hskl-betriebsfestigkeit 2 skript 1...2.2.1 stress-strain curves assuming uniaxial loading, plastic...

2 Tensile loading of notches

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2.1 Introduction The tensile test is the most important method for mechanical material testing. It provides essential results for strength, ductility as well as stiffness for uniaxial, bending-free loading of slender bars. The measured material parameters serve to dimension statically loaded components. Relatively rapid basic assertions can be made about the mechanical properties for developing materials or components. This is enormously important, for instance, for lightweight constructions which must support more and more load using de-creasingly less material (mass). Lengthy development times for lightweight ma-terials often possessing complex structures are associated with high costs. With the aid of the tensile test, the development costs can be limited by means of rec-ognising and quickly eliminating material variants having unfavourable proper-ties. Of course, it is indispensible for both material manufacturers and material proces-sors to inspect the tensile deformation behaviour of their products as part of their quality assurance, particular for mechanically highly loaded materials. The notched tensile test is employed to examine the material behaviour under more stringent static loading conditions. For this purpose, a notch is introduced into the middle of the specimen. The material and the material's state, the loading as well as the notch geometery affect the notch's deformation behaviour. Mainly, only the notch tensile strength is measured, which one also uses to assess the ma-terials toughness. Both strength and toughness are essential for the load carrying capacity of components. A tensile test on a component-like specimen provides the best approximation to the operating behaviour. However, this involves high costs and is therefore rarely carried out. Loading, which deviates from tensile tests, can induce considerable changes in the material's reaction so that other investigations may be necessary; for instance, for fatigue, creep or corrosion behaviour. These are, as a rule, time consuming and expensive. The assignment of the laboratory test consists of performing tensile loaded mate-rial tests on differently notched steel specimens and assessing the notch's tensile deformation behaviour.


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2.2 Elastic-plastic deformation of tensile loaded notches

2.2.1 Stress-strain curves Assuming uniaxial loading, plastic deformation occurs in the notch when the max-imum stress, the yield stress, is attained (cf. Fig. 2.1) (2.1) The highly stressed edge region in the notch root begins to flow and propagates deeper into the specimen with increasing load. Here, the internal regions, which are still elastically stressed, are mainly stressed from their supporting effect, that is, they bear the loading together. Fig. 2.1 shows the stress-strain curves in the notched ligament’s cross-section us-ing an example of a work-hardening material. Fig. 2.1: Stress-strain curves in the notched ligament’s cross-section for uniaxial

loading The total strain curve is elastic in the core of the notched ligament’s cross-section. At a distance x = xeS, the total strain then assumes the value

(2.2)

For x > xeS, elastic-plastic deformation occurs. In the notch root itself, the notch root's total strain consists of elastic and plastic components (2.3) Empirically, the entire total strain curve is similar to a purely elastic deformation.

max K eSRs = s =

eSt eS

RE

e = e =

K,t K,e K,pe = e + e

x xeS

x xeS

fictitious stress s*(x)

real stress s(x)

0

ReS sn

smax = sK

eK,t

eK,p

eK,e eeS

et(x)

ep(x)

ee(x)


21

As a consequence of loading beyond the elastic limit, the definition of the stress concentration factor

(only elastic) (2.4)

is no longer valid (cf. test no. 1), thus

and the proportionality (Hooke) between real stress s(x) and notch root's total strain eK,t is lost. If one introduces a fictitious stress distribution, which plotted in Fig. 2.1, for the loading sn and an assumed elastic-plastic deformation, then

(2.5)

would be valid for the fictitious notch root's stress. The nominal stress becomes

(2.6)

with AK as the ligament’s cross-section. Stress strain diagrams are depicted in Fig. 2.2, namely

• for the smooth bar (sn, et) as well as • for the notched bar,

o as the nominal stress over the notch root's total strain (sn, eK,t), o as the real notch root's stress over the notch root's total strain

(sK, eK,t), and o as the fictitious stress over the notch root's total strain ( )

One obtains the -value via Eq. (2.5) by multiplying the nominal stress by the stress concentration factor aK. The value of the nominal stress at the beginning of plastic deformation is

