Internally circumferentially cracked cylinders with functionally graded material properties

International Journal of Pressure Vessels and Piping 75 (1998) 499–507

Internally circumferentially cracked cylinders with functionally graded material properties Chunyu Li a,*, Zhenzhu Zou b a

Department of Astronautics and Mechanics, Harbin Institute of Technology, Harbin 150001, People’s Republic of China b Department of Transportation, Shijiazhuang Railway Institute, Shijiazhuang 050043, People’s Republic of China Received 20 April 1998; accepted 26 May 1998

Abstract In this paper, internal circumferentially cracked hollow cylinders, which are assumed to be made of functionally graded materials, are studied. The conventional finite element method (FEM) is improved by introducing isoparametric transformation for simulating the gradient variations of material properties in finite elements. By using this improved FEM, the mode I stress intensity factors are calculated for cylinders subjected to uniform tension. Various types of functionally graded materials and different gradient compositions for each type are investigated. The results show that the material property distribution has a quite considerable influence on the stress intensity factors. q 1998 Elsevier Science Ltd. All rights reserved Keywords: Functionally graded materials; Stress intensity factor; Finite element method; Hollow cylinder; Crack

1. Introduction In recent years, the concept of so-called functionally gradient materials (FGMs) has been introduced and applied to the development of structural components. The advantages of FGMs are that the materials could resist corrosion, radiation and high temperatures effectively and, at the same time, the residual and thermal stresses in the materials could be relaxed significantly [1]. The interest in FGMs research is growing rapidly due to these advantages. It is expected that FGMs have promising applications in aerospace engineering, chemical engineering and nuclear power plants. From the viewpoints of applied mechanics, FGMs are nonhomogeneous solids. Due to the complexity of the problem, there are only a few papers that have studied the crack problems in FGMs. Erdogan and co-workers [2–5] studied some problems of nonhomogeneous elastic materials with cracks. Wang and co-workers [6,7] studied some crack problems in composites with nonhomogeneous interlayer. An important conclusion given by these researches is that the nature of the stress singularity at a crack tip in nonhomogeneous materials would remain to be the standard square-root type as homogeneous solids provided that the * Corresponding author at Shijiazhuang Railway Institute, Department of Architectural Engineering, Shijiazhuang 050043, People’s Republic of China

spatial distribution of the material property is continuous near and at the crack tip. However, the analytical approach used by these workers can only deal with some simple distribution of material properties, such as an exponential form [2–5] or power form [6,7]. It is difficult to investigate the influences of the actual material property distributions on the fracture mechanics parameters, although these influences are very important knowledge for the optimal design of FGMs. Therefore, numerical methods have to be developed for the analyses of a large range of practical problems. Amongst numerical methods, the most versatile is the finite element method (FEM) and it has been extensively used for computational fracture mechanics studies. However, all of these finite element approaches have mainly concentrated on homogeneous materials or piecewise homogeneous materials. The finite element formulation relating to nonhomogeneous materials with continuously varying properties has not been found in literature. In this paper, we improve the conventional FEM and use it to analyze the circumferentially cracked hollow FGM cylinders. The cylinders are assumed to be under remote uniform tension. A part-through crack is assumed to be at the inner surface of cylinder. Our main objective is to investigate the influence of the material property distribution across the cylinder wall on the stress intensity factors (SIFs).

0308-0161/98/$ - see front matter q 1998 Elsevier Science Ltd. All rights reserved PII: S0 30 8 -0 1 61 ( 98 ) 00 0 53 - 2

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2. Material property model for functionally graded materials Consider an FGM cylinder described in Fig. 1a. It is of inner radius R i, outer radius R o and length 2H (z-direction). Assume that the cylinder is fully ceramic at R i and gradually changes to fully metal at R o, the volume fractions of both phases vary along r-direction and the composition in any cylindrical face is held constant. The local volume fractions in r-direction can be represented by the equations   r ¹ Ri p , Vc (r) ¼ 1 ¹ Vm (r) (1) Vm (r) ¼ t where t is the thickness of cylinder-wall, V m is the volume fraction of metal, V c is the volume fraction of ceramic and p is called gradient exponent. Using Eq. (1), a great range of FGMs composition profiles can be examined by varying the exponent p, as shown in Fig. 2a. Due to the changes in relative proportions of ceramic and metal, the material mechanical properties vary across the cylinder-wall. From the linear mixture rule [8], the elastic modulus E and the Poisson’s ratio n can be considered to vary with volume fractions as following   r ¹ Ri p (2) E(r) ¼ Em Vm þ Ec Vc ¼ Ec þ (Em ¹ Ec ) t

Fig. 1. Internal circumferentially cracked cylinder subjected to uniform tension and the finite element division on its axisymmetric face: (a,b).

 n(r) ¼ nm Vm þ nc Vc ¼ nc þ (nm ¹ nc )

r ¹ Ri t

p (3)

where the subscripts m and c refer to the metal and ceramic, respectively. Some variations of the elastic modulus E with volume fractions are illustrated in Fig. 2b,c.

