RESEARCH SUMMARY .......... .

Microstructure-Based Modeling of Deformation Processes ___________ 0. Richmond

Traditional mathematical models for deformation processes usually assume material behavior independent of thermomechanical history. Their prima­ry concern is the relationship of defor­mation forces to changes in geometry. This paper describes a more complete' approach called microstructure-based modeling. The material behavior is assumed to relate to a small number of history-dependent parameters represent­ing the most significant aspects of microstructure. The models are con­cerned not only with changes in geometry, but also with changes in product properties resulting from the changes in microstructure. Thus, the to­tal approach involves the union of solid mechanics and materials science to pro­vide a quantitative basis for the design of deformation processes to achieve con­trolled properties as well as shapes.

Strain Rate O.S/Second 100 I I I I

80 I- -.. 3000 C Q.

~ 60 -.. 3500 C ..

f 40 ~ 4000 C -iii 4500 C

20 f-

~. ~ L ,5000 C

0 0 0,2 0,4 0,6 0.8 1,0

Strain (a) Effect of Temperature al Fixed Strain Rate

Temperature soooe 50

I I I I

40 I- 10,05.C·1, -5.0 sec·1 ' .. 2,0 5.c·l\ ~" ~ 30 I- -..

~ f 20 0.5 seC-1 iii 0.05 sec-1

10

0 I I I I

0 0.2 0.4 0,6 0,8 1.0 Strain

(b) Effect of Strain Rate at Fixed Temperature

Figure 1. Stress-strain behavior of aluminum at elevated temperatures, (a) Effect of tem­perature at fixed strain rate, (b) Effect of strain rate at fixed temperature,

16

INTRODUCTION

Until now, models of deformation processing have employed primarily ideal materials like perfectly plastic solids and nonlinear viscous solids whose current response to applied stresses is completely independent of prior thermomechanical history. In a few cases, they have employed ideal strain hardening materials where the entire effect of history is contained in a single function of prior accumulated effective strain. The primary concern of such process models has been the prediction of changes in product geometry resulting from the application of specified forces or, conversely, the prediction of tool forces required to achieve certain speci­fied changes in geometry.

A new thrust in deformation modeling takes specific account of the fact that most shape changes are accompanied by changes in material structure. The new model, therefore, is overtly concerned with changes in structure, and thus, product properties, as well as tool forces and product geometry. Of course, it has long been recognized in an implicit sense that deforma­tion processes cause changes in both microstructure and geometry. What is new is the development of quantitative constitutive equations describing the relationships of processing and product histories to changes in material structure. These relationships can be used to design and control processes to achieve both the desired properties and geometries.

This new thrust assumes that the principal properties of a material can be reasonably represented by a short list of microstructural (or internal) parameters, including some measure of damage such as porosity. It further assumes that the effect of arbitrary thermomechanical histories on these parameters can also be quantitatively described.

NEAR HOMOGENIETY

No deformation process or product is perfectly homogeneous, even on a macroscopic scale. Yet, to develop constitutive equations that describe the relationship of thermomechanical history to changes in geometry and microstructure, such states should be approached as closely as possible in laboratory experiments. Also, it is easiest to describe the concept of microstructure-based modeling of deformation processes when such processes are homogeneous. To simplify discussion, the stress states will be limited to simple tension and compression and the microstructural parameters will be limited to the isotropic case.

The properties, Pi, of a product resulting from a homogeneous process may be assumed to be functions of a small list of microstructural parameters, Sj. Thus

(1)

where the properties might be, for example, tensile strength and tensile ductility, and the microstructural parameters might be porosity and disloca­tion density.

The microstructural parameters, Sj, in turn, may be assumed to be func­tionals of the prior history of strain rate, t, and temperature, T. That is,

(2)

It is really not practical, however, to describe all potential thermomechanical histories in this fashion. Thus it is assumed alternatively that the current stress, cr, is a function of the current values of the microstructural parame­ters as well as of the current strain rate and temperature. Thus,

(3)

This constitutive equation is called an equation of state. To complete the description of the material behavior, it is also assumed that the current

JOURNAL OF METALS· April 1986

rate of change, 8j, of the microstructural parameters can be described by a set of constitutive equations called equations of evolation:

81 = hi (0", t, T, 81, 82, ... ) 8 = h2 (0", t, T, 81, 82, ... ) (4)

Equations 3 and 4 are the constitutive equations for a given material. They can be integrated for arbitrary processing histories to determine resulting changes in strain and microstructure. Thus, they are the basis process for structure relations of the form given by Equation 2. They can also be used to incorporate the effects of arbitrary loading histories on changes in product strain and microstructure. Hence, they also can be used to determine structure/property relations of the form given by Equation 1.