(2.7)

the notch yield stress (here: uniaxial!).

maxK

n

sa =

s

n

maxK s

s¹a

*(x)s*Ks

*K

Kn

sa =

s

K

K

d2

n KK K d

2

F 1 (x)dAA A

-

s = = sò

t,K*K ,es

*Ks

1( ) eSn K,eS

K

RRs = =a


22

Figure 2.2: Stress-strain curves smooth and notched bar geometry (schematic)

2.2.2 Neuber's equation Since the notch root's stress for material behaviour beyond the elastic limit can no longer be described by the stress concentration factor (see above), the real notch root's stress sK and the notch root's total strain eK,t can not be simply determined. With the aid of Neuber's equation , (2.8) the stress state in the notch root can, nevertheless, be approximately established for a given stress concentration factor aK, given nominal stress sn and the associ-ated nominal strain en = et. Here, the nominal stress is computed according to Eq. (2.6). One obtains the relationship between nominal stress and nominal strain (sn, en- or sn, et-curve) from tensile tests on a smooth comparative specimen. The following stress shape factor is still needed

(2.9)

together with the strain shape factor

. (2.10)

K s ea = a a

K

n

real notch root 's stressnominal stresss

sa = =

s

K,t

n

notch root 's strainnominal straine

ea = =

e


23

Owing to the plastically deformed material's sn, et-relationship – tensile work-hardening curve, smooth comparative specimen – it is clear that, relative to the linear-elastic region, the stress and strain is disproportionately above and below, respectively. For the notched bar, it follows that . (2.11) For elastic deformation processes, the following must be true . (2.12) By employing Neuber's equation (2.8) and the eqs. (2.9) and (2.10) for the case of elastic-plastic deformation

(2.13)

together with

, (2.14)

one obtains

(2.15)

This relationship is valid for elastic nominal stresses in smooth comparative spec-imens (tensile test) and sn < ReS (2.16) And, for elastic-plastic deformations in the notch root (notched tensile test) and . (2.17) Moreover, if one uses the maximum fictitious stress value in the notch root

(2.5) and the fictitious notch root's strain

, (2.18)

or inserting

into equation (2.15), the following is given

. (2.19)

Ks ea < a < a

Ks ea = a = a

K,t 2KK

n ns e

esa a = × = a

s e

nn n,e E

se = e =

22 n

K K.t K n K n,e K Es

s e = a s a e = a

n n,e e,Se = e < e

K,t e,Se > e K e,SRs >

*K K ns = a s

** KK E

se =

*K n,e* K K n

K K n,eE

E E Ea es a s

e = = = = a e

2* * 2 n

K K,t K K K Es

s e = s e = a


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That is, the right-hand side of Eq. (2.19) is specified by the bar's geometry (aK), the loading (sn) and the material (E). The left-hand side represents the notch root's loading, that is, the product of the notch root's total strain ( ) and the real notch root's stress ( ), or is, as the case may be, the fictitious notch root's stress state

.

Figure 2.3: Determining the elastic-plastic notch root's stress state with the aid

of Neuber's hyperbola As illustrated in Fig. 2.3, one approximately obtains the required notch root's load-ing using Neuber's hyperbola (2.19) together with the -diagram (smooth comparative specimen). The case of purely elastic deformation

< ReS and , (2.20)

Eq. 2.15 becomes . (2.21) For this case, one thus obtains, which is also inevitable, the defining equation of the stress concentration factor aK. Material and material’s state have no influence on the behaviour of the elastic notch's deformation (cf. test 1).

K,te

Ks* *K Ks × e

n t,s e

KsK

K,t K,e eSEs

e = e = < e

K K ns = a s


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2.2.3 Determination of notch strengths During the simplest notch tensile test, one measures the nominal stress values

(2.6)

and the extension over a gauge length L0, within which the notch is located. Pre-cise assertions about the elastic-plastic notch root's deformation behaviour require additional strain analyses in the notch root. Strength values from the notch tensile test are measured with the aid of the nom-inal stress - notch root's total strain diagram (cf. Fig. 2.4). That is, the notch yield stress

, (2.22)

at which incipient plastic deformation begins in the notch root, the notch tensile strength

(2.23)

is determined from the maximum force and the notch's yield stress RK,px. For in-stance, one obtains the 0.2%-off-set notch yield stress

(2.24)

for a residual strain in the notch root of eK,r = 0.2 %. Figure 2.4: Strength values of the notch tensile test, schematic

nK

FA

s =

K,eSK,eS

K

FR

A=

maxK,m

K

FRA

=

0 20 2

K,p .K,p .

