Anisotropic Criterion Research Articles

Biaxial tensile behavior of stainless steel 316L manufactured by selective laser melting

In this study, miniaturized cruciform biaxial tensile specimens were optimized by finite element simulation software Ansys to vary five geometric parameters. The optimized specimens were utilized to characterize the biaxial tensile properties of 316L stainless steel fabricated through selective laser melting (SLM), with the two loading directions being vertical (X) and parallel (Y) to the building direction. It was discovered that at load ratios of 4:2 and 2:4, the yield strengths along X and Y orientations reached their respective maxima. By comparing the experimentally obtained yield loci against predictions by theoretical criteria including Mises, Hill48 and Hosford, it was found that the Hill48 anisotropic criterion corresponded most closely with the experimental results, while the other two criteria exhibited considerably larger deviations. Therefore, Hill48 was concluded to most accurately describe the yielding behaviors of SLM 316L under complex loading conditions.

Numerical simulation of ideal and non-ideal under-expanded supersonic jets with adaptive grids

This paper numerically investigates the effect of flow non-ideality on the Mach disk pattern occurring in axisymmetric under-expanded supersonic jets. The explored conditions are found in the non-ideal gas region close to the liquid–vapor saturation curve and the critical point, where involved thermodynamics cannot be accurately described by the ideal gas law. Numerical simulations of steady inviscid compressible flows are carried out by adopting the polytropic ideal gas model and improved Peng–Robinson thermodynamic model, via the open-source computational fluid dynamics code SU2. To properly resolve the Mach disk position and radius, a mesh adaptation procedure is used. An assessment of the most suitable mesh adaptation criteria for the considered ideal and non-ideal flow configurations is carried out, considering isotropic and anisotropic mesh adaptation criteria. The best mesh adaptation criterion for ideal and non-ideal under-expanded jets results to be the anisotropic criterion based on the Hessian of the density and Mach number field. Then, the effects of the flow non-ideality on the Mach disk are investigated by simulating different operating conditions of under-expanded nozzle jets of five different vapors. For nitrogen, oxygen, carbon dioxide, the Mach disk moves downstream and has a smaller radius with a higher level of non-ideality. Conversely, for siloxanes MM and MDM, the Mach disk moves upstream and has a larger radius as the non-ideality of the flows increases.

3D numerical simulations and microstructural modeling of anisotropic and tension–compression asymmetric ductile materials

In the present work, we have analyzed the effect of anisotropy on void growth and stress–strain behavior for materials that exhibit remarkable tension–compression asymmetry (i.e., zirconium alloys). For that purpose, we have performed finite element simulations using a cubic 3D cell with a spherical void inside and subjected to periodic boundary conditions. Nonlinear kinematic constraints are also imposed as boundary conditions in order to maintain the values of macroscopic ratios constant during the whole loading history of the cell and account for a general (3D) stress state. The behavior of the matrix material is described by the CPB06 anisotropic criterion developed by Cazacu et al. (2006). The numerical results are compared to those considering 3D homogeneous (without void) cell with the same initial porosity as the voided one and governed by the anisotropic porous yield criterion developed by Stewart and Cazacu (2011). To investigate the influence of prescribed stress, strength differential parameter and strain hardening exponent on stress–strain behavior and void growth in the non-homogeneous (with void) and the homogeneous (without void) cells, we have used several stress ratios, three strength differential parameters and three strain hardening exponents. Finite element results obtained from different stress ratios show the strength differential parameter significantly affect void growth in both homogeneous and non-homogeneous cells. Moreover, comparison of two cells proves that both stress–strain behavior and porosity evolution are in good qualitative agreement for all three values of strength differential parameter. In contrast, as the value of strain hardening exponent increases, the agreement between results obtained from homogeneous and non-homogeneous cells is worse. An heuristic extension of the Stewart and Cazacu (2011)’s model is proposed in this work in an attempt to improve the accuracy of the model.

