APPROXIMATION OF THE STRENGTH SURFACES OF A TRANSVERSELY ISOTROPIC MATERIAL Yu. G. Melbardis and A. F. Kregers

UDC 539.4:678.067

Strain and strength properties are basic characteristics which determine the expediency and range of application of new composite materials. Significantly different mechanical properties between the reinforcing fibers and matrix result in a high degree of anisotropy of both the strain and strength properties of the composite. Mechanical property anisotropy is particularly evident in the case of unidirectional reinforcement. Practical use is made of this scheme of reinforcement only in structural elements that are to be subjected to uniaxial tension, but it attracts the interest of investigators because it is the basic element of analysis for the study of both multiple-layer (multilaminate) plane schemes of reinforcement and three-dimensional schemes. With the use of several hypotheses, theories founded at this structural level can predict the behavior of a composite with a complex structure from known properties of its monolayers. Such single layers are assumed to be transversely isotropic, and their mechanical properties are known and are assigned analytically or from a table. Direct empirical determination of strength surfaces is complex and time-consuming, so that investigators resort to various methods to analytically describe these surface involving the use of data on the strength of the material obtained experimentally for certain characteristic loading paths. If a purely formal geometric method of describing the limiting surfaces is used as the basis for analysis of the strength of the material, then in principle any surface can be described, having selected the appropriate function. It is necessary only to ensure the satisfaction of the symmetry conditions inherent to the given material. A geometric approach was proposed in [i] to describe the limiting surfaces in the form of a tensor-polynomial series from oij. With the retention of the first two terms in this series and the imposition of certain conditions on the numerical values of strengthsurface tensor components Pij and Pijkl, we obtain the equation of the surface of an ellipsoid in a six-dimensional stress space:

pijoij+pijklOijOkl= 1.

(i)

The strength properties in Eq. (i) will pertain to a transversely isotropic material if we introduce in the same equation arguments of elastic potential W corresponding to this class of mechanical property symmetry [2]:

W=W(ll,12,13, Ou, olkohl)=W(oll,.q22+~3, o222+6332+20232,al~2+o13z,]a), where

i,],k=1,2,3;

ll=aq~ij;

I2=~ij~i; I3=~ija~k~hi.

To this end, let us write the equation of the strength surface in the principal axes of material-property symmetry in the form of a polynomial series of the above arguments to the quadratic term, inclusively:

bl~ll+b2(o22+oa3) +b3ol12+b4(o22+ff3a) 2 + bsffll(022+~33)+b6(~22+~2+20232) +b7(~122+~132) = | .

(2)

It is evident that the material has seven independent parameters. Comparing Eqs. (1) and (2), we can establish the relationship between individual components Pij and Pijkl:

pu=bl; p2222=P3~33=b4+b6; 2p2~23=b6; p~2=p33=be; 2puz~=2pu33=bs; Pml=b~; P2233= b4; 4p1212= 4pla3 = b7.

(3)

Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Kompozitnykh Materialov, No. 3, pp. 436-443, May-June, 1980. Original article submitted November i, 1979.

