Normal Stress Components Research Articles

Implicit nonlinear elastic bodies with density dependent material moduli and its linearization

We develop an implicit constitutive relation to describe the response of a compressible elastic solid, based on physical considerations, that captures all the characteristics exhibited by the popular Blatz–Ko model, but in addition presents some interesting novel features. The fact that the Cauchy stress appears linearly in the implicit constitutive relation between the stress and the left Cauchy–Green strain with the material moduli depending nonlinearly on the deformation gradient, allows us to capture several characteristic features of the response of rubber-like elastic solids. Interestingly, in the nonlinear implicit model that we develop, we find that it is possible to have the normal stress components of the stress influence the shearing motion at second order, when considering weakly nonlinear waves, that only occurs at third order within the case of the classical nonlinear Cauchy elasticity theory. Linearization of the constitutive relation under the assumption of small displacement gradient reduces the constitutive relation to one whose material moduli can depend on the trace of the linearized strain and hence the density in virtue of the balance of mass, such a feature is not possible within the context of the Blatz–Ko constitutive relation, or for that matter any Cauchy elastic body, as linearization leads to the classical linearized elastic constitutive relation that has constant material moduli.

Ice adhesion behavior of heterogeneous bituminous surfaces

The phenomenon of icing, and the derived processes for its mitigation, are of great importance in many applications, ranging from transportation and energy to food and refrigeration. This phenomenon has been studied mostly with respect to its manifestation on rigid, homogeneous surfaces, with soft materials being the topic of more recent investigations. Although, icing often occurs on substrates that are chemically and mechanically heterogeneous, e.g., widely used asphalt concrete, which consists of rigid aggregates embedded in soft bitumen, to date, ice adhesion behavior on such substrates needs to be better understood. Here, we study ice adhesion stresses—the stresses necessary to remove ice—of ice blocks on heterogeneous materials, juxtaposing the behavior of the two main constituents of asphalt concrete, the rigid aggregates (modeled by Macor®) and bitumen, to the behavior of bitumen-Macor® composites. We show that the ice adhesion shear stress on Macor® is almost twice as large as that on bitumen, whereas the ice adhesion normal stress and the normal and shear components of composite stress are in a similar range. We synthesize composite substrates that consist of bitumen stripes on Macor® and find that increasing bitumen width leads to lower ice adhesion stress, while the stripe direction with respect to the applied force direction has a minor effect. Based on our findings, we then coat the most ice-adhesive component (Macor®) with a thin superhydrophobic coating and show that this can reduce ice adhesion stress on the heterogeneous substrates. We also find that for ice formed half on bitumen and half on Macor®, if Macor® is first and bitumen second with respect to the applied force direction (material order), then the measured ice adhesion stress is less compared to the reverse case in material order.

Effect of Load Combinations on Distortional Behaviors of Simple-Span Steel Box Girder Bridges

When eccentric live loads are applied on the deck overhang of steel-box girder bridges, torsional moments, comprising pure torsional and distortional moments are generated on the box sections. The torsional moment on the bridge girders distorts the box girder cross-sections, inducing additional normal stress components and causing instability of the box girder sections in severe cases. Hence, it is essential to install intermediate diaphragms in the box sections to minimize distortional behaviors. Although the applied live loads are critical parameters that influence intermediate diaphragm spacings, the effects of live load combinations have rarely been addressed in the design of intermediate diaphragm spacings. Thus, load combinations should be evaluated to design the intermediate diaphragm spacing of the box girder bridges more thoroughly. In this study, the load combination effects on the distortional behavior and adequate intermediate diaphragm spacing were evaluated through a finite element analysis (FEA). Composite rectangular box girder bridges with different cross-sectional aspect ratios (H/B) and spans (L) were analyzed in the parametric study. It was found that the truck load, which represents the concentrated load, significantly influences the distortional warping normal stress, normal stress ratio, and intermediate diaphragm spacing. In addition, the FEA results showed that the controlling load combinations could be varied with the span.

