--- ,I AD-A226 060Materials Sciences Corporation

STRUCTURAL DEVELOPMENT OF MICRO-STRUCTURALLY

TOUGHENED METAL MATRIX COMPOSITES

Technical Final Report

MSC TFR 2111/8602

July, 1990

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11. TITLE (Include Security Classification)

Structural Development of Micro-Structurally Toughened Metal Matrix Composites

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Edward C.J. Wung and Kent W. Buesking13s. TYPE OF REPORT 13b TIME COVERED 14. DATE OF REPORT (Year, Month, Day) 1S. PAGE COUNT

1 nal Report FROM 89SEP01 TO 90JUN30 90JUN30 69

16. SUPPLEMENTARY NOTATION

17. COSATI CODES .. UBJECT TERMS b " " eff u iay bWt number)FIELD GROUP SUB-GROUP discontinuously Reinforced Metals, Microstructure Toughening

Mechanisms, Silicon Varbide/Aluminum, Impact Damage,Structural Applicati 4s, Metal Matrix Composites

19. ABSTRACT (Continue on reverse if necessary and identify by block number) This study addressed the developmentcf a new class of discontinuously reinforced metals that possess substantially improveddamage tolerance relative to the currently available materials. The novelty is to allowspecially designed toughening mechanisms to be operative in the composites. Technicaleffort of this study focused on developing a fundamental understanding of the toughening-echanisms at the micromechanics level, and identifying possible SDI structural componentst.3t can be improved through the use of this material. This goal was achieved by the

v.elopment of two micromechanical material models. The first model addressed the stress---rain ccrsiltutive modeling, and the second model addressed the modeling of fracture-ugochness of the new composites. Modeling and data correlation studies have suggested-veral guidelines for material strength design as well as fcr toughness desicn..-:narine hull and an aircraft lower longeron were selected as candidate struztural

:-71onents that miaht be improved through the use of this new material. A preliminaryi-si;n of the longeron was performed to demonstrate a design methodology for these new-r<isites.

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PREFACE

This report presents the results of a study performed .inder the Strategic

Defense Initiative Organization (SDIO), Contract N00014-14-89-C-0210, during

the period of September 1, 1989 through June 30, 1990. The program was funded

as a Phase I Small Business Innovative Research Contract. The program manager

for Materials Sciences Corporation (MSC) was Mr, Kent W. Buesking. The

principal investigator for MSC was Dr. Edward C.J. Wung. Mr. Dave Diverha of

a-.al Surface Weapons Center (NSWC) served as the Technical Monitor for the

co: tract.

W

I--

A7 7 roved by:

Y : .:.Bucksking ,

STATEMENT "A" per Dr. Steven Fishman

ONR/Code 1131TELECON 8/27/90 VG

ii

'4," -4 '

TABLE OF CONTENTS

V

Pa Re

:>TR7ODUCTION . 1

45. ECT IVES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3

AFFROACH ................ ................................ 4

Stress - Strain Constitutive Modeling ....... ............... 4

Analysis of Laminated MT Composites ........ ................ 9

Modeling of Impact Damage and Work of Fracture ... .......... ... n

Z.ructural Component Selection ........ .................. 15

RESULTS ............... ................................ 18

Material Properties ........... ........................ 18

Stress-Strain Constitutive Relations ....... ................ .18

Laminated MT Composites .......... ...................... 20

Analyses of Impact Damage and Work of Fracture .... ........... .21

Optimization of Impact Damage Resistance ...... .............. .23

Preliminary Component Design ........ ................... 24

V CCXCLUSIONS AND RECOMMENDATIONS ......... ..................... .27

REFERENCES .............. ............................... 29

TXELES ............... ................................. 31

F -__ ...UR.E.S ............. ................................ 36

AFFENDIX - Thermoplastic Response of a Homogeneous Solid ........... .62

iii

INTRODUCTION

Discontinuously reinforced metals are a class of coftptsites 'which have

:nessed extensive development over the past several years '1-4]. Most of the

research has focused on the development of silicon carbide (SiC) particulate

reinforced aluminum alloys, but reinforced magnesium, titanium and nickel based

s--r-,eralloys are also receiving attention. Although the discontinuously

reinforced metal matrix composites exhibit some favorable attributes such as

t*e enhanced stiffness and reduced CTE, their one principal drawback is the

s.'.:stantially reduced damage tolerance relative to the matrix material.

A novel fabrication approach that introduced a toughening mechanism into

t're discontinuously reinforced metals has been developed recently at United

Terhnologies Research Center (UTRC) [5]. A schematic microstructure of such a

"n:icrostructurally toughened" (MT) composite is shown in Figure 1. The energy

a-sorption capability of the MT composites is enhanced by the following two

mezhanisms. First, when cracks encounter interfacial areas between the

reinforced and toughening regions, debonding at the interface can result in

crack tip blunting. Second, the inherently higher energy absorption capability

of the toughening region inhibits crack propagation.

A MT composite consisting of silicon carbide particulate reinforced 6061

a xminum, with either aluminum or titanium tubings as the toughening region

material was selected as a model system to evaluate the merits of the

t:2ghening approach. Note that the MT composite system with aluminum tubings

as the toughening region material has been shown to have a fracture toughness

rlnging frcm 5 to 12 ft-lbs, as compared to 0.8 ft-lbs for the particulate

reinforced material without toughening mechanism [5].

The primary goal of this study was to support the ongoing development of

the MT composites at UTRC through better understanding of the merits of the

t:.:ghening mechanism at the micromechanics level, and to develop a design

::-hodology for MT composites that can take full advantagE of the unique

c' irac'fEristics of these materials. This goal wvs met b' compiling a d-sen

d--abase of "T composite material properties; analyzing th. material properties

using micromechanical and composite material models to ':.n±rstand the measurE~d

n-erial stress-strain behavior and in.pact damage resista:<e' reviewing SDI

L

stictualcompol~ents and requirements to select attracti-.- components, and

d&e-eloping preliminary material and structural designs.

OBJECTIVES

The 'T composites developed recently at UTRC have successful>v

decOnstrated that the fracture toughness of the discontiniously reinforced

metals can be greatly improved if certain toughening mechanisms can be

introduced into the material microstzuctures [5]. The primary goal of this

study was to support the ongoing development of the MT composites through

better understanding of the merits of the toughening mechanism at tht.

micromechanics level, and to develop a design methodology for SDI structural

apzlications that car take full advantage of the unique characteristics of MT

conposites.

