Three-dimensional Strength.pdf

International Journal of Fracture 45: 35-50, 1990. ©1990 Kluwer Academic Publishers. Printed in the Netherlands.

35

A new approach of three-dimensional strength theory for anisotropic materials SENG C. TAN WRDC/MLBM, US Air Force Materials Laboratory, Wright Patterson AFB, OH 45433-6533, USA Received 20 October 1988; accepted in revised form 1 June 1989

Abstract. A three-dimensional failure criterion is developed in a form of quadratic tensor polynomial with the

fundamental strength function expressed in a sine series. This criterion has the operational flexibility of any desired degree of accuracy. It can be shown that the present theory is a generalization of the Von Mises yield criterion. One of the main features of this criterion is that it does not require shear strength properties. The flexibility and accuracy of this theory are justified through its comparison with the experimental data for three material systems under uniaxial tension, uniaxial compression and biaxial loading. Comparisons are also made with several existing failure criteria.

1. Introduction

The theoretical study of composite materials started at the beginning of the twentieth century. In 1921, Hankinson [1] proposed the first well known one-dimensional empirical formula for the off-axis uniaxial compressive strength of wood, Appendix A. Investigators have shown that this formula is quite accurate for several different species of wood. The applicability of the Hankinson's formula in the wood industry has led a few researchers to derive the same equation from tensor theory. For instance, Cheng [2] derived the formula using a trigonometric function which is a complete one-dimensional theory. The theory is further pursued in a numerical-graphical scheme for biaxial loading [2]. However, no specific function and correlation are used to relate the stress components. In 1979, Cowin [3] showed that Hankinson's formula can be derived using the linear term of the Tsai-Wu criterion. Liu [4] showed that Hankinson's formula can be derived from the Tsai-Wu criterion through a special expression of the stress interaction term. The existing failure criteria for anisotropic materials, surveyed by Sandhu [5], include the maximum stress, maximum strain, Hill-type and Tsai-Wu, etc. Among the many criteria, the Tsai-Wu [6] quadratic failure criterion is a most general theory which probably received the most attention and further studies. One difficulty in applying the Tsai-Wu theory is the evaluation of its stress interaction terms. Wu [7] recommended biaxial testing methods to determine these terms. However, some concerns over the scattering of the stress interaction values still remain. The comparisons of the predictions and the experimental data [3] as well as a preliminary study by the author of this paper show that the existing failure criteria may not be flexible enough to predict accurately the strength of some highly orthotropic materials such as bovine and human bones. In this paper, a three-dimensional general failure criterion with any desired degree of accuracy is developed. The criterion does not require shear strength properties of a material. Furthermore, the Hankinson's formula can be derived easily from

36

S.C. Tan

XI

(~XlX l

Z

(a)

Cb)

Fig. 1. Coordinate systems:(a) laminate axes and; (b) principal stress axes of a fibrous anisotropiclaminate.

this theory. In this paper, a fundamental strength function is assumed first to describe the uniaxial strength characteristic of off-axis laminates. Solutions are given for the strength function with two to four strength parameters. Then a general strength theory is developed, which includes fibers lying on plane and three-dimensional oriented fibers. The present theory is examined using strength data with three material systems.

2. Fundamental strength function Without loss of generality, the coordinate system of a unidirectional laminate is illustrated in Fig. la where ~rij, i, j = x', y', z' and i = j, denote the applied normal stresses and aij ( = zij), i, j = x', y', z' and i ¢ j, denote the shear stresses. The lamina angle 0 designates the fiber orientation from the 1-axis. The principal stresses o-x, ay and o-z are in the x - y - z system, Fig. lb, and q~ is the fiber orientation from the x-axis. The stress transformation is discussed in Appendix B. Consider a unidirectional laminate, Fig. lb, subjected to a uniaxial loading, ~x, the tensile and compressive strengths (X~ and X~) of the laminate with fibers oriented at an angle, q5 degrees, from the x-axis are proposed by the following strength function

X~ =

[

A0-

~ n = 1,2,3...

