Constitutive Equations

  • Victor A. Eremeyev
  • Leonid P. Lebedev
  • Holm Altenbach
Part of the SpringerBriefs in Applied Sciences and Technology book series (BRIEFSAPPLSCIENCES)


For an arbitrary part of the body, Eqs. (3.30) and (3.31) express the balance equations for the moment and the moment of momentum. These six scalar equations contain 18 unknown quantities that are the components of tensors \(\mathbf{ T} \) and \(\mathbf{ M} \). The dependence of \(\mathbf{ T} \) and \(\mathbf{ M} \) on medium deformations is determined by the constitutive equations or constitutive relations that depend on the material properties. They are determined experimentally. The constitutive equations must obey some principles that restrict their form, see [1].


Strain Energy Density Reference Configuration Micropolar Fluid Cosserat Continuum Micropolar Elasticity 
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  1. 1.
    C. Truesdell, W. Noll, The nonlinear field theories of mechanics. in Handbuch der Physik, vol. III/3, ed. by S. Flügge (Springer, Berlin, 1965), pp. 1–602Google Scholar
  2. 2.
    A.C. Eringen, Nonlocal Continuum Field Theories (Springer, New York, 2002)MATHGoogle Scholar
  3. 3.
    C.B. Kafadar, A.C. Eringen, Micropolar media—I. The classical theory. Int. J. Eng. Sci. 9(3), 271–305 (1971)MATHCrossRefGoogle Scholar
  4. 4.
    K.C. Le, H. Stumpf, Strain measures, integrability condition and frame indifference in the theory of oriented media. Int. J. Solids Struct. 35(9–10), 783–798 (1998)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    W. Noll, A mathematical theory of the mechanical behavior of continuous media. Arch. Ration. Mech. Anal. 2(1), 197–226 (1958)MATHCrossRefGoogle Scholar
  6. 6.
    A.I. Murdoch, On objectivity and material symmetry for simple elastic solids. J. Elast. 60(3), 233–242 (2000)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    A.I. Murdoch, Objectivity in classical continuum physics: a rationale for discarding the ‘principle of invariance under superposed rigid body motions’ in favour of purely objective considerations. Continuum Mech. Thermodyn. 15(3), 309–320 (2003)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    A.I. Murdoch, On criticism of the nature of objectivity in classical continuum physics. Continuum Mech. Thermodyn. 17(2), 135–148 (2005)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    W. Muschik, L. Restuccia, Changing the observer and moving materials in continuum physics: objectivity and frame-indifference. Technische Mechanik 22(2), 152–160 (2002)Google Scholar
  10. 10.
    R.S. Rivlin, Material symmetry revisited. GAMM Mitteilungen 25(1/2), 109–126 (2002)MathSciNetMATHGoogle Scholar
  11. 11.
    R.S. Rivlin, Frame indifference and relative frame indifference. Math. Mech. Solids 10(2), 145–154 (2005)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    R.S. Rivlin, Some thoughts on frame indifference. Math. Mech. Solids 11(2), 113–122 (2006)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    A. Bertram, B. Svendsen, On material objectivity and reduced constitutive equations. Arch. Mech. 53(6), 653–675 (2001)MathSciNetMATHGoogle Scholar
  14. 14.
    A. Bertram, B. Svendsen, Reply to Rivlin’s material symmetry revisited or much ado about nothing. GAMM Mitteilungen 27(1), 88–93 (2004)MathSciNetMATHGoogle Scholar
  15. 15.
    B. Svendsen, A. Bertram, On frame-indifference and form-invariance in constitutive theory. Acta Mechanica 132(1–4), 195–207 (1999)MathSciNetCrossRefGoogle Scholar
  16. 16.
    A. Bertram, Elasticity and Plasticity of Large Deformations: An Introduction, 2nd edn. (Springer, Berlin, 2008)MATHGoogle Scholar
  17. 17.
    R.P. Feynman, R.B. Leighton, M. Sands, The Feynman Lectures on Physics, vol. 1, 6th edn. (Addison-Wesley, Reading, 1977)Google Scholar
  18. 18.
