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Identification of flow instabilities in the processing maps of AISI 304 stainless steel
Journal of Materials Processing Technology 166 (2005) 268–278
Identification of flow instabilities in the processing maps of AISI 304 stainless steel S.V.S. Narayana Murtya , B. Nageswara Raob,∗ , B.P. Kashyapc a Materials and Metallurgy Group, Vikram Sarabhai Space Centre, Trivandrum 695022, India Structural Analysis and Testing Group, Vikram Sarabhai Space Centre, Trivandrum 695022, India Department of Metallurgical Engineering and Materials Science, Indian Institute of Technology, Bombay, Mumbai 400076, India b
c
Received 27 November 2000; received in revised form 1 September 2004; accepted 1 September 2004
Abstract Among the various existing instability theories, a simple instability condition based on the Ziegler’s continuum principles as applied to large plastic flow, is found to be more appropriate for delineating the regions of unstable metal flow during hot deformation. It can be used to any flow stress versus strain rate curve. This criterion has been validated using the flow stress data of AISI 304 stainless steel with microstructural observations. © 2004 Elsevier B.V. All rights reserved. Keywords: Flow instabilities; Processing maps; AISI 304
1. Introduction The characterization of mechanical behavior of a material by tension testing measures two different types of mechanical properties: strength properties (such as yield strength and ultimate tensile strength) and ductility properties (such as percentage elongation and reduction in area). Similarly, the evaluation of workability involves both the measurement of the resistance to deformation (strength) and determination of the extent of possible plastic deformation before fracture (ductility). Therefore, a complete description of the workability of a material is specified by its flow stress dependence on processing variables (for example, strain, strain rate, preheat temperature and die temperature), its failure behavior and the metallurgical transformations that characterize the alloy system to which it belongs. However, the major emphasis in workability is on measurement and prediction of limit of deformation before fracture. One of the requirements for process modeling is a knowledge of the material flow behavior for defining deformation ∗
Corresponding author. Tel.: +91 471 256 5640; fax: +91 471 270 4134. E-mail address: [email protected] (B. Nageswara Rao).
0924-0136/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2004.09.089
maps that delineate ‘safe’ and ‘non safe’ hot working conditions. These maps show the processing conditions for stable and unstable deformation in the processing space (that is on axes of temperature and strain rate). The ultimate objective is to manufacture components with controlled microstructure and properties, without macro or microstructural defects, on a repeatable basis in a manufacturing environment. The input to generate a processing map is the experimental data of flow stress (σ) as a function of temperature (T), strain rate (˙ε) and strain (ε). As the map generated will be only as good as the input data, it is important to use the accurate, reliable and yet simple experimental technique for generating them. While hot tensile, hot torsion or hot compression techniques may be used for this purpose, hot compression test has decisive advantages over others. First of all, in a compression test on a cylindrical specimen, it is easy to obtain a constant true strain rate using an exponential decay of the cross-head speed, and conduct the test under isothermal conditions. In hot forming of metals at temperatures above recrystallization temperature, the influence of strain on flow stress is insignificant, and the influence of strain rate (i.e. rate of deformation) becomes increasingly important. Conversely, at room temperature (i.e. in cold forming), the effect of strain rate on
S.V.S. Narayana Murty et al. / Journal of Materials Processing Technology 166 (2005) 268–278
flow stress is negligible, and the effect of strain on flow stress (i.e. strain hardening) is most important. The degree of dependency of flow stress on temperature varies considerably among different materials. Therefore, temperature variations in a forming operation can have quite different effects on load requirements and on metal flow for different materials. Bulk metal working processes, such as rolling, forging and extrusion are performed on a variety of machines. The strain rates achievable depend on the speed of the machine, the geometry of the deformation zone and the geometry of the workpiece. To define the processing windows in the desired range of strain rates, workability parameters should be optimum in the specified temperature domain. Frost and Ashby [1] were the first to consider the materials response in the form of deformation mechanism maps. These are plots of normalized stress versus homologous temperature showing the area of dominance of each flow mechanism, calculated using fundamental parameters. The emphasis in these maps has been essentially on the creep mechanisms applicable to lower strain rates and they have proved to be very useful for alloy design. However, mechanical processing is done at strain rates orders of magnitude higher than those observed during creep deformation and therefore involves different microstructural regimes. Considering strain rate as one of the direct variables and temperature as the other, Raj [2] extended the concept of Ashby’s maps to construct a processing map. Raj maps represent the limiting conditions for two damage mechanisms: (a) cavity formation at hard particles in a soft matrix occurring at lower temperatures and higher strain rates, and (b) wedge cracking at grain boundary triple junctions occurring at higher temperatures and lower strain rates. At very high strain rates, a regime representing adiabatic heating was identified. In principle, there is always a region, which may be termed ‘safe’ for processing where neither of the two damage mechanisms nor adiabatic heating occurs. Using an atomistic approach and on the basis of fundamental parameters, processing maps were developed by Raj for pure metals as well as dilute alloys. The evaluated expressions for strain rate are for conditions of steady state and are most valid for pure crystalline materials and simple alloys. The deformation behavior of an alloy under a given set of conditions will depend upon its current microstructure and its prior thermomechanical history. Therefore, the locations of the boundaries on the map may vary. A number of material constants must be determined for the construction of Raj map. In some cases, the response of complex engineering alloys in processing is complicated and is not easily described by simple mechanistic models. Semiatin and Lahoti [3] have correlated the flow softening with material properties, such as the normalized flow softening rate (γ) and the strain rate sensitivity parameter (m), by a parameter, α = (−γ/m), for plane strain compression. On the basis of microstructural observations of the flow localization in titanium and its alloys, they proposed a condition: α > 5 to the occurrence of flow localization in the material during hot deformation [4,5].
269
Recognizing the practical difficulty in making use of deformation, fracture and processing maps based on atomic mechanisms, Prasad et al. [6] suggested the dynamic materials modeling (DMM) approach for describing the material behavior under processing conditions. This approach was reviewed by Gegel et al. [7] and Alexander [8]. Although DMM is essentially a continuum model, its development is based on three major areas: (a) the continuum mechanics of large plastic flow where the material acts as a dissipator of power (and not as a storage element) as described by Ziegler [9]; (b) the general principles of physical systems modeling as described by Wellstead [10], where the power content and co-content concepts for dissipator elements are applied to mechanical, electrical and magnetic systems. The DMM extends these concepts to metallurgical systems; (c) the concepts of irreversible thermodynamics describing the stability and selforganization of chaotic systems proposed by Prigogine [11], where the rate of entropy production is used to characterize the behavior of irreversible phenomena. Rajagopalachary and Kutumbarao [12] have introduced the polar reciprocity model (PRM) adopting the Hill’s associated flow rule of plasticity [13]. In these models, a dimensionless parameter, η called the dissipation efficiency parameter used in DMM and the intrinsic hot workability parameter, ς used in PRM are defined in such a way that they can be easily evaluated from laboratory generated flow stress data. When the material exhibits ideal plastic flow, it was claimed [14] that η and ς sum to unity and the predictions of PRM are identical to those of DMM. The associated flow rule of plasticity, on which the PRM rests, is based on the convexity of the potential surfaces. The harden˙ H increases monotonically to attain a high ing power term W value and ς tends to 1 at very large strain rates where the asso˙D ciated flow rule fails. The dissipative power component, W exhibits in a cusp catastrophe under these conditions and the material shows a macroscopic instability in flow behavior. Thus, it was claimed by Rajagopalachary and Kutumbarao [14] that a ς value approaching 1 is definitely a condition of instability and no separate stability condition on the ς maps is required for delineating the regimes of unstable flow. As per the DMM approach, stability criteria are essential in order to delineate the unstable regions in the η maps. In view of the linear relation between η and ς, stability conditions are essential for ς maps also. Montheillet et al. [15] have examined the dissipator power co-content approach in the dynamic material model (DMM). It is noted from their investigation that the efficiency of power dissipation (η) as derived from the dissipator 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 parameter (m). In his reply to the above aspects, Prasad [16] 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 dissipation (η) for the analysis of metal forming problems is confirmed by extensive microstructural investigations
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including those of instabilities in a wide range of materials. The strain rate sensitivity parameter, m, is one of the most important control parameters, and the physical meaning behind it and its effect on the intrinsic hot workability must be understood. For complex alloy systems, at a given deformation, m varies as a function of strain rate and temperature. Strictly speaking, η and m are dimensionless parameters which are to be evaluated from σ − ε˙ curve of the material at any ε and T. η needs integration of σ with respect to ε˙ whereas m requires differentiation of σ with respect to ε˙ . The strain rate sensitivity parameter, m gives information regarding the change in J with respect to G and does not provide information related to the amount of power partitioned between J and G when it varies with ε˙ . The value of η explicitly expressed in terms of m based on power law assumption: η = 2m/(m + 1) by Prasad et al. may create confusion of the maximization of η and m. If m is function of ε˙ , expressing η as 2m/(m + 1) is incorrect. It is known from the test results on several materials that the maximization of η and m reduce the tendency for flow localization. In fact, all the qualitative stability conditions suggested by Gegel et al. [7], Alexander [8] and Prasad [17] for delineating unstable regions in the processing map of a material, depend upon the strain rate sensitivity parameter, m. Thus, m plays an important role in the development of processing map, by providing the information regarding the allowable temperature and strain rate range. The range of T and ε˙ for stable material flow, which is useful in the development of process control algorithms should be obtained from the nature of any physical quantities, which can be evaluated from the test data. As such, there is no unique instability theory to delineate the regions of unstable flow during hot deformation, which is applicable for all the materials. The designer has to establish a suitable theory on the basis of microstructural observations of the flow localization in the intended materials. The purpose of this study is to examine the applicability of various existing instability theories by considering a test data on AISI 304 stainless steel [18] with microstructural observations.
