Determination of Young's Modulus by Using. Initial Data for Different Boundary Conditions

Transcription

1 Applied Mathematical Sciences, Vol. 11, 017, no. 19, HIKARI Ltd, Determination of Young's Modulus by Using Initial Data for Different Boundary Conditions Feda Ilhan Department of Mathematics Abant Izzet Baysal University, Bolu, Turkey Zahir Muradoglu Department of Mathematics Kocaeli University, Kocaeli, Turkey Copyright 017 Feda Ilhan and Zahir Muradoglu. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract It is well known that modulus of elasticity, i.e. Young's modulus measures stiffness of an elastic material. This paper proposes a rapid inverse analysis approach based on a measured deflection at some interior point on the middle surface of an elastic plate. An algorithm is developed to identify an unknown modulus of elasticity for a given maximal deflection and the corresponding force. Computational results with noise free and noisy data show effectiveness of the proposed approach. Keywords: Elasticity modulus, elastoplastic plate, bending problem, nonlinear biharmonic equation, finite difference method. 1 Introduction Most mathematical problems in science, technology and medicine are inverse problems [1]. Studying such problems is the only complete way of analyzing experimental results. Often these problems concern the determination of properties of some inaccessible regions from observation on the boundary of that region, as in geophysics and medicine. Further, the automatization of physical processes leads necessarily to inverse problems, which absolutely must be solved. Inverse problems have a practical importance among the pressing problems of

2 914 Feda Ilhan and Zahir Muradoglu current mathematical research. The determination of unknown coefficients in nonlinear partial differential equations of different types from additional conditions that is boundary or surface measured data are well known in the literature as inverse coefficient problems ([-6] and references therein). Hasanov and Liu studied an inverse problem associated with a nonlinear parabolic initial value problem [7] with an unknown leading coefficient k: = k( u )of the equation u t (k( u ) u) + q(x, t)u = f(x, t). They analyzed the class of admisssible coefficients and proved the coefficient stability and then obtained a result on the existence of a quasi-solution of the inverse coefficient problem. Hasanov and Mamedov studied an inverse problem related to nonlinear biharmonic equation by using finite element method in [8]. After determining the elastoplastic properties of a plate from an experimentally given relation between middle surface deflections and loads at some points on the plate, they presented the main existence theorem for the inverse problem by reformulating the inverse problem as a minimization problem for a certain functional. Numerical study of the nonlinear biharmonic equation for elastoplastic bending plate based on finite difference method has been presented in [9] and Hasanov constructed a general variational approach for solving the nonlinear problem for elastoplastic bending plate via the monotone potential operator theory [10]. The present study deals with an inversion algorithm for determination of elastic modulus of a bending plate from limited number of loading experiments. This problem is associated with the following nonlinear boundary value problem: Aw x [g(ξ (w)) ( w 1 x + 1 w 1 x )] + w [g(ξ (w)) ] x 1 x x 1 x + x [g(ξ (w)) ( w x + 1 w x )] = F(x 1, x ), x = (x 1, x ) Ω R (1.1) 1 with clamped boundary condition w(x) = w (x) = 0 (1.) n or simply supported boundary condition w(x) = w n = 0 (1.3)

3 Determination of Young's modulus by using initial data 915 The domain Ω = {(x 1, x ) R : 0 x α l α, α = 1,} is the region where the plate is placed in and it is known from the deformation theory of plasticity [1,11] that w = w(x) is the deflection of a point x Ω on the middle surface of the plate. The function on right hand side is F(x) = 3q(x)/ h 3 and q = q(x) is the intensity of the loads applied on the plate. In the first part of the study an algorithm is constructed for determination of the unknown Young modulus (elastic modulus) E from deflections measured at some points on the middle surface of the plate. This inverse problem was first formulated in [9]. An analysis of the corresponding nonlinear direct problem, based on J -deformation theory of plasticity, was given in [11]. Numerical study of the nonlinear biharmonic equation for elastoplastic bending plate based on finite difference method has been presented in [10]. Within the range of the considered physical model, the function (ξ (w)), where ξ (w) = w x1 x 1 + w x x + w x1 x + w x1 x 1 w x x is the effective value of plate curvature, defined to be the plasticity function, determines elastoplastic properties of a homogeneous isotropic plate. For many engineering materials the plasticity function g = g(ξ ) is assumed to be in the following form [9]. G, 0 ξ ξ 0 g(ξ ) = { G ( ξ 0 κ ), ξ ξ 0 < ξ ξ M, κ (0,1) (1.4) Here, the value ξ 0 is assumed to be the elasticity limit of a material. When ξ ξ 0, the plate is said to be undergoing plastic deformation. In pure elastic case, (ξ ) = G, where G = E (1+ ν) is the modulus of rigidity. According to the theory of plasticity [1], G depends on E and the Poisson constant ν > 0. For pure elastic deformations i.e. (ξ < ξ 0 ) the function g(ξ (w)) is a linear function and it is sufficient to find E due to ν > 0 is assumed to be known. In case of plastic deformations the plasticity function g = g(ξ ) must be obtained. The aim of the study is to find the elasticity modulus E > 0 for a given deflection which occured by the effect of a certain load. The iteration scheme for the nonlinear bending problem To apply any numerical method for solution of the direct problem, as well as to study the inverse problem, one needs to perform a linearization of the nonlinear problem (1.1)-(1.). The iteration scheme given in [1] permits to solve the non-

