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Abstract

This article presents a crystal plasticity methodology to evaluate the AA1050 sheet formability. In order to determine the orientation distribution of the crystals, initial texture of the material is measured through X-ray diffraction technique. Also, the stress–strain behavior of the material is determined by performing tensile test. In order to simulate the path-dependent crystal plasticity behavior of body-centered cubic crystal structures, a UMAT subroutine that employs the rate-dependent crystal plasticity model along with the power law hardening was developed previously by the authors and linked to the finite element software ABAQUS. This subroutine was further developed to simulate face-centered cubic crystal structures. The second-order derivative of sheet thickness variations with respect to time is considered as the instability factor, and forming limit diagram of the material is predicted. In order to assess the validity of formability prediction results for face-centered cubic materials, forming limit diagram of AA1050 sheet is also experimentally extracted by conducting hemi-spherical punch test. It is observed that the predicted forming limit diagram is in agreement with the experimental results. Finally, the prediction accuracy in different regions of forming limit diagram is discussed and some suggestions for further improving the accuracy are made.

Keywords

X-ray diffraction texture measurement AA1050 sheet formability prediction hemi-spherical punch test forming limit diagram

Introduction

Aluminum alloy 1050 is a popular grade of aluminum which is used for general sheet metal work where moderate strength is required. It has excellent corrosion resistance, high ductility and high reflective finish and is used in applications such as chemical process plant equipment, food industry containers and lamp reflectors.

Various sheet metal forming processes (such as hydroforming,^1,2 incremental forming^3,4 and conventional stamping methods⁵) are employed to manufacture different sheet metal parts. For simulation of these processes, forming limit diagram (FLD) of the employed material is required which can be determined experimentally⁶ or numerically.⁷ Numerical FLD predictions can be performed based on macroscopic (phenomenological) or microscopic (physical) approaches. Macroscopic approaches cannot predict the effects of initial material texture on its formability. In order to evaluate the effects of initial texture on the material formability, simulation should be done using the crystal plasticity models which are a branch of microscopic approaches. As some examples, Yang et al.⁸ simulated the effects of different pre-strains on the formability limits of Al5052 by the crystal plasticity approach. Steglich et al.⁹ investigated the mechanical behavior of magnesium sheet alloy AZ31 by means of mechanical testing and numerical simulation. They performed a biaxial test using a cruciform specimen and a hydraulic bulge test in addition to the common tensile tests. Hoc et al.¹⁰ conducted some experimental studies on interstitial-free (IF) steel sheets at different scales to study the effects of strain path on IF steel formability.

Zhuang et al.¹¹ performed some experiments to study the occurrence of localized thinning in the hydroforming of stainless steel micro-tubes. They also investigated the effects of microstructure and grain orientation on the formability of stainless steel micro-tubes in this forming process. Xu et al.¹² simulated the anisotropy behavior and texture evolution of two ferritic stainless steel sheets AISI409L and AISI430 by phenomenological and crystal plasticity–based constitutive models and tried to better understand the plastic behavior of these two materials. Hoffmann et al.¹³ suggested some micro scale experiments to validate the crystal plasticity model for deep drawing steel DC04 in addition to macro scale experiments.

Signorelli et al.¹⁴ studied the role of crystallographic texture on the forming limit strains of an electro galvanized steel sheet experimentally. Green et al.¹⁵ used a biaxial testing apparatus to study the elastic–plastic behavior of a 1145 aluminum sheet alloy and compared their observations with the crystal plasticity simulation results. Narayanasamy et al.¹⁶ studied the relevance of various crystallographic textures—resulted from annealing at different temperatures—on the formability of AA5052 sheet and determined the temperature which leads to the best formability. In other similar works, Savoie et al.¹⁷ and Wu et al.¹⁸ performed some experiments on three differently annealed aluminum sheets and measured and compared their initial textures and FLDs. Palumbo et al.¹⁹ studied the forming limits of Mg alloy AZ31 in warm conditions at constant equivalent strain rate by experimental and numerical methods.

All the reviewed crystal plasticity–based simulations require the initial material texture in the form of orientation distribution function (ODF) and a stress–strain curve as the input experimental data. There are few experimental works like Ramos et al.^20,21 and Hajian and Assempour²² which completely provide the required experimental input and verification data for a specific material. This article provides the complete set of required experimental input data for the crystal plasticity formability simulation of AA1050 sheet. In addition, forming limits of AA1050 sheet are predicted by the rate-sensitive crystal plasticity approach. Finally, experimental and numerical FLDs are compared and some suggestions for further improving the accuracy are made.