The effect of fibers under compression on the simple shear response of a soft unidirectional fiber-reinforced composite

The aim of this work is to investigate the mechanical behavior of a soft matrix reinforced by a single family of parallel fibers subjected to simple shear. Specimens with fibers oriented at −45° with respect to the applied shear load were tested. The shear and normal stress components as well as the amount of shear were simultaneously measured. Due to the initial orientation, fibers were subjected to compressive loads, resulting in buckling. The positive Poynting effect was observed when soft materials were sheared. The off-axis stiffness theory and the Murphy model at small deformations were used to predict the mechanical behavior of the soft composites. In general, there is a reasonable correlation between model predictions and measured data by adjusting only one empirical parameter. However, the model based on the off-axis stiffness shows better prediction than the Murphy model for relatively higher fiber volume fractions. From universal relations for simple shear, the results indicate that the studied composites are of a coaxial type in a weak sense.

Effect of inlet shear on turbulent flow past a wall-mounted finite-size square cylinder

The present study analyzes the effect of incoming shear flow on the wake behind a wall-mounted finite-length square cylinder of aspect ratio (AR=7) using large eddy simulation (LES). All the simulations were performed at a Reynolds number (Re) of 10000, based on cylinder width (d) and average inlet velocity (uavg), with different shear parameters, K=0.0,0.025,0.050 and 0.075. The shear-improved Smagorinsky model Lévêque et al. (2007) is used to account for the subgrid scale stress. It is found that the incoming flow shear has a profound effect on the modes of shedding. With increasing shear parameter anti-symmetric modes of shedding tends to dominate along the cylinder height, except close to the free-end where both symmetric and anti-symmetric modes of shedding co-exist. The incoming shear flow has also significant influence on the downwash flow from the free-end, the strength of which decreases with an increase in shear parameter. The resolved turbulent kinetic energy (TKE) associated with the cylinder wake is found to have maximum contribution from the transverse component of normal stress compared to the other components of normal stresses.

The Viscoelastic Fluid Flow in a Confusor

The planar flow of a viscoelastic fluid is studied in tapered channel (confusor) with different contraction rates (2:1, 4:1, 6:1). A polymer melt is considered as a specific liquid, rheological behaviour is described using the four mode Giesekus model. The effect of the contraction rate of a confusor on the axial velocity profiles and the components of normal and shear stresses is investigated. Approbation of numerical studies was carried out by means of the example of the planar flow of a viscoelastic fluid in a channel with abrupt contraction.

An improved methodology of constructing inter-fiber failure criteria for unidirectional fiber-reinforced composites

Based on Mohr’s failure theory and physical mechanisms of unidirectional (UD) fiber-reinforced composites, an improved methodology of constructing inter-fiber failure (IFF) criteria is proposed. The expression of the failure criteria is a quadratic polynomial, in which the variables are the normal and shear stress components (σn,τnl,τnt) on the potential failure plane. The coefficients of the quadratic polynomial are independent of empirical parameters and calibrated only by four basic strengths of a UD composite, i.e. transverse tensile and compressive strengths, Yt and Yc, as well as transverse and longitudinal shear strengths, S23 and S21. It avoids some deficiencies of the previous criteria for IFF in which some parameters may need to be determined by multiaxial stress tests or by empirical data-fitting operation. Compared with the theoretical prediction results and the experimental data of various UD composites, the present IFF criteria show good prediction accuracy.

Structural analyses of nano-stitched composite laminates based on FSDT using finite element approach

Here, the numerical analyses of nano-stitched graphite-epoxy laminated composite are presented. A micromechanical model is used to obtain the effective elastic properties of the nano-stitched layers by considering a very thin layer of polymer nanocomposite. The mechanical and structural characteristics are examined for thin and thick plates using the first-order shear deformation theory (FSDT) and a finite element analysis. Governing equilibrium equations and constitutive equations have been presented to investigate the out-of-plane shear stresses and in-plane normal stresses in the laminated plates. Further, transverse bending deflection and normal stresses are determined for different sets of composite laminates. Comparative studies of CNT stitched and unstitched laminated plates have been carried out for sinusoidal loading and different boundary conditions. . Results show a good agreement between present and available analytical results. The evaluation indicates that there is a significant reduction in normal and transverse stress components with the use of only 5% of vertically aligned carbon nanotubes (CNT) at the interface of the conventional laminated plates.