The constitutive Equations 3 and 4 must generally be motiyated, or at least validated, by the results of laboratory tests under controlled histories of strain rate (or stress) and temperature. A recent paper by V.M. Sample and L.A. Lalli 1 describes a limited set of constitutive equations appropriate for a significant range of hot working processes in commercial purity aluminum. These equations involve only a single microstructural parameter which correlates well with microhardness measurements and essentially represents the resistance to deformation provided by the cur­rent dislocation structure. The mechanical testing equipment and methods used to generate the model are also described. The model is given in the form:

0" = k(81)l(Z),

and 81 = tm(81Z) (5)

where Z = t exp (AfT). A is a material constant, and k, I, and mare specified material functions. Equations 5 are a special, and rather simple, case of Equations 3 and 4. Yet, they adequately represent the response of commercial purity aluminum over a very large range of thermomechanical histories, as shown in Figure 1.

Another constitutive model2 has been developed to describe the room temperature response of a metal with residual populations of pores. It is independent of temperature and strain rate but has two history-dependent microstructural parameters. One, S10 again represents the resistance to deformation of the matrix material. The other, S2, represents porosity. Since both of these parameters change with deformation history, the constitutive model includes two evolution equations. The complete model has the form:

SlP(S2)

q(SlS2)t 0"

r(S2,81)t (6)

The equation gives reasonable representation of stress-strain behavior of commercially pure iron with different residual porosities (Figure 2).

For homogeneous thermomechanical histories other than simple tension lind compression, the constitutive models in this section will, of course, include tensor functions. For anisotropic behavior, they will also include anisotropic microstructural parameters.

NONHOMOGENEOUS PROCESSES

Deformations in actual processes are generally quite inhomogeneous, although under very special conditions, e.g. perfectly-plastic materials and frictionless dies, ideal processes can be designed to produce uniform products.3 Real materials and die surfaces, of course, are neither perfectly plastic nor perfectly frictionless. Thus, the general concern is with both nonuniform processes and nonuniform products. In some cases, it may be desirable to deliberately manufacture products with inhomogeneous struc­ture in order to optimize the response to subsequent nonuniform loadings.

Figure 3 illustrates the principal aspects of a model of a nonhomogeneous deformation process. These include constitutive equations, which, as illus­trated earlier, are algebraic and ordinary differential equations describirig the response of a representative interior element to an arbitrary thermomechanical history. In addition, other equations are needed to de­scribe the response of a representative surface element to the special thermomechanical histories which it may experience. We call these latter equations tribological equations. They are especially necessary at the tool/workpiece interface where simple global friction coefficients are no longer adequate,4 and where the evolution of microstructure, including surface roughness and surface damage (cracks), should be included, just as in the constitutive models for representative interior elements.

JOURNAL OF METAL8· April 1986

350

300

250

200 .. .. ~ 1n

150

100 _ Experiment o Theory

so

oL-__ L-__ ~ __ ~ __ ~ __ -L __ ~ __ -"

o 0.04 0.08 0.12 0.16 0.20 0.24 0.28 Strain

Figure 2. Comparison of observed and mod­eled stress-strain response of porous iron.

Ldm~_~'~=~_ Path of Surface epresentatlve Element

Work Piece

Path of Interior RepresentatIve Element

:~:::::::?,,:~C"" t 'I~

Figure 3. Principal aspects of a deformation process model.

(a) Temperature Profiles

----£----30.5 31.0

(b) MlcrOhardnes. Profiles

Figure 4. Predicted temperature and micro­hardness profiles in a simple aluminum extru­sion process.

17

1.0

• Experiment o Theory

o

•

0.1 L-_--'-__ ..L...._-l. __ """'"--_--' __ -'

o 0.02 0.04 0.06 0.08 0.10 0.12

Initial Void Fraction

Figure 5. Relationship of residual porosity on tensile ductility in an iron P/M material.

