Sequential Limit Analysis Research Articles

Yield surface for void growth and coalescence of porous anisotropic materials under axisymmetric loading

Ductile fracture in metallic alloys occurs by growth and coalescence of cavities. Growth, also referred to as homogeneous yielding, refers to rather diffuse plasticity around cavities, while coalescence, also termed as inhomogenous yielding, corresponds to the localization of plasticity along some planes or directions. Coalescence can develop in various patterns; three coalescence modes have been observed experimentally: internal necking, coalescence in columns and void sheeting. Plastic anisotropy of the material is known to have a significant effect on both homogeneous and inhomogeneous yielding. Therefore, in the present study, yield criteria accounting for the transition from homogeneous yielding to inhomogeneous yielding modes in anisotropic porous materials are obtained using kinematic limit analysis on a cylindrical unit-cell with a coaxial cylindrical cavity. Two types of plastic anisotropy are considered: Hill (1948) plasticity and crystal plasticity. The proposed analytical yield criteria are compared to numerical limit analysis computations and are found to qualitatively agree with simulations. In particular, plastic anisotropy, void shape effects and their coupling are well captured, especially regarding yield stresses and deformation modes. Finally, an homogenized model for Hill porous materials is obtained by supplementing evolution laws for microstructural parameters (void aspect ratio and ligament size ratios) derived from sequential limit analysis. Proposed evolution laws are then discussed in the light of numerical results and experimental evidence.

Quantifying Residual Resistance of Light Pipelines during Large-Amplitude Lateral Displacement Using Sequential Limit Analysis

Quantifying Residual Resistance of Light Pipelines during Large-Amplitude Lateral Displacement Using Sequential Limit Analysis

Local Approach of Ductile Rupture Under Cyclic Loading Conditions

Abstract Experiments have shown that ductile failure occurs sooner under cyclic loading conditions than under monotone ones. This reduction of ductility probably arises from an effect called “ratcheting of the porosity” that consists of a continued increase of the mean porosity during each cycle with the number of cycles. Improved micromechanical simulations confirmed this interpretation. The same work also contained a proof that Gurson's classical model for porous ductile materials does not predict any ratcheting of the porosity. In a recent work (Morin et al., 2017, “A Gurson-Type Layer Model for Ductile Porous Solids With Isotropic and Kinematic Hardening,” Int. J. Solids Struct., 118–119, pp. 167–178), the authors proposed a Gurson-type “layer model” better fit than Gurson's original one for the description of the ductile behavior under cyclic loading conditions, using the theory of sequential limit analysis. A very good agreement was obtained between the model predictions and the results of the micromechanical simulations for a rigid-hardenable material. However, the ratcheting of the porosity is a consequence of both hardening and elasticity, and sequential limit analysis (Yang, 1993, “Large Deformation of Structures by Sequential Limit Analysis,” Int. J. Solids Struct., 30(7), pp. 1001–1013; Leu, 2007, “Analytical and Numerical Investigation of Strain-Hardening Viscoplastic Thick-Walled Cylinders Under Internal Pressure by Using Sequential Limit Analysis,” Comput. Methods Appl. Mech. Eng., 196(25–28), pp. 2713–2722; Leblond et al., 2018, “Classical and Sequential Limit Analysis Revisited,” C. R. Méc., 346(4), pp. 336–349.) is strictly applicable in the absence of elasticity. In this work, a proposal is made to take into account elasticity in the layer model through the definition of a new objective stress rate leading to an accurate expression of the porosity rate accounting for both elasticity and plasticity. This proposal is assessed through comparison of its predictions with the results of some new micromechanical simulations performed for matrices exhibiting both elasticity and all types of hardening. Finally, a comparison of the predictions regarding finite element modeling of pipes loaded cyclically is proposed.