Experimental procedure

Texture measurement

Experimental measurement of texture by X-ray diffraction (XRD) technique is physically based on the Bragg’s law presented in relation (1)

n λ = 2 d_{hkl} \sin θ

(1)

where $λ$ is the wavelength of incident X-ray beam (see Figure 1), $d_{hkl}$ is the distance between two neighboring hkl crystallographic planes, $θ$ is half of reflection angle and n is a natural number. By changing $θ$ angle, when Bragg’s law is satisfied, beams reflected from different atoms have constructive interference and this leads to peak observation in the detector. Figure 2 schematically illustrates a powder sample mounted on the goniometer for XRD test. By fixing $α$ and $β$ angles and changing $θ$ angle, in special $θ$ angles, the Bragg’s law is satisfied and peak intensities are observed in the detector. Peak angles can be determined analytically by using Bragg’s law or experimentally by fixing $α$ and $β$ angles of powder sample and scanning $θ$ angle for peak intensities. Here, peak angles are determined analytically. According to relation (1), assuming $n = 1$ , $λ$ and $d_{hkl}$ are needed to calculate $θ$ . In our experimental setup, Philips X’Pert XRD system, $λ = 0.15418 nm$ . Also, $d_{hkl}$ is obtained from relation (2)

d_{hkl} = \frac{a}{\sqrt{h^{2} + k^{2} + l^{2}}}

(2)

Figure 1.

Geometrical condition of radiation to and reflection from atoms in (hkl) crystallographic planes.

Figure 2.

Schematic presentation of tool and sample in XRD test.

where $a$ is the aluminum lattice constant.

AA1050 sheet is used in this work. Lattice constant $a$ for aluminum is approximately $4.05 Å$ .²³ The magnitudes of $d_{hkl}$ for each of (111), (200) and (220) crystallographic planes along with the associated peak angles calculated using relation (1) are presented in Table 1.

Table 1.

Peak angles of (111), (200) and (220) crystallographic planes for aluminum and $λ = 0.15418 nm$ .

Crystallographic plane	$d_{hkl} (Å)$	$2 θ (\circ)$
$(111)$	2.3383	38.57
$(200)$	2.025	44.81
$(220)$	1.4319	65.21

In order to determine the texture of the material, one $5 mm \times 5 mm$ sample of the sheet is polished for the XRD test. The sample is mounted on the goniometer and for each crystallographic plane, according to Table 1, $θ$ angle is set on the associated peak angle. Then, $α$ and $β$ angles are changed in $3 \circ$ steps and the reflection intensities of X-ray beam are measured for various combinations of $α$ and $β$ angles. In this way, raw stereographic pole figures of ${200}$ , ${220}$ and ${111}$ crystallographic planes are experimentally determined. Before the pole figures can be evaluated, or used for subsequent ODF calculation, background and defocusing corrections should be applied. Background intensities are caused by incoherent scattering and fluorescence in the sample. Minor contribution may also result from the interaction of the X-ray beam with any material in the path of the X-rays as well as from electronic noise.²⁴ Furthermore, increasing $α$ angle strongly decreases the reflected X-ray beam intensity independent of material texture. This is due to broadening area of the reflected X-ray beam intensity by increasing $α$ while the detector sees only a small constant area of the sample surface and is called defocusing error.²⁴ In order to correct for these measurement errors, the defocusing and background intensities for a powder sample with random texture are also measured for each stereographic plane. These correction patterns are used to diminish the background intensity and correct the defocusing error of the measured intensity according to relation (3)²⁴

I_{corr} = \frac{I_{meas} - BG}{U}

(3)

where $I_{corr}$ is the corrected intensity, $I_{meas}$ is the measured intensity, BG is the background intensity and U is the defocusing pattern. The initial texture of the material in terms of ${200}$ , ${220}$ and ${111}$ stereographic pole figures along with the associated ODF are presented in the “Results” section.

Tensile test

The tensile test is carried out to measure the stress–strain behavior of the material. For this purpose, a sample with dimensions shown in Figure 3 is prepared according to the ASTM E8-04 standard. The load–displacement curve is measured at a loading rate of $0.5 mm / s$ to a final strain of 0.23 at room temperature. The resulted experimental stress–strain data along with the calibrated simulation curve are presented in the “Results” section.

