Fracture of laminated rectangular bar after buckling/Laminuoto staciakampio strypo irimas po klupdymo.

Ziliukas, A. ; Malatokiene, A.

1. Introduction

The delamination of the composites depends on its matrix and changes mechanical characteristics of reinforced elements during deformation. The mechanical behavior of laminated composites during compression is the case when the bending moment appears besides the axial forces. The thread experiences normal stresses and shear stresses [1-3]. Similar works were done while analyzing interfaces of I-beam shelves and walls [4] columns [5] beams [6], and cases of bar buckling depending on their geometry [6, 7, 9]. J. Brewer and P. Langace, M. Fenske and A. Vizzini [9 - 11] suggested the measuring criteria of delamination. Authors [11, 12] were solving the problems of composite fracture. However, the problem of investigating composite delamination remains topical, because the investigations and evaluations of thread remain difficult.

Composite fracture measuring elasticity characteristics for separate layers is analyzed by E. Saouma [13], Z. Gurdal [14]. With mechanical characteristics of separate layers known the measuring of composite fracture is possible. This allows selecting optimal lamination materials while producing bars of significant resistance.

2. Delamination of laminated bars during buckling

In case of buckling, Fig. 1 according to Euler's formula, the critical buckling force is presented as follows

[F.sub.cr] = 4[[pi].sup.2]([EI.sub.ef]) / [L.sup.2] (1)

where [F.sub.cr] is critical buckling force; E is modulus of elasticity; L is length of bar; [I.sub.ef] is minimum moment of inertia.

[FIGURE 1 OMITTED]

The important characteristic of material is composite modulus of elasticity [E.sub.C]. It is calculated in the following way [15]

[E.sub.C] = 2[t.sub.v][E.sub.v] + 2[t.sub.m][E.sub.m] + [t.sub.f][E.sub.f] / t (2)

where t is thickness of a layer, indexes v, m and f mean cover, thread and filling respectively.

The modulus of elasticity [E.sub.v] is accepted as resin. Also the composite[E.sub.c] is received experimentally.

The limitary shear stresses [[tau].sub.lim] are calculated in the following way [15]

[[tau].sub.lim] = 1 / 2 sin2[theta][[sigma].sub.y] (3)

where [[sigma].sub.Y] is yield stress; [theta] is angle of the layers with regard to stretching axis.

The lateral displacement is calculated as follows [16]

w = [w.sub.max] / [cos 2[pi]x / L - 1]) (4)

where w is lateral displacement; x is coordinate in the longitudinal direction of the bar; [w.sub.max] is maximum lateral displacement in the middle part of the bar during delamination.

Thus, when the plate is compressed by F force, the transverse forces Q are obtained in the following way [16]

Q = Fsin[theta] = F[square root of ([tan.sup.2][theta] / [tan.sup.2][theta] + 1)] (5)

where

tan[theta] = dw / dx (6)

In order to evaluate composite strength, various criteria are applied. One of the simplest is Tresca criterion, which evaluates normal stresses and shear stresses [10]

[square root of [sigma.sup.2.sub.x] + 4[[tau].sup.2.sub.xy]] / 2 [less than or equal to [[tau].sub.lim] (7)

where [[sigma].sub.x] is normal stresses; [[tau].sub.xy] is shear stresses.

It is important that normal stresses [[sigma].sub.y] in the direction of axis y and shear stresses [[tau].sub.yz] on the plane yz are quite small and may not be considered.

Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

where A is area of cross-section; M is bending moment; [E.sub.eff] is elasticity modulus of the laminated bar.

Normal stresses are calculated in the following way

[[sigma].sub.x] = Fcos[theta] / A([V.sub.v] + n[V.sub.f]) (9)

where [V.sub.v] and [V.sub.f] are volumes of resin and reinforced elements, and n is the ratio of elasticity moduli of reinforcement and matrix.

According to the strength criterion of Mises [15]

[square root of [[sigma].sup.2.sub.x] + 3[[tau].sup 2.sub.xy]= [[sigma].sub.Y] (10)

where [[sigma].sub.Y] is yield stress.

