Plastic Anisotropy Research Articles

Anisotropic plasticity mechanisms in a newly synthesised high entropy alloy investigated using atomic simulations and nanoindentation experiments

This work used atomic simulations and nanoindentation experiments to investigate hardness, modulus alongside sub-surface crystal defects and dislocation mediated plasticity mechanisms leading to anisotropic pile up and local entropy variation in high entropy alloys. The experimental campaign began from Thermo-Calc phase prediction of Ni25Cu18.75Fe25Co25Al6.25 HEA which followed experimental synthesis of the material using arc melting method and experimental nanoindentation using a Berkovich indenter under load-controlled conditions. Through MD simulations, the value of hf/hmax in monocrystalline HEA was consistently found to be larger than 0.7 which suggested pile-up behaviour to dominate and sink-in behaviour to be unlikely. In the case of (110) and polycrystalline HEA substrates, the elastic work in the indentation hysteresis loop was seen to be larger than the (100) and the (111) orientations which explains that the (110) orientation substrate showed least elastic modulus and hardness while the (111) monocrystalline HEA showed the highest elastic modulus and hardness. From the simulations, a “lasso” type loop on the (110) orientation and cross-over of shear loops on the other orientations accompanied by dislocations of type 1/6 < 112 > (Shockley), 1/2 < 110 > (perfect), 1/3 < 001 > (Hirth), 1/6 < 110 > (Stair rod) and 1/3 < 111 > (Frank partials) were seen to manifest an early avalanche of competing plasticity events. The defects accompanying these dislocations in the sub-surface were identified to be FCC intrinsic stacking faults (ISF), adjacent intrinsic stacking faults (quad faults), coherent ∑3 twin boundary and a coherent twin boundary next to an intrinsic stacking fault (triple fault). The EBSD analysis applied to the MD data showed that the (210) orientation and the< 110 > family of directions were seemed to be preferable to plastically deform the FCC phased Ni25Cu18.75Fe25Co25Al6.25 HEA.

The mechanism of anisotropic micro-milling properties in additively manufactured Ti-6Al-4V alloy

Selective Laser Melting (SLM) technology offers great flexibility in manufacturing complex parts. However, secondary processing of selective laser melted (SLMed) parts is essential due to the poor surface morphology. This study investigated the influence of microstructural anisotropy in SLMed Ti-6Al-4V alloy on micro-milling performance. The orthogonal milling tests were conducted on the top, front, and side surfaces of specimens. Parameters including cutting force, chip morphology, burr morphology, surface quality, microhardness, and microstructure were evaluated to represent the anisotropy. The experimental findings demonstrated that the tool's orientation relative to the columnar grains influenced the slip strain modes and the density of grain boundaries in the shear band, thereby determining the correlation of the cutting force with the machined surface and feed direction. The variations in grain boundary density in different machining regions caused plastic anisotropy, which affects chip morphology, burr formation, and machined surface roughness. This research could provide valuable insights for manufacturers in selecting appropriate machining methods for precision processing.

Development of a Deep Learning Model for Capturing Plastic Anisotropy–Texture Linkage

Development of a Deep Learning Model for Capturing Plastic Anisotropy–Texture Linkage

Grain-Scale Study of SAC305 Oligocrystalline Solder Joints: Anisotropic Elasto-Plastic Constitutive Properties of Single Crystals

