Plastic Multiplier Research Articles

A trust region algorithm for finite-strain plasticity with strongly coupled hardening

As extensions to our Lagrangian finite-strain plasticity framework based on the approximate exponential integrator, we introduce two new algorithms: (i) a fixed-radius trust region root finder for the nonlinear system involving the elastic right Cauchy Green tensor and plastic multiplier (ii) a partitioned approach for the hardening variables, which are determined in a staggered form using the strongly-coupled concept typically adopted for multiphysics problems. This allows the use of intricate hyperelastic laws combined with recent yield functions, which otherwise would involve a laborious treatment, and the use of corresponding work-hardening. Work-hardening would introduce significant nonlinearities in the constitutive system if used in a fully-coupled form. For the partitioned approach, a dynamic relaxation algorithm is adopted. This allows the efficient solution of the two nonlinear equations without significant drifting. Results show that robustness and efficiency are significantly improved. Herein, algorithms are described in detail. Significant testing is performed with imposed strains up to 100 for a carbon steel. This contributes to the robustness of equilibrium iterations. Drifting is also assessed as a function of number of steps in the dynamic relaxation algorithm. Numerical experimentation is performed in the 3D tension test for an anisotropic yield function.

A novel approach for accurate development of the incremental plastic multiplier and consistent tangent operator in thermo-elasto-plastic modeling of materials

In the present research, new and accurate equations were developed for the incremental plastic multiplier (IPM) and the consistent tangent operator (CTO) to solve numerical problems in thermo-elasto-plastic (TEP) processes using the finite element method (FEM). To ensure accuracy, all material hardening coefficients were treated as temperature-dependent, and no terms and their derivatives in the analytical-mathematical solution were ignored. Two UMAT (User MATerial) subroutines with temperature-independent and temperature-dependent parameters were programmed for the EP and TEP behavior, respectively. Finite element models were created using both the Abaqus® built-in material models and the newly developed UMAT subroutines, designated as the reference and new models, respectively. In the reference model, flow stress was implemented using tabulated plastic strain and temperature data available in Abaqus®, while in the new model, the flow stress (yield function) was derived and numerically calculated based on the developed formulation. The new equations were successfully validated by comparing the results from the new model with those from the reference model. The developed IPM and CTO can be used for accurate predictions of strains, stresses, and temperatures in TEP problems, making them well-suited for industrial applications.

Elasto-Plastic Analysis of Two-Way Reinforced Concrete Slabs Strengthened with Carbon Fiber Reinforced Polymer Laminates

This study explores a technique for enhancing the punching strength of reinforced concrete (RC) flat slabs, namely carbon fiber reinforced polymer (CFRP). Four large-scale RC flat slabs were fabricated, to assess the efficacy of this strengthening method. One slab served as a reference and the three other specimens were strengthened with CFRP, as a method of external strengthening. These slabs, featuring identical overall dimensions and flexural steel reinforcement, underwent testing until failure, under the influence of concentrated patch loads. A concrete plastic damage constitutive model (CDP) was developed and employed to examine the strength of two-way RC slabs. Additionally, to enhance the strength of existing RC slabs, carbon fiber reinforced polymer (CFRP) strips are affixed to the tension surface of the sections. The research begins with the calibration of a numerical model, based on data from laboratory tests. The objective of this study is to constrain the plastic behavior of two-way RC slabs reinforced with CFRP, with a focus on establishing an optimal elasto-plastic analysis, aimed at controlling concrete damage plasticity using CFRP, and employing a plastic limit load multiplier. Subsequently, a series of numerical simulations, incorporating different variables, are conducted to investigate shear behavior. The numerical results indicate that an increase in the strengthening ratio has a significant impact on shear strength. Finite element simulations are carried out using Abaqus CAE®/2018.

