# American Institute of Mathematical Sciences

December  2017, 10(6): 1351-1373. doi: 10.3934/dcdss.2017072

## Two boundary value problems involving an inhomogeneous viscoelastic solid

 1 Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai, Tamil Nadu 600036, India 2 Department of Civil Engineering, Indian Institute of Technology Madras, Chennai, Tamil Nadu 600036, India 3 Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843, USA

* Corresponding author: Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843, USA. Ph: 1 979 8624552

Dedicated to Prof. T. Roubíček on his sixtieth birthday

Received  July 2016 Revised  November 2016 Published  June 2017

Recently, a thermodynamically consistent non-linear constitutive equation has been developed to describe the large deformation cyclic response of viscoelastic polyamides (see [17]). In this paper, two boundary value problems within the context of the above model, namely the stress relaxation of a right circular annular cylinder subject to twisting, and the inflation of a sphere are studied. In addition to solving the above problems numerically, investigation of the merits and pitfalls of studying the same boundary value problem for a special class of inhomogeneous body and its homogenized counterpart is undertaken. This study finds that for moderate strains the differences in relaxation time between the actual inhomogeneous body and its homogenized counterparts may be significant.

Citation: M. Mahalingam, Parag Ravindran, U. Saravanan, K. R. Rajagopal. Two boundary value problems involving an inhomogeneous viscoelastic solid. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1351-1373. doi: 10.3934/dcdss.2017072
##### References:
 [1] I. V. Andrianov, V. V. Danishevs'kyy and E. G. Kholod, Homogenization of viscoelastic composites with fibres of diamond-shaped cross-section, Acta Mechanica, 223 (2012), 1093-1100.  doi: 10.1007/s00707-011-0608-6.  Google Scholar [2] K. S. Challagulla, A. Georgiades and A. L. Kalamkarov, Asymptotic homogenization model for three-dimensional network reinforced composite structures, Journal of Mechanics of Materials and Structures, 2 (2007), 613-632.  doi: 10.2140/jomms.2007.2.613.  Google Scholar [3] N. Charalambakis, Homogenization techniques and micromechanics. a survey and perspectives, Applied Mechanics Reviews, 63 (2010), 030803.   Google Scholar [4] Y.-C. Chen, K. R. Rajagopal and L. Wheeler, Homogenization and global responses of inhomogeneous spherical nonlinear elastic shells, Journal of Elasticity, 82 (2006), 193-214.  doi: 10.1007/s10659-005-9031-3.  Google Scholar [5] P. W. Chung, K. K. Tamma and R. R. Namburu, A micro/macro homogenization approach for viscoelastic creep analysis with dissipative correctors for heterogeneous woven-fabric layered media, Composites Science and Technology, 60 (2000), 2233-2253.  doi: 10.1016/S0266-3538(00)00018-X.  Google Scholar [6] L. Gibiansky and R. Lakes, Bounds on the complex bulk and shear moduli of a two-dimensional two-phase viscoelastic composite, Mechanics of Materials, 25 (1997), 79-95.  doi: 10.1016/S0167-6636(96)00046-4.  Google Scholar [7] L. Gibiansky and G. Milton, On the effective viscoelastic moduli of two-phase media. i. rigorous bounds on the complex bulk modulus, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 440 (1993), 163-188.  doi: 10.1098/rspa.1993.0010.  Google Scholar [8] R. Hill, Theory of mechanical properties of fibre-strengthened materials: I. elastic behaviour, Journal of the Mechanics and Physics of Solids, 12 (1964), 199-212.  doi: 10.1016/0022-5096(64)90019-5.  Google Scholar [9] M. Hori and S. Nemat-Nasser, On two micromechanics theories for determining micro-macrorelations in heterogeneous solids, Mechanics of Materials, 31 (1999), 667-682.   