The effect of swelling on the failure mode of matter

En#nceringFracrunMe&&a Vol. 47, No. 2. PD.191-206~1999 copyright Qa 1994 El&er scieooc Ltd. Rinted in GreatBritain.All rightsruerved 0013~7P44/P4s4.00+ 0.00

THE EFFECT OF SWELLING ON THE FAILURE OF MATTER

MODE

PERICLESS. THEOCARIS NationalAcademyof Athens,P.O. Box 77230,17510 Athens,Greece Ab#rac-A study was undertakenof the mode of failure of open- or closed-Al cellular materials depending on the variation of the relative density of the material due to swelling or variation of its porosity. Cellular materials or foams were considered as transversely isotropic, an assumption which corresponds well with their mechanical behaviour. Using a tensor failure polynomial criterion, and as such the elliptic paraboloid failure criterion was considered, for the study of the mode of failure of closed-cell foams of different relative densities, it was establishedthat the failurebehaviourof the swelledmaterial

drasticallychangesits failuremodeby becominga strongtensionmaterialfrom a strongcomPressionone. Furthermore,there is a criticalvalue of the relative density for each material, where the substance passes through a quasi-isotropic state and the foam behaves like an isotropic material, where triaxiality phenomena are insignificant.

1. INTRODUCTION NATURAL CELLULARproducts like wood, cork, pitch, bone and others have been either extensively used by nature or employed by man for centuries. Furthermore, either polymeric or ceramic and metallic foams and cellular structures are actually valuable synthetic materials. Depending on the kind of failure expected for a construction, different material properties, such as elastic properties and strengths, are required as functions of the foam density, temperature and time, to ensure both safe and economic design. The measurement of these properties is an enormous task and can only be achieved by a procedure of evaluating these properties by using convenient models through which a qualitative study of the deformation and failure mechanisms of the supporting cell elements can be undertaken. Essentially, the material characteristics are separated into their elastic properties and their strengths. With regard to the strength values and failure criteria, the deformation mode of the supporting cell elements suggests that under normal stress the strengths of the closed-cell elastic foams are exclusively dependent on the strength of those struts parallel to the direction of the applied external load. Depending on the mode of the applied load (tensile or compressive), the failure mode of the foam is failure by fracture, for the first case, and an unstable failure by buckling of the struts under compressive and shear stresses. For the tensile strength the supporting polymer volume should be considered, depending on the position of the struts in relation to the stress direction, and thereby the size of the acting normal stress depends on the orientation factor of the struts. For a compressive loading failure to occur before fracture, the cross-section of the supporting material as a whole is essential for estimation of the strength, and therefore the load capacity of the whole of each strut should be considered. It has been established [l, 23 that the mode of failure of cellular materials can be satisfactorily described by a criterion of the failure tensor polynomial form and especially by the elliptic paraboloid failure criterion for general orthotropic bodies. This criterion describes the primary mode of failure for foams due to limit loading. A secondary failure mode in the compression-commpression-compression stress space may occur by a premature buckling of the foam cells much earlier than the collapse load under triaxial stress. For this mode of failure the symmetric ellipsoid failure criterion [3] complements the elliptic paraboloid failure surface (EPFS) criterion [4]. In order to define these criteria, the failure stresses in simple tension and compression must be experimentally evaluated and for the elliptic failure surface (EFS) criterion an extra value of failure by buckling in hydrostatic 3D compression is sufficient. Then, this criterion works as a failure surface, conveniently truncating the fundamental failure tensor polynomial criterion (the EPFS criterion) in order to take care of unstable failures under buckling. Experimental evidence with 191

192

P. S. THEOCARIS

various types of cellular materials has shown that their failure modes corroborate both criteria, depending on the form of structure [5]. In this paper the interesting phenomenon of the reversal of the failure mode of cellular materials from strong compression to strong tension materials with the variation of the relative density was studied. Moreover, a critical value of the density was found to exist for each foamy material, which defines the point of this reversion, where the material behaves like a quasi-isotropic material.

2. THE FAILURE SURFACE FOR CELLULAR

MATERIALS

The model shown in Fig. 1 is generally accepted and illustrates cell elements of a closed-cell foam as single elements with a cell compound consisting of regular dodecahedra. It was shown that, even for rigid closed-cell foams of low density, only the cell struts are truly supporting elements, whose directions are statistically distributed in any cross-sectional area. The microstructure of foams reflects their preparation procedure, which usually involves a continuous liquid phase which progressively solidifies, thus absorbing considerable quantities of air, or similar matter, and progressively forming cells. In this procedure, surface tension and related interfacial effects often control the foam structure. In order to minimize the surface energy in the foam structure, a hexagonal network of a honeycomb structure in various planes constitutes a logical model for investigating the mechanical properties of foams, whereas in space three adjacent films meet at angles of 120” to form a typical junction region, called the plateau border, and four plateau borders join at a tetrahedral angle of cos a = (- l/3) with a = 109.47” (see Fig. 1). When the percentage of voids created during production of the foam in an open- or closed-cell structure reaches a certain level, the body may be described as being cellular. The major characteristic of cellular materials is their low density. However, for simplicity it may be assumed that, although a tetrahedral type of unit cell containing four identical half struts joined at equal angles is the most realistic (see Fig. 1) for representing the essential microstructural features of the foam, a cubic microstructural model, introduced by Gibson and Ashby [6], for which adjacent struts or films do not align, is more convenient, since bending seems to be the essential small distortion mechanism for 3D foams. Thus, the model of Fig. 2 is introduced, where adjacent struts or films do not align, and which yields satisfactory results. Even for closed-cell foams, since the films of the faces are very thin compared to the struts at their borders, the response of these foams is expected to be identical to that of an open-cell foam. The choice of microstructural element is a noteworthy feature of all models. Depending on the type and direction of external load, these elements undergo different types of stress and deformation. It was proved that the type of stress caused within individual struts by external tensile or compressive stresses affects mainly the struts oriented in the direction of the external load, which then fail first. On the contrary, when the cells are under shear, the struts under compression located at an angle of 45” to the loading direction fail first. These observations allow the derivation of statements for the theoretical evaluation of the mechanical properties of foams as functions of foam density [6].