Extended sequential limit analysis: Addressing large deflections and shape changes in strain hardening circular membranes with a moving coordinate system

The sequential limit analysis (SLA) approach can trace load–deflection response and the corresponding plastic limit load of the structures accurately and efficiently. However, this method is limited to the structures without large shape changes in the plane stress directions. In this study, an extended SLA established on moving coordinates is proposed to investigate its further extension for strain-hardening circular membranes with large deflections and shape change behaviors. This new method features dividing entire analysis into sequential optimization problems based on moving coordinate systems to deal with the shape change in plane stress direction for the strain-hardening circular membranes subjected to large deflections. The corresponding adaptive geometry update strategy is also presented. Moreover, the elastic–plastic incremental simulations are conducted for the purpose to validate the correctness and feasibility of the simplifications during model construction and the analysis accuracy of this newly proposed method. From the contrastive results between the extended SLA and ABAQUS simulation, the accuracy of this new approach is confirmed. Furthermore, from the parametric analysis results, the fact can be concluded that the accuracy of SLA for load estimation will be improved with the increase of material Young’s modulus, and reduces with the increase of the initial yield strength and the hardening exponent and hardening coefficient. The accuracy of this method is more sensitive towards hardening exponent and initial yield strength while it is less sensitive towards hardening coefficient. In addition, this extended SLA method with the assumptions of zero bending moment can tolerate a high range variation of thickness–radius ratio, particularly in materials with higher Young’s modulus. These characteristics enables this method be adopted in wild practical application fields as a feasible tool to design the membrane structures initially and determine the structure safety factors preliminarily.

Modelling T-bar penetration in soft clay using large-displacement sequential limit analysis

T-bar penetrometers are commonly used for the measurements of undrained strength profiles for clay (e.g. intact shear strength, sensitivity, ductility). The adoption of a constant resistance factor in the absence of a full-flow failure mechanism, coupled with the strain-softening effect, usually leads to considerable underestimation of soil strength within the depth of up to several T-bar diameters. This paper presents a numerical investigation into the vertical penetration behaviour of a T-bar using a sequential limit analysis (SLA) technique. The T-bar is penetrated from the mudline to a maximum of eight diameters to capture the behaviours related to differing failure mechanisms. The robustness of the SLA is demonstrated through comparisons between lower- and upper-bound solutions and between the SLA and the coupled Eulerian–Lagrangian results, in terms of penetration resistance and failure mechanism. The influences of unit weight, strain softening and strain rate are then examined, based on which an analytical model is developed for more accurate quantification of soil strength back-calculated from T-bar penetration resistance.

Assess the effects of loading rate and interface roughness on T-bar penetration resistance

A recently developed sequential limit analysis method was employed to look into the large displacement penetration behaviour of T-bar penetrometers in soft clay, considering the effects of strain softening and strain rate. Close attention was paid to the effects of interface roughness and the strain rate on the penetration response, before reaching an ideal full-flow state, in order to shed some light on the improved estimate of the in situ strength of surficial soil. The validity of the model was confirmed through a comparison of two small-scale laboratory tests and one centrifuge test, followed by a parametric study in which a T-bar was penetrated from the soil surface to a depth of over five diameters. Analytical equations were derived to assess the effects of interface roughness, viscosity parameter and penetration rate on the T-bar penetration resistance.

An extended sequential limit analysis based on moving coordinates for pressurized spherical cap membranes with large shape change

Sequential limit analysis (SLA) is an effective and normally used method to calculate the plastic limit load of structures with large deformation. However, attribute to the assumption of neglecting the changing behavior of shape and plane stress direction, the conventional SLA method would be inaccurate in the plastic response prediction for the large-shape-change structures, especially for the pressurized spherical cap. This research develops a novel analytical model for the pressurized spherical cap based on the advanced SLA method, which features introducing the moving coordinate system and considering of the changing behavior of shape and plane stress direction into the conventional method. With the proposed method, the effects of geometry and material parameters on the plastic limit load are analyzed. Compared with the validating FE simulation results of ABAQUS software, the newly extended SLA method performances a more precise prediction of the load deflection response and plastic limit load than the normal one. Due to the limit yield degree and bending moment, the accuracy of the new model will increase with the increase of yield strength and radius. The larger the initial deflection of the pressurized spherical cap is, the smaller the relative error between the analysis results of advanced SLA and FE method is. Moreover, this newly proposed SLA remains effective and accurate within a wide range of the initial thickness-curvature radius ratio, especially for low elasticity modulus materials.

