Base force element method of complementary energy principle for large rotation problems

Acta Mech Sin (2009) 25:507 515 DOI 10.1007/s10409-009-0234-x RESEARH PAPER Base force element method of complementary energy principle for large rotation problems Yijiang Peng Yinghua Liu Received: 26 August 2008 / Revised: 11 December 2008 / Accepted: 12 December 2008 / Published online: 27 February 2009 The hinese Society of Theoretical and Applied Mechanics and Springer-Verlag GmbH 2009 Abstract Using the concept of the base forces, a new finite element method (base force element method, BFEM) based on the complementary energy principle is presented for accurate modeling of structures with large displacements and large rotations. First, the complementary energy of an element is described by taking the base forces as state variables, and is then separated into deformation and rotation parts for the case of large deformation. Second, the control equations of the BFEM based on the complementary energy principle are derived using the Lagrange multiplier method. Nonlinear procedure of the BFEM is then developed. Finally, several examples are analyzed to illustrate the reliability and accuracy of the BFEM. Keywords Base force element method (BFEM) omplementary energy principle Lagrange multiplier method Geometrically nonlinear Large rotation 1 Introduction The finite element method (FEM) based on an assumed displacement field has become a feasible choice for solving The project supported by the hina Postdoctoral Science Foundation Funded Project (20080430038) and the Funding Project for Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality (05004999200602). Y. Peng (B) Y. Liu FML, Department of Engineering Mechanics, Tsinghua University, 100084 Beijing, hina e-mail: pengyijiang@bjut.edu.cn Y. Peng Key Lab of Urban Security and Disaster Engineering, Beijing University of Technology, 100124 Beijing, hina a wide variety of problems in structural mechanics 1,2. However, some shortcomings of the displacement-based FEM have been observed in the analyses of certain classes of problems, such as the large deformation, the treatment of nearly incompressible materials, the bending of thin plates, and the moving boundary problems. Several investigations have been carried out using alternative methods that involved complementary energy principles, such as the hybrid stress method and the stress-based method. The hybrid stress method 3 5 derives element stiffness matrices by complementary energy principles using assumed equilibrating stresses in the element and displacements along the element boundary. Hybrid elements have been used for both solid and fluid linear, as well as geometric and material nonlinear problems. Hybrid elements can also be constructed to avoid the locking phenomena, which may appear in the assumed displacement method under such constraint conditions as incompressibility. However, inversion of the flexibility matrix in the hybrid method is a necessary step for generating the element stiffness matrix, which may become a computational burden, especially if higher order approximations of stress fields are required. The stress-based FE model provides an alternative approach to the problem. The method is attractive because the calculation results and the calculations using the displacement FE model provide respectively the lower and the upper limits on the productions of strain energy and the load limit, and its use allows easy estimation of error for the approximate solution. The first paper concerning the stress approach to FEM was written by Fraeijs de Veubeke 6. In the last three decades, some newer concepts have been introduced. Taylor and Zienkiewicz 7 used the penalty method to satisfy the equilibrium equations, which can be regarded as a mixed approach. Gallagher, Heinrich and Sarigul solved plane problems of elasticity using the Airy stress function

508 Y. Peng, Y. Liu and the concept of blending interpolation technique to facilitate the satisfaction of the boundary conditions 8,9. Vallabhan and Azene 10 also solved the plane problems, but used Lagrange multipliers to impose the boundary conditions. The stress-based formulation of FEM was utilized in the theory of plasticity by Rybicki and Schmit 11, and Wiȩkowski, Youn and Moon 12. The application of complementary variational principles to elastic buckling was considered by Tabarrok and Simpson 13. The stress-based finite element method utilizing the principle of complementary work or stationary complementary energy is used rather rarely in the engineering practice in comparison with the displacementbased finite element method, behind which there are two reasons: the statically admissible fields of stresses are more difficult to construct than the kinematically admissible fields of displacements; the stress boundary conditions leads to linear equations when stress functions are used, while the nodal displacements in the stress-based method are usually difficult to obtain. A new concept was introduced by Gao 14 byusingthe concept of base forces to replace various stress tensors in the description of the stress state at a point. These base forces can be directly obtained from the strain energy. For large deformation problems, when the base forces was adopted, the derivation of basic formulae was simplified 15 19. Based on the concept of base forces, precise expressions of stiffness and compliance matrices were obtained by Gao for the FEM 14. The applications of the stiffness matrix to the