K

FR

A=

eK,r

eK,t


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2.2.4 Off-set notch yield stresses and notch tensile strengths

The notch yield stresses, off-set notch yield stresses and notch tensile strengths, determined using 32NiCrMo14-5 round specimens, are plotted against various stress concentration factors aK in Fig. 2.5. As a comparison, the expected notch yield stress, subject to uniaxial loading,

(2.7)

is also depicted (dotted line). The measured notch yield stresss RK,eS (Plane Strain Condition, PSC) are somewhat larger but, as expected, fall with increasing stress concentration factor aK (cf. Fig. 2.5). The following is valid (2.25) Ductile materials demonstrate increasing gradients of the off-set notch yield stress' profile for growing plastic off-set deformation (0.05 % to 1 %) with aK. The RK,px value can become larger than the off-set yield stress (Rpx) of a smooth bar. For 1.0 % of the permitted permanent notch root's strain, the complete curve's profile of the off-set notch yield stress RK,p1.0 is even located above the 1 % off-set yield stress Rp1 of the smooth bar. Still larger plastic deformations, for the notch tensile strength, lead to their considerable increase with the stress concen-tration factor aK. Notched bars with the stress concentration factor aK ≈ 2 can exhibit about double the notch tensile strength as the tensile strength (smooth bar).

Figure 2.5: Notch yield stress, off-set notch yield stress and notch tensile strength as a function of the stress concentration factor for 32NiCrMo14-5 specimens (Source: Macherauch) The finding here are confirmed by results for various steels and cast irons in Fig. 2.6. For ductile states, the depicted relative notch tensile strength RK,m/Rm has values > 1 or ≈ 1. Although for "sharp" notches, greater than shape factors aK ≈ 3, the initial

rise in notch tensile strength then gives way to a drop. The load carrying capacity

(1) eSK,es

K

RR =a

(PSC) (1)eS K,eSK,eS K,eSR R R R> = >


27

decreases such that it is imperative to avoid very "sharp" notches even in ductile materials. Starting from a normalised (N) through to a hardened and tempered (HT) state, the decreasing material's ductility causes a fall in the notch tensile strength and, for extremely brittle hardened (H) C45 or cast materials, the curves approximate the function (cf. Fig. 2.6)

. (2.26)

2.2.5 Multiaxial stress states and notch tensile

strength As can also be seen in Fg. 2.6, the notch tensile strengths measured on Round bars (R) are considerably larger than those measured on Flat bars (F). The investigated ductile material states demonstrate relative notch tensile strengths RK,m/Rm of ap-prox. 1, for flat, and almost 2 for round bars. The circumstances here can not be simply understood by means of macroscopic supporting effects and the stress peak's reduction. An essential role is played by the different multiaxiality of the respective stress states. A uniaxial and a biaxial, plane stress state (PSS), dominates in the tensile loaded, flat notched bar. In the round bar, it can be assumed that at the notch root's surface 2, and below the sur-face, 3 stress components exist; a multiaxial stress state (MSS). One also speaks of plane strain conditions (PSC) because the strain in the transverse direction is contrained.

