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International Deep Drawing Research Group 

IDDRG 2009 International Conference 

1-3 June 2009, Golden, CO, USA 

CONSTITUTIVE RELATIONS FOR PLASTIC DEFORMATION IN 

AA5754 SHEET 

L. Hu1, S. Banovic2, T. Foecke2 M. Iadicola2, and A.D. Rollett1

1

 Department of Materials Science and Engineering 

Carnegie Mellon University 

5000 Forbes Avenue, Pittsburgh, PA 15213, USA 

e-mail: LinHu@andrew.cmu.edu , rollett@andrew.cmu.edu 

2

 NIST Center for Metal Forming 

National Institute of Standards and Technology 

Gaithersburg, MD 20899-8553, USA 

e-mail: stephen.banovic@nist.gov , tfoecke@nist.gov, mark.iadicola@nist.gov 

ABSTRACT 

Constitutive equations for the multiaxial stress-strain behavior of aluminum alloy 5754 

sheets have been developed, based on crystal plasticity. Both a Taylor-based polycrystal 

plasticity code (LApp) and a self-consistent viscoplastic code (VPSC) have been used to fit a 

single slip system hardening law to the available data for tension, plane strain and biaxial 

stretching. The fitting procedure yields good agreement with the monotonic stress-strain data. 

When the developed hardening law is used to model tests involving strain path changes, 

however, the agreement is less good. Furthermore, the simulated texture evolution is too rapid 

when compared to the experiments. These discrepancies motivate the further development of the 

constitutive relations to include such effects as grain-to-grain interactions and latent hardening. 

Keywords: Forming; Simulation; Forming Limits; Constitutive Relations; Aluminum Alloys.
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L. Hu, S. Banovic, T. Foecke, M. Iadicola, and A.D. Rollett 

1. INTRODUCTION 

5000-series Magnesium-strengthened Aluminum alloys are prospective materials for 

automotive interior surfaces. They have excellent weldability and corrosion resistance, providing 

a lower-weight alternative to steel with a boost in fuel economy, safety and driving performance 

[1]. Motivated by the need to manufacture parts efficiently from sheet aluminum metal, much 

research effort has been invested in studying basic mechanics involving metal forming. 

Specifically, modeling on the plastic deformation in aluminum sheets can provide some 

insights into the forming process. The earliest work traces back to Hollomon’s [2] empirical 

power law equation for uniaxial true stress–true plastic strain relationship of a material: 

σ = Kε p

n

 (1) 

where σ is the stress, K is the strain hardening coefficient, εp is the plastic strain and n is the 

strain hardening exponent. Later studies on dislocation slip [3] help to identify various hardening 

stages in the plastic deformation of polycrystalline material, which brings the plasticity modeling 

to the grain scale. The constitutive relation involves the evolution of the hardening on individual 

slip system. The macroscopic behavior is then derived by taking average over all the grains. 

Such constitutive relations based on slip systems have been well established for monotonic 

loading. However, they are often inadequate to predict stress-strain curves and texture 

development for processes that involve different strain modes and multi-axial loading. 

Two significant factors have drawn our attention in this study. The first factor is the intrinsic 

strain hardening due to dislocation accumulation. Various strain-hardening models have been 

proposed to describe hardening behaviors of a slip system such as isotropic hardening, latent 

hardening and kinematic hardening. However, it is still not clear whether the existing models are 

able to capture every important aspect of strain hardening, and to what extent we need to account 

for this complex dislocation interaction to be sufficiently accurate. The other factor is the 

interaction between grains, which has often been ignored in plasticity modeling. Non-uniform 

distributions of strain and stress across a deformed polycrystal [4] suggest the need for a more 

detailed description of grain interaction other than the common iso-strain [5] and iso-stress [6] 

assumptions. 

