Re1 axation of Residual Stresses

0. Vohringer, I n s t i t u t f u r Werkstoffkunde I , Universitat Karlsruhe (TH),FRG

1 . Introduction -

The residual s t resses , s e t up in components during manufacture or operatic

a re of considerable s ignif icance because of the e f fec t they may have on I

deformation and f a i l u r e behaviour of the component under mechanical load. 1

s t a b i l i t y of these residual s t resses i s thus of fundamental and practic

i n t e r e s t . Stable residual s t r e s s e s are frequently desirable , f o r example whc

compressive residual s t resses are de l ibera te ly s e t up in the surface regic

of s t ruc tura l components in order t o improve t h e i r fa t igue s t rength. On 1

other hand there are examples of the deleter ious e f f e c t s of s tab le residr

s t r e s s e s such as s t r e s s r e l i e f cracking, s t r e s s corrosion cracking , warping. If residual s t r e s s e s are relaxed under load they natural ly hc

l i t t l e i f any influence on subsequent component f a i l u r e .

Residual s t resses can be reduced or completely relieved by the de l iber ,

appl icat ion of mechanical and/or thermal energy. The e l a s t i c residual s t r a

E~ associated with the residual s t r e s s e s via Hooke's law can be converted i~

micro-pl a s t i c s t r a i n s E by su i tab le deformation processes. This transfor1 P

t i on can, fo r example, be achieved by dis locat ion s l i p , dis locat ion crel

grain boundary s l iding and diffusion creep. I f these processes occur t o onl.

l imited extent or not a t a l l , re laxat ion of residual s t resses i s a

conceivable by crack formation and propagation. Relaxation of res id

s t r e s s e s in the real case occurs by complex interact ion of a number

f a c t o r s . I t depends not only on the residual s t r e s s s t a t e i t s e l f but a l so

the s t a t e , on the condition of the load, on the geometry and the environm

of the materials.

The best known and most important techniques which bring about res id

s t ress - re laxa t ion ( c f . 1-11] are

annealing (tempering)

uniaxial deformation (drawing)

I cyclic deformation.

Also relaxat ion of residual s t resses can be caused (6,7) by

temperature cyc l ic s t ress ing

quenching

neutron bombardment

the e f fec t of a l ternat ing magnetic f i e l d s ( i n the use of

ferromagnetics)

vibration

part i a1 damage.

A di f fe ren t ia t ion i s made on the grounds of t h e i r d i f fe ren t technological

origin (11) between the residual s t resses produced during reshaping

machining

heat treatment

joi ni ng

surface treatment

casting.

In only very few cases ( c f . 7 , 8, 12-18) are the residual s t r e s s e s analysed

systematically. This i s understandable on account of the considerable experi-

mental e f f o r t i t e n t a i l s . Before the e f fec t of applied thermal or mechanical

energy on the residual s t r e s s s t a t e can be assessed, the l a t t e r must be

determined quant i ta t ively. In the case of mechanical techniques the necessar i- ly large number of measurements involve considerable expenditure of both time

and money. Most invest igat ions have therefore employed X-ray methods.

In the following review, which i s a revised and updated version of (1 I ,

current knowledge of residual s t ress-relaxat ion i s presented i n a condensed

and, as f a r as possible, systematic form. The abundance of avai lable e x p e n -

mental data nevertheless necessi ta tes a l imitat ion t o the scope of th-1s

review. Thus the f indings of residual s t ress-relaxat ion by annealing, uniaxial

and cycl ic deformation will be taken as examples which demonstrate universal ly

valid rules . Possible formulae which can be used t o quantify s t ress - re1 i e f are

discussed and the underlying structure-mechanical mechanisms considered.

2. Relaxation of residual s t resses by annealing

2.1. Thermally act ivated processes

T = 0.5 Tm [ K ]

where Tm i s the melting temperature, and then cooled slowly t o room tempc

t u r e , almost t o t a l relaxation of the residual s t resses a r i s ing from shap.

machining, heat treatment or joining operations can be achieved. The neces:

annealing time depends essen t ia l ly on the dimensions of the workpiece anc

the material s t a t e . Since an annealing temperature of 0 .5 Tm l i e s in

region of the rec rys ta l l i za t ion temperature, complete relaxation of resic

macrostresses can be expected. Residual microstresses are considerably red\

b u t not e n t i r e l y removed, since l a t t i c e defects , in par t i cu la r dis locat ic

- and in the case of heterogeneous mater ials the different expansion coel

c ien t s of the various phases a re always responsible ' f o r *ome resic

microstresses . Residual s t ress-relaxat ion by annealing i s brought about

socal led thermally activated processes f o r whsch the annealing temperature

duration are interchangeable within cer tain l imits . In order t o achi

comparable residual s t ress-relaxat ion a t a lower annealing temperature,

annealing time must be increased correspondingly.

Thermal residual s t ress-re1 axation ik fundamentally affected by the resic

s t r e s s s t a t e i t s e l f and by the material s t a t e . This i s convincingly demonst

ted by the f indings presented in Fig. 1 and 2 (18-201, which show t h e eff

of one hour anneals on the residual surface s t resses of a var iety of oric

in a variety of s tee l s measured using X-ray methods. Fig. 1 i s a bar cb

showing the residual s t resses present in the original s t a t e and a f t e r

normal indus t r ia l s t r e s s rel ieving f o r hardened components of 1 h a t 20C

The mater ials deformed in tension have undergone negligible residual s t re

re1 axation. The residual s t ress-re1 axation produced by annealing of surf

machined specimens (11 % f o r C k 45 ( c ) 17 Z fo r C k 45 (d l and 19 % f o r 103

i s the g rea te r , the greater the or iginal residual compressive s t resses .

most pronounced re1 at ive residual s t r s ss - re1 axati on i s , however, observed

the hardened mater ials (25 % fo r C k 22 and 37 % f o r C k 4 5 ) . Fig. 2 shows

re1 a t i v e residual s t resses oRs(T)/oRS(293K) f o r the same materi a1 s a! function of the homologous annealing temperature T/Tm. I t can be seen tha

one hour anneal a t 0 .5 Tm (about 600°C) resu l t s in every case in compl

relaxat ion of the macrostresses. Character is t ic , materi a1 -specif ic s t r

temperature curves are obtained. Clearly, residual s t r e s s produced If a material i s annealed f o r several hours a t a temperature of

