Characterization of ductile damage and fracture behavior under shear reverse loading conditions

This paper deals with experimental and numerical analysis of the ductile damage and fracture behavior undergoing biaxial non‐proportional shear reverse loading conditions. In the numerical analysis, an anisotropic cyclic plastic‐damage constitutive model is considered to characterize the damage and fracture behavior. Additionally, the change in the hardening rate is observed after shear reverse loading. Thus, a newly modified non‐hardening strain region method is introduced to capture the change in the hardening ratio between isotropic and kinematic hardening. Furthermore, the damage surface is assumed to translate in the direction of the damage strain increment. Hence, a softening law based on the damage strain is proposed. The modified constitutive model predicts macroscopic force‐displacement and microscopic evolution of plastic and damage behaviors under shear reverse loading with a tensile preload. Through digital image correlation technique and scanning electron microscopy, numerical results can be validated by experimental ones. Furthermore, scanning electron microscopy pictures also reveal different damage and fracture mechanisms on the micro‐level.


INTRODUCTION
Many experiments and numerical investigations have shown that stress states affect the ductile damage and fracture behavior of metals.However, in the last decades, studies have been limited to examining ductile damage and fracture caused by monotonic loading or one-axis-loaded reverse loading conditions [1][2][3][4][5].Among them, one-axis-loaded shear monotonic and cyclic tests have shown that the coalescence of the micro-shear-cracks mainly causes ductile damage, and shear reverse loading affects the evolution of damage and fracture behavior compared to the monotonic loading [4].On the other hand, limited studies have focused on reverse torsion loading under non-proportional loading conditions with a cylindrical specimen.In the case of metal sheets, biaxially loaded cruciform specimens have been recognized as a proficient means of generating complex stress states.For instance, Gerke et al. [6] proposed a biaxially loaded H-specimen that includes a tensile axis and shear axis to generate different stress states in different loading axes.Thus, the H-specimen enables non-proportional loading by changing the tensile preload to shear load during the experiment.Moreover, previous studies indicate that various preloading histories can induce different stress states and result in further damage and fracture.However, the mentioned studies have been restricted to monotonic loading conditions.To fill this gap, a series of biaxial shear reverse loading with non-proportional loading tests taking into account different tensile preloads is performed for aluminum alloy sheets.
In the present work, the H-specimen is modified by increasing the distance in the central notch part to conduct the cyclic tests, the so-called HC-specimen.The experimental methodology involves a two-step loading process.Different proportional tensile preloads, without unloading, are imposed on the tensile axis at the first loading stage to generate a range of stress triaxialities.Subsequently, in the second loading stage, a three-half-shear cyclic loading is superimposed on the shear axis until fracture.By following this approach, the experimental results allow for the investigation of the damage and fracture behavior of the material under different stress states.

CONSTITUTIVE MODELING
It is imperative to acknowledge that ductile metals primarily undergo significant plastic deformations before the manifestation of material degradation, commonly referred to as damage.As a result, Brünig [7] proposed a novel two-surface approach to capture large plastic and damage deformations for ductile metals.This innovative approach uses an anisotropic second-order damage strain rate tensor to predict damage evolution caused by various damage mechanisms on the micro-scale accurately.Recently, Wei et al. [4] incorporated combined hardening laws into the plasticity yield criterion to capture the plastic-damage behavior under reverse loading conditions.Additionally, a new kinematic softening model is proposed to characterize the translation of the damage surface.
For metals [4] and geomaterials [8], onset of plastic yielding is sensitive to hydrostatic pressure.Therefore, the Drucker-Prager yield criterion taking into account the combined hardening law is used to accurately characterize plastic behavior, where T is the effective Kirchhoff stress tensor, ᾱ represents the effective back stress tensor, and ∕ c is the hydrostatic stress coefficient.The current equivalent yield stress is given by the extended Voce hardening law with the initial yield stress  0 , the hardening moduli  1 and  2 , the additional variable function , the equivalent plastic strain  as well as the hardening exponents  1 and  2 .The rate of the effective back stress tensor with modified Chaboche nonlinear kinematic hardening components is used to capture the Bauschinger effect (BE) and changes in the yield surface under reverse loading conditions, where  1 - 6 are material constants, Ḣpl is the plastic strain rate tensor,  represents the additional variable function, and cos 2  denotes the angle parameter .Furthermore, the total hardening rate is an additive combination of the equivalent stress rate ċ and effective back stress rate α in terms of the isotropic hardening proportion  ℎ and the scalar factor  ℎ .In the previous work [4,9], the hardening ratio  ℎ is assumed to be constant ( ℎ = 1) during the whole cyclic loading process.However, some researchers [5,10,11] indicated that the hardening rate changes significantly after shear reverse loading condition.Compared to the multi-surfaces [10] and multi-kinematic-hardeningparameters approach [11], the most straightforward effective hardening function based on a cyclic equivalent plastic strain  * is introduced to predict the change in hardening after shear reverse loading [12].
On the other hand, the damage criterion  da taking into account combined softening law and stress invariants  1 and  2 is defined as where α and β are stress-state-dependent variables,  is the Kirchhoff stress tensor,  denotes the damage back stress tensor, and σ represents the initial equivalent damage stress.In addition, the rate of damage back stress tensor is formulated as where  1 and  2 are softening moduli, μ is the equivalent damage strain rate, and the damage strain rate is an additive combination of the isotropic equivalent damage stress rate ̇σ related to the size change of the damage surface and the equivalent kinematic damage back stress rate αeq corresponding to the translation of the damage surface.  is the isotropic softening proportion, which is assumed to be constant and equals the hardening proportion  ℎ .The stress triaxiality calculated as the ratio of hydrostatic stress  m and the von Mises equivalent stress  eq , as well as the Lode parameter expressed in the principal stress components   , are two key parameters used to characterize the stress states in metals.Both parameters have important applications in the study of damage and fracture mechanics.