1-1

A, sinZn~b

(1)

where superscript i is either blank (tension) o r ' (compression), and An, n = 0, 1, 2 , . . . , are the coefficients to be determined by using fundamental strength parameters. Equation (1) is the fundamental strength function of the present theory. Any on-axis or off-axis data can be utilized as the basic strength parameters. These strength parameters, or data, can be treated as boundary conditions to the fundamental strength function, (1). The more terms of these strength parameters are applied the better the strength behavior of a material can be characterized. The number of terms used for the coefficients, An, depends on the accuracy desired and the strength parameters available. Solutions are given in the following using (1) with: (i) basic

37

A new approach of three-dimensional strength theory for anisotropie materials

strength parameters which are strength along and transverse to the fibers' direction; (ii) basic strength parameters and strength for an off-axis laminate at 05 = 30°; (iii) basic strength parameters and strength for an off-axis laminate at 05 = 45°; (iv) basic strength parameters and strength for off-axis laminates at 05 = 30 ° and 05 = 60 °. All superscripts for the strength parameters, blank and ', stand for tension and compression, respectively. 2.1. Fundamental strength parameters

At least two strength data are required to characterize and to predict the uniaxial strength of off-axis laminates using (1). These minimal data required are regarded as the fundamental strength parameters. Here, the strength along and transverse to the fibers' direction of a unidirectional laminate are designated as the fundamental strength parameters. Defining that X, X' =

respective tensile and compressive strength of a lamina along the fiber direction, 05=0

°.

Y, Y' = respective tensile and compressive strength of a lamina transverse to the fiber direction, 05 = 90 °. then the coefficients in (1) can be solved with these strength parameters as Ao =

1/X i yi

A 1

-

_

X i

(2)

Xiyi

where superscript i is either blank or '. With some mathematical manipulations, the following equation is acquired Xiy

i

X~ = XZ sin205 + Yicos 205

(3)

If superscript i is ', the equation reduces to the familiar form of Hankinson's formula. 2.2. Fundamental parameters and strength for 0 = 30°

If the strength of a 05 = 30 ° laminate, U ~, is added in addition to the X ~ and Y~, the coefficients An become Ao -

1 Xi

A1 -

1 Xi

A2 = ~

(4a) 1 yi

+ yi

(4b)

~i

(4c)

38

S.C. Tan

where superscript i is either blank or '. Equations (1) and (4) can be rewritten more explicitly as

1(3

X~Y ~

3

+ y~

4) 11 i sin2 2q5

(5)

Note that the first term inside the bracket of (5) is the same as Hankinson's formula. 2.3. Fundamental parameters and strength for 0 = 45° If the strength of a q5 = 45 ° laminate, V ~, is applied rather than q5 = 30 °, then the solutions are

A0 =

1 ---= X'

A1

)(i

yi

~

+

-

A2 =

(6a)

1

1

(6b)

Vi

(6c)

where superscript i is again either blank or '. Equations (1) and (6) can be reorganized as ,

XiY i

2

+ Y'

~

sinZ2q5

(7)

The first term inside the bracket also recovers to the Hankinson's formula. 2.4. Fundamental parameters and strength for 4) = 30° and 4) = 60° If four strength parameters are utilized, the coefficients, A,, can also be solved easily. For instance, if U i and W i designate the strengths of a q5 = 30 ° and a q5 = 60 ° laminates, respectively, the following results are obtained A0 =

1 m X,

A1

=

3

A~ =

~

A3

~

=

(8a)

i

1(~ i

yi -~- U i

+ r~

1

yi

i/~ri

(8b)

U'

W~

(8c~

2

2)

~sd~

Ui -t- ~ i

A new approach of three-dimensional strength theory for anisotropic materials

39

where i is either blank or '. From the above examples, apparently an even higher order theory can be obtained easily. For N n u m b e r of strength parameters, An can be solved from a system of N unknowns in N equations.

3. Generaltheory A general strength theory is developed using the x - y - z system, as shown in Fig. lb. This criterion is based on the fact that a failure surface must be closed in an ellipsoidal space. If any stress vector is equal to or is greater than the failure surface envelope, the material would fail. Mathematically, the failure surface is described by the following quadratic form Fiai + Fijaiaj = 1,

i= x,y,z

(9a)

which can be written more explicitly as Fxa x + g a y + F~az + Fxxa2~ + Fyya2 + F~zG2 + 2Fxyaxay + 2Fxzaxaz + 2FySya~ = 1 (9b) where the strength coefficients, F~ and F~j, can be expressed as functions of the uniaxial strength parameters which are described in the previous section. The accuracy of this general theory depends on the number of strength parameters used to characterize the fundamental strength function, (1). The present theory is discussed with more details in the following subsections for the cases: (1) fibers lying on-plane and; (ii) fibers not lying on plane. 3.1. Fibers on-plane If the principal stress state exists when fibers lie on one plane of the x - y - z axes, then only one angle, qS, is needed to describe the laminate orientation. The coefficients, F~ and F~j, in (9) can be expressed as the uniaxial strength parameters, X~, Y~ and Z;. For instance, if a uniaxial tensile stress, ax, is applied to a q~ degrees laminate with failure strength, X~, then (9) can be solved by substituting a x = X+:

f=X~ + F~X, =

i

(I0)

