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Determination of complex moduli of isotropic viscoelastic materials
Polymer Testing 18 (1999) 267–279
Data Interpretation
Determination of complex moduli of isotropic viscoelastic materials Alexander Muravyov1 Department of Mechanical Engineering, University of British Columbia, Vancouver, B.C., Canada V6T 1W5 Received 10 February 1998; received in revised form 22 April 1998
Abstract A procedure for determination of complex moduli of viscoelastic (polymeric) materials is described. The complex moduli of specimen’s isotropic material are calculated from certain experimentally obtained data and with application of a 3-D finite element model of the experimental specimen. A system of two nonlinear complex equations with respect to two complex Lame’s moduli is formulated, and then solved by a successive approximation method. The convergence of the iterative procedure is observed for all cases. The complex moduli obtained for a series of polymeric materials are presented for 10–250 Hz frequency range. 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction In the present paper isotropic homogeneous viscoelastic materials are considered and it is assumed that materials are in isothermal state. Consider the constitutive stress–strain relation for the particular case when the strain is harmonic with frequency :
⑀(t) ⫽ ⑀0eit i ⫽ √ ⫺ 1 one obtains (a one-dimensional element is taken as an illustration) 1 Present address: Institute for Computer Applications in Science and Engineering, Mail Stop 403, 6 North Dryden St, NASA Langley Research Center, Hampton, VA 23681-2199, USA. Tel.: ⫹ 1-757-864-5267; fax: ⫹ 1-757-8646134; e-mail: [email protected]
0142-9418/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 2 - 9 4 1 8 ( 9 8 ) 0 0 0 2 5 - 7
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A. Muravyov / Polymer Testing 18 (1999) 267–279
(t) ⫽ E*()⑀0eit where the complex quantity E*() ⫽ E1() ⫹ iE2() is called the complex modulus. Several experimental procedures were described in the literature[1,2] with specimens having a shape of long bar. These specimens were subjected either to tension, or torsion vibration tests, with an assumption of uniform tensile, or shear stress state to be imposed. These are the traditional procedures, however the assumption of uniform stress state is not always justified. In this paper the specimen will have a 3-D cubic shape, and no simplifying assumptions about stress state will be done. As it will be seen from the procedure an arbitrary shape of specimen can be used. It is only required to have an adequate finite element model of the specimen of such shape. For a viscoelastic structure, Lame coefficients and G should be replaced by their hereditary analogues (operators).[3–6] In the case of the finite element method this implies the replacement of material constants , and G in the stiffness matrix by their viscoelastic analogues (operators). For a particular case (harmonic deformation) this leads to replacement of , G by corresponding complex moduli * and G* which are functions of frequency of deformation. Complex moduli of a polymeric material are necessary in order, for example, to obtain the steady-state response of a polymeric structure subjected to a periodic loading. Below a procedure will be presented which allows the complex Lame modulus *, and shear modulus G* of a material to be obtained from certain experimental data. Note that having two moduli *, G*, all other characteristic moduli, for example, complex Young’s modulus and complex Poisson’s ratio can be calculated. 2. A procedure for determination of complex moduli Consider a harmonic base excitation of the system shown in Fig. 1(a). A specimen of the viscoelastic material which is under consideration is assumed to have a 3-D cubic shape.
Fig. 1. Two types of base excitation test; tension type (1) and shear type (2).
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First, assume that the 3-D specimen connecting the mass m and the base is modelled as a massless one-dimensional viscoelastic spring (complex stiffness k*) with length L, and crosssection area A. Considering a single-degree-of-freedom equation with harmonic base excitation x ⫽ x0eit, one obtains: mz¨ ⫹ k*z ⫽ ⫺ mx¨ ⫽ m2x0eit
(1)
where z ⫽ y ⫺ x (see Fig. 1(a)). Assuming z ⫽ z0eit and substituting it in the equation of motion (1) one obtains ⫺ 2mz0 ⫹
A (E () ⫹ iE2())z0 ⫽ m2x0 L 1
where the term eit was eliminated in all places. Thus z0 ⫽
mx02 A ⫺ m2 ⫹ (E1() ⫹ iE2()) L
(2)
A complex transfer function is defined as: T() ⫽
yo xo ⫹ zo ⫽ xo xo
and using (2) one obtains A (E () ⫹ iE2()) L 1 T() ⫽ A ⫺ m2 ⫹ (E1() ⫹ iE2()) L
(3)
Two harmonic base excitation tests can be conducted (Fig. 1(a,b)) and transfer functions T1() ⫽ y01/x01
T2() ⫽ y02/x02
can be registered in these tests. Now instead of a viscoelastic spring model consider a finite element model which will be used to simulate the experimental specimen. Volume (3-D) finite elements will be used to mesh a given 3-D domain (Fig. 2). One side of the specimen is assumed fixed, when at the opposite side (side ⌫) some displacements will be prescribed. Two types of prescribed kinematic loadings (displacements on side ⌫) will be considered according to two experimental tests (Fig. 1(a,b)): (1) ux ⫽ 1, uy ⫽ uz ⫽ 0, and (2) uy ⫽ 1, ux ⫽ uz ⫽ 0.