Fig. 1. The dodecahedron model for the cell elements.

The failure mode of matter

193

Fig. 2. Central 3D open cubic cell model and its interconnections with the neighbouring cells for a transversely isotropic rigid-foam material @,-axis is the strong axis of the body).

It is almost certain that cellular materials permit the simultaneous optimization of stiffness, strength and overall weight in a given structure. Detailed studies of model foams have identified four distinct deformation modes: linear elasticity, nonlinear elasticity, plastic collapse and various sorts of fracture. Using hexagonal networks for the study is a good approximation, since the real deformation modes correspond to 3D foams, yet the geometry is rather simple and therefore a complete analysis is practical. However, the analysis of the 3D structure is idealized without loss of physically important features in the Gibson and Ashby model [6]. The analysis of the physical mechanisms responsible for the homogeneous deformation of 3D cellular solids has been fairly well established, relating the mechanisms of failure with different modes of deformation. The elastic properties of these materials are derived by assuming them to be general orthotropic substances, and for the models of Figs 1 and 2 they may be simplified to transversely isotropic materials. Extensive studies of model foams and cellular solids concerning their deformation modes [6-g], that is linear and nonlinear elastic behaviour and plastic collapse or various forms of fracture, have been established. On the other hand, recent studies of the failure modes of foams clearly indicated that failure under a general multiaxial state of stress depends on the triaxial loading of the bodies [ 1O-l 21. While the failure criterion introduced in these papers is much more accurate than any previous considerations, which were rather primitive, it does not comply and violates basic physical laws of mechanics. On the contrary, the tensor polynomial function and especially the EPFS are convenient anisotropic failure criteria since they are operationally simple form-invariant expressions satisfying in principle any physical laws of mechanics of anisotropic materials, needing only the accurate evaluation of the six components in tension and compression loading along the principal directions of the material. The generalization of the concept of a paraboloid of revolution failure surface with axis of symmetry the hydrostatic axis cl = a2 = cr3for the isotropic solids introduced in refs [13] and [14] to an elliptic paraboloid with symmetry axis parallel to the hydrostatic axis for the general orthotropic materials, established the concept of the EPFS criterion, which is defined, beyond the basic concepts valid for all tensor failure polynomial criteria, by an interrelation between the off-diagonal and diagonal terms of the failure fourth-rank tensor H. The tensor polynomial function, which is used extensively, is expressed in a simple invariant form. The rank of the tensors, and thus the number of terms appearing in the respective equation, may or may not seriously affect the theoretical predictions. By considering tensor polynomial coefficients up to the fourth rank, the failure function reads as follows:

P. S. THEOCARIS

194

f(u)=a~H~a+b~a--l=O.

(1)

Many anisotropic failure criteria can be written in the form of relation (1) with an appropriate definition of the Cartesian components of the fourth and second rank tensors H and b, respectively. While the stress differential effect second rank tensor h, as well as the diagonal components of failure fourth rank failure tensor H, are uniquely determined for criteria assuming the general form of eq. (l), differences arising in the definition of off-diagonal components, Hi, (i #j), express the various phenomenological conjectures of these criteria. The interrelation valid between off-diagonal and diagonal terms of the failure tensor H for EPFS is expressed by: i,j,k

Hij=f(iYM-Hii-H,,),

~3

(i #j #k).

(2)

Relation (2) constitutes a sign&ant advantage of the EPFS criterion over all other criteria based on the arbitrary or experimental definition of the off-diagonal terms. Indeed, application of the EPFS criterion to a diversity of orthotropic materials and composites indicated the soundness of this criterion. Reference [15] presents an overview and a comparison of experimental results with theory. The normal components of the failure tensors H are expressed by: 1

Hii=-

O7S Oci hi =

(i Q 3)

$ - &=(6& -

O,)Hii,

Cl

whereas for shear, components are given by: (5) hi=&-$=(a,-of)Hiie SI

(6)

