POWER-LAW CREEP BLUNTING OF CONTACTS AND ITS ...diffusional creep are dominant and in this case the...

Acra mater. Vol. 44, No. 4, pp. 1479-1495, 1996 Elsevier Science Ltd

Copyright 0 1996 Acta Metallurgica Inc. Printed in Great Britain. All rights reserved

1359-6454/96 $15.00 + 0.00

Pergamon 0956-7151(95)00279-O

POWER-LAW CREEP BLUNTING OF CONTACTS AND ITS IMPLICATIONS FOR CONSOLIDATION MODELING

R. GAMPALA, D. M. ELZEY and H. N. G. WADLEY School of Engineering and Applied Science, University of Virginia, Charlottesville, VA 22903, U.S.A.

(Received 27 February 1995; in revised form 26 June 1995)

Abstract-Densification occurs primarily by power-law creep of interparticle contacts during the initial stages of the elevated temperature consolidation of metallic powders and metal matrix composite (MMC) preforms. Past approaches to its modeling have relied upon results from indentation analyses. Here, contact deformation is viewed as a blunting process and closed-form solutions for the contact stress- displacement rate and the contact areastrain relationships are proposed for laterally constrained contact blunting by power-law creep. These solutions contain unknown coefficients which are evaluated using the finite element method. The contact stress-displacement rate relationship during blunting is found to be controlled by the degree of constraint imposed on the flow of material near the contact: initially, the loss of internal (material) constraint dominates, leading to a strain softening; this is followed by hardening in which the influence of externally imposed lateral constraint (due to the presence of neighboring contacts) becomes dominant. Strain softening decreases in importance and lateral constraint hardening occurs earlier as the creep stress exponent decreases. The blunting results are used to revise densification and fiber microbending/fracture models. It is shown that, depending on the creep stress exponent, past indentation- based models can substantially over- or underestimate the true rate of densification.

1. INTRODUCTION

Many of today’s high performance structural materials (e.g. high strength tool steels [l], titanium and nickel-base alloys [2, 31, refractory metals [4], cer- amics [5] and metal/intermetallic matrix composites [6,7]) are manufactured to near-net shape in a two- step process. First, monolithic powders [8], (tape cast) powder/ceramic fiber preforms [9] or spray-deposited monotapes containing unidirectional fibrous reinforcements [lo] are produced (Fig. 1). These are then encapsulated, evacuated and consolidated to full density using processes such as hot isostatic or vacuum hot pressing (HIP/VHP) [ 1 I]. During HIP/VHP, a combination of high pressure and temperature is used to consolidate the initially porous preforms by inelastic (plasticity/creep) flow. In the case of fiber reinforced composites, this must be done while also avoiding damage to the fibers [12]. Numerous efforts have been made to understand and model these processes [e.g. 13-161 and to use this insight to optimize the consolidation process [17].

The development of these models began by recog- nizing that at the start of the densification process the externally applied stresses are internally supported at powder particle (or surface asperity) contacts. During consolidation, high stresses rapidly develop at these contacts and the resulting contact deformation dis- places material from the contact into adjacent (interparticle) voids. When the relative density is low (typically less than about 90% of the theoretical

density), classical results from contact mechanics have been used to estimate the contact stress required for inelastic flow [13, 181 and simple “uniform redistribution” models used for estimating the contact area [19]. Since contact deformation occurs by a combination of elastic, plastic and creep mechanisms, contact mechanics solutions for each mechanism are used in the models. These fundamental results are then used to develop predictive models describing the relationships between densification/fiber damage, process conditions (temperature, pressure and time), the preform’s internal (contact) geometry and material properties.

The relative contribution of each densification mechanism depends upon temperature. When the process temperature is low (T < 0.4T,, where T,,, is the melting temperature), inelastic deformation at the contacts of metals and their alloys occurs predominantly by plasticity and the most important material property is the (temperature dependent) yield strength. At higher temperatures, dislocation and/or diffusional creep are dominant and in this case the power-law creep exponent and the activation energy are the key material properties [20]. In earlier work, this contact deformation has conventionally been treated as an indentation process, and contact mechanics results from analyses of perfectly plastic or creep indentation have been used to develop densification and fiber damage models.

Wilkinson and Ashby [21] were the first to propose a creep contact model for analyzing the densification

1479

1480 GAMPALA et al.: CREEP BLUNTING OF CONTACTS

a) Monolithic metal/alloy powders

b) Tape cast cotiposite slurries

C) Spray-deposited composite monotapes

Fig. 1. Micromechanics-based models for predicting the overall densification response of powders and composite monotapes (e.g. spray-deposited or tapecast preforms) rely on accurate analyses of the deformation behavior at repre-

sentative interparticle contacts.

of powders by hot isostatic press9g. Their approach was based on a treatment of creep indentation by Marsh [22] and Johnson [23], who observed that beneath a blunt indenter, material was displaced more or less radially from the point of first contact, and so could be analyzed as the expansion of a cavity. Fischmeister and Arzt [24] subsequently derived a model for interparticle contact deformation by liken- ing the contact to a flat punch indenting a power-law creeping half-space. Helle et al. [25] applied this to the prediction of hot isostatic pressing diagrams for powder consolidation. These diagrams have subsequently been widely used to analyze and optimize

conditions for the densification of powders under hydrostatic pressure [5,26,27]. More recently, Kuhn and McMeeking [28] have extended Helle et al.‘s result for powders to predict densification under arbitrary stress states and therefore their model also relies on indentation analysis to predict the contact deformation behavior. Likewise, indentation results have been used in analyzing the deformation of asperity contacts (between spray-deposited composite monotapes) for modeling MMC densification kin- etics and fiber microbending/fracture [15, 161.

The use of indentation theory for predicting the response of contacts in these problems is an overly simplistic approximation of the actual physical process encountered during contact deformation. For instance, the deformation field remains self-similar during indentation [i.e. the stress and strain (rate) fields change little with continued indenter penetration once fully inelastic deformation has been established]. This leads to simple indentation depth- independent relationships between the average contact stress (acting over the face of the indenter) and the rate of penetration. However, the deformation during contact blunting occurs in a finite-size body, and increasingly significant interactions occur with the body’s boundaries as deformation proceeds, Fig. 2. When the contact strain is very small, the size effect is likely to be negligible and the results of indentation may be sufficient for modeling. However, during further densification by time-independent plasticity, large contact strains are developed and the strain field-asperity surface interaction has been shown to lead to a contact strain-dependent distribution of stress and strain in the contact and a resulting displacement-dependent average contact stress [29]. The contact flow stress in this situation has been shown to drop rapidly from an initial value of 2.97~~ (where oY is the uniaxial yield stress) towards unity because of the loss of “material constraint”. Thus, indentation-based models for densification only estimate the actual consolidation behavior and may be in error, particularly as the relative density increases (i.e. when the contact strains are large).

A second limitation of the indentation approach is its inability to account for the strong influence of neighboring contacts. Again, when the contact strain is very small, surrounding contacts do not perturb the flow criterion, but as densification proceeds, ad- ditional lateral (neighboring) contacts are established, material flow at the primary contact extends through the asperity or particle and becomes more constrained, requiring higher contact stresses to sus- tain deformation. Indentation-based models do not incorporate the effect of lateral constraint upon a contact’s deformation. Recent models for the blunting of hemispherical contacts by plasticity have ac- counted for the imposed lateral constraint (see for example Gampala et al. [29], who used finite element techniques or Akisanya and Cocks [30] who used slip-line field theory and finite element analysis).

GAMPALA et al.: CREEP BLUNTING OF CONTACTS

Plastic indentation Plastic blunting of a half-space of a contact

Fig. 2. Indentation theory has often been used to predict the blunting of contacts, even though indentation and blunting are different. The most significant differences arise from the finite size of the deforming body

during blunting.

These analyses show an initial decrease in flow stress (due to a loss of material constraint) with contact strain followed by a strong increase in flow stress due to the imposed lateral constraint. No similar analysis for the high temperature (i.e. creep) response of deforming contacts is available.