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Fig. 2.
3-D finite element model of the specimen with two types of loading.
Based on application of 3-D finite elements the stiffness matrix of a viscoelastic homogeneous structure can be presented in the following form: K ⫽ *K1 ⫹ 2G*K2 where matrices K1, K2 do not depend on material constants. Such representation of the matrix K was used in other studies.[7,8] Neglecting by the mass of the specimen in comparison with the attached rigid mass, one can write the following expression for a harmonic loading with frequency , namely, [*()K1 ⫹ 2G*()K2]U0eit ⫽ F0eit
(4)
where the force vector on the right side represents the action from the attached rigid mass on the specimen. Note that the force vector F0 will include only components which correspond to degrees of freedom on the side ⌫, all other components are assumed zero. Also note that in the vector of displacements U0 the components which correspond to degrees of freedom on the side ⌫ are known (they are prescribed according to two types of loading). Assume that the size of matrices K1, and K2 is N, and the number of degrees of freedom pertaining to the side ⌫ is 3n, then the rest N ⫺ 3n (denote them as 3m) degrees of freedom will correspond to internal degrees of freedom. It is assumed here that the finite elements have three linear degrees of freedom per node. Considering the first type loading, the vector of displacements can be subdivided as follows:
Analogously one can represent the force vector
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271
Renumerating the degrees of freedom of the system and eliminating the time-dependent coefficient at both sides, one can represent Eq. (4) in the following form:
冢冤
兩
K 1⌫I
⫺⫺
⫺⫺
K 1⌫⌫
* ⫺ ⫺ K 1I⌫
兩
K
1 II
冥 冤
兩
K 2⌫I
⫺⫺
⫺⫺
K 2⌫⌫
⫹ 2G* ⫺ ⫺ K
2 I⌫
兩
K
2 II
冥冣冤 冥 冤 冥 U0⌫
⫺⫺ U0I
F0⌫
⫽
⫺⫺
(5)
0
where subvector of displacements U0I (size 3m) is unknown, and the subvector of forces F0⌫ (size 3n) is also unknown. Thus one can solve the following system of linear equations
冤
⫺I
兩
⫺⫺
⫺⫺
*K 1⌫I ⫹ 2G*K 2⌫I ⫺⫺⫺⫺⫺⫺⫺
0
兩
*K ⫹ 2G*K 1 II
2 II
冥冤 冥 冤 F0⌫
⫺⫺ U0I
⫽
⫺ *K 1⌫⌫U0⌫ ⫺ 2G*K 2⌫⌫U0⌫ ⫺⫺⫺⫺⫺⫺⫺⫺⫺⫺⫺ ⫺ *K 1I⌫U0⌫ ⫺ 2G*K 2I⌫U0⌫
冥
and subvectors U0I, and F0⌫ can be found. Then one can find the total resulting force which is necessary to apply to create the prescribed vector of displacements on ⌫. Namely, using (5) and collecting x-components, the resulting force for the 1st loading can be calculated as
Having the non-zero components of U0⌫ equal to 1, from (6) one obtains the complex stiffness Sx: Sx ⫽ *(A1 ⫹ A2) ⫹ 2G*(B1 ⫹ B2)
(7)
where A1 ⫽ RK 1⌫⌫U0⌫ A2 ⫽ RK 1⌫IU0I B1 ⫽ RK 2⌫⌫U0⌫ B2 ⫽ RK 2⌫IU0⌫
(8)
and
Introducing additional notations C1 ⫽ A1 ⫹ A2 C2 ⫽ B1 ⫹ B2 the previous expression (7) for the complex stiffness in the first type loading becomes: Sx ⫽ *C1 ⫹ 2G*C2 For the second type of loading the displacement vector will be
and the auxiliary vector R similar to one used in (8) will be
(9)
