Orthotropic Yield Criterion Research Articles

A covariant formulation for finite strain modelling of orthotropic elasticity and orthotropic plasticity with plasticity-induced evolution of orthotropy: Application to natural fibres

We introduce a rate-independent model for orthotropic elastic and orthotropic plastic material behaviour in a hyper-elasto-plastic framework at finite strains. The model is based on the postulate of covariance and does not rely on a multiplicative decomposition of the deformation gradient. Furthermore, a plastic-deformation-induced evolution of orthotropy is considered, similar to the notion of plastic spin. We propose that the orthotropic strain energy function and the orthotropic yield criterion are guided by identical structural tensors that evolve with plasticity. The modelled material behaviour is significant for natural fibres such as flax, hemp, or pulp fibres. Our formulation has three findings. Firstly, the covariant formulation of plasticity provides rate equations for the plastic variables and the structural tensors suitable for reproducing stress–strain diagrams of natural fibres. Secondly, the introduction of plastic-deformation-induced evolution of orthotropy in the proposed covariant setting results in a non-associative plasticity algorithm. Thirdly, the covariant setting allows the incorporation of suitable constitutive equations for the structural tensors to evolve orthotropy. The latter successfully models the stiffness increase in stress–strain diagrams of cyclic tensile tests of natural fibres.

An orthotropic plasticity model at finite strains with plasticity-induced evolution of orthotropy based on a covariant formulation

We present a rate-independent model for isotropic elastic–orthotropic plastic material behaviour in a hyper-elasto-plastic setting at finite strains, which is based on a covariant formulation that includes plastic-deformation-induced evolution of orthotropy. The model relies on a treatment by Lu and Papadopoulos, who made use of the postulate of covariance for an anisotropic elasto-plastic solid and derived constitutive equations of evolving anisotropies at finite strains. The latter is tantamount to the notion of plastic spin. This treatment does not rely on a multiplicative decomposition of the deformation gradient. We test our model on in-plane sheet-metal forming processes, which are governed by the evolution of pre-existing preferred material orientations. Hence, we advocate an orthotropic yield criterion directed by evolving structural tensors to describe this material behaviour. Our formulation yields two key findings. Firstly, the covariant formulation of plasticity yields suitable evolution equations for the structural tensors characterising the symmetry group of the orthotropic yield function. Secondly, the constitutive equations for the plastic variables and the structural tensors, which are both symmetric second-order tensors, give results that are in good agreement with experimental and numerical findings from in-plane sheet forming processes.

An effective parameterization of texture-induced viscous anisotropy in orthotropic materials with application for modeling geodynamical flows

In this article, we describe the mathematical formulation and the numerical implementation of an effective parametrization of the viscous anisotropy of orthorhombic materials produced by crystallographic preferred orientations (CPO or texture), which can be integrated into 3D geodynamic and materials science codes. Here, the approach is applied to characterize the texture-induced viscous anisotropy of olivine polycrystals, the main constituent of the Earth's upper mantle. The parameterization is based on the Hill (1948) orthotropic yield criterion. The coefficients of the Hill yield surface are calibrated based on numerical tests performed using the second order Viscoplastic Self-consistent (SO-VPSC) model. The parametrization was implemented in a 3D thermo-mechanical finite-element code developed to model large-scale geodynamical flows, in the form of a Maxwell rheology combining isotropic elastic and anisotropic non-linear viscous behaviors. The implementation was validated by comparison with results of the analytical solution and of the SO-VPSC model for simple shear and axial compression of a homogeneous anisotropic material. An application designed to examine the effect of texture-induced viscous anisotropy on the reactivation of mantle shear zones in continental plates highlights unexpected couplings between localized deformation controlled by variations in the orientation and intensity of the olivine texture in the mantle and the mechanical behavior of the elasto-viscoplastic overlying crust. Importantly, the computational time only increases by a factor 2-3 with respect to the classic isotropic Maxwell viscoelastic rheology.

