Failure surfaces for cellular materials under multiaxial loads—I.Modelling

Int. J. Mech. ScL Vol. 31, No. 9, pp. 635-663, 1989

0020-7403/89 $3.00+.00 ~) 1989 Pergamon Press pie

Printed in Great Britain.

FAILURE SURFACES FOR CELLULAR MATERIALS UNDER MULTIAXIAL LOADS--I. MODELLING L. J. GmsoN,* M. F. ASHBY, t J. ZHANG: and T. C. TRIANTAFILLOU1" tDepartment of Civil Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. and :Cambridge University Engineering Department, Trumpington St., Cambridge CB2 IPZ, U.K. (Received 20 October 1988; and in revisedform 7 May 1989) Abstract--Materials with a cellular structure are increasingly used in engineering. Proper design requires an understanding of the response of the materials to stress; and, in real engineering design, the stress state is often a complex one. In this paper we model the elastic buckling, plastic yield and brittle fracture of cellular solids under multiaxial stresses to develop equations describing their failure surfaces. The models are compared to data in the following, companion, paper.

NOTATION b depth of honeycomb cell c crack length, depth of neutral axis C, Ct, C2, C3 constants E* Young's modulus of bulk material E, Young's modulus of solid cell wall material F force h length of vertical members in honeycombs I moment of inertia of a cell wall K* mode I critical stress intensity factor I length of inclined members in honeycombs M bending moment MAn, Ms^ bending moments at the ends of member AB MAc bending moment at end A of member AC Mr moment at rupture of cell walls Mp plastic moment of cell walls n end constraint factor P axial force P^~ axial force along member AB Petit buckling load PI force in direction i, i = 1, 2 r distance from crack tip rotational stiffness t thickness of the cell walls U square root of axial force divided by the Euler buckling load X~ denotes direction i, i = 1, 2, 3 ct, fl angles of rotation 7, 7' constants strain £D densification strain 0 angle between the horizontal and the inclined members in honeycombs 0^, 0B angles of rotation at the ends of member AB '~'c, "le, 2p load factors v["2 Poisson's ratio of a honeycomb p* density of bulk material P, density of solid cell wall material o" stress o"a axial stress o"d deviatoric stress uniaxial elastic collapse stress of isotropic material (~*)l elastic collapse stress in direction i, i = 1, 2, 3 o'?r uniaxial tensile fracture stress of isotropic material (o*r)~ tensile fracture stress in direction i, i = 1, 2, 3 (Tfs modulus of rupture of solid cell wall material hydrostatic plastic collapse stress

636

L.J. GIBSON et al. stress in principal direction i, i = 1, 2, 3 normal stress parallel to material direction i, i = 1, 2, 3 (7I local stress field ffm mean stress uniaxial plastic collapse stress of isotropic material (~*,), plastic collapse stress in direction i, i-- I, 2, 3 yield stress of solid cell wall material ~q shear stress, i, j = I, 2, 3 plastic shear strength of isotropic material (~*,),~ plastic shear strength of orthotropic material, i , j = I, 2, 3 ~(u), 4,(u) flexibility coefficients, functions of u

1. I N T R O D U C T I O N

Materials with a cellular structure are increasingly used in modern engineering. Aluminum, paper and polymeric honeycombs are used as cores for high-performance sandwich panels; plastic and metal foams absorb energy in packaging and safety padding; and natural cellular materials--particularly woods--are widely used in structures. Micrographs showing the structure of natural and man-made cellular materials are shown in Fig. 1. We distinguish foams, which have three-dimensional, polyhedral cells (Fig. 1a-e) from honeycombs, with their two-dimensional, prismatic cells (Fig. l f). Stress-strain curves for cellular materials in uniaxial compression are shown in Fig. 2: they exhibit the three regimes of behaviour characteristic of all cellular materials. At strains less than about 5% the material is linear-elastic; the initial slope of the stress-strain curve is described by Young's modulus. As the load is increased the cells begin to collapse by elastic buckling, plastic yielding or brittle crushing, depending on the properties of the cell walls. Collapse progresses at a roughly constant load (giving a stress plateau) until the opposing walls in the cells meet and touch, when densification causes the stress to increase steeply. The tensile stress-strain behaviour is illustrated by Fig. 3. At small strains honeycombs and foams are linear-elastic in tension, as in compression. Those which have cell walls which are plastic show a yield point followed by a rising stress-strain curve (Fig. 3b): elastomeric cellular materials stiffen as the cells elongate with increasing strain (Fig. 3a) and brittle ones simply fracture (Fig. 3c). The behaviour of cellular solids in simple uniaxial loading is tolerably well understood (see, e.g. [1]). But loads in real engineering structures are often multiaxial: for instance, foams used in the cores of sandwich panels or as protection against mechanical impact are subject to a general, three-dimensional state of stress. Then it is not the uniaxial stress but the combination of stresses causing failure which is important to the designer. "Failure", of course, can mean different things, depending on the design. It could mean "excessive elastic deformation"; more usually it means "elastic collapse", "the onset of plasticity" or "fracture". Each such mechanism can be characterized by a failure surface: a surface in stress-space describing the combination of stresses which cause failure. The mechanism of failure itself may depend on stress state: a foam may be plastic in compression but brittle in tension, for instance. Then the failure surface is the inner envelope of the intersecting surfaces for the individual mechanisms. In this paper we develop approximate equations describing each mode of failure for honeycombs and foams under multiaxial stresses. Typical failure envelopes for biaxial and axisymmetric stress states, showing the intersection of the surfaces for different failure mechanisms, are plotted. The results are compared with experiments in the following companion paper [2]. 2. L I T E R A T U R E

Models for the mechanical behaviour of cellular solids seek to identify and analyse the mechanisms by which the cell walls deform and fail under load. Early models assumed that the cell wails carried axial (that is, in-plane) loading only [3-51 an assumption which inevitably leads to the conclusion, inconsistent with experimental data, that the elastic properties vary linearly with the density. The difficulty was resolved in later studies by

Failure surfaces for cellular materials--I

FIG. 1. Micrographs showing the cellular structure of(a) a flexible polyurethane foam, (b) a flexible polyethylene foam, (c) a rigid polyurethane foam, (d~ an aluminum foam, (e) a glass foam, (f) an aluminum honeycomb.

637

Failure surfaces for cellular materials--I

ELASTOMERICFOAM

~

~

b

(a}

/

l

639

IC;PoLAST'C EoL p FOAM j J b

DENSIFICATION/

LLI

w

OENSIF~

~" PLATEAU(PLASTICYIELDING) "-'LINEAR ELASTICITY(BENDING)

='---LINEARELASTICITY(BENDING) 0

E

STRAIN, E

(b)

STRAIN, E

ELASTIC-BRITTLEFOAMI

E°

(c)

I/I

b

tO hi rr tO •

1

USHINp) l

"-LINEAR EL,ASTICITY(BENDING)

ED

STRAIN, E

FIG. 2. Typical compressive stress-strain curves for cellular solids made from (a) an elastomeric, (b) an elastic-plastic material, (c) an elastic-brittle material. Cell collapse may be caused by elastic buckling, plastic yielding or brittle fracture, depending on the nature of the cell wall material. Densification occurs when opposing cell walls meet and touch (after [1]).

