AD-756 889

A SURVEY OF FAILURE THEORIES OF ISOTROPICAND ANISOTROPIC MATERIALS

R. S. Sandhu

Air Force Flight, Dynamics LaboratoryWright-Patterson Air Force Base, Ohio

January 1972

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A SURVEY OF FAILURE THEORIES OFISOTROPIC AND ANISOTROPIC MATERIALS

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A Survey of Theories of Failure of Isotropic and Anisotropic Materials,

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13. ABSTRACT

-A survey of various theories of strength fo~r isotropic and anisotropic materialsis presented. For anisotropic materials theories of failure are broadly classi-fied as theories with or without distinct failure modes. The former classincludes the criteria based upon maximum stress and strain while the latter arequadratic or biquadratic representations in which the transition from one failuremode to another is gradual.

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A SURVEY OF FAILURE THEORIES OF

>l ISOTROPIC AND ANISOTROPIC MATERIALS

R, S. SANDHU

Approved for public release; distribution unlimited.

FOREWORD

This work was conducted by Mr R. S. Sandhu, Exploratory DevelopmentGroup, Advanced Composites Branch, at the Air Force Flight DynamicsLaboratory under project 4364, "Filamentary Composite Structures:, Task436402, "Design Allowable and Criteria."

The manuscript was released by the author in January 1972. Thistechnical report has been reviewed and approved.

PHILIP A. PARMLEY '4Chief, Advanced Composi es BranchStructures Division

C,

V

Ell

TABLE OF CONTENTS

Section page

I INTPODUCTION

II THEORIES OF STRENGTH FOR ISOTROPIC MATERIALS 2

1. Galileo (1638) 2

2. Coulomb (1773) 2

3. Rankine, Lame, Clapeyron-Maximum Stress 3Theory 1858

4. Saint Venant-Maximum Principal Strain 4Theory (1837)

5. Tresca-Maximum Shear Stress Theory 5(1864-72)

6. Beltrami-Maximum Strain Energy Theory (1885) 5

7. Mohr's Theory of Strength (1900) 6

8. Distortional Energy Theory 8

9. Pressure Dependent Failure Criterion 9

III ANISOTROPIC THEORIES OF STRENGTH 12

I. Theories with Independent Failure Modes 12

(1) Jenkir (1920) 12

(2) Stowell and Liu (1961) 13

(3) Prager (1969) 15

(4) Waddoups (1966) 16

S(5) Lance and Robinson (1971) 16II. Theories without Independent Failure Modes 19

(1) Hill (1948) 19

(2) Marnn (1956) 20

Preceding page blank

i'I ° ................... • .•,,• •, ,•.• •,• •.•m • •,• • .•;• • i

TABLE OF CONTENTS (Cont'd)

Section Pa•

(3) Stassi-D'Alia (1959) 21

(4) Norris (1962) 22

S(5) Griffith and Baldwin (1962) 23

! (6) Azzi and Tsai (1965) •25! ,(7) Gol•enblat and Kopnov (1965) 25

(8) Ashkenazi (1965) 29

(9) Nalmeister (1965) 32

(I0) Hoffman (1967) ,. 34

(ii) Fisher (1967) f 35

(12) Chamis (1967) 36

(13) Bogue (1967) 37

(14) Franklin (1969) 38

(15) Tsai and Wu (1971) 39

(16) Puppo •,d Evensen (1971) 41

III SUMMARY 44

SREFERENCES 46

!

"-2

-= ]

ILLUSTRATIONS

Figure Pg

1 Mohrs' Representation of 3-Dimensional Stress 50State

2 Mohrs' Envelope 51

3 Simplified Mohrs' Envelope 52

4 Coordinate Axes 53

5 Failure Cone 53

6 Fiber Orientation 53

7 Sign Dependence of Shear Strength 54

8 Shear Stresses in Fundamental Coordindtes 54

9 Influence of y1 on the Shape of Failure Surface (51 2 0 )

TABLE'S

Table

I Strength Parameters of Theorien without Distinct Failure 56Modoo for Plane Stress Condition

I vi

." j..

SECTION I /

INTRODUCTIO,

Filamentary composite materials arelbeing used more and more in the

P construction of flight vehicles due mai4ly to their high strength and

stiffness to weight ratios. They are flrmed by embedding fibers in matrix

materials and are, therefore, nonhumog~ieous and anisotropic. It is not

expected that methods of analysis and 4esign used for structures of homo-

geneous and isotropic materials, wouldibe adequate for composites.

Unwever, they serve as the necesscary background for the new techniques

required for composite materials.

In structural application, streagth is often a characteristic of

interest. For homogeneous and isotropic materialb a number of theories

have been formulated from time to time to predict the response of the

material under general state of stress using the material properties

obtained from simple test conditions - uniaxiai tension, compression and

shear, These theories have served as the basis fo a the development of

their counterparts for anisotropic materials. The generalization has

been achieved by two approaches. In one approach, enough arbitrary

parameters are introduced in Lhe theory of strength for isotropic mater-

ials so that various failure modes can be explained. In the second

developmtnt of the theory proceeds by assuming that the matrix is isotropic

plastic material subjected to deformation constrainL- by the stiff fibers.

The aim of this report is to review the available theories of strength

with emphasis on those for anisotropic materials. Section II is devoted

to theories for isotrop 1c materials while Section Ill deals with theories

of anisotropic material.1

i•, ,%,.

SECTION II

THEORIES OF STRENGTH FOR ISOTROPIC MATERIALS

An accurate knowledge of strength characteristics of materials plays

a vital role in an efficient design of structures. It enables a designer

to determine the response of the structures under various conditions of

loading. Generation of the material properties for the entire spectrum

of loading is a very expensive procedure both in time and money. Instead

information obtained from simple test conditions is generalized to more

"complex states, and the generalized hypothesis checked for selected load

combinations, temperature, etc. Generalizations suggested by various inves-

tigators dating back to Galileo forming the subject matter of this Section

are from References 1 to 7.

1. Galileo (1638)

Subjecting stones to simple tensile tests, he observed that the strength

depended upon the cross sectional area of the bar an'1 was independent of

its length. He conciuded that a fracture would occur when the "absolute

resistance to fracture" (critical stress) was attained. He used this con-

cept to investigate resistance to fracture of a cantilever beam but placed

the neutral axis at the extreme compression fiber, which prevented him from

obtaining correct results.

2. Coulomb (1773)

During Coulomb's period, stone was the principal material of construc-

tion. It was observed that stone specimens under uniaxial compression

developed cracks which were inclined to the axis of loading. The inclina-

tion of these cracks differed from 450 which is the inclination of the plane

of maximum shearing stress under axial loading. This observation led

2

V 9 i- r' ...... ~j~am ~ ....... ....'~.. ........... ... .........

7!7

Coulomb to suggest that failure occurred when the shl ring Mtre•ol ,I, on

the failure plane was equal to the sum of the cohowivo atvoejug.h 'c' and

the frictional stress (-p on) whoro w is the coofficianL of' farctimi awd

On is the normal tensile stress on the failure plane, ie.,

IT I -C - 1101 KIOn the basis of the Equation 1, it can be shown that the inclinatio n of

the failure plane is given by

0 450 - /2 (2)

where P = tan (3)

Equation 1 implies that

(i) Compressive strength is greater than the tensile strenlgth but Lhe

predicted ratio of strength is far less than found in practice;

(ii) The failure plane makes the same angle with the direction of

the greatest principal stress for all states of stress including the one

corresponding to pure tension (which is not true);

(iii) In the case of a three-dimensional stress field, failure is

not affected by the intermediate principal stress.However, in case of soils of low cohesion, Equation 1 yields reasonably

good results.3. Rankine, Lamet , Clapeyron -Maximum Stress Theory (1858)

It is postulated in this theory that an element of a stressed body

fails when the maximum principal stress attains a value obtained from a

simple test like a uniaxial tension test, i.e.,

U2 (i~) ( (G 2)ý (4)A"00

0where 01, 02, 03 are the principal stresses and a. is the failure stress

Q ...... in a simple test.