Consistent memory surface model for plateau and hardening regions

Mild steels are frequently used in building structures because of their high strength, productivity, processability, and weldability. One key feature of mild steels is their yield plateau, which is followed by a hardening region in the stress–strain relationship. During large earthquakes, building members can experience a large plastic strain whose amplitude is difficult to predict beforehand. Constitutive equations applicable to the full-range strain field are necessary for accurate structural safety evaluation. This study provides novel constitutive equations for the plateau region, capable of reproducing a yield plateau. A set of hardening constants was proposed to satisfy the consistency condition of the plastic multiplier under any loading pattern. The closed form of the stress–strain relationship was derived to calibrate the material constants. The material constants were calibrated using two types of material test results. The results of the proposed model were consistent with the material test results under cyclic loading at various strain amplitudes. This was validated by comparing the shear buckling and lateral torsional buckling test results of an H-shaped beam with the finite element analysis (FEA) results. The numerical demonstration also highlighted the significance of not only material constitutive laws but also the introduction of residual stress and initial imperfections in reproducing the pre- and post-buckling behaviors. The convergence of the residual force for the proposed model was equivalent to those of the authors’ previous model and the Chaboche model. These results indicate that the proposed model can predict hardening phenomena at the material and structural levels in the plateau and hardening regions.

Plastic-limit probabilistic structural topology optimization of steel beams

This work presents a novel geometrically nonlinear analysis with imperfections reliability-based topology optimization (RBTO) approach, considering the volume fraction and geometric imperfections as random variables due to their crucial connections to the manufacturability of web openings in steel I-beams. The objective is achieved by controlling the plastic behavior through limit analysis, which imposes a limit on the plastic ultimate load multipliers. The suggested method is developed by imposing constraints related to the available material volume. The bi-directional evolutionary structural optimization (BESO) method is utilized to fulfill the objectives of this paper. The adequacy of the proposed technique is demonstrated by comparing the results of the algorithm with benchmark steel beams that have conventional web openings. Based on the numerical examples, it appears that the selected technique has the potential to increase the load capacity of steel beams while utilizing the same quantity of material within the design domain. Moreover, the improved beams also exhibit better performance in terms of stress levels. The results of this research suggest that the proposed technique enhances the load capacity of steel beams while maintaining the same quantity of material and improving their performance in terms of stress levels.

Effect of single and dual voids on vertical capacity of large-diameter monopiles in marine clay

This study utilizes the finite element limit analysis (FELA) method to numerically investigate the effect of single and dual voids on the vertical bearing capacity of nearshore large-diameter monopiles in marine clay. After validating the numerical model, numerous numerical analyses in predicting the behavior of the pile-void system were conducted under different geometric parameters of pile and void, different pile-void distances, and different undrained shear strengths. These numerical results can be expressed in the form of design charts for vertical capacity reduction coefficients, which can facilitate a quick and straightforward assessment of pile capacity considering single and dual voids by multiplying the capacity without voids by respective coefficients. In addition, the numerical results can further give the contours of the plastic multiplier of the pile-void system which can be used to graphically conclude several representative failure modes. In short, this study intuitively illustrates the void effect on the pile vertical bearing capacity (including failure mode) and shows the significant differences between the cases with single and dual voids.

Super/sub-loading surface model for constitutive equation of friction

In this study, an elasto–plastic constitutive model of friction incorporating the superloading surface concept was formulated as an extension of the subloading friction model. The proposed model regards the state in which the static friction coefficient is larger than the kinetic friction coefficient as the “bulkiness” of structure and the smooth transition of friction was expressed by the evolution rule of structure.The positive definiteness of the plastic multiplier, the loading condition, and a condition equation for material parameters were clearly derived. The capability of the proposed model in accordance with the subloading-friction model was confirmed in the stick–slip simulation of a mass-spring system.

Symbolic algebra integration of soil elastoplastic models

This work analyses the time integration of elastoplastic models using implementation platforms with symbolic algebra capabilities. In such platforms, the variables and state functions involved in solving boundary value problems are considered entities continuous in time, thus the time integration of stresses and hardening parameters is always based on implicit schemes. The latter confers to the computation an accuracy that is inspected in this work by analysing the integration of the Clay and Sand Model. Finally, the advantages of using integration schemes in which the linearisation of the yield function is imposed to be zero when calculating the plastic multiplier are illustrated.

