on the End of Stiff Cylindrical Cord Embedded in Rubbery Materials
KYEONGJIN YANG*
Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Science Town, Daejeon 305-701, Republic of Korea
(Received March 24, 2005) (Accepted August 24, 2005)
ABSTRACT: The general purpose of this study is the determination of J-integral when dealing with fracture of a penny-shaped crack on the end of the stiff cylindrical cord embedded in rubber matrix. The dimensional analysis is applied to derive a general equation of J-integral, and then it is assumed that the equation of J-integral can be separated into the deformation and geometry functions, the validity of which is proved by using separation parameter. The J-integral is expressed in a multiplicative form in which the geometry calibration function (or factor) is introduced in order to take into account the finite dimensions. The values of the J-integral for the rubber-cord composites with various crack and specimen radius are obtained by using finite element analysis (FEA), and these results are used for concretely determining the geometry calibration function, which, in this work, is expressed in a polynomial form of fourth order. The deformation calibration function, which is a constant for a linear elastic materials, characterizing the large deformable nonlinear effect in rubbery materials is obtained by comparing the equation of J-integral with the ones for the linear elastic deformation. As we approach the infinitesimal strain, the value of the deformation calibration function comes close to the result of linear elastic fracture mechanics (LEFM).
KEY WORDS: fracture mechanics, J-integral, separation parameter, rubber-cord composites.
INTRODUCTION
S
INCE THE APPLICATIONS of rubbery materials in various mechanical components subjected to cycling loading have been steadily increased in industries, the fracture and fatigue analysis for the rubbery materials, which is one of the major research fields dealing with the reliability of the components applied to cycling loading, have been of prime importance. As representative one among the application fields, the rubber-cord*E-mail: [email protected]
Journal ofCOMPOSITEMATERIALS, Vol. 40, No. 20/2006 1787
0021-9983/06/20 1787–14 $10.00/0 DOI: 10.1177/0021998306060172
composites find applications in automotive and aerospace engineering industries, especially for tire construction, in air springs, and in belt structures. As well known, if the J-integral as a fracture parameter can be determined for the components subjected to cycling loading, the life of the components can be easily estimated by way of fatigue analysis. For example, the fatigue life of linear elastic components with crack could be estimated by using the Paris law [1]. The failure processes of rubber-cord composites having cords embedded in rubber are usually divided into two stages. Penny-shaped cracks are initiated on the ends of cords when the composites are subjected to cycling loading.
These cracks grow until they coalesce into larger cracks. This mechanism appears to be the first stage of fatigue failure of rubber-cord composites. The next stage is the crack growth across the interply, well known as delamination. The effective fracture mechanics methods for analyzing the edge delamination problem have been well reported [2–5], but an effective study for the first stage of fatigue failure has hardly been proposed.
The fracture mechanics method for the rubbery materials was first proposed in 1953 by Rivlin and Thomas [6]. They found that energetic parameter involved in the Griffith criterion could be used to determine the tearing in rubbery materials, and for the various types of specimens, obtained the J-integral as the tearing energy for nonlinear elastic materials. Lake [7] and Gent et al. [8] have applied the energetic parameter J into the fracture and fatigue for the rubbery materials and have shown that the J-integral could be used successfully to characterize the fatigue crack growth rate. For single edge notched tension (SENT) specimens broadly used for the test of the fracture and fatigue crack growth of rubbery materials, the J-integral is given by [6–8],
J ¼2kW0c,
where W0is the uniform strain energy density in the bulk region remote from a crack, c is the crack length, and k is the deformation calibration function (or factor) dependent on the strain (k is a constant for the linear elastic materials). The strain dependence of the function k was experimentally investigated by Greensmith [9] who also employed the concept of the J-integral. Lake [10] who employed the concept of crack closure, theoretically proposed that k could be given by =1=2. And Lindley [11] expressed the strain-dependent function k as a polynomial form using the curve fitting for the numerical and experimental results. The limitation of the early works is no consideration for the effect of finite size of SENT specimen. Therefore, the equation of the J-integral proposed by them [9–11] may not be sufficient to characterize the fracture and fatigue of rubbery materials. Ghfiri et al. [12] considered the finite size effect of the specimens and employing the separation method used by Sharobeam and Landes [13], proved that the function k could be separated in the deformation and geometry function. Then they proposed the improved equation of J calibrated to the geometry and deformation functions.
All of the earlier works [6–13] were concerned with homogeneous rubbery materials.
