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On the direct determination of the right stretch tensor and the admissible region of its principal invariants
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DOI：10.1016/j.ijsolstr.2026.113971.png)
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Abstract
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This paper addresses the admissible domains and visualization of deformation and stretch tensors along with their principal scalar invariants. A unified framework is presented for both compressible and incompressible materials, with particular emphasis on the invariant-based characterization of the stretch tensor, defined as the square root of the right Cauchy-Green deformation tensor. Existing approaches for the direct computation of the stretch tensor are reviewed, including formulations that provide explicit expressions in terms of its scalar invariants. Consequently, determining these invariants becomes essential. Available procedures for their computation are examined, leading to quartic equations associated with the first and second scalar invariants. The relationships between the invariants of the deformation and stretch tensors are further explored from a geometric perspective. The admissible invariant domains associated with the square root of the right Cauchy-Green tensor are characterized graphically, enabling an unambiguous selection of the physically relevant roots of the governing quartic equations. By exploiting the concept of attainable domains, the admissible region corresponding to the selected solution - represented in the plane spanned by the first and second scalar invariants of the stretch tensor - is visualized. The proposed approach also results in the definition of a closed attainable region, which facilitates the systematic analysis and visualization of invariant-based quantities, such as principal stretches and strain energy density functions, over the entire admissible domain.
Keywords:
Cauchy-Green deformation tensor
2nd order tensor
Square roots
Stretch tensor
Continuum mechanics
Cubic and quartic equations
Attainable regions
Visualization
Principal scalar invariants and stretches
Ogden, Rivlin, Yeoh and Neo Hookean strain energy density functions
Journal
IF:
3.8
Papers:
1.1W
Citations:
3.1W
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Cited Papers