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Applying a similar derivation as for the first type of loading, one can obtain the complex stiffness for the second type of loading (Fig. 2(b)) analogous to (9): Sy ⫽ 2(*D1 ⫹ 2G*D2) where coefficient 2 stems from the fact that in experiment two specimens are used to support mass m2 (Fig. 1(b)). Note that complex stiffnesses Sx, Sy depend on coefficients C1, C2 and D1, D2 respectively, which in turn depend on the distribution of internal displacements U0I, which in turn depends on * and G*. Thus the dependence will be non-linear, namely, Sx(*,G*) ⫽ *C1(*,G*) ⫹ 2G*C2(*,G*) Sy(*,G*) ⫽ 2(*D1(*,G*) ⫹ 2G*D2(*,G*)) Now consider a base excitation test with a frequency which will correspond to the first type of loading (Fig. 1(a). Assuming that the mass at the top is m1 one can write the expression for the transfer function analogous to (3): T1() ⫽
y01 *C1(*,G*) ⫹ 2G*C2(*,G*) ⫽ x01 ⫺ m12 ⫹ *C1(*,G*) ⫹ 2G*C2(*,G*)
(10)
and similarly for the second type of loading T2() ⫽
y02 2(*D1(*,G*) ⫹ 2G*D2(*,G*)) ⫽ x02 ⫺ m22 ⫹ 2(*D1(*,G*) ⫹ 2G*D2(*,G*))
(11)
One can rewrite non-linear Eq. (10) and (11) in the following form
冋
C1(*,G*)(T1 ⫺ 1) C2(*,G*)(T1 ⫺ 1)
册冋 册 冋 *
2D1(*,G*(T2 ⫺ 1) 2D2(*,G*)(T2 ⫺ 1) 2G*
Introducing notations X ⫽ [*,2G*]T and H⫽
冋
C1(*,G*)(T1 ⫺ 1)
C2(*,G*)(T1 ⫺ 1)
⫽
T1m12 T2m22
册
(12)
册
2D1(*,G*)(T2 ⫺ 1) 2D2(*,G*)(T2 ⫺ 1)
and using (12) a successive approximation procedure then follows: Xi ⫹ 1 ⫽ [H(Xi)]−1
冋
T1m12 T2m22
册
i ⫽ 1,2,3,…
(13)
The iterations are repeated until convergence is reached 储Xi ⫹ 1 ⫺ Xi储 ⬍ ⑀, or another equivalent criteria can be used, i.e.
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273
储T1c ⫺ T1储 ⬍ ⑀ and 储T2c ⫺ T2储 ⬍ ⑀ where T1c, T2c are calculated values of the transfer functions (10), (11), and T1, T2 are experimental values of the transfer functions in the first test and second test respectively. In other words a fixed point X* of the mapping (13) is determined
冋
X* ⫽ [H(X*)]−1
T1m12 T2m22
册
The convergence of this iterative process can be analyzed from a mathematical point of view using contraction mapping principle,[9] though this analysis is omitted here.
3. Experimental transfer functions and evaluation of complex moduli For a series of polymeric materials, two tests (Fig. 1(a,b)) were conducted. The transfer functions T1, T2 for EAR C-1002 (Isodamp) material are presented as functions of frequency in Fig. 3 for test 1, and in Fig. 4 for test 2. Similar transfer functions for such materials as CR (chloroprene), NBR (acrylonitrile-butadiene rubber), PECH (polyepichlorohydrin) can be found in Muravyov[10] and are omitted here for the sake of brevity. For test 1, the value of the top mass was 1.91 kg, for the second test it was 0.54 kg. The dimensions of specimens were 0.03 ⫻ 0.03 ⫻ 0.03 m. The temperature of all specimens in all tests was 20°C. Having experimental transfer functions T1(), T2() for each material, one can substitute them in Eq. (13) in order to calculate complex moduli *, and G* for each frequency. The criteria of convergence for the iterative procedure (13) was
Fig. 3.
Transfer function in the first test for EAR C-1002.
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Fig. 4. Transfer function in the second test for EAR C-1002.
Fig. 5. Storage Young’s modulus and loss factor of EAR C-1002.
兩T2c ⫺ T2兩 兩T1c ⫺ T1兩 ⬍ ⑀ and ⬍⑀ 兩T1兩 兩T2兩 where the relative error ⑀ was assumed 0.5%. Having determined the complex moduli *, G*, one can determine E* (complex Young’s modulus) and * (complex Poisson’s ratio) according to the following formulae: E* ⫽
3* ⫹ 2G* E* G* * ⫽ ⫺1 * ⫹ G* 2G*
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275
Fig. 6. Complex Poisson’s ratio of EAR C-1002.