200°C l l h tempered

Fig. 1 : Residual s t r e s s e s oRS of s t e e l s be fo re and a f t e r e m p e r i ng ( 200°C/l h ) a ) Ck 45, normalized and t e n s i l e s t r a i n e d ( E =5 % ) (18 ) b ) Ck 22, normalized and t e n s i l e s t r a i n e d ( ~ ~ 1 5 % ) (18 ) C ) C k 45, normalized and ground ( 1 9 ) P d ) Ck 45, normalized and ground ( 1 9 ) e ) 100 Cr 6 , quenched and shot peened ( 2 0 ) f ) Ck 22, quenched (18 ) g ) Ck 45, quenched (18 )

hardening a r e re1 axed a t lower temperatures, machining s t r e s s e s a t medium

temperatures and deformation s t r e s s e s a t higher tempera tures . In order t o

achieve t h e same degree of s t r e s s - r e l i e f , f o r example, 50 %, i n one hour i n

hardened and in deformed Ck 45, a temperature d i f f e r e n c e of 150°C i s necessary

( s e e a l s o Table 1 ) . Furthermore, according t o Fig . 1 , l a r g e r i n i t i a l r e s idua l

s t r e s s e s r e s u l t i n a s h i f t of t he "s-T curve t o lower anneal ing temperature

in t h e ca se of hardened o r machined ma te r i a l .

Residual mic ros t r e s se s can be assessed with t h e help of Fig . 3 i n which the

mic ros t r e s se s remaining a f t e r a 2 h anneal i n normalised C k 45, deformed in

tens ion o r ground, a r e p lo t t ed as a funct ion of tempera ture ( 2 1 ) . The

mic ros t r e s se s were determined from t h e half widths of t h e X-ray i n t e r f e r e n c e

l i n e s using a r e l a t i o n s h i p given in (181. Again, t h e r e s idua l mic ros t r e s se s

a r e more e f f e c t i v e l y re laxed a t a higher tempera ture and analogously t o t h e

Ternperoture I" oc 0 100 200 300 LOO 500 600

Fig. 2: Inf luence of homo1 ogous tempera ture ,Tif--n r e s idua l s t r e s s r a t i o O ~ ~ ( T ) / G ~ ~ ( Z ~ ~ K ) of Ttee l s , annealed one hour. a ) t o g ) a s i n Fig . 1

Temperature in O C 0 I00 200 300 LOO 590 600 t

Fig. 3: Inf luence of homologous temperature r e s idua l m i c r o s t r e s s r a t i o of norma-

l i zvd p l a in carbon s t e e l C k 45 i n a t e n s i l e deformed (18 ) and a mi 1 l ed and ground (21 ) s t a t e , annealed two hours

residual macrostresses ( c f . Fig. 2) those produced by deformation in tension.

In comparable mater ials , residual microstresses are relieved a f t e r a longer

period or at a higher temperature than residual macrostresses ( 1 8 ) . In order

to achieve 50 % residual microstress-relaxation in t e n s i l e - deformed C k 45,

fo r example, annealed f o r 1 h, the annealing temperature must be 100°C higher

than that needed t o achieve the same degree of relaxation from residual

macrostresses ( c f . a lso Table 1 ) .

Data of residual microstresses f o r hardened materials comparable with those in

Fig. 3 are not avai lable . Indications of the residual s t ress - re laxa t ion

behaviour are however given by the analyses of X-ray in te r fe rence l ines

produced by carbon s t e e l s i n (22) . The mean l a t t i c e d i s to r t ions < E % ' / ~ , which

a re proportional t o the residual microstresses, increase with increasing

carbon content due t o the increasing number of octahedrally dissolved carbon

atoms in the body centred tetragonal martensite l a t t i c e and a l so t o the

increasing dislocation density ( 2 3 ) . The greater the carbon content t h e lower

the temperature fo r the onset of residual s t ress-relaxat ion and the grea te r

the recovery ra te . These findings indicate that fo r hardened mater ia l s and

with increasing carbon content the relaxation of residual microstresses occurs

more rapidly and a t lower temperatures than that of residual macrostresses.

Qua l i t a t ive indications as t o the behaviour of residual microstresses on

annealing can be obtained d i rec t ly from measurements of the f u l l width a t half

maximum of X-ray interference l ines . For example, fo r materials shown in Fig.

3 , half widths a f t e r grinding are g rea te r than those a f te r t e n s i l e deformatim

(18, 21 1. Since, as shown in Fig. 4 , broadening of the interference l i n e s i s frequently accompanied by an increase in hardness i t seems l i k e l y t h a t the

temperature dependence of the relaxat ion of residual microstresses i n hardened

materials i s analogous t o t h a t of residual macrostresses ( see Fig. 2 ) . In t h i s

event the relaxation of microstresses in hardened materials should occur a t

lower temperatures, in machined materi a1 s a t medium and in deformed materi a l s

a t higher temperatures.

In order t o analyse the thermally activated processes responsible f o r s t r e s s -

relaxat ion, the influence of time as well as tha t of temperature must be

known. If the time and temperature ranges are suf f ic ien t ly r e s t r i c t e d and if

0 I 0 200 LOO 600 800 I000

Vickers Hardness

Fig. 4: Full width a t half maximum of X-ray d i f f rac t ion m f s t e e l s i n d i f fe ren t s t a t e s ( a f t e r (20,21,36))

G ~ ~ ( T ) / G ~ ~ ( Z ~ ~ K ) = constant (abbreviated in the following t o

re1 a t i onship

t = to exp ( Q / k T )

I 1 should hold between annealing time t and annealing temperature T. t o i s a

constant, k i s the Boltzmann constant and Q i s the act ivat ion energy f o r

t - T range under consideration. Data in the form of oRS-lgt diagrams in

I f o r deformed and hardened s t e e l s a re not suited to a d i r e c t evaluation u Eq. 2. The e f f e c t of time on s tress-relaxat ion can be l inearised with the

of a Zener-Wert-Avrami function (24) t o

a,s/~,,,o = exp [ - ( ~ t l ~ ]

where A i s a function dependent on material and temperature

A +exp ( - Q / k T )

and m i s a numerical term dependent on the dominant mechanism (24, 25)

2 ' Y m m N -

7 m m E z! n u- .r V) C w 7 L m

7 C 0

m aJ E C .r .r +

d m> C w .r - -6 m .r w e C C u m w

C rc w 0 w E

a m 0 i u w m t o m w r .? z! m u

r-- r c + + C O O , - ,--

V) I

rn c

I , , '

l i e on almost pe r f ec t ly s t r a i g h t l i n e s . Based on Eq. ( 4 ) and ( 5 )

which can be rearranged t o give E q . ( 2 ) . This y i e l d s t he a c t i v a t i o n energy Q f o r s t r e s s - r e l a x a t i o n . As seen from Table 1 , Q depends on t h e s t a t e of t h e

mater ia l and l i e s in t he range 1.15 t o 2.07 eV f o r s t e e l s , being about 2.8 eV

f o r TiA16V4. In t h e ca se of s t e e l s t h e a c t i v a t i o n energy i s lowest f o r t h e

r e l axa t ion of res idual s t r e s s e s due t o hardening and highes t f o r those due t o

deformation. The d i f f e r e n c e between t h e Q-values obtained on shot peened

