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This study presents a model for Ti6Al4V alloy produced by applying electron beam melting as continuum media with orthotropic elastic and plastic properties and its application in total hip replacement (THR). The model exhibits three Young’s moduli, three shear moduli, and three Poisson’s ratios as elastic properties and six coefficients describing the Hill yield criterion. Several uniaxial tension and torsion experiments and subsequent data processing were performed to evaluate the properties and coefficients. The typical values obtained for Young’s moduli, shear moduli, and Poisson’s ratio were 121–124 MPa, 37–42 MPa, and 0.25–0.26, respectively. A comparison of the experimental tension and torsion curves with those obtained by a finite element analysis revealed a good correlation with a maximum error of 9.5%. The finite element simulation of a personalised pelvic implant for THR manufactured from the obtained material proved the mechanical capability of the implant to successfully withstand the applied loads.

The Fused Filament Fabrication is a 3D printing technology that allows the production of components and structures with complex geometries at low-cost, using thermoplastic materials as feedstock. This additive manufacturing technique is not yet extensively used in industries mainly due to parts’ anisotropy – which is the consequence of the deposition strategy – and residual stresses, caused by successive heating cycles. Thus, these topics have great influence on the non-linear mechanical performance of 3D printed parts. Additionally, the materials used in the Fused Filament Fabrication have distinct mechanical behaviours under tensile and compression loads. This work proposes a computational framework – using a meshless method as the numeric discretization technique, the Natural Neighbour Radial Point Interpolation method – capable to analyse the elastoplastic response of ductile materials showing distinct compressive and tensile responses, through the implementation of a modified version of the Hill yield surface. In those conditions, 3D-printed specimens of polylactic acid are tested experimentally and numerically using standard tensile and compression tests. The numerical approach is validated by comparing the numerical and the experimental curves. Then, the accuracy of the methodology is proved and the purpose of using the modified Hill yield criterion is clearly shown using a benchmark example.

Using the Kroner–Lee elastic and plastic decomposition of the deformation gradient, a differential-algebraic system is obtained (in the so-called semi-explicit form). The system is composed by a smooth nonlinear differential equation and a non-smooth algebraic equation. The development of an efficient one-step constitutive integrator is the goal of this work. The integration procedure makes use of an explicit Runge–Kutta method for the differential equation and a smooth replacement of the algebraic equation. The resulting scalar equation is solved by the Newton–Raphson method to obtain the plastic multiplier. We make use of the elastic Mandel stress construction, which is power-consistent with the plastic strain rate. Iso-error maps are presented for a combination of Neo-Hookean material using the Hill yield criterion and a associative flow law. A variation of the pressurized plate is presented. The exact Jacobian for the constitutive system is presented and the steps for use within a structural finite element formulation are described .

Fused deposition modeling has become an important technology preferred in industrial and domestic use with the development and diversification of 3D printers. However, the number of parameters that affect the strength of the printed part in 3D printers is quite high. In this study, the effect of printing angle on the mechanical properties of the specimens fabricated with 3D printer was investigated through experimental and finite elements method. PLA filament and different printing angles (0°, 45°, 90°) were used for printing the test specimens. Specimens were accepted as transversely isotropic in the finite elements analysis. The plastic behavior of the material was explained by Hill yield criterion. In the first stage of the study, results of finite elements analysis and experimental study for 0°, 45° and 90° printing angles were compared. Comparison results showed that the acceptance of transversely isotropic material and Hill yield criterion can be used to determine the strength of the printed part. In the last stage of the study, finite elements analysis was repeated for 15°, 30° 60°, 75° printing angles. The finite element analysis and experimental results showed that the mechanical properties and maximum load values of specimens increased significantly with increasing of printing angle., Eriyik yığma modelleme, 3B yazıcıların gelişimi ve çeşitlenmesi ile sanayi ve evsel kullanımda tercih edilen önemli bir teknoloji haline gelmiştir. Ancak, 3B yazıcılarda yazdırılan parça dayanımını etkileyen parametre sayısı oldukça fazladır. Bu çalışmada, 3B yazıcı kullanılarak üretilen numunelerde yazdırma açısının mekanik özelliklere etkisi deneysel yöntem ve sonlu elemanlar metodu kullanılarak incelenmiştir. Test numunelerinin yazdırılmasında PLA filament ve farklı yazdırma açıları (0°, 45°, 90°) kullanılmıştır. Sonlu elemanlar analizinde numuneler transvers izotropik malzeme olarak kabul edilmiştir. Malzemenin plastik davranışı Hill akma kriteri ile tanımlanmıştır. Çalışmanın ilk aşamasında, 0°, 45° ve 90° yazdırma açıları için sonlu elamanlar analiz sonuçları ve deneysel sonuçlar karşılaştırılmıştır. Karşılaştırma sonuçları, yazdırılan numunelerin dayanımının belirlenmesinde transvers izotropik malzeme kabulü ve Hill akma kriterinin kullanılabileceğini göstermiştir. Çalışmanın son aşamasında, 15°, 30°, 60°, 75° yazdırma açıları için sonlu elemanlar analizleri tekrarlanmıştır. Sonlu elemanlar analizi ve deneysel sonuçlar, numunelerin mekanik özelliklerinin ve maksimum yük değerlerinin yazdırma açısının artması ile belirgin şekilde yükseldiğini göstermiştir.

