Isotropic Work Hardening Research Articles

Inverse Method to Determine Parameters for Time-Dependent and Cyclic Plastic Material Behavior from Instrumented Indentation Tests.

Indentation is a versatile method to assess the hardness of different materials along with their elastic properties. Recently, powerful approaches have been developed to determine further material properties, like yield strength, ultimate tensile strength, work-hardening rate, and even cyclic plastic properties, by a combination of indentation testing and computer simulations. The basic idea of these approaches is to simulate the indentation with known process parameters and to iteratively optimize the initially unknown material properties until just a minimum error between numerical and experimental results is achieved. In this work, we have developed a protocol for instrumented indentation tests and a procedure for the inverse analysis of the experimental data to obtain material parameters for time-dependent viscoplastic material behavior and kinematic and isotropic work-hardening. We assume the elastic material properties and the initial yield strength to be known because these values can be determined independently from indentation tests. Two optimization strategies were performed and compared for identification of the material parameters. The new inverse method for spherical indentation has been successfully applied to martensitic steel.

Elucidation of the factors controlling interface decohesion and particle fracture in a friction stir alloyed Al‒Fe alloy system

In the present study, an Al‒Fe alloy was fabricated by the solid-state friction stir alloying (FSA) technique. The FSA was carried out at a constant linear speed and variable tool rotational speed ranging from 600 to 1200 RPM. The ultimate tensile strength (UTS) and total elongation (TE) of the alloy fabricated at lower tool rotational speeds (600 and 900 RPM) are ∼ 300 MPa and ∼ 10%, respectively. Whereas the UTS and TE of the alloy fabricated at higher tool rotational speeds (1200 and 1500 RPM) are ∼ 110 MPa and ∼ 20%, respectively. The transition in the fracture behavior of the alloy from brittle to ductile was observed at a tool rotational speed of 900 RPM. The brittle fracture of alloys fabricated at low tool rotational speeds is primarily associated with Al/Fe interface decohesion and coarse Fe particle fracture. Whereas the void growth and coalescence are responsible for ductile fracture in alloys fabricated at higher tool rotational speeds. The transition of the dominant strengthening mechanism from dislocation and precipitation strengthening in alloys fabricated at lower tool rotational speeds to Hall‒Petch strengthening in alloys fabricated at high rotational speeds is the primary reason for the drastic change in the fracture behavior of these alloys. The coarse undissolved Fe particles and finely dispersed Al13Fe4 contributed significantly to kinematic and isotropic work hardening, which resulted in an extremely high work hardening rate in the alloys fabricated at lower tool rotational speeds.

An equivalent work-hardening description of an interstitial-free steel sheet based on uniaxial tensile and hydraulic bulge tests

In this work, the plastic behavior of an interstitial-free steel sheet was first evaluated by standard uniaxial tensile and hydraulic bulge tests to reproduce the intermediate and high straining levels commonly observed in sheet metal forming processes. Next, a simple procedure was proposed to obtain an equivalent work-hardening curve, representing the high strain domain instead of the conventional uniaxial tensile data. This procedure involves an anisotropy parameter calculated based on Hill's quadratic criterion using the measured r-values in the rolling and transverse sheet directions. Under the assumption of an isotropic work-hardening, the equivalent curve is defined from the true stress and true thickness plastic-strain results of the hydraulic bulge test. The transformed data were fitted to usual power-law type equations to obtain an adequate work-hardening description for finite element numerical simulation purposes. Based on the goodness-of-fit measures, namely, R-squared, root-mean-square error, and root-mean-square of residual percentage, the combined Swift–Hockett–Sherby equation has proved to be the most suitable choice to describe the plastic behavior of the investigated interstitial-free steel under both intermediate and high strains conditions.

Sensitivity of friction coefficients, material constitutive laws and yield functions on the accuracy of springback prediction for an automotive part

High Strength Steel is more used in automotive industry to combine safety improvements and weight reductions of the vehicles. However, these steels raise new challenges for manufacturer, especially for springback and its prediction. Finite Element Analysis is widely used for the prediction of springback but accuracy is strongly depending on numerical as well as physical parameters. Such parameters affecting prediction are for example frictional behavior, constitutive model, loci function. In this paper the sensitivity of friction coefficients and the influence of some material laws on the springback prediction were studied and analyzed in details. Several numerical models are investigated in this work in order to allow the different aspects of materials mechanical behavior to be better described: isotropic work-hardening (Swift, Hockett-Sherby, SHS), isotropic-kinematic hardening, yield loci (Hill 48, Hill 90 and Barlat 91). In this study, we use an industrial example of the deep drawing process: the side sill closing with the material steel TRIP800. All the simulations are carried out with the commercial code Pam-Stamp 2G and comparison with the experimental results are presented.