.!I

In the above relations, the repeated index convention does not apply and the IJ~,and uCistresses express the tension (T) and compression (c) failure stresses in the i-direction. Furthermore, the ai, a; stresses express the shear strengths positive or negative in the i-plane (i > 3) and the usual contracted notation of Cartesian indices is used, meaning that index 4 corresponds to natural indices 23, index 5 to 13 and index 6 to 12. For orthotropic materials, when the coordinate system defining the failure stresses coincides with the material symmetry directions, there is no shear strength differential effect, i.e. a: = a,. For the complete definition of the EPFS, it is necessary to visualize a series of its characteristic intersections. Indeed, since intersections by the principal stress planes (ai, bi+ ,) (i = 1,2,3) consist of superficial slices of the failure locus, these intersections may be confusing, since respective intersections belonging to different criteria eventually present only imperceptible differences. Indeed, it is well established that for the experimental verification of the postulates of any failure criterion it is not suilicient to compare its predictions with experimental results from 2D failure tests. Further comparison with triaxial failure tests is necessary, and more convenient 3D failure tests should be considered, like a superposition of hydrostatic compression on uniaxial tension or compression. Such tests correspond to loading modes included in various other intersections of the paraboloid. However, for the purposes of this study the principal diagonal (bj, 6,,) intersection of the EPFS criterion sulIices. It has been shown that, for fibre-reinforced composites, the orthotropy of the materials may be satisfactorily approximated by a transverse isotropy. We therefore give in the following direct expressions for the transversely isotropic material, which are much simpler than those for the general orthotropic material [4, 16, 17’J.Indeed, for a transversely isotropic material, whose 03-axis is the strong axis, it is valid that: H,, = Hz

and

h, =h2:.

(7)

Then, the EPFS for the transversely isotropic material satisfying all the properties already described

The failure mode of matter

is expressed as follows in the (Q, , o z, a,)-principal corresponds to the strongest one [la]:

stress space, where the o,-principal

195

direction

This expression, referred to the Cartesian coordinate system Oxyz, where the Oz-axis is parallel to the hydrostatic one and the (Oxy)-plane coincides with the deviatoric plane with the Oy-axis lying on the Oa,G,,-plane, where the O&,-axis corresponds to the bisector of the right angle (a,Oa,), is expressed by:

(9) Furthermore, the deviatoric plane is defined at once by putting the value of z = 0 in relation (9). The coordinates (x,,, y,,) of the origin 0’ of the deviatoric intersection of the EPFS are given by: x0 =

3

0, Yo = up= -[h gH33

- h31.

(10)

The quantity y, = qp expresses the distance between the symmetry axis of the elliptic paraboloid surface and the hydrostatic axis, c1 = a2 = a3. The elliptic intersection of the EPFS by the deviatoric plane has as principal semi-axes the quantities a, and a,, expressed by:

(11)

The ratio of the major axis 2a, to the minor axis 2u2 of the elliptic intersection expresses the ellipticity of the paraboloid and is given by:

(12) the angle 6, subtended Oxy-system is given by [16]:

Moreover,

by the principal axes of the elliptic intersection 80 = 0.

and the (13)

For the complete study of the elliptic paraboloid surface four types of intersections are deemed necessary. These intersections are as follows: (i) The principal diagonal intersections defined by planes containing one principal stress axis and the bisector of the right angle formed by the remaining principal axes. (ii) The deviatoric x-plane which is normal to the hydrostatic axis. (iii) The principal stress plane intersections, which are convenient for the study of the mechanical properties of the anisotropic body when thin plates of the material are under plane stress conditions. (iv) The intersections of the EPFS by planes defined by the axis of symmetry of the paraboloid and either principal axis of the ellipse corresponding to the deviatoric plane. However, for the purposes of this study the expressions defining all the remaining intersections are not necessary and, therefore, may be omitted. Moreover, for rigid foams and cellular materials complementary modes of failure intervene, besides plastic yielding, or brittle crushing, or tensile fracture, described by the EPFS criterion. These modes are mainly due to the inhomogeneity of the material in the microstructure, because of its high degree of porosity. In this case internal buckling may intervene, due to an impending instability inside the material during multiaxial compression loading, or to a fast brittle fracture following the maximum tensile stress criterion. For these modes of failure the EPFS should be complemented by another convenient criterion, which should close the open side of the EPFS