The convergence of the procedure (13) was observed for all cases, though the number of iterations required to reach the prescribed level of ⑀ was different for different materials and frequencies. The fastest convergence was observed in the case of EAR C-1002 material, which yielded the values of the Poisson’s ratio between 0.35 and 0.2 (real part) in the considered frequency range. Values of E* and * for several polymeric materials are shown below. These values were
Fig. 7. Storage Young’s modulus and loss factor of CR.
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Fig. 8. Complex Poisson’s ratio of CR.
obtained with application of the described above procedure. For EAR C-1002 material the results are shown in Fig. 5, where the real part of the complex modulus E1 (storage modulus), and the loss factor E2/E1 are shown as functions of frequency. The values from Nashif and Jones[11] corresponding to the temperature 24°C (75°F) are shown for comparison purposes. One can see a good correlation with the results for EAR C-1002 material obtained in this study (note that the temperature of the specimens was 20°C). The complex Poisson’s ratio * ⫽ 1 ⫹ i2 is presented in Fig. 6. Complex moduli for CR, NBR, and PECH materials are presented in Figs. 7–12, from which one can conclude that in the
Fig. 9. Storage Young’s modulus and loss factor of NBR.
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Fig. 10.
277
Complex Poisson’s ratio of NBR.
Fig. 11. Storage Young’s modulus and loss factor of PECH.
considered frequency range (10–250 Hz) the complex Poisson’s ratio of NBR, CR materials can be considered as real and constant, because its value fluctuates between 0.4995 and 0.4991, and it has a negligible imaginary part. The same is not possible to conclude about EAR, PECH materials which demonstrate the dependence of * on frequency in the range of 10–250 Hz. One can see that the value of transfer function tends to zero (real and imaginary parts) as the frequency increases (Figs. 3 and 4). Therefore, to use its values for higher frequencies does not seem appropriate due to possible accuracy distortion of experimental data. Determination of E* and * (or * and G*) for a greater frequency range > 250 Hz requires additional experiments
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Fig. 12. Complex Poisson’s ratio of PECH.
with different geometric parameters of the specimens to provide such transfer function values which are not too small to cause any accuracy distortion. 4. Summary Application of 3-D specimens and a 3-D finite element model for the purpose of complex moduli evaluation has been presented. A numerical procedure based on the solution of two nonlinear equations (with respect to two complex Lame’s moduli) by a successive approximation method has been developed, and convergence of which has been numerically confirmed. For a considered frequency range of 10–250 Hz, the characteristic complex moduli of several isotropic polymeric materials have been determined. The value of the complex Poisson’s ratio turned to be not constant for certain materials even for a relatively narrow frequency range. References [1] Dlubac JJ, Lee GF, Duffy JV, Deigan RJ, Lee JD. Comparison of the complex dynamic modulus as measured by three apparatus. In: Sound and Vibration Damping with Polymers. Washington, (DC): American Chemical Society, 1990. pp. 49–62. [2] Sattinger SS. Direct method for measuring the dynamic shear properties of damping polymers. In Sound and Vibration Damping with Polymers. Washington, (DC): American Chemical Society, 1990. pp. 79–91. [3] Rabotnov Yu N. Elements of Hereditary Solid Mechanics. Moscow: Mir, 1980. [4] Rabotnov Yu N. Creep Problems in Structural Members. Amsterdam: North-Holland, 1969. [5] Christensen RM. Theory of Viscoelasticity. New York: Academic Press, 1982.
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[6] Ferry JD. Viscoelastic Properties of Polymers. New York: John Wiley, 1970. [7] Bagley RL, Torvik PJ. Fractional calculus—a different approach to the analysis of viscoelastically damped systems. AIAA Journal 1983;21(5):741–8. [8] Muravyov A. Analytical solutions in the time domain for vibration problems of discrete viscoelastic systems. J Sound Vibr 1996;199(2):337–48. [9] Lusternik LA, Sobolev VJ. Elements of Functional Analysis. New York: John Wiley, 1974. [10] Muravyov A. Discrete dynamic viscoelastic systems and vibration analysis of an engine supported on viscoelastic mounts. Ph.D dissertation, The University of British Columbia, Vancouver, 1997. [11] Nashif AD, Jones DIG, Henderson JP. Vibration Damping. New York: McGraw-Hill, 1985.
