Plastic zones in cracked anisotropic plates under small-scale yielding

Fibre Science and Technology 20 (1984) 25-35

Plastic Zones in Cracked Anisotropic Plates under Small-Scale Yielding

E. E. G d o u t o s a n d D. Z a c h a r o p o u l o s School of Engineering, Democritus University of Thrace, Xanthi (Greece)

SUMMARY The plastic zones developed at the tips of cracks in fibre-reinforced composite plates subjected to a uni[brm uniaxial stress perpendicular to the crack axis are determined. The composite plates are considered as anistropic bodies and the investigation takes place for smalLscale deformations. The equation of the elastic-plastic boundary separating the inside plastically deformed material from the outside elastic material is derived by using the elastic solution of the problem in conjunction with the Mises yield criterion. From the whole study the dependence of the size and shape of the plastic zones on the characteristic properties of thefibres and the matrix of the composite plate is established.

INTRODUCTION Fracture of solids takes place by a process of development of discontinuities or flaws which either pre-exist or nucleate in the material structure during deformation. In the close neighbourhood of such imperfections, most of which are usually idealised in the form of cracks, a pre-fracture stage is formed by breakage of the atomic bonds. In such a region the continuum model cannot be adopted for the description of the state of deformation and stress. This area, usually referred to as the core region, is surrounded by a zone of plastically deformed material. Determination of the size and shape of the zone of plastic deformation 25

26

E. E. Gdoutos, D. Zacharopoulos

around the tip of a crack is of major importance for the understanding of the mode of fracture of a body. The fracture may be brittle, quasibrittle or ductile depending on the size of the plastic zone. For the first case the state of affairs in the vicinity of the crack tip can be adequately described by linear elastic fracture mechanics based on the concept of stress intensity factor. Furthermore, the extent of the plastic zone influences the crack propagation velocities which are normally much less for plastic than for elastic strains. McClintock and Irwin ~ were the first to determine the plastic zone surrounding the tip of a crack in an isotropic medium for brittle fracture. Based on the elastic solution of the problem they defined the plastic zone as the region where a yield criterion is satisfied. As such a criterion, the Mises yield criterion was taken. Following McClintock and Irwin more rigorous approaches taking into account the redistribution of stresses, which may result in changes in the shape and size of plastic zones, have been developed. 2-8 A comprehensive review of the theoretical and experimental investigations concerning the plastic deformation near cracks was provided by Vitvitskii e t al. 9 In a recent paper Gdoutos 1° proved that the elastic-plastic boundary in a transparent body made of an elastic-perfectly plastic material that obeys the Tresca yield criterion can be determined photoelastically. The method was also extended to metals by using birefringent coatings. The same author ~1,~2 used two pressuremodified Mises yield criteria for the determination of the plastic zones developed around cracks in glassy polymers subjected to opening-mode or combined opening-mode and sliding-mode loading conditions. Furthermore, Gdoutos 13'14 determined the plastic enclaves at the cuspidal points of rigid inclusions or at the tips of rigid fibres embedded in an elastic plate under small scale yielding. For this reason he used the singular solution of the stress field in conjunction with the Mises yield criterion. All the above-mentioned works are concerned with the investigation of plastic deformations in isotropic materials. However, recently, fibrereinforced composite materials have become very popular in engineering applications due to their superiority to other structural materials in applications requiring high strength and lightweight components. The study of the mechanical behaviour of such materials has developed along two distinct approaches: the macro- or continuum approach and the micro- or discrete approach. In the first approach the fibre-reinforced composite material is idealised by a continuous and homogeneous

Plastic zones in cracked anisotropic plates

27

anisotropic material and the principles of anisotropic elasticity are used for the study of its mechanical behaviour. In the second approach the actual geometrical and physical characteristic properties of the constituent materials are taken into consideration. It is the objective of the present paper to study the plastic zones developed around the tips of cracks in fibre-reinforced composite materials. The investigation takes place by using the principles of anisotropic elasticity of cracked materials in combination with the Mises yield criterion. The equation of the elastic-plastic boundary separating the inside plastically deformed material from the outside elastic material is derived. The dependence of the size and shape of plastic zones on the characteristic properties of the reinforcing fibres and the matrix of the composite is demonstrated.