FOAM "•---'•TELASTOMERIC

b

(a~

/

JELASTIC-PLASTICFOAM (b)

b

-~'CE/WAL / Ll't"ALIC-NMENT W EI /~I~EAR ELASTICITY(BENDING)

~LINEAR ELASTICITY(BENDING) 0

STRAIN, E

STRAIN, E

I

ELASTIC- BRITTLEFOAM (c)

b W rr

9" :~FRACTURE E! ~LINEAR ELASTICITY( BENDINGI 0

STRAIN, E

FIG. 3. Tensile stress-strain curves for cellular materials. (a) An elastomeric foam; (b) a plastic foam; (c) a brittle foam (after [1]).

recognizing the more important contributions of cell-wall bending to the mechanical properties [6-15]. The complete set of four elastic constants describing the in-plane, linearelastic response of honeycombs, calculated from the cell-wall bending model, agree well with experiments [11"]. The modelling of the linear elastic behaviour of foams is much more complicated and, so far, less successful. The most promising approach is one which extends

640

L.J. Gtaso~ et al.

the honeycomb analysis using dimensional arguments [1, 12], but it is not capable of predicting the complete elastic modulus tensor from first principles. In compression, cells may buckle elastically, yield plastically or crush in a brittle manner [3, 8, 9, 11, 12, 16-25]. Elastic buckling gives large recoverable strains, while yielding and brittle crushing, of course, are irrecoverable. For honeycombs, an exact analysis is possible, and its results have been confirmed by experiments on specially-made elastomeric and plastic honeycombs with a wide range of geometries [11, 26]. Dimensional arguments can again be used to extend the analysis to the uniaxial collapse of foams, for all three mechanisms [1, 12]. In uniaxial tension elastic buckling is not possible. Elastomeric honeycombs and foams fail by cell-wall rupture or tearing at large strains [3]. Cellular solids made from materials that yield plastically show a yield stress in tension caused, like that in compression, by plastic bending of the cell walls. Brittle materials fracture suddenly by fast crack propagation [23, 25]. Studies of the failure of honeycombs and foams under multiaxial loads are much more limited. Klintworth and Stronge [26] compare models for elastic buckling and plastic collapse of honeycombs with models for these two processes and conclude that they interact: yielding occurs at stresses below those given by simple plastic analysis. The picture for foams is more confused. Shaw and Sata [27] measured the combination of biaxial stresses required to cause failure of a polystyrene foam which yields plastically. Their results suggest that failure is governed by the maximum principal stress, independent of the minor principal stress. At first sight, this is an unusual result: plastic solids, when fully dense, fail in a way described approximately by the yon Mises or the Tresca yield criterion, both of which involve at least two principal stresses. The later, more extensive, results of Patel and Finnie [7] and of Zaslawsky [28] (both of whom used rigid polyurethane foams which yield plastically in compression, but which may be brittle in tension) largely support Shaw and Sata's conclusion: all suggest a rectangular failure surface for foams under conditions of plane stress. The analysis developed in this paper indicates that under a general triaxial state of stress, failure depends on all three principal stresses, not just the maximum principal stress. Under conditions of plane stress, however, the failure surface can sometimes be approximately rectangular, in agreement with the data of these studies. 3. BIAXIAL FAILURE IN HONEYCOMBS We model a two-dimensional cellular solid, or honeycomb, as an array of hexagonal cells (Fig. 4). The cell edges have lengths h and l, a thickness t and a depth b. The angle between the horizontal and the inclined members is 0. Anisotropy is included by allowing the unit cell to have edges which differ in length and which meet at an angle which is treated as a variable. The solid material of which the cell wall is made has a density, Ps, a Young's modulus of Es, a yield stress (if plastic) of oy~ and a modulus of rupture (if brittle) of ate. The relative density of the honeycomb is its density, p*, divided by that of the cell wall material, p~:

p* (2+h/l) t p~ - 2 cosO(h/l+sinO) 1

(1)

or, for regular hexagons (0 = 30 ° and h = l, the isotropic structure): p* p~

2 t x/~ l"

In this section we analyse the failure of the honeycomb under biaxial loading; in the next, we extend the arguments to the more general case of the triaxial loading of foams. We develop a failure surface for each mechanism of failure (plastic collapse, brittle failui'e and elastic buckling) and superimpose them to give the complete failure envelope. To simplify the analysis we assume here that the principal stresses are aligned with the X1 and X2 axes; the more general case for principal stresses at any orientation is described by Triantafillou [29].

Failure surfaces for cellular materials--I

641

X2 I

. X1

FIG. 4. An idealized h o n e y c o m b made up of an array of hexagonal cells. Such an array deforms by the same mechanisms as a foam with a more complex geometry. The geometry of a unit cell in the two-dimensional model is shown in the upper left-hand cell.

Plastic collapse and the yield surface of a honeycomb Plastic collapse is analysed by the method of Gibson et al. l11] and Gibson and Ashby [1], recently extended to large strains under biaxial loading by Klintworth and Stronge [26]. The unit cell of a honeycomb is shown in Fig. 5. The remote biaxial stresses, cr~ and tr2 produce forces P~ and P2 on the inclined member of the unit cell, parallel to the axes Xt and X2, given by: P~ = a 1(h + l sin O)b

(2a)

P2 = a2(lc°sO)b

(2b)

and and a load on the vertical member, parallel to X2, of: P = 2a 2(/cos 0) b.

(3)

The loads P1 and P2 produce both bending moments and axial loads on the inclined members. Provided deformations are small,the contribution of the axial load to the bending moment can be neglected. Then the maximum moment, M, tending to bend the cell wall is: M = (P1 s i n 0 - P2cosO)l 2

'

(4)

when PI sin0 < P2cosO, the bending deformation is in the direction shown in Fig. 5(b), and plastic hinges form at the locations shown in Fig. 5(c). When the inequality is reversed, the moment changes sign and the bending is in the opposite sense. Both must be considered to obtain the complete failure surface. The axial stress in the inclined member, g,, is:

Ga=

(PI cos 0 + P2 sin 0) bt

(5)

Under uniaxial loading, the contribution of the axial load to yielding is small compared with that from the bending moment. However in the biaxial case it can be the dominant contribution to failure (think, for instance, of biaxial tension with P1 sin O-~P2 COS 0, when M = 0 and failure corresponds to the axial yielding of the cell walls).

642

L.J. GIBSON et al.

(a)

(b)

): FIG. 5. (a) Biaxial stresses a I and a2 acting on a unit cell of the honeycomb model. (b) The forces, Px and P2, and the moment, M, acting on the inclined member from the biaxial stresses. (c) The mode of plastic collapse, showing the location of plastic hinges in bending.