3~

Equation 4 represents a cubic surface spaced symmetrically about the

origin of coordinates. In case tensile and compressive strengths are

different, the origin of coordinates is not symmetrically located.

According to this theory if a,,= a0, a2 am 0, a3 a 0 separately4g fail-

ure is expected. However, simultaneous presence of al - a2 - a3 - 00

does not cause failure of the material as shown by experiments. It pro-

duces a volume change only.

4. Saint Venant -Maximum Principal Strain Theory (1837)

This theory states that a failure (brittle or yielding) occurs under

any state of stress when the maximum strain reaches the critical value

obtained from simple tests, i.e.,

£1- (al P(O.2 + 03)) 1-a. (5)

where a1, 02, 03 are the principal stresses, El the strain, V is the

Poissons' ratio and E is the modulus of elasticity. Equation 5 yields

SL02 + 03) +1 (6)

For a hydrostatic state of stress 01 - a2 - 03 - a, Equation 6 becomes

0 (7)

Experiments indicate that the material can sustain much higher loads than

indicated by Equation 7. In spite of its shortcomings, this theory held

sway on the Continent for several years because of the support it received

from St. Venant.

4

5. 'Tresca -M 1xinum Shear- Stress Theory (1864-1872)

Observing the flow of soft metals in extrusion tests, Tresca stated

that yielding in a material begins at a point where the maxim-um shear

stress attains the yield stress value. In general form this criterion

in terms of principal stress, 0l, o2. 03 can be expressed as

2 a ] [ 0)2 1 (- a2)2- 3 0 (8)

where is the yield stress in tension test.

The Equation 8 represents three sets of parallel plaines. Each set

is normal to one coordinate plane. These planes define a hexagonal prism

in a,, 02, and 03 stress, space with its axis making equal angles with the

coordinate axes.

It predicts satisfactorily the response of the metals such as mild

steel under complex states of stress. It expects the slip lines to be

inclined at 450 to the major and minor principal stress directions.

In case of brittle materials which have different yield stresses in

tension and compression, failure planes are different from the planes of

maximum shear.

6. Beltrami - Maximum Strain Energy Theory (1885)

This theory postulates failure in a material when the total strain

energy stored within the material reaches a value obtained from a simple

test.

Strain energy 'U' per unit volume under a general state of stress is

5

A j .L.J...,,

U U (1 + 2 (9)OO +O3l

Straln an'ray par unit voluma il a simple tension test is given by

U'2E

Uy oquating thuou oxpra~siouu, Equatlon 10 is obtained.

(+ (10)

(6,.) -101 1 2 + G203 + 03 1 1

The theory cannot be used as a criterion for the simple reason that under

high hydrostatic pressure a large amount of energy may be stored without

catutiing failure.

7. Nohn: ' Theory of Strength (1900)

Mohr devised a graphic representation of stress states and proved

that states of stress on all planes intersecting the principal planes are

represented by points lying in the shaded area (Figure 1). He used this

representation of stress states to develop his strength theory. By this

time there was enough evidence to indicate that shear stresses played an

important role in determining the response of the material to the applied

loading. Accordingly Mohr hypothesized that the plane which carries max-

imum shear stress is the weakest of all planes having the same normal

stress. For example, of all the planes having the same normal stress

4': OD (Figure 1), planes of maximum shear stresses (corresponding to the

point D2 and D4 of the stress circle) are the weakest. This theory

indicates that only the largest circle- the principal circle - of stress

needs to be considered and the criterion becomes independent of 02.

If a sufficient number of principal circles, each related to the

6

/

A'' t"';•, ,,• : i.• u•v.•,,:•,•.L:•: ,• • •:,:-. :••:• o.• ' ,<,'#,d;s• <m~'.m , : ' ';••• ••:a ' '•'-••* :•""' '::" •'""; ''..."

failure state of the material, are generated, envelopes (Figure 2) of

these circles can be drawn and used to predict the tailure state for

any stress condition. The relationship between the shear stress T and

the normal stress a on the plane of failure can be expressed as

11I F(o) (1

where F (a), a function of a, is determined experimentally and is sym-

metric about the a-axis.

One of the required characteristics of F (a) is that it cannot have

a negative root, as this would contradict the fact that materials under

hydrostatic pressure do not deform plastically. Experiments by Von

Karman indicated that for large negative values of a, Mohr's envelope

tends to bend in a direction parallel to the a-axis.

Similarly two braaehes of the envelope intersecting at a finite

slope at the positive point of a-axis lead to anomalous results. For

this reason, it was held for a long time that F (a) had no physical sig-

nificance as it approached the positive a-axis. A±ions Leon (1935),

however, pointed out that F (a) had a tangible meaning if it intersected

the positive a-axis at a right angle and had a finite radius of curvature.

Using this concept, it is possible to distinguish between cleavage and

oblique fractures produced in a material under differenc stress states.

The simplest form of F (O) is a pair of straight lines. As suggested

by Mohr, they can be obtained by drawing outer tangents to the principal

circles of stress corresponding to the failure states in tension and com-

pression tests. In case of cast iron, the predicted strength in shear

74

ti Cis which agrees satisfactorily with the experimental value. In

otthe simplified form, F (a), Figure 3, can be written as

T C- ±o (12)

where c and p are determined from the intercept which the tangent makes

with the T axis and its slope. In this form, it is the same as given

by Equation (1). Equation 12 expressed in terms of principal stresses ial and 03 becomes

"1 l (13)Sit S c

Sc 2c cost

"S 2c- si (15)

Equation (13) can also be expressed as

(01 - 03) + (0l + 03) sino - 2C coso (16)

i ~~which is similar to Equation (17) used by Guest to describe thle test i

results on tension-torsion-pressure tubes.

(01 - 03) + (01 + 03) C1 C (17)

•. ~In case tensile and compressive strengths are equal, the angle 4) is

zero and Mohr's strength theory reduces to that of Tresca.

8. Distorti.onal Energy Theory

This theory was re-discovered by several investigators before it was

accepted as being valid for ductile metals, Noting that the hydrostatic

state of stress produces only volume changes in the material, it is pos-

tulated that the material under a general state of stress yields when

ILL pel "-

r8

(i) Its energy of distortio. (Maxwell 1956, Huber 1904, Hencky 1924)

I' or

(ii) The second invariant, J of the stress deviation (Von Mises 1913);

orb (iii) The mean square shear stress (Novozhilov 1952);

(iv) The octahedral shear stress (Nadai 1933);

attains a limiting value usually obtained from the simple tension test.

All these are mathematically the same and lead to the expression of the

criterion:

½ [(Oz - oyy) 2 + (Ozz- Oxx) 2 + (a :x - )21 +

3(T2 + *2 + T2 ) a2 (18)yz xz xy o

where a's and T's are the normal and shear stresses at a point and a P,"0

the tensile yield stress.

In experimental verification, the plastic strain can be distinguished

only when it becomes measurable. This measured amount of the plastic

strain reflects more the mean square shear on all planes than the maximun.

shear stress. On this account, the experiments of Taylor and Quinney

(1931) on copper, aluminum, and mild steel tubes favor Von Mis'-i' criter-

ion (Equation 18) more than Tresca's (T. H. Lin).

9. Pressure Dependent Failure Criterion

The yield condition for isotropic ductile materials is adequately repr

sented in the stress space by Tresca's hexagonal or Von Mises' circular

cylinder with its axis directed along the mean hydrostatic stress axis.

In this fpilure condition, it is assumed that the materials have equal

yield stresses in tersion andl compression and the yield surface is convex.

9

i -:s

However, materials (cast iron, granular materials,, concrete, natural•-rocks, etc) which are pressure sensitive cannot have a yield surface

representation of a cylinder parallel to the hydrostatic axis. In gen-

eral, they do not show sixfold symmetry of equipressure cross section

for the simple reason that yield stresses of these materials in tension

and in compression are not equal. For most of these materials Mohr's

criterion with tension cut-offs is a reasonable first approximation,

but when the effect of the intermediate principal stress becomes pro-

nounced, it has to be geasralizcd.