308

0191-5665/80/1603-0308507.50

9 1980 Plenum Publishing Corporation

The remaining components are equal to zero. It is apparent from Eq. (3) that P2~22 =p2233 + 2P2323. Similarly, it may be established that there are three independent parameters in Eq. (i) for an isotropic material and 12 for an orthotropic. Results are presented in [3] of an experimental study of the effect of five different plane reinforcement schemes (including one with unidirectional reinforcement) on strength in the space of stresses o~i, o22, and o~2 and a determination is made of the numerical values of the six strength-surface tensor components Pij and PijklThe glass-plastic specimens had reinforcement laid according to the method i~o, specifically:~=0 ~ ~ = • ~ =• ~ ~=• ~ , and ~ = • ~. An attempt was made to theoretically predict the strength of a laminated composite (packet) from the known strain and strength properties of a single unidirectionally reinforced layer, taken as a component part of the packet. The principal symmetry axes of the packet were designated through ~, B = x, y, z, while the notation for the principal s~munetry axes of the layer was i, j = i, 2, 3. All of the independent strain characteristics of the transversely isotropic layer and packet were assumed to have been known (the missing characteristics in [3] were computed in accordance with [4, 5]); the strength-surface tensor components Pij and Pijkl of the unidirectionally reinforced layer were also taken from [3]. In the theoretical model, the composite laminate was loaded to fracture along a radial loading path in the space o~B, which corresponded to the direction in the experiment. We determined the length of the loading path R, i.e., the distance from the origin to the empirical value R e and calculated value R t of the strength surfaces. Three solution variants were examined. A. It is assumed that the strain is the same in all of the layers, regardless of the direction of reinforcement. With an assigned stress state o ~ , we determine packet strain ea~ = a ~ y ~ o y ~ , where ~, ~, y, ~ = x, y, z. We then determine strain in the axes of each layer eij = e~B~i=ljB (where ~i~ is the cosine of the angle between axes i and ~) and the stresses Ok~ = Ak~ijeij (where Ak~ij are components of the layer stiffness tensor); i, j, k, = I, 2, 3; the strength condition of all layers is checked in accordance with Eq. (i). For each chosen loading path in the space ~=~, we also determine its shortest length R t at which the strength surface is reached in accordance with Eq. (i) for one of the layers. B. It is assumed that the stresses are the same in all layers, regardless of the direction of reinforcement. The stresses in the layer are determined from the formula oij = oa~ia~jB; the strength condition of each layer is checked separately in accordance with Eq. (i). C.

The components of the strength tensors of the individual layers are averaged.

the averaging, we use as a weighting factor the relative volume of reinforcement in a specific layer Vr(n)/Vr, V r = Vr (:) + Vr (~) + ... + Vr (N), N is the total number of layers. The strength condition of the composite is then expressed as:

where N

P~= ZP~(')~ n = i

N

(')1~ ;

P~=

E

P ~vS(')~ ( ' ) / ~ "

~ = I

A l l t h r e e v a r i a n t s A-C may be u s e d f o r t h e c a s e o f t h r e e - d i m e n s i o n a l reinforcement of t h e c o m p o s i t e i f t h e c o n c e p t o f l a y e r s i s r e p l a c e d by t h e c o n c e p t o f t h e o r e t i c a l rods [5]. We theoretically constructed the surface of strength anisotropy in uniaxial tension along different directions in the plane y, z, which is the plane of isotropy of the strain properties of the composite (Fig. i). We examined two schemes of reinforcement in the plane y, z with pitches of 60 and 45 ~ The strain characteristics were the same in both schemes and their numerical values were determined from [3] in accordance with [5]: E r = 9"5"105 kg f/cm2; Ec = 2'9"i0~ kgf/cm2; ~r = 0.26; ~c = 0.39; ~Z = 0.57. It is apparent that the calculated values of strength in tension according to variants A-C obey the inequality RBt < Rct < RAt. Schemes a and b (Fig. i), which are isotropic in the reinforcement plane relative to the strain properties, are anisotropic relative to the strength

309

a

1z

9!11

~o

o;

Fig. i. Property anisotropy surface in the uniaxial tension of glass-plastic for reinforcement schemes a and b; 1) calculation according to variant A for scheme a; 2) same, scheme b; 3) variant C, schemes a and b; 4) variant B, scheme a; 5) same, scheme b. TABLE i. Empirical and Theoretical (variant A) Values of Length of Radius Vector R (kgf/cm 2) of Strength Surface with Different Reinforcement Schemes Strength ] characteristic

(~ = 0 ~

!