An Exact Axisymmetric Solution in Anisotropic Plasticity

A hollow cylinder of incompressible material obeying Hill’s orthotropic quadratic yield criterion and its associated flow rule is contracted on a rigid cylinder inserted in its hole. Friction occurs at the contact surface between the hollow and solid cylinders. An axisymmetric boundary value problem for the flow of the material is formulated and solved, and the solution is in closed form. A numerical technique is only necessary for evaluating ordinary integrals. The solution may exhibit singular behavior in the vicinity of the friction surface. The exact asymptotic representation of the solution shows that some strain rate components and the plastic work rate approach infinity in the friction surface’s vicinity. The effect of plastic anisotropy on the solution’s behavior is discussed.

A Novel Approach to Predict the Process-Induced Mechanical Behavior of Additively Manufactured Materials

The grain structure and texture of additively manufactured materials depend strongly on the local temperature gradients during the solidification of the material. These grain structures and textures influence the mechanical behavior, ranging from isotropy to transversal and orthotropic symmetry. In the present contribution, a cellular automaton is used to model the grain growth during selective electron beam melting. The resulting grain structures and textures serve as input for a mesoscopic mechanical model. The mechanical behavior on the mesoscale is modeled by means of gradient-enhanced crystal plasticity, applying the finite element method. Computational homogenization is applied to determine the resulting macroscopic elastic and plastic properties of the additively manufactured metals. A general orthotropic yield criterion is identified by means of the initial yield loci computed with mesoscopic simulations of representative volume elements. The numerical results are partly validated with experimental data.

Orthotropic Behaviour of Magnesium AZ31 Sheet during Strain Localization

It is known that metallic materials are characterized by anisotropy of their mechanical properties, with this being attributed to the conditions during the manufacturing process. For sheet metals, this anisotropy occurs symmetrically to the three orthogonal axes of the rolling, transverse and normal direction. This characteristic is referred to as orthotropic behaviour and manifests itself, for example, in earing during cupping tests. Therefore, orthotropic yield criteria are highly relevant for the numerical simulation of sheet metal forming processes. The Lankford coefficient, also known as the r-value, is a good experimental measure for characterizing orthotropic ductile behaviour of sheets, and can easily aid in parameter identification for yield criteria such as the Hill approaches. In the present investigations, Lankford coefficients were determined as a function of local strain in uniaxial tensile tests through high-resolution digital image correlation. The sample direction was varied between 0°, 45° and 90° to the rolling direction and the test temperature varied from RT to 350 °C at three different strain rates (0.01-1 s-1). By means of a novel backward analysis, the measuring range for the Lankford coefficients was positioned exactly in the necking area. An increase in temperatures showed a decrease in the initial Lankford coefficient. The results showed non-constant Lankford coefficients and commence the course of a natural exponential function depending on the local strain. Regardless of strain rate, the results revealed that the Lankford coefficients (r-values) at 150 °C, 250 °C and 350 °C approaches a steady-state of r = 1.14 with strains greater than 50 %.

Influence of the Replacement of the Actual Plastic Orthotropy with Various Approximations of Normal Anisotropy on Residual Stresses and Strains in a Thin Disk Subjected to External Pressure

Plastic anisotropy is very common to metallic materials. This property may significantly affect the performance of structures. However, the actual orthotropic yield criterion is often replaced with a criterion based on the assumption of normal anisotropy. The present paper aims to reveal the influence of this replacement on the distribution of strains and residual strains in a thin hollow disk under plane stress conditions. The boundary-value problem is intentionally formulated such that it is possible to obtain an exact semi-analytical solution without relaxing the boundary conditions. It is assumed that the disk is loaded by external pressure, followed by elastic unloading. The comparative analysis of the distributions of residual strains shows a significant deviation of the distribution resulting from the solutions based on the assumption of normal anisotropy from the distribution found using the actual orthotropic yield criterion. This finding shows that replacing the actual orthotropic yield criterion with the assumption of normal anisotropy may result in very inaccurate predictions. The type of anisotropy accepted is of practical importance because it usually results from such processes as drawing end extrusion with an axis of symmetry.