For idealized materials assuming that cross sections of the yield

surfaces are geometrically similar, the yield surface could be expressed

as a surface of revolution about the hydrostatic axis. Some of these•' generalizations are:

(i) Circular cone (Nadal), i.e.,

Oct [3 Co aoct - 2 (19)where Toct and aoct are octahedral shear and normal stresses and C. & C1

are the material constants;

(ii) Paraboloidal surfaces, i.e.,

(01 +02+ 3) - 31l + 9a2 [(01 - 02)2 + (02 -03 + (0• al (20)

where al, 02, 03 are the principal stresses and al, U2 are the material

constaunt; and

(iii) Criterion of Prager and Drucker

aJ1 + [ ,2 k (21)

10

where Jl is the first invariant of the stress tensor and J 2 is the second

invariant of the stress deviation and o, k are material constants. None

of these criteria, in simple forms or their generaliztiions, account for

effects of temperature, time, environments, etc. No information is

furnished about the behavior of the material subsequent to yielding in

ductile materials or growth of the failure pattern in the brittle ones.

The latter type of failure is associated with fracture of solids with

almost no accompanying inelastic deformation. A study of these materials

on microscale indicates that their strengths should be many orders more

than obtained in tensile tests. To explain this phenomenon, Griffith

(1921-24) proposed a theory that the energy required for fracture was

not evenly distributed in the brittle solid. Due to the presence of

randomly oriented cracks, high concentration of stresses occurred at

their tips results in the uneven distribution of energy. This theory and

later theoretical and experimental investigations about the formation and

propagation of cracks is the subject matter which appropriately belong

to "Fracture Mechanics" and will not be discussed in this report.

fti

SECTION III

ANISOTROPIC THEORIES OF STRENGTH

Commonly used anisotropic materials are fibrous composites fabri-

cated by embedding fibers in a suitable matrix material. This imparts

orthotropic characteristics to the material. To extend the theories of

strength of isotropic materials to composite materials,it is castomary to

treat them either as quasi-homogeneous or as a two-phase mixture of isotropic

matrix subjected to geometric constraints by the fibers. In developing

the theories, enough arbitrary parameters are introduced so that various

failure modes can be incorporated. In some of the theories gradual tran-

sition from one failure mode to another is assumed. This gives rise to a

smooth failure surface. On the other hand in certain formulations indepen-

dence of failure modes is contemplated. These theories are broadly class-

ified as theories with or without independent failu.e modes and are pre-

sented in this Section.

I. Theories with Independent Failure Modes

(1) Jenkins (19-20

Jenkins (References 8 and 9) extended the application of the

maximum stress theory to a planar orthotropic material like wood to predict

its failure. In this theory stresses acting on the orthotropic material

are resolved along the material axes (all, 022, Tl2) and it is postulated

that the failure will occur when one or all of (011, 022, T1 2 ) attain the

maximum values X, Y, S obtained by simple loading conditions, i.e., the

failure is precipitated when any one of the following conditions is satis-

fied:12

. •"i

""22 Y (22)

T1.2 S

where X and Y could be tensile or compressive stresses at failure.'

(2) Stowell and Liu (1961)

In this hypothesis (Reference 10) three failurc. modes are recog-

nized.

(i) Brittle fracture of fibers;

(ii) Yielding of matrix in shear parallel to fibers; and

(iii) Yielding of matrix in tension tansverse to the fibers.The failure of the material occurs when o11, 022a a12, normal and shear

stresses along the material axes attain limiting values, ie.e, when

Ill = Xf (23)

U22 - Ym (24)

S=2 Sm (25)

where Xf is the failure stress of fibers, Ym and Sm are the tensile and

shear strengths of the matrix.

Kelly and Davis (Reference 11) pointed out that the use of the bulk

properties of the matrix in Equation 24 and 25 ignored the observed fiber

matrix interaction in composites. It was suggested that Ym and Sm be

increased by factors of 1.15 and 1.5 respectively to incorporate the iater-

action effect.

Experiments of Jackson and Cratchley (Reference 12) to correlate the

r reDr-h and the mode of failure with the ib~r ortentation of unidirect-

ional an angle ply laminates led them to observe:

13

'4A

(i) Using experimentally determined values of Ym and Sm, Equations

23 to 25 are quite adequate to predict the behavior of off-axis speci-

mans of unidirectional laminates, particularly when the change in angle

(^.'3.5O) in testing is allowed for;

(ii) Vo strenohgtpeak is noticed when the mode of failure changes

from that of shear to the transverse one with the increasing orientation

angle; and

(iii) Equations 23 - 25 do not describe adequately the performance

of angle ply laminates.

Further studies of the Stowel and Liu criterion were made by Cooper (Ref-

erence 13). Thin sheets, in which tungsten wires were weakly bonded to

the copper matrix, were used to fabricate off-axls coupons. The experi-

mental results were in reasonable agreement with the theory except for

two zones of deviations, namely, when the angle of orientation • was:

(a) 0° < < 100

(b) 450 < 0 < 9g0

The discrepancy in the case (a) is attributed to the fibers termin-

ating at the free edge being unstressed thereby reducing the effective

width of the specimen. This edge effect diminishes rapidly with increas-

ing 0 due to the decreasing overall fracture stress and the component of

stress along the fibers.

In the second case (b) contrary to the theory, no increase of strength

was observed. In fact it slowly decreased. The explanation offered is

14

VS

'T '

that it is always possible to find a plane inclined at 450 to the loading

axis that does not cut any of the fibers.

(3) Prager (1969)

The Reference 14 states that the second and the third failure

mechanisms (Equations 24 and 25) representing the failure by plastic flow

"of the matrix are not independent of each other, but interact. This

interaction is established by assuming the matrix is a homogeneous, per-

fectly plastic solid obeying the yield condition and flow rule of Von

Mises subject to the constraint of vanishing rate of extension in the

direction of indefinitely thin reinforcing fibers. These assumptions

are used to develop a set of equations governing the behavior of unidirec-

tional and angle ply laminates. The stresses acting on the laminates are

the principal ones.

Unidirectional laminatesF

Ox Ui (26)(3(1-c) cos20 + (l+c))

4kor ox - ' - -r (l+c) - (1-c) cos26j3+ 4(i-c) 2sin2 20 (27)

warre Oy, Cox = Principal stresses acting on the laminate

1F 11 Fiber stress at failure

~ I V..,. I sheer ..... of the mritrix

Ond 0 ' gle which ox ma-es with chw occ direutioi.

Using Equations (26 and 27), the ranges of fiber failure and matrix fail-

ure modes can be determined.

15

(4) Waddoups (19661

In the application of St Venant's maximum strain theory to ani-

sotropic materials (Reference 15), it is hypothesized that the failure

is precipitated when any of the strain components associated with mat-

er~al axes attains the limitLing value. If eii •22' Cl2 are the limit-

ing strain values in a planar orthotropic material, the conditons of

failure in terms of stresses are expressed as

1 31(i) a2 -2 (P1 - all) (31)

E2 2

S(ii) a22 P22 + P12 Ell 1i (32)

(il 12 1 P2 (33)

where P e E (34)11 ~l 11

22 22 22(35)

12 " Yl2 12 (36

and Ell, E2 2 , GI2 are the moduli of elasticity of the material.

Equations 31, 32, and 33 define longitudinal, transverse, and shear

modes of failure.

(5) Lance and Robinson (1971)

The first attempt to use the maximum shear stress (Tresca) cri-

terion for anisotropic materials was by Hu (Reference 16). In developing

the theory it is assumed that material axes of orthotropy coincide with the

principal stress directions. An extension of this by Wasti (Reference 17)

takes into account the effect of the presence of reinforcement, A general

maximum shear stress based yioed theory, however, is developed in Reference

18 for a composite material consisting of stiff parallel ductile fibers

16

embedded in ductile matrix.

Yielding of the composite iS assuued to occur when the maximum ahoIa

stress on

(i) Planes parallel. to the fibers and acting ini a purpaindicular

direction; or

(ii) Planes parallel to the fibers and acting in the same dire-tioto

as fibers; or

(iii) Planes oriented at 450 to the fiber direction; attains a crit:-

ical value of stress associated with the failure planes.