Two novel alternative integration schemes for multi-invariants dependent isotropic finite deformation plasticity

Multi-invariants dependent isotropic yield function in terms of first invariant I1 of stress tensor, second J2 and third J3 invariants of deviatoric stress tensor is required for accurate description of plasticity of polycrystalline solids. In the context of multi-invariants dependent multiplicative hyperelasto-plasticity, two integration schemes (IS-1 and IS-2) are formulated and their computational costs are compared for the first time in this paper. IS-1 is fully-implicit in which backward Euler difference approximation of equivalent plastic strain rate is coupled with the exponential approximation of return mapping in principal Kirchhoff stress space and accordingly principal Kirchhoff stresses, equivalent plastic strain and plastic Lagrange multiplier need to be updated iteratively. IS-2 is semi-implicit in which plastic Lagrange multiplier evaluated at current time step and semi-implicit forward Euler difference approximation of equivalent plastic strain rate is coupled with logarithmic approximation of return mapping in Kirchhoff stress space and accordingly only plastic Lagrange multiplier needs to be updated iteratively. For J2 dependent plasticity, computation time involved in numerical simulations for both the integration schemes is nearly the same for same accuracy. For IS-1, computation time involved in simulation for multi-invariants dependent plasticity is significantly greater than that for J2 dependent plasticity due to the computation of the Jacobian matrix in return mapping involving lengthy terms of second order partial derivatives of multi-invariants dependent yield function. For IS-2, computation time involved in simulation for multi-invariants dependent plasticity is nearly the same as that for J2 dependent plasticity. This is due to the fact that the computation of the Jacobian matrix in IS-2 is not required and return mapping involves only first partial derivatives of yield function. Thus, IS-2 is computationally cost effective than IS-1.

Effect of single and dual voids on lateral capacity of large-diameter monopiles in marine clay

This study numerically examines the effect of single and dual voids on the lateral bearing capacity of nearshore large-diameter monopiles in marine clay by the finite element limit analysis (FELA) method. The ability of the numerical model to accurately predict the pile-void system behavior is evaluated by performing degenerated verification analyses on existing studies. Based on numerous numerical results, several design charts for capacity reduction coefficients are produced to provide a quick and straightforward assessment of the pile capacity with single and dual voids by multiplying that without voids by coefficients. Particularly, the coupling coefficient is further proposed to quantify the coupling effect between dual voids, and the corresponding design charts are also given. The subsequent parametric studies indicate that the void effect gets aggravated as the void is enlarged and close to piles but would be weakened as the soil around the pile gets strengthened; the void effect is positively and negatively associated with the void depth in the cases of low and great shear strength, respectively. The effect of single and dual voids (inc. coupling effect between dual voids) on the pile capacity and failure mode is further graphically explained by the plastic multiplier contour maps.

Specialized 3D Distinct element limit analysis approach for a fast seismic vulnerability evaluation of massive masonry structures: Application on traditional pagodas

A 3D limit analysis model specialized in the analysis at the collapse of massive masonry structures subjected to seismic loads is proposed. The approach belongs to the wide family of distinct element methods, because is based on a discretization of the structure by means of infinitely resistant hexahedron elements and quadrilateral interfaces where all dissipation occurs. Any failure surface -after proper linearization- can be adopted for interfaces, as for instance a Mohr-Coulomb failure criterion with tension and compression cutoffs. Having discretized the continuum in such a way that internal dissipation is allowed only between adjoining elements, the corresponding linear programming problem necessary to estimate the collapse load, can be obtained through both a kinematic and static approach. The first is more straightforward and the primal problem provides a failure mechanism, interfaces plastic multipliers and collapse multipliers. By virtue of the duality theorem, the second lower bound approach is associated with the same failure load, also providing the internal forces acting on interfaces. The model is validated on three examples of historical value, namely-three pagodas in China, all characterized by complex 3D geometries and massive walls. The results show how the procedure proposed is able to provide fast estimations of the load carrying capacity under the application of horizontal seismic actions and how the corresponding failure mechanism is rather complex, thus precluding the utilization of simplified approaches on idealized geometries.

The role of accumulated plasticity on yield surface evolution in pearlitic steel

In this paper, we analyze one specific assumption used in many cyclic plasticity models: The dependence of the yield surface evolution on the accumulated plasticity, i.e., the time-integrated plastic multiplier. We measure this scalar variable in carefully designed experiments and track the yield surface evolution. Our results show that this variable is insufficient to describe the isotropic or distortional hardening of the investigated steel, invalidating the commonly used assumption. The results highlight requirements for developing and verifying new isotropic and distortional hardening laws.