Failure occurs when the cell wall has yielded across its entire section. The stress distribution required to do this is shown in Fig. 6. The depth of the neutral axis is found by equating forces above and below the neutral axis; it is: c = ~t (\ a~,/, .+ a . ~

(6)

The plastic moment required to cause the formation of a plastic hinge is then given by the force, F, times the moment arm, t/2: M~, = Ft/2 •

bct

- - ( ~ ' - ~") 2 or

M,=

4

(o.Yl

-\~-y~,/ l"

(7)

Failure occurs when the moment, M (equation 4), equals the plastic moment, Mp. Remembering that the applied moment may be either positive or negative, depending on the

Failure surfaces for cellular materials--I

I~ys-(Ja

Oa

643

O'ys

T

T

c

t

0

-Oys-O a BENDING STRESS

O'ys

AXIAL STRESS --

+

TOTAL STRESS

FIG. 6. The bending, axial and total stress distributions across the section at the formation of a plastic hinge. The entire section has yielded causing failure. m a g n i t u d e s a n d signs of 0-1 a n d 0"2, we find:

+_[!PlsinO2P2cosO' I l

0"y~bt2F1

(PlcosO+P2sinO~21

=-~--L -\

~

/ J

or

+

(t/l)2 0"1 7 + sinO cosO+a2cosOsinO

=0"y..__z~

1 -

.

2

(8)

crys(t/l )

The equation gives two intersecting ellipses which define the failure surface for plastic yield in biaxial loading. It is plotted for regular hexagonal cells in Fig. 7.

YIELD SURFACE HONEYCOMBS 0=30 ° h/1:1 t / I :O.1 t;ys/Es:l/50

02

,;

0.06

0104

/

/ PLASTIC

0.02

-0,06

-0104

J

ELASTIC

,"

BUCKLING /

/

•

p

j•

'

002

0.04

0 06

,'-o.o2 i

i i

i ¢ • / / /pc

• /

-0.04

i ¢ • •

/,

¢fJ'

-0.06

FIG. 7. The plastic yield surface for an idealized, two-dimensional cellular solid made up of regular hexagonal cells. The surface is truncated by the elastic buckling failure surface in biaxial compression.

L.J. GIBSON et al.

644

The extreme point on the failure surface corresponds to pure axial stress in the cell walls. The ratio of applied stresses for which the bending moment cancels and only axial stresses remain is given by: PtsinO-P2cosO=O or at

cos20

a2

( ~ + s i n 0 ) sin0

=v*z

(9)

which is just equal to the Poisson's ratio v*2 for the honeycomb for loading in the X1 direction 1-11-1. For regular hexagonal cells, this ratio is 1, corresponding to equal biaxial stress. The ratio of applied stresses for which the inclined member is loaded in pure bending (that is, the axial stresses cancel) is given by:

P~cosO+P2sinO=O or

al

a2

sin0 h/l+ sin0"

(10)

For regular hexagonal cells, this ratio is - 1/3.

Brittle failure and the fracture surface of a honeycomb The combination of applied stresses at and a2 required to cause tensile failure in the cell wall is found from the applied moment and the axial stress acting on the cell walls (equations 4 and 5). The wall fractures when the extreme tensile fibre stress equals the modulus of rupture of the cell wall, afs (Fig. 8). The bending stress at failure, a f s - aa, is related to the applied moment by:

Mt

ars- aa =-~--,

(11)

where l=bt3/12 is the moment of inertia of the cell wall and the moment can be either positive or negative. Substituting for M and a~ from equations (4) and (5), and using equations (2) and (3) gives:

3[al(~+sinO)sinO--a2cos20] -t-

at(~+sinO)cosO+a2cosOsinO

(t/l)2

Jr

+

0

-(qs-Oa) BENDING STRESS

+

= afs" (12)

=S

Ofs-~

/

(t/l)

AXIAL STRESS =

-O'fs+20"a

TOTAL STRESS

FIG. 8. The bending, axial and total stress distributions across the section when the extreme tensile fibre stress reaches the modulus of rupture of the cell wall, aft. At this point the cell wall fractures.

Failure surfaces for cellular materials--I

FRACTURE SURFACE HONEYCOMBS 0=30° h/l=l ¢/1=2 t/l=0.1 crrs/E,=l/1O0

0.06

645

,~/

~ffm

,2 // 0.04 pl? • p / /

I

/

~

/ /

• /

s. I 1 t FAST BRITTLE o.o2 / j FRACTURE ~ " /I/i / t

O'fs -o'.oe

"

-o:o4

"'r°a'~/1~

E,.A T,C

-V /

/

"

ALL

o'.02

"

oo4

! 0.06

RR,TT,E

,"

/

•

I'

/

,,

/

-0.04

I

/ / / /

-0.06

FIG. 9. The brittle fracture surface for an idealized, two-dimensional cellular solid made up of regular hexagonal cells. The fracture surface is truncated by fast brittle fracture in biaxial tension and by elastic buckling in biaxial compression.

This equation defines the combination of stresses which will cause tensile failure in the cell walls. It is plotted as the dashed line in Fig. 9, but it does not define the whole failure surface. To do this, we must consider two additional failure mechanisms: crack propagation in tension and elastic buckling in compression. When a honeycomb contains flaws or defects (as it often does) the fracture strength in tensile modes of loading is less than that when loading is compressive 1-22, 23]. This is because the flaws are relatively harmless in compression, but in tension they propagate at a stress which can be calculated by using the methods of fracture mechanics. The underlying idea is that a crack of length c lying normal to a remote tensile stress, try, in an elastic solid creates a singular local stress field, a~, of:

olv/~

0"/= 2N ~

in the plane containing the crack at a distance r from its tip. Figure 10 shows such a crack in a cellular solid. Consider the first unbroken cell wall (labelled A in Fig. 10a), which we take to be half the cell width, beyond the crack tip. The moment on it is:

(h+lsinO)/2, r)dr M = fl h+'~i"°'/2(rtb(h+lsinO 2

when M exceeds the fracture moment the wall fails and the crack advances. Assembling these results gives the tensile fracture strength for loading in the XI direction: (trg)l = 2 ( ~ + s i n 0 ) 3'2x/ c \ l /

(13)

A similar analysis for loading in the X2 direction gives: (14)

646

L.J. GIBSON et aL

t

t

t

(a)

t

-

-

.

t

t

(b)

FIG. 10. Propagation of a crack through a brittle honeycomb loaded in tension. The stress concentration at tbe crack tip causes the cells walls just beyond the tip to be loaded more heavily than the others. (a) Loading in the X~ direction. (b) Loading in the X2 direction (after [1]).

Under a biaxial state of stress, brittle fracture occurs when either al exceeds (a~)l or tr2 exceeds (a~,)2: the failure surface is a box bounded by planes of constant principal stress. It gives the two lines labelled "fast brittle fracture" which close the failure surface in the tensile quadrant of Fig. 9. The failure surface closes in the compression quadrant, too. If the material of the cell wall is very easily crushed, it may fail in compression; the analysis is identical to that given above for tensile failure in the cell walls with the modulus of rupture of the wall replaced by the compressive strength. However it is more likely that it will fail by elastic buckling, which we consider next.