3. Finite element method for functionally graded materials The displacement finite element method is usually applied to stress analysis problems. Its formulation can be derived by the principle of minimum potential energy [9].

Fig. 2. The compositional profiles of FGMs and the elastic modulus distribution across the cylinder wall: (a–c).

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Consider a m-node axisymmetric element as shown in Fig. 3. The global coordinates of a point on the element at (y,h) are given by m X

r¼

Ni (y, h)ri ,

z¼

i¼1

Fig. 3. m-node axisymmetric element.

The elemental stiffness matrix [K] e for axisymmetric problem can be derived as ZZ [B]T [D][B]rdrdz: (4) [K]e ¼ Ae

where [B] is the strain matrix, [D] is the stress–strain constitutive matrix. In above formulation, only the matrix [D] has relations with the material properties. For elastic isotropic materials under axisymmetric condition, [D] may be expressed as 3 2 1 sym: 7 6 n 7 6 1 7 6 1¹n 7 6 E(1 ¹ n) 6 7 n [D] ¼ 7 6 n 1 7 (1 þ n)(1 ¹ 2n)6 7 6 1¹n 1¹n 7 6 4 1 ¹ 2n 5 0 0 0 2(1 ¹ n) (5) For homogeneous materials, E and n are constants. Now for nonhomogeneous FGMs, however, E and n are functions of the r coordinate, as expressed in Eq. (2), (3). Therefore, the variations of the material properties must be considered in the finite element analysis for FGMs. In this paper, we adopt the concept of the well-known isoparametric transformation for properly describing the variations of the material properties.

m X

Ni (y, h)zi

(6)

i¼1

where N i are the shape functions corresponding to the node i, whose coordinates are (r i,z i) in the global system and (y i,h i) in the local system. As an isoparametric element, the displacements within the element are interpolated as follows u¼

m X

Ni (y, h)ui ,

w¼

i¼1

m X

Ni (y, h)wi

(7)

i¼1

where (u,w) are the nodal displacements in the r and z directions. Now, let the material properties E and n at the point (y,h) be expressed as E¼

m X

Ni (y, h)Ei

(8)

i¼1 m X

n¼

Ni (y, h)ni

i¼1

where (E i,n i) stand for the material properties at the node i of the element. By using Eq. (8), the actual variations of the material properties in a finite element can be approximated by polynomial forms. The degree of the polynomial depends on the number of nodes in the element. Substituting Eq. (8) into Eq. (5), we obtain the elemental elastic matrix [D] e, which becomes a function of the intrinsic coordinates. Then we can calculate the elemental stiffness matrix by the standard Gaussian numerical quadrature in the intrinsic coordinates domain, that is [K] ¼ e

1 1 Z Z

[B]T [D]e [B]rdetJdydh

(9)

¹1 ¹1

After the elemental stiffness matrices [K]e and the contribution of

Fig. 4. Triangular quarter-point elements in the vicinity of crack tip.

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the nodal force vectors are combined to form a global stiffness matrix [K] and a generalized nodal force vector {F}, respectively, the vector of the global nodal displacements can be obtained by solving the following global stiffness equations [K]{u} ¼ {F}

(10)

The stresses at any point on a specified element can be obtained by the following equation {j} ¼ [D]e {«} ¼ [D]e [B]{q}

(11)

where {q} stands for the nodal displacements of the specified element.

4. Stress intensity factors computation The studies in the literature [2–7] have shown that, in nonhomogeneous materials with continuously varying properties, the nature of the stress singularity at a crack tip would remain identical to that in homogeneous solids. Therefore, the computational methods for the SIFs in homogeneous solids can be adopted for that in nonhomogeneous solids with a few modifications. There are many ways of evaluating the SIFs from finite element solution. These include the extrapolation of displacement and/or stress fields to the crack tip; Rice’s contour integral J; the strain energy approach; the virtual crack extension technique, and so on. In this paper, we use the displacement extrapolation technique. The so-called triangular quarter-point elements [10] are used as crack tip elements.p It has been verified that this kind of element results in a 1= r strain singularity within the elements as well as on the element edges. The SIFs are obtained by two-point formula. The procedure is briefly described as following.

Fig. 5. Error analyses for optimal crack-tip element size.

