Issue 
Manufacturing Rev.
Volume 6, 2019



Article Number  27  
Number of page(s)  11  
DOI  https://doi.org/10.1051/mfreview/2019024  
Published online  27 November 2019 
Research Article
Effect of crystallographic texture on the forming limit in microforming of brass
SGGS Institute of Engineering & Technology, Nanded, India
^{*} email: anilmashalkar@gmail.com
Received:
7
July
2019
Accepted:
7
October
2019
Microforming is an emerging technology to manufacture products in the light of miniaturization in several domains of industry. Plastic anisotropy is one of the material characteristics significantly affecting the micro forming process. The crystal orientation influences tensile strength, yield strength and ductility, depending on different grain sizes and principle sliding planes. The present work elaborates on the influence of the plastic anisotropic in microforming for a plane stress state condition. Yield function and constitutive equations for the anisotropic material with consideration of the crystal lattice constants and parameters of crystallography texture are proposed. The crystal orientation is considered in a subroutine VUMAT algorithm, as an internal variable based on the developed mathematical model which is implemented in Abaqus as an user material subroutine. Micro limiting dome height experimental tests for different strain paths are conducted with brass foils. The results are compared with that predicted with numerical analysis, considering critical damage and element deletions. The numerical and experimental results show a good agreement for the Alpha brass ultrathin foils, using a set of failure criterion.
Key words: Crystal orientation / stress–strain curve / anisotropy / forming limit / micro forming
© A. Mashalkar and V. Nandedkar, Published by EDP Sciences 2019
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Advances in the electronics, mobiles, energy and medical applications are influenced by developments of new materials and their processing technologies. Due to miniaturization in many sectors, micro forming has emerged as a most preferred process for sheet metal components. The development of rational, sciencebased technology in the micro forming processes is primarily concerned with the need of the detailed study of material properties. One of the specific characteristics inherent in the majority of the material is anisotropy, which is based on crystallographic structure and texture formation under high plastic strain [1]. However, the assumption of the material isotropy is still being used in the finite element analysis, though it does not actually meet the real deformation condition. In the plasticity theory of the isotropic material, shifting from the elastic condition to the plastic is usually determined on the basis of the maximum shear stress criteria developed by maximum distortion strain energy criteria by Mises [2]. The major criterion does not consider the crystallographic texture of materials and consequently the anisotropy of their physical, mechanical and plastic properties.
It should be noted that the high accuracy of the recently proposed criterions is achieved by a large amount of the anisotropy coefficients (up to 18), determination of which involves numerous mechanical tests at different stress states [3]. Though the applied anisotropy coefficients characterize the anisotropy of plastic deformations, they do not take into account the reason for anisotropy, i.e., the crystallographic texture [4]. Thus, the mentioned yield functions, on one hand, allow describing the plastic flow of anisotropic materials. On the other hand, they do not allow carrying out technical analysis of micro thin foils considering the crystallographic texture [5]. As a result, it is difficult to determine the arrangement of crystallographic texture in terms of the necessities of specific microforming processes [6,7]. However, there is short of systematic research on the micro limiting dome height [LDH] test and failure mechanism of the ultra thin foils considering the crystal orientation. The effect of texture and grain structure on strain localisation and formability is investigated experimentally and numerically for two AlZnMg alloys [8] by Lademo et al. The strongly textured materials exhibit inferior formability to the materials with weak or nearly random texture. The reason for this is attributed to the reduced workhardening capacity of the former materials and to a less degree to the plastic anisotropy. Grechnikov proposed calculation procedure which considers the crystal lattice constants and the parameters of crystallographic orientation of material. The main