Elastic buckling of a honeycomb It is now known that an elastic honeycomb buckles (giving elastic collapse of the structure) in at least two modes. Those for regular hexagons are shown in Fig. 11, Uniaxial compression in the X2 direction, and biaxial compression with a2 > al produces the first mode (Fig. l la); biaxial compression with a2 < a l produces the second. Both have been

direction (mode 1) and (b) biaxial compression (mode 2) (after [1]).

FIG. 11. Buckling of a silicone rubber honeycomb under (a) uniaxial compression in the X2

i

tm

e-"

Failure surfaces for cellular materials--I

649

analysed using a standard method, outlined in Appendix A, adapted to this special geometry 1-29, 301. The results are best understood by noting that, when a honeycomb buckles, it is because one set of cell edges is loaded axially, like a beam-column, to its Euler buckling load. The Euler load for any column of height h can be written

n2 7~2Es I Perlt--

h2

,

(15)

where I is the second moment of its area. The elastic collapse stress, (a7)2, is simply Pcrit divided by the area per cell over which tr2 acts (2 bl cos0) so that

n2 n2 Est 3 (a*02 = 2 4 c o s O l h 2 .

(16)

Central to the problem of calculating Perit for a specific network of struts is the idea of the rotational stiffness of its nodes. A node is a point where three cell edges meet. When one of the three buckles, the rotation of its ends is opposed by the bending of the other two: they exert a rotational restoring moment and it is this which determines the factor n 2 in the buckling equation. This factor depends on stress state (because a lateral compression aids the bending and reduces the rotational stiffness while a lateral tension does the opposite); and it depends on the buckling mode (because different modes have different rotations associated with them). The full analyses for the two modes of Fig. 11 are summarized in Appendix A. The results are tabulated in Table 1 and plotted in Fig. 12 where they are compared with experiment. On this figure, full symbols indicate that the first buckling mode was observed; open symbols, the second. The full lines are plots of equation (15b) using the values foi" n 2 given in the table. Simple compression in the XI direction does not cause buckling; the honeycomb folds up on a stable way. As a result, there is no intersection of the elastic buckling envelope with the trl axis; the true curve is asymptotic to the tr~ axis. That is why the very end of the mode 2

ELASTOMERIC HONEYCOMB 2.0 ""UNIAXIAL" MODE o"BIAXIAL" MODE

1.5j

~MODE 2 (FIG.11b)

1.04

a.

11a)

O.S

0.0 0.0

i

i

0.5

1.0

i

1.5

2.0

o', / FIG. 12. Data for the elastic buckling of a rubber honeycomb with regular hexagonal cells under biaxial loading. The lines indicate the calculated buckling surfaces corresponding to modes I and 2. Full symbols: mode 1; open symbols: mode 2.

650

L.J. GIBSONet al. TABLE 1. E N D CONSTRAINT FACTOR n FOR ELASTIC BUCKLING O F HONEYCOMBS ~

trl/a,

n2 (mode 1)

nz (mode 2)

0 1/3 1/2 1 2 3

0.44 0.419 0.407 0.370 0.306 0.269

-0.648 0.547 0.370 0.222 0.156

tn corresponds to the vertical cell walls of Fig. 4.

curve is shown as a broken line, and why the data points (open symbols) deviate to the right. The figure documents the way in which the buckling mode switches, at the point tr 1 = tr 2, from mode 1 to mode 2, and shows the very large difference in cr*l that this produces. At certain levels of compressive axial forces in the cell walls, the deformations of the honeycomb structure are sufficient to alter the equations of equilibrium. The level at which this occurs depends on the slenderness of the individual members of the honeycomb structure. Moreover, as the elastic buckling load is approached, small initial imperfections compatible with the buckling mode become increasingly large; if these imperfections are also compatible with a plastic collapse failure mode, the failure loads are reduced and the elastic buckling and plastic collapse modes interact. Elastoplastic interactions are discussed in Appendix B. 4. MULTIAXIAL FAILURE IN FOAMS We now examine the failure of foams under multiaxial loads. The uniaxial behaviour of foams has been successfully modelled by extending, using dimensional arguments, the analysis for honeycombs [1, 12, 22-24]. The method gives the dependence of each failure stress on density and on the cell wall properties, but it is not exact--there is always a single, unknown constant of proportionality, related to the cell geometry, which must be determined by experiment. Despite this, the method offers by far the most direct, and (thus far) successful route to the analysis of foam properties. In this section, we apply it to the analysis of the triaxial failure of foams. Since we ignore geometrical factors in this approach (they are contained in the unknown constant of proportionality), we base the analysis on a simple cubic unit cell. Plastic collapse and the yield surface for foams Consider first the isotropic, open cell of Fig. 13. Each cell wall has length l and crosssection t2; the cell has a relative density which is proportional to (t/l)2: P*/Ps = C(t/l) 2, where C is a constant near unity. The analysis for the honeycomb is extended to the foam via the following steps. The applied stress cr imposes a force F on each cell wall which is proportional to crl 2. In general, this force acts at an angle to the cell wall, thereby exerting on it a bending moment, M, which is proportional to Fl, and thus to trl a. The plastic moment required to form plastic hinges in the beam, Mp, is proportional to tryst a, where try~is the yield strength of the cell wall material. Yield occurs when the applied moment equals the fully plastic moment of the cell wall (as described for honeycombs in Section 3), giving: ( p , ~3/2 a*l=Cl \ - ~ [ / ays. (17) This equation gives a good description of data for the uniaxial yield of foams with a constant Ct, determined by experiment, of 0.3 124]. N o w consider triaxial loading. If the three principal stresses acting on the cell are all equal, the cell walls are subjected to purely axial tension or compression; then plastic collapse

Failure surfaces for cellular materials--I

651

X,, (3,

]d~

edge

/ X~,, 0"2 FIG. 13. Three-dimensional, open cubic cell under triaxial loading (after [1]).

occurs when the axial stress in the cell wall exceeds ays, or: 30"m

0.. = 0.r, - (p, /p,) ,

(18)

where am = (al + a2 + o'3)/3. The constant 3 in equation (18) arises because the stress in any one direction loads only one-third of the cell walls of the unit cubic cell. The hydrostatic loading of a random distribution of cell walls in a sphere produces the same result; it is not sensitive to cell geometry. Under any other combination of loads the cell walls suffer bending as well as axial deformation. The average bending moment is proportional to the deviatoric stress times the cube of the cell size: M oc ad 13 = X / 1 I-(0.1 -- 0"2) 2 + (0"2 -- 0"3 )2 + (0.3 -- 0.1 )2 "l 13 •

(19)

But the axial load on the cell wall modifies the moment required to bend it plastically, for the reasons discussed in Section 3. The fully plastic moment for the honeycomb wall was given by equation (7); replacing b by t for the cell-edge of the foam gives:

Mp OC0.yst3 [ l - ( 0"a l 21.

(20)

xO'ys/ _J Equating this to equation (19), using the fact that (p*/ps)oc(t/l) constants of proportionality into the single constant ? gives: add= + ?

0..