K,m

m K

R 1R

=a

Figure 2.6: Relative notch ten-sile strength as a function of the stress concentration factor aK for various ductile materials (Source: Kußmaul)

K,m

m

RR

Ka


28

2.2.6 Assessing a component's toughness The component's ability to deform primarily depends on the material and the ma-terial’s state but also on the mechanical loading state, the loading temperature and on the loading rate. Mohr's circles, depicted in Figs. 2.7 and 2.8, are used as an aid to explain the effect of the material and the loading. If relatively low resistance to dislocation glide exists, then materials fail in a duc-tile way for uniaxial loading (smooth bars). On attaining the shear yield stress

by means of (cf. Fig. 2.7, left), the maximum shear stress

, (2.27) the purely elastic deformation is replaced by elastic-plastic deformation. The prin-ciple normal stress takes effect here (2.28) Beyond the yield point ReS, the shear and normal stress increase more slowly. On reaching the shear strength , failure occurs by fracture

. (2.29) The work performed by the plastic deformation here is an important measure of the materials toughness. From the shear strength (Eq. (2.29)), one obtains, via Mohr's circle, the normal stress value (2.30) of the ductile material's tensile strength. The separation strength plotted in Fig. 2.7, left, has no effect since (2.31) Higher material strength can be obtained by means of "incorporating obstacles" to the dislocation movement in the lattice (hardening, cold working etc.). The ma-terial toughness then decreases. As a consequence of unfavourable material processing (e.g. too intensive cold working), materials can be embrittled. Then , the shear strength in the brit-

tle state (or ), is too high in relation to sT (cf. Fig. 2.7, right). Thus remains smaller than the shear strength and the separation strength is decisive for the fracture failure when the principle normal stress grows to (2.32) In this case, dislocation glide and plastic deformation seldom takes place.

(ductile)Ft maxt

(ductile)max Ft = t

(ductile) (ductile)1 eSF eS2 R Rs = t = =

(ductile)St

(ductile)max St = t

(ductile) (ductile)1 m mS2 R Rs = t = =

Ts

(ductile)T s2s > t

(brittle)St

(brittle)Ft maxt

(brittle) (brittle)1 T m SR 2s = s = < t


29

Figure 2.7: Material embrittlement in Mohr's representation

Left: Tough state Right: Brittle state If one considers a flat notched bar subject to an assumed uniaxial stress state for ductile material behaviour, then the following results for the approximate fracture failure (cf. Fig. 2.8, left) . (2.33)

The notch tensile strength is then expected to be about the same magnitude

as the tensile strength of the material (cf. section 2.2.5).

Figure 2.8: Embrittlement by means of a multiaxial stress state, Mohr's represen-

tation Left: Flat notched bar (assuming: uniaxiality) Right: Stress embrittlement for a round notched bar (MSS) Stress embrittlement, as a consequence of multiaxial stress state (MSS or PSC), can also limit the ductile behaviour of components, even quite considerably. The

(ductile) (ductile) (1)1 m K,mS2 R Rs = t = »(1)K,mR

(ductile)mR


30

causes are similar triaxial stresses beneath the notch root's surface (round notch bar). By this means, the maximum shear stress remains rather small, whereas the principle normal stresses ( ) increase (cf. Fig. 2.8). A change in the failure limit from shear to separation strength occurs. Dislocation glide

and plasticity are reduced. The multiaxial notch tensile strength of ductile

materials is considerably larger than both their tensile strength as well

as the uniaxial notch tensile strength

(2.34) The component's ductility is assessed with the aid of, among others, the measured relative notch tensile stress RK,m/Rm. Here, ideal brittle behaviour specifies the lower limit (cf. Fig. 2.6)

(2.35)

Accordingly, brittle materials have lower notch tensile strength as tensile strength (smooth bars), but do not attain the lower limit given by Eq. (2.35) 1/aK. For tough behaviour, this becomes

(Flat bar) (2.36)

or

(Round bar) (2.37)

(cf. Fig. 2.6). Thus, high notch tensile strengths arise.

maxt

321 s>s>s

Ts(MSS)K,mR(ductile)mR

(1)K,mR

(MSS) (1) (ductile)mK,m K,mR R R> »

(ideal brittle)K,m(ideal brittle)

Km

R 1 1R

= <a

(1)K,m

(ductile)m

R1

R»