A third limitation of the indentation approach is its inability to predict contact area evolution. Uniform redistribution techniques have been used to estimate the contact area evolution with density [24]. How- ever, these ignore the “piling-up” of a matrix near the contact and are expected to give inaccurate predictions of contact area evolution. Since the product of the contact area and contact flow stress is used in the models to balance the applied consolidation force, errors in the contact area evolution are as important as the flow stress relationship.

Our objective is to analyze the contact mechanics of blunting for a representative non-linear, power-law creeping hemispherical contact with laterally imposed constraint. We consider materials whose uniaxial stress (atstrain rate (6) relationship can be described by a power-law relation

0 ” g=go - 0 00 where (r. and .6,, are the reference stress and strain rate, respectively and n is the stress exponent. The two results of greatest practical interest for consolidation modeling are: (1) the relationship between the average stress acting normal to a contact and the rate of contact deformation; and (2) the relation between the contact area and the amount of blunting. Obtaining the appropriate form for these two relationships when the matrix is a non-linear creeping material plays a prominent role in the development of a blunting theory.

1481

Because the analysis of indentation experiments allows mechanical properties, such as the elastic modulus, plastic yield strength and creep exponent, to be inferred from hardness measurements [31], the analysis of indentation has received significantly more attention than that of blunting. This has motiv- ated many efforts to analyze indentation and has led to (empirical, analytical and numerical) flow stressdisplacement rate and displacement-contact area relationships for a wide variety of material constitutive behaviors and indenter geometries [32]. Blunting has received less attention. It is also a more difficult process to analyze because the size of the deformation field can be comparable to that of the asperity and is therefore significantly influenced by the proximity and shape of the asperity’s free sur- faces.

It will be shown (in Section 2.1) that the form of contact flow stress relationship for the blunting of a power-law creeping material is expected to be

where oE is the average contact stress, h/ago is the effective strain rate (in which h is the rate of blunting displacement and a is the contact radius, Fig. 2), F is a material non-linearity (i.e. stress exponent, n) dependent coefficient and a/r is the contact radius normalized by the radius of the undeformed contact.

The analysis of both indentation and blunting requires knowledge of the evolution of contact area with displacement. It is well established [33] that, for rate-independent materials, the relationship during indentation depends on the work-hardening rate (i.e. the hardening exponent), and on the stress exponent in rate dependent materials described by equation (1). It will be shown (in Section 2.2) that during the


blunting of a power-law creeping contact, the contact radius-displacement relationship can be described by

h_!- - a2 0 2c(n)’ r where c is a material non-linearity dependent coefficient.

Together, equations (2) and (3) describe blunting of power-law creeping contacts in a form that can be readily incorporated into existing densification and fiber microbending/fracture models. The unknown coefficients, F(n, a/r) and c(n), are determined by means of finite element analysis (Sections 3 and 4) and the resulting solution applied to the consolidation of monolithic powder and composite preforms (Section 5).

2. CONTACT DEFORMATION MECHANICS

2.1. Contact stress-effective strain (rate) relations

When the contact stress between a pair of elastic-plastic bodies is small (typically less than about 1. la,), both indentation and blunting occur by elastic deformation. Hertz [34] showed that the average contact stress, (a,), for an elastically compressed hemisphere displaced a distance, h, against a rigid surface is given by

8 E h -._ Q’c=j&-v2 a

where E and v are the Young’s modulus and Poisson’s ratio of the hemisphere, and a is the contact radius. Equation (4) is a linear relation between the average contact stress and a measure of the effective strain, h/a. Analogous measures of the effective strain rate during time-dependent contact deformation have also been proposed {for example, Sargent and Ashby [35] proposed the quantity /ii(&), where h is the displacement rate and A the contact area}. The Hertz solution (4) applies equally to the complementary problem of indentation of an elastic half-space by a rigid, spherical indenter of radius, r. The contact radius then refers to the projected area of the contact and h to the depth of indentation, Fig. 2. Elastic behavior is confined to small displacements (i.e. deformations a/r < 0.01) and in this regime, blunting and indentation have identical contact stress-effective strain behaviors.

For inelastically deforming media, analytical solutions equivalent to (4) are not available. Although the blunting and indentation problems may both be expressed as exact boundary value problems (at least

tThis relationship was obtained by applying conservation of volume to an incompressible, solid hemisphere of radius, r, compressed uniaxially within a rigid cylindrical die of constant radius, r, and initial height, r. The initial relative density is thus D, = 2/3. Additionally, the contact radius was related to the contact displacement, h, by h = 0.38az/r [see equation (18)].

for infinitesimal deformations), the complexity of the inelastic constitutive relations for plasticity and creep preclude analytical treatment. Approximate methods such as plane strain indentation or the blunting of perfectly plastic materials have been performed using slip-line field analysis. Solutions of this type result in simple contact plastic flow criteria of the form

Q, = Fo, (5)

where the flow coefficient, F, is a constant (independent of h/a) and by is the uniaxial yield strength. Examples are Prandtl’s solution for indentation with a flat punch [36] (F = 2.97) and Ishlinsky’s result [37] for a spherical indenter (F = 2.66). The densification models of Helle et al. [25], Kuhn and McMeeking [28] and Elzey and Wadley [15] all assume F is a constant coefficient equal to 3. Treating F as a constant (i.e. assuming the contact stress to cause flow by plasticity is a constant multiple of the uniaxial flow stress) implies self-similarity of the deformation, i.e. the conditions necessary for plastic yielding do not change with continued deformation. While this has been found to be approximately valid for indentation, it is not a realistic approximation for blunting bc- cause of the significant redistributions of stresses and strains that are likely to occur with increasing contact displacement, h.

Gampala et al. [29] have used finite element analysis to investigate the blunting of hemispherical asperities. Their results indicated that it was possible to retain the convenient form of the contact flow stress relation given by equation (5) if F is allowed to be a function of the relative density. Their results for the case of an elastic, perfectly plastic hemisphere subjected to constrained uniaxial compression could be fitted by a relationship of the form

F(D) = 340’ - 580 + 26 (6)

where the relative density, D, was related to the normalized contact radius, a/r, byt

D =[;-$>‘I]‘. (7) Equation (6) indicates F is a strong, non-monotonic function of effective strain. For an initial relative density of 213, F has an initial value of about 2.4 and then decreases with strain before exhibiting a rapid hardening. The softening arises because at first the plastic zone is able to more easily expand against a diminishing volume of surrounding elastically deformed material as the effective strain increases-a consequence of the non-self-similar deformation characteristic of the blunting process. Subsequent hardening arises from the lateral constraint imposed by adjacent contacts. Analyses with a work hardening constitutive law have shown that work hardening compensates for the softening phenomenon, and under some conditions, results in a (fortuitous) strain-independent F coefficient with an average value close to 3 [29].

GAMPALA et al.: CREEP BLUNTING OF CONTACTS 1483

The contact (indentation and blunting) deformation of work-hardening materials has also been investigated by Matthews [38]. His analysis began by assuming that the form of the contact stress solution for the linear elastic problem [equation (4)] would also be valid for power-law hardening plastic materials, i.e. those for which the uniaxial stress- strain response is given by

& = (a/&$. (8)

By matching the solution to the linear elastic and perfectly plastic (slip-line) solutions in the limits of material non-linearity, he obtained an expression for the contact stress of the form

(9)

Matthews’ relationship (9) is strictly valid only for indentation (since it was matched to indentation results), but he proposed its use for predicting the blunting of a work-hardening sphere as well.

The form of solution (9) agrees with empirical observations for indentation: for example, Meyer’s indentation experiments [39] showed that the average contact stress could be expressed as gc = k(a/2r)““, where k and n are material constants. Much later, the experimental work of O’Neill [33] and Tabor [40] demonstrated that n is just the power-law hardening

exponent in equation (8) and k is related to the hardening coefficient, cr,, in (8) by k = F(n)q,, with F dependent only on the material non-linearity.