Google Scholar [10] A. L. Kalamkarov, I. V. Andrianov and V. V. Danishevs'kyy, Asymptotic homogenization of composite materials and structures, Applied Mechanics Reviews(030802), 62 (2009), 1-20.  doi: 10.1115/1.3090830.  Google Scholar [11] R. V. Kohn, Recent progress in the mathematical modelling of composite materials, in Composite Material Response: Constitutive Relations and Damage Mechanisms (eds. G. Shi, G. F. Smith, M. I. H and W. J. J), Elsevier, 1988,155-176. Google Scholar [12] S. A. Meguid and A. L. Kalamkarov, Asymptotic homogenization of elastic composite materials with a regular structure, International Journal of Solids and Structures, 31 (1994), 303-316.  doi: 10.1016/0020-7683(94)90108-2.  Google Scholar [13] S. Nemat-Nasser, N. Yu and M. Hori, Bounds and estimates of overall moduli of composites with periodic microstructure, Mechanics of Materials, 15 (1993), 163-181.  doi: 10.1016/0167-6636(93)90016-K.  Google Scholar [14] V. Pruša and K. R. Rajagopal, Jump conditions in stress relaxation and creep experiments of Burgers type fluids: A study in the application of Colombeau algebra of generalized functions, Zeitschrift für angewandte Mathematik und Physik, 62 (2011), 707-740.  doi: 10.1007/s00033-010-0109-9.  Google Scholar [15] K. R. Rajagopal and A. R. Srinivasa, A thermodynamic frame work for rate type fluid models, Journal of Non-Newtonian Fluid Mechanics, 88 (2000), 207-227.  doi: 10.1016/S0377-0257(99)00023-3.  Google Scholar [16] K. R. Rajagopal and A. S. Wineman, Mechanical Response of Polymers -An Introduction, Cambridge University Press, 2000.   Google Scholar [17] A. Ramkumar, K. Kannan and R. Gnanamoorthy, Experimental and theoretical investigation of a polymer subjected to cyclic loading conditions, International Journal of Engineering Science, 48 (2010), 101-110.  doi: 10.1016/j.ijengsci.2009.07.002.  Google Scholar [18] E. Sanchez-Palencia, Non-homogeneous Media and Vibration Theory, Springer, 1980.   Google Scholar [19] U. Saravanan and K. R. Rajagopal, On the role of inhomogeneities in the deformation of elastic bodies, Mathematics and Mechanics of Solids, 8 (2003), 349-376.  doi: 10.1177/10812865030084002.  Google Scholar [20] U. Saravanan and K. R. Rajagopal, Inflation, extension, torsion and shearing of an inhomogeneous compressible elastic right circular annular cylinder, Mathematics and Mechanics of Solids, 10 (2005), 603-650.  doi: 10.1177/1081286505036422.  Google Scholar [21] U. Saravanan and K. R. Rajagopal, On some finite deformations of inhomogeneous compressible elastic solids, Mathematical Proceedings of the Royal Irish Academy, 107 (2007), 43-72.  doi: 10.3318/PRIA.2007.107.1.43.  Google Scholar [22] U. Saravanan and K. Rajagopal, A comparison of the response of isotropic inhomogeneous elastic cylindrical and spherical shells and their homogenized counterparts, Journal of Elasticity, 71 (2003), 205-234.  doi: 10.1023/B:ELAS.0000005547.48580.7f.  Google Scholar [23] P. Suquet, Elements of homogeneization for inelastic solid mechanics, in Homogeneization Techniques for Composite Media (eds. E. Sanchez-Palencia and A. Zaoui), Springer Verlag, Berlin, 272 (1987), 193-278. doi: 10.1007/3-540-17616-0_15.  Google Scholar [24] I. Temizer and T. Zohdi, A numerical method for homogenization in non-linear elasticity, Computational Mechanics, 40 (2007), 281-298.  doi: 10.1007/s00466-006-0097-y.  Google Scholar [25] C. Truesdell and W. Noll, The Non-Linear Field Theories of Mechanics, Springer, 2003.   Google Scholar [26] Y.-M. Yi, S.-H. Park and S.-K. Youn, Asymptotic homogenization of viscoelastic composites with periodic microstructures, International Journal of Solids and Structures, 35 (1998), 2039-2055.  doi: 10.1016/S0020-7683(97)00166-2.  Google Scholar