Abstract This paper focuses on anisotropic elastic-plastic constitutive modeling of SAC (SnAgCu) solder grains because of their importance in modeling the behavior of oligocrystalline (few-grained) micron-scale solder joints that are increasingly common in heterogeneous integration. Such grain-scale anisotropic modeling approach provides more accurate assessment of the mechanical response of solder interconnects in terms of predicting different failure modes, failure sites, and variability in time-to-failure. Anisotropic plasticity is represented using Hill–Ramberg–Osgood (RO) continuum plasticity model, which utilizes Hill's anisotropic plastic potential along with a RO power-law plastic hardening flow rule. Mechanistically motivated empirical scaling factors are proposed to extrapolate the stress–strain response for different grain sizes/shapes and for different coarseness of microstructures within each grain (generated with different cooling rates). This scaling factor can therefore also capture the effects of microstructural coarsening due to isothermal aging. This goal is achieved by first conducting monotonic tensile and shear tests on monocrystalline and oligocrystalline SAC305 solder joints containing grains of various geometries and also intragranular microscale (dendritic and eutectic) structures of various coarseness. The grain structures are characterized for each tested specimen using electron backscattered diffraction (EBSD). The Hill–RO model constants and the empirical scaling factors are then estimated by matching grain-scale anisotropic elastic-plastic finite element models of each tested specimen to the measured stress–strain behavior, using an inverse-iteration process. Grain shape is seen to influence the sensitivity of the effective stress–strain curves to the applied stress state (i.e., to the orientation of the principal stress directions) relative to (i) the material principal directions and (ii) the geometric principal directions of grains with high aspect ratio. Limitations of the current results and opportunities for future improvements are discussed.

Three-dimensional computational characterization of grain size and texture effects in magnesium alloys

This work systematically investigates the microstructure-property relationship in Mg alloys. Emphasis is placed on understanding, through high resolution crystal plasticity modeling, how grain size and texture collectively impact material strengthening and hardening, net plastic anisotropy, and tension-compression asymmetry. To achieve this, 528 fully three-dimensional finite element calculations are performed, which comprise eleven textures, four grain sizes, six loading orientations, and two uniaxial loading states (tension and compression). The grain size effect follows Hall-Petch relation that depends on both, loading orientation and initial texture. The reduction in extension twinning with grain size refinement is influenced by texture as well. Below a threshold textural strength, grain size refinement leads to an appreciable reduction in the net plastic anisotropy at yield, quantified using Hill anisotropy, and reduced tension-compression asymmetry. Using a micromechanical basis, the effect of grain size and texture on material ductility is predicted to be non-monotonic. The computational predictions serve as synthetic data sets for experimental validation and reduced-order modeling.

Comparison of anisotropic yield functions and calibrations for accurate thickness prediction in hole expansion test

The hole expansion test is an appropriate evaluation method for assessing the formability of automotive parts, where fracture prediction using simulated localized thickness is crucial. However, the accurate thickness prediction in the hole expansion test remains challenging. In this study, the selection of yield functions and calibration strategies were investigated to accurately predict the thickness of advanced high-strength steel subjected to the hole expansion test. An accurate description of the material anisotropy is important for thickness prediction because of the generation of various multiaxial stress states. To describe plastic anisotropy, a series of uniaxial tension, bulge, and in-plane biaxial tension tests were conducted on DP980 steel. Because conventional biaxial tension tests for DP980 steel showed a small strain range owing to its low ductility, a new laser-welding-reinforced biaxial tension test was developed to investigate the anisotropy of the plane strain at large strains. Differences in the ratio of the plastic strain rates were observed between the conventional and new tests. Finally, a hole expansion test and corresponding finite element simulations were conducted with different yield functions and calibrations for the Yld2000-2d yield function. The thickness strain profiles along the hole circumference and 3 mm inward were measured and predicted using simulations. In conclusion, both the yield stress and the ratio of plastic strain rates in the plane strain state were equally important for accurate thickness prediction during the DP980 hole expansion test. Furthermore, the crack location was inconsistent with the thinnest region in the experiment, which might be because of the shear fracture anisotropy encountered in DP980 steel.

