Plane Strain Deformation Research Articles

Texture Evolution of High-purity Tantalum during 90°Clock Rolling

Abstract When fabricating high-purity tantalum targets for the sputtering procedure of integrated circuit manufacturing, a through-thickness texture gradient can form in the rolled tantalum (Ta) plate. This texture gradient hinders the target sputtering performance, reducing chip reliability. This study investigated the texture evolution during the 90°clock rolling through experimental and simulation methods. Balancing the plane strain deformation with the surface shear strain deformation during fabrication and limiting the total rolling reduction to no more than 85% can weaken the through-thickness texture gradient in the Ta plate. A crystal plasticity finite element method (CPFEM) model was developed to simulate the 90°clock rolling procedure in the ABAQUS/Explicit software. The simulation and the experimental results are in agreement, with slight differences in texture type and intensity due to tantalum’s complex slip behaviors.

Effect of the deformation temperature and strain on the strain rate sensitivity of fcc medium-entropy alloys

The primary objective of the present investigation is to elucidate the operative micromechanisms influencing the strain rate sensitivity and activation volume in (FeCrNi)99Si1 and FeMnNi medium-entropy alloys. Room-temperature nanoindentation experiments at different loading rates were performed to study the evolution of the strain rate sensitivity and activation volume in (FeCrNi)99Si1 and FeMnNi medium-entropy alloys. The (FeCrNi)99Si1 samples were subjected to plane strain deformation by rolling at 77 and 300 K to study the effect of temperature on the strain rate sensitivity, while the FeMnNi and (FeCrNi)99Si1 samples were subjected to simple shear deformation by high-pressure torsion at 300 K to examine the effect of strains. Contrary to the well-documented trend observed in fcc metals and alloys, where the strain rate sensitivity typically increases with decreasing grain size, the present study reveals a distinct behavior for the current alloys. Similarly, these alloys are characterized by extremely low activation volumes of a few tens of b3 compared to 100–1000 b3 for conventional fcc metals and alloys in the microcrystalline grain size regime. Unlike conventional fcc metals and alloys, there is an insignificant change in the activation volume of the current high-/medium-entropy alloy (H/MEA) with decreasing grain size from the microcrystalline to nanocrystalline regime. The unique evolution of strain rate sensitivity and activation volume in H/MEAs is explained in terms of the evolution of distinct dislocation structures as well as synergistic operation of additional mechanisms such as twinning, phase transformation from fcc to hcp phases, cluster strengthening, and short-range ordering due to the aperiodic energy landscape existing in MEAs.

Investigation of the Influence of Boundary Conditions on the Deformability of a Model of Soil Base of limited Width in The Form of a Linearly Deformable Elastic Half-Plane

Abstract The paper examines the available approaches to the distribution of vertical compressive stresses over the depth of the soil base from a uniformly distributed load on the surface. The paper investigates the effect of boundary constraints in plan of the model of a continuous linearly deformable (elastic) layer of finite width on its distribution capability and stress-strain performance from a loaded foundation on the surface. Numerical calculations of the stress-strain state of a uniformly loaded foundation with different relative bending stiffness (flexibility), which interacts with a layer of a certain depth (compressible depth) and strict horizontal constraints in terms of width (the width dimensions of the model), were performed using the SCAD software. Based on the analysis of the results of numerical calculations for the plane problem (plane-strain deformation), the width of the model of a linearly deformable layer of finite width, which considers the distribution of compressive loads over the depth at an angle of α≈20÷25° to the vertical from the edges of the loaded foundation, was justified. In this case, the width of the model according to the angle of distribution of α≈20÷25° has virtually no effect on the average settlements and the maximum moment forces of the foundation in comparison with the increase in the width of the model.

