Derivative of a function of a nonsymmetric second-order tensor
Authors:
B. Balendran and Sia Nemat-Nasser
Journal:
Quart. Appl. Math. 54 (1996), 583-600
MSC:
Primary 73B05
DOI:
https://doi.org/10.1090/qam/1402412
MathSciNet review:
MR1402412
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Abstract: Exact explicit expressions are obtained for an isotropic tensor-valued function of a nonsymmetric second-order tensor, and its derivative, without resort to eigenvector calculations. These are then used to derive explicit expressions for the material time derivative of the general strain measures in terms of the deformation rate tensor.
B. Balendran and S. Nemat-Nasser, Integration of inelastic constitutive equations for constant velocity gradient with large rotation, Appl. Math. Comput. 67, 161–195 (1995)
D. E. Carlson and A. Hoger, The derivative of a tensor-valued function of a tensor, Quart. Appl. Math. XLIV, no. 3, 409–423 (1986)
Z.-h. Guo, Th. Lehmann, H. Liang, and C.-s. Man, Twirl tensors and the tensor equation AX — XA = C, J. Elasticity 27, 227–245 (1992)
M. E. Gurtin and K. Spear, On the relationship between the logarithmic strain rate and the stretching tensor, Internat. J. Solids and Structures 19, no. 5, 437–444 (1983)
R. Hill, On constitutive inequalities for simple materials. I, J. Mech. Phys. Solids 16, 229 (1968)
R. Hill, Constitutive inequalities for isotropic elastic solids under finite strain, Proc. Roy. Soc. London Ser. A 314, 457–472 (1970)
R. Hill, Aspects of invariance in solid mechanics, Adv. in Appl. Mech. 18, 1–75 (1978)
A. Hoger, The material time derivative of logarithmic strain, Internat. J. Solids and Structures 22, no. 9, 1019–1032 (1986)
M. M. Mehrabadi and S. Nemat-Nasser, Some basic kinematical relations for finite deformations of continua, Mech. Materials 6, 127–138 (1987)
R. S. Rivlin, Further remarks on the stress-deformation relations for isotropic materials, J. Rational Mech. Anal. 4, 681–701 (1955)
M. Scheidler, Time rates of generalized strain tensors, Part I: Component formulas, Mech. Materials 11, 199–210 (1991)
M. Scheidler, Time rates of generalized strain tensors, Part II: Approximate basis-free formulas, Mech. Materials 11, 211–219 (1991)
M. Scheidler, The tensor equation $AX + XA = \Phi \left ( A, H \right )$, with applications to kinematics of continua, J. Elasticity 36, 117–153 (1994)
T. C. T. Ting, Determination of ${C^{1/2}}, {C^{ - 1/2}}$ and more general isotropic tensor functions of C, J. Elasticity 15, 319–323 (1985)
B. Balendran and S. Nemat-Nasser, Integration of inelastic constitutive equations for constant velocity gradient with large rotation, Appl. Math. Comput. 67, 161–195 (1995)
D. E. Carlson and A. Hoger, The derivative of a tensor-valued function of a tensor, Quart. Appl. Math. XLIV, no. 3, 409–423 (1986)
Z.-h. Guo, Th. Lehmann, H. Liang, and C.-s. Man, Twirl tensors and the tensor equation AX — XA = C, J. Elasticity 27, 227–245 (1992)
M. E. Gurtin and K. Spear, On the relationship between the logarithmic strain rate and the stretching tensor, Internat. J. Solids and Structures 19, no. 5, 437–444 (1983)
R. Hill, On constitutive inequalities for simple materials. I, J. Mech. Phys. Solids 16, 229 (1968)
R. Hill, Constitutive inequalities for isotropic elastic solids under finite strain, Proc. Roy. Soc. London Ser. A 314, 457–472 (1970)
R. Hill, Aspects of invariance in solid mechanics, Adv. in Appl. Mech. 18, 1–75 (1978)
A. Hoger, The material time derivative of logarithmic strain, Internat. J. Solids and Structures 22, no. 9, 1019–1032 (1986)
M. M. Mehrabadi and S. Nemat-Nasser, Some basic kinematical relations for finite deformations of continua, Mech. Materials 6, 127–138 (1987)
R. S. Rivlin, Further remarks on the stress-deformation relations for isotropic materials, J. Rational Mech. Anal. 4, 681–701 (1955)
M. Scheidler, Time rates of generalized strain tensors, Part I: Component formulas, Mech. Materials 11, 199–210 (1991)
M. Scheidler, Time rates of generalized strain tensors, Part II: Approximate basis-free formulas, Mech. Materials 11, 211–219 (1991)
M. Scheidler, The tensor equation $AX + XA = \Phi \left ( A, H \right )$, with applications to kinematics of continua, J. Elasticity 36, 117–153 (1994)
T. C. T. Ting, Determination of ${C^{1/2}}, {C^{ - 1/2}}$ and more general isotropic tensor functions of C, J. Elasticity 15, 319–323 (1985)
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© Copyright 1996
American Mathematical Society