    W. Pietraszkiewicz, V.A. Eremeyev, On natural strain measures of the non-linear micropolar continuum. Int. J. Solids Struct. 46(3–4), 774–787 (2009)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    W. Pietraszkiewicz, V.A. Eremeyev, On vectorially parameterized natural strain measures of the non-linear Cosserat continuum. Int. J. Solids Struct. 46(11–12), 2477–2480 (2009)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    E. Cosserat, F. Cosserat, Théorie des corps déformables (Herman et Fils, Paris, 1909)Google Scholar
  21. 21.
    T. Merlini, A variational formulation for finite elasticity with independent rotation and biot-axial fields. Comput. Mech. 19(3), 153–168 (1997)MATHCrossRefGoogle Scholar
  22. 22.
    L.M. Zubov, V.A. Eremeev, Equations for a viscoelastic micropolar fluid. Doklady Phys. 41(12), 598–601 (1996)MathSciNetMATHGoogle Scholar
  23. 23.
    V.A. Yeremeyev, L.M. Zubov, The theory of elastic and viscoelastic micropolar liquids. J. Appl. Math. Mech. 63(5), 755–767 (1999)MathSciNetCrossRefGoogle Scholar
  24. 24.
    J. Chróscielewski, J. Makowski, W. Pietraszkiewicz, Statics and dynamics of multyfolded shells. in Nonlinear Theory and Finite Elelement Method (in Polish) (Wydawnictwo IPPT PAN, Warszawa, 2004)Google Scholar
  25. 25.
    V.A. Eremeyev, W. Pietraszkiewicz, Local symmetry group in the general theory of elastic shells. J. Elast. 85(2), 125–152 (2006)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    V.A. Eremeyev, W. Pietraszkiewicz, Material symmetry group of the non-linear polar-elastic continuum. Int. J. Solids Struct. 49(14), 1993–2005 (2012)Google Scholar
  27. 27.
    E. Reissner, On kinematics and statics of finite-strain force and moment stress elasticity. Stud. Appl. Math. 52, 93–101 (1973)Google Scholar
  28. 28.
    E. Reissner, Note on the equations of finite-strain force and moment stress elasticity. Stud. Appl. Math. 54, 1–8 (1975)MathSciNetMATHGoogle Scholar
  29. 29.
    R. Stojanović, Nonlinear micropolar elasticity, in Micropolar Elasticity, vol. 151, ed. by W. Nowacki, W. Olszak (Springer, Wien, 1974), pp. 73–103Google Scholar
  30. 30.
    L.M. Zubov, Variational principles and invariant integrals for non-linearly elastic bodies with couple stresses (in Russian). Mech. Solids (6), 10–16 (1990)Google Scholar
  31. 31.
    V.A. Eremeyev, L.M. Zubov, On the stability of elastic bodies with couple stresses. Mech. Solids 29(3), 172–181 (1994)Google Scholar
  32. 32.
    L.M. Zubov, Nonlinear Theory of Dislocations and Disclinations in Elastic Bodies (Springer, Berlin, 1997)MATHGoogle Scholar
  33. 33.
    E. Nikitin, L.M. Zubov, Conservation laws and conjugate solutions in the elasticity of simple materials and materials with couple stress. J. Elast. 51(1), 1–22 (1998)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    O.A. Bauchau, L. Trainelli, The vectorial parameterization of rotation. Nonlinear Dyn. 32(1), 71–92 (2003)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    J. Dyszlewicz, Micropolar Theory of Elasticity (Springer, Berlin, 2004)MATHCrossRefGoogle Scholar
  36. 36.
    A.C. Eringen, Microcontinuum Field Theory I. Foundations and Solids (Springer, New York, 1999)Google Scholar
  37. 37.
    W. Nowacki, Theory of Asymmetric Elasticity (Pergamon-Press, Oxford, 1986)Google Scholar
  38. 38.
    C. Truesdell, Die Entwicklung des Drallsatzes. ZAMM 44(4/5), 149–158 (1964)MathSciNetMATHGoogle Scholar
  39. 39.
    C. Truesdell, A First Course in Rational Continuum Mechanics (Academic Press, New York, 1977)MATHGoogle Scholar
  40. 40.