1--

Fkay,(p*/p,)

l

"

z,

and assembling the

(21)

The constant 7 is obtained from the known result for simple compression (equation 17). Setting am=0"1/3 and 0"d=al in equation (21) and substituting in equation (17) gives: yl-I -O.09(p*/p,)] =0.3. For all interesting values of P*/Ps (less than about 0.3) we find y,,~0.3. The yield criterion for foams under multiaxial stresses then becomes: a~_d= +0.3 - G. \p,y

1-

(22) la,,(p*lp,)J

J

652

L.J. GIBSONet al.

or, in terms of the uniaxial collapse stress, a'l: O.d

p*

O.ra 2

___~.~p.+ 0.81 (-~-) ( ~ )

=l.

(23)

This result is plotted in Fig. 14, in two ways. The first is for axisymmetric loading (a 2 = ira); the surface is the intersection of the two ellipses given by equation (23). The axes of the failure surface are elongated as the relative density increases. This is analogous to Fig. 7 for twodimensional honeycomb-like cellular materials. The second is for biaxial loading (a3 =0). Equation (22), for this case, gives a quartic equation relating at and tr2; it is roughly elliptical but in the tension-tension, or compression-compression quadrants, it is more box-like and could be approximated by assuming that yield occurs at a constant value of the maximum principal stress. Brittle failure and the fracture surface for foams A brittle foam starts to crush in compression when the stress on the cell walls is sufficient to break them. The uniaxial crushing strength is found by the method outlined in the last sub-section [24]. The important loading on a cell wall when the foam is loaded uniaxially is that causing bending; the bending moment is proportional to trP. The moment required to fracture the wall is proportional to trfst 3 where afs is the modulus of rupture of the material of the cell wall. Combining these expressions gives the uniaxial crushing strength:

ff*er= C2 (P* /Ps) s12 ar$.

(24)

Maiti et al. [24] use handbook data for af, and find C 2 =0.65. Reliable data for af, are difficult to obtain: it depends on the size of the largest crack within the cell wall and, because the larger the specimen the greater the probability of it having a large crack within it, the size of the cells themselves. This difficulty can be overcome by noting that the analysis for brittle crushing is identical to that for plastic collapse with Mp replaced by Mf; the other geometrical constants are identical. For a rectangular section Mf = (2/3) Mp, suggesting that C2 is about (2/3)0.3 or 0.2. For triaxial loading, the axial stress acting in the cell walls becomes more important. The argument parallels that for honeycombs: brittle crushing occurs when the extreme fibre stress in a member equals the modulus of rupture of the cell wall (equation 11): Mt trf,- tr, = 21 '

where the moment, M, can be either positive or negative. Noting, as before, that the axial stress is proportional to the mean applied stress divided by p*/p, (equation 18) and that the bending moment is proportional to the deviatoric stress times the cube of the cell size (equation 20), we find: ad

3am

//D* "~3/2 F

]

L of. r/o.)_

(25)

The constant of proportionality ),' is again found by examining the limit of simple compression (equation 24) for which % = at and a,, = a t/3. Substituting these into equation (25), solving for at/af, and equating the result to equation (24) (remembering that tensile stresses are positive) gives: ~' =

0.2 1 + 0.2(p*lp~) t/2.

(26)

For typical values of p*/p~ ~ 0.1, we find ~' m0.19. Substituting this result in equation (25) gives the failure surface for tensile failure of the cell wall:

a-Ad= + 0"19

[p* \3/2 F )

k1

30"m

1

(27)

Failure surfaces for cellular materials--I

653

YIELD SURFACE-FOAMS AXISYMMETRIC LOAD 4.

pTpl = O.l o2--o3 % s / E , : 1/50

3. "2"

I J

I

-4

I

I

4

-3

/

c&_o~

o;, -o;, l

'

-2-

BUCKLING

-4

(a)

YIELD SURFACE-FOAMS BIAXIAL LOAD

O,

pTp = 0.1 a3=O av,/E,,: L/SO ~.0

•

~ Ol 5

-1.0

- 0.5

(b) FIG. 14. Sections through the plastic yield surface for a foam under (a) axisymmetric loading and (b) biaxial loading. The section for the axisymmetric loading is truncated by the elastic buckling surface in the compression octant.

654

L.J. GIBSON et al.

FRACTURE SURFACE-FOAMS AXISYMMETRIC LOAD f:)~'~=O.l ~ = o 3 c/1=4 of,/E,= 1/10O

Oc', 6

4¸ /

/

//

2T

I

I

! .,~/.,'~ J" Y

/

//

///

//

/ / / C .FAST BRITTLE FRACTURE

I

I 4

I 6

Oc', ~/

/~-----__LTE

NSIL E

BENDING OF CELL WALLS

BUCKLING

(a)

FRACTURE SURFACE-FOAMS BIAXIAL LOAD c/1:4 o,s/E, = l/1O0

f!~'~=0.1%=0

1.0 FAST

BRITTLE

FRACTURE

/ "

-- "" I I I

0..= I

! t -0.5

/11.0 / 1

0.5

//

-0.5-

,"

i

02

cJ;

TENSILE BENDING FAILURE OF CELL WALLS

(b) FIG. 15. Sections through the brittle failure surface for a foam under (a) axisymmetric loading and (b) biaxial loading. The surface is truncated by fast brittle fracture in tension. The section for axisymmetric loading is also truncated by the elastic buckling surface in the compression octant.

Failure surfacesfor cellular materials--I

655

or, in terms of the uniaxial crushing strength:

+_~_, +0.6(P*~ 1/2 6m

oo,

\7,J

(28)

This is plotted in Fig. 15, for axisymmetric loading (a2 = a3) and for biaxial loading (a3 = 0). The axisymmetric case is analogous to that for biaxial failure in a honeycomb (Fig. 9). The fracture strength of a foam in tension, like that of a honeycomb, is governed by the propagation of a crack. The fracture toughness of a brittle foam, K,*, has been shown to be [23]: K*c = C3 afs x/~(P*/Ps) 3/2.

(29)

Maiti et al. [23] find C3 =0.65; for the reasons stated above, we adjust this to Ca =0.2. The fracture stress in simple tension is: K*c

(30)

where 2c is the length of the longest crack in the sample. Under multiaxial stresses, fracture occurs when the maximum tensile stress is equal to a~r: K~c

(31)

This boundary is included on the plot of brittle failure (Fig. 15) as a pair of lines marked "fast brittle fracture" which close the failure surface in the tensile quadrant.

Elastic buckling of a foam Like elastic honeycombs, elastic foams collapse by cell-wall bending at a critical stress, a*,. The analysis parallels that for honeycombs (Section 3). Consider the "node and four associated struts of length I shown in Fig. 16(a); the four struts meet at equal angles of 108°. Under a given stress states (aa, a2, a3) one strut carries a greater axial compression than the other three, and there is a tendency for it to buckle (strut AB, Fig. 16b). This tendency is opposed by the other three struts which determine the rotational stiffness of the node, and, through this, the constant n 2 in the. Euler equation (15). If this is known, the elastic collapse stress a*~ is found by dividing the Euler load, Pcr~t, by the area per cell over which the stress acts.