(MSS)K,m(ductile)m

R1

R>


31

2.2.7 Causes of high relative notch tensile strengths Ductile material states are to be considered for which notch tensile strengths are measured. These are the same magnitude or higher that the tensile strengths (cf. Figs. 2.5 and 2.6). Fig. 2.9 schematically depicts the stress profiles at full plastic loading for the notch tensile strength. In the flat bar (s3 0 and assuming: s2 0), the initial inhomogeneous stress profile (cf. Fig. 2.1) is practically homogenised by means of plasically reducing the stress peaks (cf. Fig. 2.9, left). In this way, macroscopic supporting effects can introduce the same notch tensile strength as the tensile strength (2.38) Figure 2.9: Full plastic loading for the notch tensile strength, schematically de-

picted for a flat bar (left) and for a round bar (right) Increased elastic-plastic load redistributions (macroscopic supporting effects) also occur within round bars possessing high stress multiaxiallity (within 3, and on the surface 2, stress components). Plastic deformation propagates from bi- to triaxially (similar) loaded material regions (cf. Fig. 2.9, right). As Fig. 2.8, right shows, 3 stress components limit the maximum shear loading . For this

» »

(1) (ductile)m mK,mR R R» =

maxt

(1) (ductile)mK,mR R»

F

F

F

F F

F

F 1s

F 2 3 0s » s »

F ns

F K,te

(MSS)K,mR

F 2s

F 3s

(ductile)mR


32

reason, the shear strength will not be easily attained so that considerably larger principle stresses are possible. The failure limit can change

from shear to the separation strength (cf. section 2.2.6). Multiaxial-lity, macroscopic supporting effects and deformation ductility thus mutually af-fect the high relative notch tensile strength for round bars (cf. Eq. (2.37)). Overall, the component's load supporting behaviour essentially depends on the material's strength and the materials toughness. Therefore, the most suitable ma-terial states are characterised by matching optimal strength with toughness. How-ever, the notch effect should, as far as possible, always be avoided!

(ductile)St

321 und, sss(ductile)St Ts

(MSS) (ductile)mK,mR / R 1>


33

2.3 Test description 2.3.1 Testing equipment and safety precautions The tensile tests are carried out on an older robust electrohydraulic driven univer-sal testing machine, manufactured in 1960 by the company Losenhausen, model UHP 40 (Fmax = 400 kN). The longitudinal displacements were measured via the traversing distance and the load was given via a pendulum weighing mechanism. Local notch root strains were not recorded. For your own safety and to avoid damage, the machine may strictly only be oper-ated by persons who have been previously instructed. Each test is to be carried out by only one person. During the clamping of the specimen bar, care must be taken that no part of the body (fingers) is ever inserted within the clamping de-vices. To guard against flying fragments and to avoid bodily entrappment between the moving traverse and the machine's base, it is imperative that the test room must be closed prior to starting the test. For this purpose, a transparent door is available through which the test's progress can be monitored. 2.3.2 Testing method To measure the applied force, the deflection pendulum is to be loaded with weights. The trailing indicator must be set to "0". The initial ligament's cross-section AK as well as the specimen bar's initial length l0 are to be determined using the available measuring device. To define the test procedure, the traversing speed is specified. The specimen bar is to be positioned between the upper and lower clamping devices and gripped. After closing the test room, the test is started. During the tensile test, one strains the specimen bar along its longitudinal axis until it breaks. Here the force F and the londitudinal displacements Dlt = lt – l0 (2.39) are recorded in a F,Dlt-diagram. lt is the measured length which is produced by the applied force F at time t (cf. Fig. 2.10).


34

Fig. 2.10: Initial gauge length l0 in the unloaded state and the measured length

lt for the applied force F To determine the strength values (cf. section 2.2.3), the corresponding nominal stress (Eq. (2.6)) is to be computed with the aid of the data from the corresponding F, Dlt-diagram.

AK l0

lt

F F


35

2.3.3 Notch tensile test Data can be taken from the F, Dlt-diagram to determine the following values Strength:

• The notch yield stress RK,eS, according to Eq. (2.22), represents the re-sistance to the incipient plastic deformation at the notch root, it concludes the purely elastic deformation. After attaining the notch yield stress, the material deforms elastic-plastically. This begins in the notch root and pro-gresses into the specimen’s interior.

• Using the curve's maximum, one obtains the notch tensile strength RK,m (Eq. 2.23).