Authors of this paper apply polynomial strength criteria [18]

F [([sigma].sub.1],[[sigma].sub.2],[[tau].sub.12]) = [R.sub.11][[sigma].sup.2sub.1] + [R.sub.22][[sigma].sup.2.sub.2] + [S.sub.12][tau].sup.2.sub.12 = 1 (11)

where [tau]12 = [[sigma].sub.1]- [[sigma].sub.2] / 2, R and S are constants, [[sigma].sub.1], [[sigma].sub.2] are principal stresses.

R and S constants are found from the boundary conditions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

Then the Eq. (11) is as follows

[[[sigma].sub.1] / [[bar.sigma].sub.1)].sup.2] + [[[sigma]2 / [[bar.sigma].sub.2]].sup.2] + [[[tau].sub.12] / [[bar.tau].sub.12]].sup.2] = 1 (13

The stresses [bar.[sigma], [bar.[[sigma].sub.2]] are obtained as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

When the strength criterion is put in the form [17]

f([[sigma].sub.1], [[sigma].sub.2], [[tau].sub.12]) = [R.sub.1] [sigma]1 + R2 [sigma] 2 + R11 [sigma] 2 +

+[[R.sup.2.sub.22][[sigma].sup.2.sub.2] + [S.sub.12][[tau].sup.2.sub.12] = 1 (15)

the boundary conditions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

We write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

According to the experimental tests [17] strength criterion Eq. (17) corresponds the experimental results better than criterion Eq. (13) and even more precisely than criteria Eqs. (7) and (11).

However, the polynomial strength criteria show formal approximation of experimental data in the coordinates of principal axes. These criteria become more complex in other coordinates. Therefore, the tensoric strength criteria are applied. For example, when the orthotropic material moves from the principal axes 1 and 2 to the turned axes 1' and 2' at the angle [omega] = 45[degrees], the strength criterion is presented in the following way

f([[sigma].sub.1], [[sigma].sub.2], [[tau].sub.12]) = [R.sub.1][[sigma].sub.1] + [R.sub.2][[sigma].sub.2] + [R.sub.11][[sigma].sup.2.sub.1] + + [R.sub.12][[sigma].sub.1][[sigma].sub.2] + [R.sub.22][[sigma].sup.2.sub.2] + [S.sub.12][[tau].sup.2.sub.12] = 1 (18)

When the boundary conditions are applied to obtaining constants, based on the Eq. (16), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

This criterion differs from the criterion Eq. (17) because new constant [R.sub.12] cannot be obtained, according to the conditions of Eq. (16).

Acording the tensoric criterion [18], which is presented in the following way

[m.sub.1][[sigma].sub.i] + [m.sub.2][[sigma].sub.0] [less than or equal to] [[sigma].sub.Y][[sigma].sub.Y,[mu]] (20)

where [m.sub.1], [m.sub.2] are constants materials; [sigma][Y.sub.[mu]] is strength limit at [[mu].sub.[sigma]] stress state; [[sigma].sub.1] is intensity of stresses (when [[sigma].sub.x] is used and [[tau].sub.xy], [[sigma].sub.i] = 1 / [square root of 2] [square root of [[sigma].sup.2.sub.s]+3[[tau].sup.2.sub.xy]); [[sigma].sub.0] = [[sigma].sub.1]+[[sigma].sub.2] / 3 = [[sigma].sub.x] / 3 is average stress.

Parameter of stress state

[[mu].sub.[sigma]] = 2[[sigma].sub.2] - [[sigma].sub.1] - [[sigma].sub.3] / [[sigma].sub.1]- [[sigma].sub.3] = 2[[sigma].sub.2]- [[sigma].sub.1] / [[sigma].sub.1] = -1 (at [[sigma].sup.x] and [T.sub.xy]), i.e. [[sigma].sub.1] = [[sigma].sub.Y,t] while stretching, and while compressing when [[sigma].sub.3] stress is used, [[mu].sub.[sigma]] = +1 and [[sigma].sub.3]=[[sigma].sub.Y,c]

Then criterion Eq. (20) is presented in the following way

1 / [square root of 2] [m.sub.1][square root of [[sigma].sup.2.sub.x+3[[tau].sup.2.sub.xy]]+[m.sub.2] [[sigma].sub.x] / 3 [less than or equal to [[sigma].sub.Y,c] (21)