practical significance of this procedure is possibility to predict the effect of crystallographic texture of rolled sheets on limiting strains and formability of material in different metal forming process [9]. Masoud Hajian studied 1010 steel sheet formability. The initial texture of sheet material was measured through Xray diffraction technique. Also, the stressstrain behaviour and FLD of the material were determined by performing simple tension and hemispherical punch tests, respectively. In order to predict the forming limits of the material by simulation, a UMAT subroutine was developed and linked to the nonlinear finite element software ABAQUS. In this subroutine, a rate sensitive crystal plasticity model along with the power law hardening was implemented. Secondorder derivative of sheet thickness variations with respect to time was used for necking criterion. The obtained FLD was compared with the experimental measurements and good agreement was found between simulation and experiment with acceptable errors between approximately 5–15% [10]. María A. Bertinetti studied the effect of the cube texture on forminglimit strains is studied using a ratedependent viscoplastic law in conjunction with the MarciniakKuczynski approach. The forming limit diagram and yield locus are determined for several spreading of grain orientations around the ideal {100} ⟨001⟩ component [11]. Fulop et al. carried simulations of the mechanical response of ultrathin ductile metal sheets. Ratedependent single crystal plasticity theory was used to implement the algorithms into a Finite Element code. A uniaxial tensile test and a threepoint bending test are computationally evaluated. The effect of the number of surface grains over the total number of grains is investigated numerically [12].
In this paper, the subroutine VUMAT is applied with the plasticity criteria and crystallographic orientation. Crystal lattice constants and crystallographic orientation parameters are included explicitly and implemented in numerical analysis of the microLDH test considering the orientation of the blanks. Also, study involves investigations on influence of crystallographic orientation on formability in microforming process and outcomes of numerical approach are validated with the experimental results.
2 Limiting dome height test
Microformability of metal foils can be measured using LDH Test. Specimen for the tensile test were designed as per ASTM E 345 standard and the properties achieved from various tests are presented in Table 1. These specimen are designed such that, the strain paths can be achieved in both drawing as well as stretching zones using uniaxial strain, plane strain and biaxial strain. Figure 1 shows the FSA M100 universal testing machine of capacity 100 KN with micro limit height dome attachment. Experiments were performed for the ultra thin foil of 40 μm thickness of alpha brass material and Figure 2 shows the microformed sample.
Properties of brass.
Fig. 1 LDH experimental setup. 
Fig. 2 Microformed alpha brass ultrathin foil. 
3 True strain measurement
Circle grid with diameter of one mm each, and center to center distance of 2 mm were printed on specimen to measure deformations. An optical microscope was used to record the minor and major axis of the ellipse. The equations for percentage true major and minor strain is given in equations (1) and (2):$$\text{major}\text{\hspace{0.17em}}\text{train}=\frac{\text{Instanteneous}\text{\hspace{0.17em}}\text{Major}\text{\hspace{0.17em}}\text{Axis}\text{\hspace{0.17em}}\text{Length}}{\text{Original}\text{\hspace{0.17em}}\text{Circle}\text{\hspace{0.17em}}\text{Diameter}}\times 100\text{.}$$(1) $$\text{minor}\text{\hspace{0.17em}}\text{strain}=\frac{\text{Instanteneous}\text{\hspace{0.17em}}\text{Minor}\text{\hspace{0.17em}}\text{Axis}\text{\hspace{0.17em}}\text{Length}}{\text{Original}\text{\hspace{0.17em}}\text{Circle}\text{\hspace{0.17em}}\text{Diameter}}\times 100\text{.}$$(2)
Uniaxial strain, plane strain and biaxial strains for 40 μm brass foil were measured from experimentation as presented in Table 2. Forming limit curves has been plotted using these values.
Maximum inplane and minimum inplane principal strain values from the physical experimentation.
4 Finite element model of the micro LDH test using vumat subroutine