In Figure 4, X coincides with the fiber direction, a. t, n are the

unit vectors. To meet the requirements of assumptions (i) and (ii),

the components of shear stress on the plane to which n (Figure 4) is

normal and acting in the directions a and i should satisfy the condi-

tions:

(i) oa aCos ( + ozx Sir 0 w Ka (37)

(ii) 1/2 (azz - oyy) Sin 2 + yz Cos 2 (p Kt (J8)

where K and Kt are critical values of shear stress in a' and t direc-

tion.

The expressions in Equations 57 and 38 attain ,.he ,'aximurn values when

(p = tan ( (39)a yz

1/2 tan (a - )/2 J (40)zz y/2yz

17

-IL

lalaiil (tit)

0 Q4k iII v (41

1/ al 2k / (0 (2

IVI owAim of 4 thin~ huato klompotita~ matIerdial W 4 otae o plallo

UI sa (43)

The~ odtiar co m 18I1Ine of atrous aire

xy 2

rCritical shuar planes correspondinig to the stresses (J4quationia 43

to 46)aj ru : :4

T ~~~Using teresults of Eutos4 o4.Eutos3,3,ad4

becomei

1011 ,, o,) 4. Qaa COO0 , (49)

I + Q 2(50

Vol, "llt~xial atroug 'ild Ow1 Squationa 48 - 51 bacoma

SIWa/Sln 2 0 (52)

2 Rt/$4i~O(53)°11 ,11w 2(54)

2 9a/Cos2 0 (55)

No intaraction butwen tihe ahoer modea is allowed for- in the criLerion.

(I ) THUO.__ S WLjTiOUT INI)rPENUIDNT VA 0 J4U. MOES

In mietals, grains are randomly oriented. It imparts isotropic

proportias to metals on wacro.'cale. This isotropic behavior is confined

to small deformations. With increasing strains., grains align themselves

in pi•er.e•ad directions. It renders the material anisotropic.

Hill (References 19, 20) suggested a yield criterion that allows for

the anisotropy. Tn the criterion it is assumed that the material poss-

easses three orthogonal planes of symmetry and there is no Bauschinger

effect. It is of the folm'

2 f(o) 1F(a22 - 033)2 + G(o3 3 a )2 + H(al- 022

+2L T + 2Hv + ZN 1 1 (56)

19

': - - r-- "

whore F, G, H, LO M, N are the parameters characteristic of the current

state of anisotropy. If X, Y, Z are tensile yield stresses and R, T, S

are the yield stresses in shear with respect to the axes of anisotropy,

Equation 56 can be expressed as

0112 (a22 2 0 33 2 1 1(-) + (-F)-- P + -7 )a 11 a22 +

T (57)

(p Z P)a2233+ P~ + Pz Y-)'11133] + (5) +

2'3 2 T 13 2(-) + (-) -

In case of isotropic materials,Na

XUY Za', and T R S (58)

Equation 57 reduces to Von Mises' criterion. For generalized plane

stress state, Equation 57 becomes

"(011)2 + (04)2 1 + 1 ) cr.a + 11 1 (59)Y 7 0-7 11022

(2) Marin (1956)

Matin (Reference 21) modified the distortion energy criterion

for orthotropic materials with different strengths in tension and compres-

sion and expressed it as

p (01( - a) 2 + (o22 b) 2 + (033 c) + q [,(li - a) ( b22 -

(60)(a b) ( 3 3 - c) + ( 3 3 - c) (Oii-a)I 6 02( 2 2 -b)3 1

where a, b, c, etc are to be determined experimentally. For plane stress

condition Equation 60 can be expressed as

20

02 +K a 2, + 0 + K G + K a K (61)3.1 1 11 22 22 2 11 3 22 4

where Kit K2$ K3 , and K 4 are constants. They can be-evaluated from tests

(i) ll XT 022 ' 0

(ii) G22 - aT Oli= 0 (62)

(iai) 1 XC 022 0

(iv) = C22 S

Substitution of Equations 62 in Equation 61 yields

K cT X

1 c X - S (Xc - XT - Xc +.Y T)KI = 2- sL (63)

2 Xc -T (64)

K'3 X c XTT/YT - T (65)

K 4 XT Y (66)

In the formulation of this criterion, it is assumed that the principal

stress directions coincide with the material axes, therefore, there is

rio way to account for shear stresses that may be present along the mater-

.. axec..

(3) Sta~ss~i .... OAlia ,.(!959)

Reference 22 gives the relationship between the stresses and the

strengths of the material as

a a102 2 2 X 0 a 0

L. 21 2 22 22 + (i- 1-) 11- + 22 + 3(-12) 1 (67)XT Xc c T T

21

In Equation 67 shear strength is not an independent quantity but is a

function of XT. This criterion assumes the material to be isotropic

with different properties in tension and compression and, therefore, it

is not valid for application to the compcsite materials.

(4) Norris (1962)

Using nine strength properties, namely three tensile, three

compressive, and three shear strengths, associated with material axes of

the orthotropic material, Norris (Reference 23) suggested an interaction

type relationship between stresses and strengths. The failure of the

material would occur if any one of the following equations is satisfied.2 2 2

.11. 022 11 022 + -12 1 (68)+2 2 2

22 + 33. 022 033 + T(69)Y' " + T Y Z T( 92 2 2

02 +_ 2 11 ii 2T) 1 (70)

where X, Y, and Z can be tensile or compressive consistent with the

nature of the stresses 011i a22, and 033

Equations 68 to 70 for the generalized plane stress condition become:

0112 0222 011022 + "-122+ (71)) +

2_2) k 1 (72)

( 1ii (73)

In Reference 25 it is stated that the shear strength S was not determined

but deduced from Equations 71 to 73 by using tension data and the results

made to fit the formulation.

22

(5) Griffith and Baldwin (1962)

Defining a 1 1 alp a22 - 02' 033 031 T2 3 '4 T'1 3

aS' T 1 2 6 a and a set of strain components e in the same fashion,

stress-strain relations of an anisotropic linearly elastic solid are

expressed as

o = -C¢ ej (ij 1 l, .. 6) (74)

e =s j (i,j 1,... 6) (75)i ij

Total strain energy UT per unit volume for orthotropic materials

(Reference 24) is given by

UT foi d 61 (76)UT

E

1/2 (S 1 02 + S2 02 ~+ S 2 +a + S aa +12 °1 2 + S13 01 3 + $23 02 3) +

(S4 4 �2 + S a2 + S66 6 ) (77)4 55 5 6

The corresponding volumetric strain energy Uv is

Uv = (a + 02 + a) (78)1 2 33

where% ((a1(Sl + Sl2 + S13 ) + a2 (SI2 + S22 + S2 3) +

Cy(S + S23 + $3))/3' (79)

The distortional energy UD for the orthotropic material can now be written

as

23

$ ' K•• i;i; >••::''• ; :.•¢.••••,`,',•.. ' . . . . ,, . <.. -. . . . . : • . .. .

.123 2 + 13 S12 -

~ 2_s S2 ] + c203[ (0

.11+ 22+ 222A2 2+13+ + -3 -j

2 [33 3 L 13

S S S + S3

11 + 12 + 23 1 [2 +

S2 J

s44 ° %555 s66 6

For plane stress condition Equation 80 reduces to

up "°[ Sl -[s 2 2 S 2~

2 322

(8e)12 2 Sl+ S2 2+ S13+ $33 +

1 2 2

SIt is assumed in this hypothesis that the incipient yielding or the

fracture will occur when UD equals a critical amount of available die-

tortion energy UD .UD is obtained by a sin~gle uniaxial test with

2 I S

Uc +

Resuls ofuedi this oneparmtheri theor are shnciieto bieding far agree

ment with the experimental data. However, it is suggested by Griffith

24 [,1 ~ ~ U D, K.IA .

required before accepting the theory based on the distortional energy.

(6) Azzi and Tsai (1965

Azzi and Tsai (Reference 25) assume that fiber reinforced com-

posites are transversally isotropic, i.e.,

Z - Y (82)

Using the relation from Equation 82 in Equation 59, the modified form

of Hill's criterion becomes

2 211)21 22(2)1+ 2 - 1) 1 (83)X Y X

Results of tests on off-axis coupons prepared from unidirectional

glass-epoxy laminates are shown to be in agreement with the predictions

based on Equation 83.

