Complex Unloading Behavior: Nature of the Deformation And

CHAPTER 3DRAFT MANUSCRIPT – Do Not Circulate

Complex Unloading Behavior: Nature of the Deformation and

Its Consistent Constitutive Representation

Li Sun and R. H. Wagoner

The Ohio State University

ABSTRACT

Complex unloading behavior following plastic straining has been reported as a significant challenge to accurate springback prediction. More fundamentally, the nature of the unloading deformation has not been resolved, being variously attributed to nonlinear / reduced modulus elasticity or to inelastic / “microplastic” effects. Unloading-and-reloading experiments following tensile deformation showed that a special component of strain, deemed here “Quasi-Plastic-Elastic” (“QPE”) strain, has four characteristics. 1) It is recoverable, like elastic deformation. 2) It dissipates work, like plastic deformation. 3) It is rate-independent, in the strain rate range, contrary to some models of anelasticity to which the unloading modulus effect has been attributed. 4) No evolution of plastic properties occurs, the same as for elastic deformation. These characteristics are as expected for straining by a mechanism of dislocation pile-up and relaxation. A consistent, general, continuum constitutive model was derived incorporating elastic, plastic, and QPE deformation. Using some aspects of two-yield-function approaches with unique modifications to incorporate QPE, the model was implemented in a finite element program with parameters determined for dual-phase steel and applied to draw-bend springback. Significant differences were found compared with standard simulations or ones incorporating modulus reduction. The proposed constitutive approach can be used with a variety of elastic and plastic models to treat the nonlinear unloading and reloading of metals consistently for general three-dimensional problems.

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To be submitted to the International Journal of Plasticity

Manuscript Date: July 23, 2010

3.1 IntroductionNTRODUCTION

The elastic response of metals related to atomic bond stretching is very nearly linear. Second-order elasticity can be expressed as follows (Powell and Skove, 1982; Wong and Johnson, 1988)

(3.1)

where , are the uniaxial stress and strain, respectively, is the initial Young’s modulus, and is a nonlinearity parameter with a value 5.6 reported for “Helca 138A” steel (Powell and Skove, 1982; Wong and Johnson, 1988)[(Powell and Skove, 1982; Wong and Johnson, 1988)]. Equation 3.1 predicts a change of modulus by approximately 3% for a dual phase (DP) steel having an ultimate tensile strength of 980 MPa, making it one of the strongest steels considered for forming applications.

The elasticity theory, which describes the mechanical behavior of material in the elastic domain, can be traced to the Hooke’s law in the seventeenth century. According to the law, the resulting deformation is proportional to the applied load as this load does not exceed the elastic limit. The coefficient of proportionality, defined as Young’s modulus, is independent external load and usually regarded as a constant value for most metals at room temperature. In recent research, howeverNonetheless, highly nonlinear unloading following plastic deformation has been widely observed [(Morestin and Boivin, 1996; Augereau et al., 1999; Cleveland and Ghosh, 2002; Caceres et al., 2003; Luo and Ghosh, 2003; Yeh and Cheng, 2003; Yang et al., 2004; Perez et al., 2005; Pavlina et al., 2009; Yu, 2009; Zavattieri et al., 2009; Andar et al., 2010)Andar et al., 2010; Caceres et al., 2003; Cleveland and Ghosh, 2002; Luo and Ghosh, 2003; Morestin and Boivin, 1996; Pavlina et al., 2009; Perez et al., 2005; Yu, 2009; Zavattieri et al., 2009, Augereau et al., 1999; Yeh and Cheng, 2003, Yang et al., 2004] . [Li – Put these in date order, and include others if you have them. Also add in other IJP references from the list I gave you and others you find from there. Just numbers are ok.], with the apparent unloading modulus reduced by up to 22% for high strength steel (Cleveland and Ghosh, 2002) and 70% for magnesium o 30% [Li – check the maxiumum value reported in the literature and put the reference in the next line] relative to the bond-stretching value (Caceres et al., 2003) [REFS]. The magnitude of the reduction depends on the plastic strain and alloy. the phenomenon of Young’s modulus degradation with accumulated plastic strain during unloading process was observed by macroscopic tensile test (Andar et al., 2010; Caceres et al., 2003; Cleveland and Ghosh, 2002; Luo and Ghosh, 2003; Morestin and Boivin, 1996; Pavlina et al., 2009; Perez et al., 2005; Yu, 2009; Zavattieri et al., 2009), ultrasonic test (Augereau et al., 1999; Yeh and Cheng, 2003) and nano-indentation test (Yang et al., 2004). The results indicated that the assumption of constant, linear Young’s modulus was not accurate enough to describe material stress-strain behavior.

In addition, the effect can differ with rest time after deformation, heat treatment and strain path (Yang et al., 2004; Perez et al., 2005; Pavlina et al., 2009)(Pavlina et al., 2009; Perez et al., 2005; Yang et al., 2004).