2. Workability parameters In order to exploit the full potential of process modeling techniques, it is essential to understand the constitutive flow behaviour of a material under processing conditions. Constitutive equations, which related the flow stress in terms of strain, strain rate and temperature: σ = f (ε, ε˙ , T )
(1)
are useful in metal forming problems and are more attractive to designers. Identification of microstructural mechanisms occurring in each temperature–strain rate domain, is essential to estimate the range of the workability parameters from the measured
flow stress data. In order to examine the various existing instability theories for delineating the regions of flow instabilities during hot deformation and recommend stable regions for the processing, the following non-dimensional parameters are determined from the measured flow stress data: The strain rate sensitivity parameter, m=
∂ ln σ ∂ ln ε˙
(2)
The flow-softening rate, γ=
1 ∂σ ∂ ln σ = σ ∂ε ∂ε
(3)
The flow localization parameter for plane strain compression [3–5] α=
−γ m
(4)
The temperature sensitivity of flow stress, s=
1 ∂ ln σ T ∂(1/T )
The efficiency of power dissipation [32], ε˙ P −G 1 J = =2 1− σd˙ε η= Jmax Jmax σ ε˙ 0 The intrinsic hot workability parameter [12], ˙H 2 ε˙ W −1= σd˙ε − 1 ς= ˙ Hmin σ ε˙ 0 W
(5)
(6)
(7)
ε˙ where P (≡ σ ε˙ ) is the power per unit volume, G = 0 σd˙ε σ the dissipator content, J = 0 ε˙ dσ the dissipator co-content, ˙ H = ε˙ σd˙ε Jmax = σ2ε˙ the maximum dissipator co-content, W ε˙ ˙ Hmin = σ ε˙ . the hardening power term and and W 2 3. Material flow instabilities during hot deformation In general, occurrence of microstructural processes in metals and alloys during hot deformation are: dynamic recrystallization (DRX), superplastic deformation, dynamic recovery, wedge cracking, void formation, intercrystalline cracking, prior particle boundary (PPB) cracking and flow instability processes. On the basis of Raj maps, the deformation characteristics of materials are interpreted as follows. In the low temperature (T ≤ 0.25Tm ), high strain rate regime (10–100 s−1 ), void formation occurs at hard particles leading to ductile fracture. In the high temperature (T ≥0.75Tm ), low strain rates (≤10−3 s−1 ) regime, wedge cracking caused by grain boundary sliding occurs (except in superplastic materials in which wedge cracking is at a minimum). In high temperature (Tm ≈ 0.75) and high strain rate regime (10−1 to 10 s−1 ), dynamic recrystallization occurs in low stacking fault energy materials. At intermediate temperatures and strain rates
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dynamic recovery process occurs. At very high strain rates (≥10 s−1 ) there is a possibility for the occurrence of adiabatic shear bands and these lead to flow localization. Out of all the above mechanisms, DRX and superplastic deformation are ‘safe’ mechanisms for hot working while dynamic recovery is preferred for warm working. All other mechanisms either cause microstructural damage or inhomogeneities of varying intensities and hence are to be avoided in the microstructure of the component. This section highlights briefly on the existing instability theories for identifying the temperature–strain rate domains of flow instabilities during hot deformation of materials. 3.1. Flow localization criterion Flow softening has been correlated with material properties by the parameter, for plane strain compression [3–5] α=
−γ m
(8)
should decrease with respect to temperature, in addition to 0 < m ≤ 1 and ∂m/∂T > 0 [21]. 3.3. Instability criterion based on Ziegler’s plastic flow theory Prasad [17] and Kalyan Kumar [22] have developed a criterion for evaluating the regimes of flow instabilities. The criterion is based on the continuum principles as applied to large plastic flow proposed by Ziegler [9] according to which instabilities occur when, D ∂D < ∂˙ε ε˙
∂J J < ∂˙ε ε˙ Assuming the power law nature of stress distribution,
α > 5,
σ = K˙εm ,
has been fixed for flow localization or fracture to occur during hot deformation of materials. 3.2. Gegel’s stability criterion Using the foundations of continuum mechanics, thermodynamics and stability theory, Gegel considered a Lyaponov function L (η, s) and suggested the following conditions for the stable material flow [20]: 0
(10)
∂η <0 ∂(ln ε˙ )
(11)
s≥1
(12)
∂s <0 ∂(ln ε˙ )
(13)
2m <η m+1
(14)
Using Eq. (5) the stability condition (13) can be written in form ⇒
∂m >0 ∂T
they get the condition for instability as m ∂ ln m+1 +m<0 ∂ ln ε˙
(15)
which indicates that the strain rate sensitivity parameter increases with temperature for stable material flow. From conditions (10) to (13), it can be concluded that for stable material flow, the σ − ε˙ curve should be convex and the flow stress
(17)
(18)
(19)
which is found to be incorrect. In the power law stress distribution, the material constant, m in Eq. (18) is independent of ε˙ and hence the first term on the left hand side of Eq. (19) becomes zero and Eq. (19) reduces to m < 0, which is not applicable for any type of σ − ε˙ curve. A simple condition for the metallurgical instability from Eq. (17) in terms of η and m derived in Refs. [23,24] is: 2m < η
From Eq. (11), a relation between η and m for stable material flow can be written as
(16)
where D (˙ε) is the dissipation function at a given temperature. In their criterion, D (˙ε) is equivalent to J co-content, which represents the power dissipation through microstructural changes and the above equation is transformed to
On the basis of microstructural observations in titanium and its alloys, a limit on the workability parameter (9)
271
(20)
Thus, for stable material flow, η < 2m and 0 < m < 1. The instability criterion (20) is valid for any type of σ − ε˙ curve. In the case of power law stress distribution as in Eq. (18), the efficiency of power dissipation, η can be written explicitly in terms of m as η=
2m m+1
(21)
which is always less than 2m for 0 < m < 1 and hence the material flow is stable. There has been much concern in the metal forming community over a number of years about the introduction of the dissipator power co-content J in the Ziegler’s criterion for identifying the temperature–strain rate domains of flow instabilities during hot deformation of materials. It is of interest to examine the condition (16) when D is equated to P, G and J, respectively.