P. S. THEOCARIS

1%

criterion and which corresponds to the buckling cut-off mode of failure of the foam. Experiments with open and closed-cell foams clearly indicate that experimental data with buckled foams and cellular materials yield a better fitting with a continuous second degree conical surface and especially an ellipsoid. It was suggested in ref. [l] that the buckling cut-off surface is defined by an ellipsoid whose major principal axis coincides with the symmetry axis of the basic EPFS. This is because the position of this axis is completely defined by the inherent anisotropy of the material expressed by the coefficient of anisotropy qp defining the parallel displacement of the symmetry axis from the hydrostatic axis. Experiments also indicate that all three components of the principal stresses participate in buckling and the limit stresses in buckling for uniaxial loading approximate the failure stresses under simple modes of loading. Moreover, accurate experiments executed in various foams indicated that the buckling limit loads cannot be conveniently described by loci defined by planes parallel to the respective principal planes. Instead, they appear to belong to some second-degree smooth conical closed surface [l 11. Buckling stresses for open-cell foams plotted in ref. [l] under multiaxial compression fit excellently on the surface of an ellipsoid having its major axis coincident with the symmetry axis of the EPFS and passing through the buckling stresses for simple compression along the principal stress axes of the material. The failure stresses for tensile cut-offs for fast brittle fracture also lie on the same surface (see, for instance, Fig. 14 of ref. [l]). For the polyurethane foams studied in ref. [2], as well as for any transversely isotropic material, this ellipsoid surface presents a symmetry along the (a,&,) principal diagonal plane, since the critical buckling stresses in the isotropic transverse plane (00, n2) should be equal. The most stable experimental values used for the definition of the ellipsoid cut-off surface are, besides the buckling stresses in uniaxial compression along the principal stress axes, the buckling and fast brittle fracture cut-offs under hydrostatic compression and tension. Therefore, the ellipsoid cut-off failure surface is defined by its major principal axis, which coincides with the symmetry axis of the EPFS, whose centre lies at the mid-distance between the buckling critical stress in hydrostatic compression and the critical stress in fast brittle fracture under hydrostatic tension. This surface also passes through the points representing the buckling critical stresses in simple compression along each principal stress axis. The theory behind the definition of symmetric EFS was presented in detail in ref. [3]. However, for the study of the influence of the relative density of the material on the mode of primary failure of cellular materials it is sufficient to evaluate and plot the respective primary failure locus, i.e. its elliptic paraboloid failure surface.

3. THE ELLIPTIC

PARABOLOID FAILURE SURFACES FOR VARIOUS FOAMS AND CELLULAR MATERIALS

As a first series of application of the EPFS criterion to cellular bodies, two distinct types of rigid foams were examined. These materials were as follows: (i) An open-cell polyacrylonitrile foam carbonized by techniques developed for preparing carbon fibres, thus creating a brittle ceramic reticulated carbon foam [18]. (ii) An open-cell aluminium rigid foam of density 135 kg/m3 [19]. All the experimental results were taken from the existing literature, which will be referred to each particular case [ll, 121. The average values of the collapse strengths of the open-cell polyacrylonitrile to the strong and weak axes of the material are [ 121: an=0.128MPa

00=0.212MPa

a,.,=an=0.138MPa

rigid foam referred

oc,=o,=0.160MPa.

(14)

The differences of the failure limits in tension and compression along the weak axes 01 and 02 were of the order of 5%, well inside the limits oft& standard deviations between these limits. Therefore, it was judged appropriate and expedier&tu mrrirb+ the material as a transversely isotropic body with the average failure limits given in (14).

The failure mode of matter

197

Table 1. The values of the characteristic quantities of the principal intersections of the EPFSs for the two different foam materials Polyacrylonitrile foam VP cl ar d d’

-0.0155

0.1185 0.1354 0.3405 0.3449

Aluminium foam -0.0577 0.675 1.4175 - 14.703 - 14.727

All quantities in MPa.

The failure limits for the open-cell aluminium rigid foam are given in ref. [l l] as follows: a, = 1.770 MPa

e0 = 0.970 MPa

cr., = eR = 0.940 MPa

bcl = ecz = 0.900 MPa.

(15)

Again, the actual differences between the failure stresses along the weak directions 01 and 02 in tension (a,, an) and in compression (a c,, a& were well inside the limits of the respective standard deviations and therefore it was judged opportune to take as valid the average values of these pairs of quantities. The characteristic quantities for the respective EPFSs of these two substances are given in Table 1. These characteristic dimensions defining completely the shape of the EPFS criterion and its intersection along the deviatoric plane are the following: the distance ‘lpbetween the hydrostatic axis and the symmetry axis of the paraboloid, the lengths of the major (u2) and minor (a,) semi-axes of its elliptic intersection by the deviatoric plane, and the distances d and d’ between the deviatoric plane and the point of piercing the paraboloid by the hydrostatic and the symmetry axes of the surface. It is worthwhile indicating that since the quantity trh = (h, + h2 + h,) for the open-cell polyacrylonitrile foam is positive the material is a compression strong one and the open side of the elliptic paraboloid lies in the compression-compression-compression octant of the stress space, whereas the same quantity for the aluminium rigid foam is negative and therefore this material is a tension strong one [ 161. However, since for both materials the ‘lpdistances are negative, their symmetry axes intersect the negative a3-axis, i.e. the origins of the elliptic intersections of their deviatoric planes lie along the negative a3 -axis. Figures 3 to 6 present the principal diagonal (asO&) plane intersections of the elliptic paraboloid failure surfaces, as well as their intersections by the deviatoric n-planes, for the open-cell

Fig. 3. The (~$5;~) principal diagonal intersection of the EPFS for the polyacrylonittile foam, together with the respective intersection of the ellipsoid representing the buckling cut-off of the material and the comparison with experimental results, especially for buckling behaviour. BFM 47/2-D

198

P. S. THEOCARIS

Open-Cell

Polyacrylonitrile

Foam

Fig. 4. The deviatoric plane intersection of the EPFS for the open-cell polyacrylonitrile foam.