The goal was achieved by the development of two micromechanical material

models. The first model addresses the stress-strain constitutive modeling of

MT composites based upon the material properties and volume fractions of the

constituents. No considerations regarding geometric shapes and dimensions of

the toughening region were given in this model. The model was used strictly to

1 predict the material stiffness or compliance matrix which was required for

stress analysis of structural components made of MT composites.

It was well documented in [5] that while the actual dimensions and shape

of the toughcning region rarely affected the stress-strain behavior of MT

co:nosites, they did influence the ability of the composites to withstand

inract damage. Therefore, a second model was developed to address the

possibility of predicting the fracture toughness of the MT composites by taking

the geometric factors into account.

A submarine hull and an aircraft longeron were identified in this study as

possible components that can be improved through the use of MT composites. A

preliminary design of the aircraft longeron was performed to demonstrate a

design methodology for MT composites.

3

APPROACH

Evidence in UTRC testing results suggested that the ess-strain behavior

the impact damage absorption capability of the MT co:.:.- tes were governed

bv different sets of material and geometric variables. T.ezefore, two

micromechanical models were developed to study these two phenomena separately.

The stress-strain model was further incorporated into an e:j:isting MSC computer

code "EPLA4M" which was developed earlier by MSC staff for elastoplastic

incremental laminate analysis [61. This revised analysis :.ves us the ability

to deal with laminated MT composites which is expected to e an important step

in the future material development. The theoretical basis for the models are

discussed in the folowing sections. A description of the structural component

selection is also incluued.

STRESS-STRAIN CONSTITUTIVE MODELING

From a microstructural standpoint, the MT composites can be treated as

continuous fiber reinforced composites, with the reinforced regions being

analogous to the fibers (fig. 2). Two existing material irdels, composite

cyl-inders assemblage model (CCA) [7], and vanishing fiber f.Lameter model (VFD)

S', have been applied successfully to describe elastoplasric behaviors of

fiber reinforced metal matrix composites. However, our past modeling

eterience indicated that CCA model might not be suitable for evaluating MT

ccmposite properties. The reason being CCA model always imposes a perfect

bcnding between constituent phases while MT composite pro;erties are strongly

affected by the bonding strength of the interface which ccld be rather low in

practice. Of course, a modified three-phase CCA model with the interface

ratzion being the 3rd phase can be constructed [9]. However, the mathematical

ccmplexity of this new model is beyond the scope of our current study.

Due to this consideration, VFD model was chosen for tie purpose of stress-

strain modeling. It should be noted that we also recomnired the fact that VFD

-::el alwavs assumes uniform transverse stress distributi- in the phases, and

-s is not suitable for modeling '-IT composites either. :--.ever, the

r-:lic'tv of the model and the fact that thE, m-odel generlly gives very

_:rate prediction in axial stress-strain response make "FD model a desirable

:.--.:ce for initial material modeling.

The existing formulation for VFD model in reference 8 was derived for

7-cc fing fibrous composites with elastic fibers embedded in ductile metal

natrix, and the materials were exposed to pure mechanical loads. Since the

reinforced region of MT composites, which was to be modeled as fibers, could be

ductile in general; therefore, analytical effort was required to extend the

existing VFD model to account for the presence of ductile fibers. In addition,

to analyze MT composites for high temperature applications, the modeling should

als include thermal effects. The mathematical formulation of this extended

model will be described next.

VFD model assumes the following equilibrium and compatibility equations:

do.. = doi = dc ' ) for ij 11, (1)

do11 = (m) + Cfdc(f) (2)m 1

dE.. = c. dE(m) + c de for ij 5 11, (3)ij m ij f ij

(f) = dE(in) (4)dE1 1 = dE11 1 1

do. . and dE.. are the uniform stress and strain increments applied to the

cczc-osite. The index (f) or (m) is used to denote fiber or matrix related

quantities. cf and c are volume fractions of the phases such that cf + Cm =

_. Fibers are assumed to be aligned along xj-dire-tion and x An d x ar- the

transverse directions.

As a step beyond the scope of existing VFD model, both fiber and matrix

-:serials are assumed to be elasticallv isotropic materials which obey Mises

-.:eld condition and Ziegler's kinematic hardening rule during plastic

_:)rmation [I0. To account for the thermal Effects, we further assume that

" elastic properties, as ..el as the viel, stresses of both constituents, are

-i erature dependent. Under such circumstances, the instantaneous_::-or-echanical responses of the constituents are dscribed by

.ef eT" .,ef (r (5)

+V.) -r

r = f for fibers. m for n:'ri:-

.eT f Mes S is- + 0

(-(r) -r rr)"

_(-(r) -r

Th- definitions of the vaiiables which appeared in the above equations are:

=~1 ~2 E 3 4 E5 ~6= " 1 1 2 2 U12 3 3 113 Y 2 ]

1 62 03 04 05 06]

= [011 022 T 1 2 633 i13 p23],

Mef = Elastic compliance at final temperature,

es=ElsiM = Elastic compliance at starting temperature,

c s Stresses at starting point.

fs6 = Change in free thermal expansion from starting

temperature to final temperature,

P= Total plastic strain increment,

M p Plastic compliance,

.pT Plastic strain increment due to temperature change.

e:.:pliit forms of rIp and ;p -ived in t,e Apendi.

Equation (5) can be recast ii:.to a i-.ore convenient form which is valid for

t-lastic and plastic loading c.ses.

.T-Cr) -(r) -(r) -(r)6

I

r f for fibers and m for matrix)

l .ef + 'tp ,M PM ±

A .eT + pT (10)1(r) = 1(r) O)"0

Substituting the constitutive relation (8) into equation (4) and makingW ( ) (M) "

:z of equations (1) and (2) to eliminate do. . and do. e:.:cept for do1] 1]

7h= resulkin form is. after rearranging the terms.

I I ) 6 (Mf)(M

dto I T B' do. + blj

11 *11 Cf.

B ( (MM< f)) / h: for j 2, 3. 4. 5, 6, (12)

(M) .T Tbi =( f) - 1iQ/h,

h = (Mli + - M11 Cf

Nri- that M . i = 1, 2, 3, 4 5, 6, are the first row components of the

defnedbhequtine().or( orin

oT the ortorof r f or ) defined by equation (10).

Now, equations (1) and (11) enable us to relate the matrix stress

Pi - ement i,_, to the overall composite stress increment j and tie temperature

ZIP , i 7 ]

(m I,1 12 ~ BM) B(m) Or)(IT 1 2 1i 3B1B 1 5 B16

0 1 0

o 0 1 0 0 0

0 0 0 1 0 0

o 0 0 0 1 0

0 0 0 0 1

b = b~m) 0 0 0 0 0-(mn) 1

Naturally, one can expect a similar relationship to exist for fibers,

!(f) = B(f) &±b f) (14

a binary material system, the local averages and overall stress and strain

i.-nrements are related by

a=c c(m + cf a( (15)- m -(in) f(f) ,

- m -(m) + cf (f). (16

S-stituting equations (13) and (14) into equation (15), one finds the

:-Alowing two relations,

c B + c B If) -Im-(;r f- -I 1

C b + c bf n. 1Q,in -(n f-f -'t

Te f is a K: identitv matrix and 0 is a 6:-I nul1 vector. Thus, B f : a1,y

can be found easily once B(m ) and b(m ) are know 'n.

