Quadrilateral isoparametric finite elements for plane ... - Springer Link

Acta Mech Sinica (2005) 21: 388–394 DOI 10.1007/s10409-005-0041-y

R E S E A R C H PA P E R

Hongwu Zhang · Hui Wang · Guozhen Liu

Quadrilateral isoparametric finite elements for plane elastic Cosserat bodies∗

Received: 9 April 2004 / Accepted: 31 March 2005 / Revised: 7 April 2005 / Published online: 13 June 2005 © Springer-Verlag 2005

Abstract 4-node, 8-node and 8(4)-node quadrilateral plane isoparametric elements are used for the solution of boundary value problems in linear isotropic Cosserat elasticity. The patch test is applied to validate the finite elements. Engineering problems of stress concentration around a circular hole in plane strain condition and mechanical behaviors of heterogeneous materials with rigid inclusions and pores are computed to test the accuracy and capability of these three types of finite elements. Keywords Cosserat model · Isoparametric elements · Patch test · Heterogeneous materials

1 Introduction The structural hierarchy of material with microstructures plays an important role in determining its macroscopic mechanical behaviors [1]. For the behavior of granular media, in classic continuum theory, the rotational degree of freedom is usually suppressed. To overcome this deficiency of the continuum models, some extended continuum models characterized by additional kinematic degrees of freedom and/or higher order deformation gradients are developed [2–7]. The kinematics of a Cosserat continuum is characterized by adding the rotational degree of freedom to the conventional displacement degrees of freedom and by adding a material constant of the dimension of length scale to the con∗ The project supported by the National Natural Science Foundation of China (10225212, 50178016, 10421002) and the Program for Changjiang Scholars and Innovative Research Team in University of China The English text was polished by Keren Wang.

H. W. Zhang (B) · H. Wang State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, Dalian 116024, China E-mail: [email protected] G. Z. Liu Xinglongtai Oil Extraction Plant, Liaohe Oil Field, Panjin 124000, China

stitutive equations [8]. Much work has been carried out on the finite element computation of the Cosserat model, such as those proposed by Adhikary and Dyskin [9] and Mori et al. [10]. However, studies on the properties of Cosserat elements in the finite element analysis are few. Provids and Kattis developed three triangular finite elements and introduced a patch test for the validation of the elements [11]. On the other hand, this paper focuses on the property and application of the plane 4-node, 8-node and 8(4)-node isoparametric elements in numerical analysis of elastic Cosserat materials. This paper is organized as follows: the second section is about the basic equations of the Cosserat theory and the corresponding virtual work principle. Because the symmetric condition of shears in Cosserat elasticity does not hold good anymore, the stress transformations among different local coordinate systems should be deduced again, as are completed in the third section. Three plane isotropic finite elements and their stiffness matrices are constructed in the fourth section. A proper method for the patch test is described and applied to verify the validity and accuracy of the finite elements in the fifth section. Finally, some practical engineering problems are computed to test the applicability of the elements investigated.

2 The Cosserat model In this paper, ui is the displacement vector in the linear elastic Cosserat model, ωi is the independent rotation vector, and χij = ωj,i is the curvature tensor. Strain tensor εij depends on both the displacement vector and the rotation vector. The stress tensor σij and the couple stress tensor mij are related to εij and χij through the constitutive equations. The kinematic equations, constitutive equations and equilibrium equations can be written, respectively, as follows: Equilibrium equations σij,i + fj = 0,

mij,i + ej ik σik + ψj = 0,

(1)

where fi is the body force vector, ψi is the body couple vector and eij k is the permutation tensor.

Quadrilateral isoparametric finite elements for plane elastic Cosserat bodies

Kinematic equations εij = uj,i + ej ik ωk ,

3 Transformation of stresses

χij = ωj,i .

(2)

The kinematic equations are reduced to the classic ones when the rotation vector is neglected. Constitutive equations σij = Dij kl εkl ,

mij = Gij kl χkl .