The aim of this work was to monitor the mechanical behavior of 316L stainless steel produced by 3D printing in the vertical direction. The material was tested in the &ldquo, as printed&rdquo, state. Digital Image Correlation measurements were used for 4 types of notched specimens. The behavior of these specimens under monotonic loading was investigated in two loading paths: tension and torsion. Based on the experimental data, two yield criteria were used in the finite element analyses. Von Mises criterion and Hill criterion were applied, together with the nonlinear isotropic hardening rule of Voce. Subsequently, the load-deformation responses of simulations and experiments were compared. Results of the Hill criterion show better correlation with experimental data. The numerical study shows that taking into account the difference in yield stress in the horizontal direction of printing plays a crucial role for modeling of notched geometries loaded in the vertical direction of printing. Ductility of 3D printed specimens in the &ldquo, state is also compared with 3D printed machined specimens and specimens produced by conventional methods. &ldquo, As printed&rdquo, specimens have 2/3 lower ductility than specimens produced by a conventional production method. Machining of &ldquo, specimens does not affect the yield stress, but a significant reduction of ductility was observed due to microcracks arising from the pores as a microscopic surface study showed.

The paper proposes a deformation and fracture model for anisotropic stratified rocks and presents theoretical and experimental data on how the rock strength and fracture geometry are influenced by principal stresses and their orientation to bedding planes. Two possible mechanisms are considered for rock fracture under true triaxial load: along bedding planes of weakness and along planes in which Mohr-Coulomb stresses reach a critical combination with cohesion coefficients and internal friction angles typical of the rock. The transition of rocks to inelastic deformation is described in the context of two criteria of which one accounts for the above fracture mechanisms and the other, being a semi-empirical analogue of the Hill yield criterion, accounts for the effect of normal stress. The experimental data presented are for the strain and strength properties of rocks sampled from the Fedorovskoye and Talakanskoye oil and gas fields and tested on an original loading system for true triaxial compression with lateral pressure (similar to the Karman scheme) and for generalized shear (three unequal and nonmonotonic principal stresses). The experimental and theoretical results, including total stress-strain curves, are in good qualitative agreement and demonstrate the possibility to evaluate the parameters entered in the model from tests of particular rocks.

It is highly appreciated the using reinforced layers in supporting metallic parts due to increased performance and formability of the sheets. Firstly, the effect of reinforcing layer will be investigated on the expanding of rings as a benchmark stability problem and then considered when the composite panels undergo the axial and biaxial stretches near instability threshold. In the ring expanding case; process will be explored by the wavelength and wave number changes through the linear perturbation analysis. The linear perturbation or linear stability is established on the elastic-visco-plastic fragmentation of the rings that includes the effect of significant deformation, geometric softening and hydrostatic pressure. Alternately, the vertex model will be used in prediction of localized necking instability that, considers the discontinuity of stress and strain rates inside the neck band. Here, the DYC-angle and quadratic Hill yield criterion help us to predict the localized necking band and consider the anisotropy effect, respectively. Also, the forming limit diagrams will be used for presenting the vertex analysis results. In the vertex theory, the non-linear reinforced layer will be considered among the neo-Hookean materials. Results will be validated with the other theoretical and experimental data available in the literature.

Short fiber reinforced thermoplastics (SFRT) are widely used for automotive components. One of the significant challenges in designing industrial SFRT components is an efficient prediction of their mechanical response under mechanical and thermal loads. In this work, an anisotropic temperature-dependent elasto-viscoplastic model is implemented using the available macroscopic material models in commercial FE solver to describe polybutylene terephthalate with 30 wt.-% short glass fibers. The elastic behavior is described by the orthotropic linear elastic model generated through the mean-field homogenization method and the anisotropy in plastic region by Hill yield criterion dependent on fiber orientation. The rate-dependent plasticity is described by the unified viscoplasticity framework of Chaboche. To describe the continuous temperature dependency, model parameters are systematically determined as a function of temperature in the typical automotive temperature range, including regions below and above glass transition temperature. Further, an optimization method based on genetic algorithm is adopted for parameter optimization. The optimized model accurately describes the anisotropic material behavior observed in tensile and stress relaxation tests in a wide range of temperature, specimen orientation, and strain rate. The model's prediction capability is validated by simulating tensile tests at three intermediate temperatures, which are not included during the calibration process.

A quadratic yield function which can describe the anisotropic behaviors of sheet metals with tension/compression symmetry and asymmetry is proposed. Five mechanical properties are adopted to determine the coefficients of each part of the yield function. For particular cases, the proposed yield function can be simplified to Mises or Hill's quadratic yield function. The anisotropic mechanical properties are expressed by defining an angle between the current normalized principal stress space and the reference direction with the assumption of orthotropic anisotropy. The accuracy of the proposed yield function in describing the anisotropy under tension and compression is demonstrated.

