Porous Plasticity Model Research Articles

Influence of textural variability on plastic response of porous crystal embedded in polycrystalline aggregate: A crystal plasticity study

Damage evolution in polycrystalline aggregates is complicated by the intricate interplay of crystallographic orientation of the porous grain and the surrounding anisotropic matrix. Therefore, formulation of design rules and damage models for polycrystalline materials proves daunting due to relative lack of thorough understanding of the underlying heterogeneity at the mesoscale. This work explores the orientation dependent void growth in a porous crystal embedded in an anisotropic polycrystalline matrix with different initial textures. Polycrystalline face-centered cubic based aggregate is simulated within the framework of crystal plasticity finite element method. Porosity is first modeled in the form of a single pre-existing spherical void in the central grain of the randomly oriented polycrystal. One-hundred crystallographic orientations of the central grain in three-dimensional Euler space are analyzed to reveal the orientation dependent trends of the porous grain. To account for textural variability, the analysis is repeated for polycrystals exhibiting preferred textures like Cube, Brass, Copper and Goss. In this manner, interesting orientation dependent trends in basic tenets of void growth like yield strength, coalescence strain and porosity evolution are unraveled across various polycrystalline textures. To account for spatial heterogeneity as well, porosity in the central grain is then re-distributed and the aforementioned analysis is repeated for all the crystallographic orientations of the central grain embedded in polycrystals with different textures. Owing to the large amount of data thus generated, statistical analysis is invoked to identify stimulating trends and key statistical variables governing the strength and toughness. Consequently, a statistical void growth model is also presented by assessing the CP simulation results and identifying suitable distribution function governing the growth of voids in polycrystals. The modeling framework is expected to inform porous plasticity models aimed at capturing damage evolution in porous grains embedded in polycrystalline materials exhibiting topological and crystallographic anisotropy.

Size-dependent yield criterion for single crystals containing spherical voids

A micromechanics-based yield criterion is derived for a porous single crystal containing a random distribution of spherical voids, using homogenization and limit analysis of a hollow spherical representative volume element obeying a strain gradient crystal plasticity model. The criterion captures the effect of plastic anisotropy due to crystallographic slip on a set of discrete slip systems, as well as the void size dependence of yielding. The yield criterion is formally similar to the well known Gurson model for isotropic porous materials, and reduces to existing results in the literature in the limiting case of large void sizes relative to the length scale in the gradient plasticity model. The yield criterion is validated by comparison with rigorous upper bound yield loci obtained using a numerical limit analysis procedure. Predictions for the size dependence of void growth are obtained by integrating the porous plasticity model, and shown to be consistent with the observations from lower scale discrete dislocation dynamics simulations. The loading path dependence of void growth in the size-independent limit are compared with finite element simulations of void growth in a single crystal using the unit cell model.

Modelling of ductile failure in materials with heterogeneous particle structure using localization analysis

In this study, we evaluate the use of strain localization analysis based on the imperfection band approach to model the effect of heterogeneous particle distribution on the ductile failure of structural metallic materials. To this end, tensile tests are performed on smooth and notched samples made of aluminium alloy AA6110 with an equiaxed grain structure and an inhomogeneous distribution of the constituent particles. The constituent particles are assumed to be the main contributors to the ductile failure of the alloy. By varying the heat treatment of the alloy, three different materials are considered with different strength, work hardening, and ductility, while the grain structure and constituent particle distribution are unaltered. Finite element simulations of the tension tests are conducted using both metal and porous plasticity models and the stress–strain histories of the elements in the minimum section of the specimen are used in strain localization analyses. In these analyses, ductile failure is assumed to occur when the strain rate inside a thin, planar imperfection band becomes infinite. The imperfection band is modelled with porous plasticity, using either a higher initial porosity than in the bulk material or by adding void nucleation. It is found that the ductility of the materials is most accurately captured by modelling the imperfection band by stress-enhanced nucleation.