Consider a crack tip region shown in Fig. 4. The displacement fields on the crack surfaces can be written as r kþ1 r w(v ¼ p) ¹ w(v ¼ ¹ p) ¼ KI (12) m 2p r kþ1 r KII u(v ¼ p) ¹ u(v ¼ ¹ p) ¼ m 2p where k ¼ (3 ¹ 4n) for plane strain and k ¼ (3 ¹ n)/(1 þ n) for plane stress, m ¼ E/2(1 þ n) is the shear modulus. For the points d and e with r ¼ L, we obtain r 2p m(r ¼ L) (w ¹ we ) (13) KI(r ¼ L) ¼ L k(r ¼ L) þ 1 d r 2p m(r ¼ L) (u ¹ ue ) KII(r ¼ L) ¼ L k(r ¼ L) þ 1 d For the points a and b with r ¼ L/4, we obtain r 2p 2m(r ¼ L=4) (w ¹ wc ) KI(r ¼ L=4) ¼ L k(r ¼ L=4) þ 1 b

(14)

r 2p 2m(r ¼ L=4) (u ¹ uc ) KII(r ¼ L=4) ¼ L k(r ¼ L=4) þ 1 b Then, the values of K I and K II at the tip o are obtained by linear extrapolation as following KI ¼ 2KI(r ¼ L=4) ¹ KI(r ¼ L)

(15)

KII ¼ 2KII(r ¼ L=4) ¹ KII(r ¼ L) It should be emphasized that the values of the material properties in above equations are the values of the corresponding points.

Fig. 6. The effect of the type of FGMs on the normalized SIFs.

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Fig. 7. The effect of the composition gradients of FGMs on the normalized SIFs.

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Table 1 The normalized stress intensity factor for internal circumferentially cracked FGM cylinders

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506

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Table 1 Continued

5. Numerical results and discussion As shown in Fig. 1a, the crack plane is a plane of symmetry and the crack problem is one of mode I. Because of the symmetry, we only consider the z . 0 part of the cylinder. The finite element division in the axisymmetric face is shown as Fig. 1b. The crack tip elements are sixnode triangular quarter-point elements. Other elements are eight-node isoparametric elements. The lengths of all element sides emanating from the crack tip are selected as 1/10 of the crack length for a/t # 0.7, and 1/20 for a/t ¼ 0.8. This selection is made from the error analyses of the computed results for edge cracked plate, which has been given analytical solution by Erdogan and Wu [5]. The finite element division of the plate is the same as described in Fig. 1b, but for x,y coordinates. The error analyses are illustrated in Fig. 5. It should be mentioned that the mid-side nodes on the

crack tip elements remain at the mid-side position for simulating the variations of the material properties in the crack tip elements. The normalized mode I stress intensity factor is defined by K FI ¼ pI j pa

(16)

In this study, we mainly investigated the influences of E(r) on the normalized SIFs. The Poisson’s ratio is assumed to be constant (n ¼ 0.3). The ratio of E m/E c is assumed as 0.2, 0.4, 0.6, 0.8, 1.2, 2.0, 5.0 and 10.0, respectively. The material gradient exponent p is selected as 0.25, 0.5, 0.75, 1.0, 1.25, 1.5, 2.0, 4.0 and 10.0, respectively. The ratio H/t is taken as 5.0. Some sample results for the normalized SIFs are shown in Figs 6–8, and the additional results are stipulated in Table 1. It should be mentioned that the results given in this study are calculated under plane strain condition.

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p the computed normalized SIFs are multiplied by 1 ¹ a=t. Clearly, the effects of the thickness of cylinder-wall on the normalized SIFs are similar to that of homogeneous solids, but the focuses, to which the multiplied normalized SIFs tend, lower gradually in position at the ordinate. 6. Conclusion The technique of isoparametric transformation has been adopted for simulating the variations of material properties. The finite element formulation developed in this paper has been verified to be very suitable for the stress analysis of FGMs. The descriptions of geometry, displacements and material properties have been achieved the same accuracy. The FGMs cylinders with an internal circumferential crack are studied by using this improved FEM. It is revealed that the material property distribution has quite considerable influence on the SIFs. Generally, the larger the elastic modulus of metal is than that of ceramic, the lower the stress intensity factors are. The composition of FGMs influences the stress intensity factors only when the difference between the elastic modulus of metal and that of ceramic is larger. This paper’s method can be extended straightforwardly to fully three-dimensional problems. The revision of the existing FEM software is very easy.

Acknowledgements This research was supported by the Natural Science Fund of People’s Republic of China. References

Fig. 8. The effect of the thickness of cylinder-wall on the normalized SIFs.

Fig. 6 shows the effect of different couple of FGM phases on the normalized SIFs. It can be seen that the parameter E m/E c has considerable influence on the SIFs. The normalized SIFs decrease with the increase of E m/E c value, especially when a/t is less. Fig. 7 shows the influences of the compositional gradient exponent p on the normalized SIFs. It can be seen that the values of p have obvious influences on the SIFs when the differences between the values of E m and E c are greater. However, the influences of p decrease with the reduction of the differences between E m and E c. When the ratios of E m/E c approach to 1.0, such as 0.8 or 1.2, the influences of p on the normalized SIFs are almost negligible. The above conclusion can also be obtained from the results in Table 1 for R i/R o equals 0.7 and 0.8, respectively. Fig. 8 shows the effect of the thickness of the cylinder wall on the normalized SIFs. For comparison with Ref. [11],

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