Transforming five different stress invariants-based yield criteria in I-J2-J3 framework into η-ξ- σ ¯ framework demonstrates that all of these yield criteria are the product formulation of the equivalent stress and the function of the normalized third invariant ξ and stress triaxiality η. The function of ξ is replaced by a polynomial function to construct a generalized yield function for describing both asymmetry and Lankford coefficients under associated flow rule(AFR) framework, in which the linear combination of a series of ξ with odd powers is utilized to model the effect of materials' strength differential (SD). Since there are more parameters in the polynomial function of ξ than the function of ξ in those existing yield functions, and the independent basis functions ξ, ξ2, ξ3 … are informative enough to describe the SD effect and Lankford coefficients, the proposed yield function shows potential flexibility. The conditions for the convexity of the proposed yield criterion are obtained by using the method of order principal minor determinant. To describe sheet metals' anisotropic characteristics in tension and compression, the proposed isotropic criterion is further generalized to orthotropy through introducing the fourth-order linear transformation to the deviatoric stress tensor. To improve the flexibility of the yield criterion through introducing more fourth-order linear transformations, several extended proposed isotropic criteria are added together. The effectiveness and flexibility of the constructed yield criterion have been verified by applying to AA2008-T4(a BCC material) and AA2090-T3(a FCC material). Comparisons with Yld2000-2d and Yoon 2014 criteria show that the proposed yield criterion has the same ability as Yld2000-2d to describe metals’ tensile properties and the proposed yield criterion can capture the SD effect like Yoon 2014, which validates that this generalized yield criterion can accurately describe not only the SD effect of metals but also the Lankford coefficients under uniaxial tensile loading and balanced biaxial tension based upon the AFR assumption. The flexibility of the proposed yield criterion is further validated by applying to zirconium through describing the evolution of yield surface for a clock-rolled plate with various levels of pre-strains during through-thickness compression. It is found that the proposed yield criterion can capture the asymmetry and the local deformation characteristic of the yield surfaces if the pre-strain is very large.

It is known that, under certain conditions, the quasi-static incremental analysis problem of elastoplastic structures with the von Mises yield criterion can be formulated as a second-order cone programming (SOCP) problem, which can be solved with a primal-dual interior-point method. Alternatively, this paper proposes to solve an equivalent unconstrained nonsmooth convex optimization problem, which has a form similar to a class of regularized least-square problems, known as group LASSO. We propose an accelerated proximal gradient method with an adaptive restart scheme for solving this unconstrained optimization problem. The algorithm is easy to implement, and free from numerical solution of linear equations unlike conventional methods in computational mechanics. Numerical experiments suggest that the presented algorithm outperforms a standard solver that implements a primal-dual interior-point method for conic optimization.

Accurate descriptions of theoretical strength and Poisson’s ratio are crucial for a successful representation of the yield behavior of porous metals. In this paper, by embedding expressions of Young’s modulus and surface energy into the well-known Griffth’s fracture model, an exponential function of theoretical strength accounting for the critical relative density is logically deduced. In order to better describe the change rule of Poisson’s ratio, a combined power function, displaying as an approximate S-type shape, is established simultaneously. With the two accurate descriptions of the theoretical strength and Poisson’s ratio, a new yield criterion for porous metals is ultimately obtained as an explicit function of porosity. According to the mathematical expression of the proposed yield criterion, its corresponding yield locus is strictly proved to be an ellipsoid. Validation of the proposed yield criterion with other criteria and available experimental data is performed and good agreement is found.

The definition of an orthotropic yield criterion presents a serious challenge in the formulation of constitutive models based on such theories as elastoplasticity, viscoplasticity, damage, etc. The need to model the behavior of a real orthotropic material requires the formulation of orthotropic yield criteria, and these may be difficult to obtain. For metals, orthotropic yield functions have been formulated by Hill [Proc. Roy. Soc. Lon. Ser. A 193 (1948) 281; J. Mech. Phys. Solids 38(3) (1990) 200], Barlat [Int. J. Plasticity 5 (1989) 51; 7 (1991) 693], Chu [NUMISHEET 93 (1993) 199] and Dutko et al. [Comput. Methods Appl. Mech. Engrg. 109 (1993) 73], but in many cases these functions do not describe the true behavior of the metal. The situation is worse when one attempts to represent a nonmetal such as a polymer, ceramic or composite. In this paper, we present a general definition of anexplicit orthotropic yield criteriontogether with a general method for definingimplicit orthotropic yield functions. The latter formulation is based on thetransformed-tensor method, whose principal advantage lies in the possibility of adjusting anarbitrary isotropicyield criterion to the behavior of ananisotropicmaterial. As example we choose the adjustment to the Hill, Hoffman [J. Comp. Materials 1 (1967) 200] and Tsai–Wu [J. Comp. Materials 5 (1971) 58] criteria, but these particular cases serve to establish the methodology for achieving the desired function adjustment for any other well-known criterion or experimental set of data obtained from laboratory.

International audience; This work aims to develop an advanced structured material [1] that enhances the physical properties in the fracture mechanics area with specimens printed by FDM. 3D printed parts are considered as complex structures and assumed to show a strong anisotropy related to deposition trajectories, air gaps, and welding lines. Therefore, a challenging problem that arises in this domain is the difficulty of predicting the mechanical behavior and controlling the physic properties of parts (tensile strength and fracture toughness…).An extrusion deposition method was proposed involving the use of a controlled deposition method to improve the fracture toughness of FDM parts [2], [3].The enhancement is based on a filament deposition optimization in order to reproduce principal stresses directions in a region of interest submitted to crack initiation around the notch tip.The overall goal of this research is to identify material constants from experimental tests. Those constants will be used then as inputs of a finite element model, which tackles to include the structural anisotropy by assigning materials references into local mesh elements [4]. Thus, the corresponding behavior was assumed to be transverse isotropic and five elastic materials constants must be identified. Hence, a set of tensile tests was performed with full-field measurements by Digital Image Correlation (DIC) for different filament orientations. Regions of interests were chosen according to the local loading state to activate specific materials constants. The experiments outcomes prove that 3D printed specimens have unexpectedly isotropic stiffness due to similar values found of longitudinal and transversal Young’s modulus 1680 MPa and 1414 MPa respectively. Although anisotropy is well highlighted when we consider tensile strength. On the light of these results, the model will be enriched by implementing a Hill yield criterion to better represent the observed plastic anisotropic behavior.The main contribution is to validate the numerical model inputs that reproduce the measured experimental fields, and later on develop an identification based on an Updated Finite Element Model Updated (UFEM).

