This paper presents a nonlinear micropolar nonclassical continuum theory (MPNCCT) for finite deformation, finite strain deformation physics of thermosviscoelastic solid medium with memory (polymeric micropolar solids) based on classical rotations
and their rates. Contravariant second Piola-Kirchhoff stress and moment tensors, in conjunction with finite deformation measures derived by the authors in recent paper, are utilized in deriving the conservation and balance laws and the constitutive theories based on conjugate pairs in entropy inequality and the representation theorem. This nonlinear MPNCCT for TVES with rheology: 1) incorporates nonlinear ordered rate dissipation mechanism based on Green’s strain rates up to order
; 2) also incorporates an additional ordered rate dissipation mechanism due to microconstituents, the viscosity of the medium and the rates of the symmetric part of the rotation gradient (of
) tensor up to order
, referred to as micropolar dissipation or micropolar viscous dissipation mechanism; 3) incorporates the primary mechanism of memory or rheology due to long chain molecules of the polymer and the viscosity of the medium by using the contravaraint second Piola-Kirchhoff stress tensor and its rates up to order
, resulting in a relaxation spectrum; 4) incorporates second mechanism of memory or rheology due to nonclassical physics, interaction of microconstituents with the viscous medium and long chain molecules by considering rates of the contravariant second Piola-Kirchhoff moment tensor up to order
, resulting in relaxation of second Piola-Kirchhoff moment tensor. This results in another relaxation spectrum for the second Piola-Kirchhoff moment tensor due to microconstituents, referred to as micropolar relaxation spectrum consisting of micropolar relaxation time constants of the material. This nonlinear MPNCCT for TVES with memory is thermodynamically and
References
[1]
Surana, K.S., Powell, M.J. and Reddy, J.N. (2015) A More Complete Thermodynamic Framework for Solid Continua. Journal of Thermal Engineering, 1, 446-459. https://doi.org/10.18186/jte.17430
[2]
Surana, K.S., Powell, M.J. and Reddy, J.N. (2015) A More Complete Thermodynamic Framework for Fluent Continua. Journal of Thermal Engineering, 1, 460-475. https://doi.org/10.18186/jte.00314
[3]
Surana, K.S., Reddy, J.N., Nunez, D. and Powell, M.J. (2015) A Polar Continuum Theory for Solid Continua. International Journal of Engineering Research and Industrial Applications, 8, 77-106.
[4]
Surana, K.S., Powell, M.J. and Reddy, J.N. (2015) A Polar Continuum Theory for Fluent Continua. International Journal of Engineering Research and Industrial Applications, 8, 107-146.
[5]
Surana, K.S., Powell, M.J. and Reddy, J.N. (2015) Constitutive Theories for Internal Polar Thermoelastic Solid Continua. Journal of Pure and Applied Mathematics: Advances and Applications, 14, 89-150.
[6]
Surana, K.S., Powell, M.J. and Reddy, J.N. (2015) Ordered Rate Constitutive Theories for Internal Polar Thermofluids. International Journal of Mathematics, Science, and Engineering Applications, 9, 51-116.
[7]
Surana, K.S., Long, S.W. and Reddy, J.N. (2016) Rate Constitutive Theories of Orders and for Internal Polar Non-Classical Thermofluids without Memory. Applied Mathematics, 7, 2033-2077. https://doi.org/10.4236/am.2016.716165
[8]
Surana, K.S., Mohammadi, F., Reddy, J.N. and Dalkilic, A.S. (2016) Ordered Rate Constitutive Theories for Non-Classical Internal Polar Thermoviscoelastic Solids Without Memory. International Journal of Mathematics, Science, and Engineering Applications (IJMSEA), 10, 99-131.
