Strength and fracture criteria application in stress concentrators areas/Stiprumo ir irimo kriteriju taikymas itempiu koncentratoriu zonose.

Ziliukas, A. ; Surantas, A. ; Ziogas, G. 等

1. Introduction

Strength and fracture parameters are very important in fracture mechanics. Hereupon when at crack tip exists stress concentration complex stress state is obtained. Here is not enough to determine highest stresses while strength and fracture are governed by equivalent effect. This effect also depends on material properties: elasticity and plasticity. In various studies [1-2] mostly brittle fracture is researched and plastic fracture studies are much more rare [3]. Fracture depend on crack size as well [4, 5]. Considering defect like round hole strength can be computed according theory of elasticity [6] and especially for fracture of brittle materials Griffiths criteria shall be applied [4]. But using this criteria it is complicated to determine crack growth and its area size. This task requires additional studies.

2. Brittle fracture criteria

Griffith's energy growth criteria

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)

where [DELTA][PI] is system potential energy variation on increased crack surface [DELTA]S; [G.sub.C] is critical energy value.

However applying this criteria becomes hard to determine increase value of [DELTA]S. Griffiths energy growth criteria is easier to calculate applying stress intensity coefficient [K.sub.I]. Then

G = [K.sup.2.sub.1]/E' (2)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where E is modulus of elasticity; u is Poisons' ratio, [K.sub.I] is stress intensity factor calculated on opening case (I moda).

Critical energy [G.sub.C] value calculated by the equation

[G.sub.C] = [K.sup.2.sub.IC]/E' (3)

where [K.sub.IC] is critical stress intensity factor.

Critical stress intensity factor [K.sub.IC] is obtained by the equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4)

where [[sigma].sub.[infinity]] is distant stresses, [2l.sub.c] is critical crack length, F is corrective function, taking into account crack and element geometry shown in Fig. 1.

[FIGURE 1 OMITTED]

For a plate with a far field uniform stress [[sigma].sub.[infinity]] we know that there is a stress concentration factor of 3 [6]. For a crack radiating this hole we consider two cases. In one the crack is short l/2a [right arrow] 0 and thus we have an approximate for field stress of 3[sigma] and for an edge crack [F.sub.1] = 1.12. Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5)

In second case the crack is long 2a [less than or equal to] 2l+2a and we can for all practical purposes ignore the presence of hole and assume that we have central crack with an effective length. Then

[l.sub.eff] = 2l + 2a/2 = l + a (6)

thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7)

In studies [7] the plate with a hole and at its edge appeared crack [K.sub.IC] is obtained by the equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)

where a is radius of the hole.

Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (9)

3. Multiparametric fracture criteria

McClintock and Leguilon [3, 4] offered strain fracture criteria when plastic zone at crack tip is taken into account and stresses [[sigma].sub.[infinity]] are calculated:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (10)

where [[sigma].sub.c] is critical stresses at crack tip.

If no hole exists (a = 0), [[sigma].sub.[infinity]] = [[sigma].sub.c].

When investigating maximum stresses at the crack tip [[sigma].sub.max] and crack tip area length [DELTA][l.sub.max] in studies [5, 8] fracture resistance stresses [[sigma].sub.coh] were obtained

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (11)

As a reference of studies [9]

[G.sub.c] = 1/2 [DELTA][l.sub.max][[sigma].sub.max] (12)

In paper [10] an obtained criterion is described

[G.sup.s.sub.c] = [h.sub.c][gamma] (13)

where [gamma] is fracture energy density (quantity of energy for unit volume), [h.sub.c] is critical crack area width dependable on stress concentration, [G.sup.s.sub.c] is specific fracture energy.

With these principles three parameters criteria are formed [10]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (14)

where [alpha] is the parameter indicating level of stress concentration (0 < [alpha] < 1), [G.sub.c] is material fracture energy with existing crack, [G.sup.u.sub.c] is fracture energy under tensile ultimate strength. Fracture energy is determined according strength and displacement tension diagram.

When fracture is specified by specific fracture energy then [alpha] = 1, and when fracture is specified by stresses, then [alpha] = 0.

4. Crack area studies

There are no analytic references in study [10] on how to specify crack area width [h.sub.c] and for parameter a assessment only the volume at crack tip analysis is proposed. But introduced analysis is approximate because complex stress and strain state exists. Therefore to determine crack area width would be the most proper by known stresses and fracture parameters equations [6]. Those equations for plane stress are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (15)

here

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (16)

This radius shows area width before crack growth and it is calculated according fracture parameters. Radius [r.sub.cr] describes area width with critical stress [[sigma].sub.c].