A numerical study on the effect of porosity distribution on ductile failure using size-dependent finite element-based representative volume elements

In this work, we use the size-dependent Monchiet-Bonnet porous plasticity model to study the influence of void size distribution on ductile fracture. The size effect implies that the void growth depends on the material intrinsic length scale in addition to the plastic deformation and stress state of the material, and smaller voids grow more slowly than larger voids. Finite element-based representative volume elements (RVEs) are built where each element is given an initial porosity and initial void size according to the specified void size distribution. The RVEs are loaded plastically to fracture under different stress states to study the influence of the void size distribution on ductility. The results show that heterogeneity can trigger a macroscopic failure mode caused by localized plastic flow. The onset of localized plastic flow is sensitive to the material heterogeneity while the stress-strain response up to the point of localization is not.

Ефективна пластична поведінка пористих матеріалів зі структурою інверсного опалу

Based on the theoretical principles of the mechanics of composites, the effective plastic behavior of a porous material with a periodic inverse opal structure under uniaxial loading was studied in detail by means of finite element modeling. The creation of such materials is based on the inversion of pores and skeleton of partially sintered dense packing of polystyrene spheres. Electrodeposited nickel was used as the skeleton of the porous material. According to the macroscopic uniaxial loading or unloading, was finding a stress-strain state at the meso-level. For this, equilibrium equations were solved at the meso-level using special boundary conditions for a periodic unit cell. Such boundary conditions relate the problem of equilibrium at the meso-level with the "effective" deformations of the composite. This made it possible to calculate macroscopic residual strains after a cycle of uniaxial loading and unloading and iteratively find the value of effective stress corresponding to residual strains of 0.2%. In this way, the yield strength of inverse opal for uniaxial loading is calculated. At the same time, as a result of finite-element calculations, the transverse deformations coefficient (plastic Poisson ratio) is determined. This coefficient, in turn, makes it possible to approximate the general plastic behavior of the metamaterial by an elliptic yield curve in the plane of invariants of the stress tensor. Invariants mean average pressure and von Mises stress. These calculations were performing for several cases of the inverse opal structure, both with and without an additional coating. Yield stresses under any type of material loading are very sensitive to porosity. In particular, the application of an additional coating, even with a thickness less than 0,05 of the diameter of the spherical pores (initial polymer particles), causes an increase in the yield strength several times. Keywords: metamaterials, inverse opal, porous plasticity model, micromechanics, theory of plasticity.

Experimental and numerical investigation of 3D-Printed bone plates under four-point bending load utilizing machine learning techniques

The fused deposition modeling (FDM) technique is widely used to produce components for various applications and has the potential to revolutionize orthopedic research through the production of custom-fit and readily available biomedical implants. The properties of FDM-produced implants are significantly influenced by processing parameters, with layer thickness being a crucial parameter. This study investigated the effect of layer thickness on the flexural properties of Polylactic Acid (PLA) bone plate implants produced by the FDM technique. Experimental results showed that the flexural strength is inversely proportional to the layer thickness due to the variation of voids in the specimens. A 3D finite element (FE) model was developed using Abaqus/Explicit software by incorporating the Gurson-Tvergaard (GT) porous plasticity model to predict the elastoplastic and damage behavior of specimens with different layer thicknesses. The characterization of the elastoplastic and GT parameters was done using a tensile test and by the calibration of a machine learning algorithm. It was shown that the FE model was able to predict the flexural behavior of 3D-printed solid plates with a maximum error of 6.13% in the maximum load. The optimal layer height was found to be 0.1 mm, providing both high flexural strength and adequate bending stiffness.

Micromechanics-based simulation of quasi-static and dynamic crushing of double-chamber 6000-series aluminium profiles

Numerical simulation is becoming an increasingly important tool in the development of crash components in the automotive industry. With accurate numerical predictions of deformation and fracture, developers may be able to optimize and utilize the energy absorption capacity of the material to its full potential. This study aims to predict the crash behaviour of such components from previous experiments by Qvale et al. (2021) using explicit finite element simulations with a fine discretization of solid elements and the Gurson–Tvergaard–Needleman porous plasticity model. How well the experimental behaviour was predicted by the simulations was evaluated through comparison to computed tomography (CT) scans throughout the entire volume of the profiles. Despite the pragmatic approach of calibrating some parameters of the constitutive model to only uniaxial tension tests, and obtaining the remaining parameters from the literature, important features, such as local details of the folding pattern and the locations of fracture initiation, were represented remarkably well, while there is still room for improvement in the dynamic simulations, as the changes from quasi-static to dynamic axial crushing were slightly underestimated.