Quantification of the influence of anisotropic plastic yielding on cold rolling force

Anisotropic mechanical properties in sheet material are of high importance, in particular for material to be used in stamping or deep drawing applications. The cold rolling process has a marked influence on the degree of anisotropy in the final product. However, the influence of material anisotropy on the cold rolling process itself has so far not been rigorously investigated. In this work, the degree of normal anisotropy is determined for two low-carbon steel grades after various degree of cold rolling reduction, both by X-ray diffraction as well as by tensile testing. The experimental work shows that material with high cold rolling reduction has non-negligible anisotropic properties; as a consequence the rolling force is seriously overestimated by cold rolling models with an isotropic yield criterion. It is therefore proposed to use the Hill48 yield criterion (instead of the von Mises criterion) in cold rolling models, this criterion takes anisotropic material behaviour into account. A comparison of cold rolling experiments with model predictions confirms that including the Hill48 yield criterion significantly improves the accuracy of a cold rolling model.

Modeling of Eyld2000-2d Anisotropic Yield Criterion Considering Strength Differential Effect and Analysis of Optimal Calibration Strategy.

Sheet metals usually experience various loading paths such as uniaxial tension, uniaxial compression, biaxial tension, and simple shear during the forming process. However, the existing constitutive models cannot always accurately describe blanks' anisotropic yield and plastic flow behavior of blanks under all typical stress states. Given this, this paper improves the Eyld2000-2d yield criterion by introducing hydrostatic pressure to the A-Eyld2000-2d yield criterion that can describe the strength differential effect of materials. Meanwhile, to control the curvature of the yield surface more effectively, the near-plane strain yield stresses were added in the parameter identification process to calibrate the exponent m, so that the exponent is no longer considered as a constant value. Taking the widely used AA6016-T4, AA5754-O, DP980, and QP980 blanks in the automotive stamping industry as an example, the effectiveness of the new model and different parameter identification methods was verified by predicting experimental data under various simple and complex loading paths. Subsequently, the new model employing the optimal parameter identification strategy was compared with four widely used asymmetric yield criteria under associated and non-associated flow rules, including CPB06, LHY2013, S-Y2004, and Hu & Yoon2021, to further verify the accuracy of the proposed constitutive model. The results indicate that parameter identification strategy with variable exponent can significantly improve the flexibility of the yield criterion in describing the plastic anisotropy of blanks. Compared to the other yield criteria examined in this work, the new model provides the best prediction accuracy for the yield stresses and plastic flows of all blanks, especially in the near-plane strain and simple shear stress states. Modeling under the concept of anisotropic hardening can more accurately capture the evolving plastic behavior of blanks than isotropic hardening.

Numerical analysis on failure of sheet metals with non-ordinary state-based peridynamics

In this paper, a general anisotropic plastic model for sheet metals is proposed in the non-ordinary state-based peridynamics. The anisotropy plasticity of metallic material is described by the Hill48 yield criterion, and the associated flow rule and non-linear mixed-isotropic-kinematic hardening are included as well. Accordingly, the Newton-Raphson method is used here, which is more efficient than the dichotomy method. Meanwhile, a widely used continuum damage model is coupled with a new bond-breaking method based on the classic Hash principles. A detailed algorithm for numerical implementation is provided for a better understanding. The proposed model is verified by several typical examples and shows its great potential in describing anisotropic plasticity-ductile deformation and failure of sheet metal. In the last, the efficiency of the proposed failure criterion is analyzed and discussed through the numerical analysis of a uniaxial tensile failure test.

Effect of evolutionary anisotropic hardening on the prediction of deformation and forming load in incremental sheet forming simulation

The incremental sheet forming (ISF) process typically accompanies large plastic strain without fracture exceeding uniform elongation or ultimate tensile strength (UTS) in the uniaxial tensile test. Therefore, ISF features improved formability compared to the conventional stamping process. Meanwhile, numerous studies have focused on modeling and simulation of formability in ISF where anisotropic yield function is one of the key constitutive laws for predicting sheet deformation. However, the conventional yield function is defined based on anisotropy of initial yielding, while the ISF modeling requires the deformation behavior at large strains beyond the UTS. In this study, an evolutionary anisotropic plasticity model is investigated based on Hill's 48 yield function combined with the non-associated flow rule; i.e., e-NAFR Hill's 48. The e-NAFR Hill's 48 model is implemented to the vectorized user-defined material subroutine in the commercial finite element software ABAQUS/Explicit. Then, the proposed FE model is applied to the forming of truncated cone, pyramid, and clover shaped single point incremental forming (SPIF) of an aluminum alloy 6014-T4 sheet. The simulated part profile, thickness variation, and forming force are compared with those of experiments. The evolutionary constitutive model shows an average accuracy of 99.15% in thickness prediction and 94.22% accuracy in forming force, demonstrating the importance of evolutionary sheet anisotropy in the SPIF process.