Fused filament fabrication (FFF) is a low-cost 3D printing technology that allows the production of components and structures with complex geometries, which cannot be achieved by traditional manufacturing processes. Nonetheless, this additive technique is not extensively used in high-value industrial sectors, mainly due to parts’ anisotropy related to deposition strategy and residual stresses caused by successive heating cycles. These features have a great influence on the mechanical performance of 3D-printed parts, in particular, thermoplastics with nonlinear behaviour (such as PLA or PA). Thus, engineering approaches to predict these elasto-plastic responses are demanded. In this work, the tensile and compression behaviours of FFF thermoplastics are investigated using a yield criterion that accounts, simultaneously, the presence of tensile and compressive loads applied on the material (a modified Hill yield criterion). The developed elasto-plastic algorithm, which uses the incremental-iterative Newton–Raphson method, is implemented within the formulation of a meshless method. Despite the use of the finite element method (FEM) for engineering applications have become widespread, new accurate and efficient discrete advanced numerical techniques—such as meshless methods (Belinha in Meshless methods in biomechanics: bone tissue remodelling analysis. Springer International Publishing, Porto, 2013 [1])—can handle the same kind of problems as the FEM and being, in some cases, even more efficient. To discretize the problem domain, meshless methods only require an unstructured nodal distribution. The numerical integration of the Galerkin weak form is performed using a background integration mesh, the nodal connectivity is enforced using the influence-domain concept and then the interpolation shape functions are obtained. In order to validate the numerical proposed algorithm, standardized tensile and compressive specimens were printed and tested. Then, using the material properties extracted from the experimental tests, a benchmark example was studied for the sake of model proof.

For finite strain plasticity, we use the multiplicative decomposition of the deformation gradient to obtain a differential-algebraic system (DAE) in the semi-explicit form and solve it by a half-explicit algorithm. The terminology HERK is synonym of Half-Explicit Runge-Kutta method for DAE. The source is here the right Cauchy-Green tensor and an exact Jacobian of the second Piola-Kirchhoff stress is determined with respect to this source. The system is composed by a smooth nonlinear first-order differential equation and a non-smooth algebraic equation. The development of a half-explicit constitutive integrator is the content of this work. The integration makes use of an explicit Runge-Kutta method for the flow law complemented by the yield constraint. The flow law is a first-order differential equation and the yield constraint (including the loading/unloading conditions) is seen as the invariant of system. A half-explicit method is adopted to ensure satisfaction of the invariant. The resulting scalar equation is solved by the Newton-Raphson method to obtain the plastic multiplier. We make use of the elastic Mandel stress construction, which is power-consistent with the plastic strain rate. Iso-error maps are presented for a combination of Neo-Hookean material using the Hill yield criterion and a associative flow law. Two complete numerical examples are presented.

The present study is devoted to the boundary value problem of coupled thermoelastoplasticity. The temperature depended yield criterion and Duhamel Neumann constitutive equation was used. The material subjected to uneven heat treatment under plane strain frameworks was considered. The new analytical solution of the problem of uneven heat treatment of the thermoelasticplastic hollow cylinder was constructed within of the thermal stresses and plastic flow theories. The three different yield criteria were used. The original numerical scheme for calculations of temperature stresses and plastic strains in the frameworks of the von Mises yield criterion was developed and implemented. The characteristics of the plastic flow in the heating domain according to the yield criterion selection are eliminated. Constructed solutions of the boundary value problems were compared and analyzed for yield criteia of von Mises, Tresca and Ishlinsky-Ivlev.

A mathematical model is developed to analyse creep in a variable thickness functionally graded rotating disc yielding according to Tresca criterion. The results obtained are compared with those estimated using von Mises criterion. The tangential and radial stresses, using Tresca criterion, are observed to be slightly higher, except those noticed near the inner and outer disc radii. The Tresca criterion yields lower values of effective stress as compared to Mises criterion. However, the Tresca criterion estimates higher values of strain rates in the disc as compared to Mises criterion, with maximum variation noticed at the inner radius. Thus, the FGM disc designed using Tresca yield criterion is more conservative than that designed according to von Mises criterion.

The most popular isotropic yield conditions, verified for many ductile metals, were proposed by Tresca in 1864 (maximum shearing stresses theory) and von Mises in 1913. The von Mises yield theory (von Mises in Mathematisch-physikalische Klasse 582–592, 1913), also known as maximum distortion energy criterion, finds considerable experimental support, especially for very ductile materials and plane stress (Banabic et al. in Int. J. Mater. Form. 3:165–189, 2010). For this reason, and for its simplicity, it is common in design. During 100 years, this theory has been developed and improved systematically by Hosford, Christensen, Tsai-Hill, etc. The modified von Mises hypothesis combines the theories of maximum strain energy and maximum distortion energy, and it involves the Poisson ratio. It ensures a smooth transition from the von Mises to the Beltrami criterion. The results obtained by this new yield hypothesis are compared with those obtained both by the classic von Mises criterion and by experiments on different metallic materials. A quite good concordance is observed between these results.

