The failure of a strain-softening material: I. Analytical approach for a double cantilever beam specimen

Theoretical and Applied Fracture Mechanics 11 (1989) 59-64

59

T H E FAILURE OF A S T R A I N - S O F T E N I N G MATERIAL: I. ANALYTICAL A P P R O A C H FOR A D O U B L E CANTILEVER B E A M S P E C I M E N E. SMITH Joint Manchester University/UM1ST, Materials Science Centre, Grosvenor Street, Manchester M1 7HS, U.K.

The crack tip stress intensity K A associated with the attainment of a fully developed softening zone in a strain-softening material is determined for a double cantilever beam (DCB) specimen, using a simple Euler-Bernoulli representation of the specimen. For the situation where the initial crack size and the fully developed softening zone size are large compared with the beam height, K A is the same for a DCB specimen as it is for a semi-infinite crack in a remotely loaded infinite solid, assuming the same softening law in the two cases. On the other hand, the size of the fully developed softening zone is very dependent on the beam dimensions. The Dugdale-Bilby-Cottrell-Swinden (DBCS) force law, where the stress is constant within the softening zone, and the linear softening law are considered as special cases. The results are in general accord with other workers' numerical results for a specific material.

1. Introduction

If a crack exists in a material such as a concrete, soil, or a ceramic, and the material is stressed, a situation develops whereby the crack extends and its faces become bridged by unbroken ligaments whose average behaviour can be represented by a decreasing stress versus increasing crack face opening relationship; in this sense, the material can be regarded as being "strain-softening". This state of affairs can be simulated by a model in which there is a softening zone of material, coplanar with the crack, in which the behaviour is nonlinear, the form of the nonlinearity being governed by the strain-softening characteristics of the material under consideration. As the solid is progressively loaded, the softening zone size increases, and this increase is associated with an increase in the crack tip stress intensity K, as measured at the leading edge of the softening zone. Several workers have recognised the importance of the value of K, i.e. KA, which is associated with the attainment of a fully developed softening zone, and also the size R of this zone. K A is the crack tip stress intensity arising from the applied loadings, and this is equal to the intrinsic fracture toughness Kic of the matrix material plus a contribution due to the restraining effect of the ligament material bridging the crack faces. For example, with fibre cements, K I C is the toughness of the reinforced matrix material, while in 0167-8442/89/$3.50 © 1989, Elsevier Science Publishers B.V.

ceramics, it is given by either the transgranular or intergranular toughness. Using this type of model, Foote, Mai and Cotterell [1] have used a computational approach to give R and K A for a specific fibre cement composite having a double cantilever beam (DCB) geometry (Mode I plane strain deformation). Their main conclusions, assuming a linear softening law, are that while R is very sensitive to the beam's d i m e n s i o n s , K A is essentially independent of these dimensions and is approximately equivalent to the value associated with a semi-infinite crack in a remotely loaded infinite solid. This paper approaches the problem in an analytical manner. Using the Euler-Bernoulli beam representation for the DCB geometry, and focusing on the situation where the initial crack size and fully developed softening zone size are large compared with the beam height, Section 2 shows that K A is independent of the beam height, and for the same general softening law has the same value as for a semi-infinite crack in a remotely loaded infinite solid. On the other hand, the size R of the fully developed softening zone is very dependent on the beam dimensions. Sections 3 and 4 consider respectively the DBCS force law [2,3], where the restraining stress is constant within the softening zone, and the linear softening law, as special cases. The results are in general accord with the numerical results obtained by Foote, Mai and Cotterell [1]. However, as will be demonstrated in

E. Smith / The failure of a strain-softening material." 1

60

Part II [4], they contrast sharply with results for other crack configurations, where a general study of the effect of geometrical and loading parameters on crack growth in a strain softening material (DBCS force law) shows that these parameters have a marked effect on both K A and R.

a

Fig. 2. T h e l o w e r a r m of the b e a m w i t h a d i s c r e t e r e s t r a i n i n g l o a d P.