plane problems of elasticity using the 4-node plane element, and the polygonal element were considered by Peng and Jin 20,21. Based on the concept of base forces, Peng and Jin 22,23 presented an explicit expression of the compliance matrix for arbitrary base forces element meshes. Using the complementary energy principle, the detailed control equations of the BFEM expressed by the base forces and incorporated with the Lagrange multiplier method, the plane problems of elasticity were solved using the 4-mid-node plane element and the polygonal element. The objective of the present paper is to present a new geometrically nonlinear formulation of the BFEM for the analysis of large displacement and large rotation problems. In the present formulation of BFEM, the base forces are treated as unknown variables, the basic equations are constructed via the complementary energy principle and the element equilibrium conditions are fulfilled by using the Lagrange multiplier method. Interpolation of the displacement field or the stress field is not used in the present method. Explicit expressions for the control equations are provided, and a procedure is developed for implementing the present method. A number of highly geometrically nonlinear problems are solved using the present formulation, and the results are compared with corresponding analytical solutions. Fig. 1 Base forces 2 Basic formulations onsider a three-dimensional domain of solid medium, where P and Q are the position vectors of a material point before and after deformation, respectively, and x i (i = 1, 2, 3) denotes the Lagrangian coordinate system. Two sets of local triads can be defined as: P i = P x i, Q i = Q x i. (1) In order to describe the stress state near point Q, a parallel hexahedron with edges dx 1 Q 1, dx 2 Q 2, dx 3 Q 3 is shown in Fig. 1. The three forces acting on the front surfaces of the hexahedron are denoted by dt i. Define T i 1 = dx i+1 dx i+2 dt i, dx i 0, (2) where 4 = 1, 5 = 2 for indexes. Quantities T i (i = 1, 2, 3) are called the base forces at point Q in the coordinate system x i. The quantities T i are independent of the values dx i but depend on the coordinates, and T i can also be understood as the stress fluxes. In order to further explain the meaning of T i,letσ i denote the stress vectors on ith coordinate plane. According to Eq. (2), T i = A i σ i, (3) where A i = Qi+1 Q i 1 (4) and the quantity A i is called base area. In Eq. (3), summation should not be made because the index i is located on the same height. According to the definitions of various stress tensors, the relations between the base forces and various stress tensors

Base force element method of complementary energy principle for large rotation problems 509 can be given. The auchy stress is σ = 1 T i Q V i, (5) Q where is the dyadic symbol, and the summation rule is implied. The first Piola Kirchhoff stress is τ = 1 T i P i, (6) V P where V P is the original base volume V P = (P 1, P 2, P 3 ). (7) The second Piola Kirchhoff stress is = 1 T i P V P i, (8) P in which T i P = F 1 T i, (9) where F is the triads transfer tensor and is called the deformation gradient tensor. It has an inverse F 1 = P i Q i, (10) where Q i is the conjugate vector of Q i. The complementary energy per unit mass can be written as: W = 1 T i u i W, (11) ρ 0 V P in which W is the strain energy per unit mass. The traditional elastic laws give the relationship between the stress tensor and strain tensor. Now, the new form of elastic law that directly links the base forces T i with displacement gradient u i is as follows: T i W W = ρv Q = ρ 0 V P. (12) u i u i Equation (12) directly expresses T i in terms of strain energy, and thus, u i is just the conjugate variable of T i. It can be seen that the mechanics problem can be completely established by means of T i and u i. 3 Formulation of the geometrically nonlinear BFEM 3.1 Expressions of the complementary energy The deformation gradient tensor F is: F = Q i P i. (13) Further, F can be expressed by polar decomposition F = F d F r, (14) where F d = Q i M i, F r = M i P i, (15) and M i is a medium triad so that F r represents the rotation while F d represents the pure deformation. Further, M i = P i + u ri, Q i = M i + u di, (16) where M i M j = P i P j, (17) then u i = u ri + u di, (18) where u ri and u di are the rotation part and the deformation part of the displacement gradient u i, respectively. According to Eq. (18), take the complementary energy per unit mass as: W = W d + W r, (19) in which W d is called the deformation part of the complementary energy and W r is called the rotation part of the complementary energy. According to Eq. (11), W d = 1 T i u di W, W r = 1 T i u ri, (20) ρ 0 V P ρ 0 V P note that potential energy W is a function of u di only, so the W d represents the frame-indifferent part of W while the W r related with rotation is not frame-indifferent. For isotropic and linear materials, there is a simpler form for W d W d = 1 (1 + ν)j 2T ν J1T 2, (21) 2ρ 0 E in which E is Young s modulus, ν is Poisson s ratio, J 1T and J 2T are the invariants of T i,so J 1T = 1 V P T i M i, J 2T = 1 V 2 P where ( T i T j) m ij, (22) m ij = M i M j. (23) Taking the first of Eq. (22), then J 2T = 1 VP 2 (T i T j )p ij, p ij = P i P j. (24) Now, consider an arbitrary polyhedral element of the BFEM as shown in Fig. 2. Letα,β,γ,... denote its faces, α g,β g,γ g,...the geometric centers of each of the faces and T α, T β, T γ,...the force vectors acting on each of the faces.

