Processing Maps for the Hot Forming of Polycrystalline Metallic Materials Using the Garofalo Equation

Abstract

Processing maps have been built for > 30 years to predict the stable conditions for metal forming at high temperatures and strain rates. These maps, which also include efficiency conditions, are based on the stability criterion of Ziegler–Prasad (Z–P) within the framework of the dynamical materials model (DMM). In the present work, this criterion is drastically modified by introducing a Garofalo equation in the expression of the dissipative co-content, J, maintaining its definition in the framework of the DMM. This modification applies to its derivative with respect to the strain rate and also to the definition of efficiency, η. This new parameter developed in this work, named rho, ρ, is different from the stability parameter, xi, ξ, of the Z–P criterion since it is based on different constitutive equations. Rho is based on a Garofalo equation and xi on a power law equation. Both are obtained from experimental data and give rise to different stability maps that can be applied to any polycrystalline metallic material. In this work, a well-documented Ti-10V-4.5Fe alloy has been chosen to contrast the differences and similarities between the two methods of characterizing the stability conditions. Special emphasis is given to the predictions for the optimum forming temperatures.

Appendix

1.1 Stability Expression of Prasad from the Concepts of the DMM and the Use of a Power Law Equation

Using the power law equation:

$$ \dot{\varepsilon } = A_{\text{PL}} \cdot e^{{ - \frac{{Q_{PL} }}{R.T}}} \cdot \sigma^{{n_{PL} }} $$

(A1)

and the concepts of the DMM, Prasad et al. obtained the following equations (where the subindex PL indicates that it has been obtained by adjusting the data to a power law constitutive equation):

$$ G = \frac{{P \cdot n_{\text{PL}} }}{{n_{\text{PL}} + 1}} = \frac{P}{{m_{\text{PL}} + 1}} $$

(A2)

$$ J = P \cdot \frac{{m_{\text{PL}} }}{{m_{\text{PL}} + 1}} = \frac{P}{{n_{\text{PL}} + 1}} $$

(A3)

Here, $\left\{ {A_{\text{PL}} ,\,Q_{\text{PL}} ,\,n_{\text{PL}} } \right\}$ are constant values (parameters) in the equation, for a given value of strain. The fit of Eq. [A1] to the experimental set of data values $\left\{ {\dot{\varepsilon },\sigma ,T} \right\}$ (N values of the mentioned sets of three variables) results in obtaining values of the constants for a given value of strain.

The maximum value of J is $J_{ \hbox{max} } = \frac{P}{2}$, which is reached for m = 1. Furthermore, for this value $J_{ \hbox{max} } = G_{ \hbox{min} } = \frac{P}{2}$. This has two consequences. It is valid strictly when the power law is justified, which is in short ranges of the thermodynamic variables of the system. It is worth noting that it would be recommendable to use the total power P as the denominator for the normalization of the efficiency instead of $J_{ \hbox{max} }$. This would make the efficiencies of J and G comparable and would rationally normalize the concept of efficiency. In addition, this would allow distinguishing the power used in changing the form from that in modifying the microstructure.

According to the DMM, the efficiency η is an important parameter that synthesizes the information of the dissipative co-content and the relations of the plastic creep with the evolution of the microstructure. It is defined as:

$$ \eta_{{J/J_{\hbox{max} } }} = \frac{J}{{J_{\hbox{max} } }} $$

(A4)

Therefore, the efficiency of the dissipative co-content referred to its maximum performance using the values of m obtained from the power law is the following:

$$ \eta^{\text{PL}}_{{J/J_{\text{max} } }} = \frac{{P \cdot \frac{{m_{\text{PL}} }}{{m_{\text{PL}} + 1}}}}{{\frac{P}{2}}} = \frac{{2 \cdot m_{\text{PL}} }}{{m_{\text{PL}} + 1}} $$

(A5)

Continuing with the application of the power law in the framework of DMM, from the stability criterion in its simpler form it is obtained that:

$$ \frac{{\partial {\text{Ln}}\left( J \right)}}{{\partial {\text{Ln}}\left( {\dot{\varepsilon }} \right)}} > 1 $$

(A6)

and using J obtained from the power law:

$$ J = \frac{{m_{\text{PL}} }}{{m_{\text{PL}} + 1}} \cdot \sigma \cdot \dot{\varepsilon } $$

(A7)

All these expression from Eqs. [A1] to [A7] are given in References 33, 34 and many other works.