Assessment model of pipe–soil interaction during large-amplitude lateral displacements for deep-water pipelines

The controlled buckling design of deep-water pipelines requires comprehensive understanding of pipe–soil interactions subject to large-amplitude lateral displacements. This paper explores this issue using a large displacement sequential limit analysis (SLA) method. The elevation of the pipe invert is kept constant during the lateral displacement to consider the situation when strong constraints from adjacent sections (e.g. intermittent rock dampers, anchors, etc.) are present and the validity of the numerical model is demonstrated by comparisons with five published high-quality centrifuge tests. Parametric studies on the effects of pipe embedment and strength profile are performed, leading to the development of analytical models to predict both lateral and vertical soil resistances during lateral displacement. The evolution of V–H yield envelopes, if presented in a normalised form, is found to be independent on the lateral displacement and can be satisfactorily captured by a binomial equation model.

Sequential limit analysis for clamped circular membranes involving large deformation subjected to pressure load

In this study, we investigate the extended sequential limit analysis for clamped circular membranes involving large deformation subjected to pressure load. Moving coordinate systems were adopted, and a shape and geometry update strategy was proposed to deal with the consideration of plane stress direction change. A sequence of optimization problems formulated on the upper-bound theorem was solved to perform the nonlinear cumulative sequence of load-deflection curves. In the formulation, the rigid-perfectly-plastic behavior were used. Then, some numerical simulations by ABAQUS were conducted to assess the parameters and simplification effect on the sequential upper-bound formulation derived.The results show that the sequential limit analysis gives a good estimation of the plastic limit load and load-deflection behavior. The collapse mechanism assumption and the update strategy of shape and geometry cause little error. Meanwhile, the limit state yield level and bending moment are the main factors that cause the relative error of limit load estimation to decrease as the elastic modulus, yield strength, and thickness increase and the radius decreases. Synthetically, the sequential limit analysis allow a wide range of thickness-radius ratios, especially for low-elastic-modulus materials, and the relative errors can be controlled within 17%, even for materials with extreme parameters.

Classical and sequential limit analysis revisited

Classical limit analysis applies to ideal plastic materials, and within a linearized geometrical framework implying small displacements and strains. Sequential limit analysis was proposed as a heuristic extension to materials exhibiting strain hardening, and within a fully general geometrical framework involving large displacements and strains. The purpose of this paper is to study and clearly state the precise conditions permitting such an extension. This is done by comparing the evolution equations of the full elastic–plastic problem, the equations of classical limit analysis, and those of sequential limit analysis. The main conclusion is that, whereas classical limit analysis applies to materials exhibiting elasticity – in the absence of hardening and within a linearized geometrical framework –, sequential limit analysis, to be applicable, strictly prohibits the presence of elasticity – although it tolerates strain hardening and large displacements and strains. For a given mechanical situation, the relevance of sequential limit analysis therefore essentially depends upon the importance of the elastic–plastic coupling in the specific case considered.

Sequential limit analysis of pipe–soil interaction during large-amplitude cyclic lateral displacements

A newly developed sequential limit analysis (SLA) technique is used to perform large-displacement numerical simulations relevant to thermally induced lateral buckling of untrenched subsea pipelines. A rigid plane-strain pipe segment, partially embedded in undrained clay, is subjected to cyclic lateral displacements with amplitudes of up to eight pipe diameters. A constant vertical dead load is applied to the pipe during each lateral sweep, but in some analyses this load is varied from one sweep to the next. The SLA method directly models the evolution of the soil surface profile, including active and dormant berms, and incorporates strain softening behaviour caused by soil remoulding. Comparisons of the numerical results with published centrifuge model test data, for a range of loading cases, are provided in terms of the pipe invert trajectory and lateral soil resistance. There is good overall agreement between the numerical and experimental results, demonstrating the suitability of SLA for solving such problems. Detailed aspects of the cyclic loading behaviour are discussed with reference to soil failure mechanisms and bearing capacity failure envelopes for combined vertical and horizontal loading. Finally, two brief parametric studies are used to explore the effects of the initial pipe embedment and vertical loading history on the subsequent lateral loading behaviour.