Combining Meyer’s result for a, with that of O’Neill and Tabor, it is possible to write the contact flow stress-effective strain relation in the form

where the empirical results of Norbury and Samuel [41] have been used to relate the indentation depth, h, to the contact radius, a [see equation (19) of Section 2.2 below]. While analyzing the case of a spherical indenter penetrating a work hardening plastic solid, Hill et al. [42] were able to reproduce Meyer’s law as well as Tabor’s results, thus validating equation (10) as the correct form of the contact stress-effective strain relationship for plastic indentation. Their numerical computations were valid for deformations upto a/r = 0.8. We see that equation (10) is a gener- alization of equations (4), (5) and (9) for the indentation of elastic (i.e. n = l), perfectly plastic (n = co) and power-law hardening materials, respectively. Ex- pressions for the form of the flow coefficient, F(n), for each case are summarized in Table 1. Equation (10) provides a valid description of blunting only when the effective strain is small (u/r < 0.1); to be valid for large deformation blunting, it would need to be

Material behavior Theory

Table 1. Summary of contact mechanics models

Normalized Effective Model contact strain Flow coefficient basis’ stress (rate) Fb

Area coefficient,

cc

Elastic Hertz [34]

Slip-line field [37]

Perfectly plastic

Gampala

et al. [29]

Plastic/work hardening

Linear

viscous

Power-law creep

Matthews

[381

Lee and

Radok 1431

Elzey/Wadley

V51

Matthews

[381

Bower et al. [45]

I, B 2 (1 - 02) h _ a

I “c 1 %

B

h

a

8 3n

1 - z 0.7 Js

2.66

1.15

6n 16 I/”

0 2n + 1 9n

Tabulated F(n) Tabulated c(n)

aI = Indentation, B = Blunting. bF = normalized contact stress/effective strain (rate)““. cc = a@z.


modified by writing F as a function of both n and the effective strain, a/r [e.g. as represented by equation (6) for the perfectly plastic case].

Equation (10) is valid for rate-dependent material behavior as well, but with the effective strain, h/a replaced by the effective strain rate, h/a& and n now representing the stress exponent in Norton’s power- law creep relation (1). The simplest case to analyze is that of a linear viscous material [i.e. n = 1 in equation (l)]. Lee and Radok [43] showed that for this situation, the indentation contact flow stress-effective strain rate relationship has the form

where et, and &, are the reference stress and strain rate, as in equation (1). This is just the linear viscous analogue of the elastic result given by equation (4). By ensuring that the linear viscous solution was recovered in the limit, n = 1, and the perfectly plastic solution in the limit as n -+co, Matthews [38] generalized the Lee and Radok result to obtain the contact stress-effective strain rate relation for a power-law creeping material:

By employing Hill’s similarity principle [44] to trans- form the creep indentation of a half-space by a rigid indenter into that of a non-linear elastic half-space indented to a unit depth by a rigid flat punch of unit radius, Bower et al. [45] have recently shown that the contact stress-effective strain rate relation during the indentation of a power-law creeping half-space is generally of the form

5 = F&-j”“. (13) ’ ‘\a$)

. , 00

Lee and Radok’s solution [43] was recovered for the limiting case of a linear viscous solid (n = 1) and Prandtl’s slip-line field solution [36] was obtained for a rigid, perfectly plastic solid (n = CD). Bower et al. [45] used finite element analyses to determine the function F(n) for intermediate cases of material non-linearity. Figure 3 compares Bower et aZ.‘s results for F(n) (given in [45] in tabular form) with Matthews’ result (12). Although the detailed FEM analyses show the flow parameter, F, to vary slightly with indentation depth (i.e. effective strain), they concluded, in agreement with Hill [44], that F can be regarded as constant for small deformations (a/r < 0.4). A recent analysis of creep indentation due to Storakers and Larsson [46] also leads to a relation of the form given by equation (13).

Given the simplicity of equation (13) and its com- patibility with expressions used in densification models, it would be very convenient if the results describing the contact stress-effective strain rate relationship for blunting of power-law creeping materials could be cast in a similar form. A simple,

0.0 0.2 0.4 0.6 0.6 1.0

l/n Fig. 3. Matthews’ [38] interpolated estimate for the contact Row coefficient, F(n), for creep indentation is a good approximation to the more exact, numerical results of

Bower et al. [45].

yet accurate relation is important since the model describing the behavior of a single contact is after- wards inserted into rather complicated expressions for the relation between applied stress and macroscopic densification rate [l&28].

Because it can be shown that for small displacements, blunting and indentation are both described by the same set of equations, including boundary conditions, the results of Bower et al. [45] [equation (13)] must apply to blunting as well, at least for small displacements. However, due to the non-self-similar nature of blunting, it is unlikely that the function F in equation (13) will be independent of the effective strain beyond very small displacements. By rewriting (13) in the form

(14)

[i.e. equation (2)] the approach might be generalized to describe finite strain blunting. Equation (14) should recover the result of Gampala et al. [29] in the perfectly-plastic limit (i.e. as n + co), in which case, F (n-co, u/r) becomes identical to the plastic flow coefficient given by equation (6).

2.2. Contact radius-displacement relation

In order to use expressions of the form given in equation (14) for modeling densification processes, it is necessary to obtain an expression for the contact radius, a (Fig. 2). The area of a contact is obviously related to the displacement, h. Thus the contact radius, a, identified in equation (14) will not be independent of h and must be determined. The Hertz analysis of a hemispherical linear elastic constant [34] shows that the displacement, h, and contact radius, a, for a hemispherical body of radius, r, are related by

h,a2. r (15)


Equation (15) is valid for both indentation and blunting, but is restricted to very small displacements (a/r < 0.01).

Calculation of the displacement-contact radius relationship for inelastic deformations is more difficult. Fischmeister and Arzt [24] and subsequently, Helle et al. [25], Kuhn and McMeeking [28] and Elzey and Wadley [15], have avoided the detailed calculation by assuming the contact area to be given by a conservation of volume principle. They took the volume of material plastically displaced from a contact and uniformly redistributed it over the remaining (free) surface of the particle. When applied to a laterally constrained, hemispherical, blunting contact [29], the resulting contact radius-displacement relationship is given by

(16)

It is interesting to note that if equation (16) is expanded and only terms of order h are retained,

r ,

Although this is an approximate result (with an error of less than lo%), it is similar in form to that of Hertz [34]. Compared to the linear elastic contact of equation (1 S), equation (17) predicts twice the contact area for a prescribed displacement. Nonetheless, the uniform redistribution model underestimates the true contact area during blunting of a perfectly plastic material. Detailed finite element calculations [47] show that the relationship between laterally constrained blunting displacement and contact radius for a perfectly-plastic material is (to within 5%) given by

h ~0.385 r (18)

Therefore earlier consolidation models based upon the redistribution approximation have significantly underestimated the rate of growth of contacts.

Extensive experimental observations of contact area during indentation have led Norbury and Sa- muel [41] to suggest the existence of a displace- mentcontact radius relationship of the form

(19)

where c is a material constant. Equation (19) is a general result, describing the elastic and perfectly plastic indentation behavior with values of the contact radius coefficient, c, given in Table 1.

Rearran ement of (19) to give an expression for c( = a/ Jg- 2rh) allows a physical interpretation of the coefficient during indentation: it represents the ratio of the true to nominal contact radii, where “nominal”

refers to the radius of a section through a hemispherical indenter located a distance, h, from the point of initial contact (cf. Fig. 2). The results above are consistent with a more rapid “piling-up” of material adjacent to the contact than predicted by the uniform redistribution model. A similar physical meaning can be ascribed in the case of blunting (Fig. 2), (with some subtle differences as discussed below).