    C.C. Wang, C. Truesdell, Introduction to Rational Elasticity (Noordhoof Int. Publishing, Leyden, 1973)MATHGoogle Scholar
  41. 41.
    R.S. Rivlin, Material symmetry and constitutive equations. Ingenieur-Archiv 49(5–6), 325–336 (1980)MATHCrossRefGoogle Scholar
  42. 42.
    A.C. Eringen, C.B. Kafadar, Polar field theories. in Continuum Physics, vol. IV, ed. by A.C. Eringen, (Academic Press, New York 1976), pp. 1–75Google Scholar
  43. 43.
    A.J.M. Spencer, Isotropic integrity bases for vectors and second-order tensors. Part II. Arch. Ration. Mech. Anal. 18(1), 51–82 (1965)MATHCrossRefGoogle Scholar
  44. 44.
    A.J.M. Spencer, Theory of invariants, in Continuum Physics, vol. 1, ed. by A.C. Eringen (Academic Press, New-York, 1971), pp. 239–353Google Scholar
  45. 45.
    Q.S. Zheng, Theory of representations for tensor functions—a unified invariant approach to constitutive equations. Appl. Mech. Rev. 47(11), 545–587 (1994)CrossRefGoogle Scholar
  46. 46.
    S. Ramezani, R. Naghdabadi, S. Sohrabpour, Constitutive equations for micropolar hyper-elastic materials. Int. J. Solids Struct. 46(14–15), 2765–2773 (2009)MATHCrossRefGoogle Scholar
  47. 47.
    Q.S. Zheng, A.J.M. Spencer, On the canonical representations for Kronecker powers of orthogonal tensors with application to material symmetry problems. Int. J. Eng. Sci. 31(4), 617–635 (1993)MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    H. Xiao, On symmetries and anisotropies of classical and micropolar linear elasticities: a new method based upon a complex vector basis and some systematic results. J. Elast. 49(2), 129–162 (1998)MATHCrossRefGoogle Scholar
  49. 49.
    A.I. Lurie, Theory of Elasticity (Springer, Berlin, 2005)CrossRefGoogle Scholar
  50. 50.
    R.W. Ogden, Non-linear Elastic Deformations (Ellis Horwood, Chichester, 1984)Google Scholar
  51. 51.
    A.I. Lurie, Nonlinear Theory of Elasticity (North-Holland, Amsterdam, 1990)MATHGoogle Scholar
  52. 52.
    A.O. Varga, Stress-Strain Behavior of Elastic Materials: Selected Problems of Large Deformations (Interscience, New York, 1996)Google Scholar
  53. 53.
    C. Truesdell, Rational Thermodynamics, 2nd edn. (Springer, New York, 1984)MATHCrossRefGoogle Scholar
  54. 54.
    J.K. Knowles, E. Sternberg, On the ellipticity of the equation of nonlinear elastostatics for a special material. J. Elast. 5(3–4), 341–361 (1975)MathSciNetMATHCrossRefGoogle Scholar
  55. 55.
    J.K. Knowles, E. Sternberg, On the failure of ellipticity and the emergence of discontinuous deformation gradients in plane finite elastostatics. J. Elast. 10, 255–293 (1980)MathSciNetCrossRefGoogle Scholar
  56. 56.
    P. Rosakis, Ellipticity and deformation with discontinuous gradients in finite elastostatics. Arch. Ration. Mech. Anal. 109(1), 1–37 (1990)MathSciNetMATHCrossRefGoogle Scholar
  57. 57.
    L. Zee, E. Sternberg, Ordinary and strong ellipticity in the equilibrium theory of incompressible hyperelastic solids. Arch. Ration. Mech. Anal. 83(1), 53–90 (1983)MathSciNetMATHCrossRefGoogle Scholar
  58. 58.
    J.L. Lions, E. Magenes, Problèmes aux limites non homogènes et applications (Dunod, Paris, 1968)MATHGoogle Scholar
  59. 59.
    G. Fichera, Existence theorems in elasticity. in Handbuch der Physik, vol. VIa/2, ed. by S. Flügge, (Springer, Berlin 1972), pp. 347–389Google Scholar
  60. 60.