Data Interpretation
Determination of complex moduli of isotropic viscoelastic materials Alexander Muravyov1 Department of Mechanical Engineering, University of British Columbia, Vancouver, B.C., Canada V6T 1W5 Received 10 February 1998; received in revised form 22 April 1998
Abstract A procedure for determination of complex moduli of viscoelastic (polymeric) materials is described. The complex moduli of specimen’s isotropic material are calculated from certain experimentally obtained data and with application of a 3-D finite element model of the experimental specimen. A system of two nonlinear complex equations with respect to two complex Lame’s moduli is formulated, and then solved by a successive approximation method. The convergence of the iterative procedure is observed for all cases. The complex moduli obtained for a series of polymeric materials are presented for 10–250 Hz frequency range. 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction In the present paper isotropic homogeneous viscoelastic materials are considered and it is assumed that materials are in isothermal state. Consider the constitutive stress–strain relation for the particular case when the strain is harmonic with frequency :
⑀(t) ⫽ ⑀0eit i ⫽ √ ⫺ 1 one obtains (a one-dimensional element is taken as an illustration) 1 Present address: Institute for Computer Applications in Science and Engineering, Mail Stop 403, 6 North Dryden St, NASA Langley Research Center, Hampton, VA 23681-2199, USA. Tel.: ⫹ 1-757-864-5267; fax: ⫹ 1-757-8646134; e-mail: [email protected]
0142-9418/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 2 - 9 4 1 8 ( 9 8 ) 0 0 0 2 5 - 7
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A. Muravyov / Polymer Testing 18 (1999) 267–279
(t) ⫽ E*()⑀0eit where the complex quantity E*() ⫽ E1() ⫹ iE2() is called the complex modulus. Several experimental procedures were described in the literature[1,2] with specimens having a shape of long bar. These specimens were subjected either to tension, or torsion vibration tests, with an assumption of uniform tensile, or shear stress state to be imposed. These are the traditional procedures, however the assumption of uniform stress state is not always justified. In this paper the specimen will have a 3-D cubic shape, and no simplifying assumptions about stress state will be done. As it will be seen from the procedure an arbitrary shape of specimen can be used. It is only required to have an adequate finite element model of the specimen of such shape. For a viscoelastic structure, Lame coefficients and G should be replaced by their hereditary analogues (operators).[3–6] In the case of the finite element method this implies the replacement of material constants , and G in the stiffness matrix by their viscoelastic analogues (operators). For a particular case (harmonic deformation) this leads to replacement of , G by corresponding complex moduli * and G* which are functions of frequency of deformation. Complex moduli of a polymeric material are necessary in order, for example, to obtain the steady-state response of a polymeric structure subjected to a periodic loading. Below a procedure will be presented which allows the complex Lame modulus *, and shear modulus G* of a material to be obtained from certain experimental data. Note that having two moduli *, G*, all other characteristic moduli, for example, complex Young’s modulus and complex Poisson’s ratio can be calculated. 2. A procedure for determination of complex moduli Consider a harmonic base excitation of the system shown in Fig. 1(a). A specimen of the viscoelastic material which is under consideration is assumed to have a 3-D cubic shape.
Fig. 1. Two types of base excitation test; tension type (1) and shear type (2).
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269
First, assume that the 3-D specimen connecting the mass m and the base is modelled as a massless one-dimensional viscoelastic spring (complex stiffness k*) with length L, and crosssection area A. Considering a single-degree-of-freedom equation with harmonic base excitation x ⫽ x0eit, one obtains: mz¨ ⫹ k*z ⫽ ⫺ mx¨ ⫽ m2x0eit
(1)
where z ⫽ y ⫺ x (see Fig. 1(a)). Assuming z ⫽ z0eit and substituting it in the equation of motion (1) one obtains ⫺ 2mz0 ⫹
A (E () ⫹ iE2())z0 ⫽ m2x0 L 1
where the term eit was eliminated in all places. Thus z0 ⫽
mx02 A ⫺ m2 ⫹ (E1() ⫹ iE2()) L
(2)
A complex transfer function is defined as: T() ⫽
yo xo ⫹ zo ⫽ xo xo
and using (2) one obtains A (E () ⫹ iE2()) L 1 T() ⫽ A ⫺ m2 ⫹ (E1() ⫹ iE2()) L
(3)
Two harmonic base excitation tests can be conducted (Fig. 1(a,b)) and transfer functions T1() ⫽ y01/x01
T2() ⫽ y02/x02
can be registered in these tests. Now instead of a viscoelastic spring model consider a finite element model which will be used to simulate the experimental specimen. Volume (3-D) finite elements will be used to mesh a given 3-D domain (Fig. 2). One side of the specimen is assumed fixed, when at the opposite side (side ⌫) some displacements will be prescribed. Two types of prescribed kinematic loadings (displacements on side ⌫) will be considered according to two experimental tests (Fig. 1(a,b)): (1) ux ⫽ 1, uy ⫽ uz ⫽ 0, and (2) uy ⫽ 1, ux ⫽ uz ⫽ 0.