TiA16V4 f o r t h e r e l axa t ion of macro and mic ros t r e s se s i s i n s i g n i f i c a n t . Table

1 a l s o shows t h e temperatures necessary f o r 50 ?: s t r e s s - r e l a x a t i o n a f t e r

a one hour anneal. Comparison with t h e corresponding a c t i v a t i o n ene rg i e s

proves unequivocally t h a t s t r e s s - r e 1 axat ion i n s t e e l s occurs by severa l

Ck 45 s t r a i n e d ( 5 % ) (macro-RS)

s t r a i n e d ( 5 % ) (mi cro-RS)

hardened (macro-RS)

C k 22 s t r a i n e d (5%) (macro-RS)

hardened (macro-RS)

TiA16V4 shot peened (macro-RS)

shot peened (micro-RS)

i[K] a t t 1 h and

jRS(T) /~Rs(293K)

: 0 . 5

ma te r i a l s t a t e

Table 1 : Resul ts of i n v e s t i g a t i o n s of thermal r e s idua l - s-relaxation ( eva lua t ed from ( l 8 , 2 3 ) )

oRS( 293K)

[~/mm* I

processes , t h e a c t i v a t i o n ene rg i e s of which inc rease with temperature. I

t i tanium a l loy c l e a r l y only one process i s ope ra t ive .

The a c t i v a t i o n ene rg i e s Q obta ined f o r s t r e s s - r e l i e f a r e shown i n Fi

r e l a t e d t o t h e a c t i v a t i o n ene rg i e s Q f o r s e l f d i f fu s ion o r high temper 0

creep as func t ion of t h e homologous temperature T1/2/Tm. With Qo = 2.6 e'

a-Fe (27 ) and 2.51 eV f o r a-Ti (281, Q/Qo l i e s i n t h e range 0 .4 t o 1

s i m i l a r range i s observed in recovery experiments (29 , 3 0 ) . These

a t t r i b u t e d t o a number of f r e q u e n t l y i n sepa rab le sof tening processes .

processes c l e a r l y a l s o dominate s t r e s s - r e l a x a t i o n and must t h e r e f o r

dependent on t h e ma te r i a l and t h e r e s idua l s t r e s s s t a t e .

C h a r a c t e r i s t i c s t r u c t u r a l changes occur dur ing t h e deformation, machinir

hardening of s t e e l s . Typ ica l ly , an inc rease i n t-he - d i s l o c a t i o n dens i ty change in t h e d i s l o c a t i o n arrangement a r e obser ted . In t h e ca se of hardc

t h e concentra t ion of d i s so lved i n t e r s t i t i a l atoms d i f f e r s from t h e e q u i l i t

va lue , and t h i s , t oge the r with t h e presence of d i s l o c a t i o n s , has a dec.

F ig . 8: Ac t iva t ion energy r a t i o of r e s idua l s t r e s s - r e l a x a t i o n of C k 22, C k 45 and TiA16V4 as a func t ion of homologous annealing tempera- t u r e a t t = 1 h and oRS(T)/o (293K) = 0.5 ( Q = a c t i v a t i o n energy of sB?f d i f f u s i o n , lmO= sol idus tempera ture)

i nf 1 uence on the residua

out in the temperature

dis locat ions adopt arral

1 s t ress f i e l d s . If a s ing le heat treatment i s carried

range corresponding t o recovery ( T < 0.5 T m ) the

~gements of lower energy by elementary processes such

as gl ide and cross s l i p by screw dis locat ions and gl ide and climb by edge

dis locat ions. I n the case of hardened mater ial , diffusion of carbon atoms

dependent on the annealing temperature o r s tage and accompanied by the

formation of character is t ic carbides i s superimposed on these processes.

The r a t e determining process, with the exception of the ear ly stages in the

annealing of hardened s tee l s (121, i s c lea r ly the thermally activated climb of

edge d i s loca t ions (31 1. If diffusion of matrix atoms occurs along the edge dis locat ions in the dislocation core, the ac t iva t ion energy should be Q = 0.5 Cb If bulk diffusion predominates the Q value f o r climb i s determined by the

act ivat ion energy of self diffusion. In the real case both procerses occur

simultaneously but to different degrees. The dis locat ion density and

arrangement are of considerable importance. In the case of randomly

dis tr ibuted dislocations or tangles of extremely high d e n s i t y , p t , f o r example

in hardened s t e e l s (Pt = lo1* t o ~ m - ~ ) s t ress - re laxa t ion i s expected t o involve di slocation-core diffusion-control led climb by Pdge dis locat ions.

Predominantly bulk diffusion will determine recovery i f the dis locat ion

configurations are relat ively s tab le and cons i s t of c e l l walls, sub grain and

low angle grain boundaries. This recovery process probably occurred in the

shot peened titanium alloy. In view of the act ivat ion energies, the

s t ress-re1 axation in the deformed s t e e l s can probably be c lass i f i ed between

these two extremes and occurred by two recovery mechanisms in competition with

one another.

According t o (12) stress-relaxation in hardened an t i - f r i c t ion bearing stee'l

during the ear ly stages of annealing i s due t o the diffusion of carbon t o

dis locat ions and the formation of Cottrel l clouds. This process, which i :;

associated with a reduction in the t e t ragona l i ty of the martensite l a t t i c e

should possess an activation energy Q.(Q. = act ivat ion energy f o r the 1 1

diffusion of i n t e r s t i t i a l atoms, f o r C in Fe,Qi = 0.9 eV).

With the exception of the early stages in the annealing of hardened s t e e l s ,

s t ress-relaxat ion in the temperature range corresponding t o recovery i s ii

consequence of the rearrangement of dis locat ions by g l ide , cross s l i p and

climb. The estimated value of the exponent of the Avrami function m = 0.3 ( Table 1 ) f o r s t e e l s and TiA16V4 should be considered in t h i s context. m-val

of the same magnitude are observed f o r the annealing of hardened s t e e l s (

and the formation of GP I zones in AlCu alloys ( 2 4 ) . In (25) t h i s m-value

a t t r ibu ted t o the e l a s t i c in te rac t ion between diffusing carbon atoms and pl

dis locat ion networks or walls. In the case of the examples under discussion

can be assumed from the v i r t u a l l y ident ical m-values tha t in the temperatu

time range under consideration a similar mechanism i s responsible

dis locat ion rearrangement and hence f o r s t ress-relaxat ion regardless of

condition of the mater ials . Since t h i s mechanism must be compatible with cl

by edge dis locat ions discussed above i t i s conceivable t h a t t h e proc

predominantly involves the el a s t i c interact ion of vacancies with e

dis locat ions capable of climb in ce l l walls.

