Elastic Behaviour of Spherical Particles Reinforced

Elastic Behaviour of Spherical Particles Reinforced Metal-Matrix Composites J.D. Botas

1, a*

, A. Velhinho

1,2,b

1,2,c

and Rui J.C. Silva

1

DCM – Materials Science Department, Faculty of Science and Technology, New University of Lisbon, Quinta da Torre, 2829-516 Caparica, PORTUGAL 2

CENIMAT – Materials Research Centre, New University of Lisbon, Quinta da Torre, 2829-516 Caparica, PORTUGAL a

b

c

[email protected], [email protected], [email protected]

Keywords: Composite mechanical properties, Composite stiffness, Spherical reinforcements dispersions, Micromechanics, Damage assessment.

Abstract. Current technology provides means of fabrication of spherical micro-particles, either hollow or compact, for all engineering materials. Such spherical particles can be further embedded into another material to build-up either random dispersions or close-packed arrays, according to the production route and the degree of anisotropy intended for the ultimate composite material. In this study, a simple analytical formula for the composite stiffness is derived from an early micromechanics model, to describe the actual reinforcement of ductile matrices by a random dispersion of uniform spherical ceramic particles. Predictions from this model are checked against some other relevant models, and specific features arising from its theoretical derivation are pointed out. Introduction Reinforced metal composite materials (MMCs) were the first to be studied in a systematic way, giving rise to many of the models now currently used for other reinforced matrix systems. They evolved from the utilization of continuous fibre reinforcement, leading to low ductility materials where yield and fracture were virtually simultaneous processes, to the more widespread use of shorter fibres reinforced materials where plastic flow in the metal matrices precedes fibre fracture. Subsequent attempts to lower production costs and approach more metal-like levels of toughness and ductility, gave rise to particulates produced either by casting or powder metallurgy routes. The use of a casting process to incorporate ceramic particles into light alloys (e.g. aluminium alloys) still remains the cheapest process for MMC production, typically resorting to 10 – 20 % volume fractions of ceramic particles around 15 μm in size. Increased strength can be further obtained by creating within the metal matrices some fine dispersions of nanometer-size particles (oxides, carbides, borides), around 0.1 μm in diameter at volume fractions below 15%, aimed at impeding the motion of dislocations. More recent trends sought instead to improve the composite reinforcements by reducing their sizes (as for nano-sized particles and tubes) or exploiting their shapes (e.g. spherical particles, either compact or hollow). This latter approach in particular relies upon the ability of spherical surfaces to avoid stress concentrations, thus minimizing the matrix cracking likelihood (at the expense of minor contributions to strengthening due to poor load-transfer intrinsic to curved interfaces). Hollow ceramic spheres can also be embedded in metal matrices to the purpose of reducing the (otherwise high) composite density, thus yielding an increase of their specific elastic moduli and fracture toughness capabilities. The prediction of a composite elastic constants can be achieved by a number of micromechanical techniques, namely the so-called “mechanics of materials approaches”, “variational principles”,

“self-constant models”, “exact solutions”, “statistical (averaging) methods”, and “empirical approaches”. For sake of brevity the specific details of the above micromechanics stiffness approaches shall be omitted, as there exist elsewhere [1-3] good accounts of their fundamentals, and will preferably be focussed on overviews [4-7] related to particulate composites stiffness. Representative models of the diverse micromechanical techniques have already produced elastic stiffness estimates for an ensemble of uniformly-sized elastic spheres, randomly embedded in a isotropic elastic matrix. Yet, interfacial effects have mostly been neglected, analytical relationships are often cumbersome [8], and many models lack sensitiveness towards reinforcement morphologies. So, in order to overcome these drawbacks, a central model will be selected in this work for its simple formulation and applicability to various material forms. This central model, thereafter termed Paul’s model [9], presents a simple strength-of-materials approximation for the composite modulus of a material with cube-shaped particles, and also assumes both matrix and particles are subjected to the same strain and have the same Poisson’s ratios. This particular model was further reformulated in the present work so as to be capable of predicting the stiffness of spherical particulates rather than that of cube-shaped particle composites. Theoretical predictions from the above models were then compared with those obtained by two other widely used models [10, 11] in engineering design applications, as well as those provided by the upper and lower bounds on composites moduli, respectively known as “Voigt (constant strain)” and “Reuss (constant stress)” bounds. The Paul model This model assumes that the states of macroscopic stress and strain imposed on a particulate by an external tensile stress can be reproduced in a typical unit volume which consists of a single particle embedded in a unit cube of matrix. Adhesion is also assumed to be maintained at the particle/matrix interface, when the unit cube of matrix becomes strained by an internal tensile force along the x direction. The above conditions yield the elastic modulus of the composite (Ec): 1 1 dx (1) =∫ 0 E + (E − E ) A( x ) Ec m p m where Em and Ep are the “matrix” and the “particle” moduli, and A(x) is the “particle morphology distribution” function. For a cubic-shaped particle within the matrix cubic cell, the integration gives:

⎡ 1 + (m − 1) V p2 / 3 ⎤ Ec = Em ⎢ (2) ⎥ 2/3 ⎣⎢1 + (m − 1) V p − V p ⎦⎥ where Vp is the volume fraction of cubic particles in the composite, and m = (Ep / Em) is the modulus ratio of the particle to the matrix.