    L. Hörmander, Linear Partial Differential Equations, A Series of Comprehensive Studies in Mathematics, vol. 116, 4th edn. (Springer, Berlin, 1976)Google Scholar
  61. 61.
    M. Agranovich, Elliptic boundary problems. in Partial Differential Equations IX: Elliptic Boundary Problems. Encyclopaedia of Mathematical Sciences, vol. 79, ed. by M. Agranovich, Y. Egorov, M. Shubin (Springer, Berlin, 1997), pp. 1–144Google Scholar
  62. 62.
    L. Nirenberg, Topics in Nonlinear Functional Analysis (American Mathematical Society, New York, 2001)MATHGoogle Scholar
  63. 63.
    E.L. Aero, A.N. Bulygin, E.V. Kuvshinskii, Asymmetric hydromechanics. J. Appl. Math. Mech. 29(2), 333–346 (1965)MATHCrossRefGoogle Scholar
  64. 64.
    A.C. Eringen, Theory of micropolar fluids. J. Math. Mech. 16(1), 1–18 (1966)MathSciNetGoogle Scholar
  65. 65.
    N.P. Migoun, P.P. Prokhorenko, Hydrodynamics and Heattransfer in Gradient Flows of Microstructured Fluids (in Russian) (Nauka i Technika, Minsk, 1984)Google Scholar
  66. 66.
    G. Łukaszewicz, Micropolar Fluids: Theory and Applications (Birkhäuser, Boston, 1999)MATHCrossRefGoogle Scholar
  67. 67.
    A.C. Eringen, Microcontinuum Field Theory. II. Fluent Media (Springer, New York, 2001)Google Scholar
  68. 68.
    A.C. Eringen, A unified continuum theory of liquid crystals. ARI Int. J. Phys. Eng. Sci. 73–84(2), 369–374 (1997)Google Scholar
  69. 69.
    A.C. Eringen, A unified continuum theory of electrodynamics of liquid crystals. Int. J. Eng. Sci. 35(12–13), 1137–1157 (1997)MathSciNetMATHCrossRefGoogle Scholar
  70. 70.
    A.C. Eringen, A unified continuum theory for electrodynamics of polymeric liquid crystals. Int. J. Eng. Sci. 38(9–10), 959–987 (2000)MathSciNetMATHCrossRefGoogle Scholar
  71. 71.
    E.A. Ivanova, A.M. Krivtsov, N.F. Morozov, A.D. Firsova, Description of crystal packing of particles with torque interaction. Mech. Solids 38(4), 76–88 (2003)Google Scholar
  72. 72.
    R.S. Lakes, Experimental microelasticity of two porous solids. Int. J. Solids Struct. 22(1), 55–63 (1986)CrossRefGoogle Scholar
  73. 73.
    R.S. Lakes, Experimental micro mechanics methods for conventional and negative Poisson’s ratio cellular solids as Cosserat continua. Trans. ASME J. Eng. Mater. Technol. 113(1), 148–155 (1991)CrossRefGoogle Scholar
  74. 74.
    R.S. Lakes, Experimental methods for study of Cosserat elastic solids and other generalized continua. in Continuum Models for Materials with Micro-Structure, ed. by H. Mühlhaus (Wiley, New York, 1995), pp. 1–22Google Scholar
  75. 75.
    H.C. Park, R.S. Lakes, Cosserat micromechanics of human bone: strain redistribution by a hydration-sensitive constituent. J. Biomech. 19(5), 385–397 (1986)CrossRefGoogle Scholar
  76. 76.
    J.F.C. Yang, R.S. Lakes, Experimental study of micropolar and couple stress elasticity in compact bone in bending. J. Biomech. 15(2), 91–98 (1982)CrossRefGoogle Scholar
  77. 77.
    R.D. Gauthier, W.E. Jahsman, Quest for micropolar elastic-constants. Trans. ASME J. Appl. Mech. 42(2), 369–374 (1975)CrossRefGoogle Scholar
  78. 78.
    R.D. Gauthier, W.E. Jahsman, Quest for micropolar elastic-constants. 2. Arch. Mech. 33(5), 717–737 (1981)MATHGoogle Scholar
  79. 79.