510 Y. Peng, Y. Liu and using Gauss theorem, Eq. (32) becomes: W e r = S 0 T u r ds 0, (33) Fig. 2 A polyhedral element of the BFEM Further, the complementary energy of the base forces element can be decomposed into a deformation part and a rotation part, that is W e = W e d + W e r. (25) After substitution of Eq. (22) into Eq. (21), the deformation part of the complementary energy of an base forces element is as follows: W e d = 1 + ν 2EV 0 (T α T β )p αβ ν 1 + ν (T α M α ) 2, (26) where V 0 is the original volume of the element, M α is the medium position vector of point α g, p αβ is the dot product of position vectors P α and P β of points α g and β g. Further, p αβ = P α P β, (27) M α = P α + u rα, (28) and M α can be given as: M α = M P α, (29) where M is the coordinate rotation tensor. For the perpendicular coordinate system, M can be written as: M = cos θ(e 1 e 1 + e 2 e 2 ) + sin θ(e 1 e 2 e 2 e 1 ) + e 3 e 3, (30) in which θ is the rotation angle of the base force element. The rotation part of the complementary energy of an base forces element can be written as: W e r = V 0 ρ 0 W r dv 0. (31) Substituting the second of Eq. (20) into Eq. (31) yields Wr e = 1 T i u ri dv 0, (32) V P V 0 in which S 0 is the boundary of an arbitrary polyhedral element, T is the traction on S 0 per unit undeformed area, and u r is the rotation part of the displacement on S 0. When the arbitrary polyhedral element is small enough, it is assumed that the stress is uniformly distributed on each face. Then, Eq. (33) can be reduced to: W e r = T α u rα, (34) where the summation rule is implied, and u rα can be written as: u rα = M α P α. (35) Substituting Eq. (29) into Eq. (35) yields u rα = (M U )P α, (36) in which U is the unit tensor. Substituting Eq. (36) into Eq. (34), we get the rotation part of the complementary energy W e r = T α (M U )P α. (37) 3.2 Governing equations The complementary energy of the base forces element D is defined as: e (T,θ)= ρ 0 W dv 0 ū T 0 ds 0, (38) D S u in which W is the complementary energy per unit mass, dv 0 and ds 0 are the volume and boundary areas of element D, ū is the specified displacement on the boundary, S u, T 0 is the traction on S u per unit undeformed area. Here, T 0 is not an independent variable and should be determined by the equilibrium condition of the boundary. When the element D is small enough, it is assumed that T 0 is uniformly distributed on each faces of an element. Then, Eq. (38) is reduced to: e (T,θ)= W e ū α T α 0, (39) where ū α and T α 0 are the specified displacement and force vectors acting on the center on the boundary face α of element D, respectively. The summation rule is implied in Eq. (39). In order to simplify the expression, let T α denote T α 0, and then according to Eq. (25), e (T,θ)= W e d + W e r ū α T α. (40)

Base force element method of complementary energy principle for large rotation problems 511 Substituting Eqs. (26) and (37) into Eq. (40) yields e (T,θ) = 1 + ν (T α T β )p αβ ν 2EV 0 1 + ν (T α M α ) 2 + T α (M U)P α ū α T α. (41) When the specified displacement ū α = 0 on the boundary S u,eq.(41) becomes: e (T,θ) = 1 + ν (T α T β )p αβ ν 2EV 0 1+ν (T α M α ) 2 + T α (M U)P α. (42) In the above derivation, the following equilibrium conditions were used: T α = 0, T α Q α = 0, (43) α where Q α is the current position vector of point α g. It should be noted that the second condition of Eq. (43) did not give any restriction on T α, because Q α is not given. The total complementary energy of the elastic system can be written as: ne = ne ( W e ū α T α). (44) According to the complementary energy principle 24, ne takes stationary value under equilibrium conditions (43) and stress boundary conditions for the real displacement and stress state. However, the equilibrium conditions can be released by the Lagrange multiplier method. For