Figure 3.

Dimensions of the tensile test sample (in).

FLD

Hemi-spherical punch test is performed for samples with different aspect ratios to measure forming limits of AA1050 sheet of 1 mm thickness. The initial geometry of the sheet samples used in the test is presented in Figure 4.

Figure 4.

Initial geometry of the sheet samples of FLD test.

The square geometry—aspect ratio = 1—is designed to produce a strain path at the right-hand side of the FLD. In this region, appropriate lubrication improves the loading symmetry and enables us to approach to the more equi-biaxial strain paths. The high aspect ratio samples are designed to produce a strain path at the left-hand side of the FLD. These samples are prone to tear from blank holder edge. In order to prevent sample edges from tearing, an amount of the material is removed from two middle sides of the samples. In this way, the central part of the sample is weakened and necking occurs in this region.

In order to measure the principal strains after necking, a pattern of circle grids of 0.1 in diameter is etch-printed on the initial sample surfaces. The circles deform to some ellipses and the diameters are used to measure the average principal strains in each circle. The samples are drawn with a punch travel rate of $0.5 mm / s$ . The employed press is a 30-ton hydraulic press with 250 mm ram stroke and table size of $600 \times 450 mm$ . The complete set of employed hydraulic press, die, punch and blank holder is shown in Figure 5. On the initiation of necking, the machine load decreases and loading is stopped.

Figure 5.

The employed hydraulic press with the complete set of die, punch and blank holder.

The sheet samples after necking are presented in Figure 6. The strains of each sample at the initiation of necking is measured and represented on the plane of principal strains to construct FLD. The resulted experimental FLD along with the numerical prediction are presented in the “Results” section.

Figure 6.

Sheet samples after necking.

Constitutive relations

Stress–strain relations

The total deformation of the material at the crystal scale is considered to be composed of two parts²⁵

F = F^{*} \cdot F^{P}

(4)

in which

F^{P} = I + \sum_{α = 1}^{n_{s}} γ^{α} (s_{0}^{α} \otimes n_{0}^{α})

(5)

and so, the rate of deformation tensor $D$ , according to its definition,²⁶ will be

D = D^{*} + \sum_{α = 1}^{n_{s}} P^{α} {\overset{\cdot}{γ}}^{α}

(6)

where $P^{α}$ is the symmetric part of Schmid tensor

m^{α} = s^{α} \otimes n^{α}

(7)

The interested reader can refer to Hajian and Assempour²² for detailed description.

Also, the rate-dependent relation between shear stress in the slip system and the corresponding shear strain rate in the form of power law is

{\overset{\cdot}{γ}}^{α} = {\overset{\cdot}{γ}}_{0} sgn (τ^{α}) {| \frac{τ^{α}}{g^{α}} |}^{b}

(8)

where ${\overset{\cdot}{γ}}_{0}$ is the reference shear rate, $g^{α}$ is the current strength of slip system $α$ , $b$ is the strain rate sensitivity exponent and $τ^{α}$ is the resolved shear stress on the slip system $α$ which is calculated from the following relation

τ^{α} = m^{α} : σ

(9)

where $σ$ is the Cauchy stress and $m$ is the Schmid tensor.

In which the variations of the function $g^{α}$ are governed by the hardening law

{dg}^{α} = q \cdot h \cdot \sum_{β = 1}^{n_{s}} d γ^{β}

(10)

where the hardening rate is

h = h_{0} \cdot {(\frac{h_{0} \cdot γ}{τ_{0} \cdot n} + 1)}^{n - 1}

(11)

and the total cumulative shear strain is

γ = \sum_{α = 1}^{n_{s}} \int_{0}^{t} | {\overset{\cdot}{γ}}^{α} | dt

(12)

To extend the stress–strain relation from rigid plastic to the elastic–plastic state, Hill and Rice²⁷ suggested the following hypo-elastic equation for the elastic behavior at the crystal scale

{\overset{\circ}{σ}}^{*} + σ tr (D^{*}) = C : D^{*}

(13)

where $C$ is the fourth-order tensor and represents the elastic moduli and $σ^{*}$ is the Jaumann rate of Cauchy stress