Formation of wavy boundaries at the open end of a drawn cup is referred to as earing. Plastic anisotropy is the main parameter contributing to formation of ears. Therefore, in order to eliminate the undesirable ears in sheet metal parts, it is essential to predict the deformation behavior of sheet metals using advanced anisotropic yield criteria. Accordingly, the objective of this article is that to use BBC2003 yield criterion, to predict earing in deep drawing of cylindrical cups from AA3105 aluminum alloy. First, a new and simple approach is introduced to calculate the eight constants of this yield function. In this approach, only the uniaxial and plane strain tension tests results, were used. Nonlinear governing equations of these constants were derived and solved by means of well-known steepest descend method. In order to investigate performance and accuracy of this method, the calculated constants were utilized to predict normalized uniaxial yield stress, anisotropy parameters and plane stress yield surface. Then, based on this yield criterion and the associated flow rule, a VUMAT subroutine was developed and employed in the commercial finite element software ABAQUS/Explicit dynamics. The simulation of drawing process was conducted for anisotropic AA3105 aluminum alloy sheet. Cup height (earing profile) and thickness distribution were obtained from experiments and corresponding numerical simulations. Comparison of these results demonstrates great potential of BBC2003 yield criterion in accurate prediction of earing.

Herein, a one-parameter plasticity model proposed by Sun and Chen [Sun CT and Chen JL. A simple flow rule for characterizing nonlinear behavior of fiber composites. J Compos Mater 1989; 23: 1009–1020] demonstrates features that make it highly attractive for characterizing non-linear responses of fibrous composites. However, a detailed exploitation of the model’s potential has been halted by unresolved problems that include tension–compression asymmetry in stress–strain curves, FEM implementation as well as optimal parameters determination, which are addressed in this investigation as well as proposed solutions are presented. The major focus in Part I of this three-part study was devoted to developing a simple model for predicting the tension–compression asymmetry in stress–strain curves for fibrous composites, which was based on Sun and Chen’s one-parameter plasticity model. A generalized Hill yield criterion was proposed from combinations of the Drucker–Prager yield criterion that considers the effect of hydrostatic pressure for isotropic materials and the Hill yield criterion suitable for anisotropic materials. By incorporating the yield strength-differential effect on the plastic flow rule in composite laminates, the one-parameter plasticity model was extended to a strength-differential effect-incorporated model. The improved model has been calibrated and validated by off-axis tension and compression tests on unidirectional carbon/epoxy (IM600/Q133) composite laminates. Results verified that the proposed model captured the complex tension–compression asymmetry in observed non-linear responses of stress–strain curves.

Experimental and numerical analyses are presented concerning of compression tests parallel and perpendicular to the grain, three-point bending, and double-shear joints in compliance with the relevant test standards (ASTM D2395, BS 373, EN 383 and EN 26891). Woods of Norway spruce (Picea abies L. Karst.) and European beech (Fagus sylvatica L.) were tested to describe their non-linear behavior. Elasto-plastic material models were the basis for the finite-element (FE) analyses with the input of own experimental data and those of the literature. The elasto-plastic material model with non-linear isotropic hardening was applied based on the Hill yield criterion in regions of uniaxial compression. The material characteristics were first optimized and validated by means of basic 3D FE models under the same conditions as applied for the experiments. Afterwards, the validated material models were implemented into the solver with more complex numerical analyses of wooden dowel joints. Concurrently, the digital image correlation (DIC) served for verification of the numerical wooden joint models. A good agreement (with a relative error up to 16%) was found between numerically predicted and experimentally measured data. The differences may be mainly attributed to some natural characteristics of wood which were not considered in the proposed material models. The proposed elasto-plastic material models are capable of predicting the wood’s ultimate strength, and therefore could contribute to a more reliable design of wood structures and their performance.

The Drucker-Prager yield criterion is combined with an equilibrium equation to provide the elastic-plastic stress distribution within rotating annular hyperbolic discs and the residual stress distribution when the angular speed becomes zero. It is verified that unloading is purely elastic for the range of parameters used in the present study. A numerical technique is only necessary to solve an ordinary differential equation. The primary objective of this paper is to examine the effect of the parameter that controls the deviation of the Drucker-Prager yield criterion from the von Mises yield criterion and the geometric parameter that controls the profile of hyperbolic discs on the stress distribution at loading and the residual stress distribution.

It is shown that the fourth-order homogeneous polynomial yield function first considered by Gotoh (Int J Mech Sci 19:505–512, 1977) for sheet metals can be recast into a particular general form in intrinsic variables. It leads to a method of generating many specific representations of a fourth-order Hill’s 1979 yield function (in Math Proc Camb Philos Soc 85:179–191, 1979) generalized for off-axis loading. One may use the generalized fourth-order Hill’s yield function for modeling sheet metals of various degrees of planar anisotropy with ease. Furthermore, a sufficient condition on the positivity and convexity of a calibrated Hill’s yield function can also be verified straightforwardly. For six sheet metals whose nine material constants in a Gotoh’s yield function have been reported in the literature, their generalized Hill’s 1979 yield functions are obtained subsequently. The proposed sufficient condition on the positivity and convexity of Hill’s yield function is found to be satisfied by five of these six sheet metals. An approximate Hill’s 1979 yield function for the sixth sheet metal has also been identified, so the proposed sufficient condition on its positivity and convexity is also met. Generalizing fourth-order Hill’s 1979 yield functions may thus be used as an alternative approach for formulating a class of positive and convex yield functions with up to nine material constants for various orthotropic sheet metals.