It is easy to see that ${ \ln }\left( J \right) = { \ln }\left( {\frac{m}{m + 1}} \right) + { \ln }\left( \sigma \right) + { \ln }\left( {\dot{\varepsilon }} \right)$ and with the calculation of derivative operations of Eq. [16], it is obtained directly that $\frac{{\partial \left( {{ \ln }\left( {\frac{m}{m + 1}} \right)} \right)}}{{\partial { \ln }\left( {\dot{\varepsilon }} \right)}} + \frac{{\partial { \ln }\left( \sigma \right)}}{{\partial { \ln }\left( {\dot{\varepsilon }} \right)}} + \frac{{\partial { \ln }\left( {\dot{\varepsilon }} \right)}}{{\partial { \ln }\left( {\dot{\varepsilon }} \right)}} < 1$; therefore, it is obvious that $\frac{{\partial \left( {{ \ln }\left( {\frac{m}{m + 1}} \right)} \right)}}{{\partial { \ln }\left( {\dot{\varepsilon }} \right)}} + \frac{{\partial { \ln }\left( \sigma \right)}}{{\partial { \ln }\left( {\dot{\varepsilon }} \right)}} < 0$. Now, using a power law, where m = $\frac{{\partial { \ln }\left( \sigma \right)}}{{\partial { \ln }\left( {\dot{\varepsilon }} \right)}}$, the stability expression, named the Prasad or xi equation, which is well known to all researchers in this field, is finally obtained:

$$ \xi \left( {\dot{\varepsilon }} \right) = \frac{{\partial \left( {{ \ln }\left( {\frac{m}{m + 1}} \right)} \right)}}{{\partial { \ln }\left( {\dot{\varepsilon }} \right)}} + m < 0 $$

(A8)

1.2 Algorithms Used for Determination of the Stability and Efficiency Equations

The determination of the dissipator co-content, J, was carried out, as mentioned in the text, using the equation $J = \mathop \smallint \limits_{0}^{\sigma } \dot{\varepsilon } \cdot d\sigma$ and the variable change $x = \alpha \cdot \sigma$. This value of J with this variable change, or algorithm, was named J(1). Notably, the generalized Leibniz rule was applied to reach these expressions since it was necessary to derive inside the integrals. If a transformation from the variable $\sigma$ to the variable $\dot{\varepsilon }$ is chosen, and the expression $\alpha \cdot \sigma = \sinh^{ - 1} \left( {\theta \left( {\dot{\varepsilon },T} \right)^{G} } \right)$ where $\left( {\theta \left( {\dot{\varepsilon },T} \right)^{G} } \right) = \left( {\frac{{\dot{\varepsilon } \cdot e^{{\frac{{Q^{G} }}{R \cdot T}}} }}{{A^{G} }}} \right)^{{\left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {n^{G} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${n^{G} }$}}} \right)}}$ is used, the following expression $J /\dot{\varepsilon }$ is obtained:

$$ J /\dot{\varepsilon } = \left( {\frac{1}{{\alpha^{G} \cdot \left( {\theta \left( {\dot{\varepsilon },T} \right)^{G} } \right)^{{n^{G} }} }}} \right) \cdot \mathop \smallint \limits_{0}^{{\sinh^{ - 1} \left( {\theta \left( {\dot{\varepsilon },T} \right)^{G} } \right)}} \left[ {\sinh \left( x \right)} \right]^{{n^{G} }} \cdot {\text{d}}x $$

(B1)

being

$$ J = \left( {\frac{{A^{G} \cdot e^{{ - \frac{{Q^{G} }}{R \cdot T}}} }}{{\alpha^{G} }}} \right)\mathop \smallint \limits_{0}^{{\sinh^{ - 1} \left( {\theta \left( {\dot{\varepsilon },T} \right)^{G} } \right)}} \left[ {\sinh \left( x \right)} \right]^{{n^{G} }} \cdot {\text{d}}x $$

(B2)

From this expression, substituting directly in the definition of efficiency, Eq. [17] in the text, we obtain:

$$ \eta^{G} = \frac{{2 \cdot \mathop \smallint \nolimits_{0}^{{\sinh^{ - 1} \left( {\theta \left( {\dot{\varepsilon },T} \right)^{G} } \right)}} \left[ {\sinh \left( x \right)} \right]^{{n^{G} }} \cdot {\text{d}}x}}{{\left( {\theta \left( {\dot{\varepsilon },T} \right)^{G} } \right)^{{n^{G} }} \cdot \sinh^{ - 1} \left[ {\theta \left( {\dot{\varepsilon },T} \right)^{G} } \right]}} . $$