Modeling the effect of notch geometry on the deformation of a strongly anisotropic aluminum alloy

In this study, an elastic-plastic model with yielding described by a newly proposed orthotropic yield criterion was used to model the unusual deformation of a strongly textured AA6060 alloy. Available experimental data from tension tests and results of crystal plasticity simulations were used to determine the anisotropy coefficients involved in the yield criterion. Virtual material tests using a recent polycrystalline model were performed to obtain flow stresses for loadings where experimental data were not available. The capability of the elastic-plastic model to account for the distinct anisotropy of the material is demonstrated through comparison of finite element simulations and experimental tests on both smooth and notched axisymmetric specimens of the AA6060 alloy. Specifically, for the smooth specimen, the model predicts that the minimum cross-section evolves from a circle to an ellipse while for the notched specimens, the minimum cross-section evolves from a circular shape to an approximately rectangular, or rhomboidal shape, respectively as observed in the experiments. This model can be easily implemented in finite element codes, requires reduced CPU time compared to crystal plasticity finite element simulations, and can be applied in simulations of large-scale structural applications.

Predictive Capabilities of Non-Quadratic Orthotropic Criteria

Very recently, it was shown that the orthotropic yield criteria of Barlat et al. [1] and Karafillis and Boyce [2] are in fact simple polynomials (see [3], [5]). Using these new mathematical expressions, the anisotropy coefficients involved in these criteria can be obtained using analytical formulas involving only experimental flow stresses, or formulas involving only Lankford coefficients for three orientations. In this paper, the benefits brought about by these new expressions in terms of fundamental understanding of these criteria and their predictive capabilities are discussed. Moreover, the implications in terms of ease of the identification of the anisotropy coefficients are demonstrated. Specifically, it is shown that using the new analytical identification procedures it becomes possible to obtain a reliable representation of a given material’s anisotropy even if the available mechanical characterization data are limited.

Forming of Materials with Cubic Crystal Structure

Generally, the plastic behavior of orthotropic metallic materials with cubic crystal structure is modelled with yield criteria that predict the same response in tension and compression. One drawback associated to using orthotropic yield criteria for simulation of forming processes is related to the uncertainties introduced by the methods used for parameter identification. Very recently, it was shown that advanced non-quadratic criteria such as Yld91 can be represented as polynomials in terms of stress components. Using this equivalent expression, the anisotropy coefficients involved in Yld91 can be obtained using analytical formulas in terms of only experimental flow stresses, or alternatively in terms of only Lankford coefficients for three orientations. In this paper, the predictive capabilities of Hill, Yld 91 and Cazacu calibrated using different identification procedures, either analytical or by numerical minimization are discussed. Moreover, simulation results of deep drawing are presented.

New expressions and calibration strategies for Karafillis and Boyce (1993) yield criterion

New expressions in terms of stress components of the Karafillis and Boyce (1993) yield criterion are derived. First, it is demonstrated that the isotropic version of this criterion is a homogeneous polynomial in J2 and J32, where J2 and J3 are the invariants of the stress deviator. Moreover, it is shown that irrespective of the value of the interpolation parameter, it has the same mathematical expression as Hershey–Hosford isotropic yield criterion. Making use of these results, new expressions of the Karafillis and Boyce (1993) yield function for orthotropic BCC and FCC materials are derived. This enables to demonstrate that Barlat et al. (1991) and Karafillis and Boyce (1993) orthotropic criteria have the same mathematical form. Most importantly, the benefits of using these new expressions of the Karafillis and Boyce (1993) orthotropic yield criterion for the evaluation of the anisotropy coefficients are demonstrated. It is shown that these coefficients can be evaluated by solving simple algebraic systems of equations involving only experimental flow stresses, or alternatively involving only Lankford coefficients for three orientations. This makes the parameterization of an advanced non-quadratic criterion reliable even when the available mechanical test data are limited.