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A. Muravyov / Polymer Testing 18 (1999) 267–279
Fig. 2.
3-D finite element model of the specimen with two types of loading.
Based on application of 3-D finite elements the stiffness matrix of a viscoelastic homogeneous structure can be presented in the following form: K ⫽ *K1 ⫹ 2G*K2 where matrices K1, K2 do not depend on material constants. Such representation of the matrix K was used in other studies.[7,8] Neglecting by the mass of the specimen in comparison with the attached rigid mass, one can write the following expression for a harmonic loading with frequency , namely, [*()K1 ⫹ 2G*()K2]U0eit ⫽ F0eit
(4)
where the force vector on the right side represents the action from the attached rigid mass on the specimen. Note that the force vector F0 will include only components which correspond to degrees of freedom on the side ⌫, all other components are assumed zero. Also note that in the vector of displacements U0 the components which correspond to degrees of freedom on the side ⌫ are known (they are prescribed according to two types of loading). Assume that the size of matrices K1, and K2 is N, and the number of degrees of freedom pertaining to the side ⌫ is 3n, then the rest N ⫺ 3n (denote them as 3m) degrees of freedom will correspond to internal degrees of freedom. It is assumed here that the finite elements have three linear degrees of freedom per node. Considering the first type loading, the vector of displacements can be subdivided as follows:
Analogously one can represent the force vector
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271
Renumerating the degrees of freedom of the system and eliminating the time-dependent coefficient at both sides, one can represent Eq. (4) in the following form:
冢冤
兩
K 1⌫I
⫺⫺
⫺⫺
K 1⌫⌫
* ⫺ ⫺ K 1I⌫
兩
K
1 II
冥 冤
兩
K 2⌫I
⫺⫺
⫺⫺
K 2⌫⌫
⫹ 2G* ⫺ ⫺ K
2 I⌫
兩
K
2 II
冥冣冤 冥 冤 冥 U0⌫
⫺⫺ U0I
F0⌫
⫽
⫺⫺
(5)
0
where subvector of displacements U0I (size 3m) is unknown, and the subvector of forces F0⌫ (size 3n) is also unknown. Thus one can solve the following system of linear equations
冤
⫺I
兩
⫺⫺
⫺⫺
*K 1⌫I ⫹ 2G*K 2⌫I ⫺⫺⫺⫺⫺⫺⫺
0
兩
*K ⫹ 2G*K 1 II
2 II
冥冤 冥 冤 F0⌫
⫺⫺ U0I
⫽
⫺ *K 1⌫⌫U0⌫ ⫺ 2G*K 2⌫⌫U0⌫ ⫺⫺⫺⫺⫺⫺⫺⫺⫺⫺⫺ ⫺ *K 1I⌫U0⌫ ⫺ 2G*K 2I⌫U0⌫
冥
and subvectors U0I, and F0⌫ can be found. Then one can find the total resulting force which is necessary to apply to create the prescribed vector of displacements on ⌫. Namely, using (5) and collecting x-components, the resulting force for the 1st loading can be calculated as
Having the non-zero components of U0⌫ equal to 1, from (6) one obtains the complex stiffness Sx: Sx ⫽ *(A1 ⫹ A2) ⫹ 2G*(B1 ⫹ B2)
(7)
where A1 ⫽ RK 1⌫⌫U0⌫ A2 ⫽ RK 1⌫IU0I B1 ⫽ RK 2⌫⌫U0⌫ B2 ⫽ RK 2⌫IU0⌫
(8)
and
Introducing additional notations C1 ⫽ A1 ⫹ A2 C2 ⫽ B1 ⫹ B2 the previous expression (7) for the complex stiffness in the first type loading becomes: Sx ⫽ *C1 ⫹ 2G*C2 For the second type of loading the displacement vector will be
and the auxiliary vector R similar to one used in (8) will be
(9)