polyacrylonitrile rigid foam and for the open-cell ahnninium rigid foam. Exceptionally, in Fig. 5, besides the intersection of the EPFS for the altinium foam by the (q06,,) principal diagonal plane, the respective intersection of the paraboloid by the (x’O’z’)-plane normal to the (a,06,J and the deviatoric cross-section was plotted, which completes the description of the form of the elliptic paraboloid. The appearances of these paraboloids for materials with weak anisotropies resemble the form of a squeezed cigar. Figures 5 and 6 clearly indicate that the open-cell aluminium rigid foam, which is a material of very high porosity, behaves like a tension strong material, as was indicated previously. The plots for both porous substances, as derived by applying the failure theory for the EPFS, were compared with experimental results given by meticulous tests run in refs [l l] and [12]. These results are plotted as open squares in Figs 3 and 4 for the Crst type of material and in Figs 5 and 6 for the second type of material. It can be concluded at once that the existing experimental evidence corroborates excellently the respective EPFS criteria. On the contrary, some proposed failure criteria, based solely on curve fitting processes not derived from the application of some physical law, present considerable discrepancies from the respective experimental tests and may be accepted only as crude approximations [l 1,121. The elliptic paraboloid failure criterion belongs to the type of phenomenological criteria, which are based on particular assumptions derived from long observations and experimental practice, and therefore it succeeds in interrelating failure with basic concepts and principles of mechanics. The EPFS criterion for foams contains two different failure regimes, either of which corresponds to a particular mechanism of failure. The primary part of the criterion, described by

The failure mode of matter

199

the EPFS, is associated with a progressive brittle crushing of the individual cells of the material after cell wall bending, followed by a densification through compression of the solid wall material along the compressive part of the stress space. This main body of the criterion is ended in the tensile part of the stress space by a failure mode, also caused by cell wall bending terminated by fast brittle fracture through propagation of initiated cracks. The part of the criterion described by the EPFS proper defines the brittle failure surface. This surface is truncated at the open side of the paraboloid by an ellipsoid which cuts-off the paraboloid and represents the other mode of failure by elastic buckling for the compressive octant or by fast brittle fracture in the tensile octant. This buckling cut-off mode of failure does not concern this study and it will not be discussed here. Details on the subject can be found in ref. [I]. Most of the orthotropic compact materials (with relatively high densities) are compression strong materials, since they are capable of sustaining large compressive stresses without failure. However, there is a restricted category of materials which are of low density and they behave like tension strong ones. Such a typical tension strong material is oriented polypropylene [16]. In these substances the solid material is distributed in little columns or beams forming the cell edges in cellular open-cell foam like solids, where clusters of microvoids intervene between molecules or crystals, thus creating structures which may be either isotropic or more frequently orthotropic. Most polymers can be readily foamed and techniques exist for doing the same thing with ceramics and glasses, and even with metals in modem technologies. The mechanical properties of foamed materials reflect to some extent the mode of distribution of the solid material, which depends on the relative density of the solid phase constituting the foam-like material. Thus, for high void fractions of the total volume structures, a lower than linear decrease in flow strength with a decrease of density was exhibited, indicating that bending stresses within the foam structure are progressively a less important feature of the collapse load. Then the structure behaves like a tension strong material, presenting a higher strength in the tension-tension-tension octant of the stress space until a collapse by internal buckling of the structure takes place. Oriented polypropylene has a geometric conformation of the individual polymer molecules in the unit cell of a polymer crystallite, depending strongly on the repulsion of the methyl groups in the planar zig-zag conformation. The molecules assume positions at 120” out of the plane of the

Fig. 5. The (~;a;,) principal diagonal plane intersection of the EPFS for an open-cell aluminium rigid foam and its transverse intersection by the (Ox’+plane, together with the deviatoric intersection of the same material.

200

P. S. THEOCARIS

Open-CdlAlumlnlum

devlatorlc plane

Fig. 6. The deviatoric plane intersection for the open-cell aluminium rigid foam.

chain and therefore form helical conformations within the unit cell of their crystallites. Therefore, low-density oriented polypropylene influenced by the spring-like cells presents a tension strong behaviour, as has already been proved in ref. [20]. Similarly, paper sheets consist of networks of fibres placed in different random arrangements, The properties of the individual fibres and the nature and frequency of bonds between them inlbtence the properties of the paper sheet. Moreover, fibres are actually filament-wound composite systems, whose cell walls are composed of a number of different layers, the fibrils, which are aggregates of cellulose molecules with cellobiose as the basic repeating unit, arranged either in an orderly or in a random fashion. The fibrils themselves are arranged in a regular fashion, differing within the various layers of the cell wall, and they are held together by the hemicellulose and lignin matrix material. Such conglomerates contain a great number of microvoid clusters distributed randomly inside the structure, and this considerably reduces the specific density of the material. The bonded fibres act like largely interconnected springs. It is therefore to be expected that low density paper sheets behave like tension strong anisotropic materials, as is clearly indicated in ref. [21]. Then, there is a clear indication that the internal structure of the materials, indicated by their relative density, plays a predominant role in the mode of their failure. However, since the examples presented above concern different materials, this fact may cast some doubts about the causes of different failure behaviours of materials with different relative densities. In order to prove the preponderant role of the relative density of a material on its mode of failure, we shall study three different types of the same material, polyurethane rubber, which was processed in different ways in order to drastically change its relative density.