With criterion Eq. (21) given in nonlinear form

[M.sub.3][m.sub.1]([[sigma].sup.2.sub.x + 3[[tau].sup.2.sub.xy]]) + [m.sub.4][[sigma].sup.2.sub.x][less than or equal to [[sigma].sup.2.sub.Y,c] (22) we obtain

([m.sub.3] + [m.sub.4]) [[sigma].sup.2.sub.x]+ [m.sub.3][[tau].sup.2.sub.xy [less than or equal to [[sigma].sup.2.sub.Y,c] (23)

In order to solve the delamination problem of a composite, authors of the paper apply strength criterion Eq. (23). Considering Eqs. (3) and (23), the strength criterion is presented in the following way

[[sigma].sup.2.sub.x][[m.sub.3]+[m.sub.4]+1 / 2[m.sub.3][sin.sup.2]2[theta]][less than or equal to][[sigma].sup.2.sub.Y,c] (24)

With the angle [theta] = 45[degrees], we obtain net shear and [[sigma].sub.x]=[[tau].sub.lim]=[[sigma].sub.Y,c] / 2, and in the case when the angle is [theta] = 0 , we obtain axial compression and [[sigma].sub.x] = [[sigma].sub.Y,c] . Then the constants [m.sub.3] and [m.sub.4] in the Eq. (23) must be calculated using these equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

From where [m.sub.3] = 4 ; [m.sub.4] = -3 .

Thus, the strength criterion Eq. (23) is presented in the following way

[[sigma].sup.2.sub.x](1+[sin.sup.2]2[theta])[less than or equal to][[sigma].sup.2.sub.Y,c] (26)

or

[[sigma].sub.x][less than or equal to][square root of [[sigma].sup.2.sub.cr,b] / 1+[sin.sup.2]2[theta]] (27)

Applying strength criterion in buckling the following value is calculated

[[sigma].sub.x][less than or equal to][square root of [[sigma].sup.2.sub.cr,b] / 1 + [sin.sub.2]2[theta] (28)

where [[sigma].sub.x,b] b is buckling stresses; [[sigma].sub.cr,b] is critical buckling stresses.

However, in order to observe fracture case while buckling the following values are necessary as [[sigma].sub.cr,b] = [[sigma].sub.Y,c] . That way considering Eq. (9) after taking buckling force from the Eq. (1) and performing the operations, the following formula is obtained

[L.sup.4.cr]=16[[pi].sup.4][([EJ.sub.ef]).sup.2](1+[sin.sup.2]2 [[theta].sub.cr]) [cos.sup.2][[theta].sub.cr] / [A.sup.2][([V.sub.r]+n[Vsub.f]).sup.2][[sigma].sup.2.sub.Y,c]

This formula determines the relation between values of length Lcr and shear angle 9cr with straight bar or bar made from composite being buckled.

3. Regularities of spreading interlayer fracture

Interlayer of laminar material suffers normal ayy and tangential Txy stresses in Fig. 2.

[FIGURE 2 OMITTED]

Referring to studies of Victor E. Saouma [13], in case of flat deformation relative fracture energy G is calculated as follows

G = (1 / [bar.[E.sub.1]] + 1 / [bar.[E.sub.2]])([K.sup.2.sub.1] + [K.sup.2.sub.2] / 2[cosh.sup.2]([pi][epsilon]) (30)

where

[bar.[E.sub.1]] = [bar.[E.sub.1]/(1 - [V.sup.2.sub.1]), [bar.[E.sub.2]] = [E.sub.2]/(1 - [V.sup.2.sub.2] ) (31)

[E.sub.1], [E.sub.2] are moduli of layer elasticity; [v.sub.1], [v.sub.2] are Poisson's ratios for the layer; [K.sub.1], [K.sub.2] are intensity ratios for layer stresses; [epsilon] is variable calculated as follows

[epsilon] = 1 / 2[pi] ln[1 - [beta] / 1 + [beta]] (32)

[beta] is parameter of elasticity loss calculated as follows

[beta] = [[mu].sub.1](1 - 2[v.sub.1]) - [[mu].sub.2](1 - 2[v.sub.1]) / 2 [[mu].sub.1](1 - [v.sub.2]) + [[mu].sub.2](1 - [v.sub.1]) (33)

where [[mu].sub.1], [[mu].sub.2] are shear moduli for layers.