Re

I Rt *

{p = "+"1 if'

~p = + 20 ~

Rt

Re

Rt

12 763 266 5 845 1 274 407 1 694 746 365 1759 1 834

5669 603 2793 2061 1462 3870 3166 1621 3043 2294

6650 282 2551 1275 455 2203 968 365 1607 2013

Re

Cp= ~ 30 ~

(P = • 45 ~

Re [ Rt

Re

Rt

40111 3002 748 336

1959 1959

756 756 1308 1308 792 3862 lf72 747

i i

R~00 R020

R100 R020

J~126 Rt2G R006 Rl20 120 RI2o

r ~, %

r,%

14 066 114 033 8088 262 363 214 9 511 9 501 7796 1 276 1695 1 148 448 704 498 1 786 1 359 2574 748 1960 658 750 368 1430 1814 I 2895 2 700 1 787 1848 1 559 23 23

24 44

18 49

1901 1457 17181 1293 18661 596 38481 2860 2440 1250 1484 394 2500 1369 2830 2519 17 45

2377

2377 6576 5752 3403 1758 1782 3565

747

4Sol 25 56

Notes. R t*, from the approximation; all R t, from calculations according to variant A; r*, r, standard deviation of Eq. (i) approximation and variant A calculations. properties according to variants A and B and isotropic according to variant C. In the latter case, the same line of a circle is obtained for both schemes a and b. For variants A and B, the strength properties under schemes a and b are not only anisotropic, but different in value in the same given direction. In accordance with variants A-C, we determined the expected strength of materials with reinforcement schemes ~ = • • • • ~ [3]. The quality of the strength-surface approximation for scheme ~ = 0 ~ and the accuracy of the predictions of surfaces for the remaining reinforcement schemes were checked in accordance with the standard deviation r:

r= ~/--M~.. [(Rit-R~/R~'. IO0%,

(4)

where Ri t and Ri e are the theoretical and experimental distances from the origin to the strength surface (the limiting length of the loading vector); M, total number of empirical loading paths. The best results were obtained with variant A. It is clear from Table I that the prediction error is roughly twice as great as the approximation error. Such a ratio of errors is also seen in predicting creep with a change in the type of stress state. It should be noted that the overwhelming majority of the calculated strengths (37 out of 40) were lower than the empirical values. This is due to a number of undetermined factors (physical nonlinearity etc.) in the mathematical model and to the fact that we took as a theoretical point of the stress surface the stress state at which one of the layers reaches its own

310

limiting state -- an event which generally does not signify macroscopic fracture of the composite as a whole. One way of improving the accuracy of strength predictions for laminated plastics on the basis of strain and strength characteristics for a unidirectionally reinforced material is improving the quality of the approximation of the monolayer strength surface. The approximation of Eq. (i) always gives a convex strength surface, something that is not always seen in experiments, particularly for unidirectionally reinforced materials [6, 7]. There are studies [7, 8] which have used certain terms of cubic series (i). The application of this relation is complicated by its implicit form relative to the length of radius vector R and the need to check the theoretical strength surface for closure and identity. Of promise in connection with this is the method of approximating the surface by expanding the function on a unity sphere [9]. The construction of limiting strength surfaces for isotropic and orthotropic materials in a plane stress state was examined in [i0, ii]. We will further examine features of the application of this method for the case where the material is transversely isotropic in its strength properties. According to [9], we may represent any limiting surface in space as scalar function f(~) on the unit sphere. The corresponding expansion of this function in the space of symmetrical stress tensors has the form !

[(~) = lira ~ I2-h/2plh(• h=O

[(~) = •

'A ,

(5)

where Plk (• is in the general case a homogeneous polynomial of degree k relative to stress tensor component Omn; at the same time, it represents the polynomial of scalar base invariants ~ of the omn tensor relative to the required orthogonal symmetry group of the properties of the medium. In particular, if the medium is transversely isotropic with the plane of isotropy 2, 3, then Plk = P~k(l,, 12, 13, oiI, O~kOk~), where I i represents the stress tensor invariants. It is easy to see that, actually, the presence of 12 in Plk leads only to the appearance of similar terms in (5). Expansion (5) is therefore conveniently written thus: I