(

)

The modified Paul model While the Paul’s model dealt with a cubic-shaped inclusion embedded in a cubic matrix cell to assess the stiffness of a particulate, the modified model seeks the same objective for a compact spherical-shaped particle embedded in a similar cubic matrix cell. Therefore, a new volume element is required: as Fig. 1 shows, A(x) = π r2; r2 + (R-x)2 = R2. Eq. 1 can now be utilized to obtain the elastic stiffness (Es) of mono-spherical particles reinforced composite, by splitting the integration interval [0; 1] into [0; 2R[ where A(x) = (π) x2 + (2 π R) x, and ]2R; 1] where A(x) = 0. This yields

2R Em 1 = (1 − 2 R ) + ∫ Es π (1 − m ) 0

dx

(3) 1 x − (2 R ) x + π (1 − m ) where the integral term can be evaluated by means of an integration standard form. The above expression then leads to

Em = (1 − 2 R ) − Es

R − R2 +

1

β

R + 2

2

1

β

ln R+ R + 2

1

β 1

+C

(4)

β

where β = π (m-1), C is an integration constant, and the particle radius R is related to the reinforcement volume fraction Vp within the unit cell, since by definition Vp = [(4 π R3 / 3) / (1 × 1 × 1)] so that R = 0.62 Vp1/3.

Figure 1 – The new volume element, containing a single spherical particle of radius R. The inset diagram shows the geometrical relationship between R, the position x and the secant length r.

By setting now an appropriate boundary condition: the composite modulus must be equal to the matrix modulus (Es = Em) if the reinforcement is absent from the volume element (Vp = 0 ⇒ R = 0), then C = 0 and Eq. 4 becomes: −1

⎡ 1 ⎤ R − R2 + ⎥ ⎢ β ⎥ 1 E s = E m ⎢(1 − 2 R ) − . (5) ln ⎢ 1 1 ⎥ 2 2 ⎢ β R + R+ R + ⎥ β β ⎥⎦ ⎢⎣ Although Eq. 5 is not as tractable as Paul’s Eq. 2, it nevertheless can set up – unlike Paul’s model and all the other models selected in this work for comparison purposes – a theoretical validity range for the composite reinforcement volume fraction Vp. Such constraint is imposed by the volume element, whose dimensions cannot be exceeded by the diameter of the enclosed spherical particle. Therefore, there must exist a maximum reinforcement volume fraction value

(V )

p MAX

= (V p )

2 R =1

= 0.52

(6)

which is only compatible with a simple cubic spherical particles fictitious pattern throughout the bulk composite. Such condition is absent from the other models referred in this work, whose reinforcement volume fractions are allowed to range from “zero” to “one”. (This inaccuracy was addressed in one single case [12], leading to the “modified Halpin-Tsai equations”, which were not considered for results comparison purposes due to the uncertain “packing fraction” parameters available for spherical particles dispersions). High volume fractions ranges also imply a growing tendency for particles denser packing accommodation, which generate “particles lattice” clusters along with a network of compact planes and preferential directions [8]. As it happens the particles

dispersion within the composite ceases to be random and the overall material ceases to be isotropic. Then the composite mechanical behaviour can no longer be described by two elastic constants solely, due to its emerging anisotropy, which in turn entrains poor predictive capability to the models that explain a anisotropic material by means of isotropic approaches.

The modified model applications The capability of the “modified model” to predict the stiffness of a particulate composite was checked against the estimates of other models, including Paul’s original model, as plotted in Fig. 2. A set of experimental results pertaining to a tungsten carbide / cobalt cermet, quoted in Paul’s work, served as a common reference source. The sensitivity of Paul and “modified Paul” models to increasing reinforcement content was investigated next, the results being shown in Fig. 3.

Figure 2 – Young moduli of a dispersion-stiffened composite material (Co/WC cermet), as predicted by a variety of models (Em (Co) = 206.9 GPa; Ep (WC) = 703.4 GPa), and compared against experimental results. Experimental values cited in [9].

Figure 3 – Composite stiffness slopes dependency on reinforcement content, as assessed from Eqs. 2 and 5.