    R. Mora, A.M. Waas, Measurement of the Cosserat constant of circular-cell polycarbonate honeycomb. Philos. Mag. A. Phys. Condens. Matter Struct. Defects Mech. Prop. 80(7), 1699–1713 (2000)Google Scholar
  80. 80.
    V.I. Erofeev, Wave Processes in Solids with Microstructure (World Scientific, Singapore, 2003)Google Scholar
  81. 81.
    P. Neff, J. Jeong, A new paradigm: the linear isotropic Cosserat model with conformally invariant curvature energy. ZAMM 89(2), 107–122 (2009)MathSciNetMATHCrossRefGoogle Scholar
  82. 82.
    J. Jeong, P. Neff, Existence, uniqueness and stability in linear Cosserat elasticity for weakest curvature conditions. Math. Mech. Solids 15(1), 78–95 (2010)MathSciNetMATHCrossRefGoogle Scholar
  83. 83.
    P. Neff, J. Jeong, A. Fischle, Stable identification of linear isotropic Cosserat parameters: bounded stiffness in bending and torsion implies conformal invariance of curvature. Acta Mechanica 211(3–4), 237–249 (2010)MATHCrossRefGoogle Scholar
  84. 84.
    D. Besdo, Towards a Cosserat-theory describing motion of an originally rectangular structure of blocks. Arch. Appl. Mech. 80(1), 25–45 (2010)MATHCrossRefGoogle Scholar
  85. 85.
    D. Bigoni, W.J. Drugan, Analytical derivation of Cosserat moduli via homogenization of heterogeneous elastic materials. Trans. ASME J. Appl. Mech. 74(4), 741–753 (2007)MathSciNetCrossRefGoogle Scholar
  86. 86.
    S. Diebels, A micropolar theory of porous media: constitutive modelling. Transp. Porous Media 34(1–3), 193–208 (1999)CrossRefGoogle Scholar
  87. 87.
    F. Dos Reis, J.F. Ganghoffer, Construction of micropolar continua from the homogenization of repetitive planar lattices. in Mechanics of Generalized Continua, Advanced Structured Materials, vol. 7, ed. by H. Altenbach, G.A. Maugin, V. Erofeev (Springer, Berlin, 2011), pp. 193–217Google Scholar
  88. 88.
    W. Ehlers, E. Ramm, S. Diebels, G.D.A. d’Addetta, From particle ensembles to Cosserat continua: homogenization of contact forces towards stresses and couple stresses. Int. J. Solids Struct. 40(24), 6681–6702 (2003)MathSciNetMATHCrossRefGoogle Scholar
  89. 89.
    S. Forest, Mechanics of generalized continua: construction by homogenizaton. J. de Physique IV France 8(PR4), Pr4-39–Pr4-48 (1998)Google Scholar
  90. 90.
    S. Forest, K. Sab, Cosserat overal modelling of heterogeneous materials. Mech. Res. Commun. 25(4), 449–454 (1998)MathSciNetMATHCrossRefGoogle Scholar
  91. 91.
    S. Forest, R. Sievert, Nonlinear microstrain theories. Int. J. Solids Struct. 43(24), 7224–7245 (2006)MathSciNetMATHCrossRefGoogle Scholar
  92. 92.
    A.S.J. Suiker, A.V. Metrikine, R. de Borst, Comparison of wave propagation characteristics of the Cosserat continuum model and corresponding discrete lattice models. Int. J. Solids Struct. 38(9), 1563–1583 (2001)MATHCrossRefGoogle Scholar
  93. 93.
    A.S.J. Suiker, R. de Borst, Enhanced continua and discrete lattices for modelling granular assemblies. Philos. Trans. Royal Soc. A 363(1836), 2543–2580 (2005)MATHCrossRefGoogle Scholar
  94. 94.
    R. Larsson, S. Diebels, A second-order homogenization procedure for multi-scale analysis based on micropolar kinematics. Int. J. Numer. Methods Eng. 69(12), 2485–2512 (2007)MathSciNetMATHCrossRefGoogle Scholar
  95. 95.
    R. Larsson, Y. Zhang, Homogenization of microsystem interconnects based on micropolar theory and discontinuous kinematics. J. Mech. Phys. Solids 55(4), 819–841 (2007)MathSciNetMATHCrossRefGoogle Scholar
  96. 96.