Figure 3 shows the numerical model of the micro LDH test. The dimensions and geometry of the model corresponds to the standard test for stretch forming. In all cases, the blank thickness is 40 μm. The finite element model has been discretized using S3R and S4R shell element with reduced integration point over the thickness, in order to reduce the number of elements. S4R is the linear, finitemembranestrain, quadrilateral shell element and is robust in nature. S3R is the linear, finitemembranestrain, triangular shell element applied to capture bending deformations or high strain gradients because of the constant strain approximation in the elements. The combination of these two will ensure complete discretization of the model and capture all deformations with minimal error. Between the tool and blank contact, pairs were prescribed, and the friction obeys the Coulombs law. The tool was assumed absolutely to be rigid. The model of anisotropic elastoplastic material considering crystallographic texture is used to describe the blank material behavior of alpha brass. In order to assess the influences of crystallographic structure on the formability, it was modeled anisotropic material, the texture of which is represented by crystallographic orientation. The characteristic for the rolled material deformation orientation of brass is{110} ⟨112⟩, rotated cube {100} ⟨011⟩, cube{100} ⟨001⟩ and isotropy. The orientation parameters are given in Table 3 [13].
Fig. 3 Modeling of the process tool with Abaqus for LDH. 
Crystallographic orientations and their parameters.
5 Continuum damage model
Damage is addressed as one of the output measure and its evolution law is given as a general function of other state variables such as stress, plastic strain, temperature and so on. From a general point of view, the damage variable should be described using a tensor formulation [14]. From the physical point of view, damage variable indicates the progressive material deterioration due to nonreversible deformation processes and can be expressed by the reduction of the nominal section area of a given reference volume element (RVE) as a result of microvoids formation and growth. Let's consider the damage due to growth of microcavities, atomic bond breaking, discontinuous surface.$$\text{Damage}\left[D\right]=\frac{{A}_{D}}{A}\text{,}$$(3)where A − overall area of the damage body, A_{D} − damaged area.
Therefore, D is scalar and values between 0 and 1$$\overline{\sigma}}=\frac{F}{(A{A}_{D})}\text{.$$(4) $$\overline{\sigma}}=\frac{F}{A\left(1\raisebox{1ex}{${A}_{D}$}\!\left/ \!\raisebox{1ex}{$A$}\right.\right)}$$(5)where D is the overall damage variable and $\overline{\sigma}$ is the effective (or undamaged) stress tensor computed in the current increment. $\overline{\sigma}$ are the stresses that would exist in the material in the absence of damage. The material has lost its loadcarrying capacity when D = 1.$$\overline{\sigma}}=\frac{\sigma}{\left(1D\right)}\text{.$$(6)
True stress was replaced by effective stress [15]$$\sigma ={E}_{0}(1D)\epsilon \text{.}$$(7)
6 Damage evolution
Based on Swift's law [16] the damage evolution equation is$${\sigma}_{y}=K{({\epsilon}_{0}+p)}^{n}$$(8) $$\text{Damage}\text{\hspace{0.17em}}(D)=\left(\frac{{D}_{C}}{{\epsilon}_{\text{PR}}{\epsilon}_{\text{PD}}}\right)\times \left[\frac{2}{3}(1\nu )+3(12\nu )\left(\frac{{\sigma}_{H}}{{\sigma}_{eq}}\right)\right]{({\epsilon}_{0}+p)}^{2n}p\text{,}$$(9)where ε_{PD}, plastic strain under which the damage evolution is negligible; ε_{PR}, plastic strain at rupture; Dc, damage parameter at rupture called the critical value of the damage; σ_{H}, hydrostatic stress; σ_{eq}, equivalent stress; σ_{y}, yield stress; ε_{0}, strain value at yield; K and n are isotropic hardening coefficients; p, equivalent plastic strain.$${\sigma}_{eq}=\sqrt{\frac{{\eta}_{12+}{\eta}_{31}}{2}.{\sigma}_{12}^{2}+\frac{{\eta}_{12+}{\eta}_{23}}{2}.{\sigma}_{22}^{2}{\eta}_{12}{\sigma}_{11}{\sigma}_{22}+\left(52{\eta}_{12}\right){\sigma}_{12}^{2}}\text{.}$$(10)
Here σ_{ij} is stress tensor [4]. The generalized anisotropic parameters η_{ij} are defined by$${\eta}_{ij}=1\frac{15\left({A}^{l}1\right)}{3+2{A}^{l}}\left(\begin{array}{c}\hfill {\mathrm{\Delta}}_{i}+{\mathrm{\Delta}}_{j}{\mathrm{\Delta}}_{k}\frac{1}{5}\hfill \end{array}\right)\text{,}$$(11)where A^{l} is the anisotropic parameter of the crystal lattice Δ_{ i }. Are the orientation factor of crystallographic orientation.$${\mathrm{\Delta}}_{i}=\frac{{h}_{i}^{2}{k}_{i}^{2}+{k}_{i}^{2}{l}_{i}^{2}+{l}_{i}^{2}{h}_{i}^{2}}{{\left({h}_{i}^{2}+{l}_{i}^{2}+{k}_{i}^{2}\right)}^{2}}\text{,}$$(12) h_{i} , l_{i} , k_{i} are Miller indices defining the eighth direction in crystal with respect to a coordinate system associated with the blank. The rolling direction set along the xaxis.
Strain tensor for 3D is written as$$p=\sqrt{\frac{2}{3}{\epsilon}^{p}:{\epsilon}^{p}}\text{.}$$(13)
Strain equivalence principle yielding occurs$$\tilde{\sigma}}=\frac{\sigma}{1D}={\sigma}_{y}=K{({\epsilon}_{0}+p)}^{n}\text{.$$(14)