2. Theoretical analysis The double-cantilever-beam (DCB) specimen is shown in Fig. 1. The height of each arm is h, whilst the beam has unit thickness. The specimen contains a crack of initial depth a 0, which extends as a result of the application of the loads F to a length a = a 0 + R, the length of the softening zone being R. It is assumed that the softening zone is fully developed in the sense that the cohesion at the initial crack tip just vanishes, this occurring when the relative displacement of the two arms of the beam at the initial crack tip attains a critical value 8c. The simplest possible representation of the specimen, which will be used throughout this paper, is where the two arms behave as Euler-Bernoulli beams and are built-in at the crack tip; in the ensuing theory, because of symmetry, it is necessary to consider only one arm and we will therefore concentrate on the lower arm. The objective in this section is to focus on the size R of the fully developed softening zone and the K value, i.e. K A, that is associated with the attainment of this state, for a general softening law, i.e. for a general relationship between the restraining stress and the relative displacement between the beam arms. In addressing this problem, consider initially the case where there is a concentrated load of magnitude P acting at a distance t from the crack

F 2h

.

TI

tip (Fig. 2). The moment and reaction at the simulated crack tip (x = 0) are respectively M and Q. With the Euler-Bernoulli beam theory, the deflection y of the lower arm as a function of the distance x behind the crack tip is given by the second-order differential equation: d2y = M - Qx E°Idx--5 =M-Qx-P(x-t)

ao

(1)

(x > t),

where E 0 is equal to E l ( 1 - u2), with E being Young's modulus and ~ Poisson's ratio, it being assumed that the beam behaves as a plate; I, the second moment of area, is equal to h3/12. Equilibrium of forces and moments requires that Q = F - P,

M= ra-

Pt.

(2)

Furthermore, the crack tip stress intensity due to the applied loadings is [5]: K A = F a ld--S = K,c + ld-- ~ Pt,

(3)

with the moment M at the crack tip being related to K~c via the expression: Kic=

M [12 V h3 .

(4)

Using the boundary conditions d y / d x = y = O when x = 0 and the conditions that d y / d x and y are both continuous at x = t, integration of eq. (1) gives: E°ly =

Fig. 1. A d o u b l e c a n t i l e v e r b e a m s p e c i m e n c o n t a i n s a c r a c k o f initial d e p t h a 0 w h i c h e x t e n d s to a d e p t h a d u e to the a p p l i c a t i o n o f the forces F. T h e r e is a fully d e v e l o p e d s o f t e n i n g z o n e o f size R = a - a 0 b e h i n d the tip o f the c r a c k .

( x < t)

Mx2 2

EoIy = Pt 3 6

Qx3 6

( x < t ),

PtZx x2 T +(M+Pt)-~-

(P+Q)x3 6

(x>t)

(5)

E. Smith / Thefailureof a strain-softeningmaterial:I It then follows from relations (5) that if u ( = 2 y ) is the relative displacement of the beam arms, then

u(

= tSt[2M- Qt],

t--~)]

(6) where u(t) is the relative displacement at x = t, whereupon with

ou u(,÷ )

In order to see whether or not there is a corresponding conclusion for the parameter R, consider the situation where a restraining stress is distributed along the softening zone, when eqs. (1) and (2) are replaced, respectively, by the equations:

EoI

d2u

2

dx 2

~0

In the case of a long crack, i.e. equation simplifies to:

t/a

(8)

M = Fa - IRto(t) -"1o

whereupon eq. (12) becomes:

E°I2 ddzu x2

M(1-X)+xfor(1-t)°(t)dt

(9)

- foX(x- t)o(t) dt

whereupon eqs. (3,), (4), and (9) give:

8u

KEc + (1 - 1)2) 8t "

(10)

Expression (10) for K a refers to the case of a discrete restraining load P, but it is readily generalized to the case where there is a restraining stress o acting throughout the fully developed softening zone by noting that o = P/St and integrating over the softening zone, whereupon expression (10) becomes: K?c + (1 -Eu 2 )

(13) dt,

is small, this

EolSu = 2MtSt + PtZ~t,

KA =

(12)

Q = F - ~Ro(t) dt,

EolSu= 2MtSt[1- ~a ] + pt28t[1- t ].

EP

M - Qx - foX(X- t )o( t ) dt

for 0 < x < R, and

)

eqs. (2), (6) and (7) give:

KA =

61

fo,,O(u) du "

(11)

This expression for KA, which is valid for all softening laws, i.e. for all o-u relations, is independent of the beam height h, and is the standard result for a semi-infinite crack in a remotely loaded infinite solid. However, it is important to appreciate that expression (11) is strictly valid only for the case where the fully developed softening zone R is small compared with the crack depth, since its derivation depends on the simplifications that are introduced by proceeding from eq. (8) to eq. (9). Furthermore, since Euler-Bernoulli beam theory is used, any conclusions refer only to the situation where both the initial crack size and the fully developed softening zone size are large compared with the beam height.