the stress boundary condition and the corresponding condition of the face forces between elements, computational techniques can be applied to achieve a rotation. Using the Lagrange multiplier method, a new function for an element can be introduced as follows: ( ) e (T,θ,λ) = e (T,θ)+ λ T α, (45) α in which arbitrary vector λ = λ 1 e 1 + λ 2 e 2 + λ 3 e 3 is the Lagrange multiplier. For two dimensions, there are λ = λ 1 e 1 + λ 2 e 2. For the elastic system: ne = ne e (T,θ,λ), (46) and by means of the complementary energy principle δ ne = ne δ e (T,θ,λ) = 0. (47) Further, Eq. (47) can be expressed as: ne (T,θ,λ) T ne ne = 0, (T,θ,λ) = 0, (48) θ (T,θ,λ) = 0. λ Equations (48) are the compatibility equations and displacement boundary conditions for the elastic system. These are the governing equations of the geometrically nonlinear BFEM. Detailed explicit expressions of the nonlinear governing Eqs. (48) will now be developed. From the first of Eq. (48) and by referring to Eqs. (45), (40), (37) and (26), it can be determined that W e d T α = 1 + ν EV 0 p αβ U where M β is given as: ν 1 + ν M α M β T β, (49) M β = MP β, (50) Wr e T α = (M U)P α, (51) ( λ α T α) T α = λ. (52) Then, according to the second of Eqs. (48) and by referring to (45), (40), (37) and (26), it can be derived that: Wd e = ν ( (T α M α ) T β M β θ EV 0 θ ), (53) in which M β θ = M θ P β, (54) M θ = cos θ(e 1 e 2 e 2 e 1 ) sin θ(e 1 e 1 + e 2 e 2 ), (55) Wr e = T α M θ θ P α, (56) ( λ α T α) = 0. (57) θ Further, according to the third of Eqs. (48), and by referring to (45), (40), (37) and (26), it can be determined that: W e d λ W e r = 0, (58) = 0, (59) λ ( λ α T α) = T α. (60) λ α

512 Y. Peng, Y. Liu The procedure for implementing the present method can be developed from the above equations, and the force vectors T acting on each of the faces of polyhedral elements can be obtained by means of nonlinear numerical calculation. 3.3 Displacements of nodes An explicit expression for displacements can be developed based on the governing equation of element. For an arbitrary polyhedral element, the governing equation can be written as: δ α = e (T,θ,λ) T α. (61) Substituting Eq. (45) into Eq. (61), we obtain: δ α = e (T,θ) T α + ( λ α T α) T α. (62) Now consider Eq. (42), the explicit expression for displacements of nodes can be written as: p αβ U δ α = 1 + ν EV 0 ν 1 + ν M α M β T β + (M U )P α + λ. (63) 4 Numerical examples The present formulation of geometrically nonlinear BFEM was used to solve problems with two-dimensional large displacement and large rotation. Gauss Newton iterative solution procedure was employed, and four examples were given to demonstrate the characteristics and performance of the present model. 4.1 onstant stress patch test The present BFEM was applied to a constant stress patch test. A plane stress panel with size 10 10 and unit thickness is shown in Fig. 3. The panel is modelled by four irregular 4-mid-node plane elements, for which a pure stretch patch test was considered. Fig. 4 Analysis of small deflection of a cantilever beam The numerical result was σ x = 10.0000, which showed accordingly that the present BFEM was capable of reproducing the constant stress states. 4.2 Analysis of small deflection of a cantilever beam The formulation was applied to a case of small rotations and displacements. In this case, it should recover the full linear response and demonstrate the effects of the aspect ratio of an element on its performance in the geometric nonlinear range. The structure analyzed is a two-dimensional cantilever beam loaded with a moment M at its free end, as shown in Fig. 4. The beam has length L = 10 and cross-sectional dimensions b = 1 and h = 2. The material has elastic modulus E = 1.5 10 8 and a zero Poisson ratio. The full load M = 1 is applied in one increment step. The analytical solutions for