Up t o now discussion has centred on s tress-relaxat ion a t temperatures T <

0.5Tm, i.e. tha t brought about by typical recovery processes. In t h i s c

mechanical parameters such as hardness and y ie ld point are not s ignif ican

a l te red . b r i n g a rec rys ta l l i za t ion anneal a t T > 0.5 Tm the d i s loca t density rapidly assumes very small values as a resu l t of the growth of

grains. This leads t o complete removal of residual macrostresses and t o v

small residual microstresses but i s associated with pronounced changes i n

mechanical propert ies . If extensive reduction i s required in the resid

s t resses i n a component without any s ignif icant change in y i e l d point

s t rength, the annealing temperature and time must be chosen t o correspond k

the recovery stage and not with recrystal l i za t ion .

2.2 Resistance t o s t ress - re laxa t ion

Stress-relaxat ion by heat treatment i s fundamentally impossible i f i

predominantly uniaxial residual s t r e s s s t a t e oRS i s smaller than the cr

resis tance. This res i s tance , designated in the following by R e c I t ,

characterized by l i y R Ep P , E P / ~ '

t h a t i s by the creep l imi t a t vanishir

small p l a s t i c deformation. As shown schematically in Fig. 9, RecIt decrei

with increasing temperature and load time. In contrast t o indus t r ia l CI

conditions only very short times are necessary f o r s t ress-relaxat ion by cr

processes (dis locat ion creep, grain boundary gl ide or diffusion creep) . I

0,

GRS

4, Ttz Tt, Ti3 Ti,

Fig. 9: Influence of temperature and time on resis tance of material RecIt resp. on creep proof s t r e s s R p E

P

increasing T and/or t, RecI t approaches a local ised residual s t r e s s peak of

magnitude oRS. For T = T t, . , cRS equals Re,/t associated with the local ised onset of creep deformation. Further increases in temperature or time r e s u l t in

an increasing and measurable microplast ic creep s t r a i n . As i l l u s t r a t e d i n Fig.

9, residual s t ress-relaxat ion begins a t higher temperatures Tti t h e smaller

the load time ti or the residual s t r e s s oRS and the greater the creep resis tance R of the mater ial . Changes in RecIt can be achieved tiy

del iberate a l t e ra t ions i n the s t a t e of the mater ial . All thermally s tab le

obstacles which have an additional work hardening e f fec t (32) s h i f t the onset

of stress-re1 axation and the e n t i r e oRS -T curve t o higher temperatures.

Stress- relaxation i s a1 so affected by the superposition on a loca l i sed

residual s t ress oRS of an external ly applied s t r e s s oL in the same d i rec t ion .

Increasing oL values s h i f t the onset of residual s t ress-relaxat ion t o shorter

times and/or lower temperatures. Applied s t r e s s e s acting in opposition t o the

residual s t resses in various regions of the material delay the onset of

microplastic deformation i n these regions. Since the d i s t r ibu t ion of residual s t resses in a component i s always inhomogeneous there will be other regions in

which superposition of oRS and d L i n the same sense promotes pl deformation. Thus an overall f a c i l i t a t i o n of residual s t ress - re laxa t ion c

expected even f o r applied s t resses in the opposing sense. If the res

s t r e s s aRS has been par t i a l ly reduced from i t s original value oRS,,

r e s u l t of microplastic s t r a i n , E the same arguments can be applied as P '

using the new value of the creep resis tance R P , E P / ~

( c f . Fig. 9 ) .

A t t h i s stage i t i s interest ing t o explain the r e s u l t shown i n Figs. 1

where residual s t ress-relaxat ion in a deformed sample of C k 45 i s delay

comparison t o that . - in C k 22 despite the compressive residual s t r e s s e s h

double the magnitude. Due t o i t s higher carbon concentration Ck 45 conta

g rea te r number of ferr i te-cementi te phase boundaries than C k 22. These

boundaries represent s tab le obstacles t o dis locat ion gl ide. They apbar

r e s t r i c t the p o s s i b i l i t i e s for movement of dis locat ion t o arrangement

lower energy required by the recovery process t o a greater degree than

promoted by the increased driving force of the greater residual s t r e s s

delayed residual s t ress-relaxat ion in deformed C k 45 i s thus *explained

increase in the creep resistance R e c / t '

In the case of a multiaxial residual s t r e s s s t a t e , the residual s t res !

employed in the arguments given above, must be replaced by an effel

residual s t r e s s oeIRS. If the principal components are o 1,RSY ' 2 , ~ s and

i t can be formulated t o

on the basis of the shape change energy hypothesis. The e f fec t ive res -

s t r e s s i s thus dependent o f ' t h e differences between the principal res .

s t r e s s components. These in the i r turn are proportional t o the shear strc

acting on dis locat ions in the s l i p systems. Stress-relaxat ion does

therefore occur f o r oe,RS < R e c / t but does fo r oe,RS 1. R e c / t . The case I Rec/t, obtains only br ie f ly since the immediate onset of creep deformi

attempts t o res to re the condition oe,RS = RecIt

2.3 The e f f e c t of the magnitude of the residual s t resses on s tress-relaxal

The data in Figs. 1 and 2 show c lear ly tha t the residual s t ress - re laxa t ic

machined or hardened s t e e l s occurs more quickly or a t lower temperatures,

greater the magnitude of the residual s t resses themselves. This i s a

consequence of temperature, time and s t r e s s dependent processes s imilar t o

those observed in socalled primary microcreep. Empirical re la t ionsh ips of the

form

can be establ ished f o r constant applied s t r e s s o ( 7 , 33) . B i s a quantity

depending on the temperature and the condition of the mater ial . For the

powers, p >> 1 and 0 < q < 1. The creep s t ra in increases with applied s t r e s s

and with time. Residual- stress-relaxation cannot, however, be compared

d i rec t ly t o a creep t e s t . I t i s much more l ike a s t ress - re laxa t ion experiment.