Uniform elastic field inside an elliptical inhomogeneity with Neuber’s nonlinear stress–strain law under generalized plane strain deformations

We prove the uniformity of the in-plane and anti-plane elastic field of stresses and strains inside an incompressible nonlinear elastic elliptical inhomogeneity embedded in an infinite linear isotropic elastic matrix subjected to uniform remote in-plane and anti-plane stresses. The elastic material occupying the elliptical inhomogeneity obeys Neuber’s special nonlinear stress-strain law. The original boundary value problem is finally reduced to a single non-linear equation for the constant effective strain within the inhomogeneity, which is rigorously proved to have a unique solution. Once the non-linear equation is solved numerically, we establish the uniform elastic field within the elliptical inhomogeneity and the non-uniform elastic field in the linear elastic matrix.

Forming limit and failure characterization of as-received and bi-axial prestrained Cu-Cr-Zr-Ti alloy sheets in the context of multistage forming

The characterization of forming limit utilizing Cu-Cr-Zr-Ti alloy sheets during the multistage forming process is imperative for successfully fabricating thrust chamber liners used in satellite launch vehicles. Therefore, the present work studies the effect of prestrain on the forming limit and failure behaviour of this alloy under different deformation paths. A stretch-forming setup was utilized to induce 8% equi-biaxial prestrain (EBP), and subsequently, these prestrained specimens were deformed till the onset of failure using the same setup. After prestraining, a shift in the failure strains was observed in the principal strain space, whereas it was found to be absent in the polar effective plastic strain, principal stress and effective plastic strain-stress triaxiality loci. The cup height along different deformation paths was marginally decreased in the prestrained specimens because of the reduction in failure strains, and the fracture occurred in the prestrained region. Further, failure analysis revealed intergranular cracks in the specimens deformed along plane strain and uniaxial deformation paths due to the presence of micron-size Cr-rich precipitates along grain boundaries. However, intergranular and transgranular cracks were present in the prestrained specimens, which were deformed equi-biaxially similar to the as-received specimens. The presence of nano-size Cr-rich precipitates inside the grains acted as nucleation sites for voids during equi-biaxial stretching, leading to transgranular cracks. It was also observed that failure in the prestrained and as-received specimens occurred at a similar effective plastic strain value due to the combined effect of void density and void size. Based on experimental observations, it was inferred that the forming limit of Cu-Cr-Zr-Ti sheets under a two-stage forming process was primarily affected by the presence of micro and nano-size Cr-rich precipitates.

A finite-strain plate model for the magneto-mechanical behaviors of hard-magnetic soft material plates

In this paper, we propose a plate model, within the framework of finite-strain elasticity, to study the magneto-mechanical behaviors of hard-magnetic soft material plates. The 2D vector plate equation is derived from the 3D governing equations through a series expansion and truncation approach. The displacement and traction boundary conditions on the edges of plate samples are also proposed. Compared with the other plate or shell theories, the current plate model naturally incorporates the elastic incompressibility and the magneto-mechanical coupling effect of the material, which doesn't involve any a priori hypotheses on the geometry and deformation of the plates. Thus, it is applicable for modeling the large deformations of hard-magnetic soft material plates. To show the efficiency of the plate model, we study a benchmark problem, i.e., the plane-strain bending deformations of plate samples induced by magnetic and mechanical loads. Some analytical solutions of the plate equation system are derived, which provide accurate predictions on the magneto-mechanical response of the plate samples with different thickness-length ratios and magnetization directions. To promote applications of the plate model, we further derive a simplified plate equation system and write out its weak form, which is then implemented into the finite element software COMSOL Multiphysics. The efficiency of the simplified plate equation system is also verified through some typical examples. It is found that the numerical results obtained from the 2D plate model show good consistency with those of the 3D volume model, while the plate model exhibits obvious advantages in the aspects of numerical efficiency.