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Applying a similar derivation as for the first type of loading, one can obtain the complex stiffness for the second type of loading (Fig. 2(b)) analogous to (9): Sy ⫽ 2(*D1 ⫹ 2G*D2) where coefficient 2 stems from the fact that in experiment two specimens are used to support mass m2 (Fig. 1(b)). Note that complex stiffnesses Sx, Sy depend on coefficients C1, C2 and D1, D2 respectively, which in turn depend on the distribution of internal displacements U0I, which in turn depends on * and G*. Thus the dependence will be non-linear, namely, Sx(*,G*) ⫽ *C1(*,G*) ⫹ 2G*C2(*,G*) Sy(*,G*) ⫽ 2(*D1(*,G*) ⫹ 2G*D2(*,G*)) Now consider a base excitation test with a frequency which will correspond to the first type of loading (Fig. 1(a). Assuming that the mass at the top is m1 one can write the expression for the transfer function analogous to (3): T1() ⫽
y01 *C1(*,G*) ⫹ 2G*C2(*,G*) ⫽ x01 ⫺ m12 ⫹ *C1(*,G*) ⫹ 2G*C2(*,G*)
(10)
and similarly for the second type of loading T2() ⫽
y02 2(*D1(*,G*) ⫹ 2G*D2(*,G*)) ⫽ x02 ⫺ m22 ⫹ 2(*D1(*,G*) ⫹ 2G*D2(*,G*))
(11)
One can rewrite non-linear Eq. (10) and (11) in the following form
冋
C1(*,G*)(T1 ⫺ 1) C2(*,G*)(T1 ⫺ 1)
册冋 册 冋 *
2D1(*,G*(T2 ⫺ 1) 2D2(*,G*)(T2 ⫺ 1) 2G*
Introducing notations X ⫽ [*,2G*]T and H⫽
冋
C1(*,G*)(T1 ⫺ 1)
C2(*,G*)(T1 ⫺ 1)
⫽
T1m12 T2m22
册
(12)
册
2D1(*,G*)(T2 ⫺ 1) 2D2(*,G*)(T2 ⫺ 1)
and using (12) a successive approximation procedure then follows: Xi ⫹ 1 ⫽ [H(Xi)]−1
冋
T1m12 T2m22
册
i ⫽ 1,2,3,…
(13)
The iterations are repeated until convergence is reached 储Xi ⫹ 1 ⫺ Xi储 ⬍ ⑀, or another equivalent criteria can be used, i.e.
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储T1c ⫺ T1储 ⬍ ⑀ and 储T2c ⫺ T2储 ⬍ ⑀ where T1c, T2c are calculated values of the transfer functions (10), (11), and T1, T2 are experimental values of the transfer functions in the first test and second test respectively. In other words a fixed point X* of the mapping (13) is determined
冋
X* ⫽ [H(X*)]−1
T1m12 T2m22
册
The convergence of this iterative process can be analyzed from a mathematical point of view using contraction mapping principle,[9] though this analysis is omitted here.
3. Experimental transfer functions and evaluation of complex moduli For a series of polymeric materials, two tests (Fig. 1(a,b)) were conducted. The transfer functions T1, T2 for EAR C-1002 (Isodamp) material are presented as functions of frequency in Fig. 3 for test 1, and in Fig. 4 for test 2. Similar transfer functions for such materials as CR (chloroprene), NBR (acrylonitrile-butadiene rubber), PECH (polyepichlorohydrin) can be found in Muravyov[10] and are omitted here for the sake of brevity. For test 1, the value of the top mass was 1.91 kg, for the second test it was 0.54 kg. The dimensions of specimens were 0.03 ⫻ 0.03 ⫻ 0.03 m. The temperature of all specimens in all tests was 20°C. Having experimental transfer functions T1(), T2() for each material, one can substitute them in Eq. (13) in order to calculate complex moduli *, and G* for each frequency. The criteria of convergence for the iterative procedure (13) was
Fig. 3.
Transfer function in the first test for EAR C-1002.
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Fig. 4. Transfer function in the second test for EAR C-1002.
Fig. 5. Storage Young’s modulus and loss factor of EAR C-1002.
兩T2c ⫺ T2兩 兩T1c ⫺ T1兩 ⬍ ⑀ and ⬍⑀ 兩T1兩 兩T2兩 where the relative error ⑀ was assumed 0.5%. Having determined the complex moduli *, G*, one can determine E* (complex Young’s modulus) and * (complex Poisson’s ratio) according to the following formulae: E* ⫽
3* ⫹ 2G* E* G* * ⫽ ⫺1 * ⫹ G* 2G*
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Fig. 6. Complex Poisson’s ratio of EAR C-1002.