Later (Reference 41), Equation-83 is generalized to account for diff-•

erent tensile and compressive strengths by stating that X and Y should

be consistant wi-h the nature of stresses ol1 and 022a

(7) Goldenblat and Kopnov (1965)

Reference 26 marks the first attempt to develop a general strength

theory for anisotropic material. It lists some basic eequirements which

the theory must fulfill. They are:

(i) It must be capable of predicting the strength of the material

under complex states of stress for which no experimental data is available;

(ii) Besides having the stress tensor characterizing the stress state,

the expression for strength should incorporate in some way the strength

properties of the material;

25

K. . • ,.!-; :,,.:... ....... .... .... . ....... . .... ... . . . .... ... ..

(iii) It should include transition from one coordinate system to

anothier including simple state of stress in any coordinate system;

(iv) It should have the structure of an invariant formed from

stress tensor and strain tensor components characterizing the strength

properties of the material;

(v) The relationships between material constants should be inde-

pendent of the coordinate system;

(vi) It should have a capability to account-for difference in

tensile and compressive strengths which resalts in shear strength

being dependent upon the sign of the tangential stresses;

(vii) The failure surface is assumed to satisfy a heuristic prin-

ciple whichiasserts that the growth of strength indices is possible

only when the new failure surfaces include the old ones so that there is

no intersection of the surfaces.

The failure criterion satisfying the conditions (i) to (vii) can be

written as:

(Fik oik) + (F p a)8 + (F a a +.-.< 1 (84)

(Fpqnm pq nm rstlmn rs tl nm

Equation 84 reduces to the strength of material formulas a 1,

1 1/2 and Y 1/3 etc.

Selecting the first two terms of the expansion, the criterion becomes

F a + F a a < 1 (i,k,p,q,rs 1,...3) (85)Fik oik + pqrs Opq Ors_

The domment made in Referefte 26 is that the final selection of the expres-

sion depends upon its being verified by experiments. If the results of

26

experiments requiremore terms can be included. The strength tensors

Fik and Fpqrs satisfy the following symmetry conditions:

F F -F F F ; F ;F Fik ki; pqrs qprs pqrt pqsr; pqrs rspq (86)

Using the symmetry conditions, and condensing the tensors of stress and

strength, Equation 85 can be written as

F~ a~ + 1 o <1 (i,j 1,2...6) (87)

If Fi and Fij are the parameters of strength in fundamental coordin-

ates, i.e., a system pertaiL4ing to the material axes of the orthotropic

body, Equation (87) becomes

F i i + lJ ;i 0j < 1 (88)

0 0where F and F are evaluated from test conditions.

(i) Stress conditions of pure tension and compression along the

material axes, yield

2

F 1/2 FOI 1/4 1 (89)

20 11 , lF2 1/2 ( ;F22 1/4 1 ) (90)

2o= / ( 11 o 11/ +c(lF3 1/2 ZT( ZC ;F 3 3 1/4

(ii) Stress conditions (Figures 7a and 7b)

1 = T6 (45), 2 = T 6 (45)+

a, T6 (4 5 ), a2 , 6(45)

and similar conditions for other planes yield

27

F 0 2 1/8 1( i+ 1)2 +Z)

1C T YC T6(45)+ 6(45)-

L ~ r ~ T C 4 ( 4 5 ) + T4 4 )F 23 - 1/8 [•( + _)2 + ( + ) 2 . ( 4_. _ + (94)

To evaluate FI2 , Fl 2,two tests eac hav been emlyd Toreovany cnrdcinrsligtherefrom, additional conditions required are:

"1 T ! ____ 1 (95)

6(45) 6(45)-

TXC + + -4(45)_ (96)

Ty Cc Zc4)+ 5(45)-

(iii) Conditions of pure shear (Figure 8)

- T6(0) +, a r C 06(0) T 6 5(

yield

XT (98)

T66 c 46(0)+ (9

T44 a4(0)+ )29)

6( 5(0)

0and F 0 are zero in fundamental coo(d9na)es as

28

I>!• .. . ...

45(00)'04(0)+ ' 4(0- (0) 0) ' %50- ad0(0)+ o6 6(0)-

All other F's are zero in fundamental coordinates as there is no

coupling between shearing stresses and the normal stresses in that coor-

dinate system.

Having computed F0 and F±0 in the fundamental system, Fi, Fij in any

other coordinate system can be determined by transformation.

In case of generalized plane stress condition in fundamental eystem,

strength criterion becomes

al 0o + F22a? 2 F606 02<1ý (101)1 1 2 2 /11 1 + 2F.1212 F2 2 66 -

Experiments on tubular specimens of fiber glass under biaxial

states of stress were found to yield satisfactory results (References

26,27).

(8) Ashkenazi (1965)

To formulate a theory of failure of orthotorpic materials,

Ashkonazi (Refej'ences 28, 29, 30, 31) assumes that

(W) The material is uniform, continuous, compact and anisotropic;

(ii) The factors like time, temperature, humidity, specimen size and

shape, etc., can be ignored;

(iii) The strength properties are tonsorial in character and can be

represented by a tensor of the fourth order, i.e.,

"" 2 3 441krll4 k So 'iko (102)

where (', i, k', k, olp o P p W l,..3) (103)

in carLosiau coordinates; and (iv) it is postu~latel that

a-ikop aikop (104)

29

where a is the strength characteristic of the material.

On the basis of these assumptions, the author derives expressions to

determine the normal and shear stresses which the orthotropic material can

sustain in any arbitrary direction. These expressions in the case of a

plane problem reduces to

1 01 4 1 1 2 " - (2flo)1 os + ( Sin2 a Cos 2 +Snab X 717 Y

1 Cos 2 2a 4 Sin2 a Cos 2c(T• " S + 45(106)TbS ~ 45

where

0b' tb "normal and shear stresses in x; y' coordinate system

making an angle a with the material axes.

45 45X , S - normal and shear strength obtalued from a coupon cut at

045 to the material axes.

Equations 104 and 105 can be wriuten as

! b Coj4a .+ b min2 2a + C Sin•,74a,I1

To~~Sn2 (108)b ~ FýP 2T S, t -s lC5

t•,.• i when b -- -

X ~(109)

and C

In case a sixth order tensor is used to raprasent the strength pro-

purties of the material, Equation 106 becomes

0300

b". 30

where Cu

and bX 3,'0 4

Experimental results appeat' to agroio With 010 cIculated v'iluoa m~iig

Equation 106. The use of Equation 109 insteaid of EqUatioll 1WO dooa 1%0('

yield any more satisfactory results.

The Equations 106 and 107 are applicable only if ub 1)o alto isit

acting on the element of the material, For complox sutatot of tou

Ashkenazi considers the possibility of uLsing Von Misals plastioity fuinc-

tion of the form

12 Fikop a kaop cnt"

which represents a surface in stress space,

For the plane stress state, Equation 1.11 becomes

12 + 2 2+ ( 2 + 2F 1 0 U 1 (l11)

In case the. stvess state is given by

11 22 12 2-

the value of F evaluated from Equation 112 is12

4. 11

2F (114)12 45 2 2 212 [ 45 2 S

Using Equation 114, Equation 112 can be wri~tteu as0Y11 2 + (0222 2 Tf --J 2 4 1 _ 1A

S(X x Y S

011022 1(15

31

~ i 1quilhM 1!~OwU aiOi Oft-dktl At bmhra4ow i,~i Atgl

100 iop .tik (ApporL hv thw i g drWA I'mii WuC 04u.LQ lia-rp

P.1w~o tooI'Lh dogruoo 4 tho tom Ot Wor eatimh

orU4ia+ l 1.15kl Aquty *.1gO .-. o IJ)OWV~ +ilo owlp

x~ Y

+ J2 ~ +

Vt2 (a~ + 0~ (0~ + 0 + j +

w~itIrl X, W, and p are to bfj dataiJlwitkd Qp~ivillleltally for' three biAxial

hquatiotA 116 itioludus NquatioIIM 106 And 107 au particular cases.