{\overset{\circ}{σ}}^{*} = \overset{\cdot}{σ} - W^{*} \cdot σ + σ \cdot W^{*}

(14)

where $\overset{\cdot}{σ}$ is the material time rate of $σ$ and $W^{*}$ is the material spin. In relations (13) and (14), the parameters $D^{*}$ and $W^{*}$ are only related to the elastic part of deformation. By further calculations, the stress–strain relation can be written in terms of total quantities $D$ , $W$ and ${\overset{\cdot}{γ}}^{α}$ as

\overset{\circ}{σ} + σ t r (D) = C : D - \sum_{α = 1}^{n} [C : P^{α} + Q^{α} \cdot σ - σ \cdot Q^{α}] {\dot{γ}}^{α}

(15)

where

\overset{\circ}{σ} = \overset{\cdot}{σ} - W \cdot σ + σ \cdot W

(16)

Solution method

Finite element model

In order to simulate the crystal plasticity behavior of the material, a user material was developed by Hajian and Assempour²² in the form of a Fortran subroutine. This subroutine was linked to the non-linear finite element software ABAQUS. This work enabled us to simulate an arbitrary plastic deformation considering the initial texture of the material and its texture evolution. In this subroutine, the rate-dependent crystal plasticity constitutive equations are solved for the magnitudes of slips on 12 slip systems. To do this, a semi-implicit time integration method was employed as follows

d γ^{α} = \frac{dt}{2} [{\overset{\cdot}{γ}}_{t}^{α} + {\overset{\cdot}{γ}}_{t + dt}^{α}]

(17)

where using relation (8) and the Taylor expansion method, ${\overset{\cdot}{γ}}_{t + dt}$ can be calculated from the following

{\overset{\cdot}{γ}}_{t + dt}^{α} = {\overset{\cdot}{γ}}_{t}^{α} + \frac{\partial {\overset{\cdot}{γ}}^{α}}{\partial τ^{α}} d τ^{α} + \frac{\partial {\overset{\cdot}{γ}}^{α}}{\partial g^{α}} {dg}^{α}

(18)

where ${dg}^{(α)}$ is given from relation (10) and $d τ^{α}$ is obtained from the combination of relations (15), (16), (9) and its time rate. By substituting ${\overset{\cdot}{γ}}_{t + dt}^{α}$ into relation (17), $d γ^{(α)}$ in each increment can be achieved. The accuracy of the solution is assured by checking the combination of relations (8) and (17) at the endpoint of each increment and elimination of the existent error by the Newton–Raphson method. In each increment, $d γ^{(α)}$ is calculated for all the Gauss points and in each Gauss point for all the crystals. The Taylor model is used for scale transition among different crystals. After convergence, the strength of each slip system, stresses and crystal orientations are updated and the calculated data are delivered to the ABAQUS software for later increments.

FLD calculation

The algorithm of calculation of FLD points is presented in Figure 7. In this algorithm, a rectangular sheet geometry is simulated by employing our developed user subroutine. In a manner like Marciniak–Kuczynski (M-K) method, a groove is added to the sheet to enable the simulation to capture necking of the material. The sheet is discretized into 702 elements and is loaded in two in-plane directions by applying displacement-type boundary conditions. In order to produce various strain paths on the FLD, the loading ratio $β$ is changed to produce deformation modes from uniaxial tension to equi-biaxial tension. In each loading ratio, the groove angle $θ$ is changed from 0 to $π / 2$ to find the weakest orientation of groove for that loading ratio. In order to determine the necking point, in each simulation, the magnitude of second derivative of thickness variations in this element with respect to time²⁸ has been used as a factor to identify the instability condition. More detailed description can be found in the study by Adams and Jeswiet.²²

Figure 7.

FLD calculation process.

Results and discussions

The complete set of required experimental input and verification data for crystal plasticity–based prediction of FLD of AA1050 sheet along with the results of numerical FLD calculation is presented in this section. To this end, the initial texture of the material was measured in terms of ${200}$ , ${220}$ and ${111}$ stereographic pole figures with the XRD technique. The measurement errors due to background intensity and defocusing phenomena were corrected as described in section “Texture measurement.”Figure 8 illustrates three corrected stereographic pole figures. The obtained stereographic pole figures were used to calculate the material ODF. The resulted ODF is presented in Figure 9 in terms of $ϕ_{2}$ sections where the x- and y-axes represent the transverse and rolling directions, respectively. This texture distribution is approximately discretized with a combination of 125 grains for using in the simulation.