If a compressive stress is applied for the same laminate, then (9) is solved by substituting ax =

--X;"

FxxX$,z - FxX$ -- 1

(11)

Solving (10) and (11) yields Fx

=

F=

-

1

1

1

x x;

(12)

S.C. Tan

40

The coefficients E~ and Fxx are uniquely determined for any given lamination angle ~b. If the uniaxial strength parameters X+ and X£ were characterized using a limited number of data, the coefficients Fx and Fxx for the other lamination angles, which are not. coincident with the data points, can be calculated using the X4 and X~ predicted by (1). The coefficients Fy, Fyy, Fz and F~z can be solved following the same procedure and the results are 1

Fyy

1

(13)

m

and 1 Fz

--

F~z -

Z~ 1

(14)

Z,~ Z,~

where Y~ and Z~, i = blank or ', are the respective uniaxial strengths in the y and z directions for a 4~ degrees laminate. Since fibers are on the x-y plane, g~ can be related to the X; by: =

X~o_4

(15)

Substituting (15) into (1) yields

Y; =

=

[

A0 -

1-1

A~ sinZn(90 - 4~) n = 1,2,3,...

[A 0 - A 1 cosZ~b - A 2 sin22~b - A 3 cos23q5

-

A 4

sinZ4q5 . . . ] - 1

(16)

where coefficients A~ have been given in any one of (2), (4), (6) or (8). The strengths in the z-axis can be expressed in a form similar to (1):

Z;

=

B 0 --

~

B~ sin 2 nq~

(17)

n=1,2,3,...

where the strength coefficients can be solved from the following conditions: Z~ =

yi,

atq~ = 0 °

Z~ =

Z i,

atq5 = 90 °

(18)

A new approach of three-dimensional strength theory for anisotropie materials

41

where yi has been defined before and Z i (i = blank or ') denote the tensile and compressive strengths in the direction with fibers going out-of-plane. A higher order theory in the z-axis can be derived in a similar way by introducing more strength parameters, such as those for q~ = 30 °, 45 °, 60 ° laminates etc., as mentioned in the previous section. In the case of transversely isotropic materials, Z~ = Yi for ~b = 0 °, 90 °. The stress interaction terms, Fxy, Fxz and Fy~ in (9) are assumed, in a general form, as

E

4"1

[o0

19a,

....

(19b)

(19c) n=1,2,3,...

From the consideration of biaxial strength symmetry (x-y plane) with respect to ~b = 45 °, the term inside the bracket, (1q~] - 45), is assumed. The terms Fxx, Fyy and F~ have been given in (12)-(14). The coefficients (C,, D,, E,, n = 0, 1, 2, . . . ) can be determined by substituting biaxial strength data into (9). For instance, the coefficients of Fxy are obtained by substituting a x and % into (9) Co -

~

C. sin2n(]~bl - 45) = Fxax + Fy% + Fxxa~ + Fyya~ -

n = 1,2,3 . . . .

1

(20)

2 0 " x O'y

The coefficients of Fxz and Fyz can be determined by substituting respective biaxial strengths, a x with % and O-ywith az into (9). For highly orthotropic materials, more than one set of biaxial strength data may be needed to solve the coefficients C,, D, and E,. However, it is expected that only the first terms, Co, Do and E0, are needed for most cases. When C 0, D o and E 0 are equal to 0.5 and Cn = D~ = E~ = 0 for n ~> 1, (9) is a generalization of the Von Mises yield criterion. In the case where shear stresses exist, the tensor transformation rule can be applied to rotate the laminate stresses into principal stresses. The transformation rule is shown in the Appendix.

