Deep Indentation of Soft Solids: A Limit State
Abstract
Indentation is a practical technique for probing the mechanical properties of soft materials such as elastomers and gels when standard tests are difficult, destructive, or simply cannot be performed in situ. Here, deep indentation of a half space made from an isotropic, incompressible one term Ogden hyperelastic solid is analysed for an indenter in the form of a rigid circular rod of radius R and that is either flat-bottomed or has a hemispherical tip. Numerical simulations reveal that a limit state is achieved when the indent depth D much exceeds the radius R of the rod and the indentation force F scales as F = \b{eta} E D^2, where \b{eta} is a dimensionless coefficient and E is the Young s modulus of the solid. The magnitude of \b{eta} increases with the degree of strain stiffening of the hyperelastic solid but is almost insensitive to the headshape of the rod and to the level of Coulomb friction at the interface with the solid. A high level of strain stiffening of the Ogden solid at finite strain implies that the level of strain at the indenter tip is small up to a large value of D/R, thereby protecting the solid from failure and penetration. Directly ahead of the indenter tip the contours of strain energy density are approximately spheroidal in shape and far ahead the Boussinesq solution is recovered. The limit state of deep indentation is compared to that of cavity expansion; whilst several qualitative features are shared, the two limit states differ in their kinematic fields and in their sensitivity to the degree of strain stiffening in the Ogden solid.
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BibTeX
@misc{shojaeifard2026deep,
title = {{Deep Indentation of Soft Solids: A Limit State}},
author = {Mohammad Shojaeifard and Zhenwei Ma and K Jimmy Hsia and Norman Fleck and Mattia Bacca},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.30971},
eprint = {2606.30971},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.30971},
}
RIS
TY - UNPB
TI - Deep Indentation of Soft Solids: A Limit State
AU - Mohammad Shojaeifard
AU - Zhenwei Ma
AU - K Jimmy Hsia
AU - Norman Fleck
AU - Mattia Bacca
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2606.30971
UR - https://arxiv.org/abs/2606.30971
AB - Indentation is a practical technique for probing the mechanical properties of soft materials such as elastomers and gels when standard tests are difficult, destructive, or simply cannot be performed in situ. Here, deep indentation of a half space made from an isotropic, incompressible one term Ogden hyperelastic solid is analysed for an indenter in the form of a rigid circular rod of radius R and that is either flat-bottomed or has a hemispherical tip. Numerical simulations reveal that a limit state is achieved when the indent depth D much exceeds the radius R of the rod and the indentation force F scales as F = \b{eta} E D^2, where \b{eta} is a dimensionless coefficient and E is the Young s modulus of the solid. The magnitude of \b{eta} increases with the degree of strain stiffening of the hyperelastic solid but is almost insensitive to the headshape of the rod and to the level of Coulomb friction at the interface with the solid. A high level of strain stiffening of the Ogden solid at finite strain implies that the level of strain at the indenter tip is small up to a large value of D/R, thereby protecting the solid from failure and penetration. Directly ahead of the indenter tip the contours of strain energy density are approximately spheroidal in shape and far ahead the Boussinesq solution is recovered. The limit state of deep indentation is compared to that of cavity expansion; whilst several qualitative features are shared, the two limit states differ in their kinematic fields and in their sensitivity to the degree of strain stiffening in the Ogden solid.
ER -