Theoretical analysis of the interaction between blasting stress wave and linear interface crack under high in-situ stress in deep rock mass

This study theoretically investigates the scattering characteristics of blasting stress waves, treaded as elastic waves, at a linear interface crack in a deep rock mass with an even compressive in-situ stress field. First, the basic equations of the deep rock mass theoretically treaded as a homogeneous isotropic medium with an initial stress field are derived based on the linear elastic, finite deformation, plane strain, isotropic, and small deformation hypotheses, and incremental stress theory. The linear interface crack is theoretically treaded as two displacement-free boundaries, of which the interpolation displacements are described by introducing two dislocation density functions. Next, the mathematical model of the blasting stress wave scattering at the linear interface crack is established through a set of singular integral equations of the first type for these two dislocation density functions with Cauchy kernels. Finally, the dislocating and opening displacements of the linear interface crack and the displacement and effective stress fields of the scattered stress wave are numerically calculated. The principal stress field of the scattered stress wave is compared with the experimental data to qualitatively verify the rationality and correctness of the mathematical model. Theoretically, it is found that even compressive in-situ stress with an amplitude of tens of megapascals significantly affects the displacement and effective stress fields of the scattered stress wave near the linear interface crack. In conclusion, considering even compressive in-situ stress is greatly significant when studying the wave and scattering behaviors of deep rock masses. This mathematical model provides a theoretical guidance for the study of stress wave energy transformation and crack propagation in the process of rock blasting and mineral mining.

Explicating tensile response of AA6061-T6 sheet post single point incremental forming: Two camera-DIC strain measurement and texture analysis

Incremental sheet forming (ISF) produces 3D components imposing localized deformation by spherical-ended tool moving on a predefined path without requiring component-specific die/punch. The formability achieved in ISF is much higher than the conventional sheet forming processes; however, the deformation imposed during ISF alters the mechanical properties of the component. The present work investigates the tensile behavior and strain localization during the uniaxial tensile test of the AA6061-T6 component after single point incremental forming (SPIF). The influence of incremental depth (Δz) and tool diameter (dt) on the tensile response and strain localization is illustrated by a digital image correlation setup with two cameras. The tensile strength of SPIF components increases with Δz and mildly decreases with dt. The changes in scallop heights/undulations formed due to tool movement at higher Δz and dt influence the localized strain accumulation. Regions of localized strain are more when Δz is high, but strain accumulation is more even with an increase in dt. Texture analysis showed increased dislocation density due to the strain hardening after SPIF. The presence of brass {011}<211>, copper {112}<111>, and S {123}<634> texture indicates the plane-strain deformation in the wall region of the component formed by SPIF.

A parametric study into the influence of Taylor-type scale-bridging artifacts on accuracy of multi-level crystal plasticity finite element models for Mg alloys

Meshing grain structures explicitly to incorporate microstructure dependent material behavior in modeling tools for metal forming processes and structural components is unfeasible. Grain-to-polycrystal meso-level homogenization theories are employed to embed polycrystal plasticity constitutive laws in finite element (FE) frameworks, which hence relate the meso-scale to the macro-scale response. A Taylor-type polycrystal plasticity (T-CP) theory is often used at the meso-scale within FE frameworks (T-CPFE) owing to its simplicity and computational efficiency. The theory relies on the iso-strain constraint imposed over constituent grains at each integration point. The constraint can be relaxed by spreading orientation distributions over finite elements in a mesh. This work seeks to establish an optimal number of crystal orientations to spread over integration points for accurate modeling of Mg alloys. Quasi-static and high strain-rates simulations of tension, compression, simple-shear, and plane strain deformation conditions are performed and post-processed to reveal the differences in predicted mechanical fields between T-CPFE and an explicit polycrystalline grain microstructure of an AZ31 Mg alloy. Moreover, several flow stress curves of the alloy are simulated and compared with measured data. A range of meshes with variable degree of relaxed constraints at integration points are suitably designed and used in the simulations. The performance of the six crystals per integration point is established to be an optimum in smoothing local deformation while allowing heterogenous deformation over the mesh. As such, the relaxed Taylor homogenization can provide tractable part level simulations of Mg alloys.