3.2. 3-D oriented fibers If the fibers of a laminate do not lie on any plane of the principal stress axes, the fibers are defined as three-dimensional oriented fibers with respect to the principal stress axes. A laminate with 3-D oriented fibers can exist due to: (1) fabrication and (2) the existence of the out-of-plane shear stresses, the tensor transformation rule is applied to obtain the principal stress state. Out-of-plane shear stresses usually occur near the free edges of a multidirectional laminate under a simple or combined loading. In addition to the laminate coordinates system and principal stress axes of Fig. 1a-b, we assume 1-2-3 as the fiber coordinate system of which 1 is along the fibers' direction, Fig. 2.

42

S.C. Tan

Y

Fig. 2. Coordinate systems for the fiber axes, 1-2-3 and the principal stress axes, x-y-z.

The respective qS, c~and 7 are the angles between the fiber axis 1 and the principal stress axes x, y and z. The following direction cosines can be obtained between the fiber axes and the principal stress coordinate system. 11 =

cosq~

=

cos(x,

1)

12 =

cose

=

c o s ( y , 1)

/3 =

cos~

=

cos(z,

(21)

1)

The other direction cosines are: m I = cos (x, 2) and nl = cos (x, 3) and so forth. The orthogonality of these two coordinate systems are governed by 1112 + mlm2 + nln2

=

0

1213 + mzm3 + nzn3

=

0

1113 + mlm3 + nln3

=

0

(22)

The procedures of applying the present theory are: (1) transforming the 3-D laminate stresses into fiber coordinates; (2) calculating the principal stresses and the direction cosines, see Appendix, between the fiber axes and the principal stress axes and; (3) utilizing (9)-(14) of which X; has been given in (1), and Y; and Z; can be related to X; by:

I

Y~

Z~

A0

=

i X;_~

=

2

A, sin 2 n~

n= 1,2,3,...

Ao

A~ sin 2 n7 n= 1,2,3,...

1' (23)

A new approach of three-dimensional strength theory for anisotropic materials

43

The stress interaction terms, F~z and Fyz, can be related to the F~y, (19a), by

Fxz

:

=

-

[Co

Cn sin2n(l~l -

n=1,2,3,...

Fyz =

F~y(q~7)

=

45)]

n=1,2,3,... ~ Cnsin2n(171-45)] ~

- [C0-

(24)

where the parameters have all been defined previously.

4. Plane stress problem

Plane stress problem is of particular importance in practice. It can be solved from (9) as

Fxax + Fyay + Fxxa2~ + F,ya~ + 2F~yax% = 1

(25)

Assuming that ax and ay are applied in a ratio of a to b, then they can be written as O"x

=

ay =

ao"

b~

(26)

By substituting (26) into (25) and solving for a, it yields

a = ( - B + x/B 2 + 4A)/(ZA)

(27)

where a2

h2

A -

XeX~ + ~

+ 2F~yab

B •

a

+ b

I?;

(2a)

The term F~y has been given in (19); X~ and Y~ (i = blank or ') have been given in (1) and (16), respectively. The substitution of (27) into (26) gives the allowable stresses ax and %. Again, if shear stress is not zero, the transformation rule of the laminate stresses into principal stresses is given in the Appendix.

5. Comparison with experimental data

The accuracy and flexibility of the present criterion are examined through the use of experimental data for bovine bone, human bone and paperboard under uniaxial and combined loading conditions.

44

S.C. Tan BOVINE FEMUR, COMPRESSION

| ~."~ t '~",,

"

l

STRENGTH_ MPo x = 144

¥,~

s+D t

Y =

,,',,-.~

z

', C, -\",,,,\:,

I

~o.

\ ",,,\:,,

•

~

:

,,

u

=

99

u'=

% ",5: (~-

.6

DATA

19o

.....

~ ••

_.. OO,,N-A,E . . . . . . . .

....

TSAI-WU

*'-...

PRESENT

I

\~o

.

i . i , i 10 20 30

0

i ' i . i , I , i , 40 50 60 70 80 90

FIBER ANGLE,

@ (DEGREE)

Fig. 3. Comparison of the predicted off-axis strength and the data [3] for bovine femur under compressive loading. HUMAN FEMUR, TENSION \~D

2

jl

I

i

I

i

I

,

I

~ I

,,,,.o-I - \ \ t -,XX

3:" 1 I-- ,,o0 o -I

\

Ld

~- ~

* •

.

!

,

=

~

S U

=

100

U'

=

173

.

.

,

=

67

\ \ \

DATA

j.l~

._: ?W'#,;#O~EDIow,N_2 ....