Tlhe ultimtae raintiwo (if uaturiials cut bt dotormuiud possibly

by usiag at8a1se, strains o OLi te anevy axp)tiuded to reach the fail~ure

status. Malmaistar (Reforuitco 232) bas~es Ohw failure evitarion an the stress

States slid po~tluaLOR Lila failureu to occur when Lthe stross path rupreaorltocl4

by title ray frowa the origi~n of coorditiatos of Stressa Space terminates on the

32

fatluve ourf a~d giveii by

+~ (ay 4 4XI (117)

whro Fi V, 1 vly iaa t miw gLth Conuorm of second, fourth and

In caua of |pla atr¢o cunditiotib, Uquation 117 with two ýCrms only

Can be written as

F' Q +1 F v1 q2 +F cj F 2 +F~ ~ yy Oyy + xy x•y + xx~x xx yyyy yY

X4% ÷ 1( yy 0 X y Y XXXX xy yyyx ayy xy

vy + 2(o 0 a F +TF 4 r 1 (118)

whou\ x, y are arbitrarily located carteo ui -oordinatoas Equation 118 has

nine coafficintas which need to be determined from nine tasts, namely two

tanuile, two compressive, one shear, and four biaxial tests. Equation 118

iu cousidurably simplified for orthot~ropic material. On condensing the

atra•gtlh and stress tensors Equation 118 1-elated to material axes becomes

, C° +Fa 4 a + 02 .F 02 + F • a' -2a1 1 2 +2 06 F6 +•1 F• 2222 2 66 2F 12

0102 1 (119)

by subjecting the material to two tension and two compressive and sheartests, coefficients Fi, Ftj become

X 11 1T C XT--

F2 2 1 TY- (120)

F + -S--; F6 +S- 6 6

33

Similar olcprass.ona can be obtained for s3, V , , in case of

t~hreo dimensional orthotropic material.

To determine F.I2 utca variety of combinations of al, and v2 can be

used. Malmeister also suggesta the use of other criteria based on

(a) Ultimate strtalns; and

(b) Both ultimate stresses and strains.

They are

86~ **~ and (121)rE+E6B 8 6 + . .. a1 (.122)

Ua a•Ota8 + - Ca + FSy6 O(XO oy6 + 1$86 y6 CY6 + 1 (122)

(10) Hoffman (1967)To determine the brittle strength of orthotropic material, ]

Hoffman (Reference 33) proposed that a fracture condition is reached when

Equation 123 is satisfied, i.e.,

01(o22 33C 3 (i -322) + 4 11

131+ C5 22 + C6 33 + C7 23 8 +09 2 1 (123)

The constants Cis can be expressed in terms of three tensile strengths,

three compressive strengths and three shear strengths, i.e.,(1=12 1 1 1T1 T~

2 1/2 T + TT

C3~ ~YY = 1/z Tc ZZ 141 C 341W1

C3 1/2 +jX T (124)

rip 34

04

T I

(14

In case of platw QStress Couditiol And aguiutIR after R en"ce 25,

tht z~ 4 rT, RO E~quation 123 becomes

0.2'2 3,11(12 2 + -"" + - -'- - ' ;' 0•, + ,-,[-%y 22"TC YTYC N- Yc 11ý 22C

~ 2(125)

(11) Flisher- (1967)

Using the Pstrength criterion of Reference 23, Fisher (Reforenice

34) derived expressions to evaluate the strength of isotropic laminateos

with N laminas(N "" 3)and inclined %t an angle of L angle to each other.

The expressior is:

, 1 1 2 a22 2 122 1 1 2 2 - 1 (126)+ (-) +

35

12 Q14i (192U)

(CIhumim (1Rofaraiw 35i) aiisulmwi tht

K I (i) UU011 ply is SOVIurailly ovLilocropic, a11d is 1.illourly ausatic;

(W±) TIIQ ply ia ooIU~VALiVQ ultdor liQadt iU, 010 LORSor Of' clatic

(iv) Tito ply axorp ±ucou ouly nxtuxi~i±oal deformuation undur frouu

chermal1 luading; and

(v) Tito diatortional onorgy of each ply ruiiains itivariant under

votatiounal tvIAforu atioll.

Following Roferncte 24, thu diatortional enurgy, indapandent of Lhartual

affuctc, can bo expvoaaod as U K1 02 + K02 +a + K 02 + K a a12 3 3 4 1 2

+ K aG + K a~ + K u a52 3 + 1 613 7 4 + 9 6 0~28)

For simiple load couditions,

U -oKz2 KT2 KR2 K 8 (129)D 2 3 7 8 9

Usiug 1Lquatiun 129 in Equation 128 yiaJds

36

,4. + ; + 2 +

K~. (130)1 3 2 XY 23 'iZ 13 xz

who•2o K I23, a"d K1 3 aro t:ho t ombinod 0( ' 'u h coofficiant which

are rhoeun in such a twanor that predietod and experimontal rusultv

ale W good agrutatit, Por a plata s•russ eodiLtioa, Equatimo 130 assumes

the f oim°115 -2,2)2 ,-•1 ,.(JL) + ( .. (" - .(33."" 8 K12 XY

Equation 131 does not distilnuish bOavwaoj the tensile alid coalpros-

sive behavior. A distiitction can easily be allowod for by permitting X

and Y to assume values X or X Y or Y consistent with tle stresses 03

ad02.

(133) Cosu,) (967

In the formulation of a yield criterion for orthot-ropic materials

the assumptions made in Reference 36 are:

(i) The yield criterion depends upon the present stress state only,

i.e., rho stress or strain history does not affect the yield strength;

(ii) The yield criterion is a scalar quantity, i.e., some scalar

combination of stresses determines when yield would occur;

(iii) Only the devatoric stresses affect the yield;

(iv) The material has three planes of anisotropy.

The deviatoric stress components arei i okA

a- o6 - (i,j,k - 1,2,3) (132)

37j j J.

U TruiluaLin tile Vuaerat tensor polynomial af Lr the first cubic term and

specialiuing to orthooonal materials (not necessarily cartasian), the

yield condition is writtaen ag

I~T + j T 2 +. (I r 3 (TI T2 2 1~t - 3)21 11 2 2 3'3 '1k2"I1 2 + 1n" 32

+2 1 (,(2 T4 T1. T3 T2 TTý) I0+32 + +1 31 23 3 2

do-1 (133)

whore (0I + a2 + a3) 0 (134)

Equation 133 has ten constants which can be evaluated from three tension,

three compression, three shear tests and using the Equation 134.

If al M a2 a3 3 C 0, Equation 133 reduces to Hill's six con-

stant theory.

(14) Franklin (1969)

Marin's criterion assumed that the material axes coincided with

the principal stress direction. Franklin (Reference 37) generalized it to

include shearing stresses on material planes. The expression suggested by

him for plane stress condition is

Ka 2 +K 1 +K 2 + o + o + 2 (135)K1 11 + K2ll a22 3 22 4 11 K5 022 + K6 12

Constants K's are determined from two tensile, two compressive and one

shear test. Using the experimental values strengths obtaLned under simple

loading conditions, Equation 135 becomes,

38

++ • ++++• !+;+ +,+ + ++ ... + + + -+ , +. .. .. + ,- A+ .. , A'++=, - -+ ........ +. ....

2 02 K 2 X XT Y -YT

+ 22.21122_ + T2 +.22T"C YTYC XTXC XTXC YTYc 22

+ (•12)2- 1 (136)S,

If K 1, Equation 136 reduces to that by Hoffman (Equation 125).2

The constant K is a floating constant. Its value may be different2

in different quandrant of stress space. It can be evaluated from biaxial

stress states.

(15) Tsai and Wu (1971)

Assuming that there exists a failure surface in the stress space,

Tsai and Wu (Reference 38) propose a scalar function

f(0K) = Fioi + F a a 1 (i,j 1,2 .... 6) (137)i i ii i j

subject to the constraint

F F - FiJ2 > 0 (i,j 1,2...6) (138)ii jii i -

In Equation 138 summations over i and j are not implied.