Based on Norbury and Samuel’s data [41], Matthews [38] proposed that the displacement- contact radius relation during the indentation of a power-law hardening material could be written in the form

(20)

Equation (20) in the limit as the material non-linearity n --*co, gives h -m’/er = 0.37a2/r, which is almost identical to the recent result for blunting [equation (18)]. The elastic result [equation (IS)] is also recovered when n = 1. Equation (20) is also valid when n refers to the stress exponent of a power-law creeping material [38]. Then, if the factor, [2n/(2n + l)]2@‘- ‘) is set equal to 0.5, the uniform redistribution model (17) will give a correct prediction for the contact area when n = 3.8 (which is in the middle of the range observed for many power-law creeping metals and alloys). Matthews’ result suggests that the coefficient, c, in equation (19) is a function of n (see Table 1). Bower ef aZ.‘s numerical results [45] for the evolution of contact area during indentation of a power-law creeping material were also found to obey equation (19) with c = c(n), (see Fig. 7 later).

From the preceding discussion (and from Table l), it is evident that regardless of the material behavior (linear or non-linear), the contact radius, a, is related to displacement, h, by an equation of the form

1 h=

a2 2c(n)2? 0

(21)

While equation (21) has been shown to hold for the blunting of linear elastic and perfectly plastic contacts, it has not yet been shown to be a valid representation for the blunting of a power-law creeping material (although this is implied by Matthews [38]). Below, we will assume its validity, calculate the contact radius coefficient, c(n), using finite element analysis and demonstrate that (21) well approximates power-law blunting for a/r < 0.8.

3. DETERMINATION OF F AND c COEFFICIENTS

Micromechanics-based approaches to the prediction of densification in both powder compacts and composite preforms relate behavior at individual contacts to the overall behavior by invoking equilibrium between applied and contact stresses, and by requiring that the densification rate be determined by the sum of the deformation rates of all the contacts.


In general, the applied stresses lead to macroscopic deformations comprised of dilatational (volume- changing) and shear (shape-changing) components, resulting in both normal and shearing displacements at individual contacts. Except for the very earliest stages of densification, in which rearrangement of particles often plays an important role, densification (i.e. the volume-changing strain component) during consolidation occurs primarily by displacements normal to interparticle contacts. Our analysis considers a single hemispherical contact subjected only to normal stresses and displacements. Since the zone of deformation at the contact is considered to be much larger than any microstructural features, no size dependence enters into the problem. The single contact model may therefore be regarded as a unit cell analysis which can be incorporated within a micromechanics model for an aggregate containing a spectrum of contact sizes.

We could identify a representative hemispherical asperity and analyze its isolated behavior when flat- tened between two frictionless parallel plates. How- ever, the contacts encountered in particulate consolidation problems are not free to laterally de- form in this way (except perhaps in the very earliest stages of densification). Instead, the deformations of neighboring particle contacts constrain lateral flow as densification proceeds. The severity of this will de- pend on the number and area of the adjacent contacts. Since this varies from particle to particle, it results in varying degrees of constraint on the flow associated with a given contact. In addition to the constraint of neighboring contacts, monotape surface asperities are also laterally constrained by their con- nection with the monotape [see Fig. l(c)]. Because of the stochastic nature of the internal geometric features of these aggregates like those in Fig. 1, the precise conditions under which any given contact deforms is not known and may at best be specified only as a probability. To include the influence of this “imposed” constraint on the creep blunting behavior,

L

Fig. 4. The representative contact blunting problem is taken to be a hemispherical, power-law creeping solid subjected to

uniaxial compression within a rigid cylindrical die.

we consider a hemispherical solid compressed uniaxially within an encircling, rigid cylindrical die (Fig. 4). We realize this is a significant simplification; in reality, some contacts will experience a more severe and some a lesser constraint than this. The simpler problem formulated here incorporates a representative contribution of the increasing incompressibility as the relative density of the cell approaches unity (a/r + 1). The analysis of the unconstrained problem would substantially underestimate u, (which would approach (TV and not co, as a/r -+l). Subsequent studies should explore this issue in more detail.

Our objective is to apply the power-law creep blunting theory, as given by equations (2) and (3), to determine the contact stress-effective strain rate response and the relationship between blunting displacement (h) and the contact area for the situation shown in Fig. 4. Our approach uses the finite element method to calculate the evolution of blunting displacement (h) and contact radius (a) with increasing unit cell density. Equation (3) is then used to determine the dimensionless contact radius coefficient, c, for each value of it. Since the temporal evolution of h and a is then known, h’ can be calculated and the flow coefficient, F(n, a/r), determined from equation (2). The results of c(n) and F(n, a/r) are then fitted to polynomial functions, which can be reinserted back into equations (2) and (3) to provide approximate closed form solutions to the laterally constrained creep blunting problem.

The axisymmetry of the FEM problem allows the solution to be obtained by analysis of a plane radial quadrant of the cell shown in Fig. 4. Contact deformation is assumed to occur by power-law (i.e. steady state) creep with a uniaxial stress-strain behavior given by equation (1). Elastic loading contributions are included [their significance is addressed separately (Section 4.3)], but since contacts typically undergo large inelastic strains during consolidation processing, the elastic deformations are negligible and more- over, do not contribute to permanent densification (although their calculation might be important in determining residual stresses in incompletely den- sified materials).

The FEM calculations were conducted by first elastically loading the hemisphere and then allowing it to creep under a constant applied load, L. The strain-displacement relations used are those for large deformations. The elastic deformations were determined assuming an isotropic, linearly elastic solid (with Young’s modulus, E = 70 GPa and a Poisson’s ratio, v = 0.3). The particular values of E and v have no effect on the results for the flow and contact radius coefficients, F(n, a/r) and c(n) and are in that sense arbitrary. The creep strain rate components, iii, were calculated using a power-law creep relation for a material under a general state of stress


where S, ( =aij - ~,,6,~/3) are the deviatoric stress components, a, (=J&%&

is the Mises equivalent stress and 6, and o,, are the reference strain

rate and stress, respectively. The functions, F(n, a/r) and c(n), are independent of the reference stress and strain rate, a,, and $, and so they were assigned arbitrary values of 10.0 MPa and 0.01 SK’, respectively

Two frictional contact conditions (Fig. 4) were analyzed:

@,, = Is23 = 0 Ir,l < a (frictionless) (23)

i, = tii, = 0 1~~1

1488 GAMPALA et al.: CREEP

Table 2. Numerical solutions for the coefficients c and F

Max HT0r

l/n% Cb Fb (%) air’

1.0

0.7

0.5

0.2

0.1

0.05

0

0.867

0.898

0.923

1.059

1.112

1.139

1.1691

0.86 + 12.95

not determined

2.2-4.49(t)+ 11.94(;)

2.83 -4.41(f) + 5.28(;)

2.86 - 3.92(;) + 3.84(f)

2.95-3.46($+2.81~)

2.95 - 3.46(;y + 2.81(f)

15

4.4

2.0

2.2

-

2.0

0.2

-

0.43

0.62

0.7

-

0.2

“For all n > 20, changes in the dependence of Fan a/r are negligible. bAverage of frictionless and no-slip contact results. cNormalized contact radius at which the maximum error occurs.

deformation at the contact with increasing n, is the piling-up of material at the contact. This resulted in a higher rate of increase in contact radius (for a given h/r-value as n + co). For example, when the blunting displacement is small (h/r = 0.08), the normalized contact radius a/r = 0.32 for n = 1 whereas for the same displacement, a/r = 0.43 when n = 10. These three effects are all reflected in a dependence of the coefficients F and c on the stress exponent.

4.1. Contact stress relations

The normalized mean contact stress (a,/a,) and normalized effective strain rate (h/a&) were determined from the FEM analysis, and their ratio com- puted to obtain the flow coefficient, F, at various values of effective strain, a/r. These data were then fitted to a quadratic function of the effective strain for each n-value (best fit expressions are given in Table 2). For the n + cc case, equation (2) agrees with the results of Gampala et al. [29] for perfectly plastic blunting, i.e F(n+oo, a/r), as given in Table 2, is identical to the plastic flow coefficient, fi(a/r), in Ref. [29]. A similar check in the n = 1 limit cannot be made since there are no available solutions for the constrained linear viscous blunting problem (though it does tend to the solutions of Matthews [38] and Bower et al. [45] as a/r-r0 for indentation).