.

\i~

"\~,

' "

i

×'= 187 Y = s8

Z __

•

"jr,w~..

"''~

TSAI-WU PRESENT

' ,'o ~'o ~'o ,'o ~'o go 7'o ~,'o ~o FIBER

ANGLE,

,:I:, ( D E G R E E )

Fig. 4. Comparison of the predicted off-axis tensile strengths of human femur and experimental data [3].

The theoretical predictions of Cowin, Hankinson, Tsai-Wu and the present author (using (5)) are compared with the uniaxial compressive strength data of bovine Haversian bone [3]. The comparison is shown in Fig. 3 where ~b denotes the angle between the loading-axis and the grain direction. Cowin [3] has three predictions due to three different expressions of the stress interaction term. Only the best and the worst of his results are illustrated. The Tsai-Wu criterion [6] is applied with the normalized stress interaction term, F~ = - 0 . 5 (corresponding to F12 = - - 3 . 1 9 X 10 -5 MPa-2). The respective comparisons of the uniaxial tensile and the compressive strengths of a human femur along an axis off the grain axis are illustrated in Figs. 4 and 5. The normalized stress interaction term, F* = - 0 . 5 (F12 = --3.64 x 10 -5 MPa-2), is applied for the Tsai-Wu criterion. In Figs. 4 and 5, the predictions between Tsai-Wu criterion, Hankinson, and one case of Cowin criteria are

A new approach of three-dimensional strength theory for anisotropic materials

45

HUMAN FEMUR, COMPRESSION

n° ~@ ~E

"%

"%~

,+\ \,., '\

~

~" '.

z

N

X' =

187

Y'

132

=

S U

~%

=

67 100

=

N 09 OO

0 ,,50

- -- COW,N-C:~Se,J

,,K.. ~ -

---

HANNINSON, C O ~ I N - 2

--

PRESENT i

,

I0

i

20

FIBER

,

"" ..... "* i

30

,

i

40

ANGLE,

,

i

50

,

i

so

,

i

,

70

i

80

,

90

~5 (DEGREE)

Fig. 5. C o m p a r i s o n o f the predicted off-axis compressive strengths o f h u m a n femur and data [3].

indistinguishable. These theories do not correlate well with the data. On the other hand, the strength characteristics of these materials can be correlated very well with the present theory, using (1), with coefficients A, given in (8). The biaxial strength of a paperboard has been studied experimentally and compared to the predictions by using several failure criteria by Rowlands and Suhling et al. [8, 9]. F r o m the results of these studies, we see that Hill-type failure criteria give unsatisfactory prediction at high level of shear stress. As the component of shear stress approaches the uniaxial pure shear strength, the failure surfaces shrink toward a x = ay = 0; which contradicts the experimental data. The predictions of the present criterion under in-plane loading condition can be computed using (25). The coefficients, A,, of X; and Y~ are given in (4). The predictions are compared to the experimental data, and they are shown in Figs. 6a-d for lamina angle, ~b = 0, 20, 30 and 40 degrees. The data for 4, = 20, 30 and 40 degrees were transformed from ~b = 0 ° with non-zero shear component (which includes 6.9, 10.3 and 15.9 MPa). The prediction with the stress interaction term, F+y = 0 of (19), agrees reasonably well with the data. If we consider only the first term of (19), i.e., C O ¢ 0 and C, = 0 for n >~ 1 and evaluate Fxy individually for each stress quadrant, the correlation between the prediction and the data is excellent. The variation of the Fxy (Table 1) for the four quadrants is much smaller than that of the Hill-type failure criteria [8]. Using the Fxy values listed in Table 1, the strength of the paperboard under general combined loading can be predicted using their principal stresses. The predicted strength ratio was then transformed back into the original axes. As shown in Table 2, the data agree much better with the present theory and the Tsai-Wu criterion than with the other criteria. The strength data shown in Table 2 was performed under combined loading with the shear component approximately equal to or greater than the pure shear strength. 6. Discussions and conclusions

A new approach is developed in this paper for the strength theory of fibrous anisotropic plates. Unlike all the other existing theories, the present theory can absorb all or parts of

46

S.C. Tan PAPERBOARD

STRENGTH,

8o• -----,

~

=

0 o

I DATA STRESS INTERACTION TERM IS GIVEN IN TABLE STRESS INTERACTION TERM = 0.0

4o

v STRENGIH MPa X = 55.92 X' = 20.46 Y. __= 30..82

2o

0

• ",% "~ ~1 •e '

-20 -40

-30

-20

1o

-lO

(a)

20

X,

SIGMA

30

4O

50

60

70

{

=

200

o', ( M P a )

PAPERBOARD STRENGTH, 60 •

DATA STRESS INTERACTION T E R M STRESS ~NTERACTION T E R M

IS IN TABLE 1 = 0.0

v

@ 20-

0 0

•

*,°

-2o

,

--40

(b)

-50

-2O

--10

,

0

.

