^{1}

^{2}

^{1}

^{2}

Stiffness estimates of unloaded isotropic particulates are made by a new analytical model, when reinforcements are either compact or hollow spheres. A statistical extension of this model is described when stiffness predictions involve loading of syntactic composites. A simple experimental routine is also proposed for monitoring the microballoons fracture upon brittle syntactic metal-matrix composites tensile loading.

Composite materials have managed to stand in the forefront of engineering materials, mainly because of their structural abilities. Among composite materials rank the particle composites (“particulates”) the cheapest of them all. The spherical particulates emergence has been dictated by the spherical reinforcements surface ability to avoid stress concentrations, thus minimizing the cracking likelihood of the surrounding matrices. More recently, the replacement of compact spherical reinforcements by hollow ones (“balloons”) led to a novel class of foam-like materials, the so-called “syntactic composites.” The hollow spheres homogeneous morphology as well as their distribution modes once embedded in various matrices impart to the overall composite structure some relevant properties—for example, low density, good energy absorption, and enhanced fracture toughness over their compact reinforcements composite counterparts. Among other current techniques, dispersions of ceramic balloons in metal matrices may be achieved by infiltrating molten metal around a preform of uniform hollow spheres, whose packing modes may eventually range from a random distribution to an ordered pattern. In accordance with the balloons extreme packing modes, a random balloons distribution will imply a fully isotropic syntactic composite whereas an hexagonal close-packed array will determine an extreme anisotropic composite behaviour. In practice, an anisotropy-free composite can be obtained as long as its balloon concentration does not surpass a certain upper bound value. Such condition ensures no significant balloon clusters formation, thus preventing “crystal-like patterns” to occur within the composite. This preventive measure implies to disqualify for design purposes any empirical formula or approach where the concentration restriction may not be accounted for (as in the conventional rule of mixtures).

Experimental data on syntactic composites [

In this form, the present work derives from an early micromechanics model [

The modified model also enables density forecasts for either compact or hollow sphere particulates, and its capability was extended so as to predict the unsteady stiffness of fragile hollow sphere particulates when subjected to uniaxial tensile loading. Since the ceramic balloon composites and the balloons themselves can be regarded as brittle solids, their failure at a particular applied tensile stress depends on the statistical existence of surface flaws or cracks. The proposed model for the stiffness of loaded syntactic composites is therefore a statistical model. Gathering mechanical information adequate enough for predicting the elastic stiffness of an isotropic solid is just a part of a broader theoretical frame aiming at determining the two independent elastic constants which define that solid elastic behaviour. As regards the isotropic composites, that search requires a statistically isotropic (random) distribution of their (also elastic isotropic) constitutive reinforcements—which are assumed in the present analysis to be an ensemble of uniform spheres, embedded in a continuous and isotropic elastic matrix. Some current advanced approaches aiming at computing the isotropic composite independent constants span from analytical models [

The earliest of the above referred analytical models is the Mori-Tanaka mean-field approximation [

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. Additional approximations regarding the particulate constituents further assume both matrix and particles are subjected to the same strain and have the same Poisson’s ratios—a mechanical condition which best suits metal/ceramic combinations. 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

The above conditions yield the elastic modulus

In predicting just a single composite stiffness value regardless the stiffness values arising from other material directions, this early model implicitly acknowledges that the particulates under assessment are to be isotropic composites. From which follows that the above stiffness predictions are valid providing the composites reinforcement content which lies in a safe concentration range, where nucleation of particle patterns does not occur significantly.

Equation (

Paul’s analysis approached an actual particles composite as if its overall properties matched those of a representative volume element. Such volume element was a cubic matrix cell enclosing a solid cubic-shaped virtual particle, whose size was adjusted by the actual particles volume fraction. This reasoning is maintained in the modified model [

When metal matrices become reinforced by ceramic microspheres, so that

However, when metal matrices become reinforced by ceramic thin-walled microballoons so that

The fundamental equations of this modified model (i.e., (

The stiffness of spherical particulates, as predicted by (

The stiffness trends displayed in the above plot by all microsphere-reinforced composites seem to be in general agreement with experimental observations, whereas the microballoon-reinforced (syntactic) composites trends show instead a severe stiffness decay as an increasing number of hollow spheres is added to the matrix material.