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Case I: When D = P, dP P ∂σ < ⇒ ε˙ +σ<σ ⇒m<0 d˙ε ε˙ ∂˙ε
(22)
The instability condition preferred by Montheillet et al. [15] is obtained in Eq. (22) using the Ziegler’s instability criterion. Case II: When D = G, ∂G G < ⇒ σ ε˙ < G ⇒ P < G ⇒ J < 0 ∂˙ε ε˙
(23)
Case III: When D = J ∂J J J < ⇒ mσ < ⇒ mP < J ∂˙ε ε˙ ε˙
(24)
When the flow stress obeys the power law (Eq. (18)), substituting the value of J given in Eq. (21) in Eqs. (23) and (24) give the condition, m < 0 as obtained in Eq. (22). If the flow stress does not obey the power law, the condition, J < 0 in Eq. (23) holds good even for m > 0, and the condition, J > mP in Eq. (24) 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, J/P (≡ η/2) and m are non-dimensional parameters. J needs integration of σ with respect to ε˙ whereas m requires differentiation of σ with respect to ε˙ . J accounts for the behaviour of the material up to ε˙ (global) whereas m gives the information at ε˙ (local). It is more appropriate to use the condition in Eq. (24), because, when η < 0, Eq. (24) gives m < 0. When η > 0, and m < 0 or m is a small positive value, the condition in Eq. (24) yields comparatively larger region of instability in the processing map. Thus η and m are important parameters to use Eq. (24) for identifying the domains of flow instabilities. At any constant ε and T, the maximum value of η with respect to ε˙ becomes 2m/(m + 1) only when ∂m/∂(ln ε˙ ) < 0. The value of η expressed explicitly in terms of m based on power law assumption: η = 2m/(m + 1) in the DMM of Prasad [6] may create confusion on the maximization of η and m. If m is a function of ε˙ , expressing η as 2m/(m + 1) is incorrect. It is known from the test results on several materials that the maximization of η or m will reduce the tendency for flow localization. But the condition in Eq. (20) or (24) for delineating flow instabilities involves η and m. Relying on the positive values of m may not always ensure stable material flow for all materials. The parameters η and m are important in delineating the regions of flow instabilities. 3.4. Alexander’s stability criterion Using the Lyaponov function stability criteria, Alexander considered a Lyaponov function L (m, s) and suggested the following conditions for the stable material flow [8]. 0
(25)
∂m <0 ∂(ln ε˙ )
(26)
s≥1
(27)
∂s <0 ∂(ln ε˙ )
(28)
The above conditions indicate that for stable material flow, the σ − ε˙ curve should be convex, the flow stress should decrease with respect to T, and m should decrease with ε˙ and increase with T. 3.5. Instability criterion in the polar reciprocity model The associated flow rule of plasticity on which the polar reciprocity model (PRM) rests is based on the convexity of ˙H the potential surfaces [13]. The hardening power term, W increases monotonically to attain a high value and the intrinsic workability parameter, ς as derived in Eq. (9) tends to 1 at very large strain rates where the associated flow rule ˙ H and W ˙ D are referred to as G and J, fails. In the DMM, W respectively. One can write a relation between η and ς as η+ς =1
(29)
It was claimed in Ref. [14] that a ς value approaching 1 is definitely a condition of instability and no separate stability criterion on the ς maps is required for delineating the regimes of unstable flow. As per the relation (29), when the value of ς approaches unity, η approaches zero. This implies J = 0 and G = P. In this case all of the power would be dissipated by heat and this could lead to plastic instability by a continuum process such as adiabatic shearing. The qualitative stability condition, ς → 1 proposed in PRM is inadequate to delineate all the regions of unstable flow. The above discussion lead to a qualitative check for the stable material flow by η>0
or ς < 1
(30)
As per the DMM approach, stability criteria are essential in order to delineate the unstable regions in the η maps. In view of the linear relation between η and ς, as given by Eq. (29), additional stability conditions are essential for ς maps also.
4. Results and discussion Applicability of various instability theories presented in the preceding sections is examined using the flow stress data of AISI 304 stainless steel with microstructural observations. Venugopal, Mannan and Prasad [18] have discussed the hot deformation characteristics of AISI 304 stainless steel in the temperature range 600–1250 ◦ C and in the strain rate range of 0.001–100 s−1 . They presented the data on flow stress (σ) with strain (ε), strain rate (˙ε) and temperature (T) generated from hot compression testing of solid cylinders using servohydraulic testing machine capable of imposing constant true strain rates on the specimen. The test data is corrected for adiabatic temperature rise. The specimens were compressed to 50% of their initial height and the load–stroke curves obtained in the hot compression were converted into true stress–true
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plastic strain curves by subtracting the elastic portion of the strain and using the standard equations for the true stress and true strain calculations. One specimen was tested to obtain each result. Interpretation of the experimental data for the development of processing maps depends upon the accurate evaluation of the workability parameters such as strain rate sensitivity parameter (m), the efficiency of power dissipation (η), the normalized flow softening rate (γ) etc. While carrying out the numerical computation, the data reported in Ref. [18] is initially transformed into logarithmic scale, in order to reduce the tenth order magnitude of strain rate to first order, to avoid any excessive round off error. With this transformation, the first derivative of the spline fit, gives directly the strain rate sensitivity parameter, m at the generated intermediate data points. The normalized flow softening rate (γ) is obtained by transforming initially the flow stress data in to natural logarithmic scale and finding it directly from the first derivative of the spline fit to the data on ln σ – ε data, at the specified ε˙ and T. Following Ref. [20] the efficiency of power dissipation (η) is obtained. The value of the workability parameter (α) is obtained from Eq. (8) by substituting the value of γ and m. The temperature sensitivity of flow stress (s) is obtained by finding the first derivative of the spline fit to the data on ln σ – ln T. In Gegel’s stability criterion as well as Alexander’s stability criterion, the derivatives of m, η and s with respect to ln ε˙ are obtained by finding the first derivatives of the spline fit to the m – log ε˙ , η – log ε˙ and s – log ε˙ data. By subtracting the values of η from unity, the workability parameter, ζ in the polar reciprocity model (PRM) is obtained. Using these workability parameters, the instability parameters for unstable material flow for the described instability theories are defined as follows. The instability parameter based on the flow localization concepts: ξ1 = 1 −
α <0 5
(31)
The instability parameters based on Gegel’s stability criterion are: ξ2 = m < 0 ξ3 = −
∂η = (1 + m)η − 2m < 0 ∂ ln ε˙
ξ4 = s − 1 < 0 ξ5 = −
∂s <0 ∂ ln ε˙
(32) (33) (34) (35)
In the case of Alexander’s criterion, ξ6 = m < 0 ξ7 = −
∂m <0 ∂ ln ε˙
ξ8 = s − 1 < 0
(36) (37) (38)
ξ9 = −
∂s <0 ∂ ln ε˙
273
(39)
The instability parameter based on the simplified metallurgical stability condition (20): ξ10 =
2m −1<0 η
(40)
which is valid for m>0
(41)
The instability parameter based on PRM (Eq. (30)), ξ11 = 1 − ζ < 0
(42)
The curves ξ i = 0 bifurcate the stable and unstable regions in the processing maps for the above five criteria considered. The regions where ξ i < 0 in the map correspond to the unstable flow (microstructural instabilities) in the material. From the determined values of the workability parameters, the instability parameters ξ i from Eqs. (31)–(42) are calculated for a strain of 0.5 at different values of ε˙ and T. The power law assumption in Eq. (18) to represent the dynamic constitutive behaviour in continuum mechanics of large plastic flow for viscoplastic materials is generally considered for the purpose of simplifying complex mathematical analysis procedures to understand the physics of the problem. This assumption was followed for analyzing the hot deformation behaviour [18]. With this assumption, the efficiency of power dissipation, (η) can be expressed explicitly in terms of the strain rate sensitivity parameter (m) as given by Eq. (21). For complicated alloy systems, the flow stress (σ) with respect to ε˙ does not obey power law and the computation of η based on power law assumption (Eq. (18)) gives erroneous results [19]. If the material obeys power law for specified strain and temperature, the strain rate sensitivity parameter (m) is independent of ε˙ and the derivative of m with respect to ε˙ becomes zero. Based on the power law assumption, the instability parameter defined by Prasad [17]: ∂ ln(m/(m + 1))/∂ ln ε˙ + m reduces to ξ = m and ξ = m < 0 becomes the instability condition. It can be verified easily from the simplified instability condition (20) by using Eq. (21) for η. Fig. 1 shows the variation of flow stress (σ) versus strain rate (˙ε) at different temperatures at a strain of 0.5. The stress–strain curves at different temperatures and strain rates presented in Ref. [18] indicate strain hardening behaviour and most of the curves show steady state region beyond a strain of 0.4. It should be noted that, when strain rate insensitivity occurs at large strains, J increases to a certain extent and remains constant, whereas G increases with ε˙ . In such a situation, J is small compared to the total power P and is not equal to zero, whereas m → 0 which implies η > 2 m and the flow is unstable. This may be the reason why most of the materials during hot deformation show instabilities at large strain rates. The idea of developing various theories of flow instabilities is to trace the limits of the strain rate and temper-
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Fig. 1. Variation of flow stress with strain rate for AISI 304 stainless steel at a strain of 0.5.