Figure 6 shows a plot of the flow coefficient, F(n, a/r) versus the effective strain, a/r for a wide range of creep exponents. It is apparent that F is a strong function of both n and the normalized contact radius, a/r. For small effective strains (a/r < 0.4), F is lower for materials with a low stress-sensitivity (i.e. low value of n). These materials will experience a greater blunting velocity (for a given contact stress) than their higher-n counterparts. We also see that the curves for n-values greater than 1 are all character- ized by a minimum. Initial deformation in these cases is accompanied by strain softening. This is a geo-

BLUNTING OF CONTACTS

metric or “shape” effect, in which the “material” constraint imposed by surrounding, elastically deformed or more slowly creeping material on the deforming region near the contact is lowered as the effective strain increases. This occurs because as a hemisphere is deformed by blunting, its geometry gradually approaches that of a right circular cylinder which, in the absence of friction and lateral constraint, deforms uniaxially [i.e. e = &,(c/o,,)n for which F equals unity] with homogeneously distributed stress and strain rate.

The subsequent increase in F with effective strain (the start of which depends sensitively on the value on n) is a direct manifestation of the lateral (externally imposed) constraint. As material fills the void between the hemisphere and its surrounding constraint, the system becomes less compressible. In the limit as a/r + 1, incompressibility requires F+ co. Materials with low n-values (for which deformation is the most homogeneous, see Fig. 5) almost immediately experience the imposed constraint (note that for the n = 1 case, lateral constraint hardening is evident even at the smallest deformations). As n-rco (i.e. in the perfectly plastic limit), deformation is concentrated at the contact and is little affected by the constraining die wall until the blunting deformation is quite large (a/r > 0.8). Overall, the influence of the external constraint is very strong: note that all the curves in Fig. 6 would approach 1 as a/r + 1 in the absence of this constraint. External constraint of this type is inevitable during consolidation and its consequences must be included if a model is to be obtained which is reasonably accurate at large deformations. The error in predicted response if external constraint is not included will be most severe when modeling the behavior of linear viscous (low-n) materials.

Predictions for F based on Matthews’ indentation model (12) are shown for comparison as dotted lines in Fig. 6. The indentation creep results of Bower et al.

3’5------l 3.0

t

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I . . . . . . . . . . . ml

e 2.5

z t

...................................... : y

............ ,o ... “‘z.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 .hh_._.2?. 4 ....... ...... E 2.0 .g E $! 1.5-

z P LL l.O- y”=’

/ . . . . . . . . . . ” = , 1

0.5 t

. . . . . Matthews (equation 12) i

0.0 1, 0.0 0.2 0.4 0.6 0.8

Normalized contact radius, a/r

Fig. 6. The contact flow coefficient, F(n, a/r), characterizing the power-law creep blunting response of a single

contact for various creep stress exponents.


Relative density, D 0.667 0.70 0.75 0.30 0.85 0.30

0.8

m 0.6

5 e F

G s c 0.4

8

z 2

2 6 0.2

z

o.ov I I I I I I I 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

+s-E

Fig. 7. FEM results for the relationship between the contact radius and blunting displacement. The curves may be ap- proximated as straight lines which, according to the blunting theory [equation (3)], should have constant slope, equal to

the contact radius coefficient, c.

[451 are very similar to those predicted by Matthews less accurate, but closed form solution and could equally well have been shown. We observe that the flow coefficient for indentation (dashed curves in Fig. 6) is independent of a/r because there is no loss of material constraint during indentation (a semi-infinite body of material always surrounds the deformation zone). It can be seen that for perfect plasticity (n --f co), the Matthews solution (i.e. the slip-line field result for indentation with a flat punch) overestimates F (and hence the contact flow stress) required to cause blunting, whereas for n = 1 or 2, the flow stress is underestimated.

Piling up

The difference in the predictions of blunting and indentation theory can be understood as follows: at high n, indentation is more difficult than blunting (i.e. indentation underestimates the rate of creep blunting) because of the softening associated with diminishing material (shape) constraint during blunting. (Other- wise, if no softening occurred, F would be about the same for both indentation and blunting.) At low n, where deformations are more uniform, blunting is more difficult than indentation because of hardening caused by the lateral constraint imposed by neighboring contacts; in the absence of lateral constraint the behavior during blunting and indentation would again be quite similar. Thus the softening due to diminishing material constraint dominates at high n and hardening due to external constraint at low n. Without these effects, F would exhibit little dependence on a/r and indentation models would apply equally well to blunting. As shown below (Section 5), these differences in the predicted flow coefficient can, depending on the value of n, lead to substantial error in the expected overall densification rate.

4.2. Contact radius

Equation (3) predicted that a lot of normalized contact radius, a/r, against P- 2h/r, results in a straight line for each n-value) with a slope, c. Both a/r and /- 2h/r can be determined from the finite element analyses for each creep exponent, n. Figure 7 shows that roughly straight line behavior is exhib- ited by the numerical results when n -+ cc. It is a less accurate description of materials with low n-values. The results show that the contact radius grows most slowly for materials with low n-values. The shaded region of Fig. 7 indicates that regardless of n, the

Sinking in

Blunting Piling up N- 4 Sinking in Hunting

Indentation indentation

4a

‘r 1 _

-i LEX

0.0 0.2 0.4 0.6 0.8 1.0

l/n

Fig. 8. The contact radius coefficient, c, as a function of creep stress exponent. The c > 1 case corresponds to “piling up” of material near the contact and occurs when n is high; c < 1 indicates “sinking in”. The

behavior during blunting is very similar to that of indentation except at low n-values.


coefficient, c (representing the average slope), always lies in the narrow range between 0.8 and 1.2.

The best straight line has been fitted through the results shown in Fig. 7 and the average slope, c, determined. These results are given in Table 2. The dependence of c on n during blunting can be approxi- mated (to within 2%) by

c(n) = 1.17 -0.64@ + 0.34($ (25)

The observation that c depends negligibly on the displacement (i.e. depth of penetration) during indentation is not surprising in light of the self-similar nature of the indentation process, but it is at first sight surprising that the same is true for blunting where finite size effects are so significant.

Figure 8 shows a plot of c against l/n. For comparison, the predictions of Bower et al. [45] and of Matthews [38] [equation (20)] for indentation are also shown. When c > 1, the true contact radius exceeds the nominal, indicating the “piling up” of material adjacent to the contact. From Fig. 8, this is seen to occur for materials with a high n-value (n > 3-4) and is a result of the stress and strain concentrations near the contact (Fig. 5). Figure 8 shows a slightly greater tendency for this pile up to occur during indentation. For intermediate n-values, the indentation results of Bower et al. [45] and those for blunting are seen to be almost identical. However, when n < 3, c < 1, indicating that “sinking in” occurs, i.e. material is pressed into the blunting asperity (Fig. 8). Bower et al.% [45] and Matthews’ [38] work suggest that this phenomenon occurs more easily during indentation than blunting.

