Proceedings of the 5th International Conference on Integrity-Reliability-Failure, Porto/Portugal 24-28 July 2016

Editors J.F. Silva Gomes and S.A. Meguid

Publ. INEGI/FEUP (2016)

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PAPER REF: 6206

INFLUENCE OF PLASTIC DEFORMATION INDUCED THROUGH

TORSION UPON OTHER MECHANICAL CHARACTERISTICS

Viorel Goanta(*)

Technical University „Gheorghe Asachi”, Bd. D. Mangeron 67, Iaşi, Romania (*)Email: vgoanta@tuiasi.ro

ABSTRACT

The mechanical characteristics of materials are determined by laboratory tests performed on

samples, and also by standard procedures which reproduce, as much as this is possible, the

working conditions during their exploitation, such as: simple or composed stresses, time of

load application, environmental conditions, etc. When the stress applied exceeds the elasticity

limit, modifications of some mechanical characteristics occur. Plasticity may be induced by

the stresses produced during exploitation, as part of the various technological processes. In

the present study, several cylindrical samples have been subjected to loading at torsion,

different degrees of plastic deformation being recorded, as a function of the differentiated

rotation angle of each sample. The hardness tests performed on these samples evidenced

variations of this parameter with the degree of plastic deformation. Further on, the same

samples were subjected to tension up to breaking, three parameters being especially followed:

the apparent yield limit, the apparent ultimate tensile stress and the energy remaining for

plastic deformation for tensile.

Keywords: Failure, plasticity, strain, hardness, torsion, tensile.

INTRODUCTION – GENERAL BACKGROUND

When loading exceeds the yield strength, materials usually suffer different plastic

deformations – weaker or stronger. The degree of plastic deformation depends on the size and

type of stress, as well as on the behavior of the material: fragile or ductile. Anyway, even if

yield point is exceeded, not the whole volume of the material is affected with plastic

deformations. Consequently, the energy accumulated in the material in view of deformation is

composed of the energy utilized for plastic deformation and the energy utilized for elastic

deformation. Whereas elastic deformation is reversible, plastic deformation causes

modification in the form and sizes of the piece. Generally, the energy of plastic deformation is

lost during microstructural dissipative processes, such as the sliding produced at the

boundaries between grains (Mohammad-Ali, 2013).

In the case of components working in elastic domains, failure may be defined as a loss of

material’s ability to store the elastic strain energy. The moment in which this happens

depends on the behavior of the material, evidenced during loading. However, even after the

yield point the material has still resources to operate, both by its capacity of taking over the

elastic deformation and by the energetic consumption for plastic deformation. Under such

circumstances, and depending on the degree of plastic deformation attained, the following

question arises: how large are still these resources and how were transformed certain

mechanical characteristics, essential for a certain degree of plastic deformation?

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Several theories have been put forward and different criteria have been proposed for

estimating materials’ failure (Liu, 2001). According to the theory of Christensen (Christensen,

2014), failure represents the end of elastic behavior, but not of plastic behavior, while the

theory of Andrianopoulos (Andrianopoulos, 2014) failure is defined as the loss of material

ability of storing elastic deformation energy, which is evidenced as early as the beginning of

loading. This second theory may be especially applied to materials with a preponderantly

fragile behavior, for which the plastic deformation energy is practically equal to zero - and yet

are produced the material failure. According to other assertions, plastic deformation is

actually a failure, so that it cannot be fully responsible for the final failure (Banabic, 2010).

The relation resulting from the Mohr-Coulomb failure criterion represents a quantification of

the normal stress (σ) and tangential stress (τ) contributions to the failure:

� = � ∙ ���∅ + � (1)

where φ represents the slope of the surface, determined with relation (1), while c is a constant

deduced from the intersection of the surface with τ axis, (Hernas, 2001).