ature at which no flow localization or fracture occurs during hot deformation. Table 1 gives the instability parameters defined in Eqs. (31)–(42) for the microstructural observations reported on AISI 304 stainless steel [18] at different temperatures and strain rates at a strain of 0.5. Table 2 gives the comparison of
the various existing instability theories with the reported microstructural observations. The instability parameters ξ 1 and ξ 11 defined in Eqs. (31) and (42) based on the flow localization concept and the polar reciprocity model correctly predicted the stable/unstable microstructural observations of deformed compression specimens on AISI 304 stainless steel. Though
Table 1 Instability parameter ξ i defined in Eqs. (31)–(42) for the microstructural observation reported on AISI 304 stainless steel [18] at different temperatures and strain rates at a strain of 0.5 T (◦ C)
Strain rate ε˙ (s−1 )
m
ξ1
ξ2 , ξ6
ξ3
ξ4 , ξ8
ξ5 , ξ9
ξ7
ξ 10
ξ 11
600 600 1000 1100 1100 1200 1200
0.001 10 0.001 0.01 100 0.001 0.1
→0 <0 0.277 0.220 0.028 0.137 0.237
>0 >0 >0>0 >0 >0 >0 >0
→0 <0 >0 >0 >0 >0 >0
→0 >0 →0 <0 >0 →0 →0
<0 <0 >0 >0 >0 >0 >0
>0 >0 >0 <0 <0 >0 <0
→0 <0 →0 <0 →0 →0 >0
→0 >0 >0 >0 <0 >0 >0
→0 >0 >0 >0 >0 >0 >0
Table 2 Comparison of the various existing instability theories with the reported microstructural observations on AISI 304 stainless steel [18] T (◦ C) Strain rate ε˙ (s−1 )
Flow localization theory Eq. (31)
Gegel’s instability theory Eqs. (32)–(35)
Alexander’s instability theory Eqs. (36)–(39)
Simplified instability Instability theory theory Eq. (40) due to PRM Eq. (42)
Microstructural observations
600 600 1000 1100 1100 1200 1200
Unstable Unstable Stable Stable Stable Stable Stable
Unstable Unstable Unstable Unstable Unstable Unstable Unstable
Unstable Unstable Unstable Unstable Unstable Unstable Unstable
Unstable Unstable Stable Stable Unstable Stable Stable
Flow localization Flow localization Dynamic recrystallization Dynamic recrystallization Dynamic recovery Dynamic recrystallization Dynamic recrystallization
0.001 1.0 0.001 0.01 100 0.001 0.1
Unstable Unstable Stable Stable Stable Stable Stable
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275
Fig. 2. Processing map for AISI 304 stainless steel at a strain of 0.5 with the microstructural observations, ( ) stable and (×) unstable [18].
the condition (40) indicated the material flow at 1100 ◦ C and 100 s−1 to be unstable, the microstructural observations indicated dynamic recovery. The value of the strain rate sensitivity parameter, m at the temperature and strain rate of 1100 ◦ C and 100 s−1 is noted as 0.028 and the value of the efficiency of power dissipation, η, is about 0.1, which is greater than 2m. Hence, the predictions using the instability condition (40) based on the Ziegler’s continuum principles as applied to large plastic flow become conservative which will give more confidence to the process designer. Fig. 2 shows the marked regions of instability using the condition (40),
which is based on the Ziegler’s instability criterion, along with the microstructural observations reported in Ref. [18]. Fig. 3 shows the contour map for strain rate sensitivity parameter, m and the contour map for efficiency parameter, η is shown in Fig. 4. The hatched region in Fig. 3 corresponds to the condition in Eq. (22) when the dissipation function is considered as the total power (D = P). The hatched region in Fig. 4 corresponds to the condition in Eq. (23) when the dissipative function is considered as the dissipator power content (D = G). The instability condition (42) also gives the same hatched region as given in Fig. 4 for the unstable region, be-
Fig. 3. Contour map for strain rate sensitivity parameter (m) for AISI 304 stainless steel at a strain of 0.5.
276
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Fig. 4. Contour map for efficiency of power dissipation (η) for AISI 304 stainless steel at a strain of 0.5.
cause, η < 0 implies ς > 1. The hatched regions in Figs. 3 and 4 are found to be within the unstable regions identified by the instability condition (40). When η > 0 and m is a small positive value, the condition (40) predicts comparatively larger region of instability in the processing map. As such there is no approximation or fixing a value based on experiments is involved in this instability condition. Uncertainties if any on the boundaries between the ‘stable’ and ‘unstable’ regions in the processing maps, will be mainly due to the flow stress data of the material. It should be noted that the constant value of
5 in the instability condition (31) has been fixed for flow localization or fracture to occur during hot deformation of the material based on microstructural observations in titanium and its alloys. Its validity for other materials will be known only after examining the microstructural observations of the deformed compression specimens. To demonstrate the above aspects further, a comparative study is made on the flow localization concept and the simple instability condition, by utilization the flow stress data of commercial grade titanium alloy Ti–6Al–4V [25]. Fig. 5
Fig. 5. Contour map based on the flow localization concept (Eq. (31)) for commercial grade Ti–6Al–4V at a strain of 0.5 with microstructural observations of Ref. [25], ( ) stable; (×) unstable.
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277
Fig. 6. Contour map of the simple instability condition (Eq. (40)) for commercial grade Ti–6Al–4V at a strain of 0.5 with microstructural observations of Ref. [25], ( ) stable; (×) unstable.
shows the condition for flow localization (31) which disagrees with the predicted locus for microstructural observations of commercial grade titanium alloy Ti–6Al–4V [25], because this condition predicts experimentally known unstable regions as stable. However, these microstructural observations of flow instabilities are found to be in the unstable regions of Fig. 6, which validates the simple instability condition (40). This comparative study indicates inconsistency in the flow localization concept (31) even for titanium alloys for which it was proposed. MATLAB (Math Works Inc., USA) software was utilized for generating the 2D contour maps. It is noted from Figs. 3 and 4 for AISI 304 stainless steel that the maximum value of m is 0.277 and the maximum value of η is 0.434 which occur at T = 1000 ◦ C and ε˙ = 0.001 s−1 . From Figs. 2–4, it is preferable to conduct the hot working in the temperature range of 900–1200 ◦ C and in the strain rate range of 0.001–0.1 s−1 . In an industrial practice, highest possible strain rates and lowest possible stresses are preferred from productivity viewpoint. Due to this reason, high strain rate region in which high values of m and η are present should be selected for bulk working operations and the lower strain rate regions for secondary metal working operations. Since, these optimum strain rates are too low for commercial processing, the limits of flow instability will have to be established so that the processing rates are pushed up as high as possible. Taking advantage of the effect of prior processing history on the processing map, connected processes may be designed. Each step of the process may refine the grain structure and expand the workability domain for the next step of processing while moving it to higher strain rates without the onset of instabilities to facilitate higher productivity.
5. Concluding remarks The various existing instability theories were reviewed. A simple instability condition based on the Ziegler’s continuum principles as applied to large plastic flow, is found to be more appropriate for delineating the regions of unstable metal flow during hot deformation. It can be used for any type of the flow stress versus strain rate curve. This criterion has been validated using the flow stress data of AISI 304 stainless steel with microstructural observations. Since there is no unique stability theory to delineate the regions of unstable flow during hot deformation, which is applicable for all materials, the designer has to establish a suitable theory on the basis of microstructural observations of the flow localization in the intended materials.
References [1] H.J. Frost, M.F. Ashby, Deformation machanism maps, in: The Plasticity and Creep of Metals and Ceramics, Pergamon Press, London, 1982. [2] R. Raj, Development of a processing map for use in warm forming and hot forming process, Metall. Trans. A 12 (1981) 1089–1097. [3] S.L. Semiatin, G.D. Lahoti, The occurrence of shear bands in isothermal hot forging, Metall. Trans. A 13 (1992) 275–288. [4] G.E. Dieter, Evaluation of workability–introduction, in: Metals Handbook, vol. 14, ASM International, Ohio, 1988, pp. 363–372. [5] S.L. Semiatin, J.J. Jonas, Formability and Workability of Metals: Plastic Instability and Flow Localization, American Society for Metals, Metals Park, Ohio, USA, 1984. [6] Y.V.R.K. Prasad, H.L. Gegel, S.M. Doraivelu, J.C. Malas, J.T. Morgan, K.A. Lark, D.R. Barker, Modelling of dynamic material behaviour in hot deformation: forging of Ti-6242, Metall. Trans. A 15 (1984) 1883–1892.
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[7] H.L. Gegel, J.C. Malas, S.M. Doraivelu, V.A. Shende, Modelling techniques used in forging process design, in: Metals Handbook, Forming and Forging, vol. 14, ASM International, Ohio, 1988. pp. 417–438. [8] J.M. Alexander, Mapping dynamic material behaviour, in: J.G. Lenard (Ed.), Modelling of Hot Deformation of Steels, SpringerVerlag, Berlin, 1989, pp. 101–115. [9] H. Ziegler, Some extremum principles in irreversible thermodynamics, with application to continuum mechanics, Progress in Solid Mechanics, 4, Wiley, New York, 1963, 93–193. [10] P.E. Wellstead, Introduction to Physical Systems Modelling, Academic Press, London, UK, 1979. [11] I. Prigogine, Time, structure and fluctuations, Science 201 (1978) 777–785. [12] T. Rajagopalachary, V.V. Kutumbararao, Recent developments in the modelling of metallic materials, Scripta Mater. 35 (1996) 311–316. [13] R. Hill, Constitutive dual potentials in classical plasticity, J. Mech. Phys. Solids 35 (1987) 23–33. [14] V.V. Kutumbarao, T. Rajagopalachary, Intrinsic hot workability map for a titanium alloy IMI 685, Bull. Mater. Sci. 19 (1996) 677– 698. [15] F. Montheillet, J.J. Jonas, K.W. Neale, Modelling of dynamic material behaviour—a critical evaluation of dissipator power cocontent approach, Metall. Trans. A 27 (1996) 232–235. [16] Y.V.R.K. Prasad, Author’s reply for modelling of dynamic material behaviour—a critical evaluation of dissipator power cocontent approach, Metall. Trans. A 27 (1996) 235–237.