Finally, if one writes the macroscopic composite stiess-strain relations

in the usual manner

S=M6+T

arf note that with the help of equations (8), (13), and (14). equation (16) can

be rewritten in a form comparable to equation (19). Hence, The composite

cc:7pliances M and iT can be identified as

M Cm -(m) -(m) + cf M(f) B(f) 20)

4E =c(M + +E )c (M b (21)m ((m) -(m)

+ -(m) f-(F) -(f) +(f)

In the case of pure elastic deformation of either phases, equations (20)

an- (21) are still applied providing that the compliances of the elastically

deformed phase or phases be set to zero. This is equivalent to letting H =

in equations (A-13) and (A-14) in the Appendix.

Equations 19 to 21 completely describe the instantaneous thermomechanical

st:ress-strain relations of the MT composites. However, despite the simplicityT

of the mathematical expressions, the compliances M and T have to be evaluated

nzzerically from equations (20) and (21). Because the plastic compliances of

thC phases, (M T as well as the factors (f and -(fim) are-(f,in)' (f (~i) B an)

de-endent upon the unknown stress averages of the phases at the final stress

state.

X;kLYSIS OF LAMINATED MT COMPOSITES

As noted previously, a computer code "EPLAM" was developed earlier by MSC

s:-ff to perform incremental elastoplastic stress analyses of laminated plates

To analyze laminated MT composites. the extended VFD model derived in the

:vious section -as programred into a subroutine that could provide the iciny

-:-ram "EPL '" with the instantaneous stress-strain relation of each micro-

_-cturallv toughened lamina. EPL.AM was then able to perform a nonlinear

:.inate analysis to determine the overall behavior of a laminated 'IT compesite

e.

9

The significance of this analysis is that it not onl' gives us the ability

t Ieal with laminated MT composites which is expected in the future plan of

- terial development, but also provides us with a convenie:.: tool to analvze a

::: composite with sparsely populated touhening tubes. In this case, the MT

ccposite can be idealized as a system of alternating layers of toughening

tubes and particulate reinforced region (fig. 2b).

M, ELING OF IMPACT DAMAGE AND WORK OF FRACTURE

The high fracture toughness is the most important advantage that the MT

ccmposites have over the conventional particulate reinforced metal matrix

ccposites. The impact energy absorption data of the MT composites were

o-:ained in [5] by conducting a series of instrumented Charpy impact tests.

shape and dimensions of the MT composite specimens used in the test are

sn'-n in Figure 3. A detailed description of the testing procedure can be

fc nd in [5].

The impact energy absorption data of SiCp/6061 Al based MT composites are

tabulated in Table 2 which includes three different toughening approaches,

nacely, (1) SiCp/6061 Al rods toughened by continuous 6061 Al region (fig. 4a),

(. SiCp/6061 Al continuous region toughened by 6061 Al tubings (fig. 4b), (3)

S --p/6061 Al continuous region toughened by CP titanium tubings (fig. 4c).

Table 2 clearly indicates that changing the material composition as well

as varying the geometric shapes and dimensions of the toughening region can

have profound effects on the impact damage absorption capabilities of the MT

cczposites. Therefore, modeling effort should account for both the material

pr perties and the geometric details. This is in contrast to the usual

pr ctice of modeling fibrous composites in which the geometric details other

t =n the volume fractions of the phases are ignored. To make the matter even

F -e complicated, the impact damage is dynamic in nature and there is no doubt

the modeling of dynamic effects on elastoplastic hetezogeneous materials

I entail tremendous difficulties.

In this study. we have chosen a different approach. ".-e have made an

-: elpt to find a quantitative criterion for .-:hether or no- the touhening

-- hanism could become operative during impact damage process. i'e believe that

x--s criterion will be essential in achieving a successful MIT composite design

10

that the toughening effects provided by the toughenir.7 region materials

be fully utilized. The treatment is by no means rigor- s but it has the

n antage of producing a quantitative criterion in terms c- the material and

,e netric variables of the MIT composites. This modeling a-tempt will be

described next.

An extremely simple case of a simply supported composite beam under

plastic collapse load is considered in Figure 5a. Recall that both phases of

the MT composite are ductile materials, therefore, the collapse load as

ca=-culated by plastic limit analysis of beams can be written as [111:

F = 2M / . (22)clp y

:s one-half of the beam span, and M is the yield moment which, if both theYreinforced and toughening regions are uniformly distributed over the cross-

sectional area, can be calculated as:

ab2 V a(r) + V (t)y 4 (r) ult (t) ult(23)

where a,b = dimensions of the cross-sectional area as shown

in Figure 5b,

V(r), V(t ) = volume fractions of the reinforced and toughening

regions, respectively,

ultr)' u(t) = the material ultimate strengths of the reinforced

and toughening regions, respeztively.

it should be noted that in this section the indices (r) anri (t) are used to

_-.ote the reinforced region and the toughening region, rE-_pectivelv.

Assuming the final failure occurs when the mid-span c-flection of the heai

r .ches a maximum value L. then, the energy absorbed by t-. beam up to the ti:re

of failure is equal to the external work done by the colli-se load, i.e..

11

W= F *A (24)

:he principle assumption in this treatment is that we ass-_- the same beam will

have absorbed the same amount of energy at failure should :- be subject to

impact loads. At present, there is no experimental evider.:e available to

support or dispute this assumption. However, by judging fr-m the fact that the

strength of most structural materials increases at high strain rates, the MT

composites possibly can absorb more energy in impact than in static loads.

As a first order estimate of the maximum beam deflec: n A at mid-span. we

adopt the strain-deflection relation that can be derived f:om an elastic

deflection curve, i.e.,

3bE = 3 A , at center (z=O). (25)max 22

21

Making use of equation (25), one can define two limiting 7id-span deflections,

A (r) 3b) (26)A =E /( )max 22

A 2 (t) 3 2) (27)max 22

where E (r) and (t) are the tensile failure strains of the reinforced andmax max

toughening region materials, respectively.

It can be understood that A1 is the mid-span deflecti:n at which the

outermost reinforced region material begins to fail, and L- is the mid-span

deflection at which the outermost toughening region material begins to fail.