EXPERIMENTAL INVESTIGATION OF TURBULENCE PRODUCED BY STATIONARY GRIDS

This work attempts to analyze the shear and normal stress components of a grid downstream under the influence of a rigid boundary. The correlation coefficient of Reynolds stresses is analyzed to inspect the fraction of turbulent motions contributing to fluctuating momentum flux. The mean turbulence dissipation rate is estimated downstream of the grid for three different mesh sizes and compared to a case without a grid. Based on the analysis, it is found that the flow approaches isotropy in the main flow region. In contrast, near the boundary, some residual inhomogeneity is observed, even at the far-field region of the grid. The Reynolds shear stress peak value decreases with the variation in mesh size of the grid. The negative and positive values of the stress depend on the spatial configuration in a vertical direction, occurrence of locations of the intersection of wires, and hole openings of different mesh sizes of the grid. The paper also analyzed the Kolmogorov complexity for various mesh sizes under the consideration of rigid boundaries.

Modeling the role of local crystallographic correlations in microstructures of Ti-6Al-4V using a correlated structure visco-plastic self-consistent polycrystal plasticity formulation

This paper presents a multi-level crystal plasticity-based simulation framework for modeling mechanical response and microstructure evolution of Ti-6Al-4V with α-lath/lamellar microstructures. The model is a correlated structure visco-plastic self-consistent (CS-VPSC) formulation linking three scales: a single crystals micro-scale, a lath/lamellar colony meso-scale, and a lath/lamellar aggregate macro-scale. A selected hardening law for the evolution of critical resolved shear stress per slip system used in CS-VPSC is phenomenological. However, it adjusts the resistances of basal and prismatic slip systems based on the geometry of slip transfer between adjacent lamellae. Consistent with experimental evidences, the resolved shear stress on the pyramidal slip planes is dependent not only on the stress in the direction of slip but also on the two orthogonal shear stress components and the three normal stress components (non-Schmid effects). Electron backscatter diffraction (EBSD) data in conjunction with a procedure relying on α→β phase transformation is used to construct paired variants of α-laths/lamellae satisfying their local crystallographic correlations. The procedure fits volume fractions of individual laths/lamellae with the experimental EBSD data and selects a distribution of habit planes between adjacent variants with respect to the loading direction. The simulation framework is applied to interpret the deformation behavior in tension and compression along two sample directions of Ti-6Al-4V fabricated via laser powder bed fusion. Moreover, the model is used to simulate texture evolution during rolling of the material to large strains. It is demonstrated that the model is capable of predicting plastic anisotropy/asymmetry and the concomitant texture evolution. While the model reveals a significant effect of habit plane inclination with respect to the loading direction on yield stress, the comparison of the data and model predictions shows that a random distribution of habit planes fits the flow response. It is further inferred that the tension-compression asymmetry arises from the non-Schmid effects.

Application of machine learning techniques to predict rupture propagation and arrest in 2-D dynamic earthquake simulations

SUMMARYSimulating dynamic earthquake rupture propagation is challenging due to uncertainties in the underlying physics of fault slip, stress conditions and the fault frictional properties. A trial and error approach is often used to determine the unknown parameters describing rupture, though running many simulations usually requires human review to determine how to adjust parameter values and is thus inefficient. We leverage machine learning approaches to reduce computational cost and improve the ability to determine reasonable stress and friction parameters. Two models have been developed using neural networks and the random forest to predict if a rupture can break 2-D geometrically complex fault. We train the models using a database of 1600 dynamic rupture simulations computed numerically. Fault geometry, stress conditions and friction parameters vary in each simulation. Both models distinguish the underlying complex data patterns that reflect the physics of the rupture. For example, our models identify in a quantitative fashion, how higher normal and shear stress components and low static and dynamic friction can be tied to the probability of rupture propagation. Both models are efficient in predicting rupture propagation such that 400 unseen examples are predicted in a fraction of a second, leading to potential applications of dynamic rupture that have previously not been possible due to the computational demands of physics-based rupture simulations.

