Constitutive behavior of as-cast AA1050, AA3104, and AA5182
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INTRODUCTION A. DC Casting Models
IN the direct-chill (DC) casting process of aluminum, the primary and secondary cooling impose strong thermal gradients on the ingot. Stresses and strains caused by both solidification shrinkage and thermal contraction may lead to distortion of the ingot shape (e.g., butt curl, butt swell, or rolling face pull-in) or to hot tearing and cold cracking. In order to understand these phenomena and optimize the casting process accordingly. DC casting simulation models were developed.[1–5] Critical input to these models is the material behavior under the prevalent conditions, i.e., low strains, low strain rates, and temperatures ranging from casting temperature to room temperature. Therefore, the thermomechanical behavior is governed successively by power-law creep, power-law breakdown and low temperature plasticity. Such behavior can be described by many different constitutive equations, varying from equations that result from simple hardening models to complex equations with internal variables to describe the microstructural evolution of the alloy. B. Constitutive Equations To describe the material behavior in the as-cast condition in the complete temperature range encountered during casting, the extended Ludwik equation has been used,[1,6–10] which is a phenomenological equation of the form
⫽ K nL
mL
冢冣 ˙ ˙ 0
[1]
W.M. VAN HAAFTEN, Researcher, is with Corus R,D&T, 1970 CA IJmuiden, The Netherlands. B. MAGNIN, Technical Director Extrusion, is with the Pechiney Centre de Recherche de Voreppe, 38340, Voreppe, France. W.H. KOOL, Senior Scientist, and L. KATGERMAN, Professor, are with the Laboratory of Materials, Delft University of Technology, 2628 AL Delft, The Netherlands. Contact e-mail: [email protected] Manuscript submitted July 5, 2001. METALLURGICAL AND MATERIALS TRANSACTIONS A
where is the stress, K is a material constant (i.e., the stress at ⫽ 1 and ˙ ⫽ 1 s⫺1), is the plastic strain, nL is the strain-hardening coefficient, ˙ is the strain rate, ˙ 0 is a constant taken equal to 1 s⫺1, and mL is the strain-rate sensitivity. It fits stress-strain curves from room temperature to solidus temperature at the strain rates applicable to DC casting. The temperature dependency of the stress comes into the equation through K, mL , and nL . For the higher temperature regime of the casting process, where the material deforms mainly by creep, experimental data[3,11–13] have been fitted to the Garofalo or Sellars– Tegart equation:[14,15]
冋
冢 冣册
ss ˙ ⫽ A sinh 0
nH
exp
冢 RT 冣 ⫺Q
[2]
where A, 0, and nH are material constants, ss is the steadystate stress, Q is the activation energy, R is the universal gas constant, and T is the temperature (K). This equation fits the data from creep tests in the temperature range 400 ⬚C to 600 ⬚C and the strain-rate range 10⫺6 to 10⫺2 s⫺1 very well. Below 400 ⬚C, the Sellars–Tegart law overestimates the stress, as at low temperatures more strain is required to reach steady-state conditions.[12] The advantage of Eq. [2] over
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