A double incremental variational procedure for elastoplastic composites with combined isotropic and linear kinematic hardening

We investigate the nonlinear behavior of elasto-plastic composites with isotropic and linear kinematic hardening. We first rely on the incremental variational principles introduced by Lahellec and Suquet (2007a). We also take advantage of an alternative formulation, recently proposed by Agoras et al. (2016) for visco-plastic composites without hardening, which consists in a double application of the variational procedure of Ponte-Castañeda. We extend in this paper this approach to elasto-plastic composites with combined linear kinematic and isotropic work-hardening. The first application of the variational procedure linearizes the local behavior, including hardening, and leads to a thermo-elastic Linear Comparison Composite (LCC) with a heterogeneous polarization field inside the phases. The second one deals with the heterogeneity of the polarization and results in a new thermo-elastic LCC with a per-phase homogeneous polarization field, which effective behavior can then be estimated by classical linear homogenization schemes. We develop and implement this new incremental variational procedure for composites comprised of linear elastic spherical particles isotropically distributed in an elasto-plastic matrix. The predictions of the model are compared with results available in the literature for cyclic proportional and non-proportional loadings. New results for elasto-plastic composites with combined isotropic and kinematic hardening are also provided. They are in good agreement with the numerical computations we carried out, at both local and macroscopic scales.

A global-local approach for the elastoplastic analysis of compact and thin-walled structures via refined models

A computationally efficient framework has been developed for the elastoplastic analysis of compact and thin-walled structures using a combination of global-local techniques and refined beam models. The theory of the Carrera Unified Formulation (CUF) and its application to physically nonlinear problems are discussed. Higher-order models derived using Taylor and Lagrange expansions have been used to model the structure, and the elastoplastic behavior is described by a von Mises constitutive model with isotropic work hardening. Comparisons are made between classical and higher-order models regarding the deformations in the nonlinear regime, which highlight the capabilities of the latter in accurately predicting the elastoplastic behavior. The concept of global-local analysis is introduced, and two versions are presented - the first where physical nonlinearity is considered for both the global and local analyses, and the second where nonlinearity is considered only for the local analysis. The second version results in reasonably accurate results compared to a full 3D finite element analysis, with a twofold reduction in the number of degrees of freedom.

<b>Influence of damping ratio on a structural optimization design considering a dynamic analysis within the time domain

Structural optimization has received increasing attention in several different areas of engineering and has been identified as the most challenging and economically rewarding task in the field of structural design. In this context, the current paper proposes a methodology based on Evolutionary Structural Optimization (ESO) that corresponds to an evolutionary procedure applied for topological optimization in which the finite elements with the lowest stress levels are progressively removed from a structure. The optimization studies are applied for structures subjected to a transient dynamic response where different damping ratios are applied in the physical models, since its determination is extremely hard and can even change the structural stiffness in case of elastoplastic regime. Thus, a nonlinear behavior is considered to evaluate the effects for each damping ratio, and elastoplasticity theory for small strains is extended for a von Mises material with linear, isotropic work-hardening. For this purpose it is possible to evaluate a combination of different optimal topologies for the different damping ratios through an algorithm developed in the Python programming language. The stress levels present such a difference for each linear and nonlinear response, which characterizes a marked change in the structural stiffness of each analyzed model.

An adapted yield criterion for the evolution of subsequent yield surfaces

In numerical analysis of sheet metal forming processes, the anisotropic material behaviour is often modelled with isotropic work hardening and an average Lankford coefficient. In contrast, experimental observations show an evolution of the Lankford coefficients, which can be associated with a yield surface change due to kinematic and distortional hardening. Commonly, extensive efforts are carried out to describe these phenomena. In this paper an isotropic material model based on the Yld2000-2d criterion is adapted with an evolving yield exponent in order to change the yield surface shape. The yield exponent is linked to the accumulative plastic strain. This change has the effect of a rotating yield surface normal. As the normal is directly related to the Lankford coefficient, the change can be used to model the evolution of the Lankford coefficient during yielding. The paper will focus on the numerical implementation of the adapted material model for the FE-code LS-Dyna, mpi-version R7.1.2-d. A recently introduced identification scheme [1] is used to obtain the parameters for the evolving yield surface and will be briefly described for the proposed model. The suitability for numerical analysis will be discussed for deep drawing processes in general. Efforts for material characterization and modelling will be compared to other common yield surface descriptions. Besides experimental efforts and achieved accuracy, the potential of flexibility in material models and the risk of ambiguity during identification are of major interest in this paper.