Effect of nearby strip loading on lateral capacity of offshore large-diameter monopiles in clay slope

This study is devoted to investigating the effect of nearby strip loading on the lateral bearing capacity of offshore large-diameter monopiles at the crest of marine clay slope by the finite element limit analysis (FELA) method. After the successive validations by cases of rigid piles and strip footings at the clay slope crest, the safe domains for the laterally loaded large-diameter monopiles are defined by the failure envelopes relating the dimensionless lateral bearing capacity H u/(c u D 2) to dimensionless strip loading q/c u. Particularly, all failure envelopes are depicted by closed-form expressions and design charts can then be given for a quick and straightforward assessment of dimensionless lateral bearing capacity for a wide range of frequently-encountered cases. In addition, the interaction between the pile bearing capacity and nearby strip loading is characterized by introducing the relationship of normalized parameters H u/(c u D 2) and q/c u, then graphically explained by plastic multiplier contour plots. Several kinds of representative curves are given successively to further clarify the effects of dimensionless factors on the interaction.

Asymptotic numerical method for finite plasticity

This paper discusses the efficiency of an algorithm based on the Asymptotic Numerical Method (ANM) to solve large strain plasticity problems. In the framework of ANM, the non-smooth constitutive law has to be replaced by a smooth one in order to be able to represent the solution path in the form of Taylor series. For this purpose, we propose to generalize the approach used in the small strain case. To achieve this, we introduce a regularized stress–strain relation that considers smoothed elastic–inelastic transitions. This regularized law is a relationship between two scalars, namely the yield function and the plastic multiplier. Classical uniaxial traction benchmarks permit to assess the procedure and to adjust the parameters of the algorithm: regularization parameters, truncation orders and parametrization. • This paper discusses the efficiency of an algorithm based on the Asymptotic Numerical Method (ANM) to solve large strain plasticity problems. • The non-smooth constitutive law has to be replaced by a smooth one in order to be able to represent the solution path in the form of Taylor series. • Classical uniaxial traction benchmarks permit to assess the procedure and to adjust the parameters of the algorithm.

A consistent algorithm for finite-strain visco-hyperelasticity and visco-plasticity of amorphous polymers

Shared rheology elements of soft tissues and additive-manufactured amorphous thermoplastics benefit from a common visco-hyperelastic and visco-plastic constitutive framework. For finite-strain and general visco/elastic/plastic constitutive laws, allowing full anisotropy, we use the Kröner–Lee decomposition of the deformation gradient combined with Mandel stress-based yield function. Relatively weak conditions are necessary for specific laws to be incorporated in this framework. For example, stress-like variables present in visco-elasticity are removed from the list of constitutive unknowns. Constitutive iteration is performed for the elastic Cauchy–Green tensor Ce and the plastic multiplier increment Δλ. The source is here the right Cauchy–Green tensor provided by any discretization. For the integration of the flow law we adopt a scaled/squared series approximation of the matrix exponential. The exact Jacobian of the second Piola–Kirchhoff stress is determined with respect to this source, consistent with the integrator. The resulting system is produced by symbolic source-code generation for each yield function and hyperelastic strain-energy density function with each of the viscous terms added. The constitutive system is solved by a damped Newton–Raphson algorithm with the corresponding stress error being extensively assessed with stress error maps. A cellular beam structure is analyzed and results compared with experiments from our group. Some of the symbolic calculations and closed-form solutions are beyond what is manually achievable and therefore the symbolic sources for this work are made available in a source code repository.

Reliability based geometrically nonlinear bi-directional evolutionary structural optimization of elasto-plastic material

The aim of this paper is to integrate the reliability-based analysis into topology optimization problems. Consequently, reliability-based topology optimization (RBTO) of geometrically nonlinear elasto-plastic models is presented. For purpose of performing (RBTO), the volume fraction is considered reliable since that the application of (RBTO) gives different topology in comparison to the deterministic topology optimization. The effects of changing the prescribed total structural volume constraint for deterministic designs and changing the reliability index for probabilistic designs are considered. Reliability index works as a constraint which is related to reliability condition added into the volume fraction and it is calculated using the Monte-Carlo simulation approach in the case of probabilistic design. In addition, bi-directional evolutionary structural optimization (BESO) method is utilized to study the effect of geometrically nonlinear elasto-plastic design. The plastic behavior can be controlled by defining a limit on the plastic limit load multipliers. The suggested work's efficiency is demonstrated via a 2D benchmark problem. In case of elastic material, a 2D model of U-shape plate is used for probabilistic design of linear and geometrically nonlinear topology optimizations. Furthermore, a 2D elasto-plastic model is considered for reliability-based design to demonstrate that the suggested approach can determine the best topological solution.