Since c (t) > f(r) is always true for MT composites, the ou:ermost reinforcedmax max

rezion material will fail first. At this initial stage of failure, the energy

absorbed by the beam is

W= F • (28)clp1

12

Next, we should investigate how the initial failure c:: the reinforced

rezion could affect the integricy of the outermost toughering region material.

A schematic representation of this interaction between thV- tLo regions is

sh.wn in Figure 6. If two regions are totally disbonded as shown in Figure 6a,

the failing of reinforced material can result in a elastic unloading and the

reinforced region will shrink by an amount equal to This motion of

shrinkage is toward the end-support of the beam and it does not drag the

toughening region to go along because the two regions are totally disbonded.

However, if the two regions are bonded, an interfacial shear zone will

de.elop over a distance 2o along the interface (fig. 6b). The length of the

shrear zone can be determined by a simple shear lag model a7-d by assuming the

shear stress in the shear zone is uniform and is equal to r . If one chooses

to match the axial displacements U(r ) and U(t ) at z= o, on e has

rS2o

(r) - s (E-20)(r) ult A(r) Er)

(29)r S.o

S(t) A ES (t)(r)

where

S perimeter of the interface bo.ndary

A (r) A(t cross-sectional

areas of the reinforced and

toughening regions, respectiv'ely

Equating U(r ) with U(t ) gives

(r) ()T(r)_o__C (h + ~ (30)

r e

C = S / A(t). 31)

Si:plified expressions of C for three simple cases can be found in Figure

Sr.ce o can not be exceeding , therefore, xo is the mriLum value of the t:.o

13

()E2o Min. r ult.(_ L . -1 32)s7- E(t) (r)

Consequently, the interfacial shear stress causes tzn toughening region to

shorten by an amount equal to

Cr2 = E [oo( -)](33)

(t) 2

To maintain the integrity of the toughening region, an adtfitional plastic flow

must be generated in the toughening region material in order to compensate for

this shortening. Since for z>2o, the toughening region will be unloaded back

to elastic state by the interfacial shear stress, the plastic flow can occur

only within the O<z<2o segment. The average plastic strain increment within

this segment is thus

Cr (r)s (r) ult Ro <

(t) E t 2(V (rE +V E )Ae = A2/2o = (t) r)(r) (t)E(t) (34)P Cr 2

s if 2o =

2E(t)

It is now clear that the initial failure of the outermost reinforced

region material would result in an additional plastic strain increment, i.e.,

Lf(t) in equation (34), in the outermost toughening region material.p

Ccnsequently, the toughening region material could fail a: once if

(r) + (t) (t)E + ~ . (35)max p max

If this is the case, the beam will fail when the outermos- reinforced region

,n3:erial starts to fail and, therefore, no toughening effE:ts can be realized.

Tie fracture energy of the beam in this case is equal to (equation 28'

On the other hand, the beam will continue to deform the toughening

rion is able to withstand this additional straining. i.e.

(r) (t) (t)max p max

14

instead of reaching a mid-span deflection A 2 as calculated b-. equation

(2-), the outermost toughening region material will fail when the mid-span

deflection of the beam reaches

(t) (t)) 3b . 3A = ( max '(p 23

The fracture energy of the beam in this case is then

W= F clp A (38)

where A is given by equation (37).

Equation (36) is the criterion for toughening mechanism to become active

during impact damage process. If it does become active, the fracture toughness

of the MT composite will increase from W, (equation 28) to W (equation 38).

STRUCTURAL COMPONENT SELECTION

Two structural components were identified as possible designs that can be

improved through the use of MT composites. The selection was based upon a

survey of structural components that are of interest to SM1, and also upon

interviews with SDI, ONR and UTRC personnel. These selected structural

components are a submarine hull and the lower longeron for the Supportable

Hybrid Fighter Structure (SHFS).

Aircraft Longeron

The lower longeron, shown in Figures 8 and 9, is part of a fighter

a:'rcraft fuselage which is located inmmediately forward of the wint carry-

t .r-ough structure. It is loaded primarilv yben.ing ab-: is designed

tc meet limit load requirements, ultimate load requiremerts, and a specified

-ligue lifetime. The component geom.etry and loading cor.iLti .s -ere provided

- General Dynamics, Fort Worth Division.

15

The present design calls for the structure to be made from hc:ogeneous

Si: Al material. This may prove to be an attractive applicant for MT

.c:o7--sites because of the life time requirements and the fact that it is a

re13tively large component. The preliminary design of this component will be

discussed later.

Submarine Hull

Another area of current Navy research has focused on the application of

com-osite materials to submersible structures. These structures can be

classified, by loading environment, as pressure hull and non-pressure hull

structures. Both classes of structure are subjected to a seawater environment,

ho;;ever the pressure hull structure defines the water tight envelope and must

support the external pressure induced by submergence. Candidate pressure hull

materials must have a very high modulus to preclude elastic stability failures.

Graphite epoxy materials are emerging as the leading candidate for composite

submersible pressure hull applications.

Non-pressure hull structures are free-flooding or water backed structures

subjected to working loads other than submergence pressure. These structures

provide support for external propulsion and sonar systems. and ser-e as ballast

tank (bow and stern) structures. This class of structure also includes the

hydrodynamic appendages, or bow and stern steering and diving planes.

Microstructurally toughened SiC/Al composite materials may be a high pay-off

candidate for non-pressure hull structure applications.

Non-pressure hull structures have traditionally been designed to include

the effects of post-yield strength of the material. The high yield strength

and excellent ductility of the microstructurally toughened SiC/Al composites

mar-e these materials very attractive for non-pressure hull applications.

',.:ei-t is a major design constraint because of the location (at the extreme for

aft ends of the submersible) of non-pressure hull structures. The low

dez _tv of the SiC/Al composite (less than half of that of steel) :ould result

- structural weight reduction which could translate to increased equipment

-oad. speed or diving depth. Maintainability is a paramount concern in the

-st tank area, therefore the non-corrosive nature of Sic is a dist!_.

d'--ntage over steel. The non-magnetic signature of SiC would also be a

16

b~<fitif acoustic detectability or stealth were a consideration (i.e.

cc--'atant submarines).

17

RESULTS

The results presented here include discussions of the material properties:

predictions of stress-strain constitutive relations; the analyses of

laz.inated MT composites; the estimations of impact damage and work of fracture;

t'-e optimization of impact damage resistance; and the initial results of the

preliminary design for the SHFS lower longeron.