On the influence of stress state on ductile fracture of two 6000-series aluminium alloys with different particle content

Tension-torsion tests were conducted on two 6000-series aluminium alloys with different area fraction of constituent particles. The two alloys, denoted alloy A and B, have previously been characterized and found to have similar matrix material, albeit the three times higher area fraction of constituent particles in alloy B than in alloy A. Single notch tube specimens of the two alloys were subjected to fifteen proportional load paths by varying the ratio of axial force and twisting moment, probing stress states from torsion to plane-strain tension. The overall failure strain in the notch was estimated analytically based on the experimental data, whereas finite element simulations were used to determine the stress and strain fields within the notch region and to estimate the local failure strain. The experiments showed that the increased particle content led to a reduction in the local failure strain of alloy B compared with alloy A that varied from 16% to 60%, depending on the stress state, with an average reduction of 39%. While the overall trend was an increasing failure strain with decreasing stress triaxiality, significant influence of the Lode parameter was observed, and thus the increase was not monotonic. Applying a porous plasticity model, localization analyses were conducted to examine the underlying mechanisms for the complex variation of the failure strain with stress state.

Use of modified Madou–Leblond model to predict crack initiation in low alloy steel specimens with different stress states

This paper aims to predict the ductile fracture in low alloy steel specimens with pure tension, pure shear and combined shear–tension type loading using modified Madou–Leblond (MML) porous plasticity model, proposed by the present authors in a recent work. This model includes the effect of localised mode of plastic deformation in a narrow band through the definition of localised porosity level under shear dominated loading conditions. The damage parameter includes contributions due to void growth process under high stress triaxiality level as well as that due to shear localisation process under low stress triaxiality conditions. Experiments have been performed on the round tensile specimen, flat type shear specimen and combined shear–tension specimen. The numerical simulations results obtained using MML model has been compared with those of experiments. It has been shown that the MML model performs well in the prediction of the material softening behaviour and critical fracture location along with the fracture profile in all three types of specimens.

Lode-dependent second porosity in porous plasticity for shear-dominated loadings

Ductile fracture of metallic alloys in shear-dominated loadings is usually modeled either with uncoupled failure criteria or with porous plasticity. In the first approach, criteria depending on the third invariant of the stress tensor: the Lode variable, have been developed. In porous plasticity, a second porosity was added to the GTN yield criterion. In this work, the two approaches are combined with a very simple equation for the second porosity evolution depending on the Lode variable. The new model involves two material parameters. They determine the intensity of the Lode effect and could be calibrated with several specimen geometries in which the Lode variable ranges from zero to some intermediate value (positive or negative). Finite element calculations of a notched KAHN specimen are performed with the Rousselier porous plasticity model (polycrystalline version). The model without the Lode effect first includes the Coulomb-Rousselier-Luo failure model at the slip system scale for ductile damage in shear, in agreement with the experimental load-notch opening displacement curve of an aluminum alloy sheet and with observations of the fracture mechanisms at the micro-scale. Then the Lode effect is also included. The damage mechanisms involved in the different resulting cracks are discussed: flat tensile crack in the notch plane, shear lips on the two specimen surfaces and slant crack, in relation with the local Lode variable. The load-displacement curve softening depends on the model parameters.