Impact of distortional hardening and the strength differential effect on the prediction of large deformation behavior of the Ti6Al4V alloy

The ability of three plasticity models to predict the mechanical behavior of Ti6Al4V until fracture is presented. The first model is the orthotropic yield criterion CPB06 developed by Cazacu et al. (Int J Plast 22:1171–1194, 2006) with a distortional hardening, allowing for the description of material anisotropy and the strength differential effect. The second model is the anisotropic Hill’48 yield criterion with distortional hardening, describing the material anisotropy with quadratic functions but is unable to model the strength differential effect. Finally, the third model is the classical Hill’48 yield locus with isotropic hardening. Distortional hardening is modeled through five yield surfaces associated with five levels of plastic work. Each model is validated by comparing the finite element predictions with experimental results, such as the load and displacement field histories of specimens subjected to different stress triaxiality values. Tensile tests are performed on round bars with a V-notch, a through-hole, and two different radial notches; compression tests are performed on elliptical cross-section samples. The numerical results show that none of the models can perfectly predict both the measured load and the sample shape used for validation. However, the CPB06 yield criterion with distortional hardening minimizes the global error of the model predictions. The results provide a quantification of the influence of mechanical features such as hardening phenomenon, plastic anisotropy, and tension–compression asymmetry. The impact of these features on the prediction of the post-necking deformation behavior of the Ti6Al4V alloy is explored.

On the impact of modelling tension-compression asymmetry on earing and thickness predictions

ABSTRACTThe metallic sheets used in sheet metal forming processes typically exhibit orthotropic behaviour, due to their crystallographic structure and the rolling process from which they are obtained. The yield criteria commonly adopted in the numerical analysis of these processes, assume that the yield surface possesses a point-symmetry with respect to its centre. However, even metallic materials with a cubic structure can present asymmetry between the tensile and the compressive plastic behaviour. Thus, it is important to evaluate the impact of considering the tension-compression asymmetry in the accuracy of the numerical results. The cylindrical cup example is known for being sensitive to both the material yield stresses and r-values in-plane directionalities. In this work, the behaviour of a AA2090-T3 aluminium alloy is described with an orthotropic yield criterion that also enables the description of tension-compression asymmetry. The numerical simulations of a cylindrical cup drawing were performed and the main parameters analysed are the earing profile and the thickness distribution, along the rolling and the transverse directions. The comparison of the experimental and numerical results shows that considering the compression yield stresses is critical to obtain accurate predictions, particularly for the thickness distribution.

New mathematical results and explicit expressions in terms of the stress components of Barlat et al. (1991) orthotropic yield criterion

In this paper are provided new mathematical results and equivalent expressions of the orthotropic yield criterion of Barlat et al. (1991) for body-centered cubic (BCC) and face-centered cubic (FCC) materials, respectively. Specifically, it is shown that this criterion is expressible as a simple polynomial in terms of the Cauchy stress components. Using these new explicit expressions, it becomes possible to establish new correlations between the anisotropy coefficients and mechanical properties. It is shown that the anisotropy coefficients involved in Barlat et al. (1991) are expressible solely in terms of uniaxial yield stresses, or alternatively only in terms of Lankford coefficients, and can be easily determined by solving simple algebraic systems of equations. Illustrative examples are presented for textured steel and aluminum alloys.

Stress-state dependent fracture characterisation and modelling of an AZ31 magnesium sheet alloy at elevated temperatures

Abstract Due to a high specific strength, magnesium alloys have a high potential to be considered for lightweight solutions in automotive industry. For the numerical design of forming processes, it is important to describe the yielding as well as the fracture behaviour of a material as precisely as possible. In order to fully characterise the fracture behaviour of an AZ31 magnesium sheet alloy at elevated temperatures, a heated test setup for uniaxial tensile machines was developed. The setup allows an adjustment of the load application angle whereby a stress variation is achieved in the centre of the specimen. In order to determine the fracture strain for different temperatures and for varying stress states, a shear stress specimen (also known as butterfly specimen) was considered to perform mechanical experiments by means of this setup. Using numerical simulations, the specific stress development and strain value in the fracture zone, which is needed to calibrate stress state fracture models, was determined for each loading angle and temperature. For this purpose, an orthotropic yield criterion CPB06, which is suitable for depiction of the particular flow behaviour of magnesium alloys (e. g. compression-tension asymmetry), was used. By this means, sufficient data for the calibration of common stress state based fracture models could be provided and the MMC- (Modified Mohr-Coulomb) fracture model was parameterised.