Radial normal stresses around the hole [[sigma].sub.rr] are calculated

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (17)

Circular normal stresses are calculated

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (18)

Tangential stresses are obtained

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (19)

With angle [theta] = 0, [[sigma].sub.r[theta]] = 0, circular stresses [[sigma].sub.[theta][theta]] are the main stresses [[sigma].sub.1] and radial stresses are the main stresses [[sigma].sub.2].

Adopting [[sigma].sub.c] = [[sigma].sub.11] = [[sigma].sub.[theta][theta]] radius [r.sub.[alpha]] is calculated from Eq. (18) taking into account [theta] = 0. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (20)

Parameter [alpha] will be obtained

[alpha] = [r.sub.cr]/[r.sub.[alpha]] (21)

when [r.sub.cr] = 0, [alpha] = 0 fracture with no crack exists.

Taking into account that [theta] = 0, radius [r.sub.cr] for plane stress calculated by Eq. (16)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (22)

For plane strain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (23)

Then three parameters criteria (14) can be obtained taking into account:

1) [G.sub.C] which is calculated from Eq. (3);

2) fracture energy [G.sup.u.sub.c] obtained by plate tension with

no crack and hole in F-[DELTA]u coordinates till ultimate strength;

3) parameter [alpha] which is obtained from Eq. (21).

5. Experiment

For the experiment three different steel grade (Table 1) specimens were chosen: 1) Steel 45, 2) Steel 15XCH[??], 3) Steel 35XrCA.

Round hole was drilled through the plate and cracks were made inside the hole with the help of laser cuts.

Energy [G.sup.u.sub.c] was calculated and expressed by tension diagram area till ultimate strength. For the plane stress plate dimensions--2x50 mm and diameter of holes--2; 5 mm. When 2 mm plate is broken by tension force no significant plate cross-section shrinkage was noticed. That's why this deformation is considered as a plain stress.

Fracture characteristics [[sigma].sub.[infinity]], [[sigma].sub.1,c], [l.sub.c] and [K.sub.IC] were obtained during the experiment and calculative area [r.sub.[alpha]] are shown in Table 2. Calculative fracture parameters values [r.sub.cr], [alpha], [G.sub.C], [G.sup.u.sub.c] and [G.sup.S.sub.c] are shown in Table 3.

For the plain strain, plate dimensions--4x50 mm and diameter of holes--2; 5 mm. When 4 mm plate is broken by tension force, plate cross-section shrinkage is noticed. That's why this deformation is considered as a plain strain.

Fracture characteristics [[sigma].sub.[infinity]], [[sigma].sub.1,c], [l.sub.c] and [K.sub.IC] were obtained during the experiment and calculative area [r.sub.x] is shown in Table 4. Calculative fracture parameters values [r.sub.cr], [alpha], [G.sub.C], [G.sup.u.sub.c] and [G.sup.S.sub.c] are shown in Table 5.

Obtained fracture energy results show reliance on size of hole and crack relation and deformation state.

6. Conclusions

1. Fracture analysis indicates that structural elements made of plastic and semiplastic materials with stress concentration around holes edges generate plastic deformations and fracture with crack growth. For strength and fracture assessment is necessity to apply multiparametric fracture criteria.

2. As yet two parameters fracture criteria were used, therefore three parameters criteria describes fracture process in more details. Those parameters are: material fracture energy with existing crack, fracture energy obtained when ultimate strength is reached and parameter indicating stress concentration level.

3. Evaluating crack area width and deformation width of stress concentrator edges, similarly calculated fracture energy with and without crack--certain specific fracture energy calculations are provided. Strength and fracture characteristics determined from experimental data by testing plates with holes specimens.

Received March 05, 2010

Accepted June 03, 2010

References

[1.] Leonavicius, M., Bobyliov, K., Krenevicius, A., Stupak, S. Superoid graphyte cast iron specimiens with defects cracking investigation. -Mechanika. -Kaunas: Technologija, 2008, Nr.1(69), p.13-18.

[2.] Bazant, Z.P., Pijaudier-Cabot, G. Non-local continuum damage, localization, instability and convergence. -Journal of Applied Mechanics, 1988, v.55, p.287-293.

[3.] McClintock, F.A. Ductile fracture instability in shear. -Journal of Applied Mechanics, 1958, v.25, p.582-588.

[4.] Leguilon, D. Strength or toughness? A criterion for crack onset at a notch. -European Journal of Mechanics--A/Solids, 2002, v.21, No.1, p.61-72.

[5.] Hutchinson, J.W., Evans, A.G. Mechanics of materials: top--down approaches to fracture.-Acta Materialia, 2000, v.48, No.1, p.125-135.

[6.] Ziliukas, A. Strength and Fracture Criteria.-Kaunas: Technologija, 2006.-208p. (in Lithuanian).

[7.] Newman, J.C. NASA Report, TND-6367, 1971.