Springback, which is generally defined as the elastically-driven shape deviation after removing the sheet metal from tooling, is essential for designing sheet forming process since it greatly affects the accuracy of forming parts. Nowadays numerical simulations in springback prediction based on finite element methods (FEM) are remarkably utilized in order to replace time-consuming trial-and-error method in tooling design. Accuracy of springback calculation is strongly dependent on the exact evaluation of stress-strain curve during forming process. It is indicated that appropriate material’s hardening model and numerical tolerance are needed to accurately predict springback in sheet metal forming (Chun et al., 2002; Chung et al., 2005; Gau and Kinzel, 2001; Geng and Wagoner, 2000, 2002; Li et al., 2002a; Yoshida and Uemori, 2003a; Yoshida et al., 2002). Isotropic hardening model, which does not make allowance for Bauchinger effect, is too rough to catch material behavior during reverse loading. The nonlinear kinematic hardening model, Chaboche model (Chaboche, 1986, 1989) was widely used in recent research since a smooth variation of stress-strain curve was obtained by introducing linear and nonlinear backstress. The mixed law combining isotropic and nonlinear kinematic hardening model, which considers Bauchinger effect in reverse loading, can be regarded as an effective method for precise prediction of springback (Eggertsen and Mattiasson, 2009; Gau and Kinzel, 2001; Morestin et al., 1996; Taherizadeh et al., 2009; Tang et al., 2010; Zang et al., 2007). Young’s modulus controlling the material unloading behavior after forming process is an important parameter and should also be considered in springback prediction. Generally, springback is found to be nearly inversely proportional to the Young’s modulus. Thus, accurate description of elastic modulus is of vital importance in terms of computational simulation of springback.

The evolutions of Young’s modulus undergoing unloading-loading test for Dual Phase steels DP780 and DP980 are shown in Fig.1(a) and Fig.1(b), respectively. They present that the tangent unloading modulus is linear at the beginning and then nonlinearity gradually increases as the magnitude of stress decreases during unloading procedure. The similar phenomenon can be observed in the reverse loading process. The hysteresis loops form during this loading-unloading process and the difference between traditional constant Young’s modulus 208 GPa and measured average Young’s modulus 145 GPa is close to 30% for DP980, which is shown in Fig.1(c).

NThe nonlinear unloading behavior has been variously attributed to ity of Young’s modulus can be referred to several causes such as higher-order elasticity due to lattice distortion (Powell and Skove, 1982; Wong and Johnson, 1988), non-uniform distribution of residual stress (Hill, 1956), time-dependent anelastic strainity (Zener, 1948; Lubahn, 1961)(Lubahn, 1961; Zener, 1948), damage evolution (Yeh and Cheng, 2003; Halilovic et al., 2009)(Halilovic et al., 2009; Yeh and Cheng, 2003, Augereau et al., 1999, Vrh et al., 2008), twinning or kink bands in HCP alloys (Caceres et al., 2003; Zhou et al., 2008; Zhou and Barsoum, 2009, 2010)(Caceres et al., 2003, Zhou and Barsoum, 2009, 2010; Zhou et al., 2008)), and variation of dislocation structurepiling up and relaxation of dislocation arrays (Morestin and Boivin, 1996; Cleveland and Ghosh, 2002; Luo and Ghosh, 2003; Yang et al., 2004)(Cleveland and Ghosh, 2002; Luo and Ghosh, 2003; Morestin and Boivin, 1996; Yang et al., 2004), damage evolution (Halilovic et al., 2009; Yeh and Cheng, 2003), and twinning (Caceres et al., 2003). Based on the finite deformation theory, the second-order elasticity can be expressed as follows (Powell and Skove, 1982; Wong and Johnson, 1988)

(3.1)

where , are the uniaxial stress and strain, respectively. is the initial Young’s modulus and is the nonlinearity parameter. The value for steel (Helca 138A) is 5.6 and it leads that elastic error is approximately 3% for DP980, which is much smaller than 30% difference in Fig.1(c).

Microcrack or void evolution is another view to represent gradual Young’s modulus degradation under loading. Damage is considered as cavity inclusion and effective modulus is calculated by equivalent inclusion method (Augereau et al., 1999; Halilovic et al., 2009; Vrh et al., 2008; Yeh and Cheng, 2003). For HCP metals, the pseudo-elastic behavior was explained as mobile movement of twin boundaries or the formation of incipient kink bands (Caceres et al., 2003; Zhou and Barsoum, 2009, 2010; Zhou et al., 2008).