In the l a t t e r case t h e t o t a l s t ra in remains constant while e l a s t i c s t r a i n i s

transformed in to p l a s t i c s t r a i n . The following expressions hold t rue : -

and

If oRS,O i s based on real values of the residual s t r e s s , p l a s t i c s t r a i n of a t

maximum several t en ths of a percent are possible a f t e r completed residual

s t ress-relaxat ion ( o - O), i . e . the deformation i s in the microcreep range. RS -

Equations ( 8 ) and ( 9 ) form the basis fo r quant i t a t ive estimates of the

residual s t ress - re laxa t ion (33) . Mu1 t i ax ia l and inhomogeneous residual s t r e s s

s t a t e s are neglected or excluded. Although Eq. ( 8 ) cannot be subst i tuted

d i rec t ly in Eq. , ( 9 ) on account of the variable value of the s t r e s s ( a = o R S )

i t can be seen qua l i t a t ive ly from a combination of the two expressions that

increasing oRS values lead t o a more effect ive residual s t r e s s - relaxat ion. As

a resu l t of the g rea te r driving force, shorter times and/or lower temperatures

are necessary. This i s in agreement with the experimental r e s u l t s presented.

3. Residual s t ress - re1 axation by p las t i c deformation

3.1 Unidirectional deformation

a1 Character is t ic examples

In ce r ta in cases in pract ice, unidirectional deformation i s often employed in

addition t o s t r e s s f r e e annealing t o rel ieve residual s t r e s s e s . For example in

the case of shaping, the residual s t r e s s e s can be reduced by a second shapir

s tage using a smaller reduction in cross sectional area. This can be achiev~

by redrawing, re ro l l ing, repressing, s t raightening, mechanical polishing a1

shot peening (34, 35) . Apart from the l a s t , however, these techniques can I

used only on simply shaped components with a uniform cross section. In tl case of welded seams a unidirect ional load i s applied t o reduce I

r e d i s t r i b u t e internal s t resses ( 6 ) . When a c r i t i c a l value of the s t r e s s

exceeded directed dis locat ion movement transforms the e l a s t i c s t r a i n associ

ted with the residual s t r e s s in to p l a s t i c s t r a i n .

Several typical examples wi 11 serve t o i l l u s t r a t e the re1 axation of residu

s t r e s s e s due t o shaping, machining, heat treatment and joining by unidire

t ional deformation. Fig. 10 shows t h e e f f e c t of drawing ( 4 ) on the behaviol

of residual longitudinal compressive s t resses a t the surface of harden1

AlCuSiMg a1 loy ( type 2014-T6) and AlZnMg a1 loy (type 7075-T6). Considerab

residual s t ress-relaxat ion i s achieved up t o a deformation of 0.5 %. A residl

remains which i s unaffected by fur ther drawing.

Strain in %

Fig. 10: Relaxation of r o l l i n g residual s t resses - o f a n C u S i M g alloy (2014-T6) and an AlZnMg .alloy (7075-T6) due t o drawing ( 4 )

- L O O -300 -200 -100 0 -100 +ZOO *3W +LO0

Residual Stress G R S 1x1 in ~lrnrn '

Fig. 11 : Distr ibut ion of longitudinal residual s t resses a ( x ) over bending height z of a quenched aRi tempered p la in carbon s teel ( C 125) a f te r bending deformation (residual surface 1 ayer s t ra in 1~~ = 0,56 % ) and a f t e r reversed bending deformation (change of residual surface layer s t ra in I A E I = 0,57 % ) ( 3 7 )

R,r

AlCuSMgZ, shot peened A I

*i I

Stress In Nlrnrn2

Fig. 12: Relaxation of shot peening residual s t resses of an mCuMg al loy (AlCu5Mg2) due t o t e n s i l e resp. compressive 1 oadi ng

Straightening can also bring about residual s t ress-relaxat ion and pos8

red is t r ibu t ion . The dotted l i n e in Fig. 11 shows the residual s t r e s s prl

in a quenched and tempered carbon s teel C 125 produced by edge bending

residual edge s t r a i n of 0.56 % (37) . The specimen was subsequc

s traightened, i . e . deformed in the reverse direct ion with an opposing mc R unt i l a l i n e in the surface was op t ica l ly s t ra igh t ( I A E , I = 0.57 % ) . Thc

residual s t r e s s p r o f i l e i s shown by experimental t e s t points in Fig. 11

residual s t resses have been reduced t o about one t h i r d of t h e i r 1

following the primary deformation.

Fig. 12 shows an example of the relaxation of residual s t r e s s e s du

machining ( 3 8 ) . Shot peened specimens o f a precipi tat ion hardening A1Ct

al loy in an optimally h&dened condition with coherent p rec ip i ta tes bet

d i f f e r e n t l y under tensi 1 e and compressive load. T h e residual compre:

s t resses oRSS of about - 300 ~ / m m ' in the surface were reduced by about

of the y ie ld point of the non-peened mater ial , ReS = 390 ~/mm\ by a ter

load and by about 46 % of the y ie ld point by a compressive load, Resi

s t ress-relaxat ion occurs much more rapidly in tension than in compress

After a t o t a l s t r a i n E~ = 2 % the residual surface s t resses are comple

removed ( see l e f t hand diagram in Fig. 13). If E~ > 2 % the sign of

changes due t o work hardening. In the case of a compressive load the res i

AICuSMg2 shot peened

Fig. 13: Residual s t resses and f u l l width a t 6lfmaximum of shot peened AlCuMg2 as a function of to ta l deformation E~ due t o t e n s i l e resp. compressive loading (38)

Fig. 14:- Quenching residual s t r e s s r a t i o of s t e e l s and fcc bure metals as a function of r a t i o G / R _ , ,, ( a = t e n s i l e s t r e s s , R p O . O l = 0.01 % proof strfR$Y' ( a f t e r 4 1 3 ) )

s t ress-relaxat ion i s incomplete. Minimum values are observed in the f u l l width

a t half maximum of the X-ray interference l ines ( c f . r i g h t hand 'diagram in

Fig. 13) where the s t r e s s - r e l i e f r a t e i s greatest . The minimum of f u l l width

a t half maximum i s more pronounced in the case of compressive s t r e s s t h a n

t ens i le s t r e s s and occurs a t smaller E~ values.

Fig. 14 i l l u s t r a t e s the behaviour of pure fcc metals and th ree carbon s t e e l s

deformed in tension t o rel ieve residual s t resses due t o quenching ( 1 3 ) . The

re la t ive residual s t r e s s G ~ ~ / G ~ ~ , ~ i s plotted against the t e n s i l e s t r e s s G

relative t o the 0.01 % proof s t r e s s G / R ~ ~ . ~ ~ . The absolute values of the residual compressive surface s t resses resul t ing from quenching without

transformation aRS,O and the RpO.O1 values of the unquenched material are also

recorded. With a t e n s i l e load of u p t o the 0.01 % proof s t r e s s the pure

fcc metals are residual s t ress-re1 axed re1 at ively rapidly and completely. In

the s t e e l s , however, residual s t ress-relaxat ion i s only p a r t i a l and becomes

l e s s the greater the carbon content and the higher the y i e l d point. Similar

observations are reported in (39) . Loads greater than R p O . O 1 lead t o residual

deformation s t resses which are manifested as increases in the residual

compressive s t r e s s i n the surface. They ,an be at t r ibuted t o a macroscopic

inhomogeneity in the work hardening between specimen surface and core ( 1 3 ) .