4 916 Feda Ilhan and Zahir Muradoglu linear problem (1.1)-(1.) via a sequence of linearized problems. By using this iteration linearized bending equation can be presented as below: x [g(n 1) (ξ ) ( w (n) x 1 w (n) x )] + x 1 x [g (n 1) (ξ ) w (n) x 1 x ] + x [g(n 1) (ξ ) ( w (n) + 1 w (n) x x )] = F(x 1, x ), x = (x 1, x ) Ω (.1) 1 By using a modification of Samarskii-Andreev finite difference scheme [15] we obtain the most appropriate approximation of the nonlinear bending equation (1.1): (g (n 1) h (ξ (n) h )(y x1 x y x (n) x )) (n) x + (g (n 1) h (ξ h )( y x x y x1 (n) x 1 )) x {(g (n 1) h (ξ (n) h ) y x1 )x1 + (g (n 1) h (ξ h ) y x1 x x (n) x )x1 x } {(g (n 1) h (ξ (n) h ) y x1 x )x1 + (g (n 1) h (ξ h ) y x1 x x (n) )x1 x } (.) Here (x ij ) = 3q(x ij )/h 3, g h (n 1) (x) = g(ξ h (y (n 1) (x))), and ξ h (y) = y x1 x 1 + y x x + 0.5(y x1 x + y x1 x ) + y x1 x 1 y x x (.3) is the finite difference approximation of the effective value of the plate curvature ξ(w). Adding to (.) the discrete analogue of the corresponding boundary condition, we obtain the finite difference approximation of the linearized problem (.1). Subsequently, this problem will be referred to as the discrete problem. To estimate the order of approximation for this discrete problem, let us denote the solution of the nonlinear problem (1.1)-(1.) by w(x ij ), the solution of the linearized problem (.1)-(1.) by w (n) (x ij ) and the solution of the discrete problem by y ij (n). To find the order of approximation of the finite difference scheme (.), for a given ε w > 0 and for each n = 1,, 3, we repeat the iteration until the condition max w (k) (x ij ) w (k 1) (x ij ) ε w is satisfied. The proposed inversion method is based on a finite number of output measured

5 Determination of Young's modulus by using initial data 917 data because in engineering practice only a limited number of discrete values of deflections can be given. Therefore all these data can only be given as finite dimensional vectors during a quasistatic process of loading, given by the increasing values of the deflection parameters. The aim is to find the elasticity modulus of the plate by using the value (F, w k ) which is obtained by experiment. Here w k are the maximal deflections corresponding to the loads F. It is known from the theory that, for two different materials under the influence of same force, the smaller elasticity modulus implies the larger deflection i.e. E 1 > E implies w 1 < w. 3 Inversion algorithm for finding elasticity modulus E The maximal deflection w 0 which is generated by a load of known intensity F is the solution of the direct problem and it is used as initial data for the algorithm that is constructed to find elasticity modulus E. Assume a priori E = E 1 is given. By the effect of E 1 and F the maximal deflection occured on the surface of the plate is represented by w h (E 1 ; F). Then the solution of the direct problem w 0 and w h (E 1 ; F) are compared. (i1) a) If w h (E 1 ; F) > w 0 then E = E 1 + δe 1. (Here δ > 0 is the increment of E.) b) If w h (E 1 ; F) < w 0 then by taking E = E 1 δe 1 the direct problem can be solved and w h (E ;F) and w 0 are compared. (i) The processes in the first step are continued until w 0 remains in between the maximal deflections w h (E k, F) and w h (E k+1, F) corresponding to consecutive E k and E k+1,respectively. Finally w h (E k, F) < w 0 < w h (E k+1, F) (or w h (E k, F) > w 0 > w h (E k+1, F)) will be satisfied. (i3) After obtaining suitable E k and E k+1, by using bisection method E = (E k + E k+1 )/ is found. By using E the direct problem is solved and the maximal deflection w h ( E, F) is found. The condition max w h ( E, F) w 0 ε E is checked whether it is satisfied or not. If this condition is satisfied then E = E is taken. Here ε E > 0 represents the required accuracy value. (i4) If the condition max w h ( E, F) w 0 ε E is not satisfied then while w h ( E, F) > w 0, E k = E k, E k+1 = E ; otherwise when w h ( E, F) < w 0, E k = E, E k+1 = E k+1 and (i3) and (i4) steps are repeated until the condition