show all references

##### References:
 [1] I. V. Andrianov, V. V. Danishevs'kyy and E. G. Kholod, Homogenization of viscoelastic composites with fibres of diamond-shaped cross-section, Acta Mechanica, 223 (2012), 1093-1100.  doi: 10.1007/s00707-011-0608-6.  Google Scholar [2] K. S. Challagulla, A. Georgiades and A. L. Kalamkarov, Asymptotic homogenization model for three-dimensional network reinforced composite structures, Journal of Mechanics of Materials and Structures, 2 (2007), 613-632.  doi: 10.2140/jomms.2007.2.613.  Google Scholar [3] N. Charalambakis, Homogenization techniques and micromechanics. a survey and perspectives, Applied Mechanics Reviews, 63 (2010), 030803.   Google Scholar [4] Y.-C. Chen, K. R. Rajagopal and L. Wheeler, Homogenization and global responses of inhomogeneous spherical nonlinear elastic shells, Journal of Elasticity, 82 (2006), 193-214.  doi: 10.1007/s10659-005-9031-3.  Google Scholar [5] P. W. Chung, K. K. Tamma and R. R. Namburu, A micro/macro homogenization approach for viscoelastic creep analysis with dissipative correctors for heterogeneous woven-fabric layered media, Composites Science and Technology, 60 (2000), 2233-2253.  doi: 10.1016/S0266-3538(00)00018-X.  Google Scholar [6] L. Gibiansky and R. Lakes, Bounds on the complex bulk and shear moduli of a two-dimensional two-phase viscoelastic composite, Mechanics of Materials, 25 (1997), 79-95.  doi: 10.1016/S0167-6636(96)00046-4.  Google Scholar [7] L. Gibiansky and G. Milton, On the effective viscoelastic moduli of two-phase media. i. rigorous bounds on the complex bulk modulus, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 440 (1993), 163-188.  doi: 10.1098/rspa.1993.0010.  Google Scholar [8] R. Hill, Theory of mechanical properties of fibre-strengthened materials: I. elastic behaviour, Journal of the Mechanics and Physics of Solids, 12 (1964), 199-212.  doi: 10.1016/0022-5096(64)90019-5.  Google Scholar [9] M. Hori and S. Nemat-Nasser, On two micromechanics theories for determining micro-macrorelations in heterogeneous solids, Mechanics of Materials, 31 (1999), 667-682.   Google Scholar [10] A. L. Kalamkarov, I. V. Andrianov and V. V. Danishevs'kyy, Asymptotic homogenization of composite materials and structures, Applied Mechanics Reviews(030802), 62 (2009), 1-20.  doi: 10.1115/1.3090830.  Google Scholar [11] R. V. Kohn, Recent progress in the mathematical modelling of composite materials, in Composite Material Response: Constitutive Relations and Damage Mechanisms (eds. G. Shi, G. F. Smith, M. I. H and W. J. J), Elsevier, 1988,155-176. Google Scholar [12] S. A. Meguid and A. L. Kalamkarov, Asymptotic homogenization of elastic composite materials with a regular structure, International Journal of Solids and Structures, 31 (1994), 303-316.  doi: 10.1016/0020-7683(94)90108-2.  Google Scholar [13] S. Nemat-Nasser, N. Yu and M. Hori, Bounds and estimates of overall moduli of composites with periodic microstructure, Mechanics of Materials, 15 (1993), 163-181.  doi: 10.1016/0167-6636(93)90016-K.  Google Scholar [14] V. Pruša and K. R. Rajagopal, Jump conditions in stress relaxation and creep experiments of Burgers type fluids: A study in the application of Colombeau algebra of generalized functions, Zeitschrift für angewandte Mathematik und Physik, 62 (2011), 707-740.  doi: 10.1007/s00033-010-0109-9.  Google Scholar [15] K. R. Rajagopal and A. R. Srinivasa, A thermodynamic frame work for rate type fluid models, Journal of Non-Newtonian Fluid Mechanics, 88 (2000), 207-227.  doi: 10.1016/S0377-0257(99)00023-3.  Google Scholar [16] K. R. Rajagopal and A. S. Wineman, Mechanical Response of Polymers -An Introduction, Cambridge University Press, 2000.   