- L O O 1 ' I 0 L O O 800 1200

Stress in ~ l r n r n ~

Fig 15: Relaxation of welding residual s t resses of a marageing s teel ( X 2 NiCoMo 18 8 5) due t o t e n s i l e loading ( a f t e r (40, 4 1 ) )

a ) Transversal residual s t resses in the welding seam center of an electron beam welded jo in t

b) Transversal residual s t resses in a distance of 3 mm of the welding seam center of a TIG welded jo in t

In an evaluation of the e f f e c t on the strength of residual s t resses set

during welding, i t i s important t o know the s t a b i l i t y of these s t resser

loading the weld. Fig. 15 i l l u s t r a t e s the considerable reduction in intc

s t r e s s accompanying t e n s i l e loading of an electron beam and TIG welded j~

in t h e martensi t ic precipi tat ion hardening s teel X2 NiCoMo 18 8 5 (40, 41 )

the electron beam welded specimen i t begins a t about 0.7 ReS , where ReS i s

y i e l d point of the unannealed weld. Residual s t ress-relaxat ion i s

complete. In the case of the T I G weld noticeable residual stress-relaxai

f i r s t occurs above R e S . By the time the t e n s i l e strength Rm i s reacl

residual s t ress - re laxa t ion i s v i r t u a l l y complete.

Residual s t ress-relaxat ion by unidirectional deformation begins a t r e l a t i t

small loads or p l a s t i c s t r a i n s . I t may be complete or p a r t i a l , both the del

and r a t e of residual s t ress-relaxat ion depending on the type and s t a t e of

material as well as on the nature of the applied load.

Stress-relaxation in shot-peened components under t e n s i l e load ( c f . Figs. 12

and 13) can qu i te eas i ly be understood with the help of the s t r e s s - s t r a i n

curve in Fig. 16c. Apart from the higher than usual yield point in the surface

region, as a r e s u l t of which the curves f o r surface and core i n t e r s e c t , the

same requirements are met as in Fig. 16b. The onset of p l a s t i c deformation

overall i s determined by the core. A modification of Eq. (10) holds f o r the

overall y ie ld point

When the t o t a l s t r a i n E ~ * has been reached and the specimen unloaded, the

original residual s t resses have been completely removed. Beyond E ~ * , however,

fur ther residual s t resses are s e t up as a resu l t of inhomogeneous deformation,

namely t e n s i l e in tHe surface regions and compressive in the core.

Stress-relaxation under compressive load can be understood by analogy.

Stress-s train curves are plotted in Fig. 16d using the same values f o r aRSS, C C

cRS and R ~ ~ ~ , ReS as in Fig. 16c. P las t i c deformation f i r s t occurs in the surface region a t

In contrast t o the t e n s i l e experiment, residual s t ress-re1 axation i n t h i s case

i s only par t i a l . This finding i s in qua l i t a t ive agreement with the r e s u l t s

represented in Figs. 12 and 13.

I f residual s t ress-relaxat ion i n tension and compression i s analysed, then,

knowing t h e core yield point R e S C and the overall yield points determined by

X-ray and mechanical methods f o r t e n s i l e loads ?ieS,t and compressive loads - ReS,c, the mean areal f ract ion of the work hardened surface layer A S and the

surface yield point ReSS can be estimated d i rec t ly using Eq. (12) and Eq. (13)

respectively. The areal f rac t ion of the surface region i s obtained from the

equilibrium condition f o r residual macrostresses

"RS s' AS + ORSC. AC = 0

taking in to account the secondary condition AS + AC = 1 ( A S and A' a re

( 1 4 )

the

corresponding mean cross sectional ;.reas) as

or using Eq. (12) as

Using the r e s u l t s f o r shot peened AlCu5Mg2 given in Figs. 12 and 13, Eq. ( 1

gives ReSS 465 ~ / m ' f o r the y ie ld point in the surface region. This val

i s about 20 % higher than the y ie ld point f o r unpeened material Re,

Furthermore, using Eq. ( 1 5 ) a value AS = 16 % i s obtained f o r the mean arE

f rac t ion of the work hardened surface region with i t s residual compressi

s t r e s s e s . This value agrees rea l ly well with the resu l t s of other experimen

in which the residual s t resses were determined by surface etching ( 3 8 ) .

Table 2 gives the overall yield point TieS i n terms of the resis tance residual s t ress - re laxa t ion f o r applied t e n s i l e and compressive load and f

sign reversal of the residual s t resses . Assuming that AS ' AC, Eqs. (12) e (13) are val id f o r both surface hardening (ReSS > ReSC) and softening ( R E

Surface compr. res . s t r e s s e s and core t e n s i l e res . s t r e s s e s

surface t e n s i l e res . s t r e s s e s and core compr. r e s . s t r e s s e s

Overall y ie ld point

f o r applied

'Table 2: Overall yield point 'ReS insterms of the resis tance E F E 7 Z u a l s t ress-relaxat ion ( A < A C , ReSS ; R e S C )

t e n s i l e load compressive load I

Actual materials usually possess a continuous residual macrostress p rof i l e

over the cross section of the specimen. I f , f o r example, residual compressive

s t resses associated with work hardening a re present in the surface region and

tens i le s t resses are present in the core, oRSS ( c f . Fig. 16b) represents the

maximum residual compressive s t r e s s in the surface region and oRSC represents

the maximum t e n s i l e s t ress in the core. Taking t h i s i n t o account, then in the

case of a predominantly uniaxial internal s t r e s s s t a t e , Eqs. (12) and (13) can

be used t o assess the onset of residual s t ress - re laxa t ion and Eq. (11) i t s C completion. In the l a t t e r s i tuat ion R e S i s replaced by ReSS or ReS and l o R S S l

C by laRS 1 . In agreement with the experimental findings presented above, C residual s t ress-re1 axation begins a t a t e n s i l e load (compressive load) f o r oRS

C > 0 ( G ~ , - - < 0) in the range 0 < < l R e S C l or f o r c j R S S > 0 ( G < 0 ) in

the range 0 i < 1 R ' 1 . I t i s complete under the condition aRSr: < 1 Ress 1 - - eS or 1aRSL( < ( R e S L ( according t o Eq. (11) a t J E ~ * J < 2 lReS51/~ or j ~ ~ * l

C 5 < 2 l R e S J / E .