Plasma-sprayed coatings: Identification of plastic properties using macro-indentation and an inverse Levenberg–Marquardt method

Plasma-sprayed coatings are widely used for thermal protection and wear stability of structural components. These coatings feature an anisotropic porous structure as a result of the thermal spraying process. Although current literature provides methods for the identification of elastic properties of these materials, to our knowledge little research is dedicated to describing their plastic behavior, especially when these coatings are subjected to macro-scale contacts encountered in industrial applications. In this work we present a novel inverse method for the identification of plastic properties of thick plasma-sprayed coatings by means of macro-indentation and finite element simulations coupled to a Levenberg–Marquardt optimization. This optimization aims to fit numerically generated residual indentation profiles to the experimentally obtained ones. For the description of the coatings’ plastic behavior we made use of the Gurson–Tvergaard plasticity criterion coupled to a linear isotropic work hardening of the matrix. This criterion is appropriate for ductile porous solids as it takes into account the hydrostatic pressure, and it is readily implementable in commercial finite element software. The constitutive parameters to be identified include the yield strength σy0 and the work hardening coefficient K of the solid matrix as well as two dimensionless fitting parameters q1 and q2 and the void fraction f. For the given plasma-sprayed coating we could show that the proposed method is capable of identifying these parameters after as little as three iterations where σy0=1624MPa, K=33,340MPa, q1=3.29, q2=4.60 and f=4.2%.

Determination of a Set of Constitutive Equations for an Al-Li Alloy at SPF Conditions

Uniaxial tensile tests of aluminium-lithium alloy AA1420wereconducted at superplastic forming conditions. The mechanical properties of this Al-Li alloy were then modelled by a set of physicallybased constitutive equations. The constitutive equations describe the isotropic work hardening,recovery and damage by dislocation density changes and grain size evolution. Based on a recent upgraded optimisation technique, the material constants for these constitutive equations were determined.

SIMULAÇÃO NUMÉRICA DO PRIMEIRO PASSE DE LAMINAÇÃO A FRIO DE UMA TIRA DE AÇO C-Mn

This work presents the modeling by the finite element method of the first cold rolling pass of a four-high mill at the CSN steel plant. For this purpose, a two dimensional model is proposed considering the work-roll as a rigid surface together with the assumptions of plane-strain state for the strip and material isotropy with nonlinear isotropic workhardening. A C-Mn steel strip with initial thickness of 3.5 mm was first characterized by means of uniaxial tensile tests in specimens cut along the rolling direction to evaluate the work-hardening parameters. In the finite element model, the contact is defined by means of the Coulomb’s law to describe the friction between the work-roll and the strip surfaces. Moreover, the industrial back and forward tensions are also taken into account as prescribed conditions in the finite element model. The numerical predictions of the rolling load show a good agreement with the measured data and the forecasted value is determined with the slab-method.

On the Stiffness of the Tangent Modulus Tensor in Elastoplasticity

In finite element analysis of pressure vessels undergoing elastoplastic deformation, low stiffness of the tangent modulus tensor will engender low stiffness in the tangent stiffness matrix, posing a risk of computational difficulties such as poor convergence. The current investigation presents the explicit tangent modulus tensor in an elastoplastic model based on a Von Mises yield surface with isotropic work hardening, and the associated flow rule. The stiffness of the tangent modulus tensor is assessed by deriving explicit expressions for its minimum eigenvalue using both tensor diagonalization and Rayleigh quotient minimization. The derived expressions are validated computationally. Using the minimum eigenvalue, the stiffness is found to depend on the current path in stress space. The results of the current investigation suggest a way of following a stress path, which bypasses low stiffness, while attaining the prescribed load.