[17] Y.V.R.K. Prasad, Recent advances in the science of mechanical processing, Ind. J. Technol. 28 (1990) 435–451. [18] S. Venugopal, S.L. Mannan, Y.V.R.K. Prasad, Processing maps for hot working of commercial grade AISI 304, Mater. Sci. Eng. A 117 (1994) 143–149. [19] S.V.S. Narayana Murty, M.S. Sarma, B. Nageswara Rao, On the evaluation of efficiency parameter in processing map, Metall. Trans. A 28 (1997) 1581–1582. [20] H.L. Gegel, Synthesis of atomistics and continuum modelling to describe microstructure, in: R.J. Arsenault, J.R. Beeler, D.M. Easterling (Eds.), Computer Simulation in Material Science, ASM International, Metals Park, Ohio, USA, 1987, pp. 291–344. [21] S.V.S. Narayana Murty, B. Nageswara Rao, On the Gegel’s stability criterion in processing maps, Metall. Trans. A 28 (1997) 2170–2173. [22] A.K.S. Kalyan Kumar, Criteria for predicting metallurgical instabilities in processing maps, MS Thesis, Indian Institute of Science, Bangalore, India, 1987. [23] S.V.S. Narayana Murty, B. Nageswara Rao, On the development of instability criteria during hot working with reference to IN 718, Mater. Sci. Eng. A 254 (1998) 76–82. [24] S.V.S. Narayana Murty, B. Nageswara Rao, Instability map for hot working of 6061A1–10vol% Al2 O3 metal matrix composite, J. Phys. D: Appl. Phys. 31 (1998) 3306–3311. [25] T. Seshacharyulu, S.C. Medeiros, W.G. Frazier, Y.V.R.K. Prasad, Hot working of commercial Ti–6Al–4V with an equiaxed ␣– microstructure: material modelling considerations, Mater. Sci. Eng. A 284 (2000) 184–194.
Identification of flow instabilities in the processing maps of AISI 304 stainless steel S.V.S. Narayana Murtya , B. Nageswara Raob,∗ , B.P. Kashyapc a Materials and Metallurgy Group, Vikram Sarabhai Space Centre, Trivandrum 695022, India Structural Analysis and Testing Group, Vikram Sarabhai Space Centre, Trivandrum 695022, India Department of Metallurgical Engineering and Materials Science, Indian Institute of Technology, Bombay, Mumbai 400076, India b
c
Received 27 November 2000; received in revised form 1 September 2004; accepted 1 September 2004
Abstract Among the various existing instability theories, a simple instability condition based on the Ziegler’s continuum principles as applied to large plastic flow, is found to be more appropriate for delineating the regions of unstable metal flow during hot deformation. It can be used to any flow stress versus strain rate curve. This criterion has been validated using the flow stress data of AISI 304 stainless steel with microstructural observations. © 2004 Elsevier B.V. All rights reserved. Keywords: Flow instabilities; Processing maps; AISI 304
1. Introduction The characterization of mechanical behavior of a material by tension testing measures two different types of mechanical properties: strength properties (such as yield strength and ultimate tensile strength) and ductility properties (such as percentage elongation and reduction in area). Similarly, the evaluation of workability involves both the measurement of the resistance to deformation (strength) and determination of the extent of possible plastic deformation before fracture (ductility). Therefore, a complete description of the workability of a material is specified by its flow stress dependence on processing variables (for example, strain, strain rate, preheat temperature and die temperature), its failure behavior and the metallurgical transformations that characterize the alloy system to which it belongs. However, the major emphasis in workability is on measurement and prediction of limit of deformation before fracture. One of the requirements for process modeling is a knowledge of the material flow behavior for defining deformation ∗
Corresponding author. Tel.: +91 471 256 5640; fax: +91 471 270 4134. E-mail address: [email protected] (B. Nageswara Rao).
0924-0136/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2004.09.089
maps that delineate ‘safe’ and ‘non safe’ hot working conditions. These maps show the processing conditions for stable and unstable deformation in the processing space (that is on axes of temperature and strain rate). The ultimate objective is to manufacture components with controlled microstructure and properties, without macro or microstructural defects, on a repeatable basis in a manufacturing environment. The input to generate a processing map is the experimental data of flow stress (σ) as a function of temperature (T), strain rate (˙ε) and strain (ε). As the map generated will be only as good as the input data, it is important to use the accurate, reliable and yet simple experimental technique for generating them. While hot tensile, hot torsion or hot compression techniques may be used for this purpose, hot compression test has decisive advantages over others. First of all, in a compression test on a cylindrical specimen, it is easy to obtain a constant true strain rate using an exponential decay of the cross-head speed, and conduct the test under isothermal conditions. In hot forming of metals at temperatures above recrystallization temperature, the influence of strain on flow stress is insignificant, and the influence of strain rate (i.e. rate of deformation) becomes increasingly important. Conversely, at room temperature (i.e. in cold forming), the effect of strain rate on
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flow stress is negligible, and the effect of strain on flow stress (i.e. strain hardening) is most important. The degree of dependency of flow stress on temperature varies considerably among different materials. Therefore, temperature variations in a forming operation can have quite different effects on load requirements and on metal flow for different materials. Bulk metal working processes, such as rolling, forging and extrusion are performed on a variety of machines. The strain rates achievable depend on the speed of the machine, the geometry of the deformation zone and the geometry of the workpiece. To define the processing windows in the desired range of strain rates, workability parameters should be optimum in the specified temperature domain. Frost and Ashby [1] were the first to consider the materials response in the form of deformation mechanism maps. These are plots of normalized stress versus homologous temperature showing the area of dominance of each flow mechanism, calculated using fundamental parameters. The emphasis in these maps has been essentially on the creep mechanisms applicable to lower strain rates and they have proved to be very useful for alloy design. However, mechanical processing is done at strain rates orders of magnitude higher than those observed during creep deformation and therefore involves different microstructural regimes. Considering strain rate as one of the direct variables and temperature as the other, Raj [2] extended the concept of Ashby’s maps to construct a processing map. Raj maps represent the limiting conditions for two damage mechanisms: (a) cavity formation at hard particles in a soft matrix occurring at lower temperatures and higher strain rates, and (b) wedge cracking at grain boundary triple junctions occurring at higher temperatures and lower strain rates. At very high strain rates, a regime representing adiabatic heating was identified. In principle, there is always a region, which may be termed ‘safe’ for processing where neither of the two damage mechanisms nor adiabatic heating occurs. Using an atomistic approach and on the basis of fundamental parameters, processing maps were developed by Raj for pure metals as well as dilute alloys. The evaluated expressions for strain rate are for conditions of steady state and are most valid for pure crystalline materials and simple alloys. The deformation behavior of an alloy under a given set of conditions will depend upon its current microstructure and its prior thermomechanical history. Therefore, the locations of the boundaries on the map may vary. A number of material constants must be determined for the construction of Raj map. In some cases, the response of complex engineering alloys in processing is complicated and is not easily described by simple mechanistic models. Semiatin and Lahoti [3] have correlated the flow softening with material properties, such as the normalized flow softening rate (γ) and the strain rate sensitivity parameter (m), by a parameter, α = (−γ/m), for plane strain compression. On the basis of microstructural observations of the flow localization in titanium and its alloys, they proposed a condition: α > 5 to the occurrence of flow localization in the material during hot deformation [4,5].