Stress intensity ratios [K.sub.1] and [K.sub.2] calculated as follows [18]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (34)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (35)

Marked cos([epsilon]log2a)= B; 2[epsilon] sin(slog2a) = C; sin([epsilon]log2a) = D ; 2[epsilon]cos([epsilon]log2a) = H; cosh ([pi][epsilon]) = J .

The following equations are presented

[K.sub.1] = [sigma][B + C] + [tau] [D - H] / J [square root of a] (36)

[K.sub.2] = [tau][B + C] - [sigma] [D - H] / J [square root of a] (37)

Considering that buckling presents critical stresses calculated after the Eq. (30)

[[sigma].sub.x] = [[sigma].sub.c] = [square root of [[sigma].sup.2.sub.cr,b] / 1 + [sin.sup.2] 2[theta]

Tangential stresses calculated after Eq. (3)

[[tau].sub.xy] = [[tau].sub.c] = 1 / 2 sin2[theta][[sigma].sub.x].

Eqs. (36) and (37) presented as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (38)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (39)

Therefore, values Gc considering Eq. (30) are obtained as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (40)

After certain operations in Eq. (40) the following equation is obtained

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (41)

Eq. (40) has a short form presented as the following equation

[G.sub.c] = Z([[sigma].sup.2.sub.cr,b] / 1 + [sin.sup.2] 2[theta] + 1 / 4 [sin.sup.2] [theta][[sigma].sup.2.sub.x)a (42)

where

Z = (1 - [v.sup.2.sub.1) / [E.sub.1] + 1 - [v.sup.2.sub.2] / [E.sub.2])[[(B + C).sup.2] + [(D - H).sup.2]] / 2[J.sup.4] (43)

Further, dependence of fracture energy on angle [theta] is analyzed.

Several edge cases are:

[theta] = 0[degrees], [G.sub.c] = Z[[sigma].sup.2.sub.cr,b]a;

[theta] = 30[degrees], [G.sub.c] = Z([[sigma].sup.2.sub.cr,b] / 1.25 + 1 / 16 [[sigma].sup.2.sub.x])a;

[theta] = 45[degrees], [G.sub.c] = Z([[sigma].sup.2.sub.cr,b] / 1.5 + 1 / 8 [[sigma].sup.2.sub.x]a.

4. Determination of strength and fracture characteristics

In order to perform experimental tests, the composite bar of thickness 12 mm was chosen. Laminated by [t.sub.v] = 0.5 mm cover, resin thickness [t.sub.m] = 2mm , and fiberglass thickness [t.sub.f] = 7 mm. This makes relative volume of filling [V.sub.f] = 0.62 , and one of matrix [V.sub.r] is 0.35. Modulus of elasticity are the following: filling is [E.sub.f] = 45 GPa, resin is [E.sub.m] = 11GPa , cover [E.sub.v] = [E.sub.m] = 11GPa. Thus, total modulus of elasticity received from the Eq. (2) makes E = 30.89GPa. [E.sub.f] and [E.sub.m] proportion is n = [E.sub.f]/[E.sub.m] = 4.09. According to ASTM D 638, sample width is 12.7 mm, [[sigma].sub.Yc] = 3000 MPa .

Cross-section area is

A = 152.4*[10.sup.-6] [m.sup.2].

Area moment of inertia

[I.sub.ef] = [I.sub.min] = b[h.sup.3] / 12 = 2.048*[10.sup.-9][m.sup.4].

Strength limit of compression

[[sigma].sub.cr,c] = [[sigma].sub.Y,c] = 3000MPa

and [EI.sub.ef] = 63 N*[m.sup.2].

Entered the values of experimental and calculated parameters into the formula (29) the following is obtained:

[L.sub.cr] = 1.28[4th root of 1 + [sin.sup.2] 2[[theta].sub.cr]], [square root of cos[[theta].sub.cr] (44)

According to Table, maximum critical length of the bars is received with the delamination angle 30[degrees].

With this angle maximum resistance stratification is obtained, and minimum resistance stratification with [theta] = 45[degrees].

Minimum critical length given by [theta] = 45[degrees], and critical value of fracture energy are applied in this case. Further, fracture regularities are analyzed.