[9]
Surana, K.S., Joy, A.D. and Reddy, J.N. (2017) Non-classical Continuum Theory for Solids Incorporating Internal Rotations and Rotations of Cosserat Theories. Continuum Mechanics and Thermodynamics, 29, 665-698. https://doi.org/10.1007/s00161-017-0554-1
[10]
Surana, K.S., Joy, A.D. and Reddy, J.N. (2017) A Finite Deformation, Finite Strain Non-Classical Internal Polar Continuum Theory for Solids. Mechanics of Advanced Materials and Structures, 26, 381-393.
[11]
Surana, K.S., Mysore, D. and Reddy, J.N. (2018) Non-Classical Continuum Theories for Solid and Fluent Continua and Some Applications. International Journal of Smart and Nano Materials, 10, 28-89. https://doi.org/10.1080/19475411.2018.1530700
[12]
Surana, K.S., Mysore, D. and Reddy, J.N. (2018) Ordered Rate Constitutive Theories for Non-Classical Thermoviscoelastic Solids with Dissipation and Memory Incorporating Internal Rotations. Polytechnica, 1, 19-35. https://doi.org/10.1007/s41050-018-0004-2
[13]
Surana, K.S., Shanbhag, R. and Reddy, J.N. (2018) Necessity of Law of Balance of Moment of Moments in Non-Classical Continuum Theories for Solid Continua. Meccanica, 53, 2939-2972. https://doi.org/10.1007/s11012-018-0851-1
[14]
Surana, K.S., Long, S.W. and Reddy, J.N. (2018) Necessity of Law of Balance/Equilibrium of Moment of Moments in Non-Classical Continuum Theories for Fluent Continua. Acta Mechanica, 229, 2801-2833. https://doi.org/10.1007/s00707-018-2143-1
[15]
Surana, K.S., Long, S.W. and Reddy, J.N. (2018) Ordered Rate Constitutive Theories for Non-Classical Thermoviscoelastic Fluids with Internal Rotation Rates. Applied Mathematics, 9, 907-939. https://doi.org/10.4236/am.2018.98063
[16]
Surana, K.S., Joy, A.D. and Reddy, J.N. (2018) Ordered Rate Constitutive Theories for Thermoviscoelastic Solids without Memory Incorporating Internal and Cosserat Rotations. Acta Mechanica, 229, 3189-3213. https://doi.org/10.1007/s00707-018-2163-x
[17]
Surana, K.S. and Carranza, C.H. (2020) Dynamic Behavior of Thermoelastic Solid Continua Using Mathematical Model Derived Based on Non-Classical Continuum Mechanics with Internal Rotations. Meccanica, 56, 1345-1375. https://doi.org/10.1007/s11012-020-01221-2
[18]
Surana, K.S. and Kendall, J.K. (2020) Existence of Rotational Waves in Non-Classical Thermoelastic Solid Continua Incorporating Internal Rotations. Continuum Mechanics and Thermodynamics, 32, 1659-1683. https://doi.org/10.1007/s00161-020-00872-6
[19]
Surana, K.S. and Long, S.W. (2020) Ordered Rate Constitutive Theories for Non-Classical Thermofluids Based on Convected Time Derivatives of the Strain and Higher Order Rotation Rate Tensors Using Entropy Inequality. Entropy, 22, Article 443. https://doi.org/10.3390/e22040443
[20]
Surana, K.S. and Carranza, C.H. (2022) Nonclassical Continuum Theories for Fluent Media Incorporating Rotation Rates and Their Thermodynamic Consistency. ZAMM—Journal of Applied Mathematics and Mechanics/Zeitschrift für AngewandteMathematik und Mechanik, 103, e202200079. https://doi.org/10.1002/zamm.202200079
[21]
Surana, K.S. and Mathi, S.S.C. (2022) Thermodynamic Consistency of Nonclassical Continuum Theories for Solid Continua Incorporating Rotations. Continuum Mechanics and Thermodynamics, 35, 17-59. https://doi.org/10.1007/s00161-022-01163-y