Parameter determination of anisotropic yield function using neural network-based indentation plastometry

Understanding the impact of plastic anisotropy on the formability of sheet metals is crucial for their industrial application and high-precision forming simulation. Existing multiple uniaxial tensile tests for measuring the plastic flow of anisotropic materials, however, are costly, time-consuming, and destructive. Therefore, it is more efficient to focus on indentation plastometry, a simple, nondestructive test that can quickly extract tensile properties. The objective of this work is to directly derive the parameters of the advanced Poly6 yield criterion and hardening, which describes strong plastic anisotropy, using indentation plastometry basd on a neural network (NN) system. The identification process for these parameters through conventional tensile tests is inherently complex, thereby determining the parameters directly from indentation data presents an unprecedented challenge. We trained NNs using a database generated from verified finite element (FE) simulations of spherical indentations. To systematically iterate these FE simulations, we designed a strategy generating a set of input anisotropic parameters that ensure the convexity of the yield function. We considered the radial and vertical displacement fields around the indentation mark along with the load-depth curve as indentation responses. Through a comprehensive analysis of the correlation between displacement profiles, we have proposed an optimal feature extraction method for NN training. The developed FE-NN model was evaluated by comparing the predicted parameters from the indentation responses of the target materials with those measured from tensile tests. These parameters were expressed as the yield locus and directional properties. The results demonstrated that the FE-NN modeling approach is robust and can accurately capture the anisotropic plastic flow from indentation responses.

Quantitative representation of directional microstructures of single-crystal superalloys in cyclic crystal plasticity based on neural networks

Nickel-based single-crystal alloys undergo microstructural degradation induced by thermal exposure. The directional rafting of microstructures significantly affects the mechanical properties and makes the material anisotropic. For structural design, establishing a quantitative description of microstructural effects in a constitutive model becomes essential and is still a tough research topic in multi-scale materials modeling. In the present work, the fabric tensor was correlated with the anisotropic cyclic crystal plasticity of nickel-based single-crystal alloys with the help of neural networks. The microstructural representative volume elements with various single-crystal morphologies were generated by the phase-field method and the deformation behaviors were studied under different crystal orientations and loading configurations. The neural network analysis confirmed that the fabric tensor can present anisotropic single-crystallographic microstructural features and describe mechanical behavior under both monotonic and cyclic multi-axial loading conditions. The history-dependent anisotropic cyclic hardening or softening behavior of the material can be captured by the introduced microstructural state variable. A principal component analysis (PCA) aided gradient-based attribution method was proposed to evaluate the importance of input variables. The characterization of different material components and their contribution to the stress–strain relationships are investigated and validated. The fabric tensor was verified to be an effective microstructural indicator for the continuum plasticity of single-crystal alloys.

Constitutive modeling of clay shales in undrained conditions and its experimental verification for Opalinus Clay