4.3. Infiluence of elasticity

The ratio of true to nominal contact radius, c, and the flow coefficient, F, relating the contact stress, u,, with blunting velocity, /i, were calculated above for the case in which the elastic displacements are much smaller than the inelastic deformation. When applying these results to consolidation, it has been assumed that elastic contributions to the contact area can always be ignored. The likelihood that this assump- tion will lead to appreciable error can be assessed using an idea of Bower et al. [45]. The normalized blunting displacement of an incompressible linear elastic hemisphere, he/a, can be expressed as

h’ Qc qF(n,a/r) h’ ‘in -=EF(l,a/r)=EF(l,a/r) z . 0

(26) a

Following Bower et al. [45], we can define a parameter, A = F(n, a/r)h/F(l, a/r)he, which is a measure of the significance of elasticity (note here h is the actual, i.e. elastic plus inelastic, displacement). Using the definition of F(n, a/r) as given by equation (2) and solving equation (26) for F(l, a/r), A can be written as

(27)

Thus A can be interpreted as the ratio of the stress required to cause blunting of an elastic cylinder of gage length, a, by an amount, h, to the stress required to blunt a cylinder by power-law creep at a rate, h’. Large values of ,4 indicate that the elastic strains are small relative to the total deformation. Elastic effects are negligible for A & 1 and can be ignored when A > 50.

predominantly Plastic

0.3 0.4 0.5

Effective strain, a/r Fig. 9. The factor, A, which reflects the importance of elastic effects during blunting, is plotted against a/r for various values of the stress exponent, n; A increases quickly with a/r in all cases, indicating the

insignificance of elastic effects during high temperature blunting.


Figure 9 shows a plot of A as a function of normalized contact radius, a/r, for various power-law exponents. Elastic contributions to the effective strain (or contact radius) are seen to be most significant when a/r is very small and deformation is perfectly plastic. For this case, deformation is concentrated near the contact, and much of the blunting solid has not exceeded the yield strength, i.e. it is elastically deformed. The effective strain (u/r) beyond which elastic effects are negligible, even for the perfectly plastic case, is about 0.12. For a hemisphere within a cylindrical die, this corresponds to a relative density of 0.67. This is only slightly above the initial relative density of 0.66, and so for all practical purposes the elastic strains can be neglected during densification by power-law creep blunting.

This detailed analysis of blunting leads to the conclusion that the contact stress-effective strain rate relationship for blunting is quite different from that of indentation, even though similar relations can be used to define the flow coefficient, F [cf. equations (2) and (13)J The contact mechanics of blunting are strongly influenced by the finite size (material constraint) and lateral constraint imposed on the blunting body, whereas during indentation, deformation is remote from all but the indented surface and is thus predominantly free from edge effects. For this reason, the contact flow coefficient is a function of the effective strain during blunting, and must be included in any accurate effective constitutive model describing densification by contact blunting.

5. APPLICATIONS

The blunting model developed above accounts for two effects which could not be considered by an analysis based on indentation: (i) the effect of the finite size (i.e. the diminishing materal constraint) of the blunting body as a/r -+ 1; and (ii) the influence of externally imposed lateral constraining forces (arising either from nearby contacts or from the substrate to which the blunting body is attached). The significance of incorporating these blunting mechanics results into consolidation models can be evaluated by using these new blunting results to predict densification and fiber fracture and compare the new predictions with those of previous indentation-based models [15, 16,281.

5.1. MMC monotape derkjication

At relative densities below about 0.9, densification of spray-deposited MMC monotapes during either hot isostatic or vacuum hot pressing has been modeled to occur by contact deformation of surface asperities [15]. The size and location of the asperities

TThe factor a, given in Ref. [15] for the indentation-based model contains an error and should read a = 0.34 (@)‘ -“2”*-“. All calculations reported here are based on the corrected factor.

were assumed to bc statistically distributed and the resulting development of new contacts, together with the continued deformation of ones formed earlier, were incorporated into the model. By realizing that at any instant the externally applied consolidation force must be in equilibrium with the sum of forces acting on all existing contacts, a relation between the applied stress, C, and the contact forces, L, was developed

X s Oc’ 1 exp( --Ir)L(H, r, z, i) dr dH (28) 0 where, z, the compacted monotape thickness, is related to the relative density of the monotape, D, by z = zoDo/D, z. is the initial (undeformed) monotape thickness, I is the area1 density of asperities, r and H are the radius and undeformed height of a particular asperity, respectively, and a,, Z? and 1 are statistical parameters characterizing the distribution of asperity sizes.

The force, L, acting on a contact can be obtained for the case of power-law creep from either equation (13) (indentation model) or from equation (2) (blunting model), by noting that the contact stress, u= = LIza’. For the blunting case, equation (3) can be used to eliminate the contact radius, a, in equation (2) giving

L = nuOF [2rhc(n)2]((2”-‘)‘2n) 0

I; I’“. - (29) 80

The overall rate of compaction, i, (equal to the asperity displacement rate, h’), is obtained by substituting the contact force (29) into the statistical model (28) and solving for I;. Since i = zo(Do/D2)d, the densification rate obtained using the blunting model is

d,= a(n)Z”D2

ZODO

x exp[ -fr$)]dH

s a, --n X 3.r’ -w’~) exp( - Ar) dr (30) 0 where H - z is just the displacement, h, and TV is given by

cr(n) = ~o(2c(n)2)“2-n

(%)n .

Examples of the ratio of the densification rates obtained using the blunting (8,) and indentation {equation (19) in Ref. [15]t} (d,) models are plotted as a function of relative density in Fig. 10 for two example materials (Cu and Ti-24Al-1 lNb, an a2 + fi titanium alloy) chosen to have widely different


Monotap S-Lamha Power-law Creep Mechanism

0.50 ’ I I I I 0.4 0.5 0.6 0.7 0.6 0.9

Relative Density, D

Fig. 10. Ratio of densification rates for spray-deposited MMC monotapes as predicted by blunting and indentation analyses of contact deformation. For materials with high n-values (e.g. Cu with n = 4.8), the indentation model underestimates the densification rate, while for materials with low n-values (n = 2.5 for Ti-24Al-1 lNb), indentation

theory overestimates the densification rate.

creep properties, Table 3. The steady state creep behavior of Cu [49] was described by

g = A {D,,,, exp[ - Q,/RT]) FT 0

i ” (32)

with material parameters as defined in Table 3. The temperature-dependent shear modulus was taken to be G,(l -0.54(T -300/T,)), where GO, the room temperature shear modulus, was 42.1 GPa.

The power-law creep behavior of the a* + /I titanium alloy was described by

8 = A % 0

‘exp(-QJRT)

with the values of material parameters as defined in Table 3. The relative density shown in Fig. 10 refers to that of the surface roughness layer (designated “S-1amina”) and not to the composite laminate as a whole (see Fig. 3 in Ref. [15] for its definition). Hence, depending on the standard deviation of the surface roughness distribution, the starting density is around 0.4 (as compared to the initial overall density of around 0.65). Since both densification models have the same dependence on pressure and temperature, these terms cancel and the ratio of densification rates is a function of only the relative density.

Figure 10 shows that, depending on the material’s creep parameters and relative density, earlier indentation-based models in some cases overestimate, and in others underestimate, the densification rate. Indenta- tion analysis underestimates the blunting rate for

materials like Cu for all densities greater than about 0.47. On the other hand, the densification rate of materials like Ti-24Al-1lNb is always overestimated on the basis of indentation. The differing results for these two materials can be understood in terms of the single-contact behavior: Fig. 6 shows that for materials with a relatively high n-value, such as Cu, the coefficient, F, for indentation is greater (implying lower densification rate for a given stress) than that for blunting when a/r > 0.25. For Ti-24Al-llNb, which has an n-value of around 2, the opposite is true: F for indentation is generally lower (greater densification rate for a given stress) than that for blunting. Both curves are also seen to exhibit a maximum. This arises because at first &/B, increases due to the softening associated with blunting (the shape effect associated with loss of material constraint). This is followed by a decreasing &/d, as the blunting response hardens due to the increasing influence of neighboring contacts (laterally imposed constraint) at large strains.

5.2. Fiber fracture during densljication of monotape

Ceramic fibers in metal matrix composite monotapes are susceptible to microbending and fracture during consolidation processing [12]. During consolidation, bending occurs because the asperity contact stress, uc, results in localized forces distributed ran- domly along the length of the fibers. The evolution of the bending and fracture has been modeled by consid- ering a statistical distribution of unit cells, each consisting of a segment of fiber undergoing three- point bending due to forces imposed by contacting hemispherical asperities [16]. The contact force was obtained using an approximate, indentation-based model. By substituting the power-law creep blunting prediction for the contact force into the fiber bending unit cell of Ref. [16], the evolution of fiber damage can be compared with that of the original model.