the horizontal stress component at point A of the beam and for the deflection at the free end B of the beam are 1.125 and 0.5 10 6, respectively. In order to demonstrate the effects of the aspect ratio of an element and the effects of distorted element meshes in the geometric nonlinear range, the structure was discretized by using 4-mid-node plane elements and changing their element geometry, as shown in Fig. 5. The computational results of the vertical deflection u By at point B of the beam and the horizontal stress component σ Ax at point A of the beam with change in the aspect ratio of theelementsareshownintable1. The results of the vertical deflection u By at point B of the beam and the horizontal stress component σ Ax at point A of the beam with change in the distorted parameter of the elements are shown in Table 2. The numerical results were compared with the results obtained by using the Q4 model (with 3 3 Gauss quadrature) of the displacement-based FEM, and the comparison reveals good performance of the proposed method. 4.3 Analysis of large rotation of a straight beam Fig. 3 Pure stretch patch test The performance of the present formulation for problems with large displacements and rotations was further studied by analyzing a straight cantilever beam loaded with a moment M at its free end, as shown in Fig. 6. The beam has length L = 12 m, cross-sectional dimensions b = 1 m and h = 1 m, and elastic modulus E = 1, 800 N/m 2,

Base force element method of complementary energy principle for large rotation problems 513 Fig. 5 Meshes of FEM. a Aspect ratio of 2, b aspect ratio of 4, c aspect ratio of 10, d distorted parameter d = 0, e distorted parameter d = 2, f distorted parameter d = 4 Table 1 Effect of element aspect ratios on precision Element aspect ratio 2 4 10 BFEM σ Ax 1.0000 1.0000 1.0000 u By 1.0000 1.0000 1.0000 Q4 model σ Ax 0.8889 0.6732 0.2500 u By 0.9038 0.7094 0.2420 Table 3 Load-deflection of the straight beam k u/l v/l BFEM Analytical BFEM Analytical 0.2 0.0164 0.0164 0.1558 0.1558 0.4 0.0645 0.0645 0.3040 0.3040 0.6 0.1416 0.1416 0.4375 0.4374 0.8 0.2432 0.2432 0.5501 0.5500 1.0 0.3635 0.3634 0.6367 0.6366 Table 2 Effect of mesh sensitivity on result Mesh distorted parameter d 0.0 2.0 4.0 BFEM σ Ax 1.0000 1.2084 1.1068 u By 1.0000 0.9542 0.6908 Q4 model σ Ax 0.2500 0.1727 0.2116 u By 0.2420 0.0979 0.0504 Fig. 8 Shape of the straight beam after deformation Fig. 6 Analysis of large rotation of a straight beam Fig. 7 Meshes of a cantilever beam with the 4-mid-node elements while the applied total load level is M = 0.5π EI/L = 19.6350 N m. The calculated specimen was divided into 4-mid-node plane elements as shown in Fig. 7. In the calculation, the full load is applied in one increment step. The displacement ratios u/l and v/l at the tip of the beam for different value of load parameter k = 2ML/(π EI) are given in Table 3. The results obtained by the present model were compared with those provided by the analytical solution 25, and the deformed shapes of the straight beam are shown in Fig. 8. The figure shows that the free end of the straight cantilever beam will rotate 90 under the total load k = 1.0. The results in Table 3 show that the present results are in good agreement with those of the analytical solution, and thus the present method has demonstrated good performance for the cases of large displacement and large rotation. 4.4 Analysis of large rotation of a curved beam This analysis investigated the performance of the present formulation for a curved beam with large rotation. The curved