F and F are related to the strengths of material obtained fromi ii

tests conducted with reference to the coordinate system. Three tension

and three compressive tests yield XT, YT' ZT, and XC, YC, ZC. The use

of the results of the tests in Equation 137, yields

F 1 1 ! F

12 c = X

2 - 1 22 F (19

•,-1 1 1

39

Similarly six pure shear tests (three positive, and three negative shear)

are used to.obtain

F1 1 F 1F4 if¥ -7 F4 4

1 1 1F5 R- - F55 " - (140)

F 1 F6 IF6 S " - F66 "S-

To evaluate the rest of Fij, combined stress tests are required, for

example, F12* To determine F1 2, variety of stress combinations can be

used. Some of them are

(i) 01 i M 2 = PT

(ii) 45-degree specimen in tensionUT

1 02 -6 2'

(iii) 45-degree specimen in compressionUC

1 2 0 6 2'

(iv) -2 VT;

(V) 01 02 VC

(vi) a 02 -P

Plots of PT' PC etc. as a function F2 indicates that not all the tests

are suitable for determining FI 2 . A small inaccuracy in the value of

U P*UT' T' C produces a large change in the value of FI 2 . For orthotropic

materials, results get highly simplified. The number of independent constants

which are non-zero, are 12. This reduces Equation 137 to

40

p

F1 ?1 + F2 "2 + 33 I + 2F 31 22+ 2F1 3 01 o 3 +

+~ ~ F ++oF 1 1 1

F +2 F+ o + (141)22 2 +2 23 ~2 03 33 3

F 02 + F 02 *+F 0244 4 55 5 66 6 1

In case of the generalized plane stress, Equation 141 becomes

F t Fa F 02+ F Cra + F 02 + F 02 -1 (142)F1 1 +F2 2 1+F1x + 2F 1 2 01 02 22 2 66 6

(16) Puppo and Evensen (1971)

It is observed in References 39, 40 that the behavior of ortho-

tropic materials appears to lie between two cases marked by the responses

of the isotropic ductile materials and non-interacting fabric like materials.

To bring about the transition between the two extreme cases, the concept of

interacting factors is introduced. Defining the factors as

al YZ

.• i R 2XZ (143)

x3 S2

a failure criterion

aR o= 1 (i 1,2,3) (144)

is postulated, where a and RMi) are

T (alO2,"3,04,a5,o6) (145)

_) , 3 (i - 1,2,3) (146)

41

I_ Yi _I.

X2 2y2 2Z2

-A(1) - _ 1 (147)

Y2 2X2"

Sym8z2

Y1_2,x 2 2X 2 2y 2

(2) 1 1A y 2 2z 2 (148)

Z 2 Z2

_1 1 B

X2 2Z2 2X 2

A y2 2y 2 (149)

Sym. 1

1 0

T2

1 (150)R2

0 0 1S2

42

A,{

ALv

In case of plane stress, Equation 144 becomes

- 022 (j 2 )12.1Y x + 22 2~

011 2 y i 22 0y22 2 (52- ~ (Y '11 j- + + 12 .

0o 1 Yl I n and Equations 151 and 1523

yield identical results.

Figure 9 indicates the effect of interacting factor y, nte hpof failure surface fdr T 1 2 onte.hp

T.

43

SECTION IV

SUMMARY

In Sections II and III various failure theories for isotropic and

anisotropic materials have been surveyed. The criteria discussed in

Section III are either distinct failure-mode-dependent or have gradual

failure mode transitions. In the first category, failure is precipi-

tated when any one or all of longitudinal, transverse, and shear stresses/

strains (References 10, 15) exceed the limits determined by tests.

Tests (References 12,13) conducted on off-axis coupons of a thin compo-

site material, however, did not indicate any peak when the mode of fail-

ure changed from shear to transverse. It appears to indicate that shear

and transverse failure modes of composites with stiff fibers are not

independent but interact (Reference 14). If the fibers are stiff but

ductile, the failure condition of the composite, according to Reference 18,

may be predicted by considering three shear modes of failure.

The second class of criteria are essentially different expressions of

a quadratic form with or without linear terms. They are either generaliza-

tions of Von Mises' criterion (References 19, 20, 21, 22, 23, 24, 25, 33,

34, 35, 37) or have been developed explicitly in quadratic form using the

stress tensor (Reference 26, 32, 38, 39) or the stress deviation (Reference

36). In the latter case one cubic term is also included.

If differences in tensile and compressive strengths are not explicitly

allowed for, the expression for the failure criterion can be written as

44

j j11 (~

or

a TFo

where a Tis the transpose Of a.

in case differences in strength are epressedly ~accountad for. Lil Qvi-

tarion is

F ay + F a~ a~ 1 (i,j 1,2 ... 6) X5j J iji

or

-T T(F + a F) a (1)

If F and F are known in one coordinate system, Uley can be determinod in

any other system by suitable transformation of coordinatea. The coof-

ficients F, F strength parameters of various theories for thie ortho.

tropic sheet in a state of generalized plhne stress are sumumari~red in

Table I.

45

Jl z I

REFERENCES

1. Timoshenko, S. P., "History: of Strengthof Materials", Mcgraw-HillBook Company, 1953.

2. Nadai, A., "Theory of Flow and Fracture of Solids," Volj and II,Mcgraw-Hill Book Company.

3. Liebowitz, H., "Frachure," Vol II,,Academic Press, 1968.

4. McClintock, F. A., Argon, A. S., "Mechanical Behavior of Materials,"Addison-Wesley Publishing Company, Inc., 1966.

5. Jaeger, J. C., "Elasticity, Fracture and Flow," Methuen and Co.Ltd, London, 1962.

6. Bert, C. W., "Biaxial Properties of Metals for Aerospace Structures,"AIAA Launch and Space Vehicle Shell Structures Conference, PalmSprings, California, April 1-3, 1963.

7. Timoshenko, S. P., "Strength of Materials," Part II, D. Van NostrandCompany, Inc., 1956.

8. Jenkins, C. F., "Report on Materials of Construction Used in Air-craft and Aircraft Engines," Great Brittain Aeronautical ResearchCommittee, 1920.

9. Kaminski, B. E., and Lantz, R. B., " Composite Materials: Testingand Design, ASTM STP 460, ASTM, 1969.

10. Stowell, E. Z., and Liu, T. S., "On the Mechanical Behavior ofFiber-Reinforced Crystalline Materials," J. Mech. Phys. Solids,Vol 9, 1961

11. Kelly, A. and Davies, G. J., Metall. Rev. 10, 1, 1965.

12. Jackson, P. W., and Cratchley, D.,"The Effect of Fiber Orientationon the Tensile Strength of Fibre-Reinforced Metals," J. Mech. Phys.Solids, Vol 14, 1966.

13. Cooper, G. A., "Orientation Effects in Fibre-Reinforced Metals,"J. Mech. Phys. Solids, Vol 14, 1966.

14. Prager, W., "Plastic Failure of Fiber-reinforced Materials," Trans.American Society of Mechanical Enginecrs, E 36, September 1969.

15. Waddoups, M. E., "Advanced Composite Material Mechanics for theDesign and Stress Analyst," General Dynamics, Ft Worth Division,Report FZM-47ri, 1967.

46

16, UW, I, W., "Modifted Tr.gid-'o Yi±ld Coumli:ion dad ARRociAttd FlowWo 5gw fr Wouroplt MACi'ali and Appleiction," ourn4 of the

17. Ww~t4 8. T.0 "Thm Ptautiu, lifidili of Tr nUversely AaiaoL'ropiCCixuu1a plauld" luat, J. Muh. Scioauce, Vol 12, 1970,

IU, Lmu vIu, PIt. , lobii.iOn, ) . N. , "A M'Aximum Shear Stross Theory ofVIA8.110 vilol Of Fib- orued Matorials," J. Moch. Phys.Solid@, Vol h9. 1971A

19, 1111.l0 it., "A Thoory of ,he Yihlding and Plastic Flow of Anisotropicmotl.4," Procoodigo of tho Royul Sociuty,, Sotris A, Vol 193, 1948.