CRACK TIP STRESS FIELD Consider an elastic orthotropic plate whose principal axes of elastic symmetry coincide with the axes of a system of Cartesian coordinates xy (Fig. 1). The plate has a crack of length 2a along the x-axis and it is subjected to normal and shear loads at infinity. A small thickness of the plate is taken so that conditions of generalised plane stress prevail in the plate. In the xy-coordinate system the material of the plate is described by the following constitutive equations:

Ex

1

V12

E1

E1

Y21

E2 ~xy

0

0

1 --

E2

0

(7 x

tTy

(1)

1

o

-6

_'~xyJ

where ex, ey, 7xy and a~, ay, z~,ydenote the components of the strain and stress tensors respectively, E 1 and E 2 are the moduli of elasticity along the x and y axes, v~2 and v21 are the Poisson's ratios associated with directions x and y and G is the shear modulus. The elasticity matrix in eqn (1) is symmetric, implying: v12E 2 -- VzlE 1

(2)

28

E. E. Gdoutos, D. Zacharopoulos

I::a ~Y

I=,,I

Oy

'l i

1

I

I

ff

Fig. I.

A cracked anisotropic plate one of the principal directions of elastic symmetry of the plate coincides with the crack axis.

From relations (I) and (2) it is established that for the description of the elastic behaviour of an orthotropic material three independent constants are needed. As such constants we will take El, E 2 and v~2 = vl. The problem of a cracked anisotropic plate subjected to in-plane loading has been considered by Sih et al. ~5 who gave the following expressions for the stress components o-~, ~, and vx~.in the neighbourhood of the crack tip for symmetric applied loads: K I

a.,. - (2r)¥7 ~- Re A K l

cr.~,- (2r)l 2 Re B K I

z,3.- (2r)1, 2 R e C

(3)

Plastic zones in cracked anisotropic plates

29

where

W - - -

//1 --//2

(4) //1 - / / 2

-f

with Z 1 = (COS 0 + / / 1 sin 0) 1/2

z 2 = ( c o s O + //2 sin O) 1/2

In the above relations Re denotes the real part of the corresponding function; r and 0 are the polar coordinates of the point considered;//1 and P2 are the roots of the fourth-order characteristic equation /14. + 2fl0//2 + O~0 = 0

(5)

with E1 /~o - 2 6

vl

E1 ~z2 = E-2-

(6)

A s shown by Lekhnitskii 16 the roots ofeqn (5) are either complex or purely imaginary and cannot be real. Thus, the four roots separate into two sets of distinct complex conjugates. The parameters ]Amand #2 are those roots that have a positive imaginary part. The parameter K, is called the opening-mode stress intensity factor and for the case when the applied loading consists of a uniform uniaxial stress normal to the crack axis, K i is given by: 15 K I = 6 a 1/2

(7)

PLASTIC ZONES For a progressively applied stress a and for brittle fracture behaviour involving small plastic deformations in the neighbourhood of the crack tip the elastic-plastic boundary can be determined by inserting the stress components ax, a r and Zxy given from relations (3) into an appropriate fracture criterion. The more widely used fracture criterion for the

30

E. E. Gdoutos, D. Zacharopoulos

description of yield behaviour of materials is the Mises criterion. According to this criterion yielding of an element of a material under a multiaxial state of stress takes place when the deviatoric strain energy absorbed by the element is equal to that corresponding to simple tension. This condition for the case of an anisotropic material is expressed mathematically by the following relation: 17

with

,(1

H = ~ ~

,

+ y2

,) 22

(9)

In these relations a 1 and a 2 are the principal stresses of the plane state of stress and X, Y and Z are the tensile strengths in the principal directions of anisotropy (X and Y are the strengths along the axes x and y and Z along the axis z perpendicular to the plane of the plate). The principal stresses (9"1, 2 are obtained from the stress components o:,, ay and Z~yas the roots of the following second order equation: a 2 - (a~ + ay)a + (a~% - z~y) = 0

(10)

Introducing now the stresses a x, ay and ~xy from relations (3) into eqn (10) the principal stresses al, 2 are determined. Inserting the stresses a~, 2 into relation (8) we obtain the following relation for the radius r of the elastic plastic boundary: r .~a.