    E. Pasternak, H.B. Mühlhaus, Generalised homogenisation procedures for granular materials. J. Eng. Math. 52(1), 199–229 (2005)MATHCrossRefGoogle Scholar
  97. 97.
    O. van der Sluis, P.H.J. Vosbeek, P.J.G. Schreurs, H.E.H. Meijer, Homogenization of heterogeneous polymers. Int. J. Solids Struct. 36(21), 3193–3214 (1999)MATHCrossRefGoogle Scholar
  98. 98.
    S. Forest, K.T. Duy, Generalized continua and non-homogeneous boundary conditions in homogenisation methods. ZAMM 91(2), 90–109 (2011)MathSciNetMATHCrossRefGoogle Scholar
  99. 99.
    V. Kouznetsova, M.G.D. Geers, W.A.M. Brekelmans, Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme. Int. J. Numer. Methods Eng. 54(8), 1235–1260 (2002)MATHCrossRefGoogle Scholar
  100. 100.
    P. Neff, S. Forest, A geometrically exact micromorphic model for elastic metallic foams accounting for affine microstructure. Modelling, existence of minimizers, identification of moduli and computational results. J. Elast. 87(2–3), 239–276 (2007)MathSciNetMATHCrossRefGoogle Scholar
  101. 101.
    I. Cielecka, M. Wozniak, C. Wozniak, Elastodynamic behaviour of honeycomb cellular media. J. Elast. 60(1), 1–17 (2000)MATHCrossRefGoogle Scholar
  102. 102.
    P. Neff, K. Chełminski, W. Müller, C. Wieners, A numerical solution method for an infinitesimal elasto-plastic Cosserat model. Math. Models Methods Appl. Sci. 17(8), 1211–1239 (2007)MathSciNetMATHCrossRefGoogle Scholar
  103. 103.
    P. Neff, K. Chełminski, Well-posedness of dynamic Cosserat plasticity. Appl. Math. Optim. 56(1), 19–35 (2007)MathSciNetMATHCrossRefGoogle Scholar
  104. 104.
    M.B. Rubin, On the theory of a Cosserat point and its application to the numerical solution of continuum problems. Trans. ASME J. Appl. Mech. 52(2), 368–372 (1985)MATHCrossRefGoogle Scholar
  105. 105.
    M.B. Rubin, Cosserat Theories: Shells Rods and Points (Kluwer, Dordrecht, 2000)MATHGoogle Scholar
  106. 106.
    B. Nadler, M. Rubin, A new 3-D finite element for nonlinear elasticity using the theory of a Cosserat point. Int. J. Solids Struct. 40(17), 4585–4614 (2003)MATHCrossRefGoogle Scholar
  107. 107.
    M. Jabareen, M. Rubin, An improved 3-D brick Cosserat point element for irregular shaped elements. Comput. Mech. 40(6), 979–1004 (2007)MathSciNetMATHCrossRefGoogle Scholar
  108. 108.
    M. Jabareen, M. Rubin, Modified torsion coefficients for a 3-D brick Cosserat point element. Comput. Mech. 41(4), 517–525 (2008)MathSciNetMATHCrossRefGoogle Scholar
  109. 109.
    V.A. Eremeyev, L.M. Zubov, Principles of Viscoelastic Micropolar Fluid Mechanics (in Russian) (SSC of RASci Publishers, Rostov on Don, 2009)Google Scholar
  110. 110.
    R.E. Rosensweig, Magnetic fluids. Ann. Rev. Fluid Mech. 19, 437–461 (1987)CrossRefGoogle Scholar

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© The Author(s) 2013

Authors and Affiliations

  • Victor A. Eremeyev
    • 1
  • Leonid P. Lebedev
    • 2
  • Holm Altenbach
    • 3
  1. 1.Halle-WittenbergMartin-Luther-UniversitätHalleGermany
  2. 2.Universidad Nacional de ColombiaBogotáColombia
  3. 3.Fakultät für Maschinenbau, Lehrstuhl für Technische Mechanik, Institut für MechanikOtto-von-Guericke-Universität MagdeburgMagdeburgGermany

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