Structural optimization based on the shakedown theory is a powerful and promising technique. However, due to the nonlinearities of physical materials and the number of variable loads in real structures, it is computationally complex and time-consuming. To simplify the occurring non-linear, non-convex optimization problems, the paper suggests reducing the number of yield conditions. The so-called a yield criterion of the mean (integral yield condition) is analysed and explained in detail, which allows taking into account one yield condition for the entire finite element instead of multiple point-wise conditions. This approach shows promising results in numerical application to the optimization of a circular plate, considering a possibility of employing the yield criteria of the mean or pointwise yield conditions in different areas of the plate in particular. The methods applied are based on the assumptions of perfect plasticity and small deformations.

Summary The static formulation of elastic shakedown analysis, based upon Melan's lower-bound theorem, can essentially be viewed as a robust optimization problem. This paper discusses an advantage that is enjoyed by taking this perspective. Specifically, we assume the von Mises yield criterion and an ellipsoidal load domain. In this setting, the shakedown analysis problem, which is viewed as robust second-order cone programming, can be recast as semidefinite programming. Copyright © 2016 John Wiley & Sons, Ltd.

Industrial sheet metal forming processes often involve complex deformation modes and it is necessary to consider nonlinear loading path effects when predicting forming limit curves. Moreover, the yield criterion plays a critical role in the accuracy of predicted forming limits.In this work the MK analysis was modified to relate the initial imperfection factor to a physical property such as the surface roughness, and the orientation of the imperfection was also allowed to vary. This model was used to predict the strain-based and stress-based forming limit curves (FLC and SFLC) of sheet materials that are subject to either linear or non-linear strain paths.Two different yield criteria were employed in this study, Hill's 1948 quadratic yield criterion and Hosford's 1979 non-quadratic yield criterion. The theoretical model was validated by comparing predicted FLC and experimental FLC curves obtained from the literature. FLCs and SFLCs predicted with these two yield criteria were compared for both linear and nonlinear loading paths.Results showed that both the quadratic and non-quadratic yield criteria predict the FLC with acceptable accuracy however on the whole the non-quadratic yield criterion generally provides a slightly better correlation with experimental data, especially on the right side of the FLC. Keywords: Forming limit curve, Formability, Yield criterion, MK analysis, Sheet metal, Loading path

Rolling contact fatigue often initiates in the highly deformed surface layer of railway rails. However, the behavior of pearlitic rail steels, subjected to such large shear strains, is not well known. Due to buckling, it is not possible to obtain the large shear deformation with tubular test bars. We have, therefore, developed a novel experimental methodology. Large shear strains (up to 1.13) were obtained by twisting cylindrical solid test bars under a compressive axial load. After that, these bars were re-machined into thin-walled tubular test bars. The re-machined bars were then used to characterize the behavior of the deformed material. Changes in both the elastic and plastic responses are observed, quantified, and discussed. We evaluate the ability of different yield criteria to model the experimentally measured yield surfaces. Three anisotropic yield criteria are considered: Hill (1948), Barlat's yld2004-18p, and Karafillis and Boyce (1993). The accuracies of these criteria are evaluated and compared to the estimated experimental uncertainty. Furthermore, cross-validation is used to investigate the predictive abilities of the yield criteria. It turns out that the Hill yield criterion fits and predicts the experimental data very well. Finally, the evolution of yield surfaces is found to be strongly dependent on the amount of plastic deformation defining the yield limit.

A new linear yield criterion, referred to as average apothem (AA) yield criterion is developed in this paper based on the Tresca and Twin Shear Stress (TSS) yield criteria. This criterion is a piecewise linear function and its locus on the π-plane in Haigh Westergaard stress space is an equilateral and non-equiangular dodecagon. Comparison with the Tresca and Mises yield criteria shows that the locus of the proposed yield criterion lies between those of the Tresca and Mises loci and is much closer to that of Mises yield criterion. Further, its specific plastic work rate is developed. As an application of the proposed criterion, the AA yield criterion is applied to the analysis of a generalized worked example of strip forging and an analytical solution of the stress effective factor is obtained. Comparison with the results by Tresca, Mises and TSS yield criteria shows that the proposed criterion provides an intermediate solution between the results of Tresca and Mises criteria.

The first yield criterion for anisotropic materials was proposed by von Mises (1928). Compared with other classic anisotropic criteria, it is seldom mentioned in practical applications in literature. Here, the suitability of the anisotropic criterion of von Mises to characterize initial yielding of Particulate Metal Matrix Composites (PMMCs) is studied with a computational micromechanics approach. First, a calibration procedure is described for experimental quantification of the yield parameters. Then, an actual calibration is performed by means of numerical experimentation using FEM analysis of multi-particle cells. The cells used in the numerical experiments consist of a representative volume element (RVE) of the material, including a number of reinforcing particles, randomly distributed inside the RVE. Particles’ aspect ratio and orientation are controlled in such a way that the microstructure of the model material shows an explicit geometric anisotropy. The developed scheme allowed to qualitatively describe the relation between the anisotropy of the microstructure and the anisotropy of the mechanical behavior. Results show that the anisotropic criterion of von Mises is a good candidate to characterize initial yield of particulate composites. However, some further work needs to be done in order to extend its applicability to a wider range of mechanical loading.