Furthermore, Fig. 3 slopes enabled the detection of points of inflection (Vp = 0.12 in Paul’s model; Vp = 0.16 in the “modified” model). The nature of this deviation, although unclear from a theoretical viewpoint, may not be relevant in practice given the numerical values proximity. This is not the case of the estimates in Fig. 2, where Es (Vp) consistently provides larger moduli values than those displayed by Ec (Vp). In order to assess theoretically the magnitude of this deviation, a factor f was defined as the ratio between the stiffness estimates achieved, for a common Vp, by the “modified model” to those obtained by the “original model”, and both curves were subsequently treated as straight lines with different slopes. Such factor is then given by ⎧⎪⎛ E ⎞ ⎛ E ⎞ ⎛ slope E s (V p ) ⎞⎫⎪ ⎛E ⎞ ⎟⎬ , (7) f ≡ ⎜⎜ s ⎟⎟ = ⎨⎜⎜ m ⎟⎟ + ⎜⎜1 − m ⎟⎟ × ⎜ E c ⎠ ⎜⎝ slope E c (V p ) ⎟⎠⎪⎭ ⎝ E c ⎠V p ⎪⎩⎝ E c ⎠ ⎝ The “modified model” may also be useful for assessing the elastic moduli of thin-walled hollow spheres reinforced composites, often termed syntactic foams, because the external morphology of uniform-sized fully compact spheres is indistinguishable from that of an equal number of hollow spheres, as long as the latter are made of identical (ceramic) material and keep the same outer radius. Provided these conditions are met both reinforcement types, hereafter termed as (micro)”spheres” and (micro)”balloons”, will displace the same volume once they are embedded into the matrix – so permitting an identical reinforcement volume fraction to be obtained. At this stage both sets of reinforcements only differ by their densities (ρ), such that ρb = ρs [1 – (r0 / R)3]

where the subscripts “b” and “s” stand respectively for a balloon (i.e. hollow sphere) and a solid sphere (i.e. fully compact sphere); R and r0 represent a balloon’s outer and inner radii. The above description no longer applies when the composites are subjected to increasing tensile forces because extensive reinforcement breakage is triggered, leaving behind growing amounts of minute debris pieces. The spherical particles destruction rate is greater for the balloon-reinforced matrices [13], leading to cumulative damages from balloon shells cracking along with pores formation. Such debris pieces become loosely bound to the surrounding matrix and/or are far too small to allow any load transfer process to occur at the debris/matrix interfaces. Therefore, the debris no longer behave as “reinforcing units” but rather like non-reinforcing “inclusions”, and ought not to be taken into account in further reinforcement volume fraction evaluations. The remaining sound balloons, still acting as actual composite-stiffners or composite-strengthners, will define an “effective reinforcement volume fraction” (Veff) – which can be related to the nominal volume fraction Vp prevailing at the stage where no tensile forces existed, by a simple relationship Veff = Vp (1 – n / N) where n is the estimated number of broken balloons in a population of N original balloons. Eq. 5 can thus be applied to syntactic composites if n is previously estimated, eventually by fitting to its evolution a statistical (e.g. Weibull’s) distribution to assess the number of surviving balloons at each applied stress level. Conclusions A new model is proposed for estimating the elastic modulus of spherical-particles reinforced composites. Approximations made on its derivation suggest best predictive ability to be achieved on ceramic reinforced metal-matrix composites. The model has built-in capability to define the validity range of reinforcement content, and can be extended so as to make similar estimates for spherical hollow particles (balloons) reinforced composites. The model makes provision, not yet quantified, for reinforcement damage assessment by means of elastic modulus measurement. References [1] A.R. Bunsell, J. Renard: Fundamentals of Fibre Reinforced Composite Materials (IOP Publishing Ltd, UK 2005). [2] J.M. Berthelot: Composite Materials: Mechanical Behavior and Structural Analysis (SpringerVerlag Inc., USA 1999). [3] R. Brooks, in: Flow-induced Alignment in Composite Materials, edited by T.D. Papathanasiou, D.C. Guell, Woodhead Publishing Ltd, UK (1997). [4] S. Ahmed and F.R. Jones: J. Mater. Sci. Vol. 25 (1990), p. 4933. [5] W.S. Johnson, M.J. Birt: J. Comp. Tech. & Research Vol. 13 (1991), p. 161. [6] D.B. Zhal, S. Schmauder, R.M. McMeeking: Z. Metallkd. Vol. 84 (1993), p. 802. [7] S.K. Mital, P.L.N. Murthy, R.K. Goldberg, NASA Technical Memorandun 107276 (1996). [8] J. Segurado, J. Llorca: J. Mech. Phys. Solids Vol. 50 (2002), p. 2107. [9] B. Paul: Trans. Met. Soc. AIME. Vol. Feb (1960), p. 36. [10] J.C. Halpin, J.L. Kardos: Polymer Eng. Sci. Vol. 16 (1976), p. 344. [11] O. Ishai, L.J. Cohen: Int. J. Mech. Sci. Vol. 9 (1967), p. 344. [12] L.E. Nielsen: J. Appl. Phys. Vol. 41 (1970), p. 4626. [13] H. Toda, in: Metal and Ceramic Matrix Composites, edited by B. Cantor, F.P.E. Dunne, I.C. Stone, IOP Publishing Ltd, UK (2004).