With [theta] = 45[degrees] Lcr Eq. (42) presents

[G.sub.c] = Z (0.6666[[sigma].sup.2.sub.cr,b] + 0.125[[sigma].sup.2.sub.x]) a (45)

with [[sigma].sub.x] = [[sigma].sub.cr,b],

[G.sub.c] = 0.792Z[[sigma].sup.2.sub.cr,b] (46)

Consequently, critical value of fracture energy is described by material characteristics Z and [[sigma].sub.cr,b], that depends on fracture length a. [G.sub.c]--a dependence for analyzed glass plastic bar presented in Fig. 3.

The obtained dependences allow measuring critical values of relative energy with various approximate thread lengths and active stress known. Therefore, practical observing thread length and known active stresses allows foreseeing after critical energy value if the fracture spreads further causing construction failure or the thread remains constant (Fig. 4).

[FIGURE 3 OMITTED]

With [G.sub.c] = const stresses [[sigma].sub.x] depend on thread a in Fig. 4.

[FIGURE 4 OMITTED]

5. Conclusions

1. Delamination of composite constructional elements is determined by normal and shear stresses in the thread.

2. Strength criteria used to evaluate composite strength are too complex because of big number of constants and their difficult determination.

3. The nonlinear strength criterion suggested by the author in case of complex state of stresses allows obtaining engineeringly simple dependency between critical delamination angles and critical bar lengths at buckling.

4. According to the experimental and calculation data, minimum critical length of the bar at buckling is obtained with the delamination angle 45[degrees].

5. Measuring elasticity characteristics for separate layers critical fracture energy is calculated after suggested formulas.

6. Having critical values of fracture energy further possibilities of fracture are foreseen after thread length and stresses.

References

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[6.] Borisov, A.V. 2010. Elastic analysis of multilayered thick-walled spheres under external load, Mechanika 4(84): 28-32.

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[8.] Shu, D.; Mai, Y.W. 1993. Delamination buckling with bridging, J. Compos Sci Technol. 47: 25-33.

[9.] Brewer, J.C.; Lagace, P.A. 1988. Quadratic stress criterion for initiation of delamination, J.Compos Mater. 22: 1141-1155.

[10.] Fenske, M.T.; Vizzini, A.J. 2001. The inclusion of inplane stresses in delamination criteria, J.Compos Mater. 35: 1325-1340.

[11.] Wang, S.S. 1983. Fracture mechanics for delamination problems in composite materials, J.Compos Mater. 17: 210-223.

[12.] Hwu, C.; Kao, C.J.; Chang, L.E. 1995. Delaminations fracture criteria for composite laminates, J. Compos. Mater. 29: 1962-1987.

[13.] Saouma, V.E. 2000. Fracture Mechanics. Dept. of Civil Environmental and Architectural Engineering, University of Colorado, Boulder.

[14.] Gurdal, Z.; Hafka, R.T.; Hajela, P. 1991. Design and Optimization of Laminated Composite Materials. Wiley, New York. 352p.

[15.] Bai, Y.; Vallee, T.; Keller, T. 2009. Delamination of pultruded glass fiber - reinforced to axial compression, J. Composite Structures 91: 66-73.

[16.] Timoshenko, S.P.; Gere, J.M. 1993. Theory of Elastic Stability (2nd ed.). Mcgraw-Hill International Book Company.

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[18.] Ziliukas, A. 2006. Strength and Fracture Criteria (in Lithuanian). Technologija, Kaunas. 208p. A. Ziliukas, A. Malatokien

Received January 25, 2011

Accepted June 27, 2011

A. Ziliukas, Strength and Fracture Mechanics Centre, Kaunas University of Technology, Kestucio St. 27, Kaunas, Lithuania, E-mail: antanas.ziliukas@ktu.lt

A. Malatokiene, Department of Building Structures, Kaunas University of Technology, Studentu St. 48, Kaunas, Lithuania, E-mail: ausrazilinskaite@takas.lt

Table
Dependencies of critical delamination angles and plate
lengths

 No.      [[theta].sub.cr], degrees    [L.sub.cr], m

  1                   0                    1.28
  2                   5                    1.286
  3                  10                    1.308
  4                  28                   1.3705
  5                  30                    1.39
  6                  32                    1.367
  7                  40                    1.103
  8                  45                    1.076