In the case of a multiaxial residual s t r e s s s t a t e with an applied s t r e s s s1

acting in the direct ion of greatest principal residual s t r e s s G , ,RS , residu31

stress-relaxat ion begins when the local value of the e f fec t ive s t r e s s ae = f ( o l

+ nl ,RS; G ~ , ~ ~ ; G ~ , ~ ~ ) reaches the y ie ld point ReS1- Based on the shape change energy hypothesis

1

a1+c1 , R S - ~ 2 , R S ) Z + ( G ~ , ~ ~ - G ~ , ~ ~ ) ~ + ( G ~ , ~ ~ - G , - G ~ , R S ) 2 . (16) If the y i e l d point i s the same for the surface region and core ( R e S i = R =

e? a R ~ ~ ~ ) and the principal component of the residual s t r e s s = G~~ ,

greater applied s t r e s s ol i s needed t o induce residual s t r e s s - relaxation t h d n

in the uniaxial case ( c f . Eq. ( 1 2 ) ) due t o inh ib i t ion of s l i p .

A quanti ta t ive description of the re1 axation of mu1 t i axial residual macro-

s t resses with a known prof i l e over the component cross section has proved to

be extremely d i f f i c u l t . Even using simple, one dimensional models the

ccmputations a re -subs tan t ia l (42, 43).

C ) The behaviour of residual microstresses

Residual microstresses behave in a complex fashion during the relaxation of

residual macrostresses by unidirectional deformation. So f a r only i so'

experimental data based on X-ray p r o f i l e analysis are avai lable reli

t o t h i s area.

When a material i s shaped or machined, both directed and inhomoger

residual microstresses are set up. Those of the f i r s t type are the resu'

backstresses due t o dis locat ion p i l e ups a t boundary surfaces and e l ;

s t ra ined second phases of heterogeneous materi a1 s . If these dis locat ions

i n the reverse direct ion during local i sed p l a s t i c deformation, the t

s t r e s s e s and hence the directed residual microstresses are i n i t i a l l y redl

Further deformation causes renewed build up of backstresses i n t h e oppc

d i rec t ion associated with dis locat ion mult ipl icat ion and hardening which

once again increase the residual microstresses. -

Dislocation arrangements in a randomly d i s t r ibu t ion or in tangles or c

give r i se t o inhomogeneous residual microstresses. Micropl a s t i c deform?

can lead t o the rearrangement of dis locat ions into arrangements of 1

energy and thus bring about residual microstress-relaxation. .If new di

cat ions are produced, a renewed build up of residual microstresses i s su

posed on the relaxation process. This provides an explanation of the behav

of the f u l l width a t half minimum f o r an aluminium alloy under unidirect i

deformation described above ( c f . Fig. 13) .

A reduction in the residual microstresses i n hardened s t e e l s i s observed

on deformation (44) and on machining (20,361. Two superimposed e f f e c t s

operate here. After hardening, a very high density of dis locat ions i s pre

e i t h e r randomly dis tr ibuted or in tangles (23,451. Micropl a s t i c deforma

brings about a rearrangement of these dis locat ions into configurations

lower d i s to r t ion energy and therefore a reduction in the residual mi

s t r e s s e s . On the other hand dissolved carbon atoms may be induced t o d i f i n t o the energet ical ly more favourable octahedral i n t e r s t i c e s in

martensi te l a t t i c e under the influence of the s t r e s s f i e l d of the mo

d i s loca t ions . This causes a reduction in the te tragonal i ty and hence

l a t t i c e d i s to r t ion due t o dissolved carbon atoms.

3.2 Deformation under cvcl ic s t r e s s

The fat igue strength of metal l ic mater ials can be influenced qui te considerab-

l y by residual s t resses . Since in cer tain cases they can be considered as

local ly variable intermediate s t resses , they can lead to substant ial increases

in the fat igue strength ( c f . f o r example, ( 1 , 3, 5, 8, 9 , 11, 4 ) ) . The

influence of residual s t resses on the processes in the precrack f r e e stage of

cyc l ic deformation i s of par t icular importance. Coupled with t h i s i s the

s t a b i l i t y of the residual s t resses .

Fig. 17: Relaxation of shot peening residual -- stresses of an AlCuMg al loy (AlCu5Mg2) due t o bending fat igue with several s t ress amplitudes aa (38)

amplitude oa = 325 N/mnz. I t should be noted tha t in the f i r s t half of t

cycle there i s a considerable reduction in the residual compressive s t r e s s

on the f ron t face of the specimen, which i s in compression, whereas in t second half of the cycle , when the reverse s ide of the specimen i s

compression, residual compressive s t resses in t h i s face are reduced t o a val

of about 200 N/mnZ. After t h e f i r s t cycle there i s l i t t l e fur ther reduction

residual s t r e s s . The l e f t hand diagram in Fig. 18, however, shows t h a t in t

case of a shot peened titanium alloy (TiA16V4), pronounced residu

s tress-relaxat ion in the f i r s t cycle i s followed by continuing s tea

reduction in residual s t r e s s u p t o f rac ture ( 3 8 ) . In both mater ials , AlCu5M I and TiA16V4, there was a c h a r a c t e r i s t i c reduction in the residual s t r e s s wi

increasing s t r e s s amplitude. The normalised representation in the r i g h t ha I I diagram in Fig. 18 provides an impressive i l l u s t r a t i o n of t h i s behaviour.

S i s divided by the y ie ld point in the surface region ReS ( c f . section 3.

obtained from compression t e s t s . The residual compressive s t resses remaini

I a f t e r f rac ture ( N = Nf 1 in two completely d i f fe ren t materials now l i e on t same curve.

I t has been shown in numerous invest igat ions ( c f . fo r example, ( 1 , 5, 8, 16,

17, 20, 21, 36, 4 7 ) ) that the e f fec t of residual s t resses decreases with

increasing s t r e s s amplitude and number of cycles as a r e s u l t of residual s t ress - re laxa t ion . This i s exemplified in Fig. 17 by a shot peened aluminium

a l loy (AlCu5Mg2) which has undergone a reverse bending fat igue t e s t ( 3 8 ) . The

residual macrostresses on the front and reverse sides of f l a t specimens a re

plot ted against the number of cycles t o f rac ture fo r three d i f fe ren t s t r e s s

amplitudes oa. The greatest residual s t ress - re laxa t ion occurs during the f i r s t

cycle in every case. I t i s par t i cu la r ly pronounced a t the greatest s t r e s s

c. - - I E V" E I -200 -

v

Ln

E - L O O . - wl

&.A AICu5Mg2 z

0.0 TlA16VL --0

I shot peened

Fig. 18: Relaxation of shot peening residual s t r e s s e s of AlCu5Mg2 and TiAl6V4 due t o bending fa t igue as a function ofSs t ress amplitude o resp. r a t i o oa/ReS ( R e S = surface yield p & n t ) (38)

Fig. 18 incorporates the three most important parameters affect ing residual

s t ress-relaxat ion. These are the yield point ReS c h a r a c t e r i s t i c of the

material (o r the y ie ld point in the surface region R e S S ) , t h e s t r e s s amplitude

appropriate t o the type of loading oa and the i n i t i a l residual s t r e s s ~ ~ ~ ( 0 ) .