In order to overcome the nonlinearity of Mises criterion, a new linear yield criterion with a dodecagon shape of the same perimeter as Mises criterion was derived by means of geometrical analysis. Its specific plastic work rate expressed as a linear function of the yield stress, the maximum and minimum strains was also deduced and compared with that of Mises criterion. The physical meaning of the proposed yield criterion is that yielding of materials begins when the shear yield stress τs reaches the magnitude of 0.594σs. By introducing the Lode parameter, validation of evolution expressions of the proposed yield criterion with those based on Tresca, Mises and TSS criteria as well as available classical yield experimental results of various metals shows that the present results intersect with Mises results and coincide well with experimental data. Moreover, further application to the limit analysis of circle plate as an example is performed to demonstrate the effectiveness of the proposed yield criterion, and the subsequent comparison of limit loads with the Tresca analytical solutions and Mises numerical results shows that the present results are higher than the Tresca analytical results, and are in good agreement with the Mises numerical results.

A non-linear triple shear energy yield criterion for salt rock is presented in this paper. It is the development of the triple shear energy yield criterion, of which the Mohr–Coulomb criterion can be seen as a special case. The main factors affecting the primary strength of salt rock, such as the mean stress and the Lode angle, are considered in the non-linear triple shear energy yield criterion. The non-linear new criterion provides the non-linear change trend of salt strength both in the I 1–J 2 stress space and in the deviatoric plane. Comparative study between the non-linear criterion predictions and experimental results of salt rock shows that the non-linear triple shear energy yield criterion fits quite well with both conventional triaxial test data and the true triaxial test data. For Maha Sarakham salt, the predictive capability of the non-linear triple shear energy yield criterion is clearly better than that of some other criteria used by Sriapai, such as modified Lade criterion, 3-D Hoek, and Brown criterion, Drucker–Prager criterion et al. The availability of the non-linear triple shear energy yield criterion can also be confirmed by comparative analysis between theoretical values and experimental values for non-salt rocks. So the non-linear triple shear energy yield criterion is a general failure criterion for rocks fractured by shear stress.

In the current research to model anisotropic asymmetric sheet metals a new non-AFR criterion is presented. In the new model, Modified Yld2000-2d proposed by Lou et al. (2013) is considered as yield function and Yld2000-2d proposed by Barlat et al. (2003) is considered as plastic potential function. To calíbrate the presented criterion, the yield function which is a pressure dependent criterion requiers ten directional yield stresses such as uniaxial tensile stresses in three directions of 0º, 45º, 90º, uniaxial compressive yield stresses in six directions of 0º, 15º, 30º, 45º, 75º, 90º from the rolling direction along with biaxial yield stress. Moreover, the plastic potential function which is a pressure indepedent criterion needs eight experimental data points such as tensile R-values in seven directions of 0º, 15º, 30º, 45º, 60º, 75º, 90º from the rolling direction and also biaxial tensile R-value. Finally with comparing the obtained results with experimental data points, it is shown that the presented non-AFR criterion predicts compressive yield stress, biaxial tensile yield stress and R-values more accurately than Modified Yld2000-2d and it would be considered as a new criterion for anisotropic asymmetric metals.

A yield criterion for porous materials, referred to as porosity-dependent (PD) yield criterion, is proposed in this paper. In order to reflect the change law between Poisson׳s ratio and material porosity, the relationship between them is established using a logarithm-fraction compound function. Then the criterion firstly weighted by Poisson׳s ratio and ultimately incorporated with material porosity is then obtained. Its physical meaning is that when the total elastic energy reaches its limit value, then the porous material yields. Its loci in the principal stress space and on the π-plane are mathematically proved to be an ellipsoid and a circle respectively. Validation with available experimental data and traditional theoretical results shows that the present results have the same change trend as those of traditional theoretical results, and are in good agreement with experimental ones.

The plastic zones for I, II, III mode crack under small scale yielding are analyzed by MY criterion, and the analytical solutions of the sizes of the plastic zones for plain stress and plain strain states are first obtained. The solutions for I and II mode crack show that the two solutions are functions of the yield stress, stress strength factor and polar angle, while the solution for III mode crack just depends on the yield stress and stress strength factor. Comparison of the plastic zone with those based on Tresca and Mises criteria shows that Tresca criterion predicts the maximum, the result by MY yield criterion lies between them, and is very close to that by Mises criterion. Besides, the relationship of plastic zone between plain stress and plain strain are also discussed.

This study deals with the description of the anisotropic behaviour of the mild steel sheet used in the reverse deep drawing process of a cylindrical cup, which was proposed as benchmark at the Numisheet’99 conference. The effect of the yield criterion on the numerical results is analysed using three yield functions, von Mises, Hill’48 and Barlat Yld’91, combined with the Swift hardening law. The anisotropy parameters of the Hill’48 model are identified using either the yield stresses or r-values, obtained from the uniaxial tensile test at three different directions. On the other hand, the anisotropy parameters of the Yld’91 are determined taking into account both the yield stresses and r-values, minimizing an objective function. The comparison between experimental and numerical results is presented, being the punch force evolution and the thickness distribution along the cup wall the principal variables under study. In both forming stages, the predicted punch force evolution is close to the experimental one, whatever the yield criterion adopted. Nevertheless, the cup wall thickness distribution is strongly influenced by the yield criteria, being clearly overestimated by the von Mises yield criterion. On the other hand, the Yld’91 yield criterion provides a thickness distribution closer to the experimental one, for both forming stages. The strain paths during both forming stages ranges from uniaxial compression, when the material flows between the die and blank-holder, to plane strain in the cup wall, whereas the important strain path changes occurs in the die radius.