.-. ERIAL PROPERTIES

The base material for thp MT composites studied in this report is the SiC

pr:ticulate reinforced Al 6061 material. Three toughening approaches were

i-.-estigated in this report (see Figure 4). They are: (1) a continuous

aiz-ninum 6061 toughening region, (2) aluminum 6061 toughening tubes, and, (3)

C? titanium toughening tubes. The properties of each individual material were

o'_ained in [5] by tensile testing. It should be noted that prior to tensile

testing heat treatment and consolidation were applied to the precursor

materials in the identical manner as in the extrusion process of the MT

ccmposites.

A summary of the material property data was provided in [5] and has been

retroduced here in Table 1. Note that there are three SiCp/AI 6061 materials

in Table 1. They differ in the volume fractions of SiC particles and also

differ in the way that they were fabricated. The specific compositions and

ge:metries of the MT composites studied in this report are tabulated in Table

2. The Table shows a total of 8 MT composites, and each material has been

designated a material number. For simplicity, the individual MT composite will

be referenced by its designated number.

ST7ESS-STRAIN CONSTITUTIVE RELATIONS

The extended VFD model :as used to calculate the axia stress-strai

reations of the MT composites. For simplicity, the nonlinear stress-stra-r

-- viors of the constituents were idealized by using an elastic-linear straln

> :dening law which could be represented by a bilinear cur-.'e as the one. that is

s'-.wn in Figure 10.

18

Figures 11 and 12 are the predicted axial stress-strain curves of the No.

2 .nd No. 4 MT composites. Both materials apparently failed when the

reinforced region materials reached their respective maxi.us strains.

T). refore, the tensile strengths of both MT composites did not benefit from the

presence of toughening region material which was Al 6061 in both cases. This

is probably due to the fact that Al 6061 has lower tensile strength than the

Sip/Al 6061 material. Hence, when the relatively brittle SiCp/Al 6061 started

to fail, the load transferring process simply overstressed the remaining Al

6CM1 material.

It is reasonable to assume that in order to improve the ductility and

static tensile strength of MT composites one has to select a toughening region

rrn-terial which has higher tensile strength than the reinforced region material.

A good example is the CP titanium toughening tubes that were used in the Nos.

6. 7 and 8 MT composites. It can be seen from Table 2 that the failure

strengths and strains of these three MT composites all shoved improvement over

thre corresponding values of the reinforced region material which was the UTRC

fabricated SiCp/Al 6061 material. In fact, the impact damage absorption

capabilities of MT composites will also be enhanced by using toughening

materials which have higher tensile strengths. This will be discussed in more

detail later.

The influence of interfacial bonding strength on the tensile strength of

th-e MT composites can be observed from Table 2 as well. Both No. 2 and No. 3

M7 composites had samples prepared by different surface treating methods. The

acid cleaning method was used to promote strong interfacial bonding while the

acid cleaning plus debonding agent was used to promote weak interfacial bonds.

The samples with strong interfacial bonds were shown to have slightly higher

tensile strengths and tensile failure strains.

The indication is that strong bonding allows for stress transferring

pr.cess to be carried out more efficiently in a cracked MT composite svstem.

T'- bridging effect works in the same manner as in the case of fiber reinforced

---osites containing matrix crncks. However. the effect is quite small in the

cZe of MT composites where the ultimate strengths of the constituents are not

-'-t much different. Consequently, a detailed study of this effect was

e-.zluded from the limited Phase I work plan.

19

Figure 13 is the predicted axial shear stress-strair. :ehavior of the No. 2

MT composite. No experimental data point was plotted wit- this curve becau,e

t'.zre is no data currently available. This illustrated -. i usefulness of

ma:erial modeling because unavailable material properties : n always be

es:mated. In addition, the effects of alternate materiail and volume

fractions can also be evaluated with ease by material modling.

For MT composites with low concentration of tougheni.- tubes, there might

be a considerable amount of reinforced region material ga:±ering in between

ad'acent tubes. An incremental laminate analysis was madc available to cope

with this situation (see Figure 2b). On the other hand, ::-s material could

also be approximated by a single cylindrical assemblage b.. with a larger O.D.

such that all the reinforced region material can be accorr=:dated within this

en-arged tube.

These two different approaches were used to analvze :he axial stress-

strain behavior of the No. 6 MT composite. But, instead cf using 0.51 for the

volume fraction of the toughening tubes, a lower volume fr=ction of 30% was

selected to better represent the case of low volume conce-tration of toughening

tues. Figure 14 shows that axial stress-strain curves F7rdicted by both

a -roaches do not have significant differences. This implies that the laminate

analysis is not needed if the No. 6 MT composite has more :han 30% volume

fraction of toughening tubes. However, this conclusion r:iht not be valid for

M7 composites with different material compositions.

L.!INATED MT COMPOSITES

As shown previously, the laminate analysis has given -s a tool to deal

with MT composites which might have low volume concentra:ion of toughening

tubes. In addition, this analysis also gives us the ability to analyze

laminated MT composites. Figure 15 shows the comparison :etween three

di.fferent laminations of the No. 2 MT composite. The vet much isotropic

-ire of the No. 2 material renders the similarity of a'_ three a.:ial stret-

st:ain curves.

It appears that there is very little to gain by r tinz the MT

c:mposites. However, as will be discussed in the folio;:>: section. The i r:-ct

d-.age resistance of MT composites can be improved by ha-irg a weaker bondin.

20

-- :ween the reinforced and toughening region materials. It is expected that a"..er interfacial bonding can severely degrade the transverse properties of

->MT composites. Therefore, lamination could .,ell be a viahie etiod for

z--ensating this degradation and, thus. allo-inc for the construction of a

hinh impact energy absorbing MT composite without sacrificing the integrity of

its transverse properties.

AN;ALYSES OF IMPACT DAMAGE AND WORK OF FRACTURE

A crude model for estimating the impact damage absorbing capability of "IT

cc:posites was developed in the Approach section. The initial goal was to

iz-ntify a quantitative criterion for whether or not the toughening mechanism

cc-ld become operative during impact damage process. Equation (36) was found

to be such a criterion. If the criterion is satisfied, the MT composite will

attain a fracture toughness which can be calculated from equation (38),

otherwise the fracture toughness will be given by equation (28).

However, the application of equation (38) requires a knowledge of the

interfacial shear strength r . Since r is unknown at present time, it can

or.v be estimated by a data correlation procedure.

The procedure is straightforward. By assuming r varies from 0 to 30 ksi.s

corresponding fracture toughness can be computed either from eq'uation (38)

o- from equation (28). A representative r can then be estimated hv

c,,rrelating the experimentally measured toughness data with the calculated

tcughness-Interfacial shear strength curve.