In the dislocation pile-up and release mechanisms, The perspective that variation of Young’s modulus is attributed mainly to dislocation pile-up was reported in recent research (Cleveland and Ghosh, 2002; Hart, 1984; Luo and Ghosh, 2003; Morestin and Boivin, 1996; Yang et al., 2004). The mobile dislocations which move along the same slip planes are until stopped by grain boundaries or other obstacles and forming dislocation pile-ups form(or similar structures such as polarized cell walls). When the applied stress is releasedreduced, the repelling dislocations move away from each other, providing additional unloading strains concurrent with strains from atomic bond relaxation. Moving dislocations dissipate work, by exciting lattice phonons (Hirth and Lothe, 1982) [REF – maybe Hirth and Loethe?]. Thus, while such pile-up-and-release strains are expected to be at least partly recoverable, they cannot be energy preserving. push against each other at first and then the contraction become loose. As a result, the degradation of Young’s modulus is observed due to the displacement of mobile dislocations. The macroscopic tensile experiments show that the measurement of unloading Young’s modulus is dependent on time, heat treatment and strain path of the pre-plastic strain (Pavlina et al., 2009; Perez et al., 2005; Yang et al., 2004) and the microscopic measurement, which is executed by nano-indentation test, found that the distribution of apparent Young’s modulus in the grain is location dependent: the magnitude near the grain boundary is smaller than center region (Yang et al., 2004). Above observations strongly support the viewpoint that mobile dislocation is the key reason to cause variation of Young’s modulus.

Many attempts have been carried out on modeling the degradation of Young’s modulus mathematically. Most of them use the average unloading Young’s modulus, or called the ‘chord modulus model’ (Fei and Hodgson, 2006; Ghaei et al., 2008; Kubli et al., 2008; Li et al., 2002b; Luo and Ghosh, 2003; Morestin and Boivin, 1996; Yu, 2009; Zang et al., 2007) , which is demonstrated in Fig.1(c). The chord modulus model simplifies the hysteresis loop forming during loading-unloading process as a straight line, and assumes unloading Young’s modulus a decreasing function of equivalent plastic strain. This method is easy to employ in FEM and improves accuracy in the simulation of springback prediction. However, the chord modulus model introduces model inaccuracy due to the existence of residual stress in springback process. One practical consequence of the changed unloading modulus is the challenge of simulating springback accurately. The general rule is that the magnitude of springback is proportional to the flow stress and inversely proportional to Young’s modulus (Wagoner et al., 2006). [R. H. Wagoner, J. F. Wang and M. Li: “Springback,” Chapter in ASM Handbook, Volume 14B: Metalworking: Sheet Forming, ASM, Materials Park, OH, 2006, pp. 733-755]. Simulations of springback are improved markedly by taking the observed unloading behavior into account (Morestin and Boivin, 1996; Li et al., 2002b; Fei and Hodgson, 2006; Zang et al., 2007; Vrh et al., 2008; Halilovic et al., 2009; Yu, 2009; Eggertsen and Mattiasson, 2010) [REFS]. Nearly all of the proposed practical approaches to incorporating complex unloading behavior rely on adopting a “chord modulus” (i.e. the slope of a straight line drawn between stress-strain points just before unloading and after unloading) (Morestin and Boivin, 1996; Li et al., 2002b; Luo and Ghosh, 2003; Fei and Hodgson, 2006; Zang et al., 2007; Ghaei et al., 2008; Kubli et al., 2008; Yu, 2009)(Fei and Hodgson, 2006; Ghaei et al., 2008; Kubli et al., 2008; Li et al., 2002b; Luo and Ghosh, 2003; Morestin and Boivin, 1996; Yu, 2009; Zang et al., 2007). The chord modulus model has conceptual and practical advantages: incorporating it in existing software is no more difficult than altering Young’s modulus in the input (and possibly as a function of strain). However, the method has limitations, namely that the real unloading is not linear and thus unloading to any internal stress otherless than zero (i.e with any residual stress) will have inherent errors. More fundamentally, the physical phenomenon is not truly elastic (i.e. energy preserving), and thus loading and unloading excursions may follow considerably different stress-strain trajectories than expected.

The plastic constitutive equation must also be known accurately for springback applications in order to evaluate the stress and moment before unloading. This is particularly true when the plastic deformation path includes reversals, as for example while being bent and unbent being drawn over a die radius (Gau and Kinzel, 2001; Chun et al., 2002; Geng and Wagoner, 2002; Li et al., 2002b; Yoshida et al., 2002; Yoshida and Uemori, 2003a; Chung et al., 2005)[Chun et al., 2002; Chung et al., 2005; Gau and Kinzel, 2001; Geng and Wagoner, 2000, 2002; Li et al., 2002a; Yoshida and Uemori, 2003a; Yoshida et al., 2002]. Nonlinear kinematic hardening (Chaboche, 1986, 1989) is widely used as an effective method for prediction of springback under such conditions (Morestin et al., 1996; Gau and Kinzel, 2001; Zang et al., 2007; Eggertsen and Mattiasson, 2009; Taherizadeh et al., 2009; Tang et al., 2010)(Eggertsen and Mattiasson, 2009; Gau and Kinzel, 2001; Morestin et al., 1996; Taherizadeh et al., 2009; Tang et al., 2010; Zang et al., 2007).