In such a context, the following explanations may be provided:

a) Whichever the behavior of the material, the first thing to occur is the initiation of a

microdefect, within which the first failure of the material is recorded. The initiation

mechanisms are quite diverse, being conditioned, among others, by the structure of the

material. There follows propagation of a macrofissure, determined by the macroscopic

behaviour of the material, up to final breaking. A microdefect from which breaking

should be initiated may preexist in the material or it may be produced by two possible

mechanisms: cleavage or shear, as a function of the capacity of the material of

permitting shifting and accumulation of dislocations;

b) The material present in the vicinity of the initiation zone, suffers smaller or larger

plastic deformation, as a function of distance from the breaking zone. At higher

distances, the material suffers only elastic deformations.

c) The primordiality of the yield mechanism: cleavage or shear is still a debatable aspect.

If, macroscopically, a material appears as having failed under the action of a normal stress, it

is highly possible that, at microstructural level, its grains or its limits should have sliding

under the action of tangential stresses (Chakrabarty, 2006).

In the case of materials with a ductile behavior, the von Mises yield criterion - assuming that

plastic deformation of materials begins when the second deviatoric stress invariant, J2, attains

a critical value – is valid:

� =�

������ − �

+ �� − ��� + ���� − ���

� + �� + � �

+ ��� (2)

where σii represents normal stresses and σij are tangential stresses.

Considering the different behavior of materials during loading, establishment of a unique

relation, which should include all possible plastic deformations, is difficult. Constitutive

equations on the yield criteria have been produced in (Hoek, 2002), being checked separately

for torsion and tension. Andrianopolos (Andrianopoulos, 2014) proposes a failure criterion on

the basis of the two parts of the density of elastic deformation energy: the distorsional and the

dilatational one. The two proposed relations are differentiated for materials with fragile

behavior (breaking through cleavage) and, respectively, for materials with ductile behavior

(breaking through sliding).

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However, the question still to be answered is: for a material with some behavior, which comes

to be broken, how much energy should be consumed for dilatational deformation and how

much energy is consumed for ”distortional” deformation? For example, for two identical

samples, one subjected to torsion and the other one to tensile, are equal the energies

consumed in these two cases, for reaching breaking? One should observe that subjecting the

material to some loading, combined with torsion-tension over the yield point helps the

material achieve grain refinement and, consequently, different characteristics are obtained.

Chepeng Wang (Wang, 2013) studied the effect of plastic deformation obtained through stress

combined by traction-torsion upon microhardness. Plastic deformation through torsion

induces a significant decrease in grain size from the central part to the surface.

The experimental tests and analyses on the behavior of polycrystalline metals subjected to

considerable deformations have been discussed in numerous studies. Exceeding of the yield

point causes the development of deformation textures, which are especially interesting, as

they acquire anisotropic properties in many forming processes, such as: stamping, extrusion,

rolling, and wiredrawing. For their subsequent modelling, determination of metals’ response

under such conditions is important.

LOADING AT TORSION

The torsion test may produce large and uniform plastic deformations, prior to their

localization or prior to breaking. Consequently, this test is recommended for determining the

work hardening characteristics of metals (Beausir, 2009).

The present chapter analyzes the results of the torsion tests applied to 8 samples made of

S275JR steel. The test was performed on a universal testing machine equipped with a device

with cogged wheels, permitting a free rotation of both ends of the sample. The samples have a

circular section and have been subjected to torsion at various levels of plastic deformation.

The loading differences were obtained on the basis of torsion angle variation. Later on, the

hardness value was determined and the energy corresponding to plastic deformation was

calculated by the below-described method.

Figure 1 plots the variation curves of torque versus the rotation angle for the 8 samples

subjected to torsion up to certain levels of plastic deformation. Sample 8 was loaded up to

breaking, while sample 1 was the least loaded of all.

Fig. 1 - T-φ curves at torsion loading for the 8 samples

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Accordingly, table 1 lists the final values of the torque, as well as the maximum angles at

which the marginal transversal sections had been rotated for each of the 8 samples.