Figure 16 shows a typical toughness-interfacial shear strength curve. The

z-terial involved is the No. 1 MT composite. This curve indicates that for an

irterfacial shear strength greater than 5 ksi, initial failure of the

reinforced region material in this MT composite will always overstrain the

z.-:fhening region material to failure at the same time. In this c the

g .ni ng region material will be una ble to i.crease the tot,'hnfst o he -,1- 1 -i . Ol the t e h tJ if 'f , . -" t :.. h , , ! ,- ,-

--i to kelow 5 ksi 6 then. the tou }henin, recio:v will Sart to Ms

sence shown by increasing the toutihnes up ', a .a:-i1r:: f'1 c11 -

A tou-huess of 3.9 ft-lhs was tr,-,sured to- -his ,t ri, . Thi> v,. . i.

e' n lower than the lowest prediction in Figure iC -.lilch is sii ht v 1,igher

21

:-.,n 10 ksi This low toughness was probably due to the r.ttch effect that ca!:i

a direct consequence of the prenotched specimen. Since the impact damage

:-:iel does not account for notch effects, the only meanicf,-l conclusion tV:t

.. can draw from this data correlation is that the actual interfacial shear

strength was greater than 5 ksi. Thus, when the reinforced region material

failed prematurely due to notch effect, the overstraining brought the

toughening tubes to fail simultaneously.

Figures 17 and 18 show the data correlation results for the No. 2 and No.

SIT composites. Note that there were two toughness data points measured for

ear<h of these two MT composites. The pair of higher tougitess values were the

r1-sult of a weaker interfacial bonding for which a debonding agent was applied

ever the SiCp/6061 Al rod surfaces before the MT composites were fabricated.

::- relation to the higher toughness values, an interfacial shear strength of

:.5 ksi can be identified from both figures. However, the pair of measured lo'..

S tughness data are again lower than the lowest predictions on both figures.

The two conclusions which can be drawn from Figures I- and 18 are:

(1) Without debonding agent, the interfacial shear strengths were higher

than 5 ksi in both cases. Therefore, when the reinforced region

failed prematurely due to the notch effect, the%- brought the

toughening regions and the MT composites to fail at the same time.

(2) By applying the debonding agent, the interfacial shear strength was

brought down to 2.5 ksi. Due to this weaker bonding, the toughening

mechanism was allowed to be operative after the initial failure of

the reinforced region material had occurred. The result was a much

improved fracture toughness for each material.

When applying the same data correlation scheme to other MT composites, the

aiditional remarks can be made:

3 ' . *, and No. , -'i T pcsl tes for .lc- * . ] .. 1'--. ... . -

toUtrhenint ren ion :aterial, the interfacial sT,:" ssi -i. F I h .as iiU-

than 5 ks i be.,!-, the oirhening mech sn.......rb1e :o aa.co,: e

np, rar i'.e and tio :v:te,-ia as a wt~oK- T eitd ':~.:- t , r-, > f,-':ced

region material failed due to the notch effect (see Figures 19 and

20).

(4) No. 6 and No. 7 MT composites for which the CP titanium tubings were

the toughening region material, the interfacial s-.ear strength can be

found to be 7 ksi in -elation to the measured toughness values (see

Figures 21 and 22).

Note that for the MT composites mentioned in the last two remarks, no

dehonding agent was applied to the tubing surfaces to pro--te w-& w interfacial

b::.ding. Nevertheless, the CP titanium tues t-ee stil ,We nr,,ise the

m.:erial toughness to a higher level while the Al '<"- :- ;,.d to lend any

tc:ghening effects to the reinforced region mW.tt r :, P . . " . ,,,in- tubes

also improved the static tensile strength of XT ,. . - tubes

aazin failed to do the same. It appears niow tr., .

ct.cice for the toughening region material.

OBTIMIZATION OF IMPACT DAMAGE RESISTANCE

What we have learned from the previous data correlatic-.: stud" is that our

cr-de model seemed to be able to furnish reasonable agreement with the test

data providing hat the interfacial shear strengths that were estimated by the

mc:el can be validated. In addition, we also learned that CP titanium is a

be-ter choice for tougnening region material than Al 6061. However, even with

C7 titanium as the toughening region material, further improvement of fracture

tc :ghness is still possible. In the following we will use the No. 6 MT

ccposite as an example to demonstrate various ways of improving its fracture

tc -ghness.

After a careful examination of equations (32) to (38) and a parametric

s-.:dv of this damage model, four possible ways of improvint the material.,hness %:ere identified. Figure 23 shows an improved fracture toughness if

7 can increase the Yoing's modulus of titaniium from 1-5.5 s to 28 "Isi. The

"--son for this improvement is that a stiffer toughening r ion material can

-ice the additional plastic straining that is brou ht upon b" e

:--erfacial shear stress (see eqlation 34).

23

Figure 24 shows similar improvement of fracture toughness if one can

i:_-rease the ultimate tensile strain of the titanium from 19% to 254. The

re-.son for this improvement is that the material can now e:.:perience larger

d-Z!ection before the final failure occurs (see equation 3-). It is worthwhile

tc point out that the titanium alloys can be tailored to achieve higher Young's

modulus on the expense of a reduced ultimate tensile strain. Therefore, it is

pcssible to use our model to find the best combination for these two variables

such that the fracture toughness of the material can be optimized.

Aside from varying material properties, it is also possible to achieve

h-her fracture toughness by varying the geometries. Figure 25 shows the

improved fracture toughness if one uses smaller but thicker titanium tubes such

tat the tube volume fraction remains unchanged. The use of thicker tubes

implies a smaller value for the parameter C (see Figure 7a). A smaller C can

reduce the additional plastic straining which would be brought upon the

tcugL,- Lng region material by the interfacial shear stress (see equation 34).

Therefore, a smaller parameter C promotes higher fracture toughness.

At the limit of a smaller but thickened tube, the tube becomes a solid

rcd. Figure 26 shows great improvement by using solid CP titanium rods instead

of" tubes. The improvement can be realized even when the interfacial shear

s:rength is higher than 30 ksi. It is interesting to point out that recent

UTRC experiments have confirmed this prediction. In their result (as shown in

Table 3), 1/8 in. CP Ti rods were shown to have given the MT composite higher

tcughness than the 1/4 in. CP Ti tubes could. This result was achieved despite

the fact that the volume fraction of the toughening region was only 25% for the

solid rods and 30% for the tubes.

PR.ELIMINARY COMPONENT DESIGN

Recall that the lower longeron for the Supportable Hybrid Fighter

S-ructure (SHFS), as shown in Figures 8 and 9. was selected as a potential

-Ictural component that can be i-t1od th-ouEh th c u: e cf MT conr:sites. A

:7'eliminarv design of this component will be described next. The component

:- retrv and loadiTg conditions were provided bv Ceneral Dv:namics. Fort Worth

I -sion.