A computational multiscale model for anisotropic failure of sheet molding compound composites

We present a holistic multiscale approach for constructing anisotropic criteria describing the macroscopic failure of sheet molding compound composites based on full-field simulations of microscale damage evolution. We use an anisotropic damage model on the microscale that directly operates on the compliance tensor, captures matrix and bundle damage via dedicated stress-based damage criteria and allows for a comparison of simulation and experimental results. To identify the damage material-parameters used in the non-linear and time-consuming full-field simulations, we utilize Bayesian optimization with Gaussian processes. We validate the full-field predictions on the microscale and subsequently identify macroscopic failure criteria based on distributions taken from experimental findings. We propose failure surfaces in stress space and stiffness-reduction triggered failure surfaces to cover both a structural analysis and a design process perspective.

Potential Roof Collapse Analysis of Tunnel Considering the Orthotropic Weak Interlayer on the Detaching Surface

The evaluation of the roof collapse in tunnels or cavities remains one of the most complex issues in geotechnical engineering. Taking the detaching surface of the tunnel roof collapse as an orthotropic weak interlayer, an analytical approach for determining the limit collapse range considering the arch effect of the tunnel is presented in this paper by the variation calculus. A discontinuity criterion moving from the anisotropic criterion proposed by the present authors is applied to the orthotropic interlayer. The phenomenon of sharp points in collapse blocks is further analyzed. Based on the proposed approach, illustrated examples are analyzed to investigate the effect of the strength parameters and the consideration of the collapse cusp, which show different influence laws on the range of collapse blocks. Those interesting conclusions can provide guidance for the prediction of the collapse mechanism of the tunnel.

Simulation of the micro Single Point Incremental forming process of very thin sheets

The purpose of this paper is to simulate a complex forming process with parameters identified from tensile and shear tests. An elastic-plastic model is retained which combines a Hill’s 1948 anisotropic criterion and plastic potential using a non-associated flow rule. Firstly, a mechanical characterization is made with homogenous tests like tensile and shear tests [1]. On the other hand a process of micro Single Point Incremental forming is simulated [2]. It consists in deforming a clamped blank using a hemispherical punch which has a small diameter compared to the blank dimensions. From a small-size sheet of 0.2 mm thick, a square-based pyramid is obtained incrementally, with a define height path and advanced speed, by a tool instrumented to measure the forming force, which deforms locally the material. It is shown that the non-associated flow plasticity model leads to a good agreement between experimental and numerical results for the evolution of the punch force during the process.

Impact of DIC biases on the selection process of a unique test for anisotropic plasticity characterization

The exploitation of field measurements with inverse identification methods may reduce the number of required tests to characterize complex material constitutive models, provided that the generated stress field is sensitive enough to the targeted material parameters. For anisotropic elastoplastic material, the objective is to generate various stress states in the specimen through a single test. In this study, the effect of Digital Image Correlation measurement biases on the selection of the most suitable specimen geometry for characterisation of a complex anisotropic plasticity criterion using a unique uniaxial tensile test is investigated. To this aim, finite element (FE) based synthetic images are generated and DIC is used on these images. The biases in DIC measurement result in biased stress states that may cause errors in identification results.

Constitutive and Fracture Modeling of Biaxially Oriented Polyethylene Terephthalate Film and Its Application to Polymer-Coated Sheet Metal Forming

AbstractPolyethylene terephthalate (PET) films produced by rolling and tentering have biaxial orientation and thus exhibit anisotropic mechanical properties such as yielding, hardening, and fracture. The anisotropy of film makes it difficult to analyze deformation and failures occurring during processing such as metal forming of polymer-coated sheet metals. In this work, the anisotropic yielding and hardening of the PET film were measured and an anisotropic constitutive model is developed to represent the anisotropic behavior. In addition, fracture limits of the PET film were measured using the Nakajima test. To describe anisotropic fracture behavior of the film, an anisotropic fracture criterion based on the strain energy density and the theory of isotropic equivalent material was proposed. For validation of the developed constitutive model and the fracture criterion, a deep drawing test of the PET-coated sheet metal was conducted. The film cracking predicted by the anisotropic fracture criterion agreed well with measurements.

A Novel Anisotropic Failure Criterion With Dispersed Fiber Orientations for Aortic Tissues.