20, IHil,0 R,, "Tho Nat~homaticl Theory of Plasticity," Oxford UnivetsityPraou, London, 1,950,

21. Harbtit J, "TihooLes of Strongth for Combined Stresscs and Non-lao.toplc Materialmi," Journal Aeronautical Sciences, 24 (4), April1957,

2., Swti•i-D)'Alia, F,0 "Limiting Couditions of Yielding of Thick WalledCylind•ra avid Sphereal Shells,:, U.S. Army European Research OfficeUVC-1351 01-4263-60, 24 November, 1959.

23. Norris, C. H,, "St:rangth of Orthotropic Materials Subjected toCombined Stress," Forest Producto Laboratory, Report 1816, 1962.

24, Griffith, J. ,.v and Baldwin, W. M., "Failure Theories for Generallyor thotropic Mterials," Developments in Theoretical and AppliedMochauics, Vol 1, 1962.

25. A~xi, V. D,, Teai, S. W., "Anisotropic Strength of Composites,"Uxper'iuntal Mechanics, September 1965.

26. Goldenblat, I. I., and V. A. Kopnov," Strength of Glass-ReinforcedPlastics in Complex Stress State," Mekhanika Polimerov, Vol 1 (1965)pago 70; English Translation• Polymer Mechanics, Vol 1., 1966.

27. Protasov, V. D., and Kopnov, V. A., "Study of the Strength of Glass-Reinforced Plasr .'s in the Plane Stress State," Mekhanika Polimerov,Vol. 1, No. 5, pp 39-44, 1965.

28. Ashkaetaii, E. K., "On the Problem of Strength Anisotropy of Construc- 3

tion Materials", The S. M. Kirov Forest Products Academy, Chair ofConstnrucion Mechanics, Leningrad.

47~-4

29. Ashkenazi, E. K., "Anisotropy in the Strength of Construction Mat-erial," C. M. Kirov Wood Technology Academy, Leningrad, Vol 31,No. 5, May 1961.

30. Ashkenazi, E. K., "The Construction of Limiting Surfaces for BiaxialStressed Condition of Anisotropic Materials," Zavodskaya Laboratoriya,Vol 30, No. 2, Leningrad Forestry Academy, February 1964.

31. Ashkenazi, E. K., "Problems of the Anisotropy of Strength," Mekhanika

Polimerov, Vol 1, No. 2, 1965.

32. Malmeister, A. K., "Geometry of Theories of Strength," Mekhanika

Polimerov, Vol 2, No. 4, 1966.

33. Hoffman, 0., "The Brittle Strength of Orthotropic Materials,"J. Composite Materials, Vol 1, 1967.

34. Fischer, L., "Optimization of Orthotropic Laminates," Journal ofEngineering for Industry, August 1967.

35. Chamis, C. C., "Failure Criteria for Filamentary Composites,"Composite Materials: Testing and Design, ASTM STP 460, ASTM, 1969.

36. Bogue, D. C., "The Yield Stress and Plastic Strain Theory for Aniso-tropic Materials," ORNL-TM-1869, Oak Ridge National Laboratory, OakRidge, Tennessee, July, 1967.

37. Franklin, H. G., "Classic Theories of Failure of Anisotropic Mater-ials," Fiber Science Technology, Vol 1, 1968.

38. Tsai, S. W.., and Wu, E. M., "A General Theory of Strength forAnisotropic Materials," Journal of Composite Materials, Vol 5,January 1971.

39. Puppo, A. H., Evensen, H. A., "Strength of Anisotropic MaterialsUnder Combined Stresses," AIAA/ASME 12th Structures, StructuralDynamics and Materials Conference, Anaheim, California, April 19-211971.

40. Puppo, A. H., "General Theory of Failure for Orthotropic Materials,"Engineering Report 69-El, Whittaker Corp., April 1969.

41. Tsai, S. W., "Str-ngth Characteri-tics of Composite Materials,"NASA CR-224, April 1965.

44 48

A ~ <*...~h. 77Z777..7i.ii. .. ~ 7~7777

........".. ..

42. Sendeckyj, G. P., "A Brief Survey of EHmprical Multiaxial StrengthCriteria for Composites,"t Testing and Dosign (Second Conference),ASTM STP 497, pages 41-51, April 1971.

43. Sendeckyj, G. P., "On Empirical Strength Theories," Presented atAS•I-NMAB Symposium on Predictive Testing, Atlantic City, N. J., Juno-July, 1971.

ii

49

.'.

VV

bL

50Jb,,Pt~oa~~ablecopyu ' ý(ro)

EI

44

oo

51

•, '. ., . h . - , I . . . -

........

IVI

ICd

pu

CL 1)6 -

CG)

:jon-

52)

~11

lbb

LL.-

L.-to0

U-;

Nt

N5

r(45)

r$(48

+ ro (48) ~(5 (48)

4. +-r4(45) (45)

(6 ) re(4b r(45

Figure 7. Sign Dependence of SL~ar Strength

C-4~ (0)

454

-A(o * a

C3.J

44.

fl\ V Call

L6

I55

14

'4 -

1 '7

it

.. I'

V '4 0

lip

ý1ý AIL

-I,, 6~~)

* Ilk

TABLE3TIRENceTH PARAMETERS OF 714r,

WITMOUT DISTIN CT AFAISUg fef. 1401 PARP.

AUT/~oAS EQUATION FtoI

HILL alp~ a.-x

MAR31 #~+ 0FT -I .

STAUX r. +a týFrF .1 -XT - - -

49JFFIrN - TA~tv CfiD rtE*Af~ IS IWO P4ATSMAi. Couai'pS4 citird~. swtgqy op Ali roi

RALtDWIA/ , ~ -c 4 OMPUX ee TRpLo 13 /4 JAr"~ ro rRS 014y'oa riovA4 ffNJr*4Y I&A rNJ/9

Ac - y Y - Yr 1 +

Q*LJUEN8J.AT- F0 trN' 8L"XW t

KOPNosf

AIM CIWRX I4~biRATIC P400m4 1,s iJU(4,fTroz. 7)71R5E c.AmT4NrASrASH SAIAItZ ARE Rs4Qe9.~D T8IE be8GMIA/9*0 F*AW RIAAIA&-. 7esrJ.

-e -T Y - -Y

MALMI. MEWT FT3 *t0'FT~ ul9 X-7cXr Y19 T X %r T8ra *rAtrraptmm

H*FFMANvOC +~ 4eFo X4 - XT --YT

F AIS JJ a -

XL ZX'Y

HO UETHIS calyrAIO.J HAS A "eoVSTAi~ C AJJO0CIATr*, WITH PET(~) FOIX 'riARBo~UE flIME~iJ*O'A. @arHoradPgc MA76fAIA L, TI/gAC ARE /'h'.E CO P49TAN TS T*6 1liE bt r

FRARKLIN Pl i'0FQ' -1(c XT ',(r 2~ X7 Xc.

TS5AI - W U F~r A'c pr isIXc-Y '~Ye, -o Yr70O eWI4N' y c 'r Xc Xr P'Ro,4a 8s'Ap/i4dr

r~~r~O Y'(gpuppo X/

'S. 0 ./ &ri a a

'Si7

TABLE3P4TH PARAM4UTERS or-T 7JIoRoc

VONC 7 PAII U*, 140044 P.rO PS.AWD ST*. $u.~ rACS v IOW

Fi A u. F4 s c IZEPARKJ

w3 )(ir a XTKC. ;I iva, XZ[ T~SXrC-3 XT-Ac,*Y

T1 C~tlrf~idv. S~e'4GY op AiSTr*Ailem

,J01Jt.04r/d#AL AEr~4Y IWAMY51S44 AfsT.

I 6

IA;'A I- rjrS7-

I

2*X F-r X4 YT Yr-

km I~ FL OA T/Nf COW M I T'4

LV~ DXL ~ ) axy -Y .

VA P NIWE COjerrqJT.S Va jw bIercimiolou.

- 3.. - 0v7ri4 CoA(JfTANr

I To $. ereigi I I

PR-r AOM MAIMJ/4. T7Sr Yr-yc.

56