269
Recognizing the practical difficulty in making use of deformation, fracture and processing maps based on atomic mechanisms, Prasad et al. [6] suggested the dynamic materials modeling (DMM) approach for describing the material behavior under processing conditions. This approach was reviewed by Gegel et al. [7] and Alexander [8]. Although DMM is essentially a continuum model, its development is based on three major areas: (a) the continuum mechanics of large plastic flow where the material acts as a dissipator of power (and not as a storage element) as described by Ziegler [9]; (b) the general principles of physical systems modeling as described by Wellstead [10], where the power content and co-content concepts for dissipator elements are applied to mechanical, electrical and magnetic systems. The DMM extends these concepts to metallurgical systems; (c) the concepts of irreversible thermodynamics describing the stability and selforganization of chaotic systems proposed by Prigogine [11], where the rate of entropy production is used to characterize the behavior of irreversible phenomena. Rajagopalachary and Kutumbarao [12] have introduced the polar reciprocity model (PRM) adopting the Hill’s associated flow rule of plasticity [13]. In these models, a dimensionless parameter, η called the dissipation efficiency parameter used in DMM and the intrinsic hot workability parameter, ς used in PRM are defined in such a way that they can be easily evaluated from laboratory generated flow stress data. When the material exhibits ideal plastic flow, it was claimed [14] that η and ς sum to unity and the predictions of PRM are identical to those of DMM. The associated flow rule of plasticity, on which the PRM rests, is based on the convexity of the potential surfaces. The harden˙ H increases monotonically to attain a high ing power term W value and ς tends to 1 at very large strain rates where the asso˙D ciated flow rule fails. The dissipative power component, W exhibits in a cusp catastrophe under these conditions and the material shows a macroscopic instability in flow behavior. Thus, it was claimed by Rajagopalachary and Kutumbarao [14] that a ς value approaching 1 is definitely a condition of instability and no separate stability condition on the ς maps is required for delineating the regimes of unstable flow. As per the DMM approach, stability criteria are essential in order to delineate the unstable regions in the η maps. In view of the linear relation between η and ς, stability conditions are essential for ς maps also. Montheillet et al. [15] have examined the dissipator power co-content approach in the dynamic material model (DMM). It is noted from their investigation that the efficiency of power dissipation (η) as derived from the dissipator 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 parameter (m). In his reply to the above aspects, Prasad [16] 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 dissipation (η) for the analysis of metal forming problems is confirmed by extensive microstructural investigations
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including those of instabilities in a wide range of materials. The strain rate sensitivity parameter, m, is one of the most important control parameters, and the physical meaning behind it and its effect on the intrinsic hot workability must be understood. For complex alloy systems, at a given deformation, m varies as a function of strain rate and temperature. Strictly speaking, η and m are dimensionless parameters which are to be evaluated from σ − ε˙ curve of the material at any ε and T. η needs integration of σ with respect to ε˙ whereas m requires differentiation of σ with respect to ε˙ . The strain rate sensitivity parameter, m gives information regarding the change in J with respect to G and does not provide information related to the amount of power partitioned between J and G when it varies with ε˙ . The value of η explicitly expressed in terms of m based on power law assumption: η = 2m/(m + 1) by Prasad et al. may create confusion of the maximization of η and m. If m is function of ε˙ , expressing η as 2m/(m + 1) is incorrect. It is known from the test results on several materials that the maximization of η and m reduce the tendency for flow localization. In fact, all the qualitative stability conditions suggested by Gegel et al. [7], Alexander [8] and Prasad [17] for delineating unstable regions in the processing map of a material, depend upon the strain rate sensitivity parameter, m. Thus, m plays an important role in the development of processing map, by providing the information regarding the allowable temperature and strain rate range. The range of T and ε˙ for stable material flow, which is useful in the development of process control algorithms should be obtained from the nature of any physical quantities, which can be evaluated from the test data. As such, there is no unique instability theory to delineate the regions of unstable flow during hot deformation, which is applicable for all the materials. The designer has to establish a suitable theory on the basis of microstructural observations of the flow localization in the intended materials. The purpose of this study is to examine the applicability of various existing instability theories by considering a test data on AISI 304 stainless steel [18] with microstructural observations.
2. Workability parameters In order to exploit the full potential of process modeling techniques, it is essential to understand the constitutive flow behaviour of a material under processing conditions. Constitutive equations, which related the flow stress in terms of strain, strain rate and temperature: σ = f (ε, ε˙ , T )
(1)
are useful in metal forming problems and are more attractive to designers. Identification of microstructural mechanisms occurring in each temperature–strain rate domain, is essential to estimate the range of the workability parameters from the measured
flow stress data. In order to examine the various existing instability theories for delineating the regions of flow instabilities during hot deformation and recommend stable regions for the processing, the following non-dimensional parameters are determined from the measured flow stress data: The strain rate sensitivity parameter, m=
∂ ln σ ∂ ln ε˙
(2)
The flow-softening rate, γ=
1 ∂σ ∂ ln σ = σ ∂ε ∂ε
(3)
The flow localization parameter for plane strain compression [3–5] α=
−γ m
(4)
The temperature sensitivity of flow stress, s=
1 ∂ ln σ T ∂(1/T )
The efficiency of power dissipation [32], ε˙ P −G 1 J = =2 1− σd˙ε η= Jmax Jmax σ ε˙ 0 The intrinsic hot workability parameter [12], ˙H 2 ε˙ W −1= σd˙ε − 1 ς= ˙ Hmin σ ε˙ 0 W
(5)
(6)
(7)
ε˙ where P (≡ σ ε˙ ) is the power per unit volume, G = 0 σd˙ε σ the dissipator content, J = 0 ε˙ dσ the dissipator co-content, ˙ H = ε˙ σd˙ε Jmax = σ2ε˙ the maximum dissipator co-content, W ε˙ ˙ Hmin = σ ε˙ . the hardening power term and and W 2 3. Material flow instabilities during hot deformation In general, occurrence of microstructural processes in metals and alloys during hot deformation are: dynamic recrystallization (DRX), superplastic deformation, dynamic recovery, wedge cracking, void formation, intercrystalline cracking, prior particle boundary (PPB) cracking and flow instability processes. On the basis of Raj maps, the deformation characteristics of materials are interpreted as follows. In the low temperature (T ≤ 0.25Tm ), high strain rate regime (10–100 s−1 ), void formation occurs at hard particles leading to ductile fracture. In the high temperature (T ≥0.75Tm ), low strain rates (≤10−3 s−1 ) regime, wedge cracking caused by grain boundary sliding occurs (except in superplastic materials in which wedge cracking is at a minimum). In high temperature (Tm ≈ 0.75) and high strain rate regime (10−1 to 10 s−1 ), dynamic recrystallization occurs in low stacking fault energy materials. At intermediate temperatures and strain rates
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dynamic recovery process occurs. At very high strain rates (≥10 s−1 ) there is a possibility for the occurrence of adiabatic shear bands and these lead to flow localization. Out of all the above mechanisms, DRX and superplastic deformation are ‘safe’ mechanisms for hot working while dynamic recovery is preferred for warm working. All other mechanisms either cause microstructural damage or inhomogeneities of varying intensities and hence are to be avoided in the microstructure of the component. This section highlights briefly on the existing instability theories for identifying the temperature–strain rate domains of flow instabilities during hot deformation of materials. 3.1. Flow localization criterion Flow softening has been correlated with material properties by the parameter, for plane strain compression [3–5] α=
−γ m
(8)
should decrease with respect to temperature, in addition to 0 < m ≤ 1 and ∂m/∂T > 0 [21]. 3.3. Instability criterion based on Ziegler’s plastic flow theory Prasad [17] and Kalyan Kumar [22] have developed a criterion for evaluating the regimes of flow instabilities. The criterion is based on the continuum principles as applied to large plastic flow proposed by Ziegler [9] according to which instabilities occur when, D ∂D < ∂˙ε ε˙
∂J J < ∂˙ε ε˙ Assuming the power law nature of stress distribution,
α > 5,
σ = K˙εm ,
has been fixed for flow localization or fracture to occur during hot deformation of materials. 3.2. Gegel’s stability criterion Using the foundations of continuum mechanics, thermodynamics and stability theory, Gegel considered a Lyaponov function L (η, s) and suggested the following conditions for the stable material flow [20]: 0
(10)
∂η <0 ∂(ln ε˙ )
(11)
s≥1
(12)
∂s <0 ∂(ln ε˙ )
(13)
2m <η m+1
(14)
Using Eq. (5) the stability condition (13) can be written in form ⇒
∂m >0 ∂T
they get the condition for instability as m ∂ ln m+1 +m<0 ∂ ln ε˙
(15)
which indicates that the strain rate sensitivity parameter increases with temperature for stable material flow. From conditions (10) to (13), it can be concluded that for stable material flow, the σ − ε˙ curve should be convex and the flow stress
(17)
(18)
(19)
which is found to be incorrect. In the power law stress distribution, the material constant, m in Eq. (18) is independent of ε˙ and hence the first term on the left hand side of Eq. (19) becomes zero and Eq. (19) reduces to m < 0, which is not applicable for any type of σ − ε˙ curve. A simple condition for the metallurgical instability from Eq. (17) in terms of η and m derived in Refs. [23,24] is: 2m < η
From Eq. (11), a relation between η and m for stable material flow can be written as
(16)
where D (˙ε) is the dissipation function at a given temperature. In their criterion, D (˙ε) is equivalent to J co-content, which represents the power dissipation through microstructural changes and the above equation is transformed to
On the basis of microstructural observations in titanium and its alloys, a limit on the workability parameter (9)
271
(20)
Thus, for stable material flow, η < 2m and 0 < m < 1. The instability criterion (20) is valid for any type of σ − ε˙ curve. In the case of power law stress distribution as in Eq. (18), the efficiency of power dissipation, η can be written explicitly in terms of m as η=
2m m+1
(21)
which is always less than 2m for 0 < m < 1 and hence the material flow is stable. There has been much concern in the metal forming community over a number of years about the introduction of the dissipator power co-content J in the Ziegler’s criterion for identifying the temperature–strain rate domains of flow instabilities during hot deformation of materials. It is of interest to examine the condition (16) when D is equated to P, G and J, respectively.