Accurate failure criteria play a fundamental role in biomechanical analyses of aortic wall rupture and dissection. Experimental investigations have demonstrated a significant difference of aortic wall strengths in the circumferential and axial directions. Therefore, the isotropic von Mises stress and maximum principal stress, commonly used in computational analysis of the aortic wall, are inadequate for modeling of anisotropic failure properties. In this study, we propose a novel stress-based anisotropic failure criterion with dispersed fiber orientations. In the new failure criterion, the overall failure metric is computed by using angular integration (AI) of failure metrics in all directions. Affine rotations of fiber orientations due to finite deformation are taken into account in an anisotropic hyperelastic constitutive model. To examine fitting capability of the failure criterion, a set of off-axis uniaxial tension tests were performed on aortic tissues of four porcine individuals and 18 human ascending thoracic aortic aneurysm (ATAA) patients. The dispersed fiber failure criterion demonstrates a good fitting capability with the off-axis testing data. Under simulated biaxial stress conditions, the dispersed fiber failure criterion predicts a smaller failure envelope comparing to those predicted by the traditional anisotropic criteria without fiber dispersion, which highlights the potentially important role of fiber dispersion in the failure of the aortic wall. Our results suggest that the deformation-dependent fiber orientations need to be considered when wall strength determined from uniaxial tests are used for in vivo biomechanical analysis. More investigations are needed to determine biaxial failure properties of the aortic wall.

Advances in anisotropy of plastic behaviour and formability of sheet metals

This paper reviews the most recent models for description of the anisotropic plastic behavior and formability of sheet metals. After a brief review of classic isotropic yield functions, recent advanced anisotropic criteria for polycrystalline materials of various crystal structures and their applications to cup drawing are presented. Next, the discussion focuses on novel formulations of anisotropic hardening. A brief review of the experimental methods used for characterizing and modeling the anisotropic plastic behavior of metallic sheets and tubes under biaxial loading is presented. The experimental methods and theoretical models used for measuring and predicting the limit strains, development of new tests for determining the Forming Limit Curves (FLC), as well as on studying the influence of various material or process parameters on the limit strains are presented.

A general failure criterion for soil considering three-dimensional anisotropy

Abstract This paper proposes a generalized three-dimensional (3D) failure criterion for cross-anisotropic soils based on the spatially mobilized plane (SMP) criterion. In the proposed criterion, the anisotropic property of soil is considered by introducing a novel modification stress tensor, in which the angle between the principal stress and the soil depositional direction serves as a basic variable. Taking advantage of the SMP criterion, the proposed anisotropic criterion is subtly extended into a 3D version. The proposed criterion only involves two friction angles, φ v and φ h , which are determined by triaxial compression test along and perpendicular to the depositional direction, respectively. To account for the effects of the depositional direction, two special modification stress tensors regarding different depositional conditions (vertical and inclined) are discussed in this study. Comparisons between the predictions from the proposed strength criterion and the observations of the true triaxial tests on four cross-anisotropic soil samples show that the proposed generalized failure criterion can well capture both the cross-anisotropic behaviors and the 3D strength of cross-anisotropic soils.

Using the Pietruszczak-Mroz anisotropic failure criterion to model the strength of stratified rocks

The strength of stratified rocks is closely related to their inherent structural anisotropy, which cannot be accurately represented by the isotropic failure criteria. In this study, the Pietruszczak-Mroz (PM) anisotropic failure criterion was used to describe the directional dependency of strength of stratified rocks. In the meridian plane, the frictional coefficient was established by incorporating an anisotropic variable defined as a joint invariant of fabric and stress tensors. In the deviatoric plane, the anisotropic failure surface was characterized by a smooth shape function of Lode's angle. The material parameters for the failure criterion can be determined using the triaxial test. The PM anisotropic criterion was utilized to predict the anisotropic strength of specific stratified rocks. Analysis of the experimental data validated the assertion that PM anisotropic criterion has excellent ability to model the anisotropic strength features of stratified rocks.