In this study, simulations were undertaken using the Los Alamos Polycrystal Plasticity 

Packages (LApp) and Viscoplastic Self-consistent codes (VPSC). Mechanical tests are 

conducted on sheet alloy samples by collaborators at National Institute of Standards and 

Technology (NIST) to compare and validate the simulation work performed at CMU. The 

objective of the study is to establish equations that properly account for strain hardening and 

grain interaction behaviors during deformation. By incorporating such equations into current slip 

system based plasticity models, the accuracy of predictions for stress-strain curves and texture 

evolutions can be improved. 

2. MATERIALS 

5754 Aluminum Alloy is an Al-Mg alloy designed for structural and weldable sheets in 

vehicles such as inner body panels, splashguards, heat shields, air cleaner trays and covers, and 

load floors. Its chemical composition is reported in Table 1. The alloy derives its strength from 

the solid solution strengthening with magnesium atoms, interacting with dislocations and 

inhibiting dynamic recovery processes during straining. The addition of Manganese addition 

provides grain size control by forming well-dispersed particles. Also, for a given strength 
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L. Hu, S. Banovic, T. Foecke, M. Iadicola, and A.D. Rollett 

requirement, the addition of manganese allows lower magnesium content and ensures a greater 

degree of stability of the alloy [7]. 

Table 1. The chemical composition of AA5754 alloy used in this study. 

Element Mg Mn Fe Si Cr, Cu, Pb, Ni, Sn, Ti, Zn Aluminum 

Weight Percent 3.75 0.29 0.24 0.06 Individually <0.05 Balance 

The material used in this investigation was industrially processed AA5754-O sheet, 

nominally 1 mm thick. Figure 1 shows the microstructure of the RD/ND section of as-received 

material as observed by optical microscopy. The grains are relatively equiaxed in the rolling 

plane and slightly elongated along the RD. This is indicative of a recrystallized microstructure 

associated with the O-temper. 

 

Figure 1. Optical micrographs of as-received AA5754 sheet material showing a nearly equiaxed 

microstructure in both RD-TD and RD-ND cross-sections [8]. 

3. EXPERIMENTAL PROCEDURE 

3.1 Mechanical tests under single strain paths 

Samples were cut to appropriate dimensions for testing and deformed in three strain states: 

uniaxial tension, plane strain and equibiaxial tension. Uniaxial deformation and plane strain 

testing were conducted with the sheet rolling direction parallel (RD-oriented) and perpendicular 

(TD-oriented) to the main stress axis. Thus, five mechanical tests were performed on the sample: 

RD-oriented and TD-oriented uniaxial tensile tests, RD-oriented and TD-oriented plane strain 

tests, and equibiaxial stretch test. Under all modes, samples were produced at approximately 

every 5 % true strain to near-failure. 

The textures of the samples were measured using X-ray diffraction technique. Three 

incomplete pole figures were measured with 0° ≤ α ≤ 65°, namely: {111}, {200}, and {220}. 

The 3-D crystallite orientation distribution (COD) and full pole figures were derived from the 

pole figures using the preferred orientation package from Los Alamos (popLA). The COD was 

used to generate a weighted set of discrete orientations that are representative of the texture. 

3.2 Mechanical tests with strain path changes 

Mechanical tests with strain path changes were performed on the same material to study 

formability and which serve to investigate behaviors such as kinematic hardening and latent 
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L. Hu, S. Banovic, T. Foecke, M. Iadicola, and A.D. Rollett 

hardening. The samples were first deformed in equibiaxial tension to a prescribed strain and then 

uniaxially tested parallel and perpendicular to the RD of the sheet respectively. 

4. PLASTICITY SIMULATION 

Two models were used to simulate the plastic response of the material. The initial texture is 

incorporated using a set of 1152 weighted orientations derived from the COD as previously 

described. The material parameters used in the actual simulations were identified by the best fit 

(root mean square error) of the experimental values and giving the same weight to all mechanical 

tests. This process is automated by using a UNIX shell script. 