Xl

Xl B

B

A

A

E

X3

X3 X,

X2 (a)

D

(b)

FIG. 16. (a) The four members at a node of a pentagonal dodecahedral cell. (b) The assumed deflected shape of the members under compressiveloading.

656

L.J. GIBSON et al.

The difficulty in calculating n 2 lies in identifying the mode of buckling. One obvious one is sketched in Fig. 16(b); it is the parallel, for the 3-dimensional array of struts, of the first mode shown by honeycombs (Fig. l la). However the buckling modes in foams have not been observed directly (unlike those for honeycombs) and we do not know whether other modes

ELASTIC BUCKLING SURFACE- FOAMS AXlSYMMETRIC LOAD

o2=o3

1.0

I

i

- 1.0

\ -1.0

(a) ELASTIC BUCKLING SURFACE- FOAMS

o,

BIAXIAL LOAD 03--0

1.0

- 1£

O2

-1.0"

(b) Fzo. 17. Sections through the elastic buckling failure surface of the members under (a) axisymmetric loading and (b) biaxial loading.

Failure surfaces for cellular m a t e r i a l s - - I

657

TABLE 2. END RESTRAINT, n, FOR ELASTIC BUCKLING OF FOAMS

L o a d condition U niaxial compression, a; = a, a2 = a3 = 0 Biaxial compression, a 2 = a 3 = a , ~rI = 0 Hydrostatic compression, al = a2 = 0 3 = o a~ = a, a 2 = a 3 = -- a/8 a I = --a/2, ~r2=aa=cr

nz

o*/o-'11'

0.41 0.36 0.34 0.42 0.37

1.00 0.88 0.83 1.02 0.90

to.. is the value of a at failure.

exist. Assuming the mode shown in Fig. 16, values o f n 2 c a n be calculated, using the method outlined in Appendix A. The results are summarized in Table 2 1-30]. Sections through the resulting failure surface for elastic buckling for axisymmetric and biaxial loading are shown in Fig. 17. This surface is included on the yield and brittle failure surfaces shown in Figs 14(a) and 15(a) for an axisymmetric applied stress (a2 = 0"3) ( f o r the conditions of Fig. 14b and 15b, for biaxial loading, the elastic buckling surface falls outside of the yield surface). In practice, the transition between the failure surfaces for elastic buckling and plastic yielding will be smooth, like that for the transition between Euler buckling and yield with changing slenderness ratio for a simple column. This smooth transition is the result of the interaction between elastic buckling and plastic collapse modes. Elastoplastic interactions are extremely difficult to quantify for foams, due to their complex structure.

Failure in anisotropic foams Consider first the yield surface of an anisotropic foam. The uniaxial plastic collapse stress, 0.'1, now depends on direction. Let the values in the three principal directions be (apl)l, (apl)2 and (0"'1)3; the yield surface intercepts the three axes at these points. We start from the result for the isotropic case (equation 22) which we write in the form:

where 0.*1= 0.30.ys(O*/ps) 3/2 and 0.* = 0.ys(p*/os)/3. We first replace the strength-normalized deviatoric stress 0.d/0.'1, by the quantity

(0.,,*,)3 .jr.I. +,r,'. Lt,.,/3(-¢,),:., 7 " '

~23

t,(0.,",)3 (0.,*,), ./ + /

31

|

(33)

i where the stresses are defined in Fig. 18 and ('t'p~l)12, ("C~1)23 and (Zpl)31 are the plastic shear strengths of the orthotropic material (see [29]). Note that 0.'1= w/3Z*l for isotropic materials. The above quantity comes directly from the definition of the deviatoric stress ad (or, better, the second invariant of the deviatoric stress tensor):

0.a =.J½1-(0., t - 0.22):z + (0.22- 0.33):: +(0.33- 0.~ ;)2 ] + 3 (-c~2+-c~3 +.c~, ). When expression (33) is equal to unity, the foam deforms plastically by a cell-wall bending mechanism. When it is zero there is no tendency to yield by this mechanism; and when this is true, the normal stresses are related by the equations 0.11

022

0"33

(0".'1), Under these conditions, plastic failure is only possible by the axial stretching of the cell walls. This suggests that the quantity am~0.* be replaced by the quantity IV'0.11

3L(0.p*),

_~_ 0.22

.{. 0.33 ]0.p~l

(0.p*O', (0"1)310" <

658

L.J. GIBSON et al.

011

"[12~ 1 3

t2

J {~"

T3 2

Xl

~,, t U

1

x,/

XI-3

FIG. 18. The state of stress in orthotropic foams.

giving a simple transformation of the axes from ax to all [apt/(api)l ] etc. Assembling these results gives the anisotropic yield criterion: o"11

\(~*,)~ (~-~,),) J

+rf Lk

v (,. ) +

y+(

= l-O.09(P*~ V(~+~+~_~

yiV

~1~.

\P, JL\(%,), (cr,,,)~ (%0321

(34)

Note that this equation reduces identically to our starting equation when all three uniaxial strengths are the same, and that it corresponds to a distortion of the original yield envelope in the ratios of the uniaxial yield strengths. The argument for brittle fracture parallels that for yield. The elastic buckling surface is roughly approximated by normalizing each stress axis by the uniaxial elastic buckling stress in that direction. 5. C O N C L U S I O N S

The failure surfaces for honeycombs and foams are more complicated than those for more conventional materials. A ductile metal (for example) fails by plastic yielding, regardless of the stress state: its failure surface is the von Mises envelope. That for a brittle metal (like cast iron) or polymer (like PMMA) may be the inner envelope of two intersecting surfaces: that for plastic yielding, truncated in tension by that for brittle fracture. The failure envelope for cellular materials is--or at least it can be--one step more complex still. It is the inner envelope of the surfaces corresponding to three distinct mechanisms: elastic collapse by cellwall buckling, plastic collapse by ceU-waU bending, and brittle crushing or fracture by cell wall-breakage. The elastic buckling surface exists only in the quadrants in which at least one principal stress is compressive. There is no buckling mode when all stresses are tensile; here other surfaces appear, which can (depending on relative density and cell-wall properties) lie inside the buckling surface in other quadrants as well. The simplest is that for plastic yielding which has the shape of a distorted ellipsoid. The brittle failure surface is a little more complicated; it