Google Scholar [17] A. Ramkumar, K. Kannan and R. Gnanamoorthy, Experimental and theoretical investigation of a polymer subjected to cyclic loading conditions, International Journal of Engineering Science, 48 (2010), 101-110.  doi: 10.1016/j.ijengsci.2009.07.002.  Google Scholar [18] E. Sanchez-Palencia, Non-homogeneous Media and Vibration Theory, Springer, 1980.   Google Scholar [19] U. Saravanan and K. R. Rajagopal, On the role of inhomogeneities in the deformation of elastic bodies, Mathematics and Mechanics of Solids, 8 (2003), 349-376.  doi: 10.1177/10812865030084002.  Google Scholar [20] U. Saravanan and K. R. Rajagopal, Inflation, extension, torsion and shearing of an inhomogeneous compressible elastic right circular annular cylinder, Mathematics and Mechanics of Solids, 10 (2005), 603-650.  doi: 10.1177/1081286505036422.  Google Scholar [21] U. Saravanan and K. R. Rajagopal, On some finite deformations of inhomogeneous compressible elastic solids, Mathematical Proceedings of the Royal Irish Academy, 107 (2007), 43-72.  doi: 10.3318/PRIA.2007.107.1.43.  Google Scholar [22] U. Saravanan and K. Rajagopal, A comparison of the response of isotropic inhomogeneous elastic cylindrical and spherical shells and their homogenized counterparts, Journal of Elasticity, 71 (2003), 205-234.  doi: 10.1023/B:ELAS.0000005547.48580.7f.  Google Scholar [23] P. Suquet, Elements of homogeneization for inelastic solid mechanics, in Homogeneization Techniques for Composite Media (eds. E. Sanchez-Palencia and A. Zaoui), Springer Verlag, Berlin, 272 (1987), 193-278. doi: 10.1007/3-540-17616-0_15.  Google Scholar [24] I. Temizer and T. Zohdi, A numerical method for homogenization in non-linear elasticity, Computational Mechanics, 40 (2007), 281-298.  doi: 10.1007/s00466-006-0097-y.  Google Scholar [25] C. Truesdell and W. Noll, The Non-Linear Field Theories of Mechanics, Springer, 2003.   Google Scholar [26] Y.-M. Yi, S.-H. Park and S.-K. Youn, Asymptotic homogenization of viscoelastic composites with periodic microstructures, International Journal of Solids and Structures, 35 (1998), 2039-2055.  doi: 10.1016/S0020-7683(97)00166-2.  Google Scholar
Targeted boundary displacement in the inhomogeneous body
Applied boundary force on the inhomogeneous body to realize the desired boundary displacement
Variation of $\mu_{mean-hom}$ with different inner to outer radius ratio $R_i/R_o$ for different spatial variations of $\mu_1$ in the inhomogeneous model obtained by correlating the variation of the torsional moment required to engender a given angle of twist
Variation in percentage relaxation time with $R_i/R_o$ for linear variation of $\mu_1$ obtained by correlating the variation of the torsional moment required to engender a given angle of twist
Variation of $\mu_{mean-hom}$ with $R_i/R_o$ for linear variation of $\mu_1$ obtained by different correlations to engender a given angle of twist
Percentage variation in relaxation time with rate of twisting and different thicknesses of the cylinder for linear variation of $\mu_1$ obtained by correlating the variation of the torsional moment
Variation of $\mu_{mean-hom}$ with $R_i/R_o$ for different spatial variations of $\mu_1$ obtained by correlating the variation of the radial component of the normal stress applied at the inner surface of the sphere to engender a given inflation
Radial variation of the cylindrical polar components of the Cauchy stress at time $t$ $=$ $0.25$ s for different radial variations of $\mu_1$ when the annular cylinder is subjected to pure twist
Radial variation of the spherical polar components of the Cauchy stress at time $t$ $=$ $0.25$ s for different radial variations of $\mu_1$ when the annular sphere is subjected to inflation
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