Table 1 - Final rotation angle and torque

Sample no. 8 7 6 5 4 3 2 1

Angle [degree] 1164.1 947.89 839.79 674.20 483.36 381.53 279.85 92.11

Torque [N·m] 55.65 54.45 45.75 45.75 43.20 30.90 34.65 24.15

One may observe that the slope of the variation curve of the torque versus the rotation angle is

the same for all 8 samples. The low variations of torque recorded are induced by the device

employed, which transmits the torque at the ends by means of two cogged wheels. In this

way, the decrease of torque occurs during shifting of the contact from one tooth to another.

For the samples subjected to torsion, also determined was the energy corresponding to plastic

deformation, from the variation diagrams of the torque versus the rotation angle - Fig. 2. It

was assumed that the energy losses induced by the utilization of the torsion device and of

other dissipative elements are low comparatively with the work provided from the outside.

Fig. 2 - Calculation of the energy corresponding to plastic deformation at torsion for sample 7 – an example

The relation of calculus for determining the energy corresponding to plastic deformation to

torsion may be written as:

��

ϕ⋅−

ϕ−ϕ⋅+= ∑ ++

2

T

2

)()TT(EP

i

i1ii1i (3)

where:

- Ti and Ti+1 represent the torques provided by the testing machine, as listed in the file

table at positions i and i+1;

- φi and φi+1 are the rotation angles calculated on the basis of shiftings di and di+1

provided by the testing machine, listed in the file table at positions i and i+1;

- Tp is the final torque, included in table 1;

- φE, the angle corresponding to the zone of elastic discharge obtained by drawing a

parallel line to the elasticity line through the final loading point.

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The first term of relation (3) provides the total energy, whereas the second term represents the

elastic energy, Ee, that may be recovered when loading is over. Consequently, the difference

between the two energies is represented by the energy corresponding to plastic deformation,

when the sample is subjected to torsion over the limit of elasticity.

DETERMINATION OF HARDNESS

Hardness tests have been used for a long time for the characterization of materials, as they are

simple, non-destructive and have the capacity to evaluate mechanic characteristics in small

volumes. Measures such as plastic strain strength, rigidity and the fracture toughness of the

material can be assessed based on hardness tests (Kim, 2008). One of the key parameters

which characterizes the plastic yield properties is the strain-hardening exponent, n, for those

materials which observe Hollomon’s equation, σ=K·εn, where σ represents true stress, ε

represents true strain and K is the strength coefficient (Gaško, 2011). Kim takes into

consideration the possibility of establishing a relation between the strain-hardening exponent,

n, and the indentation size effect (ISE) represented by the characteristic length h.

Strength, fracture toughness and the fatigue characteristics of steels may be estimated on the

basis of its hardness Smoljan, 2012). When the elasticity limit is exceeded, microstructural

transformations, which lead to the modification of the mechanical characteristics, also occur.

The mechanical properties of steel depend directly on the hardening degree. The relation

between HV hardness and the 0.2% offset yield strength (Rp0,2 [N/mm2]) is (Smoljan, 2002):

���, = �0.24 + 0.03� ! + 170� − 200

where C is the degree of hardening, defined as the ratio between hardness measured after the

loading applied almost up to breaking and the hardness value obtained on the initial, non-

loaded sample (Just, 1976).

In the present investigation, testing of loading to torsion was followed by determinations of

hardness on the 8 samples, with the Emcotest EMC10 device. For sample 1, determination

was made in the immediate vicinity of the zone in which breaking occurred; for each of the

other 7 samples, 10 determinations of hardness were performed at equal intervals, along the

sample. Also performed were determinations of the mean, maximum, minimum values, and of

the mean value, without the minimum and maximum values. All these determinations are

illustrated in the graphs plotted in Fig. 3, showing the variation of the mentioned hardness

values for all 8 samples.