(3)

Boundary conditions σij ni = f¯j , mik ni = ψ¯ k , ui = u¯ i ,

(4) (5)

ωi = ω¯ i .

For the two-dimensional continuum under the plane-strain condition, the kinematic equations and the constitutive equations can be rewritten in matrix form as ε = LU , where

σ = Eε,



∂/∂x  0  0  L=  0 ∂/∂y  0 0 λ + 2µ λ λ

   E=   

0 ∂/∂y 0 ∂/∂x 0 0 0

(6)  0 0  0   −1  , 1  ∂/∂x  ∂/∂y

λ λ λ + 2µ λ λ λ + 2µ

As shown in Fig. 2, the stresses in the new coordinate system (1 -o-2 ) are defined as σ1 1 σ2 2 σ1 2 σ2 1 M1 3 M2 3 . α is the angle between the original coordinate system and the current/inclined plane coordinate system. From the equilibrium in 1 and 2 directions, the following results are obtained: The principal direction of the normal stresses is given by σ12 + σ21 . (8) tg(2α) = σ11 − σ22 The normal and the shear stress components corresponding to the principal direction can be expressed as 1 σ1 1 or σ2 2 = (σ11 + σ22 ) 2 1 ± (σ11 − σ22 )2 + (σ12 + σ21 )2 , (9) 2 1 1 (10) σ1 2 = (σ12 − σ21 ), σ2 1 = (σ21 − σ12 ). 2 2 The expressions of the other state variables can be deduced from the above equations.

 µ + µc µ − µc µ − µc µ + µ c

4µlc2

   ,    4µlc2

where λ = E · γ /(1 + γ )(1 − 2γ ) and µ are the Lame’s constants, E is the Young’s modulus, γ the Poisson’s ratio, µc a constant similar to the shear modulus, lc the material characteristic constant. The notation rule of the above equations is shown in Fig. 1. The energy with respect to the admissible displacement/ strain field can be expressed as dσij dεij dV + dmij dχij dV − dfj duj dV = V V V − dψj dωj dV − df¯j duj d − dψ¯ j dωj d . (7) V

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Combined with Eqs. (2), taking the variation of with respect to u and ω yields the equilibrium equations and the traction boundary conditions.

Fig. 1 Two dimensional Cosserat model

4 Finite element formulations Three Cosserat elements are developed based on the above model in this section. The 4-node and 8-node elements have three degrees of freedom at each node, and the 8(4)-node element has three degrees of freedom at each corner node, two degrees of freedom at each mid-edge node, as shown in Fig. 3. The formulation of the displacement fields ui and the rotation fields ωi over the element can be expressed as ui = Niku uk , ωi = Nikω ωk , (11) where uk and ωk are the corresponding values at the element nodes. The shape functions of the 4-node isoparametric element take the following form Ni = (1 + ξi ξ )(1 + ηi η)/4, i = 1, 2, 3, 4. (12) The shape functions of the 8-node isoparametric element are Ni = (1+ξi ξ )(1+ηi η)(ξi ξ +ηi η−1)/4, i = 1, 2, 3, 4, Ni = (1 − ξ 2 )(1 + ηi η)/2, i = 5, 7, Ni = (1 − η2 )(1 + ξi ξ )/2, i = 6, 8. (13) The 8(4)-node isoparametric element shares the same interpolation function of ui with that of the 8-node isoparametric element. The interpolation function of ωi in the 8(4)-node isoparametric element takes the same form as that of the 4-node isoparametric element. Substituting Eqs. (11) into Eq. (7) gives KeU e = F e, (14) e e e where K , U , F are the element stiffness matrix, displacement vector and load vector, respectively.

