On compressible materials capable of sustaining axisymmetric shear deformations. III. Helical shear of isotropic hyperelastic materials
Authors:
Millard F. Beatty and Qing Jiang
Journal:
Quart. Appl. Math. 57 (1999), 681-697
MSC:
Primary 74B20
DOI:
https://doi.org/10.1090/qam/1724300
MathSciNet review:
MR1724300
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Abstract: A helical shear deformation is a composition of non-universal, axisymmetric, anti-plane shear and rotational shear deformations, shear states that are separately controllable only in special kinds of compressible and incompressible, homogeneous and isotropic hyperelastic materials. For incompressible materials, it is only necessary to identify a specific material, such as a Mooney-Rivlin material, to determine the anti-plane and rotational shear displacement functions. For compressible materials, however, these shear deformations may not be separately possible in the same specified class of hyperelastic materials unless certain auxiliary conditions on the strain energy function are satisfied. We have recently presented simple algebraic conditions necessary and sufficient in order that both anti-plane shear and rotational shear deformations may be separately possible in the same material subclass. In this paper, under the same physical condition that the shear response function be positive, we present an essentially algebraic condition necessary and sufficient to determine whether a class of compressible, homogeneous and isotropic hyperelastic materials is capable of sustaining controllable, helical shear deformations. It is then proved that helical shear deformations are possible in a specified hyperelastic material if and only if that material can separately sustain both axisymmetric, anti-plane shear and rotational shear deformations. The simplicity of the result in applications is illustrated in a few examples.
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A. J. M. Spencer, On finite elastic deformations with a perturbed strain energy function, Quart. J. Mech. Appl. Math. 12, 129–145 (1959)
R. S. Rivlin, Large elastic deformations of isotropic materials. VI - Further results in the theory of torsion, shear and flexure, Phil. Trans. Roy. Soc. London A242, 173–195 (1949)
D. M. Haughton, Circular shearing of compressible elastic cylinders, Quart. J. Mech. Appl. Math. 46, 471–486 (1993)
M. F. Beatty and Q. Jiang, On compressible materials capable of sustaining axisymmetric shear deformations. Part 2: Rotational shear of isotropic hyperelastic materials, Quart. J. Mech. Appl. Math. 50, 212–237 (1997)
X. Jiang and R. W. Ogden, On azimuthal shear of a circular cylindrical tube of compressible elastic material, Quart. J. Mech. Appl. Math. 51, 143–158 (1998)
Q. Jiang and M. F. Beatty, On compressible materials capable of sustaining axisymmetric shear deformations. Part 1: Anti-plane shear of isotropic hyperelastic materials, J. Elasticity 35, 75–95 (1995)
M. F. Beatty and Q. Jiang, Compressible, isotropic hyperelastic materials capable of sustaining axisymmetric, anti-plane shear deformations, Contemporary Research in the Mechanics and Mathematics of Materials, Eds. R. C. Batra and M. F. Beatty, International Center for Numerical Methods in Engineering (CIMNE), Barcelona, Spain, 1996
D. A. Polignone and C. O. Horgan, Axisymmetric finite anti-plane shear of compressible nonlinearly elastic circular tubes, Quart. Appl. Math. 50, 323-341 (1992)
D. A. Polignone and C. O. Horgan, Pure azimuthal shear of compressible nonlinearly elastic circular tubes, Quart. Appl. Math. 52, 113-131 (1994)
Y. C. Fung, On pseudo-elasticity of living tissue, Mechanics Today 5, 49-66. Ed. S. Nemat-Nasser, Pergamon Press, New York, 1980
J. D. Humphrey and F. C. P. Yin, On constitutive relations and finite deformations of passive cardiac tissue: (Part) I. A pseudostrain-energy function, J. Biomech. Engrg. 109, 298-304 (1987); (Part) II. Stress analysis in the left ventricle, Circulation Res. 65, 805–817 (1989)
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© Copyright 1999
American Mathematical Society