After yielding, plastic deformation occurs due to the deviatoric stress, which consists of unequal principalstresses. Deviatoric stress is the difference between principle stress and hydrostatic stress. Another way of representative linear elastic stressstrain is given below$$\sigma =\lambda trace(\epsilon )I+2\mu \epsilon \text{.}$$(15)
The stress equation including the damage can be written as$$\sigma =(1D)(\lambda trace(\epsilon )I+2\mu \epsilon )\text{.}$$(16)
Initially, the time step is zero and at first increment the new trial stress tensor is evaluated from the linear stress strain curve. If the new calculated trial stress value is less than the yield stress, then it is stored as new trial stress value. When the trial stress exceeds yield point and enters plastic region, damage criterion is applied, so new trail stress and plastic strain is due to damage. Also, plastic strain and yield stress values are updated and tested for the damage initiation. Crack initiation is considered when D value exceeds 0.9. ΔD is calculated when p > ε_{PD.} Δγ calculated from the below given equations. Δ is determined considering constant D during one step which is justified in the explicit calculation because of very small increments$$\mathrm{\Delta}\gamma =\frac{\sqrt{{\sigma}_{\text{new}}^{\text{Dtrial}}:{\sigma}_{\text{new}}^{\text{Dtrial}}}\sqrt{\frac{2}{3}}(1{D}_{\text{old}}){\sigma}_{\text{yold}}}{2\mu \left(1+\frac{h}{3\mu}\right)(1{D}_{\text{old}})}$$(17) h, isotropic hardening law.
7 Results and discussion
Limiting Dome Height test for 40 μm alpha brass foil was carried out numerically with ABAQUS using VUMAT subroutine and the results of maximum inplane principle strain and minimum inplane principle strain in three cases uniaxial strain, plane strain and biaxial strain are plotted below for the different orientation (orientation of brass {110} ⟨112⟩, Rotated cube {100} ⟨011⟩, Cube {100} ⟨001⟩ and Isotropy).
7.1 LDH test for {110} ⟨112⟩
The results for LDH test for {110} ⟨112⟩ are plotted below in Figures 4–6.
From the graphs it is evident that fracture for maximum principle strain in uniaxial stretching occurs at 0.16, plane strain at 0.24 and in biaxial at 0.32. Maximum inplane strain shown by average value nearer to red zone elements in Figures 4a, 5a and 6a. In case of minor principle strain necking occurs at −0.1 in uniaxial stretching, 0.015 in plane stretching and at 0.17 in biaxial stretching shown by average nearer to brown zone elements in Figures 4b, 5b and 6b.
Fig. 4 (a) Maximum inplane principle strain under uniaxial stretching. (b) Minimum inplane principle strain under uniaxial stretching. 
Fig. 5 (a) Maximum inplane principle strain under plane stretching. (b) Minimum inplane principle strain under plane stretching. 
Fig. 6 (a) Maximum inplane principle strain under biaxial stretching. (b) Minimum inplane principle strain under biaxial stretching. 
7.2 LDH test for {100}⟨011⟩
Results of limiting dome height test {100} ⟨011⟩ for 40 μm alpha brass foil are plotted below in Figures 7–9. From the numerical outcomes it is evident that fracture for maximum principle strain in uniaxial stretching occurs at 0.12, plane strain at 0.14 and in biaxial at 0.35. Maximum inplane strain shown by average value nearer to red zone elements in Figures 7a, 8a and 9a. In case of minor principle strain necking occurs at −0.06 in uniaxial stretching, −0.001 in plane stretching and at 0.1465 in biaxial stretching shown by average nearer to brown zone elements in Figures 7b, 8b and 9b.
Fig. 7 (a) Maximum inplane principle strain under uniaxial stretching. (b) Minimum inplane principle strain under uniaxial stretching. 
Fig. 8 (a) Maximum inplane principle strain under plane stretching. (b) Minimum inplane principle strain under plane stretching. 
Fig. 9 (a) Maximum inplane principle strain under biaxial stretching. (b) Minimum inplane principle strain under biaxial stretching. 
7.3 LDH test for {100}⟨001⟩
Results pertaining to limiting dome height test {100} ⟨001⟩ are presented in Figures 10–12. It is evident that facuture for maximum principle strain in uniaxial stretching occurs at 0.23, plane strain at 0.17 and in biaxial at 0.32. Maximum inplane strain shown by average value nearer to red zone elements in Figures 10a, 11a and 12a. In case of minor principle strain necking occurs at −0.13 in uniaxial stretching, 0.015 in plane stretching and at 0.1368 in biaxial stretching shown by average nearer to brown zone elements in Figures 10b, 11b and 12b.
Fig. 10 (a) Maximum inplane principle strain under uniaxial stretching. (b) Minimum inplane principle strain under uniaxial stretching. 