6 918 Feda Ilhan and Zahir Muradoglu max w h ( E, F) w 0 ε E is satisfied. By this algorithm, under the effect of two forces of different intensities (F = 00 [kn] and F = 50 [kn]) the values for elastic modulus of elastoplastic bending plates satisfying simply supported boundary condition are given in Table 3.1 and Table 3.. Keeping in mind the fact that in practice, the measured deflection value w can be given only by a measuring error, in our computational experiment the input data w is taken as a noise free data firstly and then it is treated in the form of w α = w + αw where α R is a noise parameter. In this study different noise levels α = ±0.03, α = ±0.10 are used. For all cases the relative error is defined as E h = (E E h). The problem is solved for different E,h ε E > 0 values and relative error for Ε is investigated. The Poisson constant is taken as ν = 0.5 for all computations. Table 3.1. Finding E by using deflection of the bending plate satisfying simply supported boundary condition (F=00 [kn]). F = 00 [kn] w = 3,179 [cm] E 0 Noise ε E = 0,1 δe h ε E = 0,01 δe h ε E = 0,001 δe h % , , , % 100 0, , , % , , ,111 3% , , ,091 10% , , ,0908 0% , , , % , , , % 850 0, , ,111 3% , , ,091 10% , , ,0908

7 Determination of Young's modulus by using initial data 919 Table 3.. Finding E by using deflection of the bending plate satisfying simply supported boundary condition (F=50 [kn]). F = 50[kN] w = 3,9661 [cm] E 0 Noise ε E = 0,1 δe h ε E = 0,01 δe h ε E = 0,001 δe h % , , , % 100 0, , , % , , ,111 3% , , ,091 10% , , ,0864 0% , , , % , , , % 850 0, , ,111 3% , , ,091 10% , ,0864 When we analyze Table 3.1, the maximal deflection of an elastoplastic plate satisfying simply supported boundary condition is under the effect of a force of 00 [kn] is w = [cm]. By using this noise free data, Ε is obtained 1100[kNcm ], 0975[kNcm ] and 1006[kNcm ] for the accuracies of ε E = 0.1, ε E = 0.01, ε E = respectively. These Ε values are rather close to the exact value of the Young modulus Ε = 1000[kNcm ] and while for the case ε E = 0.1 the relative error δe h = , for ε E = 0.01 and ε E = the relative errors are δe h = and δe h = respectively. That is, decreasing ε E values lead to smaller relative errors. Under the effect of the same force, when the noise parameter is α = 0.03, Ε = 1663 [kncm ] with accuracy ε E = 0.01 ; when α = 0.10, Ε = 3350[kNcm ] is obtained. It means that greater noise parameter causes more error in Young modulus. Also, it is observed from the Table 3.1 and Table 3. that; when the initial data E 0 is changed, the same results are obtained. By using these data, it is possible to say that the algorithm is useful to determine Ε for expected accuracy whatever the initial data E 0 and the force is. In addition, under the same conditions i.e. when the applied load and the maximal deflection is the same, if the boundary conditions are changed the same results are obtained.

8 90 Feda Ilhan and Zahir Muradoglu 4 Optimal Control Problem In the second part of the study, optimal control problem is studied that means by considering the maximal deflections of the plates which are made of soft (Ε = [kncm ]) and rigid materials ( Ε = 1000 [kncm ] ) the intensity of the load that may affect the plate is found. When doing that work, an algorithm similar to the algorithm that constructed for finding the E is used. Assume a plate made of rigid material (Ε = 1000 [kncm ]) and satisfying simply supported boundary condition is taken. The deflection on the surface of the bending plate that is occured by a certain load is the solution of the direct problem. The aim is to find the intensity of the applied load by using known deflection. For finding the solution of the inverse problem the deflection w is used as input data. First we take noise free w and then by taking different noise levels R, w α = w + αw is taken. Here α = ±0.03, α = ±0.05, α = ±0.10 noise levels are used. For all these cases the relative error is defined as δf h = (F F h). The,h parameter ε F > 0 is defined to determine the accuracy of obtained F. The obtained results are given in Table 4.1. F Table 4.1. Finding F by using deflection of the bending plate made of hard material satisfying simply supported boundary condition E = 1000[kN cm ], w = 3,179 [cm] (F = 00[kN]) F 0 Noise ε F = 0,1 δf h ε F = 0,01 δf h ε F = 0,001 δf h 170 [kn] 300 [kn] 0% 03,75 0, ,46 0,003 00,05, % 19,50 0, ,37 0, ,0 0,099-5% 19,50 0, ,6 0, ,03 0, % 177,50 0, ,37 0, ,96 0,100 3% 07,50 0, ,6 0,081 05,97 0,099 5% 07,50 0, ,37 0, ,96 0, %,50 0,115 0,6 0,1031 0,03 0,100 0% 0,50 0,015 00,6 0, ,03 1, % 191,5 0, ,06 0, ,06 0,097-5% 187,50 0, ,37 0, ,96 0,050-10% 187,50 0, ,50 0, ,50 0,065 3% 0,50 0,015 06,5 0, ,01 0,0301 5% 17,50 0, ,50 0, ,50 0, % 17,50 0, ,37 0, ,96 0,0998