Effect of Plastic Anisotropy on the Distribution of Residual Stresses and Strains in Rotating Annular Disks

Hill’s quadratic orthotropic yield criterion is used for revealing the effect of plastic anisotropy on the distribution of stresses and strains within rotating annular polar orthotropic disks of constant thickness under plane stress. The associated flow rule is adopted for connecting the stresses and strain rates. Assuming that unloading is purely elastic, the distribution of residual stresses and strains is determined as well. The solution for strain rates reduces to one nonlinear ordinary differential equation and two linear ordinary differential equations, even though the boundary value problem involves two independent variables. The aforementioned differential equations can be solved one by one. This significantly simplifies the numerical treatment of the general boundary value problem and increases the accuracy of its solution. In particular, comparison with a finite difference solution is made. It is shown that the finite difference solution is not accurate enough for some applications.

Anisotropic Yield Criteria

The most widely used isotropic yield criteria are defined in terms of the principal values of the stress deviator. In this paper, it is demonstrated that these yield criteria are expressible as polynomials in terms of the second and third invariants of the stress deviator. Using these new mathematical results, new, simple and explicit expression of Yld91 and Karafillis and Boyce [1] orthotropic yield criteria in terms of stress components are obtained. Moreover, it is shown that for orthotropic BCC materials, Yld 91and Karafillis and Boyce [1] criteria are particular forms of Cazacu and Barlat [2] orthotropic yield criterion.

Prediction of the torsional response of HCP metals

One of the greatest challenges that a researcher in the field of the theory of plasticity is facing is that it has to establish general mathematical relations between the stresses and strains that should be applicable to any loading, although the experimental information available is generally restricted to uniaxial tension and/or compression tests. In particular, it has been long recognized that classic yield criteria cannot accurately capture the torsional response of hcp metals. In this paper, it is shown that Cazacu et al. [1] orthotropic yield criterion, identified based on uniaxial tests, captures the unusual characteristics of the torsional response of hcp AZ31 Mg. Furthermore, for the first time, on the basis of the same criterion, it is predicted the shape of the yield surface of this material for combined tension-torsion and combined compression-torsion loadings. Most importantly, it is shown that from the analysis of the stress-strain responses in a few very simple mechanical tests, using this criterion one can predict whether axial strains develop in torsion.

Multiscale modelling of the response of Ti-6AI-4V sheets under explosive loading

The objectives of the current study are to develop a multiscale numerical modelling method to predict the plastic deformation of Ti-6Al-4V sheets under blast loading, validate the method with experiments and characterise, at room temperature, the impulsive mechanical behaviour of the alloy. The numerical modelling technique relates the microstructure of the alloy with its macroscopic behaviour taking into account anisotropic effects by combining the viscoplastic self-consistent polycrystal model (VPSC7c) with the Cazacu–Barlat orthotropic yield criterion (CPB06) as implemented in the finite element (FE) solver of LS-DYNA. Sheet specimens of two thicknesses are tested using an experimental setup which applies a planar blast load. High speed cameras and the digital image correlation (DIC) technique are used to measure the evolving strains in the specimens. In addition, an analytical model is used to calculate the maximum displacements. The obtained values are compared with the outcome of the tests and FE simulations.

A computational approach for thermo-elasto-plastic frictional contact based on a monolithic formulation using non-smooth nonlinear complementarity functions

A new monolithic solution scheme for thermo-elasto-plasticity and thermo-elasto-plastic frictional contact with finite deformations and finite strains is presented. A key feature is the reformulation of all involved inequality constraints, namely those of Hill’s orthotropic yield criterion as well as the normal and tangential contact constraints, in terms of non-smooth nonlinear complementarity functions. Using a consistent linearization, this system of equations can be solved with a non-smooth variant of Newton’s method. A quadrature point-wise decoupled plastic constraint enforcement and the use of so-called dual basis functions in the mortar contact formulation allow for a condensation of all additionally introduced variables, thus resulting in an efficient formulation that contains discrete displacement and temperature degrees of freedom only, while, at the same time, an exact constraint enforcement is assured. Numerical examples from thermo-plasticity, thermo-elastic frictional contact and thermo-elasto-plastic frictional contact demonstrate the wide range of applications covered by the presented method.