One of the simulations was done using LApp, a polycrystal plasticity code developed at Los 

Alamos National Laboratory. This model is used to calculate the stress state and texture 

evolution of applying a given strain rate to a set of orientations (grains). The basis of the 

simulation using LApp is the Taylor full constraint (FC) model where a uniform strain state is 

assumed. The strain hardening is modeled using the Voce law: 

θ = θ0 (1− τ τ V ) (2) 

where θ0 is the initial hardening rate and τ v is the saturation critical resolved shear stress. In the 

analysis, the initial critical shear stress τ 0 and τ v are respectively 65 MPa and 90 MPa. 

The other simulation was performed using the viscoplastic self-consistent (VPSC) code 

developed by C. N. Tomé and R. A. Lebensohn at Los Alamos National Laboratory. Similar to 

the LApp model, this model is used to predict the stress-strain response and texture development 

in the polycrystalline material. The model treats each grain as a viscoplastic ellipsoidal inclusion 

embedded in a homogeneous elastic matrix (HEM). The properties of the matrix (HEM) are 

adjusted self-consistently to coincide with the average of all inclusions forming the aggregate. 

The tangent method [9] is adopted as the linearization scheme in the simulation. In contrast to 

the Taylor FC model, in the VPSC method, the strain in each grain can vary as a function of 

orientation, depending on the stiffness of the interaction. The extended Voce hardening law is 

used in the simulation: 

τ = τ 0 + (τ 1 + θ1Γ)(1− exp(−Γθ0 τ 1)) (3) 

where τ 0, θ0 , θ1 , ( τ 0+τ 1) are the initial CRSS, the initial hardening rate, the asymptotic 

hardening rate and the back-extrapolated CRSS. Parameters used in the simulation are: τ 0 = 33 

MPa, τ 1 = 98 MPa, θ0= 285 MPa, θ1 =0. Withθ1 =0, the hardening equation reduces to the same 

form used in LApp. 

5. RESULT AND DISCUSSION 

5.1 Single-path mechanical test results 

Figure 2 shows the stress-strain curves for AA5754 in the aforementioned five deformation 

modes simulated by LApp as well as those determined experimentally. In the uniaxial tension 

case, the stress in the direction of applied strain is measured. The predicted curves are in good 

agreement with experimental data from Figure 2a and 2b. Figure 2c and 2d shows respectively 

the stresses in the RD and TD directions versus strains for equibiaxial tension. LApp apparently 

overestimates experimental data with a faster work hardening. Figure 2e and 2f shows stressstrain curves for both longitudinal and transverse directions in plane strain tension where LApp 

tends to underestimate the flow stress in comparison to experiment. 
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L. Hu, S. Banovic, T. Foecke, M. Iadicola, and A.D. Rollett 

(a) (b) 

(c) (d) 

(e) (f) 

Figure 2. Comparison of stress-strain curves measured in the experiments and predicted by Taylor FC 

model using the LApp code for different strain modes. (a) RD-oriented and (b) TD-oriented uniaxial 

tension; (c) along the RD and (d) TD direction for equibiaxial stretch; (e) RD-oriented and (f) TDoriented plane strain. 

Figure 3 presents the fitting of simulated curves by VPSC with experimental stress-strain 

curves and it is apparent that the fitted curves match the data better than for the Taylor model. 

Since simulations so far have been done using a same hardening behavior in both LApp and 

VPSC, therefore, this shows that incorporating the grain interaction improves the modeling of 

plastic deformation. However, for the plane strain tension case, VPSC was only able to match 

one of the stresses accurately. The non-trivial deviation of calculated stresses from experimental 

data suggests the need to further analyze the strain hardening behavior and grain interaction 

effect. 
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L. Hu, S. Banovic, T. Foecke, M. Iadicola, and A.D. Rollett 

(a) (b)

(c) (d) 

(e) (f) 

Figure 3. Comparison of stress-strain curves measured in the experiments and predicted by VPSC for 

different strain modes. (a) RD-oriented uniaxial tension; (b) TD-oriented uniaxial tension; (c) along the 

RD and (d) TD direction for equibiaxial stretch; (f) TD-oriented plane strain. 