Lithospheric plane strain deformation caused by a vertical dip-slip fault

Lithospheric plane strain deformation caused by a vertical dip-slip fault

Deformation of dissimilarly oriented aluminum sheets welded by friction stir welding under uniaxial tension perpendicular to the weld bead

This study investigates the deformation behavior of dissimilarly oriented tailor-welded blanks (D-TWBs) under uniaxial tension perpendicular to the weld beads. The D-TWBs were fabricated by friction stir welding of 1-mm-thick aluminum sheets in the rolling (0°), 45°and transverse (90°) directions. This study examined the mechanism of uniaxial tensile deformation of the aluminum TWBs by analyzing the microstructure and flow curve. In the D-TWBs, a material with a lower deformation resistance was elongated more than twice larger as much as the other material, and fracture occurred in this material. The total elongation became smaller than that for the raw materials by 0.15 to 0.35 owing to a decrease in the local elongation after necking. This resulted from the plane strain deformation near the weld bead immediately before fracture. Texture rotation occurred almost in-plane in the process-affected zone of the D-TWBs, and the method of rotation differed between materials with different orientations. However, despite the differences in texture, the study suggests that deformation was not strongly affected. The F values and n values, which were evaluated from partial deformation, did not vary significantly from those of the raw materials, but a decrease of approximately 10% was observed at the maximum. Similarly, the r values of the D-TWBs also did not vary significantly in the original orientation with low a r value; however, in the orientation with a large value, it decreased approximately 0.2 at the maximum. Finally, the study also examined mechanisms of deformation and fracture in the D-TWBs.

Inter-void shearing effect on damage evolution under plane strain deformation in high-strength aluminum alloy sheet

In this study, the ductile damage responses of high-strength 7000 series aluminum alloy (AA), AA 7075-T6 sheet samples, subjected to the plane strain deformation mode were investigated using finite element (FE) simulations. In the experiments, uniaxial tension (UT) and plane strain tension (PST) tests were conducted to characterize the plasticity and ductile damage behavior of the AA 7075-T6 sheet samples. The limiting dome height (LDH) and V-die air bending tests were conducted to evaluate the ductility of the material subjected to plastic deformation and friction between the tools, and the corresponding fractured samples were qualitatively analyzed in terms of dimples using fractography. FE simulations were performed to predict the ductility of the AA 7075-T6 sheet samples under plane strain deformation using an enhanced Gurson−Tvergaard−Needleman (GTN) model, namely the GTN-shear model. The model was improved by adding the shear dimple effect to the original GTN model. The predicted results in terms of the load–displacement curves and displacements at the onset of failure were in good agreement with experimental data from the aforementioned tests. Furthermore, virtual roll forming simulations were conducted using the GTN-shear model to determine the effect of the prediction on ductile behavior for industrial applications.

New analytical model for thermomechanical responses of multi-layered structures with imperfect interfaces

A new analytical model is developed for thermomechanical responses of multi-layered structures with an arbitrary number of layers and subjected to general thermal and mechanical loading. The formulation is based on an extended Bernoulli–Euler beam theory and a slip-interface model. The former includes Poisson’s effect and covers both the plane stress and plane strain deformations, and the latter allows slipping between two adjacent layers but no jump in the normal displacement or traction. An analytical solution for a multi-layered structure under general thermomechanical loading is derived by using a new approach that first determines one interfacial shear stress and the curvature of the deformed structure. To illustrate the newly developed model, three example problems for two-, three- and five-layer structures respectively are analytically solved by directly applying the new model. In all three cases, the solutions are obtained in closed-form expressions by considering both temperature changes and mechanical loads including body forces, distributed normal and shear stresses on the top and bottom surfaces, and normal forces, transverse shear forces and bending moments at the two ends, unlike existing ones. It is shown that the current solution for two-layer structures recovers an existing solution without considering Poisson’s effect and mechanical loading and the classical solution of Timoshenko for perfectly bonded bi-metal thermostats as two special cases. The closed-form solution for five-layer structures with imperfect interfaces is derived here for the first time. In addition, numerical results are provided for five- and seven-layer transistor stacks to quantitatively demonstrate the new model. It is found that the current results for the five-layer transistor stack agree well with those obtained by others, thereby further validating the new model.