On the other hand, the work for the growth of penny-shaped cracks on the cord-ends in rubber-cord composites was performed by Huang and Yeoh [14]. They performed the fatigue life prediction using J-integral in which the calibration function k is assumed to be a constant. But for precise prediction of the fatigue life, the strain and geometry dependences of the function k should be considered.
In this work, an equation of the J-integral for a penny-shaped crack on the end of the cord embedded in rubber matrix is proposed. To be applied for designing the rubber-cord
composites and estimating the fatigue life, both the geometry calibration function, which characterizes the effects of the finite sizes such as the cord radius and specimen size, and the deformation calibration function dependent on strain are considered. Rubber and cord are assumed as the nonlinear elastic materials represented by Mooney-Rivlin model and linear elastic material much stiffer than rubber, respectively. The dimensional analysis is applied to derive an equation of the J-integral. It is assumed that the equation of J-integral can be separated into the deformation and geometry functions, the validity of which is proved by using a separation parameter. The result for the J-integral shows that a multiplicative form of the equation of the J-integral is valid for the rubber-cord composites as well as the homogeneous rubbery materials. In addition, it is found that, when considering the crack growth near the cord, the value of the J-integral decreases with the crack growth under the constant remote load.
THEORY
A Griffith-type energy approach to the tearing of rubbery materials was first proposed by Rivlin and Thomas [6]. The tearing energy known as the J-integral is defined as
J ¼ @
@A, ð1Þ
where is the potential energy supplied by the internal strain energy and external forces, and A is the area of crack surface. If the specimens are in load control, the J-integral is represented by the rate of change in complementary strain energy U* with crack area as
J ¼dU
dA : ð2Þ
Fracture problems of materials exhibiting a nonlinear stress–strain relationship, such as many polymers, are generally approached in terms of global analysis based upon energy balance. Consider an arbitrary counter-clockwise path around a crack tip.
Rice [15] showed that the J-integral could be written as a path-independent line integral as follows:
J ¼ Z
Wdy Ti
@ui
@xds
, ð3Þ
where W is the strain energy density, Ti are the components of the traction vector, uiare the components of the displacement vector, and ds is a length increment along contour .
Begley and Landes [16] introduced the multi-specimen technique which is both the earliest and most accurate method used for the estimation of the J-integral from test records.
This method is based on the J-integral interpretation which is equivalent to the line integral of Equation (3). Ghfiri et al. [12] also applied this energetic approach in homogeneous rubbery materials.
J-integral of a Penny-shaped Crack on the End of a Stiff Cord
Consider a rubber-cord composite with a penny-shaped crack on the end of a stiff cylindrical cord, as illustrated in Figure 1. The strain energy density W at any point P can be described by the following set of variables:
W0, c, R, Z, A, B, "0
where W0 is the strain energy density in the bulk region not affected by a crack, c is a crack length, R and Z are cylindrical coordinates of the point P, A is the cord radius, Bis the specimen radius, and "0characterizing the overall level of strain is the strain in the bulk region. For linear elastic materials, the variable "0is redundant, while for nonlinear elastic materials, it is needed due to the characterization of the strain energy density W not being proportionate to W0, which was first discussed by Greensmith [9] and Lake [10]
who reported the strain-dependence of W from the experimental and analytic results, respectively. From dimensional analysis, W is expressed as
W ¼ W0f c R, c
Z,A c, c
B, "0
: ð4Þ
Figure 1. Rubber-cord composites with a penny-shaped crack on the end of a cord: B ¼ specimen radius, A ¼ cord radius, c ¼ crack radius.
Introducing reduced variables, r ¼ c=R, z ¼ c=Z, a ¼ A=c, b ¼ c=B, Equation (4) becomes
W ¼ W0f r, z, a, b, "ð 0Þ: ð5Þ The total energy change in the system due to propagation of the crack is
dU
dc ¼X dW
dc v, ð6Þ
where U is the total strain energy in the system and v is the volume element at P.
Referring the energy change to unit area of crack surface A(¼ c2), the J-integral can be obtained as
J ¼ dU
dA¼W0cX
p
1
r3z2g r, z, a, b, "ð 0Þrz, ð7Þ where the function g is used for simplifying the equation. Carrying out the summation of Equation (7) in dimensionless space, r and z, one may write the J-integral as
J ¼ k A c, c
B, "0
cW0: ð8Þ
where k(A/c, c/B, "0) is the unknown function to be determined, which reflects the effects of the finite size and deformation level on the J-integral.