,

I0

SIGMA

×,

.

,

2O o',

.

,

,

4O

.

50

60

(MP°)

PAPERBOARD STRENGTH, 60

,

3O

~

=

50 °

I

•

DATA STRESS INTERACTION TERM iS IN TABLE 1 STRESS INTERACTION TERM = 0.0

v

>.-

2o

0 o

-2o

-

-40

,

,

,

-30

-20

-10

(c)

•

,

O

SIGMA

PAPERBOARD • --

•

10 X,

a,

,

•

20

,

•

30

,

i

40

50

.

60

(MPa)

STRENGTH,

~

=

40 o

DATA STRESS INTERACTION T E R M IS IN TABLE I STRESS INTERACTION T E R M = 0,0

v3 .° 20

o

',I

~.Y -=

-2_,o._;o._io-'~o (d)

o ,5~'o

30,82 ,o.,?~ X

~'o ,'os'o

eo

S,GMA x, ~, (MPo)

Fig. 6. Correlation o f the predicted strength envelope and the experimental data [8, 9] for unidirectional paperboards: (a) ~b = 0°; (b) ~b = 20°; (c) q5 = 30 ° and; (d) q5 = 40 °. A d d i t i o n a l data for ~b = 30°: U -- 42.2 M P a and U' = 1 6 . 9 M P a .

A new approach of three-dimensional strength theory for anisotropic materials

47

Table 1. Stress interaction for the four q u a d r a n t s of p a p e r b o a r d Stress q u a d r a n t

First

Second

Third

Fourth

Fxy

- 1 . 7 2 x 10 4

- 3 . 7 3 x 10 4

- 5 . 2 3 × 10 .4

- 4 . 4 5 × 10 -4

Table 2. Strengths predicted using different failure criteria and the data [8] for a 0 ° p a p e r b o a r d under combined loading for which ~12 ~ 15.9 M P a (pure shear strength = 16.6 MPa) Strength & error

Norris

Tsai-Hill

Tsai-Wu (F* = 0.0)

Tsai-Wu (F~* = - 0.05)