(14)

for 0 < x < R, simplifying for the case of a long crack, i.e. R/a is small, to

°Id u +4oRo(t) d t - £X(x2

dx 2

t)o(t) dt

(15) M=KIc/h3~ (eq. (4)). Writing D = u/8 c, w = x / R , ~p=t/R, o=pcf(D), with Pc

with

being the maximum stress in the softening zone (at the crack tip), M , = Kic/R2pc~hY/12, and )~4= Eoh33c/24pc, the differential equation (15) becomes: ~k4 d2D R 4 dw 2 (16) with the boundary conditions d D / d w = D = O when w = 0 (these recognise that the beam is built-in at the crack tip) and D = 1 when w = 1. The satisfaction of these boundary conditions requires that R be expressed in the form: R = Xg(M,),

(17)

where the function g is dependent on the characteristics of the softening law and the parameter M,. It therefore follows that the softening zone size R, unlike KA, is very dependent on the beam

62

COHESI STRESSVE ]a

E. Smith / The failure of a strain-softenmg material." 1 COHESIVE

(21) that the fully developed softening zone size R is given by the equation:

STRESS

Pc

RELATIvEPC~

d'c D I S P L A C E M E N T

Eoh38c -4[ R + 3a 0 ] Pc -1< [ ) ~ 2 - ~ "

Fig. 3. (a) The DBCS force law relating the cohesive stress within the softening zone and the relative displacement (of the beam arms). (b) The linear softening relation between the cohesive stress within the softening zone and the relative displacement (of the beam arms).

Furthermore, the K A value associated with the attainment of the fully developed softening zone is given by eqs. (3) and (19) as

KA=

height h. For example, with the special case where K , c = 0, i . e . M . = 0, it follows from relation (17) that

[E0h38c

]1/4 R=[ 2-~p~ j g"'

(18)

3 ~

],/2

p e r 2,

where g, depends solely on the softening law characteristics.

3. Special case of DBCS force law The simplest possible description of the softening zone is the DBCS representation [2,3] whereby the cohesive stress maintains a constant value Pc within the softening zone (Fig. 3(a)). For this force law o ( t ) = P c , and eqs. (13), with M = 0 (since it will now be assumed that K l c = 0) give: peR2 2a '

(19)

(23)

with R being given by relation (22). For the special case where the initial crack depth is large, relations (22) and (23) simplify to give the results: R = 0.76h [

t

F-

(22)

RELATIVE ) d'c D I S P L A C E M E N T

E°Sc

]1/4

pch]

'

K A = [EoPcSc] 1/2.

(24)

(25)

Relation (24) shows that the fully developed softening zone size is very dependent on the beam height, in accord with the preceding section's general findings. The fully developed softening zone size for the DBCS force law with a semi-infinite crack in a remotely loaded infinite solid is R ~ = ¢rEoSJ8pc [3]. Comparison of the value of R from relation (24) and the expression for R ~ , shows that these two values can differ appreciably. On the other hand, the value of K A given by expression (25) is the same as that for the DBCS force law with a semi-infinite crack in a remotely loaded infinite solid.

4. Special case of linear softening law Integration of the differential equation (14) and imposition of the boundary conditions at x = 0 gives the solution for u, the relative displacement of the beam arms, as

E°l~u-2

Qx3 6

pox4 24 '

(21)

with x being measured away from the crack tip. Thus, with u = 8~ at the trailing edge x = R of the softening zone, it follows from relations (20) and

This section analyses the situation where there is a linear relation between the cohesive stress a and the relative displacement (u = 2 y ) within the softening zone (Fig. 3(b)), i.e.

With this force law, substitution in eq. (15) and

E. Smith / The failure of a strain-softening material." I

differentiation gives the fourth-order differential equation:

63

shows that the K A value associated with the attainment of a fully developed softening zone is:

d4o - -

- X4,o = 0,

(27)

dx 4

where h , = [2pJEISc]x/4. The solution of this equation which satisfies the boundary conditions that y = d y / d x = 0 a t x=0is: o = A cosh X , x - B sinh X,x + B sin X,x

+(pc-A)

cos X,x.

1 d2a

---xfn(

+ sinh X,R cos X,R) - X,R(1 + cosh X,R cos X,R)} × ((sinh X , R - sin X,R)} -1]

(28)

This solution must also conform to the governing eq. (15), which becomes, using eqs. (2) and (26), and with M -- 0, i.e. KIC = 0:

~k4, d x 2

× [{(cosh X,R sin X,R

(33)

with R being given by relation (32). In the case of a long crack, relation (32) simplifies to: (cosh X,R cos X,R + 1) = 0

(34)

The solution X,R is equal to (7r/2)+ 0, with 0, lying between 0 and v / 2 , being the solution of the equation:

t -1)a(t)dt

Jo ~ a

+ foX(x- t)a(t) dt.