Fig. 3 - Variation of hardness on samples previously subjected to torsion

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One may observe that two curves are quite similar, namely the mean values and the mean

values without maxima and minima. Consequently, the mean hardness values were

considered as the most representative ones for describing their variation as a function of the

degree of plastic deformation.

Under such circumstances, a correlation could be established between the energies

corresponding to plastic deformation at torsion and the (mean) hardness value obtained after

torsion for each of the 8 samples. Fig. 4 plots the variation of the energy corresponding to the

plastic deformation obtained through differentiated torsion, versus hardness after torsion. The

values of hardness listed in Fig. 4 represent the mean values of the 10 determinations made on

each sample.

Fig. 4 - Hardness versus the energy corresponding to plastic deformation at torsion

Fig. 4 shows that, as the samples were subjected to torsion with increased plastic

deformations, hardness increases, and the explanation being that, by loading the material over

the yield point, its structure gets refined and rearranged, and the grains become finer.

Accordingly, as already mentioned, hardness may be a measure of the degree of plastic

deformation (during exploitation) of the samples (pieces) loading over the yield limit.

Consequently, for sample 1, the one most subjected to torsion (Fig. 1), the energy

corresponding to plastic deformation is of 951 [J], the mean hardness being of 260 HV10. For

sample 8, the least subjected to torsion, the energy corresponding to plastic deformation is of

27 [J], while mean hardness is of 194 HV10. Important to know is whether a plastically

deformed piece (during exploitation or as a result of its technological fabrication) still

possesses sufficient reserve of resistance to further resist the (static or variable) mechanical,

thermal loadings, to the action of the environment, etc. Under such conditions, if the

exploitation of some pieces involves loadings that may induce plastic deformations, the

values of hardness registered in the plastically deformed zone may provide indications on the

degree of plastic deformation suffered by that piece.

TENSILE LOADING AFTER TORSION

N.M. Zarroug (Zarroug, 2003) made separate tests of traction and torsion upon the same

sample, using the yield criterion: σ2+3τ

2 = Y

2, where σ represents normal tension, τ represents

tangential tension and Y is the axial yield stress. Experimental and theoretical results

concerning the elastoplastic response of a circular steel rod subjected to non-proportional

biaxial loadings are reported by Arm (Arm, 1999). He studied: elastoplastic torsion followed

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by tension, keeping the initial angle of twist constant, and elastoplastic tension followed by

torsion, holding the initial axial displacement constant. Experimental results show that when

the rod is initially subjected to a torque and then, keeping the corresponding angle of twist

constant, to a gradually increasing axial load, the rod behaves as if its torque-carrying ability

has been drastically reduced without in any way affecting its axial load-carrying ability.

The present chapter analyzes the influence of the plastic deformation obtained through torsion

upon other mechanical characteristics, respectively yielding stress, ultimate tensile stress and

the energy accumulated for plastic deformation at tensile after torsion, obtained through

subsequent tensile loading.

Following the operation of subjecting the samples to torsion, and their differentiation

according to the degree of plastic deformation, they were subjected to traction up to breaking

(Fig. 5). The following observations may be made in relation with the samples presented in

Fig. 5:

- Sample 8 broke as early as loading to torsion was applied (Fig. 1), consequently it

could not be subjected to traction, any more;

- Sample 0 had not been previously subjected to torsion, but only to tensile, being

considered as a reference for the aspect of the characteristic curves to traction (Fig. 6).

- Sample 7 was subjected to previous torsion, at a large rotation angle – 1164.15 degree;

- Sample 1 was subjected to previous torsion, at a low rotation angle – 92.11 degree;

Sample 7, which had been intensely deformed through torsion, showed no bottleneck

which is specific for toughness materials subjected to tensile. All the other samples show

bottleneck on tensile, suggesting that the samples subjected to a lower deformation degree

through torsion, possess the capacity of plastic deformation through traction. Mention should

be made of the fact that the modalities of plastic deformation caused by the application of the

two types of loadings – torsion and tensile, are different. On torsion, plastic deformation

appears as a result of surfaces sliding under the action of tangential stresses, while loading to

traction causes cleavage (separation) of surfaces. Even under such conditions, it appears that

one type of deformation (sliding) consumes the second type of possible deformation

(cleavage).