Detailed and simplified non-linear models for timber-framed masonry structures

Abstract The need for improved methodologies to describe the post-elastic behaviour of existing structures in the framework of seismic vulnerability assessment has long been recognised. The study presented herein deals with the non-linear seismic response of timber-framed (T-F) masonry structures, such as those found in traditional edifices of cultural heritage. T-F masonry generally consists of masonry walls reinforced with timber elements, including horizontal and vertical elements, as well as X-type diagonal braces. Since the Bronze Age T-F buildings were common in regions where moderate-to-strong earthquakes were frequent. There is ample historical evidence that the embodiment of timber elements in masonry walls is closely related to earthquakes. The paper focuses on the description of the seismic response of T-F structures by means of a detailed analytical model. Although elastic analysis can adequately identify regions with high stresses, it fails to capture the redistribution of stresses and the ensuing failure mechanism. The simulation of T-F masonry is made here using a plasticity model. Non-linear laws for the materials, such as a trilinear stress-strain curve for monotonic loading of timber and a Mohr-Coulomb contact law for wooden members, are used to express their behaviour under moderate and high stress levels. An associated flow rule is assumed and Hill's yield criterion is adopted with isotropic work-hardening. Masonry infills are not included in the model due to their insignificant contribution after the initial elastic stage of the response. The proposed finite element model is intended for a detailed non-linear static analysis of parts of a building. A simplified model using beam and link elements with non-linear axial springs is also developed, which is appropriate for 2D non-linear analysis of common buildings. Both models are validated using experimental results of three T-F masonry walls obtained from the literature. Finally a non-linear static analysis of the facade of an existing building situated in the island of Lefkas, Greece is performed.

Evaluation of constitutive models for textured aluminium alloys using plane-strain tension and shear tests

The constitutive modelling of the strongly textured aluminium alloys AA6063-T6 and AA7003-T6 is studied. The materials were delivered in the form of flat extruded profiles. Plane-strain tension and shear tests in the plane of the flat profiles are performed. The tests are then used to evaluate a constitutive model including an anisotropic yield function, the associated flow rule and a nonlinear isotropic work-hardening rule. The parameters of the yield criterion and the work-hardening rule were identified primarily from uniaxial tension tests in different in-plane directions. It is suggested how analytical considerations and the results from the plane-strain tension and shear tests may be used to obtain a more accurate calibration of the anisotropic yield criterion. To further assess the constitutive model, finite element simulations of the plane-strain tension and shear tests are carried out and the results compared with the experimental force-elongation curves. Significant deviations in the experimental and predicted results are disclosed, and attributed partly to the parameter identification, primarily based on uniaxial tension tests, and partly to the assumption of isotropic work-hardening. Polycrystal plasticity calculations are carried out for simple shear of the AA7003-T6 material, indicating that texture evolution plays an important role in determining the response in this test already at moderate strains.

Finite element analysis of the influence of the restraining force in the draw bend test

Draw restraining force is used in sheet metal forming processes to regulate material flow and control springback in the parts. In this paper, the influence of draw restraining force on springback behaviour of dual phase steels is investigated, through numerical simulations of the draw bend test. Three grades of dual phase steels are submitted to different normalized back forces in the range from 0.5 to 1.1 of the initial yield stress. All simulations were performed with the DD3IMP in-house code. The study highlights the importance of an accurate modelling of the isotropic work hardening of dual phase steels for an accurate springback prediction.

On the application of FEM to deformation of high-density polyethylene

This paper presents a study that uses finite element method (FEM) to simulate deformation behaviour of high-density polyethylene (HDPE) when subjected to tensile loading, either without or with the presence of pre-cracks. For the former, dog-bone-shaped model of HDPE was deformed under uni-axial tensile (UT) loading beyond the initial yielding, to generate stable necking in the gauge section. The paper proposes a simple correction factor to determine the equivalent von Mises stress that is needed for the input to the FEM model, in order to generate the same loading level as that observed experimentally for neck propagation. The paper points out that such consistency in the loading level could not be generated in the past mainly because of a misconception that axial stress in the neck was regarded as the equivalent of the von Mises stress. The study also explored the consideration of crack growth in double-edge-notched tensile (DENT) specimen, and showed that the conventional von Mises yield function, with the assumption of isotropic work hardening, cannot be directly applied to simulate the deformation behaviour of DENT specimen. Instead, empirical parameters were employed to reflect the polymer orientation during the necking process. The paper shows that FEM models for both UT and DENT tests can reproduce the experimental load–displacement curves quite accurately, and concludes that with a proper yield function to reflect the deformation involved in the mechanical tests, the deformation behaviour observed experimentally can be accurately mimicked using the FEM simulation.