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Case I: When D = P, dP P ∂σ < ⇒ ε˙ +σ<σ ⇒m<0 d˙ε ε˙ ∂˙ε
(22)
The instability condition preferred by Montheillet et al. [15] is obtained in Eq. (22) using the Ziegler’s instability criterion. Case II: When D = G, ∂G G < ⇒ σ ε˙ < G ⇒ P < G ⇒ J < 0 ∂˙ε ε˙
(23)
Case III: When D = J ∂J J J < ⇒ mσ < ⇒ mP < J ∂˙ε ε˙ ε˙
(24)
When the flow stress obeys the power law (Eq. (18)), substituting the value of J given in Eq. (21) in Eqs. (23) and (24) give the condition, m < 0 as obtained in Eq. (22). If the flow stress does not obey the power law, the condition, J < 0 in Eq. (23) holds good even for m > 0, and the condition, J > mP in Eq. (24) 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, J/P (≡ η/2) and m are non-dimensional parameters. J needs integration of σ with respect to ε˙ whereas m requires differentiation of σ with respect to ε˙ . J accounts for the behaviour of the material up to ε˙ (global) whereas m gives the information at ε˙ (local). It is more appropriate to use the condition in Eq. (24), because, when η < 0, Eq. (24) gives m < 0. When η > 0, and m < 0 or m is a small positive value, the condition in Eq. (24) yields comparatively larger region of instability in the processing map. Thus η and m are important parameters to use Eq. (24) for identifying the domains of flow instabilities. At any constant ε and T, the maximum value of η with respect to ε˙ becomes 2m/(m + 1) only when ∂m/∂(ln ε˙ ) < 0. The value of η expressed explicitly in terms of m based on power law assumption: η = 2m/(m + 1) in the DMM of Prasad [6] may create confusion on the maximization of η and m. If m is a function of ε˙ , expressing η as 2m/(m + 1) is incorrect. It is known from the test results on several materials that the maximization of η or m will reduce the tendency for flow localization. But the condition in Eq. (20) or (24) for delineating flow instabilities involves η and m. Relying on the positive values of m may not always ensure stable material flow for all materials. The parameters η and m are important in delineating the regions of flow instabilities. 3.4. Alexander’s stability criterion Using the Lyaponov function stability criteria, Alexander considered a Lyaponov function L (m, s) and suggested the following conditions for the stable material flow [8]. 0
(25)
∂m <0 ∂(ln ε˙ )
(26)
s≥1
(27)
∂s <0 ∂(ln ε˙ )
(28)
The above conditions indicate that for stable material flow, the σ − ε˙ curve should be convex, the flow stress should decrease with respect to T, and m should decrease with ε˙ and increase with T. 3.5. Instability criterion in the polar reciprocity model The associated flow rule of plasticity on which the polar reciprocity model (PRM) rests is based on the convexity of ˙H the potential surfaces [13]. The hardening power term, W increases monotonically to attain a high value and the intrinsic workability parameter, ς as derived in Eq. (9) tends to 1 at very large strain rates where the associated flow rule ˙ H and W ˙ D are referred to as G and J, fails. In the DMM, W respectively. One can write a relation between η and ς as η+ς =1
(29)
It was claimed in Ref. [14] that a ς value approaching 1 is definitely a condition of instability and no separate stability criterion on the ς maps is required for delineating the regimes of unstable flow. As per the relation (29), when the value of ς approaches unity, η approaches zero. This implies J = 0 and G = P. In this case all of the power would be dissipated by heat and this could lead to plastic instability by a continuum process such as adiabatic shearing. The qualitative stability condition, ς → 1 proposed in PRM is inadequate to delineate all the regions of unstable flow. The above discussion lead to a qualitative check for the stable material flow by η>0
or ς < 1
(30)
As per the DMM approach, stability criteria are essential in order to delineate the unstable regions in the η maps. In view of the linear relation between η and ς, as given by Eq. (29), additional stability conditions are essential for ς maps also.
4. Results and discussion Applicability of various instability theories presented in the preceding sections is examined using the flow stress data of AISI 304 stainless steel with microstructural observations. Venugopal, Mannan and Prasad [18] have discussed the hot deformation characteristics of AISI 304 stainless steel in the temperature range 600–1250 ◦ C and in the strain rate range of 0.001–100 s−1 . They presented the data on flow stress (σ) with strain (ε), strain rate (˙ε) and temperature (T) generated from hot compression testing of solid cylinders using servohydraulic testing machine capable of imposing constant true strain rates on the specimen. The test data is corrected for adiabatic temperature rise. The specimens were compressed to 50% of their initial height and the load–stroke curves obtained in the hot compression were converted into true stress–true
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plastic strain curves by subtracting the elastic portion of the strain and using the standard equations for the true stress and true strain calculations. One specimen was tested to obtain each result. Interpretation of the experimental data for the development of processing maps depends upon the accurate evaluation of the workability parameters such as strain rate sensitivity parameter (m), the efficiency of power dissipation (η), the normalized flow softening rate (γ) etc. While carrying out the numerical computation, the data reported in Ref. [18] is initially transformed into logarithmic scale, in order to reduce the tenth order magnitude of strain rate to first order, to avoid any excessive round off error. With this transformation, the first derivative of the spline fit, gives directly the strain rate sensitivity parameter, m at the generated intermediate data points. The normalized flow softening rate (γ) is obtained by transforming initially the flow stress data in to natural logarithmic scale and finding it directly from the first derivative of the spline fit to the data on ln σ – ε data, at the specified ε˙ and T. Following Ref. [20] the efficiency of power dissipation (η) is obtained. The value of the workability parameter (α) is obtained from Eq. (8) by substituting the value of γ and m. The temperature sensitivity of flow stress (s) is obtained by finding the first derivative of the spline fit to the data on ln σ – ln T. In Gegel’s stability criterion as well as Alexander’s stability criterion, the derivatives of m, η and s with respect to ln ε˙ are obtained by finding the first derivatives of the spline fit to the m – log ε˙ , η – log ε˙ and s – log ε˙ data. By subtracting the values of η from unity, the workability parameter, ζ in the polar reciprocity model (PRM) is obtained. Using these workability parameters, the instability parameters for unstable material flow for the described instability theories are defined as follows. The instability parameter based on the flow localization concepts: ξ1 = 1 −
α <0 5
(31)
The instability parameters based on Gegel’s stability criterion are: ξ2 = m < 0 ξ3 = −
∂η = (1 + m)η − 2m < 0 ∂ ln ε˙
ξ4 = s − 1 < 0 ξ5 = −
∂s <0 ∂ ln ε˙
(32) (33) (34) (35)
In the case of Alexander’s criterion, ξ6 = m < 0 ξ7 = −
∂m <0 ∂ ln ε˙
ξ8 = s − 1 < 0
(36) (37) (38)
ξ9 = −
∂s <0 ∂ ln ε˙
273
(39)
The instability parameter based on the simplified metallurgical stability condition (20): ξ10 =
2m −1<0 η
(40)
which is valid for m>0
(41)
The instability parameter based on PRM (Eq. (30)), ξ11 = 1 − ζ < 0
(42)
The curves ξ i = 0 bifurcate the stable and unstable regions in the processing maps for the above five criteria considered. The regions where ξ i < 0 in the map correspond to the unstable flow (microstructural instabilities) in the material. From the determined values of the workability parameters, the instability parameters ξ i from Eqs. (31)–(42) are calculated for a strain of 0.5 at different values of ε˙ and T. The power law assumption in Eq. (18) to represent the dynamic constitutive behaviour in continuum mechanics of large plastic flow for viscoplastic materials is generally considered for the purpose of simplifying complex mathematical analysis procedures to understand the physics of the problem. This assumption was followed for analyzing the hot deformation behaviour [18]. With this assumption, the efficiency of power dissipation, (η) can be expressed explicitly in terms of the strain rate sensitivity parameter (m) as given by Eq. (21). For complicated alloy systems, the flow stress (σ) with respect to ε˙ does not obey power law and the computation of η based on power law assumption (Eq. (18)) gives erroneous results [19]. If the material obeys power law for specified strain and temperature, the strain rate sensitivity parameter (m) is independent of ε˙ and the derivative of m with respect to ε˙ becomes zero. Based on the power law assumption, the instability parameter defined by Prasad [17]: ∂ ln(m/(m + 1))/∂ ln ε˙ + m reduces to ξ = m and ξ = m < 0 becomes the instability condition. It can be verified easily from the simplified instability condition (20) by using Eq. (21) for η. Fig. 1 shows the variation of flow stress (σ) versus strain rate (˙ε) at different temperatures at a strain of 0.5. The stress–strain curves at different temperatures and strain rates presented in Ref. [18] indicate strain hardening behaviour and most of the curves show steady state region beyond a strain of 0.4. It should be noted that, when strain rate insensitivity occurs at large strains, J increases to a certain extent and remains constant, whereas G increases with ε˙ . In such a situation, J is small compared to the total power P and is not equal to zero, whereas m → 0 which implies η > 2 m and the flow is unstable. This may be the reason why most of the materials during hot deformation show instabilities at large strain rates. The idea of developing various theories of flow instabilities is to trace the limits of the strain rate and temper-
274
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Fig. 1. Variation of flow stress with strain rate for AISI 304 stainless steel at a strain of 0.5.