Failure surfaces for cellular materials--I

659

d e p e n d s o n w h e t h e r o r n o t a c r a c k l a r g e r t h a n the cell size pre-exists in the m a t e r i a l . If n o large c r a c k is present, the cell wall fails w h e n its m o d u l u s of r u p t u r e o r its c o m p r e s s i v e c r u s h i n g s t r e n g t h is reached. A p r e - e x i s t i n g c r a c k further reduces the s t r e n g t h o f the f o a m in tension; failure t h e n o c c u r s when the m a x i m u m p r i n c i p a l tensile stress reaches the f r a c t u r e t o u g h n e s s o f the f o a m d i v i d e d b y x / ~ . T h e u p s h o t o f all this is t h a t the failure e n v e l o p e for a h o n e y c o m b o r f o a m can t a k e o n a v a r i e t y of shapes, f r o m b o x - l i k e to ellipsoidal; it c a n h a v e corners; a n d its s h a p e in the tensile q u a d r a n t c a n be q u i t e different from t h a t in the p u r e l y c o m p r e s s i v e one. W i t h a little experience, o n e c a n a s s o c i a t e c e r t a i n s h a p e s with c e r t a i n m e c h a n i s m s ; a n d g u i d e d b y o t h e r e x p e r i m e n t a l o b s e r v a t i o n s (like the r e c o v e r y o f s h a p e after elastic c o l l a p s e b u t n o t after p l a s t i c failure) o n e c a n b u i l d u p a c o m p l e t e p i c t u r e of the failure process, identifying each p a r t of the e n v e l o p e with a specific m e c h a n i s m . D a t a for the failure o f c e l l u l a r solids u n d e r m u l t i a x i a l stresses are limited. F o r this r e a s o n a series o f tests o n a v a r i e t y of f o a m s u n d e r uniaxial, biaxial, a x i s y m m e t r i c a n d h y d r o s t a t i c l o a d i n g c o n d i t i o n s were p e r f o r m e d . T h e results are r e p o r t e d in the following c o m p a n i o n p a p e r a n d a r e c o m p a r e d with the m o d e l s d e v e l o p e d here. T h e m o d e l s give a g o o d d e s c r i p t i o n o f the failure surfaces of foams. Acknowledgements--We are grateful to Prof. J. W. Hutchinson of the Division of Applied Sciences, Harvard

University, for enlightening discussions on the elastic buckling of gridworks. Financial support for this project was provided by the U.S. National Science Foundation Solid Mechanics Program (Grant No. MSM 8603821), the U.K. Science and Engineering Research Council (Grant No. GR/D/00528) and the NATO Program for International Collaborative Research (Grant No. 0805/06), for which we are grateful.

REFERENCES 1. L. J. GIBSONand M. F. ASHBY,Cellular Solids: Structure and Properties. Pergamon Press, Oxford (1988). 2. T. C. TRIANTAFILLOU,A. ZHANG,T. L. SHERCLIFF,L. J. GmSON and M. F. ASHBY,Int. J. Mech. Sci. 31, 665 (1989). 3. A. N. GENT and A. G. THOMAS,J. appl. Polymer Sci. 1, 107 (1959). 4. A. N. GENT and A. G. THOMAS,Rubber Chem. Tech. 36, 597 (1963). 5. J. M. LEDERMAN,J. appl. Polymer Sci. 15, 693 0971). 6. W. L. Ko, J. Cell. Plastics 1, 45 (1965). 7. M. R. PATELand I. FINNIE, Lawrence Livermore Laboratory Report UCRL-13420 (1969). 8. M. R. PATELand I. FINNIE,J. Mater. 5, 909 (1970). 9. G. MENGES and F. KNIPSCHILD,Polymer Engng Sci. 15, 623 (1975). 10. F. K. ABD EL-SAYED,R. JONES and I. W. BURGENS, Composites 10, 209 (1979). 11. L. J. GIBSON,M. F. ASHBY,G. S. SCHAJERand C. I. ROBERTSON,Proc. R. Soc. (Lond.) A382, 25 (1982). 12. L. J. GIBSONand M. F. ASHBY,Proc. R. Soc. (Lond.) A382, 43 (1982). 13. W. E. WARRENand A. M. KRAYNIK,The effective elastic properties of low density foams. Presented at the Winter Annual Meeting of the ASME, Boston, MA, p. 123 (1987). 14. W. E. WARRENand A. M. KRAYNIK,Mech. Mater. 6, 27 (1987). 15. W. E. WARRENand A. M. KRAYNIK,J. appl. Mech. 55, 341 (1988). 16. V. A. MATONIS,Soc. Plast. Engng J. September, 1024 (1964). 17. R. CHAN and M. NAKAMURA,J. Cell. Plastics 5, 112 (1969). 18. R. C. RUSCH,J. appl. Polymer. Sci. 14, 1263 (1970). 19. P. H. THORNTON and C. L. MAGEE,Metall. Trans. 6A, 1253 (1975). 20. P. BARMA,M. B. RHODESand R. SALOVER,J. appl. Phys. 49, 4985 (1978). 21. J. S. MORGAN,J. L. WOOD and R. C. BRADT,Mater. Sci. Engng 47, 37 0981). 22. M. F. ASHBY,Metall. Trans. 14A, 1755 (1983). 23. S. K. MAITI,M. F. ASHBYand L. J. GIBSON,Scripta Metall. 18, 213 (1984). 24. S. K. MAITI, L. J. GIBSONand M. F. ASHBY,Acta Metall. 32, 1963 (1984). 25. T. KURAUCHI,N. SATO,O. KAMIGAITOand N. KOMATSU,J. Mater. Sci. 19, 871 (1984). 26. J. W. KLINTWORTHand W. J. STRONGE,Int. J. Mech. Sci. 30, 273 (1988). 27. M. C. SHAWand T. SATA,Int. J. Mech. Sci. 8, 469 (1966). 28. M. ZASLAWSKY,Exper. Mech. No. 2 February, 70 (1973). 29. T. C. TRIANTAFILLOU,Ph.D. Thesis, Dept. of Civil Engineering, Massachusetts Institute of Technology, Cambridge, MA (1989). 30. J. ZHANG,Ph.D. Thesis, Cambridge University Engineering Dept., Cambridge, U.K. (1989). 31. S. TIMOSHENKOand J. M. GERE, Theory of Elastic Stability, McGraw-Hill, New York (1961). 32. W. MERCHANT,Struct. Engr 32, 185 (1954). 33. M. R. HORNE, Proc. R. Soc. (Lond.) A274, 343 (1963).

L.J. GIBSON et al.

660

APPENDIX THE ELASTIC

BUCKLING

A

OF HONEYCOMBS

AND FOAMS

Cellular solids, at the microscopic level are a network of plates or struts which are rigidly connected at nodes. When such a network is loaded, it deflects elastically and may fail by plastic collapse or brittle fracture; if it does not, it may still fail by elastic buckling. The essence of the analysis of the elastic buckling stress, t~*~, is given here. The first step is to identify the buckling mode. Figure 11 showed the two buckling modes for a honeycomb. The symmetry, in both cases, allows the isolation of a representative unit for each mode; these are shown in Fig. Al(a) and (b). The unit, if repeated (with reflection) builds up the entire pattern. The buckling modes for the foam have not been identified by direct observation. We assume that the mode shown in Fig. Al(c) is the dominant one. In uniaxial loading one member (identified as AB in Fig. A l(a) and (c)) carries a predominantly axial load, and tends to buckle. It is restrained by the other members, which exert a restoring moment on its ends. This restoring moment increases (linearly) as the angle of rotation of the node, ~, increases--it is like a spring; the spring constant of the node is measured by its rotational stiffness SR=dMAa/dct. Any end-loaded column, even one with moments MABacting at its ends (a "beam-column") buckles when the compressive axial load P reaches a critical value, the Euler load:

n2 ~2 El P~,. =

12

,

(A1)

MBA(" B

D

MBA

M I l a n ' s ....... ..... I

~

f

MAB

I I

\l

C

(a)

(b) I

"A

J D

(c) FIG. AI. Modes ofelastic buckling. (a) H o n e y c o m b s - - m o d e 1; (b) h o n e y c o m b s - - m o d e 2; (c) foams.