Modelling large plastic deformations of cohesive soils using sequential limit analysis

SummaryThis paper introduces sequential limit analysis (SLA) as a method for modelling large plastic deformations of purely cohesive materials such as undrained clay. The method involves solving a series of consecutive small‐deformation plastic collapse problems using finite element limit analysis, thus ensuring high levels of accuracy, efficiency, and robustness. The techniques needed to develop an SLA implementation for two‐dimensional (plane strain) problems are described in detail, including model geometry updating routines, treatment of rigid bodies, interfaces and boundaries, and periodic remeshing and interpolation of field variables. A simple total stress‐based constitutive model is used to account for strain softening and strain rate effects. Extensive verifications and validations are performed using analytical solutions and physical model test results, comparing both collapse loads and failure mechanisms, to demonstrate the effectiveness of the SLA approach. Additional solution quality checks on the bracketing discrepancy between lower‐bound and upper‐bound limit analysis solutions, and on the incompressibility of the rigid‐plastic material, are also presented.

A Gurson-type layer model for ductile porous solids with isotropic and kinematic hardening

The aim of this work is to propose a Gurson-type model for ductile porous solids exhibiting isotropic and kinematic hardening. The derivation is based on a “sequential limit-analysis” of a hollow sphere made of a rigid-hardenable material. The heterogeneity of hardening is accounted for by discretizing the cell into a finite number of spherical layers in each of which the quantities characterizing hardening are considered as homogeneous. A simplified version of the model is also proposed, which permits to extend the previous works of Leblond et al. (1995) and Lacroix et al. (2016) for isotropic hardening to mixed isotropic/kinematic hardening. The model is finally assessed through comparison of its predictions with the results of some micromechanical finite element simulations of the same cell. First, the numerical and theoretical overall yield loci are compared for given distributions of isotropic and kinematic pre-hardening. Then the predictions of the model are investigated in evolution problems in which both isotropic and kinematic hardening parameters vary in time. A very good agreement between model predictions and numerical results is found in both cases.

Deformation pattern during normal faulting: A sequential limit analysis

AbstractWe model in 2‐D the formation and development of half‐graben faults above a low‐angle normal detachment fault. The model, based on a “sequential limit analysis” accounting for mechanical equilibrium and energy dissipation, simulates the incremental deformation of a frictional, cohesive, and fluid‐saturated rock wedge above the detachment. Two modes of deformation, gravitational collapse and tectonic collapse, are revealed which compare well with the results of the critical Coulomb wedge theory. We additionally show that the fault and the axial surface of the half‐graben rotate as topographic subsidence increases. This progressive rotation makes some of the footwall material being sheared and entering into the hanging wall, creating a specific region called foot‐to‐hanging wall (FHW). The model allows introducing additional effects, such as weakening of the faults once they have slipped and sedimentation in their hanging wall. These processes are shown to control the size of the FHW region and the number of fault‐bounded blocks it eventually contains. Fault weakening tends to make fault rotation more discontinuous and this results in the FHW zone containing multiple blocks of intact material separated by faults. By compensating the topographic subsidence of the half‐graben, sedimentation tends to slow the fault rotation and this results in the reduction of the size of the FHW zone and of its number of fault‐bounded blocks. We apply the new approach to reproduce the faults observed along a seismic line in the Southern Jeanne d'Arc Basin, Grand Banks, offshore Newfoundland. There, a single block exists in the hanging wall of the principal fault. The model explains well this situation provided that a slow sedimentation rate in the Lower Jurassic is proposed followed by an increasing rate over time as the main detachment fault was growing.

A simple test for nonstationarity in mixed panels: A further investigation

The test of Ng (2008) is one of the few that enables general inference regarding the proportion of non-stationary units in panel data. The current paper furthers the investigation of Ng (2008) in two directions. First, the existing sequential limit analysis is generalized to a very flexible asymptotic framework in which the number of time periods, T, can be either fixed or tending to infinity jointly with the number of cross-section units, N. Second, the test statistic is evaluated not only under the null hypothesis, but also under alternatives that can be either fixed or local.

Plastic limit pressure of spherical vessels with combined hardening involving large deformation

The paper aims to investigate plastic limit pressure of spherical vessels of nonlinear combined isotropic/kinematic hardening materials. The Armstrong-Frederick kinematic hardening model is adopted and the Voce hardening law is incorporated for isotropic hardening behavior. Analytically, we extend sequential limit analysis to deal with combined isotropic/kinematic hardening materials. Further, exact solutions of plastic limit pressure were developed analytically by conducting both static and kinematic limit analysis. The onset of instability was also derived and solved iteratively by Newton's method. Numerically, elastic–plastic analysis is also performed by the commercial finite-element code ABAQUS incorporated with the user subroutine UMAT implemented with user materials of combined hardening. Finally, the problem formulation and the solution derivations presented here are validated by a very good agreement between the numerical results of exact solutions and the results of elastic–plastic finite-element analysis by ABAQUS.