The failure mode of matter

4. THE FAILURE MODES OF POLVIJBETHANE

201

FOAMS WITH VARIABLE DENSITIES

Similar results have been found in a single type of material, a polyurethane foam, with three different densities. Typical closedcell polyurethane foams are studied, which are rigid-cell substances and present different amounts of porosity, so that their densities varied from d, = 64 to d2 = 96 to d3 = 192 kg/m3. The porosities or relative densities of these foams were defined as the ratios of the weights of the foam materials to the respective weight of the solid material of equal volume. While the uniaxial compression stresses have been derived from compression tests in small cubes, whose geometric axes on the average coincided with the material symmetry axes, the uniaxial tension stresses were derived from tests on cylindrical specimens, whose axes coincided with the strong orthotropic axis of the material. On the other hand, the biaxial loading stresses along planes containing the strong anisotropic axis (Oa,) were derived from cubic specimens and axisymmetric tests performed in triaxial cell testing devices, where cylindrical specimens were subjected to a lateral pressure in the radial direction, superimposed on an axial loading. These tests yielded the failure stresses along the principal diagonal plane (a,&). The average values of the principal failure stresses in simple tension and compression along the principal material directions for the three types of polyurethane foams are given as follows. (i) For the polyurethane

foam PUR-64 with density d, = 64 kg/m3: CY~= 0.640 MPa

or, x oTL= 0.420 MPa

trc3 = 0.630 MPa

b,-, x crc2= 0.320 MPa.

(ii) For PUR-96 (d2 = 96 kg/m3): CJ~= 1.150 MPa

cr, x an = 0.650 MPa

ac3 = 1.080 MPa

acI = aa = 0.750 MPa.

(iii) For PUR-192 ($ = 192 kg/m3): ayj = 2.440 MPa

aTIz an = 1.580 MPa

a c3 = 3.350 MPa

a=, x ac2 = 2.560 MPa.

The values of failure stresses in simple tension and compression along the principal axes Oa, and Oa, present some scatter and differ only a little for all three materials. They were averaged in order to facilitate the calculations and use the simpler formulas for transversely isotropic materials. Using these values and the theory for the elliptic paraboloid failure surface, the characteristic quantities for each type of foam were evaluated and are given in Table 2. These properties, defining completely the EPFS for each foam, are the shape of the paraboloid presented in the principal stress space by its principal diagonal (a36,&plane indicated in Fig. 7 for the polyurethane rigid foam PUR-64 (d, = 64 kg/m3). The paraboloid failure surface is further defined by its intersection with the deviatoric plane (Fig. 8). For the other two types of foams, the same configurations are given in Figs 9 and 10 for PUR-96 and Figs 11 and 12 for PUR- 192. In Table 2, the following characteristic dimensions of the three EPFSs are given. The distances Table 2. The values of the characteristic quantities of the principal intersections of the EPFSs for the three polyurethanes of different density Rigid polyurethane foams (PUR)

‘lP

aI a2

d d

d = 64 kg/m3

d = 96 kg/m’

d = 192 kg/m’

-0.079 0.274 0.524 - 1.145 - 1.172

0.088 0.521 0.912 -4.96 -5.01

0.291 0.996 2.352 2.911 2.956

All quantities in MPa.

202

P. S. THEOCARIS

Fig. 7. The (c&) principal diagonal plane intersection of the elliptic paraboloid failure surface for the closed-cell polyurethane rigid foam PUR-64 and its comparison with experimental evidence. The intersection of the buckling cut-off ellipsoid is also plotted.

‘lpare first tabulated between the symmetry axes and the respective hydrostatic axes of the EPFSs. These distances express the characteristic stress vectors which, when multiplied by the respective strain vectors, yield the amount of elastic energy consumed to overcome the anisotropy of the foams. These elastic energies are constant for each type of foam, for any amount of elastic deformation developed in the material [22]. The a,, a2 dimensions of the EPFSs yield the respective lengths of the major and minor semi-axes of the elliptic intersections of the EPFSs by the deviatoric planes, the quantities d define the distances of the points piercing the EPFSs by the hydrostatic axes from the deviatoric planes and the origins 0, whereas d’ express the distances of the vertices of the EPFSs from the deviatoric planes. For strongly anisotropic substances, where the respective EPFSs are very shallow, the differences (d’ - d) become insignificant. Here the small differences (d’ -d) for the three polyurethanes are due to the fact that qp is very small and tends to zero, especially for the intermediate type of foam PUR-96. When the positions, orientations and sizes of all these intersections of the elliptic paraboloid failure surfaces are defined, the failure loci of the foams are completely determined. It is worthwhile indicating that the quantity trb = (h, + h2 + h,) defines the orientation of the EPFS. Positive values

PUR 64

I

Deviatwic

plan?

Fig. 8. The deviatoric plane intersection for the closed-cell polyurethane rigid foam PUR-64.