Previous fiber fracture models [16] assumed the asperity response to be given by a uniaxial creep response (i.e. F = 1):

where y is the deformed height of the asperity, (r - h). The average contact stress, oc = L/na’, where L is the applied force. The force exerted on the fiber by the asperity must be balanced by the bending reaction so that L = 2k,(z - y), where k, is the fiber bend stiffness and z is the monotape thickness [so that

Table 3. Creep narameters used for model medictions

Parameter. svmbol (units) cu 1491 Ti-14Al-2lNb 1151

Creep constant, A (b-l) stress exponent, n Activation energy, Q, (kJ moleI) Young’s modulus, E (GPa) Pre-exp volume diffusion, D,,, (III* sr’) Melting temperature, T,,, (K) Bureers vector. b (ml

3.4 x 106 6.0 x 10” 4.8 2.5

197.0 285.0 140.0 - 0.12 T (“C)

2.0 x 10-5 1356

2.5 x 10-l’


2(z - y) represents the fiber deflection]. By approxi- mating the contact area as 27rr(r -v) [cf. equation (17)], equation (34) can be written as a first order ordinary differential equation in y

A similar differential equation for blunting can be obtained by rearranging equations (2) and (3)

3 = -$J2r]y - rlc(n)’

’ %(z -Y)

2xra,( y - r)c(n)‘F(n, ( y - r)/r) 1 ’ (36) where h’ = 3.

The number of fibers fractured per unit length of fiber can be calculated for any consolidation cycle by incorporating the unit cell model [either equation (35) or (36)] into the macroscopic model given in Ref. [16]. Figure 11 shows the cumulative number of fractures

‘i .c 80- Y

c -

3 60-

5 5 2 40-

LL

20 -

0

0.4 0.5 0.6 0.7 0.8 0.9

Relative density, D

(b) 240 I I I

(b) n = 4.8 (Copper) 200 - T = 543K (0.4Tm)

7 z. 160 - u.

c -

8

120-

5 ti F 60-

IL

40 -

0.5 0.6 0.7 0.8 0.9

Relative density, D

Fig. 11. Comparison of fiber damage during consolidation of spray-deposited composite monotapes predicted on the basis of indentation and contact blunting (a)

Ti-24Al-11 Nb/SCS-6; (b) Cu/SCS-6.

during densification of composite materials with matrix properties of either Cu (n = 4.8) or Ti-24Al- 11Nb (n = 2.5) and reinforcements with SCS-6 (SIC) fiber properties (the fiber properties given in Ref. [ 161). The model based on blunting mechanics predicts a higher number of fractures in both cases, with the correction being more significant in the case of Ti-24Al-11Nb. The unrealistically low value of F (= 1) used in the earlier model for this case in particular resulted in an underestimate of the fiber damage probability of around 40%.

5.3. Consolidation of metal powders due to power-law creep

The model of Kuhn and McMeeking [28] is widely used for predicting the consolidation behavior of metal powders by power-law creep. Starting with the average normal stress at an interparticle contact, cr, (given by the indentation solution of Fischmeister and Arzt [24]), they write the energy dissipation rate per unit area of contact for a pair of particles as u,h’, where h’ is now the relative normal velocity of the two contacting particles. Similarly, the creep dissipation rate for a pair of blunting particles [with a, given by equation (2)] is:

u,ti =o,F n,f r (37)

Kuhn and McMeeking [28] assumed F = 3 and also used the result of Helle et al. [25] to relate the average contact radius to the overall (average) relative density, D:

a/r = [(D - Do)/(3(1 - ~oW12, where Do is the initial relative density.

However, the material non-linearity influences the relationship between contact size and density, with a growing faster with D as n increases. For the case of blunting, combining equation (3) with the expression D = 2r/(3(r - h)) (which can be obtained from conservation of mass for a hemispherical contact subjected to constrained uniaxial compression) gives

a/r =,(,)&(I -2) (38)

where values of c(n) are given in Table 2. The next step is to relate the creep dissipation for

a single contact to the total dissipation occurring per unit volume in the powder aggregate. Following Kuhn and McMeeking [28], this is

+ 2&/3l’ + ‘in sin 4 d4 (39)

where l? and fi are the macroscopic deviatoric and dilatational strain rates, 4 locates the angular pos- ition of a contact relative to the (cylindrical) coordi- nate system, and K is given by:

K = f(,,‘?)“n&n, D)D2 2c(r~)~(D - Do) 1 - I’*”

DO > (40)


where P = F(n, a/r) with a/r given by equation (38). For a given material creep exponent, n, the coefficients c and F may be obtained from Table 2.

With the exception of the parameter, K(n, D), equation (39) is identical to the creep dissipation rate obtained by Kuhn and McMeeking [28] based on indentation analysis of single contact behavior. In the blunting approach, this parameter replaces their coefficient, C(n, D) [not to be confused with c(n) given in equation (3)]

The relation between macroscopic stress and strain- rate can be expressed in potential form so that C, = n/(n + i)awvjaE,j or inversely, _Ei, = atr/az,, where C, is the applied mean stress, & the applied effective stress and the lower bound estimate for Y is

Y =z,A+z,B- 5 W”(fi> 8). (42) The structure of the macroscopic constitutive relation (as expressed by the potentials) is unaffected by the choice of a blunting- or indentation-based analysis of a single particle contact. The only difference is in the magnitude of the factors K and C (which are independent of C,j and gij). Therefore the shape of the creep contours (i.e. constant values of !P plotted in stress- space) is the same for both models, but their magni- tudes at any point in stress-space may differ significantly, depending on the value of the stress-exponent and relative density. The factors, K(n, D) and C(n, D), given by equations (40) and (41) represent the magnitude or strength of the potential (increasing values of K or C correspond to increasing creep resistance). Thus a comparison of the macroscopic behavior predicted on the basis of blunting and indentation is obtained from the ratio of K(n, D)/C(n, D). Figure 12 shows K/C as a function of relative density for two n-values: for low n < 2,

1.6 I n = 2 tndentation overe*timates c \ blunting creep rate 1.4 A

I n = 10 tndentat~on underestimates

blunting creep rate

0.75 0.80 0.85 0.90

Relative density, D

Fig. 12. The ratio of creep resistance coefficients, K and C, indicating the relative strengths of creep dissipation during blunting and indentation, respectively; if the stress exponent is low (e.g. n < 2), then K/C > 1, indicating that the creep resistance during blunting is greater than that predicted by indentation theory. Indentation therefore overestimates the overall densification rate of powders for low n. The opposite

trend is observed for high n (e.g. n = 10).

2.5 r I I

Powder Stage I Consolidation Power-law Creep Mechanism

2.0 - a-

a”

.s E

1.5 - ” = 4.6 (Copper) /‘I p 5

‘Z z 1.0 E t

$ 0.5 n = 2.5 (Ti-24AI-11 Nb)

0.0 I I I I 0.65 0.70 0.75 0.60 0.65 0.90

Relative density, D Fig. 13. Examples of the ratio of powder densification rates predicted using contact blunting and indentation for copper

(n = 4.8) and Ti-24Al-11Nb (n = 2.5).