• The off-set notch yield stresses RK,px (Eq. (2.24)) can not be determined since no notch root strains are measured.

Deformation:

• The maximum irreversible longitudinal displacement

(2.40) is determined from the broken notched bar.

Moreover, on smooth bars, one obtains • the fracture strain as the maximum plastic deformation

(2.41)

and • the necking at fracture as the change in the cross-section in the necking

region

(2.42)

LB and AB are the measured length and the smallest cross-section of the bar

after fracture.

B B 0l l lD = -

B 0

0

L LA 100%L-

=

0 B

0

A AZ 100%A-

=


36

2.4 Assignment

Force displacement diagrams are to be recorded at room temperature from pre-pared smooth and notched specimen bars made of normalised and hardened steels S235JRG2 (RSt 37-2) and C45E using a hydraulic driven universal testing ma-chine. Prior to beginning the test, the specimens' dimensions are measured. The notch's stress concentration factors of the used bars are aK = 1; 1.58; 1.87 und 2.55. The traversing speed is to be set to 20 mm/min. The obtained notch yield stress RK,eS and notch tensile strength RK,m are to be

plotted as functions of aK. In doing this, the function is to be completed. In

another diagram, the relative notch tensile strengths are to be depicted as a

function of aK together with the limiting curve . All results should be dis-

cussed in a summary, in particular with regard to the toughness behaviour.

eS

K

Ra

K,m

m

RR

K

1a


37

2.5 Protocol Group: Date: Recorded by: Testing machine:

2.5.1 Tensile test (smooth specimen) Specimen 1 Specimen 2

Material Material state

Prior to starting the test Initial length L0 in mm Initial diameter D0 in mm Initial cross-section A0 in mm2

After completing the tests (permanent deformation in the specimen) Gauge length after fracture LB, mm Specimen's extension LB in mm Diameter after fracture DB in mm Maximum tensile force Fm (indica-tor), kN

The following are to be determined from the force-displacement diagram Lower yield point force FeL in N Upper yield point force FeH in N Yield point force FeS in N Maximum tensile force Fm in N Specimen's extension LB in mm

Results Lower yield stress ReL in MPa Upper yield stress ReH in MPa Yield stress ReS in MPa Tensile strength Rm in MPa Necking at fracture Z in % Fracture strain A5, %

D

D


38

2.5.2 Notch tensile test

Notched spe-cimen 1

Notched spe-cimen 2

Notched spe-cimen 3

Material Material state Stress concentration factor

Prior to starting the test Initial length l0 in mm Initial diameter dK in mm Initial ligament cross-section AK in mm2

After completing the tests (permanent deformation in the specimen) Gauge length after fracture lB in mm

Specimen's extension lB in mm Maximum tensile force FK,m (indi-cator), kN

The following are to be determined from the force-displacement diagram Yield point force FK,eS in N Specimen's extension lB in mm

Results

in MPa

Notch yield stress RK,eS in MPa Notch tensile strength RK,m in MPa

Limiting value

Relative notch tensile strength

Specimen's extension lB in mm

Ka

D

D

eS

K

Ra

K

1a

K,m

m

RR

D


39

Notched spe-cimen 1

Notched spe-cimen 2

Notched spe-cimen 3

Material Material state Stress concentration factor

Prior to starting the test Initial length l0 in mm Initial diameter dK in mm Initial ligament cross-section AK in mm2

After completing the tests (permanent deformation in the specimen) Gauge length after fracture lB in mm

Specimen's extension lB in mm Maximum tensile force FK,m (indi-cator), kN

The following are to be determined from the force-displacement diagram Yield point force FK,eS in N Specimen's extension lB in mm

Results

in MPa

Notch yield stress RK,eS in MPa Notch tensile strength RK,m in MPa

Limiting value

Relative notch tensile strength

Specimen's extension lB in mm

Ka

D

D

eS

K

Ra

K

1a

K,m

m

RR

D


40

Approximation of the stress concentration factors

hskl-betriebsfestigkeit 2 skript 1...2.2.1 stress-strain curves assuming uniaxial loading, plastic...

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