[ (~) = lira ,~.~ I2-k/2pzk (O11, G22 + G33, G122+ O132, I3). S i n c e f ( ~ ) i n (6) e x p r e s s e s i n s t e a d o f (6) we o b t a i n

the length of radius

vector

R in the ~ij

(6) stress

space,

then

R=R(S,),

(7)

where S i are the basis functions on a sphere:

Sl = 01112-'~; $2 = (a22+ 033)/I2-'~; $3 = (~122 + ~132)/I2-1; $4 = 1312 -'~. Let us now look at a specific limiting strength surface of a unidirectionally reinforced glass-plastic made on a base of ETsT-I binder material. In [i0, ii], it was sufficient to assume I ~ 4 to construct the corresponding limiting strength surface in expansion (5). But, as shown by our analysis, this is clearly inadequate in the case of a transversely isotropic material. In connection with the fact that an increase in I would entail a catastrophically rapid increase in the number of material parameters, we conducted an analysis of the geometric properties of arguments Si and their combinations. For convenience of analysis, Eq. (7) is rewritten in the following form:

~=ao+aiSi+aijSiSj+a~skS~SjS~+

....

where ai, aij , and aij k are parameters of the strength surface, kgf/cma; 3, 4.

(8) i, j, k = i, 2,

Each of the base functions S i and their combinations in the stress space graphically represent a certain limiting surface. Let us refer to each such surface as an elementary limiting surface (ELS). The sum of these ELS in the stress space gives us the total limiting strength surface. Let us examine certain terms of series (8) individually. First, we will examine a base function of the type Sin , where i = i, 2, 3, 4 and n = i, 2, 3, .... Function S~ in the plane oi~, ou2 takes the form S~ = o , : ( c ~ 2 + o222) -~/2.

311

fizz2 28 .....

;X . . . . . . .

'-~

Fig.

i--

~~ /

t'~"

Fig. 3

2

Fig. 2. Graph of function R t = $In on a sphere. The numbers denote values of n. Fig. 3. Graph of function S~S~ on a sphere in polar coordinates. + and -- indicate the sign of the function in the corresponding quadrant.

~ ......

~

b

a

\),, Fig. 4. Graph of function s:ns2 m in the first quadrant on a sphere: o**, o22) symmetry axes; a) n = m = 1 (i), 2 (2), 3 (3), 4 (4); b) m = I; n = 1 (i), 2 (2), 4 (3), 12 (4). If the oij stress ~ensor components, o,~ and o~2 in the present case, are expressed through polar coordinates r and ~, then o** = r cos ~; o~2 = ~ sin ~. Substituting these values in the expression for S~, in the present case we obtain S: = cos ~. Thus, any degrees of combination of S i are in essence dimensionless functions of the polar coordinates of the oij space. Figure 2 shows a graph of function S~ n at different degrees of nonlinearity n. Function S~ n is always positive, except for the case when o,i < 0 and n is odd. It is apparent from Fig. 2 that the ELS contracts relative to its symmetry axis o1~ with an increase in n, but the point of intersection with the o,~ axis does not change. This function appears similar in any other plane of stresses oij. Consequently, we can depict function S2 n in plane o**, o2~ if the entire family of curves in Fig. 2 is rotated counterclockwise by ~/2. With the substitution of o22 for o~2 (Fig. 2), we obtain fi~e ELS of function $3 n, the only difference being that the function is positive over the entire plane at any n. Function s,ns2 m in the plane oi~, o22 takes the form cos n a sinm ~; this function is shown graphically in Fig. 3 at n = i and m = I. The sign of the function in each of the quadrants is determined by the sign of the trigonometric functions in this quadrant and by whether the degrees of nonlinearity n and m are odd or even. Figure 4 shows the dependence of function sins2 m on n and m. At n = m (Fig. 4a), the four lobes are located symmetrically relative to the diagonals of the quadrants in axes o~,, o22; these axes are symmetry axes of the figures. The lobes decrease in size with an increase in degree n. At m = const and n > m, with an increase in degree n (Fig. 45) the lobes decrease in size and are rotated in the direction of axis o,:. In this case, axes o11 and o22 remain symmetry axes. It has been established that function $IS2S3 in the space oi:, o22, o~2 is represented by eight three-dimensional lobes located along the diagonals of the octants. After analyzing the properties of the individual sums of series (8) and the empirical data on the strength of a unidirectionally reinforced glass-plastic [3] (Table i), we chose the following form of theoretical equation of the limiting surface:

Rt =ao+alSy+a2S12S+a3S2+a4S22+asS3 +a6S1S22+aTS12S2+asS12S22+agSl~S~+amS~S~S~. 312

(9)

Fig. 5. Strength surface of glass-plastic according to Eq. ( 9 ) . Stresses, kgf/mm 2.

Here, for the sake of brevity, the indices of ai, aij, and aij k have been replaced by a serial-number index. We took even and odd degrees for $I and $2 to describe the difference in strength in tension and compression. Obtaining long and narrow ELS lobes in the direction of axis ~11 led to relatively low degrees of nonlinearity of f~nction S~. Equation (9) determines the strength surface in a six-dimensional stress space, so that the empirical data from [3] was insufficient for the purpose of determining parameters "a." The missing strength Roo~ was taken fromconsiderations Ro2o < Roo~ < Roo6, and is numerically equal to 455 kgf/cm 2. An attempt to approximate the strenzth surface by the method of expansion of a function on a unit sphere showed that, on the one hand, this method is extraordinarily flexible. On the other hand, it requires a large amount of initial empirical data on the strength of the material compared to (i). We therefore additionally took R12o = 960 kgf/cm 2. We found from an analysis of (9) that coefficient ao corresponds to the strength of the material with loading by stress 023. The numerical values of the remaining parameters a were found by the method of least squares relative to integral function (4). The calculations were performed on a WANG 2200V computer in accordance with the algorithm in [12]. With a relative approximation error r = 14%, we found the following values of the coefficients: ao = 455; a: = 2277.5; a2 = 11333.5; a3 = --467; a~ = 226; a5 = 406; a6 = 135; a7 = -- 1680; a~= 3214.6; a9 = 2176; alo = 1342.1 (kgf/ cm=). Since certain coefficients come out negative and certain combinations of Si are negative in the sequence of directions in the space oij, when the volume of empirical data is small it is necessary that the theoretical strength of the material in any stress state be greater than zero in the process of determining optimum values of coefficients "a." In accordance with (9), the strength surface was constructed graphically with the digital plotter of the WANG 2200V system by means of thealgorithm in [13]. An axonometric representation of the surfaces in the axes ~I~, ~2~, and ~ = , shown in Fig. 5, was obtained by constructing several sections of the plane passing through axis ~12 at an angle of 15 ~ Each section is depicted by 24 points. The strength surface in the axes ~23, ~:i is similar to the surface in the axes ~12, a1~. The section of the strength surface in the axes G2=, ~23 and ~22, o ~ is close to a circle in shape, while the section in the plane ~ 3 , c~s is close to an ellipse with its major axis along ~ 3 . In conclusion, it should be noted that approximation of the strength surface with Eq. (8) automatically ensures transverse isotropy of the strength properties of the material, as well as identity and closure of the surface. In the general case, the resulting surface is not convex, which makes it possible to describe a broader class of empirical data. LITERATURE CITED l,

2. 3.

.

A. K. Malmeister, "Geometry of strength theories," Mekh. Polim., No. 4, 519-534 (1966). A. Grin and J. Adkins, Large Elastic Strains and Nonlinear Continuum Mechanics [Russian translation], Moscow (1965). Z. T. Upitis and R. B. Rikards, "Investigation of the dependence of composite strength on reinforcement structure in a plane stress state," Mekh. Polim., No. 6, 1018-1024 (1976). G. A. Van Fo Fy, "Elastic constants and the stress state of glass strip," Mekh. Polim., No. 4, 593-602 (1966).