Fig. 5 - Samples subjected to torsion and traction

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Fig. 6 plots the specific stress-strain curves, resulted through tensile tests up to

breaking, following their previous loading to torsion with various degrees of plastic

deformation.

Fig. 6 - Characteristic stress-strain curves to traction after torsion

At first sight, considerable differences are observed among the stress-strain curves obtained

through tensile loading up to breaking upon the samples previously subjected to torsion.

Another observation was that the samples subjected to high plastic deformation to torsion

(samples 7, 6) evidence low specific deformations at tensile. On the other side, the same

samples show higher values of the yield limit and ultimate tensile stress. Mention is to be

made of the fact that involved here are apparent yield limit and apparent ultimate tensile

stress, once they are obtained for a material structurally modified through the previous torsion

test. To illustrate this, the curves from Fig. 7 – showing the variation of the yield limit and

ultimate tensile stress (obtained through subsequent traction) comparatively with the energy

corresponding to the plastic deformation obtained through differentiated torsion, and

calculated with relation 3 – were plotted.

Fig. 7 - Tensile yield strength, respectively ultimate tensile strength versus the energy corresponding to the

plastic deformation obtained through differentiated torsion

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It was observed that, at torsion test over the elastic limit, plastic deformations appear, causing

”hardening” of the material, characterized by structural re-arrangement and formation of finer

grains, which explains the increased hardness and higher values for yield limit and ultimate

tensile stress. These modifications are directly related with the degree of plastic deformation

obtained through torsion and expressed by the energy determined with relation (3) and

illustrated in Fig. 2.

Analysis of the stress curves plotted in Fig. 1 and of the characteristic ones, from Fig. 6,

shows that the larger is the area under the torsion curves (Fig. 1), the lower will be the area

under the curves characteristic to tensile (Fig. 5). This distinct observation raises the problem

of the variation of the sum of these areas, which represents the total energies of deformation,

up to breaking – the first part of loading being to torsion, and the second one - to traction. Fig.

8 presents the sum of the energies corresponding to plastic deformation, caused by the torsion

loading and the traction loading applied after torsion. Table 2 also lists the separate values for

the two energies.

Fig. 8 - Sum of energies (torsion + traction)

Table 2 - Energies of plastic deformation to torsion and traction

Sample 8 7 6 5 4 3 2 1 0

EpTensile [J] 0 93 165 168 188 206 285 335 343

EpTorsion [J] 950 732 548 439 278 210 125 27 0

EpR+EpT [J] 950 825 714 607 466 416 410 363 343

Based on the previous observation, referring to the decrease of the energy of plastic

deformation to traction test, when the energy of plastic deformation to torsion increases the

sum of the two energies of plastic deformation is expected to be approximately constant. Fig.

8 shows that the plastic energy necessary for failure through torsion is much higher than the

plastic energy necessary for failure through traction. Consequently, for pure torsion, the

energy accumulated by the sample for breaking - sample 8 – is of 951 [J], while the energy

necessary for breaking through the pure traction loading of a sample is of 343 [J] - sample 0.

In this way, the sum of energies is affected by the higher value of the energy accumulated in

the torsion sample for its breaking. In the graph plotted in Fig. 8, showing the sample

subjected to breaking through pure torsion (sample 8), the aspect of the curve is a decreasing

one.