9 Determination of Young's modulus by using initial data 91 When we examine the output data in Table 4.1, under the affect of a force of 00 [kn] the plate which satisfies simply supported boundary condition has a maximal deflection of w = [cm]. When this maximal deflection is used as noise free data, F is obtained [kn], [kn] and [kn] by the accuracy ε F = 0.1, ε F = 0.01, ε F = 0.001, respectively. These values are very close to the exact value F = 00 and the relative error is δf h = , δf h = and δf h = for the cases ε F = 0.1, ε F = 0.01 and ε F = respectively. That is, the smaller ε implies the smaller relative error. By using the deflection value w,obtained under the same intensity of load, with the accuracy of ε F = 0.01 ; F is obtained as 05.6[kN], 09.37[kN], 0.6[kN] for the noise parameters α = 0.03, α = 0.05, α = 0.10 respectively. As the noise level increases the relative error of F increases. When the initial data F 0 is changed, nearly the same results are obtained but sometimes there are small differences in relative errors. When the same calculations are made for a plate which is satisfying the same boundary condition but made of soft material, in the sense of noise parameters and calculation accuracy of F, similar results are obtained. By using these observations it can be said that the given algorithm is quite useful for obtaining F with a high accuracy. Besides, under the same conditions i.e. when the Young modulus and the maximal deflection are the same, when the boundary condition is changed in to clamped boundary condition, similar results are obtained. 5 Conclusion Depending on the different forces and different deflections elastic modulus of the material which composes the elastoplastic plate is obtained. When possible bending limit is known the maximum load to be loaded is determined. The results obtained in this study can be applied to various problems in mathematics, physics and numerical mechanics. The generated algorithm provides an important contribution to finding a numerical solution of the inverse coefficient problems. The presented numerical examples show that the algorithms allow us to determine the plasticity function with high accuracy, even for acceptable noise levels. Acknowledgements. We are thankful to Prof. Dr. Alemdar Hasanov for the formulation of the problem and his valuable guidance.

10 9 Feda Ilhan and Zahir Muradoglu References [1] G. Anger, Inverse Problems in Differential Equations, Plenum Press, New York and London, [] A. Hasanov, A. Erdem, Determination of unknown coefficient in a non-linear elliptic problem related to the elastoplastic torsion of a bar, IMA Journal of Applied Mathematics, 73 (007), no. 4, [3] L. Yang, J. Yu, Z. Deng, An inverse problem of identifying the coefficient of parabolic equation, Applied Mathematical Modelling, 3 (008), [4] Q. Chen, J. Liu, Solving an inverse parabolic problem by optimization from final measurement data, Journal of Computational and Applied Mathematics, 193 (006), [5] A. Hasanov, Identification of an unknown source term in a vibrating cantilevered beam from final overdetermination, Inverse Problems, 5 (009), [6] S. Tatar, Z. Muradoğlu, Numerical solution of the nonlinear evolutional inverse problem related to elastoplastic torsional problem, Applicable Analysis, 93 (014), [7] A. Hasanov, Z. Liu, An inverse coefficient problem for a nonlinear parabolic variational inequality, Applied Mathematics Letters, 1 (008), [8] A. Hasanov, A. Mamedov, An Inverse problem related to the determination of elastoplastic properties of a plate, Inverse Problems, 10 (1994), [9] F. Ilhan, Z. Muradoglu, Numerical solution of non-linear biharmonic equation for elasto-plastic bending plate, Applied Mathematical Sciences, 9 (015), no. 146, [10] A. Hasanov, Variational approach to non-linear boundary value problems for

11 Determination of Young's modulus by using initial data 93 elasto-plastic incompressible bending plate, Int. J. of Non-Linear Mech., 4 (007), [11] L. M. Kachanov, Fundamentals of the Theory of Plasticity, North-Holland Publishing Company, Amsterdam, London, [1] A. A. Samarskii, V. B. Andreev, Difference Methods for Elliptic Problems, Nauka, Moscow, (in Russian) Received: March 17, 017; Published: March 9, 017