Separation of J
In this work, it is assumed that the J-integral for the rubber-cord composite with a penny-shaped crack on the end of a cord may be represented in a separable form as follows:
J ¼ H Wð 0Þ G geometryð Þ: ð9Þ
Comparing Equation (8) with Equation (9), one knows that the functions H(W0) and G(geometry) can be expressed as
H Wð 0Þ ¼k2ð"0Þ W0, G geometryð Þ ¼c k1
A c, c
B
, ð10Þ
where the relation k(A/c, c/B, "0) ¼ k1(A/c, c/B)k2("0) holds. Equation (10) implies that the calibration function k(A/c, c/B, "0) may be separated in the geometry calibration function k1and deformation calibration function k2. The validity of the separation can be inspected through the separation parameter Qijdefined by
Qij¼J cð i, . . .ÞW0 J c j, . . .
W0
¼H Wð 0Þ G cð i, . . .Þ
H Wð 0Þ G c j, . . . ¼G cð i, . . .Þ
G c j, . . . , ð11Þ
where cj is a given reference crack radius. For the separation of Equation (9), to be valid the separation parameter Qijshould be independent of the strain energy density W0. If so, the equation of the J-integral becomes
J ¼ k1
A c,c
B
k2ð"0Þ W0c: ð12Þ
Using this approach, the purpose pursued in this work is as follows: firstly to propose a concrete form of the J-integral calibrated by the finite sizes such as crack length, cord radius and specimen radius, and deformation level; secondly to compare the deformation calibration factor k2("0) obtained in this work with the results obtained by Lindley [11] considering two-dimensional homogeneous specimens with through crack; and thirdly to discuss the effects of the stiff cord and the crack growth on the J-integral.
NUMERICAL ANALYSIS
Finite element analysis (FEA) for the rubber-cord composites of the various geometries under different loading levels were conducted to obtain the equation of the J-integral. The dimensions of the various types of specimens are shown in Table 1.
Regarding the axisymmetry, these are meshed with eight-noded biquadratic, reduced integration elements provided by the finite element code ABAQUS (version 6.1). Figure 2 shows a representative specimen meshing of the rubber-cord composite with a penny- shaped crack on the end of the cord (A/c ¼ 3/7, c/B ¼ 7/18). Sufficient contours in the vicinity of the crack tip, 15 contours as shown in zoomed-in meshes of Figure 2, were used for evaluating the J-integrals calculated by domain integral as implemented in ABAQUS. Numerical analysis was achieved for each value of the crack length and the specimen radius. The material behavior of rubber is modeled by the Mooney-Rivlin relationship:
W ¼ C10ðI13Þ þC01ðI23Þ, ð13Þ
Table 1. Dimensions of the rubber-cord composites with a penny-shaped crack on the end of the cord: unit of length ¼ mm, cord radius A ¼ 3 mm.
C 4 5 6 7 8 9
5 6.25 7.5 8.75 10 11.25
6 7 8.57 10 11.6 14
7.7 7.7 10 14 14 18
B 10 10 14 18 18 22
14 14 18 22 22 26
18 18 22 26 26 30
20 22 26 30 30 45
25 30 35 40
where the material constants C10 and C01 were set at 0.15 and 0.04 MPa, and I1and I2
are respectively the first and second invariants of the strain, which are related to the principal extension ratios i(i ¼ 1, 2, 3) as follows:
I1¼21þ22þ23
I2¼21 þ22 þ23 : ð14Þ The cord is assumed to be a linear elastic material much stiffer than rubber, of which the Young’s modulus and Poisson’s ratio are 90 MPa and 0.35, respectively.
RESULTS AND DISCUSSION
First, using the separation parameter defined by Equation (11), the validity of the separation form of J-integral given in Equation (9) or (12) is investigated. The values of the J-integral for the rubber-cord composites with A ¼ 3 mm, B ¼ 10 mm, and various crack lengths are obtained under the different loading level W0. Substituting the values of J-integral into Equation (11) with two reference crack lengths cj/B ¼ 6/10, 7/10, one can obtain the value of Qij under the different value of W0, of which the results are shown in Figure 3. Regardless of the reference crack length, the separation parameter Qij could be regarded as being independent of the remote strain energy density W0. And that implies that the separation form of J assumed in Equation (12) is reasonable.
Figure 2. Finite elements meshing used in the numerical work: A/c ¼ 3/7, c/B ¼ 7/18.
Therefore the calibration function k(A/c, c/B, "0) can be represented in a multiplica- tive form of the geometry function k1(A/c, c/B) and the deformation calibration function k2("0).