Clay shales are low-permeable sedimentary rocks that exhibit inherent cross-anisotropic stiffness, strength, and permeability. Accurate design of tunnels excavated in clay shales requires appropriate constitutive models that account for the anisotropic properties of the host rock, especially under undrained conditions. This paper presents a transversely isotropic elastoplastic-damage model for fully-saturated clay shales in undrained conditions. The model describes four deformation regimes observed during consolidated undrained experiments, i.e. (1) transversely isotropic elastic deformation, (2) anisotropic plastic deformation with cross-anisotropic peak strength, (3) semi-brittle post-peak softening due to both strength and stiffness degradation, and (4) residual strength state. The performance of the model was evaluated using available laboratory tests for Opalinus Clay, the selected host rock for underground disposal of radioactive waste in Switzerland. A large experimental data set, consisting of 18 undrained triaxial compression tests, was used to back-calculate the required model parameters. The adopted experimental data include the effective stress path and the stress–strain curves of specimens with bedding orientations between 0° and 90° with respect to the loading and effective consolidation stresses between 0.76 and 16 MPa. It is shown that the proposed model adequately captures the key material features of Opalinus Clay observed in the undrained experiments.

Size-dependent yield criterion for single crystals containing spherical voids

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

Plastic anisotropy of powder bed fusion maraging steel: An experimental replication study

The plastic anisotropy of additively manufactured metals can be estimated with the use of the ratio of width to thickness plastic strain, commonly referred to as R-value. Published research has indicated the suitability of the experimentally obtained R-value for the case of maraging steel produced via powder bed fusion (PBF) at different build orientations and subsequently heat treated. The present study replicates this experimental methodology for the same alloy as per published literature, which has been produced with a different PBF equipment (having a different laser power and using a different layer thickness). This replication experimental study has successfully assessed the predictability of the R-value methodology for the PBF maraging steel. In particular, the presence of a comparable level of plastic anisotropy has been discovered in the examined material. Moreover, a previously reported decreasing R-value trend with an increase of the build orientation inclination has also been confirmed.

Anisotropic damage behavior in fiber-based materials: Modeling and experimental validation

This study presents a thermodynamically consistent continuum damage model for fiber-based materials that combines elastoplasticity and damage mechanisms to simulate the nonlinear mechanical behavior under in-plane loading. The anisotropic plastic response is characterized by a non-quadratic yield surface composed of six sub-surfaces, providing flexibility in defining plastic properties and accuracy in reproducing material response. The damage response is modeled based on detailed uniaxial monotonic and cyclic tension-loaded experiments conducted on specimens extracted from a paper sheet in various directions. To account for anisotropic damage, we propose a criterion consisting of three sub-surfaces representing tension damage in the in-plane material principal directions and shear direction, where the damage onset is determined through cyclic loading tests. The damage evolution employs a normalized fracture energy concept based on experimental observation, which accommodates an arbitrary uniaxial loading direction. To obtain a mesh-independent numerical solution, the model is regularized using the implicit gradient enhancement by utilizing the linear heat equation solver available in commercial finite-element software. The study provides insights into the damage behavior of fiber-based materials, which can exhibit a range of failure modes from brittle-like to ductile, and establishes relationships between different length measurements.

A Molecular Dynamics Study on the Dislocation-Precipitate Interaction in a Nickel Based Superalloy during the Tensile Deformation.

In the present paper, the dislocation-precipitate interaction in the Inconel 718 superalloy is studied by means of molecular dynamics simulation. The atomistic model composed of the ellipsoidal Ni3Nb precipitate (γ″ phase) and the Ni matrix is constructed, and tensile tests on the composite Ni3Nb@Ni system along different loading directions are simulated. The dislocation propagation behaviors in the precipitate interior and at the surface of the precipitate are characterized. The results indicate that the dislocation shearing and bypassing simultaneously occur during plastic deformation. The contact position of the dislocation on the surface of the precipitate could affect the penetration depth of the dislocation. The maximum obstacle size, allowing for the dislocation shearing on the slip planes, is found to be close to 20 nm. The investigation of anisotropic plastic deformation behavior shows that the composite system under the loading direction along the major axis of the precipitate experiences stronger shear strain localizations than that with the loading direction along the minor axis of the precipitate. The precipitate size effect is quantified, indicating that the larger the precipitate, the lower the elastic limit of the flow stress of the composite system. The dislocation accumulations in the precipitate are also examined with the dislocation densities given on specific slip systems. These findings provide atomistic insights into the mechanical behavior of nickel-based superalloys with nano-precipitates.