ature at which no flow localization or fracture occurs during hot deformation. Table 1 gives the instability parameters defined in Eqs. (31)–(42) for the microstructural observations reported on AISI 304 stainless steel [18] at different temperatures and strain rates at a strain of 0.5. Table 2 gives the comparison of
the various existing instability theories with the reported microstructural observations. The instability parameters ξ 1 and ξ 11 defined in Eqs. (31) and (42) based on the flow localization concept and the polar reciprocity model correctly predicted the stable/unstable microstructural observations of deformed compression specimens on AISI 304 stainless steel. Though
Table 1 Instability parameter ξ i defined in Eqs. (31)–(42) for the microstructural observation reported on AISI 304 stainless steel [18] at different temperatures and strain rates at a strain of 0.5 T (◦ C)
Strain rate ε˙ (s−1 )
m
ξ1
ξ2 , ξ6
ξ3
ξ4 , ξ8
ξ5 , ξ9
ξ7
ξ 10
ξ 11
600 600 1000 1100 1100 1200 1200
0.001 10 0.001 0.01 100 0.001 0.1
→0 <0 0.277 0.220 0.028 0.137 0.237
>0 >0 >0>0 >0 >0 >0 >0
→0 <0 >0 >0 >0 >0 >0
→0 >0 →0 <0 >0 →0 →0
<0 <0 >0 >0 >0 >0 >0
>0 >0 >0 <0 <0 >0 <0
→0 <0 →0 <0 →0 →0 >0
→0 >0 >0 >0 <0 >0 >0
→0 >0 >0 >0 >0 >0 >0
Table 2 Comparison of the various existing instability theories with the reported microstructural observations on AISI 304 stainless steel [18] T (◦ C) Strain rate ε˙ (s−1 )
Flow localization theory Eq. (31)
Gegel’s instability theory Eqs. (32)–(35)
Alexander’s instability theory Eqs. (36)–(39)
Simplified instability Instability theory theory Eq. (40) due to PRM Eq. (42)
Microstructural observations
600 600 1000 1100 1100 1200 1200
Unstable Unstable Stable Stable Stable Stable Stable
Unstable Unstable Unstable Unstable Unstable Unstable Unstable
Unstable Unstable Unstable Unstable Unstable Unstable Unstable
Unstable Unstable Stable Stable Unstable Stable Stable
Flow localization Flow localization Dynamic recrystallization Dynamic recrystallization Dynamic recovery Dynamic recrystallization Dynamic recrystallization
0.001 1.0 0.001 0.01 100 0.001 0.1
Unstable Unstable Stable Stable Stable Stable Stable
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275
Fig. 2. Processing map for AISI 304 stainless steel at a strain of 0.5 with the microstructural observations, ( ) stable and (×) unstable [18].
the condition (40) indicated the material flow at 1100 ◦ C and 100 s−1 to be unstable, the microstructural observations indicated dynamic recovery. The value of the strain rate sensitivity parameter, m at the temperature and strain rate of 1100 ◦ C and 100 s−1 is noted as 0.028 and the value of the efficiency of power dissipation, η, is about 0.1, which is greater than 2m. Hence, the predictions using the instability condition (40) based on the Ziegler’s continuum principles as applied to large plastic flow become conservative which will give more confidence to the process designer. Fig. 2 shows the marked regions of instability using the condition (40),
which is based on the Ziegler’s instability criterion, along with the microstructural observations reported in Ref. [18]. Fig. 3 shows the contour map for strain rate sensitivity parameter, m and the contour map for efficiency parameter, η is shown in Fig. 4. The hatched region in Fig. 3 corresponds to the condition in Eq. (22) when the dissipation function is considered as the total power (D = P). The hatched region in Fig. 4 corresponds to the condition in Eq. (23) when the dissipative function is considered as the dissipator power content (D = G). The instability condition (42) also gives the same hatched region as given in Fig. 4 for the unstable region, be-
Fig. 3. Contour map for strain rate sensitivity parameter (m) for AISI 304 stainless steel at a strain of 0.5.
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Fig. 4. Contour map for efficiency of power dissipation (η) for AISI 304 stainless steel at a strain of 0.5.
cause, η < 0 implies ς > 1. The hatched regions in Figs. 3 and 4 are found to be within the unstable regions identified by the instability condition (40). When η > 0 and m is a small positive value, the condition (40) predicts comparatively larger region of instability in the processing map. As such there is no approximation or fixing a value based on experiments is involved in this instability condition. Uncertainties if any on the boundaries between the ‘stable’ and ‘unstable’ regions in the processing maps, will be mainly due to the flow stress data of the material. It should be noted that the constant value of
5 in the instability condition (31) has been fixed for flow localization or fracture to occur during hot deformation of the material based on microstructural observations in titanium and its alloys. Its validity for other materials will be known only after examining the microstructural observations of the deformed compression specimens. To demonstrate the above aspects further, a comparative study is made on the flow localization concept and the simple instability condition, by utilization the flow stress data of commercial grade titanium alloy Ti–6Al–4V [25]. Fig. 5
Fig. 5. Contour map based on the flow localization concept (Eq. (31)) for commercial grade Ti–6Al–4V at a strain of 0.5 with microstructural observations of Ref. [25], ( ) stable; (×) unstable.
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277
Fig. 6. Contour map of the simple instability condition (Eq. (40)) for commercial grade Ti–6Al–4V at a strain of 0.5 with microstructural observations of Ref. [25], ( ) stable; (×) unstable.
shows the condition for flow localization (31) which disagrees with the predicted locus for microstructural observations of commercial grade titanium alloy Ti–6Al–4V [25], because this condition predicts experimentally known unstable regions as stable. However, these microstructural observations of flow instabilities are found to be in the unstable regions of Fig. 6, which validates the simple instability condition (40). This comparative study indicates inconsistency in the flow localization concept (31) even for titanium alloys for which it was proposed. MATLAB (Math Works Inc., USA) software was utilized for generating the 2D contour maps. It is noted from Figs. 3 and 4 for AISI 304 stainless steel that the maximum value of m is 0.277 and the maximum value of η is 0.434 which occur at T = 1000 ◦ C and ε˙ = 0.001 s−1 . From Figs. 2–4, it is preferable to conduct the hot working in the temperature range of 900–1200 ◦ C and in the strain rate range of 0.001–0.1 s−1 . In an industrial practice, highest possible strain rates and lowest possible stresses are preferred from productivity viewpoint. Due to this reason, high strain rate region in which high values of m and η are present should be selected for bulk working operations and the lower strain rate regions for secondary metal working operations. Since, these optimum strain rates are too low for commercial processing, the limits of flow instability will have to be established so that the processing rates are pushed up as high as possible. Taking advantage of the effect of prior processing history on the processing map, connected processes may be designed. Each step of the process may refine the grain structure and expand the workability domain for the next step of processing while moving it to higher strain rates without the onset of instabilities to facilitate higher productivity.
5. Concluding remarks The various existing instability theories were reviewed. A simple instability condition based on the Ziegler’s continuum principles as applied to large plastic flow, is found to be more appropriate for delineating the regions of unstable metal flow during hot deformation. It can be used for any type of the flow stress versus strain rate curve. This criterion has been validated using the flow stress data of AISI 304 stainless steel with microstructural observations. Since there is no unique stability theory to delineate the regions of unstable flow during hot deformation, which is applicable for all materials, the designer has to establish a suitable theory on the basis of microstructural observations of the flow localization in the intended materials.
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