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Some Actual Problems of Fracture Mechanics of Materials and Structures 429

Such loading provides the gradual propagation of the crack. The dependence of

critical value of force

∗

on the crack length l is shown in Fig. 12 by dashed line 1–3.

The plate was tensioned step-wise: 1 – in the dry air; 2 – in the surface-active en-

vironment (for glass plate, for example, in the water); 3 – the same plate (after envi-

ronment removal) was again tensioned in the air. The experimental results (glass,

metal) showed that the surface-active environment decreases the static fatigue load-

ing by 10–15% as under cyclic loading. Thus the contradiction between cyclic and

static strength was removed for surface-active environment on one hand, and on the

other it was shown that the most important material characteristics is its resistance to

crack propagation in it, i.e. its crack growth resistance. It is typical of macrovolumes

and not of mesovolumes. Just the surface-active environment changes the value of

crack growth resistance, i.e. the characteristics of mesovolumes and not the macro-

volumes of the material as a whole. The present-day fracture mechanics differs from

the classical concepts of fracture mechanics. It includes the characteristics of meso-

volumes during the estimation of materials strength. The change of these character-

istics under the interaction with the surface-active environment can substantially in-

fluence the static and cyclic durability.

Evaluation of the environment effect on structural materials strength in terms of

the change of their crack growth resistance is an important problem of engineering

practice for assessment of durable-operation structural elements lifetime.

Within the frames of fracture mechanics concept the residual cyclic durability of

the materials is determined by using (

vK− ) diagram according to formula (10). As

it is known, at first the crack growth rate (

v ) diagrams versus the SIF values

(

,KKΔ ) in the environment and in the dry air were constructed. Already at the

beginning of 80s of the 20

century in the science on materials strength a distur-

bance appeared concerning the invariance of (

vK− ) diagrams constructed like this

in the given surface-active environment. If the diagrams are constructed as in the

case of dry (neutral) air, not in the air – but in the given surface-active environment,

in particular corrosive environment, one can get different (v ~ K

) diagrams if their

construction begins from different SIF values (Fig. 13). Using such an approach one

will receive different (v ~ K

) diagrams. So they are not invariant, that is they are not

characteristic of the system „deformed metal–environment”.

In the mid 80s of the 20

century in was stated by I.M. Dmytrakh, L.V. Ratych

and V.V. Panasyuk [40] that non-invariance of such diagrams is stipulated by: the

crack growth rate (

v ) in metals under loading and environments influence, in par-

ticular corrosive environment, depends not only on SIF (

Imax

K ) but also on the

physico-chemical properties of the environment itself and the metal near the crack

tip. It does not depend on their properties on the smooth surface. If, for example, it

is considered that physicochemical properties of the system „metal–environment”

are characterized by the hydrogen factor (pH) and electrode potential (

), these

values for the given system on the surface of the body

()

pH ,

⎡

⎤

⎣

⎦

and near the

crack tip

()

pH ,

⎡⎤

⎣⎦

are not equal.

430 V. Panasyuk and I. Dmytrakh

Fig. 13. Dependence of ((v–K

)-diagrams) of crack growth (v) on K

under the simultaneous

action of loading and environment, constructed using different initial

()

K : a – cyclic

loading; b – static loading [39]

Based on these facts a new model [40] of the physicochemical situation at the

corrosion crack tip in metal materials was proposed. This situation supposes that

the crack growth rate (

v ) depends on

()

,pH

K and

(Fig. 14).

Fig. 14. Scheme of the physicochemical situation near the crack tip

According to this model the ( vK− )-diagram of the corrosive crack growth is

determined as a certain function of parameters, which characterise the stress-strain

state and physicochemical situation of the system „material–environment” at the

crack tip, that is:

()

;pH ;

vfK=

, (13)

Some Actual Problems of Fracture Mechanics of Materials and Structures 431

where

K is the stress intensity factor for the cracked body;

()

and

are

the hydrogen factor and electrode potential in the vicinity of the crack tip in the

system „material–environment”.

It follows from formula (13) that to receive the invariant (

vK− ) diagram it is

necessary to provide such conditions:

()

pH const

= and const

ϕ= . (14)

Experiments show [40, 41] that the maximum rate (

max

v ) in the system „metal–

corrosive environment” will be when parameters

and

()

reach minimum

values, that is

()

(

)

min

max I

;pH ; .

vfK=ϕ (15)

Thus, diagram (15) becomes the basic diagram for estimation of the durability of

cracked structural elements in the given service environments.

The original methods for experimental investigations of metal local corrosive

fracture with the account of electrochemical processes parameters in the crack are

developed for the proposed model scheme [40, 41]. The methods of direct electro-

chemical measurements in corrosive cracks are based on the application of special

minielectrodes of different construction [41], which are placed in special holes of

the tested cracked specimen.

Such an approach was a principally new tool for the determination of the influ-

ence of surface-active and corrosive-aggressive environments on the materials

physicomechanical characteristics. These methods are effective and agree well

with well-known (but low effective) approaches of American scientists [42]. In

accordance with these approaches the enveloping curve of a great deal of experi-

mental data received under different test conditions is used as a basic (calculation

diagram (

~vK)-diagram). The (

~vK) diagram received by equation (15) prac-

tically coincides with this curve (see Fig. 15).

At the end the problem of hydrogen (or hydrogen-containing environments) in-

teraction with metals should be mentioned. This problem needs a special attention.

It is a many-sided problem of the present-day material science, and is concerned

with the phenomena of hydrogen embrittlement (or plasticization) of metals under

their deformation in the hydrogen or hydrogen-containing environments. In whole

this problem includes the following aspects:

– development of materials resistant to hydrogen embrittlement and develop-

ment of reliable criteria for estimation of the given materials service ability in hy-

drogen environment to use them in engineering structures, in particular for hydro-

gen storage and transportation;

– use of hydrogen as a technological media for effective treatment of materials

and improvement of their functional properties.