(B3)

The application of the Leibniz generalized rule is needed to obtain the expression of $dJ/d\dot{\varepsilon }$ (for a given strain and temperature) as: $dJ/d\dot{\varepsilon } = \dot{\varepsilon } \cdot \partial \sigma \left( {\dot{\varepsilon },T} \right)/\partial \dot{\varepsilon } = \left( {1/\left( {n^{G} \cdot \alpha^{G} } \right)} \right) \cdot \left( {\theta \left( {\dot{\varepsilon },T} \right)^{G} /\sqrt {1 + \left( {\theta \left( {\dot{\varepsilon },T} \right)^{G} } \right)^{2} } } \right)$. Substituting this expression in Eq. [9], it is finally obtained:

$$ \rho = \frac{1}{{n^{G} \cdot \alpha^{G} }} \cdot \left[ {\frac{{\theta \left( {\dot{\varepsilon },T} \right)^{G} }}{{\sqrt {1 + \theta \left( {\dot{\varepsilon },T} \right)^{{G^{2}}} } }} - \left( {\frac{{n^{G} }}{{\left( {\theta \left( {\dot{\varepsilon },T} \right)^{G} } \right)^{{n^{G} }} }}} \right) \cdot \mathop \smallint \limits_{0}^{{\sinh^{ - 1} \left( {\theta \left( {\dot{\varepsilon },T} \right)^{G} } \right)}} \left[ {\sinh \left( x \right)} \right]^{{n^{G} }} \cdot {\text{d}}x} \right] $$

(B4)

1.3 Construction of Efficiency and Stability Maps

An important methodological aspect is that related to software for the execution of the calculations that are necessary for the preparation of efficiency and stability maps. This is a non-trivial aspect due to the large size and complexity of multiple formulas with hyperbolic expressions, with direct and inverse functions, and all this combined with the calculations of derivatives and expressions. This can lead to the generation of different kinds of errors: truncation, rounding and cumulative.

For developing the algorithm of the efficiency and stability maps, we use the MATHCAD Professional 7.0. The program starts by defining the input parameters, which are the four adjustment coefficients of the Garofalo equation $\left\{ {A^{G} , Q^{G} , n^{G} , \alpha^{G} } \right\}$. We give an example of the use of the program for an ideal material:

INPUT DATA (example for strain ɛ = 0.2):

q: = 59527.10721895 (effective activation energy in calories/mol K).

n: = 3.7104796 (exponent of the hyperbolic sine function).

al: = 0.00338 (α = cofactor of true stress).

A = exp(29.121729) a: = 4.44023758 10¹² (coefficient A or entropy factor).

The definition of the indexes is as follows. For the temperature and the strain rate, we use the index i and j, respectively. The range goes from 0 to about 40 for the temperature (for large temperature ranges of about 300 K) and from 0 to about 80 for the strain rate (for large strain rate intervals of about 5 orders of magnitude). The mesh size factor i × j should be > 3000 in these cases to have a good resolution in the maps.

The expression in the Mathcad program to determine the interval for the strain rate is:

$$ E_{j} : = \exp \left( {x + \frac{j}{z}} \right) $$

The value of x corresponds to ln of the lowest value of the strain rate, or j = 0, and ln of j/z corresponds to the highest value, or j = 80. For an interval of 0.001 to 100 s⁻¹ and j = 0 to 80, this expression becomes: $E_{j} : = \exp \left( { - 6.9078 + \frac{j}{6.9487}} \right)$, since − 6.9078 +80/6.9487 = 4.6052 = ln100, being ln 0.001 = − 6.9078. Obviously, we have identified $\ln \left( {\dot{\varepsilon }_{j} } \right) = \ln \left( {E_{j} } \right) = \left( {x + \frac{j}{z}} \right)$, and with the result in this case E0: = 0.001 and E80: = 100.0.

The expression to determine the interval for temperatures is:

Ti: = y + w i.

The value of y corresponds to the lowest temperature in K, or I = 0, and w i for the highest, or I = 40. For instance, for an interval of 900 to 1200 K and I = 0 to 40, this expression becomes:

Ti: = 900 + 7.5 i, since 900 + 7.5 40 = 1200, and therefore T₀ = 900 K and T₄₀ = 1200 K.

With this information it is possible to build the maps in a MATHCAD Professional 7.0. program. However, in case of any complications, and by prior request, we have made the program available for those interested in drawing these new maps. The importance of the correct determination of the Garofalo equation parameters before launching the program must be noted.