A full-field crystal plasticity model including the effects of precipitates: Application to monotonic, load reversal, and low-cycle fatigue behavior of Inconel 718

This work advances a recently developed high-performance, full-field elasto-viscoplastic fast Fourier transform (MPI-ACC-EVPCUFFT) formulation to model large plastic deformation and low-cycle fatigue behavior of Inconel 718 (IN718). Specifically, the recently developed model of Eghtesad et al., 2020 [1] incorporating a strain rate, temperature, and strain-path sensitive hardening law based on the evolution of dislocation density and a slip system-level back-stress law influencing the resolved shear stress is advanced in several aspects. First, strengthening effects due to grain size and shape, solid solution, shearing of small precipitates, and Orowan looping around larger precipitates are incorporated into the initial slip resistance, which evolves with the hardening law including the latent hardening. Second, the resolved shear stress on the slip plane in the direction of slip is altered by accounting for the two orthogonal shear stress components and the three normal stress components, in addition to the slip system-level kinematic effects. The model is used to interpret the complex mechanical behavior and microstructural data for samples of alloy IN718 in additively manufactured (AM) forms before and after hot isostatic pressing (HIP) and in a wrought form. Voxel-based microstructural cells consistent with the characterization data are synthetically constructed to initialize the model setups for predicting simple compression, tension, load reversal, and low cycle fatigue behavior of the alloy. Variation in the microstructural features such as the distribution of grain size and shape, crystallographic texture, content of annealing twins, and precipitates among the samples facilitated reliable calibration and validation of the model parameters. Predicted anisotropy, tension/compression asymmetry, non-linear unloading upon the load reversal, the Bauschinger effect, reverse hardening, texture evolution as well as cyclic hardening/softening along with the mean stress relaxation during low-cycle fatigue are in good agreement with the corresponding experimental data for the alloy. Furthermore, the simulations based on the high-resolution microstructural cells facilitated the discussion of the mechanical fields induced by microstructure, and especially annealing twins.

Investigations of internal stresses in high-voltage devices with deep trenches

Deep trenches, as essential elements of silicon chips used in electronic high-power and high-frequency devices, are known as starting points for dislocation generation under the influence of internal mechanical stresses resulting mainly from the difference in the thermal expansion coefficients between silicon and silicon dioxide. Since the electrical insulation of the devices requires a sequence of mechanical, chemical, and high-temperature processes during the preparation of the deep trenches, including the formation of an amorphous SiO2 edge layer, the emergence of the internal stresses is hardly avoidable. The method of cross correlation backscattered electron diffraction in the scanning electron microscope is used here to quantitatively determine the magnitude and local distribution of internal stresses in silicon around the deep trenches after four different process steps. For this purpose, Kikuchi diffraction images are recorded of the wafer cross section areas along lines perpendicular and parallel to the deep trenches. After Fourier transformation, these images are cross correlated with the Fourier transform of the diffraction image from a stressfree reference sample site. The well-established numerical evaluation of cross correlation functions provides the complete distortion tensor for each measuring point of the line scan, from which the stress tensor can be calculated using Hooke's law. It is found that the in-plane normal stress component σ11 perpendicular to the long edges of the deep trench is larger than the other stress components. That means it essentially determines the magnitude of the von-Mises stress, which was determined as a general stress indicator for all measuring points, too. A characteristic feature is the local distribution of the stress component σ11 with maximum tensile stresses of some hundred megapascals at transition between Si and amorphous SiO2 on the long edges of the deep trench, and with even higher maximum compressive stresses immediately below the bottom of the deep trench. At a distance of about 2 μm from the edges of a single deep trench, all stress components decrease to negligibly small values so that steep stress gradients occur. The range and distribution of tensile and compressive stresses are in accordance with finite element simulations; however, the measured stresses are higher than expected for all investigated states so that dislocation formation seems to be possible. The influence of the electron acceleration voltage on the determination of the internal stresses is discussed as well.

RESEARCH OF INFLUENCE OF THE EARTH’S SUBSIDENCE OUTSIDE THE LOADS CONTOUR ON THE STABILITY OF THE HEIGHT POSITION OF THE POINTS OF LOCAL LEVELING NETWORKS

The influence of static loadings of the foundations of structures in the process of soil compaction of the base on the subsidence of the soil surface and the bench outside the loaded contour is considered in the work. The studies were performed on models of rectangular massive foundations using the method of equivalent soil layer, based on the theory of compaction of linear-deforming half-space. Using this method, the values of the average sedimentation of the models of rectangular foundations were calculated, taking into account all components of normal stresses and lateral soil extension. On the basis of the method of angular points of the equivalent layer, the values of subsidence of the soil surface outside the contour of the foundation models are calculated, depending on the distance from the contour of the foundation, the value of the average subsidence of the foundation and its parameters. To determine the areas of the soil surface most sensitive to the perception of static loads, lines of equal settlements of the soil surface outside the contour of the foundation models are constructed. On the basis of theoretical studies, probable magnitudes of sedimentation of soil benchmarks are calculated, depending on the subsidence of the soil surface and the distance from the contour of the foundation. The probable minimum distance from the contour of the structure to the places of laying of initial soil benchmarks is established to ensure the required accuracy of geodetic observations. The results of the conducted researches may find application in the design of geodetic observations of the settlements and deformations of the foundations of engineering structures and the choice of the locations of local level points.