Ductile Failure Under Combined Tension and Shear: Simulation Using A Gurson-Type Porous Plasticity Model

A modification of the Gurson–Tvergaard’s yield function based on the micromechanics analysis of the plastic deformation of the porous materials containing spherical empty voids arranged in cubic arrays was proposed by Sean McElwain et al. [2006a, Yield criterion for porous materials subjected to complex stress states, Acta Materialia, 54, 1995–2002; 2006b, Yield functions for porous materials with cubic symmetry using different definitions of yield, Advanced Engineering Materials, 8, 870–876]. The derived yield function possesses in particular the distinctiveness to explicitly depend upon the third stress invariant. On another note, a phenomenological modification of the well-known Gurson–Tvergaard–Needleman (GTN) isotropic hardening model that incorporates damage accumulation under shearing was proposed by Nahshon and Hutchinson, [2008, Modification of the Gurson model for shear failure, European Journal of Mechanics A/Solids, 27, 1–17]. This model incorporates only one extra parameter to account for damage under shear-dominated loadings. In this work, the quasi-static ductile damage and failure behavior of a butterfly specimen is numerically studied using a further extension of this model based on both of the above modifications. To this end, a stress integration algorithm based on the general backwards-Euler return algorithm has been implemented into a finite element code. The simulations show that the proposed extended damage evolution model can capture both the tension and shear failure mechanisms. As long as the softening initiation of the considered specimen is not reached, the computational fracture results highlight the similarities and almost have perfect agreement with those provided by the use of the GTN model. This observation holds for tension-dominated deformation and shear-dominated deformation as well. However, for the later loading, discrepancy shows up at the onset of softening and especially as soon as the failure of material is triggered.

A crystal plasticity model for porous HCP crystals in titanium alloys under multiaxial loading conditions

In this paper, an efficient and effective crystal plasticity model is proposed for porous HCP crystals subject to a variety of multiaxial loading conditions. These conditions include (i) uniaxial, biaxial, and triaxial, (ii) tension and compression, (iii) low and high triaxiality, and (iv) axisymmetric and non-axisymmetric loadings. The framework is based on a combination of variational homogenization, phenomenological extensions, and assumptions motivated by observations from the high-fidelity micromechanical analysis. A novel penalty-free algorithm is employed to reach and maintain a specified stress state while performing representative volume element (RVE)-based crystal plasticity finite element (CPFE) analysis with porosity. The RVE studies demonstrate that the initial porosity, crystallographic orientation, and stress states have a significant effect on the homogenized mechanical responses of microscopically porous RVEs. The proposed porous crystal plasticity model is developed and calibrated using a database generated from the results of micromechanical RVE analysis. The calibrated porous model is reasonably effective in predicting the response of porous crystalline RVEs.

ADAPT — A Diversely Applicable Parameter Identification Tool: Overview and full-field application examples

The free and open source software tool ADAPT (A Diversely Applicable Parameter Identification Tool) for the inverse parameter identification of constitutive material models is presented and applied in combination with Finite Element (FE) simulations. Different types of input data, namely integral data such as forces and field data such as displacement and strain fields, are considered within a multi-objective optimisation scheme. The effect of each of those data sets is investigated in detail for an elasto-plastic and a non-local ductile damage model applied to a dual-phase steel. To demonstrate the wide range of parameter identification applications, several examples are given. The behaviour of a soft polymer using a finite strain non-local visco-elastic damage model is predicted. It is shown that field data such as displacement fields are required to identify the material parameters of a non-local damage model. Differing from existing approaches, high resolution scanning electron microscopy (SEM) data of voids and optical strain fields are employed to identify the material parameters of a Gurson–Tvergaard–Needleman porous plasticity model. This strategy reduces the error in terms of the predicted void area fraction in the particular air bending example considered by approximately 500% compared to a parameter identification approach not using microstructural data. The applied identification scheme contributed to the achievement of a good qualitative agreement for the prediction of the void area fraction in the process chain of hot calibre rolling with subsequent forward rod extrusion of a case-hardened steel by using an uncoupled Lemaitre-type damage model with strain rate and temperature dependency.