EXPERIMENTAL SETUP AND MATERIAL PARAMETERS
The one-axis-loaded shear and HC-specimen geometries are shown in Figure 1, respectively.For the one-axis-loaded shear test, the cyclic load is imposed on the horizontal axis (axis 2, see Figure 1A), and the experimental technique can be found in ref. [4].In the case of the non-proportional biaxial shear reverse test, the loading process is divided into two stages: in the first loading stage, preload 7 kN is imposed on the tensile axis of the HC-specimen (horizontal axis, see Figure 1H), without unloading.Subsequently, a tension-compression-tension (TCT) three-half cycle shear reverse loading is superimposed on the shear axis (vertical axis) until the specimens fail.The test is labeled cyc-T7 in this paper.
The material parameters of the investigated aluminum alloy EN-AW 6082-T6 (chemical components, see ref. [4]) are summarized in Table 1.The elastic and plastic material parameters are directly determined using the uniaxial tensile monotonic and cyclic experiments data.The parameter identification technique is discussed in detail in ref. [4,9].Additionally, the material constants for modeling the damage are inversely fitted based on the experiments.

EXPERIMENTAL AND NUMERICAL RESULTS
The experimental force-displacement curves for the one-axis-loaded shear cyclic tests TCT and CTCT, plotted as the back dashed lines in Figure 2, have been published in ref. [4].The corresponding numerical results without the non-hardening correction (wo/ corr., represented by blue solid lines) can also be found in ref. [4].Moreover, the numerically predicted force-displacement curves taking into account the non-hardening correction (w/ corr.) are illustrated with the red solid lines in Figure 2. Obviously, the numerical results considering the non-hardening correction after shear reverse loading significantly improve the numerical accuracy, especially the global force-displacement behavior after reversal yielding.Again, the experimental and numerical force-displacement curves for the test cyc-T7 are shown in Figure 3.As illus- trated in Figure 3A, it is evident that the numerically predicted force-displacement curve without the non-hardening correction fails to accurately capture the experimental behavior, especially in the last tensile loading pattern.The proposed non-hardening correction can justify this behavior, providing a more accurate numerical result, as shown in Figure 3B.
In addition, the changes of the first principal strains  1 on the notch surface during the cyclic loading processes (RP1, RP2, and FP) are displayed in Figure 4A.The first principal strains  1 at reverse point 1 (RP1) and fracture point (FP) are distributed in the form of a left-sloping shear band and localized on the notch surface left-top and right-bottom with maximum values of 0.1089 and 0.265, respectively, as observed in Figure 4A.At reverse point 2 (RP2), the first principal strain  1 is distributed as an x-shape, and their maximum value equals 0.110.It indicates that the reverse loading histories change the distribution of the first principal strains and affect the associated plastic behavior.Moreover, the evolution of the first principal damage strains  da 1 during the loading processes is shown in Figure 4A.No damage appears at RP1.During the subsequent reverse loading, the first principal damage strains  da 1 are observed in the notched surface and the notch cross-section, where the maximum value  da 1 = 0.04% is numerically predicted on the bottom of the notched surface.Finally, the damage occurs and develops on the notched surface induced by a positive shear loading pattern, resulting in the maximum first principal damage strain  da 1 = 1.15% at the top edge of the notched surface.In conclusion, shear reverse loading conditions change strain states during the experiments and significantly influence the evolution of damage.
Figure 5 shows the fractured picture and the scanning electron microscopy image taken from the fractured surface.As seen in Figure 5A, the specimen breaks in the right-top and left-bottom of the notches, and the distribution of the fracture line is consistent with the distribution of the first principal strain in Figure 4A (FP).In addition, large voids and coalesced micro-shear-cracks are visible in Figure 5B since a high stress-triaxiality ( = 0.48,  = −0.65) is numerically predicted in the test cyc-T7.As pointed out in ref. [4], the one-axis-loaded shear cyclic tests generate stress triaxialities from −0.1 to 0.1.Thus, the novel non-proportional biaxial tests enable different stress states to study the influence of shear reverse loading conditions on plastic, damage, and fracture behavior.

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Numerical and experimental force-displacement curve for the test cyc-T7.F I G U R E 4The first principal strains  1 and the first principal damage strains  da 1 .
Material parameters.Numerical and experimental force-displacement curves for one-axis-loaded shear tests.
TA B L E 1