Stress State of Pretensioned Structures

Abstract Prestressed structures are used in civil engineering and other sectors of industry for the production of pressure vessels of the reactors and protective shells for nuclear plants, storage tanks for liquefied gas, etc. In such structures, pretension may reach great magnitudes depending on the manufacturing process during the fabrication or later with the addition of the operational loads. Therefore, it is necessary to address the prestressed state in order to predict optimal loading for the structural elements in the design process of these structures. In this study, the prestressed states of these structures are investigated using an analogy between the prestressed state and the thermoelasticity problems by the photothermoelasticity. Detailed information about the stress distribution in entire structures, locations, and magnitudes of the maximum stresses and bond stresses was obtained. Simultaneously, analytical relations were developed for the normal and shearing stress components between dissimilar materials in the joint zone.

The propagation mechanism of an oblique straight crack in a rock sample and the effect of osmotic pressure under in-plane biaxial compression

In this paper, the propagation mechanism of an oblique straight crack has been studied theoretically, which reveals its mechanical characteristics under in-plane biaxial compression. Firstly, the stress components away from the boundary are derived based on the superposition principle. The normal stress components are strengthened and shear stress component is restrained compared to the uniaxial condition. Then the relationship between stresses and stress intensity factors is analyzed, and the effect of stresses on the strength of cracked rocks is discussed. The analysis of wing crack growth shows that the reliable experimental results are very demanding for sample preparation. Based on Mohr-Coulomb criterion and Mohr’s stress circles, the failure mechanism of cracked rocks is analyzed, and the physical meaning of some formulas is vividly displayed. Moreover, we study the relationship between friction angle θ0 and angle β, which determines the minimum compressive strength of cracked rocks. There are evidences that the increase of crack opening width leads to β0 (a value of β) away from the theoretical value determined by sliding crack model, so that the role of stress σx can no longer be ignored. Theoretically speaking, for an initially closed crack, we find that, for the first time, both wing crack growth and shear compression failure are more likely to occur when the angle β between 22.5 and 45 degrees combining the statistical results of Barton and Choubey (Rock Mech Rock Eng 10:1–54, 1977). As for an initially open crack, the characteristics of stress intensity factors and circumferential stresses are also discussed, especially when σ1 equals σ3. Finally, we study the effect of osmotic pressure on stresses and stress intensity factors, the weakening of the properties of crack surfaces by water is also considered, and the mechanical behavior of a rock sample with an oblique straight crack changes dramatically.

Study of the influence of interface normal stresses on the bond behavior of FRCM composites using direct shear and modified beam tests

In the last decades, the increasing need for strengthening and retrofitting of existing masonry members has led the scientific community toward the development of innovative strengthening systems. Among these systems, inorganic-matrix composite materials have been gaining increasing popularity due to their effectiveness, relatively low cost, ease of the application, and reversibility of the intervention. Inorganic-matrix composites, usually referred to as fiber- (or fabric-) reinforced cementitious matrix (FRCM) composites, were successfully used as externally bonded reinforcement (EBR) of masonry members subjected to various load combinations. However, their effectiveness is strictly related to the FRCM-substrate bond behavior. Therefore, numerous set-ups were employed to investigate the bond properties of FRCM composites. Among them, single-lap direct shear test set-ups were the most commonly employed and are now recommended by available design guidelines for FRCM composites.In this paper, a modified beam (MB) test set-up is used to study the bond behavior of two different FRCM composites: a polyparaphenylene benzobisoxazole (PBO) FRCM and a carbon FRCM. The parameters studied are the dimensions of the blocks used in the MB tests and the presence of bare or embedded fibers at the gap between the blocks. The bond behavior observed is compared with that of single-lap direct shear tests of the same FRCM composites. The results obtained provide an insight on the influence of normal stress components at the matrix-fiber interface on the FRCM bond behavior, which may play an important role in the capacity of FRCM strengthened curved members.