Present criterion

Experimental data

X:Y:S %

17.4:20.5:13.9 - 12.1

17.3:20.3:13.9 - 12.6

20.0:23.6:16.1 1.0

25.1:29.5:19.9 26.8

19.6:23.0:15.7 - 1.0

19.8:23.3:15.9

X:Y:S %

30.9:14.8:12.8 - 19.5

29.8:14.3:12.3 - 22.4

38.1:18.3:15.8 - 0.8

52.7:25.3:21.8 37.2

38.5:18.4:15.9 0.3

38.4:18.4:15.9

X:Y:S %

19.9:19.9:13.8 - 13.9

19.9:19.9:13.8 - 13.9

23.3:23.3:16.0 0.9

30.2:30.2:20.8 30.7

22.9:22.9:15.8 -0.9

23.1:23.1:15.9

X:Y:S %

18.3:6.1:10.6 - 42.5

21.3:7.1:12.3 - 33.0

32.5:10.8:18.8 2.2

38.6:12.9:22.3 21.4

31.6:10.5:18.4 - 0.6

31.8:10.6:18.4

X:Y:S %

18.4:6.1:10.6 - 41.6

21.3:7.1:12.3 - 32.4

32.5:10.8:18.8 3.2

38.6:12.9:22.3 22.5

31.6:10.5:18.3 0.3

31.5:10.5:18.2

X:Y:S %

12.4:17.7:14.8 -15.1

12.3:17.6:14.7 -15.8

14.7:20.9:17.5 0.7

17.1:24.4:20.4 17.1

i4.1:20.1:16.8 -3.4

14.6:20.8:17.4

the uniaxial on-axis and off-axis strength as the basic strength parameters. Since uniaxial testing (tension or compression) is easy to perform and it provides data that shows the behavior of a material, therefore, it can be viewed as the strength characterization of a material. In the case that homogeneity and uniformity of a material are deficient, such as bones and woods, the other existing strength theories may not have the flexibility to predict the strength accurately, such as the examples shown in Figs. 3-5. It appears that only the present theory is capable of absorbing the inhomogeneity as a basic strength behavior. After the fundamental strength function is solved with two or more data points, the general strength behavior under uniaxial or combined loading is able to be predicted accurately. The fundamental theory is expressed as a sine series. Hankinson's formula can be derived by using only the first two terms of this series. Apparently, the criterion can have any desired degree of accuracy. Using the fundamental strength function, (1), and uniaxial strength parameters, a three-dimensional failure criterion is developed, which relates the principal stresses and strengths in a quadratic form. This general theory satisfies the physical condition that a failure envelope must be in a closed space. If shear stresses exist, they could be transformed into the principal stresses. After the principal strength is predicted, it can then be transformed back into the original coordinates. The stress interactions are assumed to generally exist between the normal stress components. They should be evaluated using biaxial testing results. Researchers and engineers have found that the stress interaction term varies considerably from one data point to another, even within a stress quadrant, when the existing failure criteria are applied. For instance, the stress interaction term determined from a unidirectional ~b = 30 ° laminate is usually different from that of a q5 = 45 ° laminate. The existing failure criteria, such as Tsai-Wu, only use one stress interaction term for each stress quadrant or one common term for the four stress quadrants. Therefore, it remains a concern which interaction value should be utilized. However, this problem does not occur in the present theory. The stress interaction terms, (19a-c), are

48

S.C. Tan

assumed in a series. The number of terms applied depends on the accuracy required and the dependency of the stress interaction upon the stress ratio. The application of the existing failure criteria, surveyed by Sandhu [5], needs some basic strength parameters determined from experiments. These parameters include uniaxial tensile and compresive strengths of a lamina along as well as transverse to the fibers' direction and lamina shear strength, etc. The characterization of the shear strength requires that a large proportion of the test section of a specimen be under maximum uniform shear stress state. However, most uniaxial specimens under shear testing fail prematurely due to stress concentrations near the corners of the specimens or grips. In the case when large specimens are difficult to prepare, such as ceramic composites and some bones, shear tests are difficult to conduct. Therefore, the shear strengths are generally much more difficult to determine than the other uniaxial strength parameters for all material systems. This reflects the unique advantage of the present criterion because it does not require shear strength properties. Therefore, it should be very useful for materials when shear strengths are especially difficult to evaluate. Several major conclusions can be drawn from this study: 1. The operational flexibility and accuracy of the present criterion have been assured through comparisons with a number of experimental data. Excellent correlation between theory and data has been obtained. For highly orthotropic materials, such as human and bovine bones, the advantage of applying the present criterion to obtain excellent prediction is obvious as compared to other existing criteria. 2. No shear strength properties are needed. 3. Theoretically the stress interaction terms can be determined with no difficulty even for highly orthotropic materials. Although it seems a well known fact that a higher order polynomial theory gives a more accurate result, whether the coefficients of the higher order theory can be determined is an issue. It is often the case that the coefficients of a higher order theory cannot be determined or are extremely difficult to determine. However, this problem is eliminated if the present criterion is utilized. To apply the present theory to multidirectional composite laminates, one needs to characterize the materials first. This involves evaluating the coefficients of the fundamental strength function in (1) and the stress interaction terms in (19). Secondly, one needs to know the stresses in each ply of a laminate through a stress analysis such as laminated plate theory, three-dimensional plate analysis, etc. Finally, the general strength theory, (9), is utilized that combines the fundamental strength function with the stress state Of each ply contained in a laminate.

Appendix A Hankinson's formula was developedempiricallyfrom compressivestrength tests on spruce. Using the notations described above, the compressivestrength of a wood in a direction inclinedat an angle ~bto the grain is X'Y'

X~ = X'sin 2q5 + Y'cos 2q~

(29)

Hankinson noted that this formula applies as well to the fiber stress at elastic limit in compression as to the ultimate strength in compression.