Damage degradation modelling for transverse cracking in composite laminates under low-velocity impact

This paper derives a damage evaluation law for fibres and an expression for the damage parameter function. It also proposes an approach to the matrix damage degradation law. A proposed approach to both the constitutive damage degradation model and increment law is developed to predict intralaminar damage evolution in composite laminates. Failure envelopes for different failure criteria are discussed in term of the fracture plane of matrix cracking under compressive load. The damage surface consistency condition is applied to derive a plastic multiplier as a function of the damage plastic flow so that the plastic strain is updated at each time increment and the stress–strain constitutive relationship of the damage model will also be updated. A user-defined subroutine has been adopted to implement a proposed constitutive damage degradation model. The effectiveness of the proposed method has been examined under low velocity impact. The numerical findings confirm that results obtained using the suggested approach are in good agreement with experimental results.

Combined-loading elastoplastic constitutive model for a unified description of the mechanical behavior of the soil skeleton

This paper proposes a combined-loading elastoplastic constitutive model (CL model) for a unified description of the mechanical behavior of soil materials, including clay, sand, and intermediate soil. This mathematical model combines the super-subloading yield surface Cam-clay model and the non-associated Drucker–Prager (DP) model. The first model introduces the skeleton soil structure concept into the Cam-clay model based on critical-state soil mechanics. It describes the mechanical behavior of naturally deposited clay more accurately than the existing model does. The non-associated DP model considers the frictional properties of granular materials and is widely used to describe the undrained shear behavior, particularly that of sandy soil. The newly proposed model is one in which these two independently existing models simultaneously or independently work in loading states according to the loading criterion, which can be derived as a natural result from plastic multipliers. This framework allows for a unified description of a wide range of soil materials, including naturally deposited clayey soils and sandy soils.

Nonlocal plastic softening algorithm and numerical investigation based on the Drucker–Prager criterion

Strain softening in geotechnical materials often leads to the ill condition of boundary value problems. That is, the regional governing equations or partial differential equations are no longer applicable under the original fixed solution conditions, and the over non-localization theory is used to solve the finite element equations to ensure the possibilities of the governing equations. Based on the Drucker–Prager (D–P) criterion, a nonlocal elastoplastic constitutive model and its numerical algorithm for geomaterials are established while considering the softening plasticity. Strain localization leads to high strain gradients and severe deformations in the zone. The introduction of characteristic lengths establishes the relationship between microstructure and macroscopic deformation. Accordingly, the surroundings will affect a certain point of a damaged rock material. When a Gaussian weight function is introduced, and the characterization is concerned on the damage point, the influence gradually decreases, and the characteristic length controls the influence range as the distance increases. When using the closest point projection method to calculate the stress regression of the D–P criterion, the plastic multiplier is weighted and averaged by the Gaussian weight function, and the local plastic multiplier is converted into a nonlocal plastic multiplier for numerical analysis. The results show that with the introduction of the weight function and the feature length, the localized zone width no longer depends on the grid size, that is, the grid sensitivity is overcome, and the result is more in line with the physical nature.

Elasto-Plastic limit analysis of reliability based geometrically nonlinear bi-directional evolutionary topology optimization

This paper presents elasto-plastic limit analysis of reliability-based geometrically nonlinear topology optimization. For this purpose, by reason of uncertainties the volume fraction is considered randomly during optimization. Thus reliability-based design has been considered for solving the problems. To perform reliability-based topology optimization design, the Monte-Carlo simulation method has been applied to calculate the probability of failure, thus the reliability index. Besides, bi-directional evolutionary structural optimization (BESO) method is used to consider the effect of geometrically nonlinear design for elasto-plastic analysis. Plastic behavior is controlled by applying a bound on the plastic limit load multipliers using limit analysis. The adequacy of the proposed method is exhibited by three 2D benchmark problems. 2D models of L-shape beam and U-shaped plate are considered for reliability-based design and geometrically nonlinear analysis topology optimization in case of elastic material. Additionally, 2D and 3D elasto-plastic material models have been considered to demonstrate that the proposed method can find the optimal topology of elasto-plastic models for reliability-based design and geometrically nonlinear analysis.