Two different configurations of steel-to-timber connections are tested in bending in normal conditions and under ISO-fire exposure. To observe the influence of clearances in the connection area on the fire resistance of the connections, two specimens were previously tested under cyclic loadings. These tests consist in the application of loading-unloading cycles by controlled displacements. The experimental results of connections tested in cold and under ISO-fire conditions are analyzed and commented. These results are then used to validate a finite element model. This model allows to simulate numerically the evolution of the temperatures inside the connections as well as their mechanical and thermo-mechanical behaviours. The thermal modelling is validated on the basis of the temperature-time evolutions measured during fire tests. The nonlinear modelling of the mechanical behaviour of timber is done using the Hill yield criterion in combination with the Tsaï-Wu failure criterion. The thermo-mechanical modelling allows obtaining fire resistances of the tested connections in good agreement with the experimental ones.

The accuracy of the forming limit curves reported by the Marciniak-Kuczynski model as well as the quality of the predictions provided by the FE programs depend on the yield criterion that describes the anisotropic plastic behavior of the sheet metal. Two identification procedures of Barlat89 and BBC2005 are used in this paper. The first procedure is a conventional one well described in the literature [, while the second approach is developed by adding an experimental parameter determined in plane-strain state that allows establishing the exponent of the yield criteria. The mechanical response of the IF steel shows that the limit strains are strongly influenced by a small alteration of the yield surface. The influence of the yield criterion on the thickness predictions obtained in the simulation of a square cup deep drawing simulation is also studied.

This work investigates the relation between shear stress and plastic yield considering that a crystal can only deform in a limited set of directions. The shear stress in arbitrary directions is mapped for some cases showing relevant differences. Yield loci based on mean shear stress are constructed. The Tresca yield criterion can be improved by averaging the shear stress over directions near the direction of maximum shear stress. Yield criteria based on averaging over crystallographic direction show a clear influence of the actual orientation of these direction, notably in case of few crystallographic directions. The general finding is that the higher the isotropy of a material, the lower the plane strain factor. The shape of the yield loci is comparable to those derived by the Hershey criterion with exponents lower than 3.

The classical yield criteria do not describe the yield size-dependency that has been observed in micro-scale structures ( Fleck, Muller, Ashby, & Hutchinson, 1994 ; Stolken and Evans 1998; Moreau et al. 2005 ), and may also significantly underestimate the yield loads of structures at micron and sub-micron scales ( Son et al., 2003 , Liu et al., 2013 ). The present paper suggests a specific size-dependent yield criterion based on the strain gradient theory that is aimed at addressing this inadequacy. In order to develop the strain gradient based yield criterion, the deviatoric part of the strain energy is calculated based on this theory and equated to the deviatoric strain energy of a macro-sample under tensile yielding load. For bending of microbeams and torsion of microbars, the yielding loads are derived in closed-forms and compared with the experimental results as well as the results calculated based on the classical von-Mises criterion. It is indicated that when the characteristic size of the structures (e.g., thickness of the microbeams or diameter of the microbars) is comparable to the length scale of the structure material (which are usually on the order of micrometers), there is a considerable gap between the results of the classical von-Mises criterion and the experimental observations while the results of the strain gradient based criterion are in good agreement with the experimental findings. As the size of the structure increases, the results of the new criterion converge to those of the classical von-Mises criterion.

Commonly used orthotropic Hill’s criterion of plastic flow initiation (Hill in Proc R Soc Lond A 193:281–297, 1948) suffers from some constraints and inconsistencies, which are of two different origins. Firstly, in case of high orthotropy degree, the quadratic form corresponding to Hill’s criterion may change type from convex and closed elliptic to concave and open hyperbolic in the deviatoric stress space (Ottosen and Ristinmaa in the mechanics of constitutive modeling, Elsevier, Amsterdam, 2005). Secondly, application of classical Hill’s criterion to transversely isotropic materials shows a discrepancy between Hill’s limit curves in the transverse isotropy plane and the Huber-von Mises prediction for isotropic materials (Huber 1904; von Mises 1913). The basic result of the present paper is to propose the new transversely isotropic von Mises–Hu–Marin’s-type criterion of hexagonal symmetry that is free from both constraints. The new enhanced Hu–Marin’s-type limit surface represents an elliptic cylinder, the axis of which is proportional to stress/strength, in contrast to Hill’s-type limit surface possessing the hydrostatic axis. Hence, this condition does not exhibit the deviatoricity property, which is a price for coincidence with the Huber–von Mises condition in the transverse isotropy plane, but with cylindricity ensured for an arbitrarily high orthotropy degree. The hybrid-type transversely isotropic Hu–Marin’s criterion of mixed symmetry based on additional biaxial bulge test, capable of fitting experimental findings for some complex composites, is also proposed. Application of this criterion has been verified for a unidirectional SiC/Ti composite examined by Herakovich (Thermal stresses V, Lastran Corp. Publ. Division, pp 1–142, 1999).