Next, the procedure for obtaining the unknown function k1(A/c, c/B) which characterizes the effects of the finite sizes such as the cord radius and specimen radius on the J-integral, and the function k2("0) which characterizes the overall level of strain
Figure 3. Separation parameter Qij for various crack radius: (a) reference crack radius cj/B ¼ 6/10, (b) reference crack radius cj/B ¼ 7/10.
in rubbery materials is presented. Geometry function kj(A/c, c/B) at a certain strain energy density W0jis obtained by using the following relation,
J
W0jc¼k2ð"0Þ k1 A c, c
B
¼Cjk1 A c, c
B
¼kj A c, c
B
, ð15Þ
where the constant Cjis a deformation calibration factor at a strain energy density W0j, and the function kj(A/c, c/B) might be regarded as a geometry calibration function multiplied by a constant Cj. Hereafter, subscript i and j are used to represent some constants and functions at strain energy density W0jand W0j, respectively. For a constant strain energy density W0j, the unknown values of the function kj(A/c, c/B) for various A/c, c/B can be determined from Equation (15). Using the following relation,
ðJ=W0cÞW0i
ðJ=W0cÞ ¼W0j Cik1ðA=c, c=BÞ Cjk1ðA=c, c=BÞ Ac,Bc
¼Ci Cj
, ð16Þ
where |A/c, c/B means that the values of J/W0c are compared with the same geometries, and the constant Ci is a deformation calibration factor at a strain energy density W0i, the values of kj(A/c, c/B) can be also obtained for W0ias follows:
kj A c, c
B
¼ J
W0c W0i
Cj Ci
: ð17Þ
In this work, first the value of the reference strain energy density W0j in 2 kJ/m3 is chosen, and then, for W0i¼1, 5 kJ/m3, the values of J=W0cW0iare obtained. Substituting the values of J=W0cW0iand Cj/Ciobtained by using Equation (16) (see Table 2 for detailed values) into Equation (17), one can obtain the values of the function kj(A/c, c/B).
The values of the function kj(A/c, c/B) were plotted with c/B for various A/c, as shown in Figure 4 in which the values of the function kj(A/c, c/B) are almost independent of W0
where the values of the function kj(A/c, c/B) for each A/c is fitted to a fourth order polynomial:
kj
A c, c
B
¼f0
A c
þf1
A c
c B
þf2
A c
c B
2
þf3
A c
c B
3
þf4
A c
c B
4
, ð18Þ Table 2. The values of Ci/Cjobtained by Equation (16).
C(mm)
B(mm) 4 5 6 7 8 9
10 1.0166 1.0123 1.0095 1.0078 1.0068 1.0027
14 1.0119 1.0105 1.0098 1.0120 1.0095 1.0084
18 1.0161 1.0137 1.0123 1.0195 1.0140 1.0124
22 1.0153 1.0137 1.0108 1.0120 1.0117 1.0121
26 1.0187 1.0152 1.0174 1.0060 1.0122 1.0952
30 1.0119 1.0118 1.0075 1.0105 1.0098 1.0105
Average 1.0124
Here it is noted that, when c/B is larger than 0.5, the function kj(A/c, c/B) is little dependent on A/c. That means that the effect of the cord on J-integral vanishes.
It should be noted that, as mentioned above, the function kj(A/c, c/B) of Equation (18) is a geometry calibration function only at a reference strain energy density W0j¼2 kJ/m3. Therefore, as Equation (18) is divided by a factor Cj, the pure geometry calibration function k1(A/c, c/B) can be obtained.
For linear elastic materials, the J-integral for the infinite body with a penny-shaped crack is given by [17]
J ¼6
W0c: ð19Þ
For a case involving two finite dimensions such as cord and specimen radius, the J-integral may be presented by
J ¼6
W0c k1 A c, c
B
: ð20Þ
On the other hand, for nonlinear elastic materials like rubbery material, the J-integral is given by
J ¼ kj
A=c, c=B Cj
k2ð"0Þ W0c where, kj
A=c, c=B Cj
¼k1
A c, c
B
:
ð21Þ
Figure 4. Geometry function at a strain energy density W0j¼2 kJ/m3.