The failure mode of matter

203

+ exp points 8 ref. [lg] *

buckling

cut-off points

Fig. 9. The (~~6,~) principal diagonal plane intersehon of the EPFS and the buckling cut-off ellipsoid for the closed-ctll polyurehne rigid foam PUR-96.

for the trh indicate that the vertices of the EPFS lie in the tensile octant of the (o,o& frames. It can be stated from Table 2 that both distances d and d’ are negative for PUR-64 and become positive for the other two types of polyurethanes. This implies the important fact that the more porous material (PUR-64) is a tension strong material and its paraboloid has its open end toward the tension-tension-tension o&ant, whereas the other two types of polyurethanes are typical compression strong materials. Therefore, the amount of porosity makes the material change its failure behaviour from a compression strong to a tension strong material. Therefore, a critical value of porosity exists, for which the material transmutes from a normally compression strong material to an exceptionally behaving tension strong one. The existence of such tension strong materials has already been observed in the literature for various types of paperboards, as well as for oriented polypropylene [16,21,22]. All these tension strong materials present an inherent high degree of porosity.

PUR96

Fig. 10. The deviatoric plane intersection for the closed-ce41polyurethane rigid foam PUR-96.

P. S. THEOCARIS

(E PFS)

(EFS) axp.points gmf. fill

Fig. 11. The (a,S,,) principal diagonal plane intersection of the EPFS and the buckling cut-off ellipsoid for the closed-cell polyurethane rigid foam PUR-192.

Table 2 indicates further that the negative values for d and d’ become positive in the close vicinity of the density for the PUR-96 material, where the qp characteristic distance is close to zero. Furthermore, it has been proved that for the light anisotropy of this material, its EPFS presents an oblong shape, approaching a cylindrical surface at its largest part. In this case the form of the EPFS takes a cigar shape, cut-off transversely at one of its ends. On the contrary, for highly anisotropic materials the respective EPFSs become very shallow and they look like wireless plate antennas. Then, the high values of d and d’ for PUR-96 in comparison with the respective values for PUR-64 and PUR-192 indicate clearly that this material behaves like a quasi-isotropic substance. The values of the other characteristic quantities in Table 2 giving the positions, orientations and shapes of the typical intersections of the EPFSs for all types of polyurethanes indicate that these intersections are eccentric relative to the origin of the principal stress frame. Since this happens not only for the anisotropic foams of PUR-64 and PUR-192 polyurethanes, but also for the quasi-isotropic PUR-96 foam, this is a clear indication that the strength differential effect (SDE) for the anisotropic bodies is the principal factor influencing their failure behaviour. Thus, it may be concluded that the total neglect of the influence of the SDE in the various classical yield criteria (except the Coulomb criterion for brittle materials) significantly distorts the image of yielding even for initially isotropic materials, since introduction of plasticity results in a progressive development of anisotropy. Figures 7-12 also contain the experimental points of failure stresses of the three types of polyurethane foams given in refs [lO--121. In the same figures the intersections of the buckling cut-off ellipsoids by the respective (as&,) diagonal planes were plotted and compared with the existing experimental evidence [2]. The comparison of the experimental results with the ellipses given by the theory indicates a satisfactory coincidence, thus proving the validity of the ellipsoidal failure locus. This concordance of results is much higher than in the comparison of the experiments with failure loci defined by planes parallel to principal stress planes indicated by other criteria [ 10, 111. In Fig. 13, the values of the characteristic distances qp between the symmetry axes of the elliptic

The failure mode of matter

Fig. 12. The deviatoric plane intersection for the closed-cell polyurethane rigid foam PUR-192.

205

Fig. 13. The variation of the ?p distance between the symmetry axes and the hydrostatic axes of the EPFSs for various types of polyurethane foams versus their relative density.

paraboloid failure surfaces for the various types of polyurethane foams versus their relative densities were plotted. In this plot, besides the already studied three cases of foams, a fourth one with relative density d4 = 148.2 kg/m3 was tested and its qp value was determined by using relation (10). It was found that tip = 0.205 MPa. This intermediate value assured the correct plotting of Fig. 13, indicating that a continuous increase of the characteristic distance ‘lp occurs as the density of the foam is increased. From this diagram it may be derived that for a foam with density 4 = 76.5 kg/m3, the material behaves as a quasi-isotropic one.

5. CONCLUSIONS The failure behaviour of a series of rigid-cell foams was studied in this paper. It was shown that the EPFS describes satisfactorily the failure behaviour of the rigid foam materials for the primary brittle crushing mode of failure. The validity of this powerful criterion introduced for the general orthotropic solid and already proved for a series of materials of homogeneous texture, as well as for compact composites [17’J,is now extended to materials presenting significant amounts of porosity. For rigid-cell foam materials, this criterion is complemented by an ellipsoid failure surface describing the failure modes of either elastic buckling in the compressive octant or fast brittle fracture in the tensile o&ant. This complements the definition of the failure loci for rigid-cell foams and other cellular materials. The EPFS, complemented with the EFS taking care of failure under either internal buckling in compression, or fast brittle fracture in tension, presents a high flexibility to describe the failure of any material, brittle or ductile, failing in tension or compression and shear, isotropic or