Figure 8.

The corrected stereographic pole figures of the employed AA1050 sheet.

Figure 9.

Orientation distribution function of the employed AA1050 sheet.

The crystal elastic constants of the material are presented in Table 2. The hardening parameters of the material have been extracted by fitting the simulated stress–strain curve to the experimental data as shown in Figure 10. The values of these parameters have been reported in Table 3. Also, the rate sensitivity exponent $b$ and the reference shear rate ${\overset{\cdot}{γ}}_{0}$ are considered to be $500$ and $0.001$ , respectively.

Table 2.

Crystal elastic constants of the material.

Parameter	Value (GPa)
$C_{11}$	206
$C_{12}$	116
$C_{44}$	54

Table 3.

The hardening parameters of material.

Parameter	Value	Unit
$h_{0}$	450	MPa
$n$	0.165	–
$τ_{0}$	19.5	MPa

Figure 10.

The resulted stress–strain of the material.

The thickness distribution of the material is computed for various strain paths and FLD of the material is calculated following the procedure described in section “FLD calculation.”

The experimentally measured forming limit curve including fail, safe and neck points along with our crystal plasticity–based prediction of neck points are presented in Figure 11. The experimental points at the right-hand side of FLD are obtained from sufficiently lubricated square samples. The lubrication helps us to reach a symmetrical situation in the two principal loading axes.

Figure 11.

Resulted FLD from experiment and simulation.

The percent errors of FLD prediction with respect to the experimental results are shown in Figure 12. It is observed that computational results satisfactorily predict the experimental FLD. Although the prediction accuracy seems acceptable, some differences yet exist between numerical and experimental results. There are some reasons for these differences. In this research, just slip on the slip systems of the crystal has been considered, which is the main mechanism of plastic deformation. But there are some other mechanisms that accommodate the plastic deformation, such as twinning and grain boundary sliding. Another reason is that the Taylor model has been selected for the scale transition among the microscopic and macroscopic scales.

Figure 12.

Percent errors of FLD prediction with respect to experimental results.

Adding the effects of twinning and grain boundary sliding into the model and employing more advanced models of scale transition can further improve the results. Improving the currently developed tool for crystal plasticity–based prediction of forming limits and investigating the effects of various texture components on the formability of face-centered cubic (FCC) sheet metals will be studied in a future work.

Conclusion

A crystal plasticity methodology to evaluate the AA1050 sheet formability was presented in this work. In order to determine the orientation distribution of the crystals, the initial texture of the sheet material was measured through XRD technique. Also, the stress–strain behavior of the material was obtained by conducting tensile test. The results of texture measurement by XRD and tensile test were used as the input material properties in the FLD prediction by simulation. The simulations were based on the numerical procedure developed previously by the authors in which a rate-dependent crystal plasticity model along with the power law hardening were employed in a user material subroutine to model the behavior of body-centered cubic (BCC) crystalline solids. The procedure was extended to model the behavior of FCC crystal structures. In order to assess the validity of formability prediction results for FCC materials, FLD of AA1050 sheet was also experimentally extracted by conducting hemi-spherical punch test in different strain paths. Second-order derivative of sheet thickness variations with respect to time was considered as the instability factor to extract forming limits of the material. It was observed that the numerical FLD satisfactorily predicts the experimental curve. Finally, the reasons for the existing differences were discussed and some suggestions to further improve the prediction accuracy were made.

Footnotes

Appendix 1 Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship and/or publication of this article.

References

Zhang

S-H

Cheng

. Formability improvement of austenitic stainless steel by pulsating hydroforming. Proc IMechE, Part B: J Engineering Manufacture. Epub ahead of print 9 June 2014. DOI: 10.1177/0954405414535382.

Yang

. Reliability analysis of corner filling in the tube hydroforming process. Proc IMechE, Part B: J Engineering Manufacture. Epub ahead of print 20 June 2014. DOI: 10.1177/0954405414535584.

Adams

Jeswiet

. Design rules and applications of single-point incremental forming. Proc IMechE, Part B: J Engineering Manufacture. Epub ahead of print 25 June 2014. DOI: 10.1177/0954405414531426.

Zhang

Lei

. The high-pressure liquid jet incremental forming for the aluminum sheet. Proc IMechE, Part B: J Engineering Manufacture. Epub ahead of print 30 June 2014. DOI: 10.1177/0954405414539930.