514 Y. Peng, Y. Liu 5 onclusions Fig. 9 Analysis of large rotation of a curved beam Table 4 Load displacements of the curved beam k 0.4 0.8 1.2 1.6 u/(π R) 0.0002 0.6379 1.2741 1.2732 v/(π R) 0.6366 0.9997 0.6348 0.0001 Based on the concept of the base forces, a new finite element method based on the complementary energy principle is proposed for problems with large displacements and large rotations. The complementary energy of an element under large deformation can be expressed explicitly by taking the base forces as state variables and can be separated into a deformation part and a rotation part. Explicit Eqs. (26) and (37) of the complementary energy of a polyhedral element are given for the BFEM. The nonlinear governing equations are obtained by using the complementary energy principle for large elastic deformation and incorporating the Lagrange multiplier method, in which all formulas for the BFEM are explicit expressions on the integral, e.g. explicit Eq. (63) is given for displacements of nodes of the BFEM. Typical geometrically nonlinear examples show that the explicit formulation of the base forces element method presented in this paper are valid and exhibit excellent performance in the range of high geometric nonlinearity. All the results were obtained by using a unique loading step, and the findings mentioned above are important for all nonlinear geometric problems. References Fig. 10 Shape of the curved beam after deformation beam in the shape of a half-circle with radius R was fixed at one end, and a tip moment M was applied at the other end against its curvature, as shown in Fig. 9. The bend is an arc of a circle of radius R = 100 m, and the beam cross-section is a square with sides of unit length. The material has elastic modulus E = 2, 000 N/m 2 and zero Poisson ratio. The curved beam is modeled using 40 5 4-mid-node elements, and the total loading is set to be M = 2EI/R = 3.3333 Nm. In the calculation, the full load was applied in one increment step. The different values of load parameters k = 4MR/(5EI), displacement ratios u/(π R) and v/(πr) at the free end of the curved beam are listed in Table 4, and the deformed shapes of the curved beam are shown in Fig. 10. It can be seen from Fig. 10 that the half-circular beam will bend into a straight line under the load level of k = 0.8, and it will bend into a half circle in the opposite direction under the load level of k = 1.6. This example demonstrates that the proposed formulations have promising potential of applications to geometric nonlinear problems. 1. Zienkiewicz, O..: The Finite Element Method. McGraw-Hill, New York (1977) 2. Bathe, K.J.: Finite Element Procedures. Prentice-Hall, New Jersey (1996) 3. Pian, T.H.H.: Derivation of element stiffness matrices by assumed stress distributions. AIAA J. 2, 1333 1336 (1964) 4. Pian, T.H.H., Tong, P.: Basis of finite element methods for solid continua. Int. J. Numer. Methods Eng. 1, 3 28 (1969) 5. Pian, T.H.H.: A historical note about hybrid elements. Int. J. Numer. Methods Eng. 12, 891 892 (1978) 6. Fraeijs de Veubeke, B.: Displacement and equilibrium models in the finite element method. In: Zienkiewicz, O.., Holister, G.S. (eds.) Stress Analysis. Wiley, New York (1965) 7. Taylor, R.L., Zienkiewicz, O..: omplementary energy with penalty functions in finite element analysis. In: Glowinski, R. (ed.) Energy Methods in Finite Element Analysis. Wiley, New York (1979) 8. Gallagher, R.H., Heinrich, J.., Sarigul, N.: omplementary energy revisited. In: Atluri, S.N., Gallagher, R.H., Zienkiewicz, O.. (eds.) Hybrid and Mixed Finite Element Methods. Wiley, hichester (1983) 9. Sarigul, N., Gallagher, R.H.: Assumed stress function finite element method: two-dimensional elasticity. Int. J. Numer. Methods Eng. 28, 1577 1598 (1989) 10. Vallabhan,.V.G., Azene, M.: A finite element model for plane elasticity problems using the complementary energy theorem. Int. J. Numer. Methods Eng. 18, 291 309 (1982) 11. Rybicki, E.F., Schmit, L.A.: An incremental complementary energy method of nonlinear stress analysis. AIAA J. 8, 1805 1812 (1970)

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