313

5.

6, 7. 8.

9. i0. ii. 12. 13.

A. F. Kreger and Yu. G. Melbardis, "Determination of the deformability of threedimensionally reinforced composites by the method of stiffness averaging," Mekh. Polim., No. i, 3-8 (1978). E. K. Ashkenazi, Strength of Anisotropic Wooden and Synthetic Materials, Moscow (,1966). E. M. Wu, "Phenomelogical criteria of the fracture of anisotropic media," in: Composite Materials. Vol. 2. Mechanics of Composite Materials, Moscow (1978). E. M. Wu and J. K. Scheublein, "Laminate strength -- a direct characterization procedure," in: Composite Materials: Testing and Design (Third Conf.), Am. Soc. Test. Mater., Spec. Tech. Publ., No. 546, Am. Soc. for Testing and Mater. (1974), pp. 188206. A. Zh. Lagzdin', "On the expansion of a scalar function of Sn-1 tensor components on a unit sphere," Mekh. Polim., No. I, 30-36 (1974). R. B. Rikards and A. K. Chate, "Initial surface of a unidirectionally reinforced composite in plane stress," Mekh. Polim., No. 4, 633-639 (1976). R. B. Rikards and Ya. A. Brauns, "Approximation of strength surfaces in plane stress," Mekh. P01im., No. 3, 406-414 (1974). A. F. Kreger, "Algorithm for finding the minimum of a function in many variables by the slope method," Algoritmy i Programmy, No. 2, 9-11 ( 1 9 7 4 ) . A. F. Zilauts, "Algorithm for graphic construction of an axonometric projection of a three-dimensional surface," Gosfond Algoritmov i Programm. Inv. No. P 003946.

FRACTURE KINETICS OF ADHESION LAYERS IN COMPOSITE SYSTEMS S. V. Perminov, V. S. Kuksenko, and V. E. Korsukov

UDC 539.612:678.067

Questions dealing with the adhesive strength of composite systems are among the most important and least studied problems of the strength of composites. This is due to the fact that many physical methods of studying solids are "three-dimensional" in principle. Adhesive strength, as a rule, is determined by the chemical and physical structure of thin layers on an atomic scale. In connection with this, it is particularly valuable to conduct phenomenological studies of the fracture of adhesion layers, thus making it possible to obtain information on the kinetics of parameters which characterize the energy of the bonds in these layers and their structural heterogeneity. This article presents the results of investigations of the adhesive strength of simple contact pairs: substrate-thin-film, where the substrates were plates of devitrified glass or ceramic 22KhS on which were vacuum-deposited (residual gas pressure 10 -5 bar) thin metallic films of aluminum and chromium under conditions ensuring high-strength adhesive contact. The films were deposited on a UVN-M unit. Considering the large amount of scatter of the empirical data in service-life tests of the adhesion layers, we studied at least 30-40 specimens at each fixed temperature. Figure 1 shows the setup used for direct-pull and torsion tests of the layers. A rupture (in the form of small pieces ofthe material) of the required area was obtained by means of photolithography, which ensured a high degree of reproducibility of the specimens with respect to area. The studies were conducted in a heating and cooling chamber in which the specified temperature was maintained within •176 Figure 2 shows the temperature-time dependence of the life of several adhesive joints. Despite the large amount of empirical scatter, a family of curves can be obtained for each contact pair and extrapolated to a single region, wherethey intersect at log tad = --12 • 2 if we assume the existence of a linear function log Tad(Oe x) and analyze the test data by V. I. Ul'yanov (Lenin) Leningrad Electrotechnical Institute. Translated from Mekhanika Kompozitnykh Materialov, No. 3, pp. 444-450, May-June, 1980. Original article submitted June 18, 1979.

314

0191-5665/80/1603-0314507.50

9 1980 Plenum Publishing Corporation