisation of plastic strain in narrow bands bridging the ligaments between voids can both occur as precursors to failure. In particular, localisation of plastic strain leads to coalescence through the formation of void sheets or plastic collapse of the ligament. If yielding is defined as the point in macro stress space at which the macroscopic plastic dissipation becomes large, either diffuse plasticity or localisation can cause yielding. Appropriate combinations of triaxiality T and Lode parameter L can cause localisation to occur earlier, and thereby modify the yield surface significantly. This is more likely to happen at high values of porosity. The competition between the two modes of yielding has been captured by a recently proposed multi-surface yield framework (Keralavarma, S. M., 2017, “A multi-surface plasticity model for ductile fracture simulations,” Journal of the Mechanics and Physics of Solids, 103, pp. 100–120), where the competition between the Gurson criterion for yielding by diffuse plastic flow and a criterion for localized yielding within discrete coalescence bands leads to a piecewise smooth yield locus with sharp vertices. In the present work, we generate yield surfaces computationally by using a voided, cuboidal unit cell and a computational homogenisation framework that allows for both macro deformation gradient and macro Cauchy stress control. The basic aim is to see how the multi-surface framework of yield compares with macro yield loci generated computationally using a formulation where both finite deformations and void shape changes are allowed. We show, for spherical, prolate and oblate initial voids, that localisation inevitably hastens macro yield and adds sharp vertices to the yield locus, for a wide range of L and T. The multi-surface framework, at least for spherical initial voids, is remarkably successful in capturing this competition.

Multi-surface plasticity model for concrete with 3D hardening/softening failure modes for tension, compression and shear

In this work we develop multi-surface plasticity model which can reproduce the inelastic behavior and failure modes of concrete in tension, compression and shear. The main novelty of the proposed concrete model can also capture all different phases of localized failure for massive structures, where the elastic behavior is followed by the creation of the fracture process zone with a large number of micro-cracks and subsequent final failure mode with micro-cracks coalescence into the macro-crack. The fracture process zone is represented by homogenized plasticity criterion with hardening (in particular the non-associated Drucker-Prager) since the number of micro-cracks is considered sufficiently large and their orientation random. The macro-crack is represented with a surface of displacement discontinuity, which is typical of all localized dissipative mechanisms due to the apparition and development of localization zones. The main novelty of proposed model is to provide the full set of 3D localization modes for tension, for compression and for shear, with each mode using corresponding fracture energy.

Collapse mechanism analysis of historic masonry structures subjected to lateral loads: A comparison between continuous and discrete models

The aim of this paper is to show to what extent a simple constitutive model can adequately describe the collapse mechanisms of historic masonry structures under horizontal seismic loads. Referring to block masonry, the paper presents the formulation and the numerical implementation of a constitutive relationship for modeling masonry structures regarded at a macroscopic scale as homogenized anisotropic continuum. The macroscopic model is shown to retain memory of the mechanical characteristics of the joints and of the shape of the blocks. The overall mechanical properties display anisotropy and singularities in the yield surface, arising from the discrete nature of the block structure and the geometrical arrangement of the units. The model is formulated in the framework of multi-surface plasticity. It is implemented in a FE code by means of a minimization algorithm directly derived from the Haar-Karman principle.A sensitivity analysis to mechanical and geometrical parameters was carried out to show the influence on the predicted response of small-scale wall components from the literature. The model is then used for the analysis of a numerical case study which is representative of a masonry church subjected to horizontal seismic action.The result of the model will be compared with those obtained by a rigid block model (RBM) of the same small-scale masonry prototypes and of the church building. In the paper, the results of the two models are discussed and analyzed in the way to highlight the weakness and the potentiality of the two different approaches.

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 Multi-Surface Plasticity Method for Lower Bound Shakedown Load

A new direct method for calculation of lower bound shakedown limits based on Melan’s theorem and a novel, non-smooth multi-surface plasticity model is proposed and implemented in a Finite Element environment. The load history is defined by a finite number of extreme points defining the load-envelope of a periodic load set. The shakedown problem is stated as a plasticity problem in terms of a finite number of independent yield conditions, solved for a residual stress field that satisfies a piecewise, non-smooth yield surface defined by the intersection of multiple yield surfaces. The implemented Finite Element procedure is applied to two shakedown problems and the results compared with lower and upper bound elastic shakedown solutions given by the Linear Matching Method, LMM. The example analyses show that the proposed Elastic-Shakedown Multi Surface Plasticity (EMSP) method defines robust lower bound shakedown limits between the LMM lower and upper bound limits, close to the LMM upper bound.