Based on these three parameters, i t has been possible t o achieve a systematic

c lass i f i ca t ion of much of the available data ( 1 , 5, 8 ) . A t t h i s point,

therefore, a discussion of some aspects bearing i n mind the problems mentioned

in the preceding section should suff ice.

Assuming a predominantly uniaxial residual s t r e s s s t a t e , i t i s a fundamental

pr inciple t h a t residual s t ress-relaxat ion will commence within the f i r s t cycle

( N = 1 ) of an al ternat ing load i f the s t r e s s amplitude reaches the overall

yield point (0,- = ITe@. The processes described in sect ion 3.1 will then take place. I t should, however, be noted t h a t , according t o Eq. (12) and (131,

different overall yield points apply in the f i r s t t e n s i l e and compressive - haif-cycle ( c f . a lso Table 2 ) . If oa < I%Sh the residual s t r e s s e s should remain s tab le in the f i r s t cycle. For N > 1 , residual s t r e s s - relaxat ion can

begin gradually i f a t l e a s t the overall cyc l ic y ie ld point i s exceeded (11,

46). I t i s necessary t h a t

Equations (12) and (13) are equally valid f o r the cyc l ic overall y i e l d po i~ l t C i f R~~~ and ReS are replaced by the corresponding cyc l ic y ie ld points for

surface and core. Applying the boundary conditions given in Table 2 , then

depending on whether cycl ic p l a s t i c deformation begins in the core or a t the

surface. I f no cyc l ic p l a s t i c deformation occurs, then

which ensures the existence of a s tab le residual s t r e s s s t a t e under al terna-

t ing load (11, 46).

Residual s t ress-relaxat ion when N > 1 can be a t t r ibu ted t o c y c l i c p'

deformation. In t h i s case the dis locat ion arrangements present i n the s t ;

material are rearranged t o configurations charac te r i s t i c of the fa1

condition f o r the given s t r e s s amplitude. Especially f o r cold worked matt

d i s loca t ion rearrangements are frequently associated with the sofl

processes in the precrack f ree s tage of fa t igue (48, 49).

Rearrangement of dislocations from configurations typical of the machir

heat t rea ted condition t o tha t of the fatigued s t a t e i s fundamental t

re laxat ion of macrostresses. Residual e l a s t i c s t r a i n s associated wit

residual s t resses are once again transformed in to micropl a s t i c s t r a i n

f a t i g u e softening proce;ses are involved re1 axation of microstresses car

be expected. In the case of machined and heat t reated mater ials t h i s mani

i t s e l f in decreasing hacdness and sharper X-ray interference l i n e s (20

36) . Differences in the residual s t ress-relaxat ion behaviour of cold wc

machined and heat t reated mater ials under cycl ic load can be a t t r i b u t

d i f f e r e n t dis locat ion arr.angements and dens i t i es . The closer t h e sirnil a r i

the dis locat ion configuration charac te r i s t i c of fat igue the l e s s the e x t ~

d i s loca t ion rearrangement and hence of residual s t ress-relaxat ion. '

4. Summarv and Conclusions

Residual s t ress-relaxat ion by heat treatment or under unidirectional o r c

mechanical s t r e s s i s brought about essen t ia l ly by the movement of d i s

t i ons. This converts the e l a s t i c s t r a i n s re1 ated t o residual s t r e s s e s

microplast ic deformation. The onset of residual s t ress-relaxat ion i s depe

on t h e resis tance R i t o the onset of p l a s t i c deformation. In the ca

unidirect ional deformation R i = ReS, cycl ic loading Ri = R~~~~ and

treatment R i = R ec / t ' In pr inciple , residual s t ress-relaxat ion occurs whe l i n e a r superposition of applied and residual s t r e s s reaches t h e mat

res i s tance R i assuming a predominantly uniaxial s t a t e according t o

S t r e s s - r e l i e f does not occur i f the condition

i s f u l f i l l e d .

The appl ied s t r e s s a may be ze ro , have a value which does not vary wi th t ime

( c f . thermal r e s idua l s t r e s s - r e l a x a t i o n ) , have a s t e a d i l y i nc reas ing value

( c f . un id i r ec t iona l deformat ion) o r a value varying p e r i o d i c a l l y wi th t ime

( c f . c y c l i c deformat ion) . S ince , according t o Eq. (191, t h e onse t of r e s idua l

s t r e s s - r e l axa t ion depends on t h e r e s i s t a n c e of t h e mater ia l R i as well a s on

the appl ied load o , a l l t h e parameters cha rac t e r i z ing t h e s t a t e of t h e

ma te r i a l as well a s tempera ture and t ime o r frequency make themselves f e l t .

The onset of s t r e s s - r e l a x a t i o n i s delayed by t h e presence of s u f f i c i e n t s t a b l e

obs t ac l e s t o d i s l o c a t i o n movement which inc reases R i a t t h e loads g iven. For T

< 0.4 Tm these could , f o r example, be g ra in and phase boundaries, finel!/

d ispersed incoherent p a r t i c l e s , coa r se secondary phases and, a s long a s no

d i f fu s ion can occur , d i s so lved impurity atoms and c e r t a i n d i s l o c a t i o n

arrangements. The onset and e x t e n t of r e s idua l s t r e s s - r e l a x a t i o n a r e

influenced i n a complex f a sh ion by t h e combined e f f e c t s of heat t r ea tmen t , unidi rec t ional and c y c l i c s t r e s s and by mu1 t i axi a1 appl ied and/or r e s idua l

s t r e s s s t a t e s . The k i n e t i c s of r e s idua l s t r e s s - r e l a x a t i o n a r e determined

e s s e n t i a l l y by t h e d i f f e r e n c e o + oRS - R i . The r a t e i nc reases t h e g r e a t e r t h e applied load, i .e. t h e h igher t h e temperature, t h e longer t h e t ime and t h e g rea t e r t h e magnitude and amplitude of t h e appl ied s t r e s s , p l a s t i c de fo r -

mation o r number of cyc l e s . I t o f t e n inc reases as t h e amount of r e s idua l

s t r e s s i nc reases . The r a t e i s f a s t e r t h e fewer t he s t a h l e o b s t a c l e s t o

d i s loca t ion g l i d e .

5. References

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New York (1981) .

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