Development of Uncoupled Anisotropic Ductile Fracture Criteria

Two mainstreams of transformation methods from isotropic fracture criteria to anisotropic ones are to replace the Von Mises yield function with other yield function and linear transformation of strain or stress tensor. Several typical anisotropic criteria such as the transformation methods of Park, Luo, Lou and Yoon, Jia and Bai and a new transformation method are analyzed and discussed from the aspects of required experimental data, material constant amounts, stress state range, average predictive error (Erravg), etc. The fitting results demonstrate that the prediction accuracy of the methods of Luo, Lou and Yoon, Jia and Bai and the new method are better than that of the isotropic criteria from the Erravg aspect based on the experimental results of DP590. In addition, the new transformation method linearly transforms both stress and strain tensor without increasing too many parameters.

Wellbore Stability in Layered Rocks: A Comparative Study of Strength Criteria

AbstractWellbore instability is a critical issue restricting efficient well drilling and successful development of oil and gas field. Most instability problems originate from shale formations because of their distinct laminated structures that cause significant anisotropy and moderate to high clay contents that are prone to shrinkage and swelling. To account for these influences on the mechanical responses of shales, this study aims to identify an appropriate strength criterion for stability analyses. Two anisotropic criteria including single plane of weakness and the modified Hoek–Brown criteria were compared to evaluate their suitability in characterizing the anisotropic strength of layered rocks including shale, schist, and slate under different confining pressures. Comparative case studies indicated that the single plane of weakness criterion overestimates the strength of layered rocks at some orientation angles. The modified Hoek–Brown criterion can fit well with the experimental data of layered rocks. Moreover, wellbore stability analysis models for shale gas wells were built, respectively, for each criterion and applied to in situ scenarios. The single plane of weakness and modified Hoek–Brown criteria provide similar results of collapse pressure, and the shale failure is mainly determined by the bedding plane. This further validates that the modified Hoek–Brown criterion is a good choice for wellbore stability analysis in shale formations with bedding planes. This study shows the potential of using the modified Hoek–Brown criterion to enhance the safety and efficiency of well drilling and operation in shale formations.

Prediction of the limiting redrawing ratio

Redrawing process is used for reducing the lateral dimensions and increasing the depth of the cylinder cup, which was formed by deep drawing. To determine the optimum values of the process parameters, it is necessary to know the redrawability. In this paper, a new mathematical mode is established to predict the limiting redrawing ratio (LDR), which is based on Hill’s anisotropic criteria, tensile instability in the cup wall and sheet bending/unbending theory around the die arc radius. The maximum and minimum thickness ratio (tmax/tmin) of cup wall and friction effect were considered in the model. The calculation from the model is in good agreement with the results of finite element simulation, and also was confirmed by the experiments of commercially pure titanium (TA1) cup redrawing processing. The effects of various material properties, friction coefficient μ, radius of the die arc rd, half die cone anger α, half die opening ro, cup maximum thickness tmax, cup thickness ration tmax/tmin have been discussed to understand and control redrawing process for improving redrawablity.

Anisotropic fracture modeling of sheet metals: From in-plane to out-of-plane

Sheet metals usually exhibit varying degrees of anisotropy due to formation of texture during rolling process. The grain structures and distributions in sheet normal direction may also be different from that within the sheet plane. For example, martensite banding structures are usually observed in the middle of thickness for advanced high strength steels. The particular martensite morphology and inhomogeneous distribution will greatly affect the deformation and fracture behavior under out-of-plane direction. It is necessary to model the anisotropic fracture of sheet metals from in-plane to out-of-plane. In this work, a new stress invariant based ductile fracture criterion was developed by introducing a stress triaxiality and normalized third invariant dependent function h(η, ξ) and further extended to an anisotropic form through the linear transformation of stress tensor. Parametric study showed that the new anisotropic criterion can describe the direction dependency of ductile fracture in both strain and stress spaces. A series of in-plane fracture tests including tension with a central hole, notched tension, V-bending test, Nakajima test and in-plane shear as well as a new out-of-plane shear test were conducted to study the anisotropic fracture behavior of DP980 sheet metals over a variety of stress states. Hybrid experimental-numerical method is used to determine the fracture strains and the loading paths to fracture. The results show that slight in-plane anisotropy of fracture responses exists for this DP980 sheet, while the out-of-plane shear fracture stress is approximately 11% lower than that under the in-plane shear condition. The new anisotropic ductile fracture criterion is able to predict the anisotropic fracture behaviors of this DP980 sheet with high accuracy.