[8.] Mohammed, I., Leichti, K.M. Cohesive zone modelling of crack nucleation at bimaterial corners. -Journal of Mechanics Physycs and Solids, 2000, v.48, p.735-764.

[9.] Camacho, G.T., Ortiz, M. Computational modelling of impact damage in brittle materials.-International journal of Solids and Structures, 1996, v.33, No.20-22, p.2899-2938.

[10.] Li, J., Zhang, X.B. A criterion study for nonsingular stress concentrations in brittle or quasi- brittle materials. -Engineering Facture Mechanics, 2006, v.73, p.505-523.

A. Ziliukas *, A. Surantas **, G. Ziogas ***

* Kaunas University of Technology, Strength and Fracture Mechanics Centre, K^stu?io st. 27, 44312 Kaunas, Lithuania, E-mail: [email protected]

** Kaunas University of Technology, Strength and Fracture Mechanics Centre, Kqstu?io st. 27, 44312 Kaunas, Lithuania, E-mail: [email protected]

*** Kaunas University of Technology, Strength and Fracture Mechanics Centre, Kqstu?io st. 27, 44312 Kaunas, Lithuania, E-mail: [email protected]

Table 1
Mechanical properties of chosen materials

Material           Yield strength        Ultimate strength
                 [[sigma].sub.Y], MPa   [[sigma].sub.U], MPa

Steel 45                 320                    680
Steel 15XCHA             350                    630
Steel 35XrCA             404                    730

Table 2
Strength, fracture and crack tip area parameters under
plane stress

  No.     a, mm   [2l.sub.c], mm    [[sigma].sub.     [Q.sub.1,c],
                                   [infinity]], MPa        MPa

                                       Steel 45

   1        2           1                424              1427
   2        5          1.3               372              1251

                                     Steel 15XCHA

   3        2          1.6               440              1480
   4        5           2                394              1324

                                     Steel 35XrCA

   5        2           3                490              1648
   6        5          3.7               441              1484

  No.     a, mm     [r.sub.      [r.sub.cr],  a
                  [delta]], mm      mm

                              Steel 45

   1        2          2           0.5      0.250
   2        5         3.8          0.65     0.17

                            Steel 15XCHA

   3        2          2           0.8       0.4
   4        5         3.8           1       0.26

                            Steel 35XrCA

   5        2          2           1.5      0.75
   6        5         3.8          1.85     0.49

Table 3
Values of energy and fracture parameters under plane
stress

No.       [K.sub.Ic],       [G.sub.c], kJ/   [G.sup.u.sub.c] *
        Mpa*[m.sup.3/2]       [m.sup.2]         kJ/[m.sup.2]

                                    Steel 45

 1             80                32.0               4.84
 2
                                 Steel 15XCH[??]

 3            105                55.1               8.03
 4

                                Steel 35X[GAMMA]CA

 5            160               128.0               7.05
 6

No.    kJ/[m.sup2]

 1       11.63
 2        9.45

         Steel 45

 3        26.9
 4        20.2

      Steel 15XCH[??]

 5        97.8
 6        66.3

Table 4
Strength, fracture and crack tip area parameters under
plane strain

 No.    a, mm    [2l.sub.c], mm    [[sigma].sub.       [[sigma].
                                  [infinity]], MPa   sub.1,c], MPa

                          Steel 45

  1       2           0.8               450              1512
  2       5           1.1               420              1411

                        Steel 15XCHA

  3       2           1.4               480              1612
  4       5           1.8               440              11478

                        Steel 35XTCA

  5       2           2.5               520              1747
  6       5           3.1               460              1545

 No.      [r.sub.      [r.sub.cr]   a
        [alpha]], mm      mm

                  Steel 45

  1          2           0.37     0.18
  2         3.8          0.42     0.11

                Steel 15XCHA

  3          2           0.56     0.28
  4         3.8          0.66     0.17

                Steel 35XTCA

  5          2          0.115     0.057
  6         3.8         0.146     0.038

Table 5
Values of energy and fracture parameters under plane
strain

  No.      [K.sub.IC],      [G.sub.c],     [G.sup.u.sub.c]
          Mpa*[m.sup.3/2]   kJ/[m.sup.2]     kJ/[m.sup.2]

                             Steel 45

   1           72.8             29.1             4.4
   2

                          Steel 15XCH[??]

   3           95.5             50.1             7.3
   4

                         Steel 35X[GAMMA]CA

   5           145.6           116.5             6.4
   6

  No.     [G.sup.S.sub.c]
           kJ/[m.sup.2]

            Steel 45

   1           8.94
   2           7.14

         Steel 15XCH[??]

   3           19.28
   4           14.79

        Steel 35X[GAMMA]CA

   5           12.67
   6           10.63