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Fig. 2 Transformation of stresses between different coordinate systems

Fig. 3 (a) 4-node isoparametric element, (b) 8-node isoparametric element, (c) 8(4)-node isoparametric element

5 Patch test Providas and Kattis pointed out that in the Cosserat elasticity there are certain difficulties in patch test because of the presence of the independent rotation in the definition of strain [11]. Indeed, an equilibrium state of Cosserat constant strain results when the displacements are taken to vary linearly, a constant rotation field is selected and a constant body couple is applied. If both the displacements and the rotation are chosen to be linear polynomials then linearly varying strains are produced and the equilibrium equations require the application of constant body forces and a linearly varying body couple. The method which can be found in Ref. [11] is used to carry out the patch test of the elements. The rectangular regions, as shown in Fig. 4, are covered by several arbitrarily distorted 4-node/8-node/8(4)-node elements with at least one node inside the regions. The purpose of adopting different meshes is to guarantee the variety of the patch test. These three tests will be expatiated as follows. The material parameters used in the patch tests are µ = 1.0 × 103 N/mm2 , µc = 5.0 × 102 N/mm2 , γ = 0.25, lc = 0.1 mm. Patch test 1: Rotation angle is constant. Body force and body couple force are not considered. This test is designed to examine the element when it reduces to the classic one. The

corresponding analytical solutions are given as u = 10−3 (x1 + 0.5x2 ), v = 10−3 (x1 + x2 ), = 0.25 × 10−3 , f1 = f2 = = 0, σ11 = σ22 = 4, σ12 = σ21 = 1.5, m13 = m23 = 0. Comparisons of the analytical solutions and the numerical solutions show that all elements pass the patch test 1. Patch test 2: Rotation angle and body couple force are constant, and body force is not considered. The shear stresses are not equal to each other at the adjacent edges of the element. The effect of the rotation angle is not taken into consideration. The analytical solutions are given as u = 10−3 (x1 + 0.5x2 ), v = 10−3 (x1 + x2 ), = 0.25(1 + α) × 10−3 , f1 = f2 = 0, = 1, σ11 = σ22 = 4, σ12 = 1, σ21 = 2, m13 = m23 = 0. From the numerical results it can be concluded that all the elements pass the patch test 2.


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Fig. 4 Structure and mesh used for 4/8/8(4)-node element patch test Table 1 Stresses of 4-node element in patch test 3 Element number 1 2 3 4 5

σ11

Numerical analytical 4.000 086 4.000 000 4.000 088 4.000 000 4.000 036 4.000 000 4.000 000 4.000 000 3.999 850 4.000 000

σ22


σ12

Numerical analytical 1.750 028 1.750 000 −1.249 934 −1.2500 00 −1.250 004 −1.250 000 −1.250 048 −1.250 000 1.749 963 1.750 000

Patch test 3: Rotation angle and body couple force are functions of coordinate values, and body force is constant. This is the general condition of the Cosserat element. The analytical solutions are given as u = 10−3 (x1 + 0.5x2 ), v = 10−3 (x1 + x2 ), = (0.25 + 0.5α(x1 − x2 )) × 10−3 , f1 = f2 = 1, = 2(x1 − x2 ), σ11 = σ22 = 4, σ12 = 1.5 − (x1 − x2 ), σ21 = 1.5 + (x1 − x2 ), m13 = −m23 = 2l 2 /α. The stress results of all elements in patch test 3 are shown in Tables 1–3, respectively, and it can therefore be concluded that all the elements pass the patch test 3.

σ21

Numerical analytical

M13

1.249 954 1.250 000 −1.750 055 −1.7500 00 −1.749 982 −1.750 000 −1.749 951 −1.750 000 1.250 006 1.250 000


M23


−0.040 007 −0.040 000 −0.039 986 −0.040 000 −0.039 985 −0.040 000 −0.040 004 −0.040 000 −0.040 009 −0.040 000