Anisotropic Strength of Granular Material Considering Fabric Evolution

Abstract The effect of fabric anisotropy on strength has not been considered in most strength criterions for granular material. Some criterions can describe the variation of material strength with stress-induced anisotropy, but micro-mechanism and physical meanings are undefined. Some criterions consider the effect of fabric anisotropy, but the evolution of fabric is ignored during loading by assuming a constant fabric tensor. Based on strength mechanical characteristics of granular material, in this paper, the relationship between macro stress and micro contact force of granular material is derived by micromechanics. Then, the concept of true stress tensor is proposed. An anisotropic strength criterion of granular material considering fabric evolution is established and its applicability is validated by comparing with test results for different granular materials. The analysis results indicate that the proposed anisotropic criterion can be utilized to describe the strength feature of anisotropic granular materials, which gives a way for the cause analysis of the strength of granular materials from the perspective of Microscopic mechanism.

The intersection of a hyperplane with a lightcone in the Minkowski spacetime

Klainerman, Luk and Rodnianski derived an anisotropic criterion for formation of trapped surfaces in vacuum, extending the original trapped surface formation theorem of Christodoulou. The effort to understand their result led us to study the intersection of a hyperplane with a lightcone in the Minkowski spacetime. For the intrinsic geometry of the intersection, depending on the hyperplane being spacelike, null or timelike, it has the constant positive, zero or negative Gaussian curvature. For the extrinsic geometry of the intersection, we find that it is a noncompact marginal trapped surface when the hyperplane is null. In this case, we find a geometric interpretation of the Green's function of the Laplacian on the standard sphere. In the end, we contribute a clearer understanding of the anisotropic criterion for formation of trapped surfaces in vacuum.

Using Non-linear Homogenization to Improve the Performance of Macroscopic Damage Models of Trabecular Bone.

Realistic macro-level finite element simulations of the mechanical behavior of trabecular bone, a cellular anisotropic material, require a suitable constitutive model; a model that incorporates the mechanical response of bone for complex loading scenarios and includes post-elastic phenomena, such as plasticity (permanent deformations) and damage (permanent stiffness reduction), which bone is likely to experience. Some such models have been developed by conducting homogenization-based multiscale finite element simulations on bone micro-structure. While homogenization has been fairly successful in the elastic regime and, to some extent, in modeling the macroscopic plastic response, it has remained a challenge with respect to modeling damage. This study uses a homogenization scheme to upscale the damage behavior from the tissue level (microscale) to the organ level (macroscale) and assesses the suitability of different damage constitutive laws. Ten cubic specimens were each subjected to 21 strain-controlled load cases for a small range of macroscopic post-elastic strains. Isotropic and anisotropic criteria were considered, density and fabric relationships were used in the formulation of the damage law, and a combined isotropic/anisotropic law with tension/compression asymmetry was formulated, based on the homogenized results, as a possible alternative to the currently used single scalar damage criterion. This computational study enhances the current knowledge on the macroscopic damage behavior of trabecular bone. By developing relationships of damage progression with bone's micro-architectural indices (density and fabric) the study also provides an aid for the creation of more precise macroscale continuum models, which are likely to improve clinical predictions.

Identification of Anisotropic Criteria for Stratified Soil Based on Triaxial Tests Results

Abstract The paper presents the identification methodology of anisotropic criteria based on triaxial test results. The considered material is varved clay – a sedimentary soil occurring in central Poland which is characterized by the so-called “layered microstructure”. The strength examination outcomes were identified by standard triaxial tests. The results include the estimated peak strength obtained for a wide range of orientations and confining pressures. Two models were chosen as potentially adequate for the description of the tested material, namely Pariseau and its conjunction with the Jaeger weakness plane. Material constants were obtained by fitting the model to the experimental results. The identification procedure is based on the least squares method. The optimal values of parameters are searched for between specified bounds by sequentially decreasing the distance between points and reducing the length of the searched range. For both considered models the optimal parameters have been obtained. The comparison of theoretical and experimental results as well as the assessment of the suitability of selected criteria for the specified range of confining pressures are presented.