206

P. S. THEGCARIS

anisotropic. It can be readily defined by only the six values of stresses in simple tension or compression along the principal anisotropic axes and also takes into consideration the SDE. The distance ‘lp between the hydrostatic and the symmetry axes of the EPFSs defines a characteristic stress depending exclusively on the anisotropy of the material, which is a constant independent of the externally applied load. It was shown that there is a direct relationship between this quantity and the relative density of foamy materials. As the q,, distance increases with density, the anisotropy of the cellular material changes. The transition of the q,, distance from negative to positive indicates that the mechanical behaviour of the foam changes mode of deformation. Indeed, for negative ‘lpstress, PUR-64 behaves like a tension strong material, whereas for qp x 0 it becomes an isotropic material and for positive rip it changes to a compression strong material, as happens with PUR-192. However, the sign of the r,-pstress is not a criterion to define if a material is strong in compression or tension. The only sure criterion for the classification of a material as strong in tension or compression is whether the quantity of trh is positive or negative. However, a change of sign of ‘lpfor a material with different degrees of anisotropy also indicates a change of behaviour from strong tension to strong compression and vice versa. Acknowledgements-The research work contained in this paper is supported by the Lilian Boudouris foundation. The author expresses his gratitude for this support. He is also indebted to bis secretary Mrs Army Zografaki for helping him in typing the manuscript and plotting the figures of the paper.

REFERENCES [I] P. S. Theocaris, The elliptic paraboloid failure criterion for cellular solids and brittle foams. Acta Mech. 89(2), 93-121 (1991). [2] P. S. Theocaris, Failure modes of closed-cell polyurethane foams. Inr. J. Fracrure 56(4), 353-375 (1992). [3] P. S. Theocaris, The symmetric ellipsoid failure surface for transversely isotropic body. Acta Mech. 92(l), 35-60 (1992). [4] P. S. Theocaris, The elliptic paraboloid failure surface for ZD-transtropic plates (fiber laminates). Engng Fracture Mech. 33, 185-203 (1989). [5] P. S. Theoauis, Tensor failure criteria for composites: properties and comparison of the ellipsoid failure surfaces with experiments. Ado. Polymer Technol. 11(l), 27-40 (1992). [6] L. I. Gibson and M. F. Ashby, The mechanics of three-dimensional cellular materials. Proc. R. Sot. Land. Ser. A 382(1782), 43-59 (1982). p] M. F. Ashby, The mechanical properties of cellular solids. MeraIl. Trims. 14A, 1755-1769 (1983). [8] G. Menges and F. Rnipschild, Estimation of mechanical properties for rigid polyurethane-foams. Polymer Engng Sci. M(S), 623627 (1975). [9] W. E. Warren and A. M. Rraynik, The linear elastic properties of open-cell foams. J. uppl. Me&. 55(2), 341-346 (1988). [lo] L. J. Gibson, M. F. Ashby, J. Zhang and T. C. Triantatillou, Failure surfaces for cellular materials under multiaxial loads-I. Modelling. ht. J. Mech. Sci. 31(9), 635663 (1989). [l l] T. C. Triantafillou, J. Zhang, T. L. Sherclift, L. J. Gibson and M. F. Ashby, Failure surfaces for cellular materials under multiaxial load+II. Comparison of models with experiment. Inf. J. Mech. Sci. 31(9), 665-678 (1989). [12] T. C. TriantalIBou and L. J. Gibson, Multiaxial failure criteria for brittle foams. In?. J. Mech. Sci. 32(6), 479-496 (1990). [13] N. W. Tschoegl, Failure surface in principal stress space. J. Polymer Sci., Parr C, Polymer Symp. 32,239-267 (1971). [14] P. S. Theocaris, Generalixed failure criteria in the principal stress space. Tbeor. appl. Mech. (Theoretichtta i Prilojzia Mekkanika) Bulgarian Acad. Sci. M(2), 74-W (1987). [15] P. S. Theo&s, Weighing failure tensor polynomial criteria in composites. Inf. J. Da?ncge Mech. l(l), U

(1992). [la] P. S. Theocaris, The elliptic paraboloid failure surface for transversely isotropic materials off-axis loaded. Rheol. Acfa zs(2), 154-165 (1989). [17] P. S. Theocaris, The paraboloid failure surface for the general orthotropic material. Acta Mech. 79(l), 53-79 (1989). [18] J. Wang, Reticulated vitreous carbon-a new versatile electrode material. Electrochim. Acru 26, 1721-1726 (1981). (191 P. H. Thorton and C. L. Magee, The deformation of aluminum foams. MeraIl. Trans. 6A, 1253-1263 (1975). [20] P. S. Theocaris, Properties and experimental verification of the paraboloid failure criterion with transversely isotropic materials. J. Mater. Sci. X!(5), 1076-1085 (1990). [21] P. S. Theocaris, Failure behavior of paper sheets. J. Reinf. Pkwfic Camp. 8(6), 601626 (1989). [22] P. S. Theo&s, Decomposition of strain energy density in fiber reinforced composites. Etrgng Fracture Mech. 33, 335-343 (1989). (Receioed 16 February 1993)