K/C > 1, indicating that the blunting model predicts a greater creep resistance, as expected from single- contact behavior (Fig. 6). As n increases, the softening associated with the shape effect during blunting lowers the contact creep resistance. Thus, for n = 10, the blunting-based prediction leads to lower creep resistance than on the basis of indentation (i.e. K/C < 1) for all D > 0.73. To evaluate the consequence of this for the densification rate, we note that the dilatational component of strain rate fi (= -d/D), obtained by differentiating the creep potential (42) with respect to macroscopic mean stress is

IC 1 (n + 1)/n fi = -2d,{2~(2/3)@+ n/n -n

I[( > *

+(EJ”“l”~I(%)ii. (43)

Figure 13 shows the ratio of densification rates due to blunting (6,) and indentation (6,) for Cu and Ti-24Al-1lNb powders, whose power-law creep properties are given in Table 3. The densification rate of a low n-value material such as Ti-24Al-llNb, is overestimated on the basis of indentation whereas for high n-value materials like Cu, indentation underestimates the densification rate (in this case for densities greater than about 0.76). The differences can clearly be very significant (depending on the values of n and D).

6. CONCLUSIONS

The availability of models for analyzing indentation has led to their widespread use for estimating the response of deforming contacts in situations where blunting is a more physically accurate description of the deformation. A relatively simple, yet accurate model for blunting has been proposed and a finite


element analysis of a laterally constrained, power-law creeping hemispherical contact used to calculate the strain and stress exponent dependence of a flow coefficient, F and the contact radius coefficient, c, as a function of blunting displacement. The contact stress-displacement rate relationship during blunting is controlled by the degree of constraint imposed on the flow of material near the contact: initially, the loss of internal (material) constraint as the deformation field becomes more homogeneous dominates, leading to a strain softening; this is followed by hardening in which the influence of externally imposed lateral constraint (due to the presence of neighboring contacts) become dominant. The model has been used to revise existing consolidation models for powders and spray deposited monotapes. It has been shown that the earlier models can either substantially over- or underestimate the densification rate. They also failed to incorporate lateral constraint hardening which, depending on the value of the creep exponent, can exert a strong influence on the contact flow stress even when the densification strains are low.

Acknowledgements-The authors are grateful to M. F. Ashby, R. M. McMeeking, N. A. Fleck and J. M. Duva for helpful discussions. The financial support of the Advanced Research Proiects Agency (Program Manager, W. Barker) and the National Aeronautics-and Space Administration (Program Manager, D. Brewer) through grant NAGW 1692 is gratefully acknowledged.

REFERENCES

1. L. R. Olsson and H. F. Fischmeister, Powder MetaN. 21, 13 (1987).

2. F. H. Froes and J. E. Smugeresky (editors), Powder Metallurgy of Titanium Alloys. TMS, Warrendale, PA (1980).

3. G. H. Gessinger, Powder Metallurgy of Superalloys. Butterworths, London (1984).

4. R. Eck, Int. J. Met. Powder Technol. 17, 201 (1981). 5. D. M. Dawson, in High Performance Materials in

Aerospace (edited by H. M. Flower), p. 182. Chapman and Hall, London (1995).

6. G. S. Upadhyaya, Metals Mater. Proc. 1, 217 (1989). 7. A. Bose, B. Moore and R. M. German, J. Metals 40,14

(1988). 8. R. M. German, Powder Metallurgy Science. Metal

Powder Industries Federation, Princeton, NJ (1984). 9. P. K. Brindlev. in High Temperature Ordered Inter-

metallic Alloys-H (edited by N.‘S. Stoloff et al.), p. 419. MRS, Pittsburgh, PA (1987).

10. D. G. Backman, JOM 42(7), 17 (1990). 11, R. J. Schaefer and M. Linzer (editors), Hot Isostatic

Pressing: Theory and Applications. ASM, Metals Park, OH (1981).

12. J. F. Groves, D. M. Elzey and H. N. G. Wadley, Acta metall. muter. 42(6), 2089 (1994).

13. M. F. Ashby, in Hot Isostatic Pressing of Materials: Applications and Developments, p. 1.1. Royal Flemish Society of Engineers, Antwerp, Belgium (1988).

14.

15.

16.

17.

18.

19. 20.

21.

22. 23. 24.

25.

26.

27.

28.

29.

30.

31. 32.

33. 34. 35.

36. 37. 38. 39. 40.

41.

42.

43.

44. 45.

46.

47.

48.

49.

K. J. Newell et al., in Recent Advances in Titanium Metal Matrix Composites (edited by F. H. Froes and J. Storer), p. 87. TMS, Warrendale, PA (1995). D. M. Elzey and H. N. G. Wadley, Acta metall. mater. 41(8), 2297 (1993). D. M. Elzey and H. N. G. Wadley, Acta metall. mater. 42(12), 3997 (1994). H. N. G. Wadley and R. Vancheeswaran, in Recent Advances in Titanium Metal Matrix Composites (edited by F. H. Froes and J. Storer), p. 101. TMS, Warrendale, PA (1995). D. S. Wilkinson, Ph.D. thesis, Cambridge University (1977). E. Arzt, Acta metall. 30, 1883 (1982). Y. M. Liu, H. N. G. Wadley and J. M. Duva, Acta metall. mater. 42(7), 2247 (1994). D. S. Wilkinson and M. F. Ashby, Acta metall. mater. 23, 1277 (1975). D. M. Marsh, Proc. R. Sot. 279, 420 (1964). K. L. Johnson, J. Mech. Phys. Soiicis 18, 115 (1970). H. F. Fischmeister and E. Arzt, Powder MetaN. 26, 82 (1983). A. S. Helle, K. E. Easterling and M. F. Ashby, Acta metall. mater. 33, 2163 (1985). E. Arzt, M. F. Ashby and K. E. Easterling, Metall. Trans. 14A, 211 (1983). K. Uematsu et al., in Hot Isostatic Pressing: Theory and Applications (edited by R. J. Schaefer and M. Linzer), p. 159. ASM, Metals Park, OH (1981). L. T. Kuhn and R. M. McMeeking, Int. J. Mech. Sci. 34, 563 (1992). R. Gampala, D. M. Elzey and H. N. G. Wadley, Acta metall. mater. 42(9), 3209 (1994). A. R. Akisanya and A. C. F. Cocks, J. Mech. Phys. Solids 43, 605 (1995). W. D. Nix, Metall. Trans. A 20, 2217 (1989). K. L. Johnson, Contact Mechanics. Cambridge Univer- sity Press, Cambridge (1985). H. O’Neill, Proc. Inst. Mech. Engrs 151, 116 (1944). H. Hertz, J. reine u. angew. Math. 92, 156 (1882). P. M. Sargent and M. F. Ashby, in Proc. Int. Colloq. on Mech. of Creep of Brittle Materials (edited by A. C. F. Cocks and A. R. S. Ponter), pp. 326344. Elsevier Applied Science. Amsterdam (1991). L. Prandtl. Gottinger Nachr. Math. Phvs. 74, 37 11920). A. J. Ishlinsky, FMM 8, 201 (1944). ’ . ’ J. R. Matthews, Acta metall. 28, 311 (1980). E. Meyer, Z. Ver. deutsch. Ing. 52, 645 (1908). D. Tabor, Hardness of Metals. Clarendon Press, Oxford (1951). A. L. Norbury and T. Samuel, J. Iron Steel Inst. 117, 673 (1928). R. Hill, B. Storakers and A. B. Zdunek, Proc. R. Sot. A423, 301 (1989). E. M. Lee and J. R. M. Radok, Trans. Am. Sot. Mech. Engng 27, 438 (1960). R. Hill, Proc. R. Sot. A436, 617 (1992). A. F. Bower, N. A. Fleck, A. Needleman and N. Ogbonna, Proc. R. Sot. A441, 97 (1993). B. Storakers and P.-L. Larsson, J. Mech. Phys. Solids 42, 307 (1994). R. Gampala, PhD. Thesis, p. 104, University of Virginia (1994). ABAQUS FEM Code, developed by Hibbitt, Karlsson & Sorensen, Providence, RI (1984). H. J. Frost and M. F. Ashby, Deformation Mechanism Maps. Pergamon Press, Oxford (1982).

Date post:	25-Jan-2021
Category:	Documents
Upload:	others
View:	0 times
Download:	0 times

POWER-LAW CREEP BLUNTING OF CONTACTS AND ITS ...diffusional creep are dominant and in this case the...

Documents