Surana, K.S. and Kendall, J.K. (2023) NCCT for Micropolar Solid and Fluid Media Based on Internal Rotations and Rotation Rates with Rotational Inertial Physics: Model Problem Studies. Applied Mathematics, 14, 612-651. https://doi.org/10.4236/am.2023.149037
[24]
Eringen, A.C. (1964) Simple microfluids. International Journal of Engineering Science, 2, 205-217. https://doi.org/10.1016/0020-7225(64)90005-9
[25]
Eringen, A.C. (1966) Mechanics of Micromorphic Materials. In: Görtler, H., Ed., Applied Mechanics, Springer, 131-138. https://doi.org/10.1007/978-3-662-29364-5_12
[26]
Eringen, A. (1966) Theory of Micropolar Fluids. Indiana University Mathematics Journal, 16, 1-18. https://doi.org/10.1512/iumj.1967.16.16001
[27]
Eringen, A.C. (1968) Mechanics of Micromorphic Continua. In: Kröner, E., Ed., Mechanics of Generalized Continua, Springer, 18-35. https://doi.org/10.1007/978-3-662-30257-6_2
[28]
Eringen, A.C. (1967) Linear Theory of Micropolar Viscoelasticity. International Journal of Engineering Science, 5, 191-204. https://doi.org/10.1016/0020-7225(67)90004-3
[29]
Eringen, A.C. (1968) Theory of Micropolar Elasticity. In: Liebowitz, H., Ed., Fracture, Academic Press, 621-729.
[30]
Eringen, A.C. (1969) Micropolar Fluids with Stretch. International Journal of Engineering Science, 7, 115-127. https://doi.org/10.1016/0020-7225(69)90026-3
[31]
Bringen, A.C. (1970) Balance Laws of Micromorphic Mechanics. International Journal of Engineering Science, 8, 819-828. https://doi.org/10.1016/0020-7225(70)90084-4
[32]
Eringen, A.C. (1972) Theory of Thermomicrofluids. Journal of Mathematical Analysis and Applications, 38, 480-496. https://doi.org/10.1016/0022-247x(72)90106-0
[33]
Eringen, A.C. (1978) Micropolar Theory of Liquid Crystals. In: Johnson, J.F. and Porter, R.S., Eds., Liquid Crystals and Ordered Fluids, Springer, 443-474. https://doi.org/10.1007/978-1-4615-8888-7_30
[34]
Eringen, A.C. (1990) Theory of Thermo-Microstretch Fluids and Bubbly Liquids. International Journal of Engineering Science, 28, 133-143. https://doi.org/10.1016/0020-7225(90)90063-o
[35]
Eringen, A.C. (1992) Balance Laws of Micromorphic Continua Revisited. International Journal of Engineering Science, 30, 805-810. https://doi.org/10.1016/0020-7225(92)90109-t
[36]
Eringen, A.C. (1992) Continuum Theory of Microstretch Liquid Crystals. Journal of Mathematical Physics, 33, 4078-4086. https://doi.org/10.1063/1.529859
[37]
Eringen, A.C. (1966) A Unified Theory of Thermomechanical Materials. International Journal of Engineering Science, 4, 179-202. https://doi.org/10.1016/0020-7225(66)90022-x
[38]
Eringen, A.C. (1972) Theory of Micromorphic Materials with Memory. International Journal of Engineering Science, 10, 623-641. https://doi.org/10.1016/0020-7225(72)90089-4
[39]
Eringen, A.C. (1999) Microcontinuum Field Theories I. Foundations and Solids. Springer.
[40]
Eringen, A. and Ryan, M. (2002) Microcontinuum Field Theories II: Fluent Media. Applied Mechanics Reviews, 55, B15. https://doi.org/10.1115/1.1445333
[41]
Toupin, R.A. (1962) Elastic Materials with Couple-stresses. Archive for Rational Mechanics and Analysis, 11, 385-414. https://doi.org/10.1007/bf00253945
[42]
Eremeyev, V.A. and Pietraszkiewicz, W. (2015) Material Symmetry Group and Constitutive Equations of Micropolar Anisotropic Elastic Solids. Mathematics and Mechanics of Solids, 21, 210-221. https://doi.org/10.1177/1081286515582862
[43]
Eremeyev, V.A., Lebedev, L.P. and Altenbach, H. (2013) Foundations of Micropolar Mechanics. Springer.