A numerical study of a size-dependent finite-element based unit cell with primary and secondary voids

Aluminium alloys contain various types of intermetallic particles with different sizes, such as constituent particles and dispersoids. The main mechanism of ductile fracture in these materials is assumed to be nucleation of voids around the constituent particles, which grow during plastic deformation and eventually coalesce, resulting in material failure. The role of the dispersoids is less certain, but they are assumed to contribute in the last stages of the ductile fracture process. While the constituent particles are in the range of a couple of microns, the size of dispersoids is normally one order of magnitude smaller. To disclose the possible effects of the dispersoids on the ductile fracture process in aluminium alloys, this paper presents a numerical study of a finite-element based unit cell, which consists of a single spherical void embedded in a matrix material represented by a porous plasticity model with void size effects. Accordingly, the single, primary void of the unit cell is assumed to have nucleated on a constituent particle, whereas the matrix porosity is assumed to account for secondary, smaller voids nucleated on dispersoids. The effects of the intrinsic length scale of the matrix material on the void growth and coalescence are studied for a range of stress states, while the initial primary and secondary void volume fractions are kept constant. The secondary voids have a substantial effect on the behaviour of the unit cell when their size is large compared to the intrinsic material length scale, but they were not found to influence the growth of the primary void. Instead, the growth of the secondary voids promotes strain softening and influences the coalescence process of the primary voids, which gradually changes mode from internal necking to loss of load-carrying capacity of the inter-void ligament.

Generalized Interaction Diagrams for Caisson Foundations in Layered Soil under Seismic Conditions

Caissons are rigid foundations most commonly used to support bridge piers. In this study, numerical analysis using the finite-element method based computer program ABAQUS was carried out to study the response of caissons under static and pseudostatic conditions. The caisson was embedded in layered soil comprising a saturated clay layer sandwiched between two sand layers. Sand was modeled using an elastic, Mohr-Coulomb plastic model and clay was modeled using a porous elastic and clay plasticity model to account for soil consolidation. The wall friction angle (δ) and seismic acceleration coefficients (kh and kv) were varied in this study. Calculations were made by executing a mathematical code in MATLAB to obtain the lateral soil pressure, maximum and minimum base pressures, tilt, shift, and point of rotation of the caisson. The analysis was further extended to develop three-dimensional (3D) interaction diagrams relating the combination of applied vertical load (V), lateral load (Q), and moment (M) in nondimensional form required to cause caisson failure for different wall friction angles and seismic acceleration coefficients. These diagrams can be used for the safe design of caissons under different seismic conditions and ground conditions for a variety of design loads.

Development of interpretable, data-driven plasticity models with symbolic regression

In many applications, such as those which drive new material discovery, constitutive models are sought that have three characteristics: (1) the ability to be derived in automatic fashion with (2) high accuracy and (3) an interpretable nature. Traditionally developed models are usually interpretable but sacrifice development time and accuracy. Purely data-driven approaches are usually fast and accurate but lack interpretability. In the current work, a framework for the rapid development of interpretable, data-driven constitutive models is pursued. The approach is characterized by the use of symbolic regression on data generated with micromechanical finite element models. Symbolic regression is the search for equations of arbitrary functional form which match a given dataset. Specifically, an implicit symbolic regression technique is developed to identify a plastic yield potential from homogenized finite element response data. Through three controlled test cases of varying complexity, the approach is shown to successfully produce interpretable plasticity models. The controlled test cases are used to investigate the robustness and scalability of the method and provide reasonable recommendations for more complex applications. Finally, the recommendations are used in the application of the method to produce a porous plasticity model from data corresponding to a representative volume element of voids within a metal matrix.

Numerical Simulation of Some Steel Structural Elements with Uncertain Initial Porosity