6 Numerical examples 6.1 Stress concentration around a circular hole In order to examine the applicability of elements investigated above, a stress concentration problem around a circular hole in an isotropic plane under uniform tension, as shown in Figs. 5 and 6 is computed. With the symmetry of the problem only a quarter of the structure is considered. The material parameters and the geometric data used are r = 0.216 mm, L = 16.2 mm, γ = 0.3, E = 2.0 × 105 N/mm2 . The numerical solutions of the stress concentration factor according to different values of r/ lc and µc are given in Tables 4 and 5. The analytical solutions are also shown for comparison. It can be seen from Tables 4 and 5 that for the same r/ lc , the stress concentration factor decreases with the increase of µc , and this is quite different from the result 3.0 obtained by the classic theory. For the same µc , the stress

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Table 2 Stresses of 8-node element in patch test 3 Element number

σ11

1


σ22

3.999 997 4.000 000 4.000 005 4.000 000 4.000 013 4.000 000 4.000 007 4.000 000 4.000 022 4.000 000

2 3 4 5


σ12


σ21

−2.499 983 −2.500 000 −0.549 987 −0.550 000 −1.550 021 −1.550 000 −2.499 993 −2.500 000 −0.549 999 −0.550 000

3.999 994 4.000 000 4.000 001 4.000 000 4.000 013 4.000 000 4.000 028 4.000 000 4.000 003 4.000 000


M13

5.499 979 5.500 000 3.549 981 3.550 000 4.550 000 4.550 000 5.499 973 5.500 000 3.549 982 3.550 000


M23


−0.039 999 −0.040 000 −0.039 999 −0.040 000 −0.039 999 −0.040 000 −0.040 000 −0.040 000 −0.040 001 −0.040 000

Table 3 Stresses of 8(4)-node element in patch test 3 Element number 1 2 3 4 5

σ11


σ22


σ12

Numerical analytical −5.200 047 −5.200 000 −2.000 011 −2.000 000 −3.000 015 −3.000 000 −4.000 010 −4.000 000 −0.799 946 −0.800 000

concentration factor increases with the increase of r/ lc . It can also be found from Tables 4 and 5 that all the results obtained by 4-node, 8-node and 8(4)-node elements satisfy the required accuracy in computation and the accuracy of 8-node, 8(4)-node isoparametric elements is higher than that of the 4-node element. In the case of µc = 98 900.0 and µ − µc < 0, the computer program can still give a good result. Figure 7 shows the comparison of the three different results with respect to µc and r/ lc .

Fig. 5 Circular hole in a Cosserat material

σ21


M13


M23


−0.040 000 −0.040 000 −0.039 996 −0.040 000 −0.040 000 −0.040 000 −0.040 006 −0.040 000 −0.040 007 −0.040 000

6.2 Mechanical response of heterogeneous materials To show further the capability of the computer program developed, a 40 mm × 20 mm matrix material with several inclusions, as shown in Fig. 8, is computed with the model described above. Constraint displacement and traction boundary conditions are imposed on the left and right sides of the sample. The distributed load on the right rigid side is:

Fig. 6 Mesh around a circular hole


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Table 4 Stress concentration factor at circular hole, r = 0.216, r/ lc = 1.063 µc 0.0 5 130.77 25 638.46 98 900.00 327 938.46

Analytical

4-node

% error

8-node

% error

8(4)-node

% error

3.000 2.849 2.555 2.287 2.158

2.849 2.720 2.460 2.209 2.073

5.03 4.53 3.72 3.41 3.94

2.952 2.805 2.519 2.259 2.134

1.60 1.54 1.41 1.22 1.11

2.952 2.805 2.519 2.257 2.132

1.60 1.54 1.41 1.31 1.20

Table 5 Stress concentration factor at circular hole, r = 0.216, r/ lc = 10.63 µc 0.0 5 130.77 25 638.46 98 900.00 327 938.46

