JOURNAL OF MATERIALS SCIENCE LETTERS 18 (1 9 9 9 ) 1757 – 1758 On the dynamic material model for the hot deformation of materials S. V. S. NARAYANA MURTY, B. NAGESWARA RAO Vikram Sarabhai Space Centre, Trivandrum 695 022, India Montheillet et al. [1] examined the dissipator power co-content approach in the Dynamic Material Model (DMM), which is being used extensively by Prasad and his research team for analyzing the high temper- ature forming of metals. It is noted that the efficiency of power dissipation (η) as derived from the dissipa- tor power co-content ( J ) to predict the likelihood of flow localization is not as well founded physically as the established procedure based directly on the strain rate sensitivity (m). In his reply to the above aspects, Prasad [2] explains the physical interpretations of G, the dissipator content, and J , the dissipator co-content from the thermodynamic principles, and claims that the concept of maximizing the efficiency of power dissipa- tion (η) for the analysis of metal forming problems is confirmed by extensive microstructural investigations including those of instabilities in a wide range of ma- terials. Motivated by the discussions of the above re- searchers, studies are made further to understand the modeling of dynamic material behavior. According to the DMM, the power, P (per unit vol- ume) absorbed by the work piece material during plastic flow is given by, P = G + J (1) or σ ˙ ε = ˙ ε  0 σ d ˙ ε + σ  0 ˙ ε d σ (2) In the variational procedure, the G content is the work function and the J co-content is a complementary set. From Equation 2, it follows that at any given (ε) and T , the change in J with respect to G yields the well known strain rate sensitivity parameter (m) which is,  ∂ J ∂ G  ε,T =  ∂ (ln σ ) ∂ (ln ˙ ε)  ε,T = m (3) The value of J at a given strain (ε), strain rate (˙ ε) and temperature (T ) is estimated from the flow stress (σ ). If the flow stress obeys power law: σ = K ˙ ε m (4) then, G = P 1 + m (5) J = mP 1 + m (6) The effect of J on the plastic flow of materials can be visualized if the power dissipation capacity of the work piece is expressed in terms of efficiency of power dissipation, η, which is defined η = J J max = 2 J P , (7) where J max = P /2 (8) This parameter (η) is used to create a three dimen- sional map for dissipating the dynamic behavior of the work piece material. These maps require additional information for delineating regions where fracture or defects are most likely to occur. A criterion based on continuum principles as applied to large plastic flow proposed by Ziegler [3] was de- veloped for delineating the regimes of flow instability, according to which instability occurs when, dD d ˙ ε < D ˙ ε (9) where D is the dissipation function. Prasad [2] used D = J in his formulation. It is of interest to examine the condition (9) when D is equated to P , G and J respectively. Case I. When D = P , dP d ˙ ε < P ˙ ε ⇒ ˙ ε ∂σ ∂ ˙ ε + σ<σ ⇒ m < 0 (10) The instability condition preferred by Montheillet, Jonas and Neale [1] is obtained in Equation 10 using the Ziegler’s instability criterion. Case II. When D = G, ∂ G ∂ ˙ ε < G ˙ ε ⇒ σ ˙ ε< G ⇒ P < G ⇒ J < 0 (11) Case III. When D = J ∂ J ∂ ˙ ε < J ˙ ε ⇒ mσ< J ˙ ε ⇒ mP < J (12) When the flow stress obeys the power law, (Equation 4), substituting the value of J given in Equation 6 in Equations 11 and 12 give the condition, m < 0 as ob- tained in Equation 10. If the flow stress does not obey the power law, the condition, J < 0 in Equation 11 holds good even for m > 0, and the condition, J > mP in Equation 12 also holds good for small positive values of m. Quantification of unique range of m for unstable flow for all materials is impossible. Strictly speaking, 0261–8028 C  1999 Kluwer Academic Publishers 1757