A new approach of three-dimensional strength theory for anisotropic materials

49

Appendix B If the stress components for the x ' - y ' - z ' laminate system, Fig. la, are known, one can determine the magnitudes and directions of the principal stresses by using tensor transformation rule [10]. Assuming l, m, n are the direction cosines of the angles between the x ' - y ' - z ' axes and the principal stress axes and ap, p = x, y, z, the magnitude of the principal stress, we obtain the following equations from the consideration of stress equilibrium in the x', y', z' axes, respectively (op - ax,x,)l- r~,ym - "G,~,n = 0 - G , e , I + (% - ary,)m - "cy,~,n = -Z~,z,1 - zy,z,m + (% - G,~,)n

0

(30)

= 0

The nontrivial solutions are obtained by letting the determinant of these equations equal zero. Through expanding the determinant and letting it equal zero, the following equation is obtained ~p3 __ (tTX.X" _~_ Gy'y' -[- (Tz'z')ff2p ~- ((~x'x'Gy'y"

-~- ffy'y'(~z'z"

2zx'/G'_,"crz' -- ffx'x"Ey'z'2

-- (G'x'ay'y'G'-'~ +

_

-~- (~x'x'ffz'z ' __ Tx,y ,2

ffy,y..Cx2,z, _

az,z, rx,y.)2

=

__ •x'z'2

__ +2z,)~p

0

(31)

The three roots of this equation are the principal stresses. Substituting each of these solutions into (30) and utilizing the following relationship: l 2 + m2 + n2 =

1

(32)

one can obtain three sets of direction cosines for the principal stresses. Equation (32) is necessary because of the degree of arbitrariness of the direction cosines obtained from (30). The transformation of two-dimensional laminate stresses into principal stresses can either be calculated using Mohr circle [11] or by the following formula:

a+,~,y -

(Tx'x'tUGY'Y'-] - ~/((Tx'x'--(~Y'Y")2-~ 2 2 2 z +.y.

(33)

The resulting laminate or fiber orientation, qS, with respect to the principal stress axis, x, is given by:

=

0 -

lltan

~

t .~x,y, ( 2Zx'y' )] - %,,.J

(34)

where 0 is the original lamina orientation with respect to the x'-axis.

References 1. R.L. Hankinson, Investigation of Crushing Strength of Spruce at Varying Angles of Grain, Air Service Information Circular No. 259, U.S. Air Service (1921). 2. S. Cheng, Failure Criterion for Clear Wood under Combined Stresses, Report f o r the Forest Products Lab., Madison, W1 (1982). 3. S.C. Cowin, Journal o f Applied Mechanics 46 (1979) 832-838. 4. J.Y. Liu, Journal o f Composite Materials 18 (1984) 216-226. 5. R.S. Sandhu, A Survey of Failure Theories of Isotropic and Anisotropic Materials, Air Force Technical Report AFFDL-TR-72-71 (1972). 6. S.W. Tsai and E.M. Wu, Journal o f Composite Materials 5 (197i) 58-80. 7. E.M. Wu, Journal o f Composite Materials 6 (1972) 472-489.

50

S.C. Tan

8. R.E. Rowland, D.E. Gunderson, J.C. Suhling and M.W. Johnson, Journal of Strain Analysis 20 (1985) 121-127. 9. J.C. Suhling, R.E. Rowlands, M.W. Johnson and D.E. Gunderson, ExperimentalMechanics 25 (1985) 75-84. 10. S.P. Timoshenko and J.N. Goodier, Theory of Elasticity, McGraw-Hill Co., New York (1979) 219-224. 11. S.P. Timoshenko, Mechanics of Materials, Van Nostrand Reinhold, New York (1974) 51-67.

R6sum6. On d~veloppe un crit6re de rupture A trois dimensions sous forme d'un tenseur polynomial quadratique comportant une fonction de r6sistance exprim~e par une s6rie sinusoidale. Ce crit6re pr6sente une ftexibilit6 op6rationnelle permettant tout degr6 de prbcision dbsir& On peut montrer que la th~orie pr6sente est une g6n6ralisation du crit6re de plastification de Von Mises. L'une des caract6ristiques principales de ce crit6re est qu'il ne requiert pas les propri6t6s de r6sistance au cisaillement. On rend compte de la flexibilit6 et de l'exactitude de cette th~orie en la comparant aux donn6es exp6rimentales relatives fi trois syst6mes de mat6riau soumis ~ tension monoaxiale, fi compression monoaxiale et/t sollicitation biaxiale. Des comparaisons sont 6galement faites avec plusieurs crit@es de rupture existants.