(29) sin O c o s h ( 2 + O ) = 1,

By substituting expression (28) for o into eq. (29), it follows that A = p--~ 2'

(30)

(35)

whereupon 8 = 0.30 or X,R = 1.87. The fully developed softening zone size R is therefore given by the expression: R = 0.84h[ p--~] E°Sc ]1/4,

(36)

while the corresponding K A value is given from eq. (33) by the expression:

1

'j2 I[ cos

+ cos

ta

+ X ~ 1, a (sinh X,R + sin X,R)

" (31)

Remembering that a = 0 when x = R, i.e. at the trailing edge of the softening zone, relations (28), (30), and (31) give: cos

+ .1 (cosh X,R sin X,R A,a + sinhX,R cos X,R) = 0,

(32)

an equation which gives the fully developed softening zone size R. Furthermore, relation (3)

Relation (36) shows that the fully developed softening zone size is, as for the DBCS force law, very dependent on the beam height, again in accord with the general findings in Section 2. Interestingly, the fully developed zone sizes for the DBCS and the linear force law do not differ appreciably (compare expressions (24) and (36)). The value of K A is given by expression (37) is the same as that for a linear softening law with a semi-infinite crack in a remotely loaded infinite solid. This conclusion and that concerning the dependency of R on beam height are in general accord with the numerical results [1] for a specific material with a linear softening law, i.e. a fibre cement composite for which E = 6000 MPa, Pc = 6 MPa and 8c = 0.8 mm, their computations being for specific a 0 values within the range 2.40 mm to 100 mm, and for

64

E. Smith / The failure of a strain-softening material: 1

specific h values within the range 20 mm to 80 mm.

5. Discussion

The Euler-Bernoulli representation has been used to describe the double cantilever beam type specimen, and this has allowed the formulation of a simple analytical theory which gives (Section 2) the crack tip stress intensity K A due to the applied loadings, and also the size R of the fully developed softening zone, in a strain-softening material for a general softening law; particular consideration has been given to the situation where the initial crack size is large. The DBCS force law and the linear softening law have been considered as special cases (Sections 3 and 4). Because the approach is analytical, it has been possible to draw general conclusions regarding the effect of geometrical parameters, i.e. the beam height, on the magnitudes of R and K A. In a sense, the analysis is more wide ranging than that of Foote, Mai and Cotterell [1] whose approach was numerical and thereby confined to specific values of the various geometrical parameters and to a specific material with a linear softening law. On the other hand, it should be emphasised that this paper's simple analytical expressions apply to the case where the initial crack size and the fully developed softening zone size are large compared with the beam height, and in this respect, the analysis is more restrictive than that of Foote, Mai and Cotterell [1].

The main conclusions that arise from the present analysis, and which complement and are supportive to those obtained by Foote, Mai and Cotterell [1], are that whilst the fully developed softening zone size R is very dependent on geometrical parameters, the associated K A value is insensitive to these parameters and has the same value, for the same softening law, as that for a semi-infinite crack in a remotely loaded infinite solid. It has also been shown that, for any given geometrical configuration involving a deep crack, R is relatively insensitive to the details of the force law, though very dependent on the maximum stress Pc and the maximum relative displacement 8c in the softening zone. In Part II [4], crack configurations other than the DCB geometry are examined, and it is shown that, unlike the DCB case, geometrical and loading parameters can have a marked effect on both K A and R, strongly suggesting that the results for the DCB geometry are special, and should not be used to draw general conclusions about crack growth in a strain-softening material. References

[1] R.M.L. Foote, Y.W. Mai and B. Cotterell, J. Mech. Phys. Solids 34, 593 (1986). [2] D.S. Dugdale, J. Mech. Phys. Solids 8, 100 (1960). [3] B.A. Bilby, A.H. Cottrell and K.H. Swinden, Proc. Roy. Soc. A272, 304 (1963). [4] E. Smith, Theoret. AppL Fracture Mech. 11, 65 (1989) (this issue). [5] H. Tada, P.C. Paris and G.R. Irwin, The Stress Analysis of Cracks Handbook, Del Research Corporation, Hellertown, PA (1973).