Fig. 11 (a) Maximum inplane principle strain under plane stretching. (b) Minimum inplane principle strain under plane stretching. 
Fig. 12 (a) Maximum inplane principle strain under biaxial stretching. (b) Minimum inplane principle strain under biaxial stretching. 
7.4 LDH test for isotropy
Limiting Dome Height testisotropy for 40 μm alpha brass foil was carried out numerically with ABAQUS and the results of maximum inplane strain and minmum inplane strain for three strain path cases uniaxial strain, plane strain and biaxial strain are plotted below in Figures 13–15. From the graphs it is evident that fracture for maximum principle strain in uniaxial stretching occurs at 0.1238, plane strain at 0.1466 and in biaxial at 0.3511. Maximum inplane strain shown by average value nearer to red zone elements in Figures 13a, 14a and 15a. In case of minor principle strain necking occurs at −0.081 in uniaxial stretching, −0.01 in plane stretching and at 0.17 in biaxial stretching shown by average nearer to brown zone elements in Figures 13b, 14b and 15b.
Fig. 13 (a) Maximum inplane principle strain under uniaxial stretching. (b) Minimum inplane principle strain under uniaxial stretching. 
Fig. 14 (a) Maximum inplane principle strain under plane stretching. (b) Minimum inplane principle strain under plane stretching. 
Fig. 15 (a) Maximum inplane principle strain under biaxial stretching. (b) Minimum inplane principle strain under biaxial stretching. 
7.5 Forming limit curves (FLC)
For plotting FLC maximum values of the major and the minor strain are determined by measuring the principal strains at failure state. Numerical simulation of failure limit curves is carried out using finite element analysis platform ABAQUS.
The details of maximum inplane principle strain and minimum inplane principle strain at different strain paths for thin foil (uniaxial, plane and biaxial) are presented in Table 4. For the construction of the forming limit diagram (FLD) maximum inplane principle strain is plotted on Y axis and minimum inplane principle strain on xaxis. The forming limit curves were plotted by joining limit strain coordinates and this procedure was repeated for all four crystallographic orientations. Numerically plotted FLDs are shown in Figures 16–19. It is apparent from Figures 16–19 that, as the orientation of foil changes forming limit curve changes. The area below curve presents safe zone for forming. Higher the curve on major stain axis, higher the formability. Amongst four cases investigated Brass orientation has higher formability, rotated cube is at second place, cube and isotropy orientation has almost same formability as curve resembles. Cube and isotropy orientation gives the same values of the fracture strain.
Maximum inplane and minimum inplane principal strain values from the numerical simulation.
Fig. 16 Forming limit diagram for brass orientation − numerical simulation. 
Fig. 17 Forming limit diagram for rotated cube – numerical simulation. 
Fig. 18 Forming limit diagram for cube − numerical simulation. 
Fig. 19 Forming limit diagram for isotropy − numerical simulation. 
7.6 Validation of numerical results with experimental test data
For {110} ⟨112⟩ orientation, experimental and numerical failure limit curves are presented below in Figure 20. It is observed that maximum uniaxial strain in both approaches is in full agreement within 20% error. Maximum plain strain is almost same in numerical approach with that of experimental. The state of maximum biaxial strain in both approaches is within 18% error.
When plotted both the failure limit curves as shown below they are in found to be of the same nature. It is obvious that numerical results strongly agree with experiment. For {100} ⟨001⟩ orientation, experimental and numerical failure limit curves are presented below in Figure 21. It is observed that maximum uniaxial strain in both approaches is in full agreement within 10% error. Maximum plain strain within 20% error in numerical approach with that of experimental. The state of maximum biaxial strain is in both approaches is in then 20% error. When plotted both the failure limit curves as shown below they are in found to be of the same nature. It is obvious that numerical results strongly agree with experiment. Other orientation given more than 20% error.
Fig. 20 Comparison of numerical and experimental failure limit curves for brass orientation. 
Fig. 21 Comparison of numerical and experimental failure limit curves for rotated cube orientation. 
8 Conclusions
Based on the numerical simulations and experiments on the study of the forming limit of microforming of brass, the following conclusions can be drawn:

The Brass orientation and rotating cube orientation results match with the experimental results within 20% error, and they show that the material has crystallographic orientation of brass and rotating cube orientation texture for the alpha brass C26000.

The simulation results relating to the isotropy do not match experimental results well, which means that the material obeys the anisotropic material behavior.

The results show limiting values in the FLD diagram for the different orientations, which will help the process and tool designs taking into account of the crystallographic orientation effects.
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Cite this article as: Anil Mashalkar, Vilas Nandedkar, Effect of crystallographic texture on the forming limit in microforming of brass, Manufacturing Rev. 6, 27 (2019)
All Tables
Maximum inplane and minimum inplane principal strain values from the physical experimentation.
Maximum inplane and minimum inplane principal strain values from the numerical simulation.
All Figures
Fig. 1 LDH experimental setup. 

In the text 
Fig. 2 Microformed alpha brass ultrathin foil. 

In the text 
Fig. 3 Modeling of the process tool with Abaqus for LDH. 

In the text 
Fig. 4 (a) Maximum inplane principle strain under uniaxial stretching. (b) Minimum inplane principle strain under uniaxial stretching. 

In the text 
Fig. 5 (a) Maximum inplane principle strain under plane stretching. (b) Minimum inplane principle strain under plane stretching. 

In the text 
Fig. 6 (a) Maximum inplane principle strain under biaxial stretching. (b) Minimum inplane principle strain under biaxial stretching. 

In the text 
Fig. 7 (a) Maximum inplane principle strain under uniaxial stretching. (b) Minimum inplane principle strain under uniaxial stretching. 

In the text 
Fig. 8 (a) Maximum inplane principle strain under plane stretching. (b) Minimum inplane principle strain under plane stretching. 

In the text 
Fig. 9 (a) Maximum inplane principle strain under biaxial stretching. (b) Minimum inplane principle strain under biaxial stretching. 

In the text 
Fig. 10 (a) Maximum inplane principle strain under uniaxial stretching. (b) Minimum inplane principle strain under uniaxial stretching. 

In the text 
Fig. 11 (a) Maximum inplane principle strain under plane stretching. (b) Minimum inplane principle strain under plane stretching. 

In the text 
Fig. 12 (a) Maximum inplane principle strain under biaxial stretching. (b) Minimum inplane principle strain under biaxial stretching. 

In the text 
Fig. 13 (a) Maximum inplane principle strain under uniaxial stretching. (b) Minimum inplane principle strain under uniaxial stretching. 

In the text 
Fig. 14 (a) Maximum inplane principle strain under plane stretching. (b) Minimum inplane principle strain under plane stretching. 

In the text 
Fig. 15 (a) Maximum inplane principle strain under biaxial stretching. (b) Minimum inplane principle strain under biaxial stretching. 

In the text 
Fig. 16 Forming limit diagram for brass orientation − numerical simulation. 

In the text 
Fig. 17 Forming limit diagram for rotated cube – numerical simulation. 

In the text 
Fig. 18 Forming limit diagram for cube − numerical simulation. 

In the text 
Fig. 19 Forming limit diagram for isotropy − numerical simulation. 

In the text 
Fig. 20 Comparison of numerical and experimental failure limit curves for brass orientation. 

In the text 
Fig. 21 Comparison of numerical and experimental failure limit curves for rotated cube orientation. 

In the text 
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