Comparing Equations (20) and (21), one finds Cjas follows:
Jlinear
JnonlinearW0¼2, geometry ¼k1ðA=c, c=BÞð6=Þ, W0c
k1ðA=c, c=BÞ, CjW0c ¼ð6=Þ Cj
, ð22Þ
where Jlinear and Jnonlinear mean J-integral for a linear elastic material and a nonlinear elastic material under the same remote loading and geometries, respectively. One easily knows that Equation (22) can give the value of Cjas
Cj¼6
Jnonlinear
Jlinear W0¼2, geometry
ð23Þ
The J-integral for a linear elastic material Jlinear is obtained by FEA for the matrix of the linear elastic material with Young’s modulus ¼ 1.5 MPa and Poisson’s ratio ¼ 0.4999, instead of the Money-Rivlin model. As compared with the geometries of A ¼ 3 mm, c ¼6 mm, and B ¼ 14, 18, 22, 26, the values of Cj calculated by Equation (23) for each model were 1.810, 1.812, 1.810, and 1.808 (average: 1.810). Dividing Equation (18) with Cj¼1.81, one can rewrite an equation of the J-integral as
J ¼ k2ð Þ "0 W0c f0 A c
þf1 A
c
c B
þf2 A
c
c B
2
þf3 A c
c B
3
þf4 A c
c B
4
, ð24Þ where
f0
A c
¼ 64:62 þ 472:25 A c
837:35 A c
2
þ543:45 A c
3
f1
A c
¼622:87 4346:6 A c
þ8012:2 A c
2
5203:3 A c
3
f2
A c
¼ 2036:7 þ 14160 A c
26767 A c
2
þ17442 A c
3
f3 A c
¼2742:1 19156 A c
þ36842 A c
2
24097 A c
3
f4 A c
¼ 1300:4 þ 9172:4 A c
17862 A c
2
þ11722 A c
3
:
ð25Þ
It might be worth knowing how to depend the deformation calibration function on the overall strain level "0, and that the function k2("0) is really independent of the geometric variables. Arranging Equation (24) for the function k2("0), the values of the function k2("0) with "0 can be obtained, which is shown in Figure 5. The function k2("0) is almost independent of geometries, and moreover, approaching infinitesimal strain, k2("0) approaches 1.9, of which the value corresponds to the calibration factor 6/ for a linear elastic material. The result of conversion of constant in the result of fitting of the experimental data performed by Lindley [11] for the sheet specimens, to 6/ for the penny- shaped crack is also shown in line of Figure 5. That implies that the deformation
calibration function obtained from two-dimensional sheet specimens can also be applied to the configuration with penny-shaped crack.
It might be worth to note how to depend the J-integral on the crack growth under a constant remote loading. The J-integral with crack length c/B, for A ¼ 3 mm and B ¼ 10 mm under a constant loading W0¼2 kJ/m3, is shown in Figure 6. The J-integral with crack length c/B shows the characterization as follows: firstly, for small crack radius, though the crack length increases, the J-integral decreases with the crack growth due to the effect of a cord on the J-integral; secondly, for deep crack
Figure 5. Deformation function k2("0).
Figure 6. J-integral with c/B under a constant remote loading, W0¼2 kJ/m3.
radius(c/B>0.7), the J-integral increases due to the effect of crack size. For examining the validity of Equation (24), FEA performed was under different material properties of rubber, (i) C10¼1 MPa, C01¼0.2 MPa, (ii) C10¼0.5 MPa, C01¼0.2 MPa, (iii) C10¼0.1 MPa, C01¼0.02 MPa. The equation of J-integral proposed in this work almost agreed with the results of FEA under three kinds of material properties, as shown in Figure 7, where the data points are the results of FEA, and the lines are obtained by Equation (24).
CONCLUSIONS
An equation of J-integral for a penny-shaped crack on the end of the cord embedded in rubber matrix is proposed. The dimensional analysis is applied to derive an equation of J-integral. It is assumed that the equation of J can be separated into the deformation and geometry function, the validity of which is proved using a separation parameter.
Therefore, the equation of J-integral can be expressed in a multiplicative form of the geometry and deformation calibration function. The deformation calibration function, which is a constant for the linear elastic materials, characterizing the overall level of strain in rubbery materials is obtained by comparing the results for J obtained by considering nonlinear elastic deformation with the ones obtained by the linear elastic counterpart.
As approaching the infinitesimal strain, the value of the deformation calibration function approaches a constant 6/ which is the same as one for a linear elastic material. And, comparing with the result of Lindley [11], one can know that the deformation calibration function obtained for two-dimensional sheet can be applied in the configuration with penny-shaped crack in cord-end. The equation of J-integral in Equation (24) was compared with the results of FEA for the models with three different material properties of rubber. The numerical results were in good agreement with the proposed equation of J-integral.
Figure 7. J-integral with the strain energy density under the various material properties.
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