Currently the main task for researchers in the field of fracture mechanics and materi-

als strength in hydrogen-containing environments is the consolidation of their efforts

432 V. Panasyuk and I. Dmytrakh

Fig. 15. Basic diagram of the cyclic corrosion cracking of the weld material (10ХМФТ

steel) constructed using traditional [42] approaches (doted line) and in the frames of the

proposed concept [41] (firm line) in the reactor water of boron regulation (1%

ВО

+КОН up to рН 8): ▲ – 0.017 Hz, 80°С; – 0.017 Hz, 25°С; – 0.1 Hz, 80°С;

– 0.1 Hz, 25°С;

– 0.33 Hz, 80°С; – 0.33 Hz, 25°С; – 1.0 Hz, 80°С; – 1.0 Hz,

25°С;

– 0.33 Hz, 25°С,

min

pH pH 6

. R = K

Imin

Imax

– asymmetry of the loading cycle

on the studies, which concerned to the development of future hydrogen energy infra-

structure. As typical sample of such research the recent works [43-45] can be men-

tioned where the hydrogen influence on low-alloyed pipeline steels was considered.

Here the existence of some critical hydrogen concentration in metal, which causes

the significant loss of local fracture resistance of material, was shown. For assess-

ment of local strength in presence of hydrogen the diagram “work of local fracture –

hydrogen concentration” has been proposed and verified [43].

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Some Actual Problems of Fracture Mechanics of Materials and Structures 435

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Karpenko Physico-Mechanical Institute, Lviv, in Ukrainian (1999)

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1554–1559 (2010)

Cyclic Plasticity with an Application to

Extremly Low Cycle Fatigue of Structural Steel

Dragoslav Šumarac

and Zoran Petrašković

Faculty of Civil Engineering, University of Belgrade, Serbia

Research-productive Centre System DC 90, 11000 Belgrade, Serbia

sumi@eunet.rs, dc90@eunet.rs

Abstract. In the present paper the Preisach model of hysteresis is applied to

model cyclic behavior of elasto-plastic material. The problem of axial loading of

rectangular cross section will be studied in details. Hysteretic stress-strain loop

for prescribed history of stress change is plotted for material modeled by series

connection of three unite element. All obtained results clearly show advantages

of the Preisach model for describing cyclic behavior of so called stable plastic

material. Other effects such are racheting and creep will be studied elsewhere.

In this paper extremely low cycle fatigue will also be examined. Extremly low

cycle fatigue stands for number of cycles to failure in between 10

and 20.

The stress level is larger than the yield stress and the plastic strain is of the same

magnitude as the elastic strain. In this paper it is shown that this case is of

importance to dampers applied for reconstruction of earthquake damaged

structures.

1 Introduction

In the present paper the model of elasto-plastic behavior for quasy static cyclic

loading will be explained. Special attention will be made to hysteretic behavior

and mathematical models applied for their description. In this paper stress-strain

loops will be constructed for cyclic axial loading. From this examples it is obvi-

ous that suggested (Preisach) model is simple enough and very appropriate to

describe hysteretic behavior of elasto-plastic material. In the case of racheting,

cyclic creep and fatigue some aditional improvements of Preisach model should

be applied. An interesting proposal for Very low cycle fatigue has been done by

Dufailly and Lemaitre [7]. In this case number of cycles to failure is of the order

of 10 to 20. Plastic strain is much larger than the elastic one as well as the dissi-

pative work to the elastic power. In this paper it will be show that this case is

of importance to dampers applied for reconstruction of earthquake damaged

structures.

438 D. Šumarac and Z. Petrašković

2 The Preisach Model of Hysteresis

The Preisach model of hysteresis, dealing with the problem of magnetism, is de-

scribed thoroughly in the monograph of Mayergoyz [6]. The phenomenon of hystere-

sis occurs in various branches of physics: mechanics, magnetism, optics, adsorption

etc. Application of the Preisach model to cyclic behavior of elasto-plastic material is

shown in (Lubarda et al., [3]) and extended to cyclic bending by (Sumarac and Stosic,

[5]). In this paper short outline of Preisach model of hysteresis will be presented. Ac-

cording to Mazergoyz [6], the Preisach model implies the mapping of an input of

strain ε (t) on the output of stress σ (t) in the integral form:

() ()

() ,tPGtdd

αβ

σ= α

εα

∫∫

(1)

Fig. 1. Elementary hysteresis operator

where G

α,β

is an elementary hysteresis operator given in Figure 1. Parameters α

and

β are up and down switching values of the input, while P (α, β) is the Preisach

function. i.e. a weight (Green

׳s) function of the hysteresis nonlinearity to be

represented by the Preisach model. The domain of integration of integral (1) is

right triangle in the

α, β plane, with α=β being the hypotenuse and (α

,β

= - α

) be-

ing the triangular vertex (Fig.2). History of loading corresponds to staircase line L

(t) which divides triangle into two parts (Lubarda et al., [2]). Maxima or minima

of loading history are represented by the vertices with coordinates (

α,β) on stair-

case line L(t) in such a way if the input at a previous instant of time is increased,

the final link of L(t) is horizontal, and vise versa if it is decreased it is vertical.

Therefore, the triangle is divided into two parts with the positive and negative val-

ues of G

α,β

by the interface staircase line L(t). From formula (1) it is than obtained:

() ( ) () ( ) ()

()()

At At

tPGtddPGtdd

+ −

αβ αβ

σ= α

εα

−α

εα

∫∫ ∫∫

(2)

ε(t)

-1

α,β

ε(t)