Shim

. Applications of digital tryout to stampings of hard-to-form parts. Proc IMechE, Part B: J Engineering Manufacture. Epub ahead of print 18 June 2014. DOI: 10.1177/0954405414535582.

Dariani

Liaghat

Gerdooei

. Experimental investigation of sheet metal formability under various strain rates. Proc IMechE, Part B: J Engineering Manufacture 2009; 223(6): 703–712.

Safikhani

Hashemi

Assempour

. The strain gradient approach for determination of forming limit stress and strain diagrams. Proc IMechE, Part B: J Engineering Manufacture 2008; 222(4): 467–483.

Yang

Dong

Zhou

. Crystal plasticity-based forming limit prediction for FCC materials under non-proportional strain-path. Mater Sci Eng A 2010; 527(2425): 6607–6613.

Steglich

Jeong

Andar

. Biaxial deformation behaviour of AZ31 magnesium alloy: crystal-plasticity-based prediction and experimental validation. Int J Solids Struct 2012; 49(25): 3551–3561.

10.

Hoc

Rey

Raphanel

. Experimental and numerical analysis of localization during sequential test for an IF-Ti steel. Acta Mater 2001; 49(10): 1835–1846.

11.

Zhuang

Wang

Lin

. Experimental and numerical investigation of localized thinning in hydroforming of micro-tubes. Eur J Mech A: Solid 2012; 31(1): 67–76.

12.

Barlat

Ahn

. Constitutive modeling of ferritic stainless steel. Int J Mater Form 2010; 3(2): 135–145.

13.

Hoffmann

Bertram

Shim

. Experimental identification and validation of a crystal plasticity model for a low carbon steel on different length scales. Int J Mater Form 2010; 3(1): 65–68.

14.

Signorelli

Serenelli

Bertinetti

. Experimental and numerical study of the role of crystallographic texture on the formability of an electro-galvanized steel sheet. J Mater Process Tech 2012; 212(6): 1367–1376.

15.

Green

Neale

MacEwen

. Experimental investigation of the biaxial behaviour of an aluminum sheet. Int J Plasticity 2004; 20(89): 1677–1706.

16.

Narayanasamy

Ravindran

Manonmani

. A crystallographic texture perspective formability investigation of aluminium 5052 alloy sheets at various annealing temperatures. Mater Design 2009; 30(5): 1804–1817.

17.

Savoie

Jain

Carr

. Predictions of forming limit diagrams using crystal plasticity models. Mater Sci Eng A 1998; 257(1): 128–133.

18.

Neale

Giessen

. Crystal plasticity forming limit diagram analysis of rolled aluminum sheets. Metall Mater Trans A 1998; 29(2): 527–535.

19.

Palumbo

Sorgente

Tricarico

. A numerical and experimental investigation of AZ31 formability at elevated temperatures using a constant strain rate test. Mater Design 2010; 31(3): 1308–1316.

20.

Ramos

Stout

Bolmaro

. Study of a drawing-quality sheet steel—I: stress/strain behaviors and Lankford coefficients by experiments and micromechanical simulations. Int J Solids Struct 2010; 47(17): 2285–2293.

21.

Ramos

Stout

Bolmaro

. Study of a drawing-quality sheet steel—II: forming-limit curves by experiments and micromechanical simulations. Int J Solids Struct 2010; 47(17): 2294–2299.

22.

Hajian

Assempour

. Experimental and numerical determination of forming limit diagram for 1010 steel sheet: a crystal plasticity approach. Int J Adv Manuf Tech. Epub ahead of print 25 September 2014. DOI: 10.1007/s00170-014-6339-9.

23.

Hatch

JE.

Aluminum: properties and physical metallurgy. Ohio: American Society for Metals, 1984.

24.

Randle

Engler

. Introduction to texture analysis: macrotexture, microtexture and orientation mapping. Boca Raton, FL: Taylor & Francis, 2000.

25.

Asaro

. Micromechanics of crystals and polycrystals. Adv Appl Mech 1983; 23: 1–115.

26.

Lai

Rubin

Krempl

. Introduction to continuum mechanics. New York: Butterworth-Heinemann, 2009.

27.

Hill

Rice

. Constitutive analysis of elastic-plastic crystals at arbitrary strain. J Mech Phys Solids 1972; 20(6): 401–413.

28.