DOI: 10.1090/qam/69700. Tensors associated with time-dependent stress

Tensors associated with time-dependent stress

10.1090/qam/69700

Crossref journal-article

American Mathematical Society (AMS)

Quarterly of Applied Mathematics (14)

Abstract

It is assumed that six functional relations exist between the components of stress and their first m m material time derivatives and the gradients of displacement, velocity, acceleration, second acceleration, . . . , ( n − 1 ) \left ( {n - 1} \right ) th acceleration. It is shown that these relations may then be expressed as relations between the components of m + n + 2 m + n + 2 symmetric tensors if n m n \> m , and 2 m + 2 2m + 2 symmetric tensors if m n m \> n . Expressions for these tensors are obtained.

Bibliography

Cotter, B. A., & Rivlin, R. S. (1955). Tensors associated with time-dependent stress. Quarterly of Applied Mathematics, 13(2), 177â182.

Authors 2

Barbara A. Cotter (first)
R. S. Rivlin (additional)

References 3 Referenced 96

R. S. Rivlin and J. L. Ericksen, Stress-deformation relations for isotropic materials, J. Rational Mech. & Anal. (in the press)
S. Zaremba, Sur une conception nouvelle des forces intérieures dans un fluide en mouvement. Mém. Sci. Math. No. 82, Gauthier-Villars, Paris, 1937
10.1098/rspa.1950.0035 / Proc. Roy. Soc. London Ser. A / On the formulation of rheological equations of state by Oldroyd, J. G. (1950)

Dates

Type	When
Created	8 years, 8 months ago (Dec. 14, 2016, 5 p.m.)
Deposited	10 months, 1 week ago (Oct. 25, 2024, 11:50 a.m.)
Indexed	1 month, 1 week ago (July 30, 2025, 8:34 p.m.)
Issued	70 years, 2 months ago (July 1, 1955)
Published	70 years, 2 months ago (July 1, 1955)
Published Print	70 years, 2 months ago (July 1, 1955)

Funders 0

None

BibTeX

@article{Cotter_1955, title={Tensors associated with time-dependent stress}, volume={13}, ISSN={1552-4485}, url={http://dx.doi.org/10.1090/qam/69700}, DOI={10.1090/qam/69700}, number={2}, journal={Quarterly of Applied Mathematics}, publisher={American Mathematical Society (AMS)}, author={Cotter, Barbara A. and Rivlin, R. S.}, year={1955}, month=jul, pages={177–182} }

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  "abstract": "<p>It is assumed that six functional relations exist between the components of stress and their first <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"m\">\n  <mml:semantics>\n    <mml:mi>m</mml:mi>\n    <mml:annotation encoding=\"application/x-tex\">m</mml:annotation>\n  </mml:semantics>\n</mml:math>\n</inline-formula> material time derivatives and the gradients of displacement, velocity, acceleration, second acceleration, . . . , <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis n minus 1 right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mo>(</mml:mo>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mi>n</mml:mi>\n        <mml:mo>\u2212</mml:mo>\n        <mml:mn>1</mml:mn>\n      </mml:mrow>\n      <mml:mo>)</mml:mo>\n    </mml:mrow>\n    <mml:annotation encoding=\"application/x-tex\">\\left ( {n - 1} \\right )</mml:annotation>\n  </mml:semantics>\n</mml:math>\n</inline-formula>th acceleration. It is shown that these relations may then be expressed as relations between the components of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"m plus n plus 2\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>m</mml:mi>\n      <mml:mo>+</mml:mo>\n      <mml:mi>n</mml:mi>\n      <mml:mo>+</mml:mo>\n      <mml:mn>2</mml:mn>\n    </mml:mrow>\n    <mml:annotation encoding=\"application/x-tex\">m + n + 2</mml:annotation>\n  </mml:semantics>\n</mml:math>\n</inline-formula> symmetric tensors if <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n m\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>n</mml:mi>\n      <mml:mspace width=\"mediummathspace\"/>\n      <mml:mi>m</mml:mi>\n    </mml:mrow>\n    <mml:annotation encoding=\"application/x-tex\">n \\&gt; m</mml:annotation>\n  </mml:semantics>\n</mml:math>\n</inline-formula>, and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2 m plus 2\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mn>2</mml:mn>\n      <mml:mi>m</mml:mi>\n      <mml:mo>+</mml:mo>\n      <mml:mn>2</mml:mn>\n    </mml:mrow>\n    <mml:annotation encoding=\"application/x-tex\">2m + 2</mml:annotation>\n  </mml:semantics>\n</mml:math>\n</inline-formula> symmetric tensors if <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"m n\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>m</mml:mi>\n      <mml:mspace width=\"mediummathspace\"/>\n      <mml:mi>n</mml:mi>\n    </mml:mrow>\n    <mml:annotation encoding=\"application/x-tex\">m \\&gt; n</mml:annotation>\n  </mml:semantics>\n</mml:math>\n</inline-formula>. Expressions for these tensors are obtained.</p>",
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