24

The 19 design loading cases, shown in Table 4, were used to compute the

';::mate stresses shown in Table 5. The maiximum and minir-: normal stresses

each loading case were determined by

axial load bending load (39)a(max, min) 0 (max, min) (max, min)

whereaaxial load axial load (max, min) / cross-sectional area,(40)(max, min)

bending load Mix + M2 y 41)o = I I

yp xp

M I , M 2 = the bending moments about the prin.Zipal axes

yp and xp, respectively;

I , I = the moment of inertias with restect to the

principal axes xp and y p, respe:tively.

Eq*ation (41) was used to compute the ultimate bending stresses at the four

ccrner points, A, B, C, and D of the component cross-secti:nal area where the

m.a:,:imum and minimum bending stresses of the beam were most likely to occur (see

Fizure 9). The maximum and minimum values among the ultimate stresses at the

fcur corner points were then selected as the maximum and minimum bending

stresses of the beam.

The shearing stresses are of secondary concern. With a simple assumption

that all of shear-i was resisted by the area of the bottom flange and all of

the shear-2 was resisted by the area of the right hand flange, the maximum

shearing stress of the section was evaluated for each loading case. The

results in Table 5 indicate that the shearing stresses are generally one order

ot agnitude smaller than the normal stresses.

From Table 5, the maximum tensile stress in the longe:'-n is about 83 ksi

--n- the maximum compressive stress is about 101 ksi. These strengths are

-.-:-inabIE with a 35 volume percent SiCp/2124 SiCd/Al com:-ites, reference 12.

refore. a discontinuously, reinforced metal matrix corpceite can be used to

25

c::-struct a longeron which :ill meet the requirement of u'-imate static

s:rengths.

As for the fracture toughness consideration, a tougen.ing mechanism

c'=posed of CP Ti solid rods was shown previously to have 7reatly enhanced the

fracture toughness cf the discontinuously reinforced SiCp Al 6061 materials.

T- addition, the relatively large size of the longeron beam will permit the use

of larger CP Ti rods. As can be seen from Figure 7b, the geometric parameter C

fzr toughening rods is inversely proportional to the rod diameter. Therefore,

i-creasing the size of the rod will reduce the parameter C and further increase

the fracture toughness of the material.

However, as far as static strength is concerned, the ultimate strength of

C7 titanium is 70 ksi which is lower than the required strength of the

'zngeron. Therefore, even though high strength SiCp/Al 6>11 material can be

cbtained by increasing the volume fraction of the particulate reinforcement,

t:-.e addition of CP Ti rods will decrease the material sta-ic strength and might

e-.-en make the material less ductile as explained in the seztion of stress-

strain predictions. It appears now the longeron construction will require a

r.ew toughening region material which should have a streng:h of at least 1002~.

This example demonstrated the need for further MT ma:erial development.

H:re suitable toughening region materials have to be ider:ified and tested.

Cr.e possible candidate is the titanium alloy Ti-6AI-4V which has an ultimate

strength of 163 ksi at room temperature.

26

CONCLUSIONS AND RECOM>!ENDATIONS

The results of this Phase I study have led to several important

cc:nclusions regarding the material design of MT composit,.s. This program

resulted in the creation of an extended VFD material model that was capable of

approximating the axial stress-strain response reasonably well. The model is

also capable of handling material nonlinear strain hardening, temperature

dependent material properties, and very general loading path which might

include thermal loads. The extended VFD model was further incorporated into an

existing MSC computer code which was originally developed for elastoplastic

l&Tinate analysis. This new software has given us the ability to analyze

l1aminated MT composites. The program also resulted in the development of a

crude impact damage model which was proven to be useful in the understanding of

the effects of toughening mechanisms at micromechanics level. A preliminary

design example was performed for an aircraft longeron beam, and the result

brought up the need for identifying more high strength toughening region

materials.

The stress-strain data correlation has suggested the following general

guideline for MT composite strength design. For higher ultimate strength and

mcre material ductility, the toughening region material should have higher

u timate strength as well as higher failure strain relative to the reinforced

region material. Strong interfacial bond also helps to raise the ultimate

strength and ductility of the MT composites. But the improvement is not

s: gnificant.

For achieving higher impact damage resistance, the impact damage data

ccrrelation has suggested the following guidelines for MT composite toughness

design.

(i From the material property aspect, the toughening region material should

higher ultimate strength, higher failure strain, and higher stiffness as

c Tpared to the reinforced region material.

I From the geometry aspect, the form of toughening region must eisure a small

,e-metric parameter C (see Equation 31). This implies that thicker tubes are

27

bEtter than thinner ones; solid rods are better than tubes: and larger rods are

better than smaller rods.

§ From the interfacial bonding aspect, weaker bonding tjrmotes much higher

fracture toughness. The improvement is so great that the slightly decreased

static strength and ductility of the MT composites as the results of a weak

ir.terfacial bonding can be considered justifiable.

The preliminary design of the lower longeron structural component showed

th-at a discontinuously reinforced SiC/Al 6061 material ca- be found to meet the

static strength requirements. However, an appropriate to::hening region

material for the longeron structure was found to require higher ultimate

strength than both aluminum 6061 and CP titanium can offer. Therefore, there

is a demand for identifying additional toughening region taterials with

ultimate strengths no less than 100 ksi.

The MT composites have many fascinating properties and a very promising

future. The Phase I study assessed the merits of the MT composites on a

micromechanics level. Several recommendations can be made which should be

considered for the Phase II program. Future efforts should focus on

characterizing the behavior of MT composites on a structural level. This will

include, as a minimum, studies of the effects of holes arnd other

diScontinuities, and the effects of general loading conditions including cyclic

lcads. A structural component will be selected. This cctponent might be the

longeron beam which was selected in Phase I study, or other more appropriate

cc7ponents. Detailed designs will be performed to specif.y the structure

configuration, material composition, the joining method which will be used to

attach the component to the main structure and finally. the fabrication and

testing of a subscale component to assess the design methodology.

28

REFERENCES

1. McDaniels, D. L., "Analysis of Stress-strain, Fracture and Ductility

Behavior of Aluminium Matrix Composites Containing Disc-ntinuous Silicon

Carbide Reinforcement", Metall. Trans., 16A, 1985, P.11>:5.

2. Schuster, D. M., and Skibo, M. D., "Low-Cost Discontinuously Reinforced

SiC-Al Fabricated by Casting", Proceedings of Sixth Metal Matrix Composites

Technology Conference, Naval Postgraduate School, Monterey, CA, May 1985,

p..27-1.