Ductile damage and failure of thin sheet metals: New biaxial experiments and numerical simulations

AbstractThe paper deals with the ductile damage and failure behavior of thin sheet metals under proportional biaxial loading paths. A newly designed specimen is presented and an experimental series with different loading ratios is performed. Corresponding numerical simulations including plastic anisotropy reflect the stress state and enable the interpretation of the different damage mechanisms. Due to fabrication issues the specimen geometry for this class of materials has to get by without notches in thickness direction and consequently the new specimens pre‐define the localization by a series of holes. The strain fields at the specimen surface are evaluated by digital image correlation and the fracture surface is analyzed by scanning electron microscopy. The experimental and numerical methods described here provide a possibility for the targeted investigation of the ductile damage behavior of thin sheet metals under proportional loading.

A new texture descriptor for data-driven constitutive modeling of anisotropic plasticity

Constitutive modeling of anisotropic plastic material behavior traditionally follows a deductive scheme, relying on empirical observations that are cast into analytic equations, the so-called phenomenological yield functions. Recently, data-driven constitutive modeling has emerged as an alternative to phenomenological models as it offers a more general way to describe the material behavior with no or fewer assumptions. In data-driven constitutive modeling, methods of statistical learning are applied to infer the yield function directly from a data set generated by experiments or numerical simulations. Currently these data sets solely consist of stresses and strains, considering the microstructure only implicitly. Similar to the phenomenological approach, this limits the generality of the inferred material model, as it is only valid for the specific material employed in the virtual or physical experiments. In this work, we present a new generic descriptor for crystallographic texture that allows an explicit consideration of the microstructure in data-driven constitutive modeling. This descriptor compromises between generality and complexity and is based on an approximately equidistant discretization of the orientation space. We prove its ability to capture the structure–property relationships between a variety of cubic–orthorhombic textures and their anisotropic plastic behavior expressed by the yield function Yld2004-18p. Three different machine learning models trained with the descriptor can predict yield loci as well as r-values of unseen microstructures with sufficient accuracy. The descriptor allows an explicit consideration of crystallographic texture, providing a pathway to microstructure-sensitive data-driven constitutive modeling.

Formability prediction of high‐strength steel sheet using experimental, analytical and theoretical analysis based on strain and stress forming limit curves and its application to automotive forming parts

AbstractIn this work, the high‐strength steel (HSS) sheet dual‐phase 440 (DP440) were conducted to establish the forming limit curve (FLC) and analytical forming limit stress curve (FLSC) obtained from experimental forming limit curve. First, the Nakajima stretch forming examination was carried out to obtain forming limit curve of investigated sheet. Afterwards, the theoretical Marciniak–Kuczinsky (M–K) model was developed and calculated to evaluate localized necking limits both in strain and stress spaces combination with anisotropic yield criteria. Then, the analytical forming limit stress curves were plastically calculated by using experimental forming limit curve data combination with Swift hardening model and anisotropic yield criteria namely, Hill’48 and Yld2000‐2d for representing anisotropic plastic deformation behavior on examined steel sheet. Finally, automotive stamping parts were performed in order to verify an applicability of all developed curves. It was observed that the analytical forming limit stress curves could more precisely predict the formability of automotive parts better than the forming limit curve based on strain. Particularly, the one based on Yld2000‐2d predict better than the one based on Hill’48. Simultaneously, the experimental forming limit curve and analytical forming limit stress curve were also evaluated comparing with the theoretical calculated forming limit curve and forming limit stress curve using the Marciniak–Kuczinsky model. It should be noted again that the experimental forming limit curve and analytical forming limit stress curve are the best one. Therefore, the Yld2000‐2d yield function better represented the anisotropic behavior of the high‐strength steel sheet dual‐phase 440 than Hill′ 48 yield function, and can suitable be used for the analysis prediction and design of bumper automotive parts under forming processes.