Study on the influence of work-hardening modeling in springback prediction

The main goal of this study is to evaluate the influence of work-hardening modeling in springback prediction in the first phase of the Numisheet’05 “Benchmark 3”: the U-shape “Channel Draw”. Several work-hardening constitutive models are used in order to allow the different materials’ mechanical behavior to be better described: the Swift law (a power law) or a Voce type saturation law to describe the classical isotropic work-hardening; a Lemaître and Chaboche type law to model the non-linear kinematic hardening, which can be combined with the previous two; and Teodosiu’s microstructural work-hardening model. This analysis was carried out using two steels currently used in the automotive industry: mild (DC06) and dual phase (DP600). Haddadi et al. [Haddadi, H., Bouvier, S., Banu, M., Maier, C., Teodosiu, C., 2006. Towards an accurate description of the anisotropic behaviour of sheet metals under large plastic deformations: Modelling, numerical analysis and identification. Int. J. Plasticity 22 (12), 2226-2271] performed the mechanical characterization of these steels, as well as the identification of the constitutive parameters of each work-hardening model, based on an appropriate set of experimental data such as uniaxial tensile tests, monotonic and Bauschinger simple shear tests and orthogonal strain-path change tests, all at various orientations with respect to the rolling direction of the sheet. All the simulations were carried out with the in-house FE code DD3IMP. The selected sheet metal formed component induces high levels of equivalent plastic strain. However, for the several work-hardening models tested, the differences in springback prediction are not significantly higher than those previously reported for components with lower equivalent plastic strain levels. It is shown that these differences can be related to the predicted through-thickness stress gradients. The comparative significance of both equivalent plastic strain levels and strain-path changes in the through-thickness stress gradients is discussed.

A comparative analysis of elasto-plastic constitutive models for porous sintered materials

The objective of the paper is to present a general algorithm that enables, in a simple way, the incorporation of a set of well known elasto-plastic constitutive models, for porous sintered materials, and to perform a comparative analysis of these various models. These models were modified in order to incorporate an isotropic work hardening and a porosity coupling behavior and implemented in the ABAQUS @ finite element program. The proposed algorithm considered an operator split technique which decomposed the problem into an elastic predictor followed, if necessary, by a plastic corrector step. The algorithm is presented in a general frame work and is thereafter particularized for a given set of constitutive models for porous sintered materials. In the work is also implemented some of these published elasto-plastic models that have not been implemented previously into FEM programs. Finally, some simple examples are solved with the aim of comparing the performance of these different models.

A Three Dimensional Discrete Dislocation Dynamics Analysis of Cyclic Straining in 316L Stainless Steel

The early stages of the formation of dislocation microstructures in low strain fatigue are analysed,using three-dimensional discrete dislocation dynamics modelling (DDD). A detailed analysis of the simulated microstructures provide a detailed scheme for the persistent slip band formation, emphasizing the crucial role of cross-slip for both the initial strain spreading inside of the grain and for the subsequent strain localization in the form of slip bands. A new ad-hoc posttreatment tool evaluates the surface roughness as the cycles proceed. Slip markings and their evolutions are analysed, in relation to the dislocation microstructure. This dislocation-based study emphasizes the separate contribution of plastic slip in damage nucleation. A simple 1D dislocation based model for work-hardening in crystal plasticity is proposed. In this model, the forest dislocations are responsible for friction stress (isotropic work-hardening), while dislocation pile-ups and dislocation trapped in Persistent Slip Bands (PSB) produce the back stress (kinematic workhardening). The model is consistent with the stress-strain curves obtained in DDD. It is also consistent with the stress-strain curves experimentally obtained for larger imposed strain amplitudes.

Constitutive model for the elastic–plastic analysis of porous sintered materials

It is a known fact that it is very difficult to implement the algorithm of plasticity of porous materials in FEM commercial programs for different types and characteristics. The aim of this paper is to put forward a flexible model of plasticity of porous materials by implementing it into the general purpose finite element program ABAQUS®. A modified algorithm is developed in this paper which takes into account the effect of an isotropic work hardening of the porous material. This is incorporated into the FEM program to yield a flexible solution for different yielding criterion. The present criterion is compared with other known published work and was found to be in good agreement. It was found that all the failure criteria used were successfully implemented using the newly developed algorithm. Also, the present criterion is adequately adjusted to fit all other results obtained from other criteria of failure. Finally, some of these published criteria had not been implemented previously for use in the FEM programs, but in this paper were made to do so. Other theoretical results will also be presented.