Failure surfaces for cellular materials--I

661

where I is the second moment of area and I is the column length. When the ends are pin jointed (completely free to rotate, so that SR=0) the constant n 2 = 1. When they are both clamped so that no rotation is possible (S~ = co), then n 2 = 4. If the top of the column is free to sway sideways, values of n 2 as small as 1/4 are possible. The difficulty in analysing the buckling of cells is that of calculating the rotational stiffness of their nodes; it determines the constant n 2 in the Euler equation. Once this is known, it is easy to calculate o*~:it is simply P divided by the area per cell on which the stress acts. The general approach to such problems can be found in the book by Timoshenko and Gere [31]. Figure A1 shows that, for honeycombs, three members are involved; in foams, four. We first calculate the axial loads and moments acting on each member when remote stresses a~, cr2 are applied to the material, and identify the angles in terms of the rotation ~ at the node and the angles ~-fl through which each beam is bent. We then make use of the slope-deflection theorem [31] to calculate the combination of stresses which maintain the buckled mode. For the beam AB, for instance,

MAsl

Msal

0^ = 3 - ~ - ~ ( u ) + - ~ - ~ ( u ) ,

Mn^I

(A2)

MAaI

o~ = 3--~- ¢,(,,) + - - ~ - ~b(u), where I/'(U)= 3 ( 1 u tal2u ), (A3) ~(u) = 3u(si~-2u l u ) , and

u=1_/P^B

(A4)

2Xt Et "

Here PAB is the axial load acting on the beam AB, and MAB and MBA are the moments at its ends, causing the beam to bend through the angles 0^ and 0a. Applied to each of the three members in turn, this method gives a set of equations for the angles 0. The requirement of equilibrium at the central node and equilibrium of each member gives a further set of equations with the form (for Fig. Al(b), as an example): r

2M^B + MAC= 0,

(A5)

MAB-- MBA+ Pae al = 0.

(A6)

This gives a set of equations from which the angles ~ and fl, and the moments MAc etc. can be eliminated to give a relationship between the forces--and the remote stresses tr~ and a2--required to maintain each configuration. The final results for configurations (a) and (b) of Fig. A1 for regular hexagonal cells are, for mode 1:

tan(1;-12~/31a2~tan(l_/-3~/31(3a'+~r2).~_ /3at+a2=O Ej ) \2t~/ E,t J ~t ~r2

(A7)

and, for mode 2:

;

/ 3a~ + a2 0" 2

\ 2t ~1

Ej

4 tan(1J-

)

=o.

(AS)

12~.~-tla )

Values of a, and a2 calculated from them are listed in the text.

APPENDIX B INTERACTION

BETWEEN ELASTIC BUCKLING AND PLASTIC COLLAPSE MODES IN E L A S T I C - P L A S T I C H O N E Y C O M B S

Elastoplastic interactions in structures may be calculated by estimating the combined load factor 2c as follows [32, 33]: 1

1

1

--=--+--,

(BI)

where 2, and 20 are the elastic buckling and plastic collapse load factors, respectively. The above formula gives good

L.J. GIBSON et

662

al.

estimates of the failure load when the elastic buckling and plastic collapse modes considered are geometrically similar; it gives conservative estimates when they are not. In elastic-plastic honeycomb structures under normal in-plane stresses, the plastic collapse mode (Fig. 5c) is geometrically similar to the elastic buckling mode I = shown in Fig. BI. These combine to form the elastoplastic collapse mode I =p illustrated in the same figure. Note that the buckling mode I = would never be observed in elastomeric honeycombs; it corresponds to stress levels higher than those required to produce the buckling modes 1 or 2 (Fig. 11). Therefore, it is only significant when considering the elastoplastic interaction. Mode I =is the result of the elastic buckling of the inclined cell walls due to compressive axial forces. It is described by

12(hl+sinO) c°sO n 2E,(t/l) 3

12sin0cos0 at

n 2E,(t/l) 3

a2 = 1.

(B2)

Combining equations (B2) and (8) on the basis of equation (B1) we find the following approximate expression for the failure envelope associated with the elastoplastic mode I=" of collapse:

i(h

al ~+sinO sinO-~2cos20

+2 L

I ~ra(h~+sinO)cosO+tr2sinOcosO - 12 1

1[ 2

Es(t/i)--'~-

e~ ~+sinO cosO+ezsinOcosO +

~

= 1.

Equation (B3) holds only when the axial force in the buckled cell walls is compressive, i.e. when

+ IsinO)cosO-o2blsinOcos 0 >>.O.

I

Ie

if_

leP

lap

laeP

lb p

lb ep

2P

2ep

1

2

FIG. B1. The elastic buckling and plastic collapse modes for honeycombs and their interaction.

(B3)

-tr~b(h

Failure surfaces for cellular materials--I

663

Next, we consider the interaction between the elastic buckling modes 1 or 2 (Fig. 11) and geometrically similar to the plastic collapse modes. The buckling mode 1 is geometrically similar to the plastic collapse modes la p and lb p shown in Fig. B1, which have been analysed by Klintworth and Stronge [261, using the upper-bound theorem of plastic analysis. The combined elastoplastic envelope describing mode I a "p as the result of the interaction between modes 1 and la p is found to be described as follows: 2 -~+sin0 sin0

-<"L

<',,u/o

12 -~+sin0 cos0

*

J

-<':L

(84)

The failure envelope for mode lb cp, the result of the interaction between modes 1 and lb p, is described by

12(h-l+sinO)cosO 2(h~+sinO)sinO --(71

i

~2

n z Es(t/I)3

ay,(t/I):t

5"452(h7).c°sO 2cos'0 q v ~ / = i. E,(t/I) 3 oy,(t/l) j

(85)

The buckling mode 2, combined with the plastic collapse mode I (Fig. BI) results in the following elastoplastic interaction equation which gives a conservative estimate of the failure stress 1"29]: sin0-a~cos20 +2 L

~

[

e I ~+sin0

J - ~e~ .t3l (3'751 E ,

]

+ 1963a2)

cos0+a2s}n0cos0 z

+L

~

J =

(B6)

1.

The failure envelope for elastic-plastic honeycomb structures is described by the inner of the envelopes given by the elastoplastic interaction equations. It is illustrated in Fig. B2 for isotropic honeycombs under normal in-plane stresses. Among the various elastoplastic interaction collapse modes, the ones most likely to occur are modes la 'p and lb "p.

h / l = 1, ill=0.1,

0=30

0.015 j

°

E s=50oy

(If y

F

s

Txy = 0

/7

~x

Oys

'// i'

0.0]5'

,,//ii /',/,;'

I

)015

....

1

I I I

2

_ _

[ep

~ . ~ lae~ __.__ lUep ---

2eP

FIG. B2. The elastic buckling, plastic collapse and elastoplastic interaction failure envelopes for an isotropic honeycomb under normal in-plane stresses.