3. Heimann, T. H., Gubbay, J. D., and Hall, E. J., "Discot:inuously Reinforced

MiMC for Missile Guidance Hardware", Proceedings of Seventh Metal Matrix

Composites Technology Conference, NSWC, Silver Spring, M[aryland, May 1987,

p. 1 5 -1 .

4. DiGiovanni, P. R., Franceschi, L. and Rosenberg, E. M.. "Development of a

Discontinuously Reinforced SiC/Al Tactical Missile Wint", Proceedings of

Seventh Metal Matrix Composites Technology Conference, 'NSWC, Silver Spring,

Maryland, May 1987, p.1 7 -1 .

5. Nardone, V. C. and Strife, J. R., "Development of Micrcstructurally

Toughened Composites", UTRC Annual Report For ONR Contract N00014-87-C-

0406, August, 1989.

6. Chatterjee, S. N., and Wung, E. C. J., "Software Develcpment for Matrix

Composite Material Evaluation", Technical Final Report. MSC TFR 1816/8304,

Contract No. N00164-87-C-0233, WSC, Crane, Indiana, January 1988.

.arrett, D. J. , and Buesking. K. v. , "Temperature Deperfent Nonlinear MetalI-trix Laminate Behavior". NASA Contract Report 4016 c- NASA Contract NAS1-

_7822. September, 1986.

29

8. Dvorak, G. J., and Bahei-EI-Din, Y. A., "Plasticity Analysis of Fibrous

Composites", Journal of Applied Mechanics, Vol. 49. pp. 327-3%5, June,

1982.

9. Hashin, Z., "Thermoelastic Properties of Fiber Composites with Imperfect

Interface", Mechanics of Materials, 8, pp. 333-348, 1990.

10. Ziegler, H., "A Modification of Prager's Hardening Rule", Quarterly of

Applied Mathematics, Vol. 17, pp. 55-65, 1959.

11. Hodge, P.G., Jr., "Plastic Analysis of Structures", McGraw-Hill Book

Company, Inc., New York, 1959.

12. Misra, M. S. and Rawal, S. P. "Composite Materials for Space

Applications", NASI-18230, Martin Marietta report MCR-87-636, July 1988.

30

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Figure 1. Schematic Comparison Between Conventionaland Microstructurally Toughened ParticulateReinforced Composites

36

reinforced region(analogous to fiber)

toughening tube(analogous to matrix)

(a) Stress-Strain Constitutive Model

reinforced region

Toughening tube@ 00. i+ reinforced region

(b) Idealization Of MT Composites WithDilute ConcFentration Of Toughening Tubes

Figure 2. Micromechanical Models

37

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Figure 5. A Simply Supported Beam Under Plastic Collapse Load

40

: Shear Stresss Acting AlongThe Interface r

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APPENDIX

THERMOPLASTIC RESPONSE OF A HOMOGENEOUS SCLID

Assuming a material is elastically isotropic and obeys Mises yield

:.:ction and Ziegler's kinematic hardening rule during plastic deformation

- - , the yield function can be written as

f= 1 (sij - aij)(sij - aij) - IC2 (T) = 0 (A-l)

whEre

s.j, a.. = the deviatoric components of a.. and a.., respectively;

a.. = current stress state;iJ

a.. = current center of yield surface in stress space;

K(T) = yield stress in simple shear, and is assuzed to be a function

of temperature T.

Z'-gler's kinematic hardening rule specifies the kinematic motion of the yield

s. :-face during plastic deformation as

&ij = j4 (aij - ij ), (A-2a)

or. in the deviatoric stress space,

aij = it (s ij aij) (A-2b)

,::--.ere 4 is to be determined later.

The yield function, equation (A-1), remains equal to zero during plastic

-->ormation. Therefore, the following consistency equatic- has to be satisfied

.never plastic deformation takes place,

62

f = (Of/s .. - ( f/ja. .j + (3f/9 )k

(A-3)

= (s.. - a. .)(s. a..) 2ck 0.13 l 13 13

S P.stituting equation (A-2b) for the a.. in equation (A-3) and solving for 4.

o: obtains

M = [(sij - a.j)s ij - 2Kk]/(2K 2). (A-4)

13 2.3

-z te that the relation (si - aij) (si - ai) = 2K2 , i.e., equation (A-i), has

a..o been utilized to obtain (A-4).

The plastic strain increment, from the normality rule, is

A x (af/as ) = A (s. - a..). (A-5)Ij ij i i

Tc determine A, we follow Ziegler's assumption that the projection of plastic

s:rain increment on the exterior normal vector of the yield surface and the

projection of A.. on the same normal vector differ only by a constant c, i.e.,

c(afiasij )ij = (af/Bsij )6i.. (A-6)

=-rying out the differentiations and making use of the consistency equation

,A-3), equation (A-6) becomes

c(sij - aij) j = (s. . - a.j)j - 2Kk. (A-7)

For the case of simple tension in x, direction, (A-7) takes the form

p 2, 3 k). (A-8)~11 3c -

m- magnitude uf c can now be evaluated by comparing (A-8) .;ith the known

- rcmental s~ress-plastic strain relation in simple tension, i.e.,

p 03 1c). (A-9)

11 H6

63

-e H is the instantaneous hardening parameter. Thus, or.e has

c = 2H / 3. (A-10)

Now, A can be solved by substituting (A-5) and (A-10) into (A-7). The

result is

3X 4HX 2 [(Sk2 - ak9)k2 - 2,c]. (A-11)

Accordingly, the plastic strain increment follows from (A-5), with A taken from

(A-ll),

3j 4H [(S - ak2)k2 i 2K(sjj - a..). (A-12)

*, Equation (A-12) can be written in a matrix form comparable to equation

p = Mp & + ipT (7)

T*e instantaneous compliances M p and jpT can now be identified as

p 3 * * ** ** ** **- 4Hx2 (S S1S 2 2sls 3 sls4 2s1s, 2s 1 s 6

* * * * * ** **

(S2 )2 2s2s3 S2S4 2s 2s, 2S2 S6

(A-13)

4(S3)2 2s3s4 4S3S 5 4S3S 6

(s4 )2 2s4 s, 2s4s6

4(s ,2 41s"ss

Symm. 4(s 6 )

2

64

A

pT 3k * .' * * * (1- 2HK [s 1 2 2s3 s 4 26s 2sA

[ * * * * *

S S 2 S 3 S 4 55 S6]

* * * * * *

= [s 1 1 s 2 2 S 1 2 S 3 3 S 1 3 S"J,

s.. = s.. -a... (A-15)

65