The main research purpose of this work was to study the elasto-plastic responses of some fundamental steel structural elements exhibiting stochastic volumetric microvoids. The constitutive model of the steel material was consistent with the deterministic Gurson–Tvergaard–Needleman (GTN) porous plasticity model, where some of the microvoids parameters have additionally been defined as Gaussian random variables. The iterative stochastic finite element method implemented based on the-tenth order stochastic perturbation technique was utilized in numerical experiments. An interoperability of the computer algebra system MAPLE 2019 and the finite element method system ABAQUS was employed to study the influence of the initial microvoids f0 with uncertainty in the structural steel on the statistical scattering of the resulting stresses and deformations. The basic probabilistic characteristics of the structural response were computed and contrasted with statistical estimators inherent in the Monte–Carlo simulation and also with the results obtained from the semi-analytical probabilistic method. Reliability indices according to the first-order reliability method (FORM) were also calculated. Two numerical illustrations included the (i) tension test of the round cylindrical steel rebar and the (ii) bending test of the steel I-beam restrained at both its ends. Expectations and coefficients of variation of the structural responses confirmed here the importance of the microvoids for the stochastic elasto-plastic behavior of some basic engineering structures, where tensile stress plays an important role in designing procedures.

Finite element simulation of ductile fracture in polycrystalline materials using a regularized porous crystal plasticity model

In the present study, a hypoelastic–plastic formulation of porous crystal plasticity with a regularized version of Schmid’s law is proposed. The equation describing the effect of the voids on plasticity is modified to allow for an explicit analytical solution for the effective resolved shear stress. The regularized porous crystal plasticity model is implemented as a material model in a finite element code using the cutting plane algorithm. Fracture is described by element erosion at a critical porosity. The proposed model is used for two test cases of two- and three-dimensional polycrystals deformed in tension until full fracture is achieved. The simulations demonstrate the capability of the proposed model to account for the interaction between different modes of strain localization, such as shear bands and necking, and the initiation and propagation of ductile fracture in large scale polycrystal models with detailed grain description and realistic boundary conditions.

Micromechanical modelling of ductile fracture in pipeline steel using a bifurcation-enriched porous plasticity model

In this paper, we investigate the possibility of predicting ductile fracture of pipeline steel by using the Gurson–Tvergaard–Needleman (GTN) model where the onset of void coalescence is determined based on in situ bifurcation analyses. To this end, three variants of the GTN model, one of which includes in situ bifurcation, are calibrated for a pipeline steel grade X65 using uniaxial and notch tension tests. Then plane-strain tension tests and Kahn tear tests of the same material are used for assessment of the credibility of the three models. Explicit finite element simulations are carried out for all tests using the three variants of the GTN model, and the results are compared to the experimental data. The capability of the simulation models to capture onset of fracture and crack propagation in the pipeline steel is evaluated. It is found that the use of in situ bifurcation as a criterion for onset of void coalescence in each element makes the GTN model easier to calibrate with less free parameters, all the while obtaining similar or even better predictions as other widely used formulations of the GTN model over a wide range of different stress states.

Effect of Lüders and Portevin–Le Chatelier localization bands on plasticity and fracture of notched steel specimens studied by DIC and FE simulations

The impact of the Portevin–Le Chatelier (PLC) effect on plasticity and fracture ahead of a severe notch is investigated by DIC field measurements for C–Mn steel. The PLC effect causes an intermittent activity of highly localized strain rate bands. This first high temperature investigation of the effect of a notch on the PLC localization bands is successfully reproduced numerically in terms of strain localization in 3D FE simulations. This thereby validates quantitatively the McCormick-type model under these complex conditions. It is found that, when the PLC effect is at play at elevated temperature (175°C), a flat to slant crack transition is observed. In contrast, at room temperature, when only Lüders bands are active, the crack remains mostly flat, i.e. normal to the loading direction of the SENT samples. This behaviour may be related to the early loss of symmetry and intermittency of the plastic zone that is found for PLC even affecting the initial Lüders bands at elevated temperature. During the slant fracture at high temperature, only half the fracture energy is absorbed compared to flat fracture at room temperature. The McCormick-type model simulations predict slant strain rate bands through the sample thickness, that are consistent with the slant fracture found at elevated temperature. Accordingly, no slant bands are found for simulations outside the PLC domain. Both experiments and simulations show PLC strain localization bands that are flip-flopping up and down during crack propagation. By combining the McCormick-type model with a Rousselier porous plasticity model, the flat fracture of the sample at room temperature and the associated macroscopic curve are reproduced successfully, but the toughness at elevated temperature is overestimated.