A modified dislocation density-based model of 30CrMnSiA considering the coupling effect of electroplasticity and grain refinement during the pulsed current-assisted plane strain deformation

A modified dislocation density-based model of 30CrMnSiA considering the coupling effect of electroplasticity and grain refinement during the pulsed current-assisted plane strain deformation

Inception of macroscopic shear bands during hot working of aluminum alloys

Macroscopic shear bands (MSB) may develop during hot working of metallic materials. They are well-understood as a physical manifestation of flow instability, and the processing regimes wherein they form are well-charted. However, the microstructural transitions that occur between the onset of flow instability and MSB inception are not fully understood. In order to elucidate them, several aluminum alloy specimens were subjected to strip testing in a thermomechanical simulator (Gleeble™) at 298 K and 573 K. Prominent MSB were observed along the diagonals of the strip volume of only Aluminum-6 wt% Magnesium alloy specimen deformed at 573 K. Comparing the experimental grain morphology and crystallographic textures with those from plastic flow models revealed that MSB inception occurred only after ∼0.20 homogeneous plane strain deformation. However, classical flow instability was predicted at much smaller strain. This ‘delay’ was explained experimentally by showing that clusters of neighbouring severely deforming, fragmenting, mostly soft-oriented grains gradually developed due to lattice rotations along the specimen diagonals, and that MSB inception corresponded to the formation of a percolating network of such grains spanning the specimen. Further, clear experimental evidence revealed that differential dynamic recovery between hard- and soft-oriented grains was essential for MSB formation.

EFFECT OF LOCAL STRESSES ON THE FRACTURE OF METALLIC MATERIALS WITH DIFFERENT LATTICES UNDER SINGLE LOADS

The study discusses the effect of the local stress field at the crack tip on the fracture behavior of coarseand ultrafine-grained materials with bcc, fcc, and hcp lattices under single loads (impact and static). The local stresses at the crack tip under impact and static loading were assessed by the hmax/t ratio, where hmax is the maximum depth of the plastic zone under the fracture surface and t is the specimen thickness. The depth of plastic zones under the fracture surface was determined using layer-by-layer etching of the surface followed by X-ray diffraction analysis. The study results showed that it is not always possible to establish an unambiguous relationship between the fracture mechanisms of metallic materials and the local stress field at the crack tip. Nevertheless, some particular relationships were established: (i) the cleavage, quasi-cleavage, or intergranular brittle fracture of materials, regardless of lattice type, is indicative of plane strain deformation, (ii) under plane stress, all materials, regardless of lattice type, exhibit ductile fracture with micropitting, and (iii) in the transition from plane strain to plane stress, most fcc materials fracture by a mixed mechanism.

Superposing plane strain on anti-plane shear deformations in a special class of fiber-reinforced incompressible hyperelastic materials

The purpose of the present research is to investigate how the nonlinearity and the boundary conditions have the power to influence the coupling of the various modes of deformation when a plane strain deformation is superimposed on anti-plane shear deformation for a class of fiber-reinforced incompressible hyperelastic materials described by a strain energy density W. Attention is confined when W depends only on the first invariant of the strain tensor and on the square of the stretch in the direction of the fibers.We are able to write down the governing equations of equilibrium as a coupled system of three nonlinear partial differential equations for three displacement fields. Two displacements are the in-plane components and one displacement is the anti-plane state. The system that we are able to deduce in a compact form is always compatible at variance with the case in which the anti-plane shear problem is analyzed. As explicit example of our findings we study the problem of the helical shear and we investigate into details the coupling of its axial and azimuthal components.