Analytical

4-node

% error

8-node

% error

8(4)-node

% error

3.000 2.956 2.935 2.927 2.923

2.849 2.808 2.778 2.734 2.680

5.03 5.01 5.35 6.59 8.31

2.952 2.910 2.890 2.882 2.878

1.60 1.57 1.53 1.54 1.54

2.952 2.909 2.889 2.880 2.877

1.60 1.59 1.57 1.61 1.57

Fig. 9 Heterogeneous material with pore inclusions

Fig. 7 Stress concentration factor versus r/ lc

Fig. 8 Heterogeneous material with rigid inclusions Fig. 10 Displacement in x direction versus the volume fraction of inclusions

qx = 1 N/mm, qy = 1 N/mm. The structure is discretized with 1 020 nodes and 1 053 elements and is assumed in the plane stress state. The volume fractions of inclusions in the matrix material are set to be 5%, 10%, 15% and 20% in the computation. The Young’s modulus and the Poisson’s ratio of the matrix material and the inclusions are taken, respectively, as: Em = 2.0 × 105 N/mm2 , γm = 0.25; Ec = 4.0 × 105 N/mm2 , γc = 0.25. Cosserat parameters are µc =

4.0 × 104 N/mm2 , lc = 0.1 mm. The case in which the inclusions are pores, as shown in Fig. 9, is also computed for the better understanding of the Cosserat model mentioned in the previous section. The displacement results according to different volume fractions and inclusion properties are given in Figs. 10 and 11,

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is presented. Unambiguous test data and computation results are presented. It is indicated that all the three types of elements pass the patch test and can be used to obtain good accuracy in the simulation of stress concentration problems. The application of the element to predict the mechanical behaviors of heterogeneous materials with inclusions/pores shows the practicability of the elements and the computer program developed.

References Fig. 11 Displacement in y direction versus the volume fraction of inclusions

u and v are the displacements in horizontal and vertical directions at the middle point of the right side of the sample. It is found that the material stiffness increases for the rigid inclusions and decreases for the pore inclusions with the increase of the volume fraction of the inclusion. It is observed also from the numerical results that the Cosserat model has not as strong influence on the results for the pure tension case, as for the shear deformation case. The similar conclusion was reported by several authors in their early work[3]. This demonstrates again the validity of the numerical model and the computer program developed.

7 Conclusions Three types of finite elements, i.e. 4-node, 8-node, and 8(4)node isoparametric elements are used for the finite element analysis of Cosserat material model. The detailed procedure for the patch tests of these three kinds of Cosserat elements

1. Lake, R.: Materials with structural hierarchy. Nature, 361(11), 511– 515 (1993) 2. Vardoulakis, I., Graf, B.: Calibration of constitutive models for granular materials using data from biaxial experiments. Geotechnique, 35(3), 299–317 (1985) 3. de Borst, R.: Simulation of strain localization: a reappraisal of the Cosserat continuum. Eng. Comput., 8, 317–332 (1991) 4. Cosserat, E., Cosserat, F.: Theorie des Corps Deformables. Hermann Paris 1909 5. Mindlin, R.D.: Stress functions for a Cosserat continuum. Int. J. Solids Struct., 1, 265–271 (1965) 6. Aifantis, E.C.: On the microstructural original of certain inelastic models. Journal of Engineering Materials and Technology, 106, 326–334 (1984) 7. Zhang, H.W., Zhang, X.W.: A combined parametric quadratic programming and precise integration method based dynamic analysis of elastic-plastic hardening/softening problems. Acta Mechanica Sinica, 18(6), 638–648 (2002) 8. Huang, W., Bauer, E.: Numerical investigations of shear localization in a micro-polar hypoplastic material. Int. J. Numer. Anal. Meth. Geomech., 27, 325–352 (2003) 9. Adhikary, D.P., Dyskin, A.V.: A Cosserat continuum model for layered materials. Comput. & Geo., 20(1), 15–45 (1997) 10. Mori, K., Shiomi, M., Osakada, K.: Inclusion of microscopic rotation of powder particles during compaction in finite element method using Cosserat continuum theory. Int. J. Numer. Meth. Eng., 42, 847–856 (1998) 11. Providas, E., Kattis, M.A.: Finite element method in plane Cosserat elasticity. Comput. & Struct., 80, 2059–2069 (2002)