The convergence of the procedure (13) was observed for all cases, though the number of iterations required to reach the prescribed level of ⑀ was different for different materials and frequencies. The fastest convergence was observed in the case of EAR C-1002 material, which yielded the values of the Poisson’s ratio between 0.35 and 0.2 (real part) in the considered frequency range. Values of E* and * for several polymeric materials are shown below. These values were
Fig. 7. Storage Young’s modulus and loss factor of CR.
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Fig. 8. Complex Poisson’s ratio of CR.
obtained with application of the described above procedure. For EAR C-1002 material the results are shown in Fig. 5, where the real part of the complex modulus E1 (storage modulus), and the loss factor E2/E1 are shown as functions of frequency. The values from Nashif and Jones[11] corresponding to the temperature 24°C (75°F) are shown for comparison purposes. One can see a good correlation with the results for EAR C-1002 material obtained in this study (note that the temperature of the specimens was 20°C). The complex Poisson’s ratio * ⫽ 1 ⫹ i2 is presented in Fig. 6. Complex moduli for CR, NBR, and PECH materials are presented in Figs. 7–12, from which one can conclude that in the
Fig. 9. Storage Young’s modulus and loss factor of NBR.
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Fig. 10.
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Complex Poisson’s ratio of NBR.
Fig. 11. Storage Young’s modulus and loss factor of PECH.
considered frequency range (10–250 Hz) the complex Poisson’s ratio of NBR, CR materials can be considered as real and constant, because its value fluctuates between 0.4995 and 0.4991, and it has a negligible imaginary part. The same is not possible to conclude about EAR, PECH materials which demonstrate the dependence of * on frequency in the range of 10–250 Hz. One can see that the value of transfer function tends to zero (real and imaginary parts) as the frequency increases (Figs. 3 and 4). Therefore, to use its values for higher frequencies does not seem appropriate due to possible accuracy distortion of experimental data. Determination of E* and * (or * and G*) for a greater frequency range > 250 Hz requires additional experiments
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Fig. 12. Complex Poisson’s ratio of PECH.
with different geometric parameters of the specimens to provide such transfer function values which are not too small to cause any accuracy distortion. 4. Summary Application of 3-D specimens and a 3-D finite element model for the purpose of complex moduli evaluation has been presented. A numerical procedure based on the solution of two nonlinear equations (with respect to two complex Lame’s moduli) by a successive approximation method has been developed, and convergence of which has been numerically confirmed. For a considered frequency range of 10–250 Hz, the characteristic complex moduli of several isotropic polymeric materials have been determined. The value of the complex Poisson’s ratio turned to be not constant for certain materials even for a relatively narrow frequency range. References [1] Dlubac JJ, Lee GF, Duffy JV, Deigan RJ, Lee JD. Comparison of the complex dynamic modulus as measured by three apparatus. In: Sound and Vibration Damping with Polymers. Washington, (DC): American Chemical Society, 1990. pp. 49–62. [2] Sattinger SS. Direct method for measuring the dynamic shear properties of damping polymers. In Sound and Vibration Damping with Polymers. Washington, (DC): American Chemical Society, 1990. pp. 79–91. [3] Rabotnov Yu N. Elements of Hereditary Solid Mechanics. Moscow: Mir, 1980. [4] Rabotnov Yu N. Creep Problems in Structural Members. Amsterdam: North-Holland, 1969. [5] Christensen RM. Theory of Viscoelasticity. New York: Academic Press, 1982.
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[6] Ferry JD. Viscoelastic Properties of Polymers. New York: John Wiley, 1970. [7] Bagley RL, Torvik PJ. Fractional calculus—a different approach to the analysis of viscoelastically damped systems. AIAA Journal 1983;21(5):741–8. [8] Muravyov A. Analytical solutions in the time domain for vibration problems of discrete viscoelastic systems. J Sound Vibr 1996;199(2):337–48. [9] Lusternik LA, Sobolev VJ. Elements of Functional Analysis. New York: John Wiley, 1974. [10] Muravyov A. Discrete dynamic viscoelastic systems and vibration analysis of an engine supported on viscoelastic mounts. Ph.D dissertation, The University of British Columbia, Vancouver, 1997. [11] Nashif AD, Jones DIG, Henderson JP. Vibration Damping. New York: McGraw-Hill, 1985.