A Deterministic Methodology to Calibrate Pressure-Independent Anisotropic Yield Criteria in Plane Strain Tension Using Finite-Element Analysis

The yield strength of materials under plane strain deformation is often not characterized experimentally due to difficulties that arise in interpreting the results of plane strain tensile tests. The strain and stress fields in the gauge region of these tests are inhomogeneous, making it challenging to extract the constitutive response from experimental measurements. Consequently, the plane strain yield stress is instead predicted using phenomenological plasticity models calibrated using uniaxial and biaxial tension data. To remove this uncertainty, a simple finite-element based inverse technique is proposed to determine the arc of the associated yield locus from uniaxial-to-plane strain tension using a constrained form of Vegter’s anisotropic yield criterion to analyze a notch tensile test. The inverse problem is formulated under associated deviatoric plasticity and constrained such that only a single parameter, the major principal yield stress under plane strain deformation, needs to be identified from the finite-element simulations. The methodology was applied to two different automotive steel grades, an ultra-high strength DP1180 and a DC04 mild steel. The predictive accuracy of the constitutive models was then evaluated using an alternate notch geometry that provides an intermediate stress state between uniaxial and plane strain tension. By performing notch tensile tests in three sheet orientations, three arcs of the yield surface were obtained and employed to calibrate the widely used Yld2000 yield function. The study shows that for DP1180, the normalized plane strain yield stress was in the range of 1.10 to 1.14 whereas for DDQ steel, the normalized plane strain yield stress was notably stronger, with values ranging from 1.22 to 1.27, depending on the orientation. The proposed methodology allows for a wealth of anisotropic plasticity data to be obtained from simple notch tests while ensuring the plane strain state is accurately characterized, since it governs localization and fracture in many forming operations.

An experimental study of morphological formation in bilayered tubular structures driven by swelling/growth

Circumferential wrinkling in soft tubular tissues is vital in supporting normal physiological functions. Most existing literature was dedicated to theoretical modeling and finite element simulations based on a specific growth model. This paper presents an experimental investigation on pattern formation and evolution in bilayered tubular organs using swelling deformation of polydimethylsiloxane (PDMS) and aims at supplying a thorough comparison with theoretical and finite element results. To create a twin model in modeling and simulation, the shear modulus in the incompressible neo-Hookean material is estimated via uni-axial tensile and pure shear tests. Five bilayered tubes with different material or geometrical parameters are fabricated. Swelling experiments are carried out for these samples in an individual experimental setup where a plane-strain deformation is guaranteed, and several surface patterns and the associated mode transformations are observed, namely, creases, wrinkles, period-doubling profiles, wrinkle-to-crease transition, and wrinkle-to-period-doubling transition. In particular, an interfacial wrinkling pattern is also observed. To make comparisons, a buckling analysis is conducted within the framework of finite elasticity by means of the Stroh formulation and a refined surface impedance matrix method. In addition, a finite element analysis (FEA) is performed to trace the evolution of surface instabilities. It turns out that the experimental findings agree well with the theoretical predictions as well as the finite element results. From our experiments, it is found that creasing mode may appear instead of wrinkling mode when both layers share a similar mechanical property. It is expected that the current work could provide novel experimental insight into pattern formation in tubular structures. In particular, the traditional impedance matrix method has been adapted, which enables us to resolve eigenvalue problems with displacement boundary conditions, and the good agreement among experimental, theoretical, and simulation consequences supplies strong evidence that a phenomenological growth model is satisfactory to reveal mechanisms behind intricate surface morphology in tubular tissues.

Hyperelastic behaviors of closed-cell porous materials at a wide porosity range

In this paper, the constitutive model proposed in the previous work for neo-Hookean materials with spherical voids and an explicit homogenization solution named Shrimali-Lefèvre-Lopez (SLL) model for isotropic porous elastomers are adopted to theoretically predict the macroscopic effective hyperelastic responses of the closed-cell porous materials with rubber-like incompressible neo-Hookean matrix at finite deformations. Representative volume element (RVE) models with randomly distributed non-overlapping polyhedral voids are employed to numerically validate the theoretical models at a wide porosity range from 0.05 to 0.9. Various deformations including hydrostatic deformations, pure shear deformations, uniaxial deformations and plane strain deformations are applied to the theoretical and RVE models. The results show that the theoretical models can well estimate the effective hyperelastic properties of the porous materials under all deformation conditions considered, and our theoretical model is complementary with the SLL model for predicting different effective properties under given deformation. The constitutive models are also validated experimentally by comparing with the experimental data from the literature.