Topic_B: Experimental Mechanics

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Loading to torsion up to the plastic domain leads to the modification of hardness, namely to

its increase with the increase of the degree of plastic deformation. During exploitation, the

degree of plastic deformation through torsion may be obtained from the determination of

hardness, by means of the curve of correlation between hardness and the energy of plastic

deformation, plotted in Fig. 4. For example, if, during exploitation, for a component which

worked under torsion, a hardness value of 230 HV is registered, there results that the

respective piece had been plastically deformed, with an energy of approx. 698 J,

corresponding to a stress occurring somewhere between curves 6 and 7 (Table 2 and Fig. 1),

more close to curve 7. Under such conditions, if the respective component is to be subjected

to traction, as well, it will not be capable to accumulate a large amount of energy of plastic

deformation up to breaking. To provide a more precise solution to this issue, the variation

curve of hardness versus the plastic energy accumulated on loading to traction up to break,

applied after the torsion loading, was plotted (Fig. 9). In this way, the component whose

hardness, determined after plastic deformation to torsion, is of 230 HV, may accumulate up to

breaking at traction only approx. 125 J, whereas a sample subjected from the beginning only

to traction loading may accumulate 343 [J].

Fig. 9 - Hardness after torsion versus the plastic energy accumulated on loading to

traction (after torsion) up to breaking

CONCLUSIONS

The present study analyzed the influence of plastic deformation obtained through torsion upon

other mechanical characteristics, such as: hardness, the apparent yield limit, the apparent

ultimate tensile stress and the energy remaining for plastic deformation for tensile.

To this end, 8 samples with full circular section, they were loaded first to torsion, at different

levels of plastic deformation. The differences of loading to torsion were obtained on the basis

of torsion angle variation. The same aspect is observed for all the 8 curves representing the

variation of the torque versus the rotation angles, the differences referring to the value of the

torsion moment being due to the device with cogged wheel used with which loading had been

performed.

In a subsequent stage, for the samples subjected to torsion, there have been determined both

the energy corresponding to plastic deformation, from the variation diagrams of the torsion

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moment versus the rotation angle, and Vickers HV10 hardness. Calculation of the energy

corresponding to plastic deformation to torsion made use of relation (3), whose parameters are

plotted graphically in Fig. 2.

Representation of the variation of the energy corresponding to plastic deformation, obtained

through differentiated torsion, versus hardness after torsion (Fig. 4) shows that, when the

samples have been subjected to torsion with higher plastic deformations, hardness increased.

Consequently, hardness may be a measure for determining the degree of plastic deformation

(during exploitation) of the components loaded over the limit of elasticity through torsion.

Therefore, for sample 8, the one most subjected to torsion (Fig. 1), the energy corresponding

to plastic deformation is of 951 [J], hardness being of 260 HV10. For sample 1, the least

subjected to torsion one (Fig. 1), the energy corresponding to plastic deformation is of 27 [J],

mean hardness recording a value of 194 HV10.

Following their loading to torsion, the samples - differentiated by their degree of plastic

deformation – were subjected to traction, up to breaking. Fig. 6 shows significant differences

among the characteristic curves obtained through traction up to breaking applied to the

samples previously subjected to torsion. The observation made was that the samples suffering

important plastic deformations to torsion show low specific deformations at traction, sample 7

from Fig. 5. On the other side, the same samples show the higher values of the yield limit and

ultimate tensile stress, the higher was the plastic deformation obtained through torsion.

Therefore, during torsion tests, plastic deformation causing ”hardening of the material” occur,

which explains both the increased hardness and the increased of the yield limit and ultimate

tensile stress. On the other side, it was observed that, the larger is the area under the torsion

curves (Fig. 1), the smaller will be the area under the curves characteristic to traction (Fig. 5).

This is an obvious observation, involving the variation of the sum of these areas, which

represent energies of total deformation up to breaking – the first part of loading being to

torsion, while the second is to traction. Fig. 8 shows that the energy necessary for breaking

through torsion is much higher than the plastic energy necessary for breaking through

traction. Accordingly, for pure torsion, the energy accumulated by the sample for failure is of

951 [J], while the energy necessary for failure through loading to traction of similar sample is

of 343 [J]. Consequently, the sum of the energies is affected by the higher value of the energy

accumulated in the torsion sample in view of its break.

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