= D2 +

D2-ZHX2D,D2

(11)

with D1'2 --

ReA + R e B 2 + ½[(ReA - R e B ) 2 + 4Re C] L'2

(12)

For the case now of a matrix material reinforced with fibres along the xaxis the strengths Y and Z are equal and eqn (11) takes the form: r ~-a~daj=D2 +

D 2 -D,D 2

(13)

Equation (13) defines the radius r of the elastic-plastic boundary at the tip of a crack in a fibre-reinforced material. This boundary corresponds to

Plastic zones in cracked anisotropic plates

31

the limiting curve separating the inside plastic material from the outside elastic material at the crack tip.

RESULTS A fibre-reinforced material which was simulated by an orthotropic m e d i u m is characterised by its four elastic constants E l, E 2, G and v 1 and the two tensile strengths X and Y along the reinforcing and the perpendicular directions. F o r the determination of the stress field in the n e i g h b o u r h o o d of the crack tip only the two composite material constants % and flo are needed, while for the description of its yield behaviour the ratio X/Y also enters. Thus, for the description of the elastic-plastic b o u n d a r y the three quantities %, flo and X/Yare needed. A further reduction of the dependence of the elastic-plastic b o u n d a r y on the material constants is made by making the assumption that ~2 = (El~E2) = (X/Y). This means that the strengths along the principal directions are proportional to the corresponding moduli of elasticity, a condition which is satisfied with an adequate degree of accuracy in m a n y materials. Under these conditions the elastic-plastic b o u n d a r y was determined for many combinations of the elastic constants % and flo of the anisotropic material. It is worthwhile to observe that tbr an isotropic material ( E 1 = g 2 and 2 G = E ~ / ( v + I ) ) the constants ~o 2 and flo given from relations (6) are equal to one. Figure 2(a)-(c) presents the elastic-plastic boundaries for/30 = 1.1 and ~o2 = l'0(a), 0-9(b) and 0.8(c). The numerical values on the x and y axes centred at the crack tip correspond to the nondimensional quantity r(2XZ/aZa). The curve of Fig. 2(a) ( ~ = 1.0, /30 = 1.1) is very close to the corresponding curve for the isotropic plate (%2 = 1, flo = 1) (Reference 11, Fig. 2). However, the isotropic case cannot be obtained from eqn (13) since eqns (3) and (4) of the crack-tip stress field of the anisotropic plate do not give as special case the equations of the isotropic cracked plate. F r o m Fig. 2 it is observed that for a fixed value of /30 the elastic-plastic b o u n d a r y increases in the direction perpendicular to the crack axis as ~o2 = E 1 / E 2 = X/Y decreases. This result should be expected since as the direction along the crack axis becomes weaker than the other the material in the stronger direction deforms more than in the weaker. The elastic-plastic boundaries for/30 = 2 and % = 1.1, 1"4, 1-6 and 2.0

E. E. Gdoutos, D. Zacharopoulos

32

1.:

x

(a)

(~

(c) Fig. 2. Elastic-plastic boundaries at the tips of cracks in an anisotropic body with /3o = 1-1 and ~2 = 1-0 (a), 0.9 (b) and 0.8 (c). The numerical values on the axes x and y centred at the crack tip correspond to the non-dimensional quantity r(2XZ/a2a).

and for flo = 6 and ~o = 1.0, 2.0, 4.0 and 9.0 are presented in Figs 3 and 4. F r o m these figures similar observations as for Fig. 2 can be made. Thus, we see that plastic deformation is more pronounced along the stronger direction of the material. The plastic zones become more oblong along the direction of the crack as ~o increases. We further observe the strong deviation of the plastic zones from the case of the isotropic body (Fig. 2(a)). Both the size and shape of the plastic zones are greatly influenced by the degree of anisotropy of the material. It is therefore concluded that the anisotropic character of fibrereinforced materials should be seriously taken into account for the determination of the plastic zones around cracks. This is of major significance in studying failure of anisotropic bodies in the presence of a

33

Plastic zones in cracked anisotropic plates

(b) (a)