In view of the state of understanding of the unloading modulus effect, a few revealing experiments were performed using a a high-strength steel, DP 980, chosen to accentuate the deviations from bond-stretching elasticity. DP 980 isis aa dual-phase steel with nominal ultimate tensile strength of 980 MPa . Based on inferences drawn from these results, a consistent, general (3-D) constitutive model was developed this paper, a new Quasi-Plastic-Elastic (QPE) model was developed to represent the observed variation of Young’s modulus, and required parameters were determined. Dubbed the QPE model (Quasi-Plastic-Elastic), it was implemented in Abaqus/Standard (ABAQUS) [Abaqus] and compared with tensile tests with unloading/loading cycles at various pre-strains, with reverse tension/compression tests, and with draw-bend springback tests. QPE introduces a third component of strain (similar to one envisioned elsewhere in 1-D form (Cleveland and Ghosh, 2002) [REF: Cleveland]) that is Compared with the existing chord modulus models which generally considered elastic modulus as a decreasing function of equivalent plastic strain, the QPE model introduced a third component of strain that was recoverable (elastic-like) but energy dissipative (plastic-like) and in a natural way produced nonlinear loading and unloading curves. The new model was implemented into user material routines for ABAQUS/Standard (ABAQUS) in springback predictions of draw-bend test and the comparison between the simulation results and experimental data has been performed. .

3.2 Experimental ProceduresEXPERIMENTAL PROCEDURES

3.2.1 Materials

The materials studied in this paper are Dual-Phase (DP) steel-DP780 and DP 980 steel wasas selected for testing because dual-phase steels have large, numerous islands of hard martensite phase in a much softer ferrite matrix. The islands serve to strengthen but also to lower the yield stress by providing stress concentrators. DP 980 is the strongest alloy typically considered for forming applications, where springback is likely towould be a significant issue. Because of the high strength and large, numerous obstacles, DP 980 was expected to maximize accentuate the unloading modulus effect, assuming a dislocation pile-up and release mechanism as the principal source. which have a microstructure combining a soft ferrite matrix and a hard martensite phase “islands”.

The DP DP 980 alloy used in this study had previously been characterized to obtain an accurate 1-D plastic constitutive equation (Sung, 2010) along with standard mechanical properties [Sung 2010] as shown in Table 3.1. The thickness of the sheet steel is 1.4mm and 1.43mm for DP780 and DP980, respectively. The mechanical properties are listed in Table 3.1 (Sung, 2010). The standard tensile tests represented in Table 3.1 were carried out at General Motors North America (GMNA, 2007) according to ASTM E8-08 at a crosshead speed of 5mm/min and conducted at General Motors North America (GMNA, 2007). The normal plastic anisotropy parameters anisotropies r1 and r2 refer to results from alternate test procedures applied to sheets of original thickness and thinned reduced thickness (Sung, 2010)for balanced biaxial testing, respectively.[Sung 2010]. In either case, the r values are close to 1 and do not different greatly with testing direction, justifying an assumption of plastic isotropy as a first approximation.

A similar but weaker alloy (with fewer, smaller islands of martensite), DP 780, was used for limited comparative testing. It is the same alloy that has been characterized to obtain an accurate 1-D plastic constitutive equation (Sung, 2010)along with standard mechanical properties [Sung 2010] as shown in Table 3.1..

3.2.2 Tensile Testing

SThe standard parallel-sided tensile specimens (ASTM-E646) with a gage length 75mm and width 12.5mm were cut in the rolling direction andwere performed in used for uniaxial tensile testing. Unless otherwise stated, a nominal strain rate of 10-3/s was imposed. The axial load is provided byAnn MTS 810 testing machine and an Electronic Instrument Research LE-05 laser extensometer is were used to measure strain according to displacement between two fixed points on the specimen.

Compression / Tension Testing

[Li – I just noticed that you didn’t say anything about C/T testing here. You need to add this section, sufficient to understand the test and the correction procedures for biaxial stress and friction. Need to give references for more complete treatments. I have given you a starting point below.]

Compression/tension testing was performed using methods appearing in the literature (Boger et al., 2005). [Boger 2005]. Two flat backing plates and a pneumatic cylinder system were used to provide side force to constrain the exaggerated dog-bone specimen against buckling in compression. Side forces of 3.3512 kN [Li – check what you used] were applied and a laser extensometer (Boger et al., 2005)(Boger et al., 2005) was used to measure specimen extension directly.