ТЕОРЕТИЧНЕ ДОСЛІДЖЕННЯ НАПРУЖЕНО-ДЕФОРМОВАНОГО СТАНУ БАЗОВИХ ПЛИТ УЗРП-16

Theoretical studies of the stress-strain state of base plates, which are the base of the UZRP-16 universal collapsible machines have been done. These machines are used for welding works in machine building industry. The finite element method was applied to solve the problems. Nature of influence of strength and structural parameters on the base plate stress-strain state in operation has been determined, namely: the relations between displacements and stresses arising in the base plates and the bending moment magnitudes have been recorded; the stress-strain state pattern of the base plates has been obtained depending on the place of bending moment application; influence of the conditions for bearing and fastening the plates on their stress-strain state has been investigated; influence of the plate geometric parameters on stress and displacement has been studied; The stress-material and displacement-material relations have been obtained for the plates. Based on the theoretical study results of the base plate stress-strain state, the following have been obtained: stress plots and patterns of deformed surfaces, which are symmetrical with respect to the plate central axes; maximum values of normal and tangential stress components arising in the field of bending moment application; stress on the rectangular base plate surface is 2.1 times higher than the stress on square plate surface under the same conditions of bearing and loading; stresses acting on the plate surfaces and being tensile stresses within the range of 10 to 70 MPa. It was found that the square shape of the plates, according to the stress state, is predominant in relation to the rectangular shape. The optimal condition for bearing is fixing the plates at nine points. For the first time, graphs were drawn for choosing the base plate thickness under action of various operational loads.

Thermally induced stress in a nanoconfined gas medium.

Molecular dynamics simulations of static argon gas at three different levels of rarefaction are conducted for a channel of 5.4nm height to investigate the simultaneous effect of the wall force field and the gas temperature on the stress distribution along the channel height. Using the interactive thermal wall model, different temperatures are applied on the channel walls to be able to investigate the effect of the wall temperature and the induced heat flux through the gas medium on the stress distribution. Considering the monoatomic neutral argon gas, the kinetic, particle-particle virial, and surface-particle virial are considered for computing the stress distribution along the channel height. The normal stress components in the bulk gas region are distributed isotropically regardless of the gas density, temperature, and induced heat flux through the domain, while an anisotropy is observed due to the presence of the surface-particle virial. As the gas becomes hotter, the velocity of the gas atoms increases, and thus the kinetic stress component also increases. Besides, the gas density in the wall force field region reduces which eventually attenuates the surface-particle and particle-particle virial stress within 1nm from each wall. This effect was also observed as the gas becomes cooler. It is shown that the combination of gas density, wall temperature, and induced heat flux are the main parameters which determine the distribution of stress within the gas medium especially in the wall force field region where repulsive and attractive interactions exist.

Enhanced non-linearity during unloading by LPSO phase in as-cast Mg-Zn-Y alloys and slip-dominated non-linear unloading mechanism

The non-linear unloading behavior and the underlying deformation mechanism of as-cast Mg-Zn-Y alloys with the long period stacking ordered (LPSO) phase were investigated. To evaluate the stress-strain behavior, compression-unloading tests under stress control were performed for as-cast alloys with six different volume fractions of the LPSO phase, which were produced using the high-frequency induction melt casting technique in an Ar atmosphere. The experimental results showed that the non-linearity during unloading was more significant for alloys with a higher volume fraction of the LPSO phase. The LPSO single-phase alloy, in which deformation twinning was hardly activated, exhibited the most significant non-linear unloading behavior, which originated from the small linear elastic region and the steep decrease in the instantaneous hardening modulus during unloading. Crystal plasticity finite-element analysis for the compression-unloading of the LPSO single-phase alloy indicated that the activity of the basal slip system during unloading led to non-linear unloading behavior. The heterogeneous stress distribution that developed during compressive deformation led to the activity of the basal slip system during unloading owing to the local stress tensor, whose normal stress component in the axial direction had opposite sign to that of the macroscopically imposed stress. Comparisons between numerical results with different implemented deformation modes, i.e., the basal slip system and the second-order pyramidal <c+a> slip system, indicated that strong plastic anisotropy contributes to the development of a pronounced heterogeneous stress field and an insignificant increase in the critical resolved shear stress before unloading, which leads to a higher non-linearity during unloading.