Thickness improvement in single point incremental forming deduced by sequential limit analysis

Multistage forming is one of the most practical solutions to avoid severe thinning in single point incremental forming (SPIF). A successful implementation of multistage SPIF is strongly dependent on an appropriate deformation path. In this paper, firstly, a simplified modeling technique is proposed using sequential limit analysis. It is shown that sequential limit analysis can predict the thickness distribution faster than an equivalent model in a commercial finite element modeling code like Abaqus can. The reliability of the model is assessed by comparing experimental and simulated results for single-stage and multistage SPIF cones. This model is utilized to study the effect of various deformation paths on the thickness distribution. As a result, a new multistage strategy is designed and implemented to form a 70° wall angle cone in three stages. The thickness distribution of the cone is improved significantly compared to cones formed by a single-stage and a conventional three-stage strategy. Besides this improvement, the new multistage SPIF can be carried out in much less time.

Predicting orogenic wedge styles as a function of analogue erosion law and material softening

The evolution of a compressive frictional wedge on a weak, frictional and planar décollement, subjected to frontal accretion, is predicted with a two step method called sequential limit analysis. The first step consists in finding, with the kinematic approach of limit analysis, the length of the active décollement and the dips of the emerging ramp and of the conjugate shear plane composing the emerging thrust fold. The second step leads to a modification of the geometry, first, because of the thrust fold development due to compression and, second, because of erosion. Erosion consists in removing periodically any material above a fictitious line at a selected slope, as done in analogue experiments. This application of sequential limit analysis generalizes the critical Coulomb wedge theory since it follows the internal deformation development. With constant frictional properties, the deformation is mostly diffuse, a succession of thrust folds being activated so that the topographic slope reaches exactly the theoretical, critical value. Frictional weakening on the ramps results in a deformation style composed of thrust sheets and horses. Applying an erosion slope at the critical topographic value leads to exhumation in the frontal, central, or rear region of the wedge depending on the erosion period and the weakening. Erosion at slopes slightly above or below the critical value results in exhumation toward the foreland or the hinterland, respectively, regardless of the erosion period. Exhumation is associated with duplexes, imbricate fans, antiformal stacks, and major backthrusting. Comparisons with sandbox experiments confirm that the thickness, dips, vergence, and exhumation of thrust sheets can be reproduced with friction and erosion parameters within realistic ranges of values.

An investigation into thickness distribution in single point incremental forming using sequential limit analysis

Single point incremental forming (SPIF), needing no dedicated tools, is the simplest variant of incremental sheet metal forming processes. In the present work, a simplified model of SPIF of a truncated cone, capable of predicting the thickness distribution, has been developed using sequential limit analysis (SLA). The obtained results were validated experimentally and compared with thickness predictions obtained from an explicit shell FE model implemented in Abaqus. It is shown that SLA is capable to solve the thickness prediction problem more accurately and efficiently than the equivalent FEA approach. As an application of the proposed model, the effect of the diameter of the hemispherical tool tip and the step down on the thickness distribution and the minimum thickness in a 50° cone is studied using SLA. By introducing bending and stretching zones in the wall of the cone, variations of the minimum thickness by changing the tool diameter and the step down are discussed.

Exact solutions of sequential limit analysis of pressurized cylinders with combined hardening based on a generalized Hölder inequality: Formulation and validation

The paper aims to investigate nonlinear combined isotropic/kinematic hardening cylinders under internal proportional pressure by sequential limit analysis. The Armstrong–Frederick kinematic hardening model is adopted and the Voce hardening law is incorporated for isotropic hardening behavior. In particular, we establish the kinematic formulation of sequential limit analysis from the corresponding static formulation by a generalized Hölder inequality. Especially, it is found that the derived kinematic formulation involving combined isotropic/kinematic hardening is equivalent to that by the bipotential concept. Further, exact solutions of plastic limit pressure were developed analytically by performing both static and kinematic limit analysis. Finally, the problem formulation and the solution derivations presented here are validated by a very good agreement between the numerical result of exact solutions of the present work and the upper bounds from the kinematic formulation available in literature.