References 1. V.M. Sample and L.A. Lalli, "Effects of Thermo· mechanical History on the Hardness of Aluminum," Aluminum Technology '86, March 11-13, 1986, London. 2. O. Richmond and R.E. Smelser, "A Constitutive Model for Ductile Materials Containing Voids," to be published. 3. O. Richmond and H.L. Morrison, "Streamlined Wire Dra.wing Dies of Minimum Length," J. Mech. Phys. Sollds, Vol. 15, 1967, pp. 195·203. 4. C.Y. Lu, E.J. Appleby, R.A. Rao, M.L. Devenpeck, P.K. Wright and O. Richmond, "A Numerical Solu­tion of Strip Drawing Employing Measured Boundary Conditions Obtained with Transparent Sapphire," Numerical Methods in Industrial Forming Process, Proc. Int. Conf., Swansea, 1982. 5. R.E. Smelser, "A Thermomechanically Coupled Analysis of Streamlined Die Extrusion Including Hard­ness Predictions," Aluminum Technology '86, March 11-13, 1986, London. 6. W.G. Fricke, Jr., M.A. Przystupa and F. Barlat, "Modeling Mechanical Properties from Crystallograph­ic Texture (ODF) of Aluminum Alloys," Aluminum Technology '86, March 11-13, 1986, London.

ABOUT THE AUTHOR"""""". O. Richmond received his Ph.D. in engineer­ing mechanics from Pennsylvania State Uni­versity in 1958. He is currently Corporate Fel­low at Alcoa Laboratories, Alcoa Center, Pennsylvania.

If you want more Information on this subject, please circle reader service card number 30.

18

Because the constitutive equations for both interior and surface elements are local in nature, they are either algebraic or ordinary differential equa­tions with time as the independent variable. To treat the nonhomogeneous fields of actual processes, these equations must combine with general physi­cal laws expressing conservation of mass, momentum and energy. Thus, in a general process model, the interior constitutive equations appear togeth­er with the classical conservation laws to form the field equations having up to four independent variables representing space and time, and a num­ber of dependent variables representing stress, velocity, temperature and various microstructural parameters. The exterior tribological equations ap­pear as part of the boundary conditions along with the various geometric and surface constraints which distinguish a particular process.

R.E. Smelser5 described a complete model of extrusion through stream­lined dies. Here, the constitutive equation, Equation 5, includes a single microstructural parameter representing microhardness or dislocation resistance. The field equations also include the energy conservation equa­tion to couple the solution of the temperature distribution with those of the stress and velocity distributions. For simplicity, the die surface was assumed to be frictionless and insulated. To illustrate the power of the new approach, microhardness distributions were predicted as well as die forces and stress, velocity and temperature distributions. An illustration of temperature and microhardriess distributions is shown in Figure 4. This is one of a number of process models under current development at Alcoa Laboratories.

STRUCTURE/PROPERTY RELATIONS

As mentioned earlier, the complete microstructural approach to process modeling must include a knowledge of the relationship of product micro­structure to product properties, i.e. Equation 1. The development of these relationships also involves constitutive equations, as illustrated by the case of essentially homogeneous products subjected to essentially homogeneous product histories. The first illustration involves using constitutive equations, Equation 6, to model tensile ductility in a strain hardening metal with residual porosity. The model of tensile ductility2 includes the effect of necking on stress triaxiality and includes a slight inhomogeneity in porosi­ty to initiate plastic flow localization and subsequent fracture. Figure 5 shows the predicted relationship of tensile ductility to residual porosity and compares these predictions with experimental values. This illustrates the specific use of constitutive equations, in this case, Equation 6, to generate a microstructure/property relation in the form of Equation 1 where the specific microstructural variable is porosity and the specific property is tensile ductility.

A second example,6 illustrates the prediction of yield surfaces and forming limits for sheet products as a function of measured crystallite orientation distribution functions. Hence, the principal property of interest is forming limits, and the principal microstructural aspect is crystal orientation. Fig­ure 6 shows predicted changes in crystallographic texture. This is an example of using an anisotropic microstructural parameter to predict an anisotropic property.

5 4 ~ e c 3

~ 2 GI ,. .. c !!! l-

x - Copper Texture

.: 0 1------+------#-1'----1 ::: e iii -1

-2 '-----'----'----'"--'---'---'---' -2 o 2 4

Stress in Rolling Direction

(a)

GA·184526

0.5 ....----,---,---,--,----"

5 0.4 ~ e c 0.3 0> :§ "0 rr. 0.2 .: c "! iii 0.1

0.1 0.2 0.3 0.4 0.5 Strain in Transverse Direction

(b)

Figure 6. Relationship of crystallographic texture to forming limits in aluminum. (a) Yield Locus. (b) Forming Limit Diagram.

JOURNAL OF METALS· April 1986