Figure 4 shows the recalculated {111} pole figures from the as-received materials for both 

RD- and TD-oriented samples. The sheet displays a low degree of texture with a combination of 

recrystallization (cube and S) and retained deformation (Brass and copper) components. 

Figure 4. Recalculated {111} pole figures of as-received materials. 
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L. Hu, S. Banovic, T. Foecke, M. Iadicola, and A.D. Rollett 

(a) (b) 

(c) (d) 

Figure 5. Evolution of {111} pole figures with strain level from (a) experiments and simulation by (b) 

LApp (c) and VPSC; in each figure, the strain levels 5%, 10% and 15% going from left to right, except for 

equibiaxial tension which also has 20%; (d) texture strength as a function of strain. 

Figure 5a shows the development of measured {111} pole figures as a function of 

deformation mode and strain level. Figure 5b and 5c are the corresponding pole figures 

simulated by LApp and by VPSC respectively. Pole figures from experiments and simulation are 

compared at the following equivalent von Mises strain levels: 5%, 10%, and 15% for uniaxial 

tension and plane strain; and 5%, 10%, 15% and 20% for equibiaxial tension. LApp and VPSC 

both predict correctly the sharpening of major as-received texture components. The plots of 

texture strength, Fig. 5d, show that VPSC predicts a weaker texture at low strains but a higher 

strengthening rate compared to LApp. In general, simulated textures strengthen significantly 
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L. Hu, S. Banovic, T. Foecke, M. Iadicola, and A.D. Rollett 

with strain whereas the experimental textures barely change. This may be due to the neglect of 

the local neighborhood of individual grains in both models, which adds shears and rotations in 

order to maintain local compatibility and equilibrium. 

5.2 Multipath mechanical test results 

Figure 6 compares the experimental and theoretical stress-strain curves in the direction of 

applied uniaxial tension. For Figure 6(a), (c), (e) and (g), samples are loaded in RD-oriented 

uniaxial tension with a 5%, 10%, 15% and 20% biaxial prestrain, respectively. For Figure 6(b), 

(d), (f) and (h), samples are loaded in TD-oriented uniaxial tension with a 5%, 10%, 15% and 

20% biaxial prestrain, respectively. The calculations were performed with the VPSC code using 

the same parameters fitted to the single path (monotonic straining) mechanical tests. In addition, 

isotropic hardening behavior and tangent type grain interaction are enforced. The VPSC code 

overestimates the flow stress for the beginning of uniaxial tension, especially as the prestrain 

increases. The softening phenomenon in the material when subjected to strain path changes may 

be associated with anisotropic hardening behaviors during the process. As currently 

implemented, the calculated stress-strain response shows a small increase in flow stress at the 

change in strain path, which is opposite to the observed decrease. 

(a) (b) 

(c) (d) 
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L. Hu, S. Banovic, T. Foecke, M. Iadicola, and A.D. Rollett 

(e) (f) 

(g) (h) 

Figure 6. Comparison of stress-strain curves measured from experiments and simulated by VPSC for RDoriented tension after (a)5% (c) 10% (e) 15% (g) 20% biaxial prestrain; TD-oriented uniaxial tension 

after (b) 5% (d) 10% (f) 15% (h) 20% biaxial prestrain. 

6. CONCLUSIONS 

Plastic deformation under various strain paths for AA5754 aluminum alloy has been 

calculated with both the LApp and VPSC codes. The single slip system hardening parameters 

have been fitted by comparison with the experimental stress-strain curves. Texture development 

has also been compared with experiments. The results indicate that taking grain interaction into 

consideration refines the polycrystal plasticity modeling. However, the failure to predict the 

mechanical response for multiaxial loading and strain path changes points out the need to find 

more effective ways to describe accurately the strain hardening and grain interaction. 

7. ACKNOWLEDGEMENTS 

Support from the National Institute of Standards and Technology is gratefully acknowledged. 

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