In order to further investigate this peculiar stiffness behaviour, predictions from the “modified model”

For sake of a correct comparison, the same balloon moduli

Syntactic composites stiffness as predicted by this model (

As shown above, either model defines similar stiffness decay trends and both are sensitive to cell-porosity exponents (i.e., larger porosity-exponents lead to smaller composites stiffness). Yet Toda’s model anticipates larger stiffness decay rates than this model and predicts, unlike (

As previously referred, a complementary feature of this modified model is the assessment of composites

The theoretical density of

As for the microballoons-reinforced (syntactic) composites, a similar reasoning leads to

Since the microstructures of most syntactic composites incorporate balloon-size distributions rather than single-size balloon dispersions, and the ceramic microballoons strength becomes adversely affected by large balloon diameters, the weakest microballoons become prone to fracture as the composites are subjected to increasing external loading (as in tensile or fracture toughness testing). Once fractured at a given applied stress level, these particles cease to act as “reinforcing units” and rather behave as nonreinforcing “inclusions”, so they ought not to be taken into account in further reinforcement volume fraction evaluations. Moreover, a new composite density assessment is necessary because there is a difference between each balloon displaced volume as compared to the debris pieces volume following the balloon fracture.

Since the fracture of microsphere-reinforced composites is mainly due to “matrix cracking” operating mechanisms, whereas the fracture of microballoon-reinforced composites is rather ascribed to extensive “particle cracking” [

Such a correction implies to define an “effective reinforcement volume fraction” (

The corrective (

Linear dependence of

The reinforcement concentration correction, by means of which the nominal balloons volume fraction is replaced by an “effective” value, also implies a similar correction of the virtual particle radius

Curvilinear dependence of

As both “effective” variables

Virtual particle size, within the representative volume element, plotted versus intact balloons volume fraction.

The above diagram seemingly comprises three distinct particle growth regimes: a first high-curvature segment (up to

The combined first and second stages spread over a concentration range, where a balloon’s dispersion remains isotropic, as certified by a simulation study [

Figures

A compounded view of the “effective” variables and the unbound balloons ratio.

Following the “effective” variables definition, the formerly derived (

From the above, it is clear that (

The ensuing treatment will resort to the Weibull statistics as such a distribution was formerly designed to suit the fracture of brittle solids (e.g., ceramic microballoons, and derived composites). The Weibull’s approach aims at predicting brittle solids tensile strengths, based on (widely scattered) tensile strengths gathered from a batch of ~20–30 nominally identical composite samples. Once these experimental tensile strengths are arranged in order of increasing failure stresses and a failure stress probability is assigned to each failure by an adequate estimator [

The sole relevant information for the present modelling analysis would, in principle, just be the tensile strength data provided by the composite specimens batch. Nevertheless, it may be referred for sake of completeness that a certain “probabilistic strength” can be assigned to the whole lot of brittle composites following after a

The above mentioned composite specimens tensile strengths must now be related to the corresponding balloon applied tensile stresses. Should the tensile tests be conducted on a batch of continuous fibre composites, whose elastic fibres are loaded along their axes (as assumed by Voigt “constant strain” model) and whose matrices are also elastic, the stress acting on each fibre would be calculated from theoretical stress ratios as per

The ceramic balloons are brittle solids, so their strength must comply with a two-parameters Weibull distribution similar to that described by (

Since

The statistical model for stiffness of loaded syntactic composites arises from

Equation (

Equation (

A theoretical analysis has been conducted on the elastic stiffness of spherical-particles reinforced composites. Composites were assumed to have thin particle-matrix interfaces; both particle and matrix constituents were further supposed to have identical Poisson’s ratios and to be subjected to the same strain upon composite loading. While spherical (either compact or hollow) reinforcements were not significantly cracked by external tensile stresses, the composites predicted stiffness was simply derived from the states of macroscopic stress and strain within a representative unit volume element. This was not the loaded syntactic composites case, whose stiffness predictions might otherwise be affected by severe microballoons destruction due to moderate-to-high applied tensile stresses. A complementary approach was therefore devised for this specific case, which involved the definition of some parametric variables related to the number of surviving balloons at a given stress level, as well as assigning brittle solids strength statistics to ceramic microballoons and their syntactic composite materials. The overall output of the above contributions is a statistical stiffness model for microballoons-reinforced tensile loaded composites. This model reliability can seemingly be confirmed from theoretical predictions (e.g., (

The two independent elastic constants of an isotropic random ensemble of elastic spherical inclusions embedded in a continuous, isotropic, and elastic matrix can be provided by numerical (finite element) methods as well as by three current analytical models: the Mori-Tanaka mean-field analysis [

Much like the previous analytical models, this micromechanics model [

Designing with syntactic composites depends on these materials performance when tested for a particular property, while balloons enhanced annihilation is imposed by growing applied loads. The “elastic stiffness” property behaviour, for example, is described in Figure

As a final remark, all the engineering properties cited above (and stiffness in particular) are to behave as “variable parameters” for as long a period as the loading test duration. Hence, such variability also applies to any other elastic constant eventually bound to stiffness by any conceivable fundamental mechanical relationship, providing it is valid for isotropic composites.