[44]
Yang, F., Chong, A.C.M., Lam, D.C.C. and Tong, P. (2002) Couple Stress Based Strain Gradient Theory for Elasticity. International Journal of Solids and Structures, 39, 2731-2743. https://doi.org/10.1016/s0020-7683(02)00152-x
[45]
Smith, G.F. (1965) On Isotropic Integrity Bases. Archive for Rational Mechanics and Analysis, 18, 282-292. https://doi.org/10.1007/bf00251667
[46]
Smith, G.F. (1970) On a Fundamental Error in Two Papers of C.-C. Wang “on Representations for Isotropic Functions, Parts I and II”. Archive for Rational Mechanics and Analysis, 36, 161-165. https://doi.org/10.1007/bf00272240
[47]
Smith, G.F. (1971) On Isotropic Functions of Symmetric Tensors, Skew-Symmetric Tensors and Vectors. International Journal of Engineering Science, 9, 899-916. https://doi.org/10.1016/0020-7225(71)90023-1
[48]
Spencer, A.J.M. (1971) Theory of Invariants. In: Eringen, A.C., Ed., Mathematics, Elsevier, 239-353. https://doi.org/10.1016/b978-0-12-240801-4.50008-x
[49]
Spencer, A.J.M. and Rivlin, R.S. (1958) The Theory of Matrix Polynomials and Its Application to the Mechanics of Isotropic Continua. Archive for Rational Mechanics and Analysis, 2, 309-336. https://doi.org/10.1007/bf00277933
[50]
Spencer, A.J.M. and Rivlin, R.S. (1959) Further Results in the Theory of Matrix Polynomials. Archive for Rational Mechanics and Analysis, 4, 214-230. https://doi.org/10.1007/bf00281388
[51]
Wang, C.C. (1969) On Representations for Isotropic Functions, Part I. Archive for Rational Mechanics and Analysis, 33, 249-267. https://doi.org/10.1007/bf00281278
[52]
Wang, C.C. (1969) On Representations for Isotropic Functions, Part II. Archive for Rational Mechanics and Analysis, 33, 268-287. https://doi.org/10.1007/bf00281279
[53]
Wang, C.C. (1970) A New Representation Theorem for Isotropic Functions: An Answer to Professor G. F. Smith’s Criticism of My Papers on Representations for Isotropic Functions. Archive for Rational Mechanics and Analysis, 36, 166-197. https://doi.org/10.1007/bf00272241
[54]
Wang, C.C. (1971) Corrigendum to My Recent Papers on “Representations for Isotropic Functions”. Archive for Rational Mechanics and Analysis, 43, 392-395. https://doi.org/10.1007/bf00252004
[55]
Zheng, Q.S. (1993) On the Representations for Isotropic Vector-Valued, Symmetric Tensor-Valued and Skew-Symmetric Tensor-Valued Functions. International Journal of Engineering Science, 31, 1013-1024. https://doi.org/10.1016/0020-7225(93)90109-8
[56]
Zheng, Q.S. (1993) On Transversely Isotropic, Orthotropic and Relative Isotropic Functions of Symmetric Tensors, Skew-Symmetric Tensors and Vectors. Part I: Two Dimensional Orthotropic and Relative Isotropic Functions and Three Dimensional Relative Isotropic Functions. International Journal of Engineering Science, 31, 1399-1409. https://doi.org/10.1016/0020-7225(93)90005-f
[57]
Surana, K.S. (2015) Advanced Mechanics of Continua. CRC/Taylor and Francis.
[58]
Surana, K.S. (2021) Classical Continuum Mechanics, Second Edition. CRC/Taylor and Francis