'

1'.2~

(c)

/d)

Fig. 3. As in Fig. 2 for flo = 2 and ~ = 1.1 (a), 1.4 (b), 1.6 (c) and 2.0 (d). crack. Failure of such a body results either from extensive plastic deformation or from crack propagation. The presence of yielding prevents the crack from extension, and the crack has the tendency to propagate through the elastic material. Furthermore, for extensive plastic deformation surrounding the whole area around the crack tip, the crack decelerates in the plastic area and usually propagates in a stable manner. Thus, the determination of the plastic enclaves around cracks also enlightens the possible crack propagation behaviour and therefore describes the complete failure p h e n o m e n o n of the material which m a y result from excessive plastic deformation, crack extension or both.

E. E. Gdoutos, D. Zacharopoulos

34

y

2,4

0~ 2.0 2,

1.

(b) y

0.8

(c/ Yl

G8 0.4

1.2 =

(a) Fig. 4.

As in Fig. 2 for flo = 6 and eo2 = 1 (a), 2 (b), 4 (c) and 9 (d).

REFERENCES 1. F . A . McClintock and G. R. Irwin, Plasticity aspects of fracture mechanics, Fracture Toughness Testing and its Applications, A S T M S T P 381 (1965) pp. 84~113. 2. J . R . Rice, Mechanics o f crack tip deformation and extension by fatigue, Symposium on Fatigue Crack Growth, A S T M S T P 415 (1967) pp. 247-309.

Plastic zones in cracked anisotropic plates

35

3. J.R. Rice and G. F. Rosengren, Plain strain deformation near a crack tip in a power-law har~tening material, J. Mech. Phys. Solids, 16 (1968) pp. 1-12. 4. J. W. Hutchinson, Plastic stress and strain-field at a crack tip, J. Mech. Phys. Solids, 16 (1968) pp. 337-48. 5. P.D. Hilton and J.W. Hutchinson, Plastic intensity factors for cracked plates, Engng Fract. Mech., 3 (1971) pp. 435-41. 6. P.S. Theocaris, Experimental solution of elastic-plastic plane-stress problems, Trans. A S M E J. Appl. Mech., 29 (1962) pp. 735-43. 7. P.S. Theocaris and E. Marketos, Elastic-plastic strain and stress distribution in notched plates under plane stress, J. Mech. Phys. Solids, 1I (1963) pp. 411-28. 8. P.S. Theocaris and E. Marketos, Elastic-plastic analysis of perforated thin strips of a strain-hardening material, J. Mech. Phys. Solids, 12 (1964) pp. 377-90. 9. P.M. Vitvitskii, V.V. Panasyuk and S. Ya. Yarema, Plastic deformation around crack and fracture criteria, Engng Fract. Mech., 7 (1975) pp. 305-19. 10. E.E. Gdoutos, A photoelastic determination of the elastic-plastic boundary, J. Phys. D, Applied Physics, 12 (1979) pp. 1317-20. 11. E.E. Gdoutos, Crack-tip plastic zones in glassy polymers under small scale yielding, J. Appl. Polym. Sci., 26 (1981) pp. 1919-30. 12. E.E. Gdoutos, Plastic zones at the tips of inclined cracks in glassy polymers under small scale yielding, J. Appl. Polym. Sci., 27 (1982) pp. 879-92. 13. E.E. Gdoutos, Spread of plasticity from cuspidal points of rigid inclusions, Acta Mechanica, 44 (1982) pp. 91-105. 14. E.E. Gdoutos, Yield loci around fiber inclusions, J. Franklin Inst., 312 (1981) pp. 361-71. 15. G.C. Sih, P.C. Paris and G. R. Irwin, On cracks in rectilinearly anisotropic bodies, Int. J. Fract. Mech., 1 (1965) pp. 189-203. 16. S.G. Lekhnitskii, Anisotropic Plates, Translated from the Russian edition by S. W. Tsai and T. Cheron, Gordon and Breach Sci. Publ., New York, 1968. 17. B. Paul, Microscopic criteria for plastic flow and brittle fracture, in Fracture 11, edited by H. Liebowitz, Academic Press, London and New York, 1968, pp. 348-54.