Revisiting Viscoelastic Characterization: Surface Tension Effects in Spherical Indentation of Biological Materials

The mechanical behavior of biological materials (cells, tissues, and hydrogels) is important to understand how microstructural organization governs macroscopic function. Indentation testing, especially at the microscale, has become one of the most powerful techniques for probing viscoelastic properties because it requires only minimal sample preparation and yields local mechanical parameters with high precision. However, most classical indentation analyses, such as those derived from Hertzian or Sneddon’s contact theories, treat the material as purely elastic and ignore surface tension. This approach can lead to inaccurate characterization in soft biological systems when contact radii approach micrometer scales.

The main challenge is viscoelastic indentation, one infers elastic moduli from load–depth relations that implicitly assumption that surface tension is absent. In reality, the deformation and contact behavior of cells and tissues are significantly influenced by surface tension, which also affects the apparent relaxation or creep behavior measured experimentally. Ignoring surface tension leads to systematic misestimations of viscoelastic parameters—often interpreting the material as stiffer or more compliant than it truly is, depending on the experiment type. Despite numerous studies on contact mechanics incorporating viscoelasticity or surface elasticity independently, few have attempted to couple these effects within a unified framework capable of quantifying their interplay. To this account, new research paper published in Mechanics of Materials  and conducted by Professor Yue Ding, Professor Wei-Ke Yuan, Professor Xuan-Ming Liang and Professor Gang-Feng Wang from the Xi’an Jiaotong University alongside from the Professor Xinrui Niu from the University of Hong Kong, the researchers developed two new interrelated models: a finite element framework that couples the standard linear solid viscoelastic law with constant surface tension, and an analytical load–depth relation that extends the classical Hertzian theory through an intrinsic length parameter.

The researchers first implemented a finite element model of a rigid spherical indenter pressing into a viscoelastic half-space governed by the standard linear solid law, parameterized by instantaneous modulus , long-term modulus , and relaxation time . Surface tension was incorporated as a constant surface energy term using a user-defined subroutine in ABAQUS, which modified nodal forces and stiffness matrices of surface elements. The indenter radius was 10 µm, and material parameters spanned kPa, , and s—values typical of soft tissues and cell aggregates. Simulations were conducted for both relaxation (constant displacement) and creep (constant load) protocols, with surface tensions from 0 to 0.05 N m⁻¹. When the team compared apparent moduli derived from indentation data with and without surface tension, a striking asymmetry emerged. During relaxation, surface tension elevated the apparent relaxation modulus throughout the time course, making the material seem stiffer and slowing its stress decay. Conversely, in creep tests, surface tension suppressed the apparent compliance, flattening the time-dependent increase in indentation depth. Quantitatively, neglecting surface tension produced overestimations of both instantaneous and long-term elastic moduli by factors that grew with γ; for γ = 0.05 N m⁻¹,  was misestimated by more than 40%. The team showed the classical Hertzian relationship between contact radius and indentation depth breaks down once surface effects are included. The contact radius becomes smaller than its nominal geometric value and evolves dynamically during relaxation or creep, decreasing with time as viscoelastic softening amplifies the influence of surface tension. This coupling generated deviations that could not be captured by existing models assuming constant contact geometry. Moreover, the researchers established an explicit modified load–depth relation that embeds surface tension as a correction term scaling with an intrinsic length . The refined equation successfully collapsed simulation data across different material parameters and experimental modes, demonstrating its generality. Importantly, they proposed a procedure to extract both viscoelastic constants and surface tension simultaneously: by performing two indentation experiments at distinct depths, one can solve coupled equations for , , , and γ. This approach overcomes long-standing difficulties in independently measuring surface and bulk responses in biological samples. The combined numerical and analytical framework thus delivers a consistent interpretation of how surface tension modulates viscoelastic relaxation and creep in soft, incompressible materials.

In conclusion, the new models designed by Professor Yue Ding and colleagues illustrate how surface tension modifies relaxation and creep under spherical indentation. The combined approach enables simultaneous determination of viscoelastic constants and surface tension from experimental data, offering a precise and unified tool for characterizing soft biological materials. The new findings reveal that neglecting surface tension becomes invalid once material stiffness drops to the kilopascal range. Surface tension alters the very shape of the stress field beneath the indenter, introducing an apparent stiffening during relaxation and a retarding effect during creep. Consequently, parameters derived from standard Hertzian analyses cannot be trusted in soft regimes unless surface contributions are explicitly included. The methodological advance is equally important because the authors provided an analytical bridge between continuum viscoelasticity and surface thermodynamics by embedding surface tension directly into the load–depth relation. Their formulation introduces an intrinsic length scale that quantifies the competition between surface and bulk effects. This explains previously inconsistent measurements across laboratories and also offers a practical calibration pathway for atomic-force or nano-indentation studies of cells and hydrogels. Additionally, the proposed dual-depth protocol further enables simultaneous extraction of bulk viscoelastic parameters and surface tension from a single experimental setup, eliminating the need for independent surface characterization. Furthermore, the new study showed how microscale mechanical tests can capture emergent physical interactions at the interface between solids and fluids. In biological contexts, such interactions govern phenomena as diverse as cell adhesion, tissue morphogenesis, and tumor invasion—all of which rely on a delicate interplay between internal viscosity and cortical tension. The study thus extends beyond contact mechanics into the biomechanics of living matter, offering quantitative insight into how mechanical cues at the cell surface influence relaxation dynamics. We believe there is broad implications and the theoretical framework can be adapted to dynamic or oscillatory loading, multiple indenter geometries, or layered systems representing cellular membranes over cytoplasmic substrates. It also paves the way for designing biomimetic materials whose apparent stiffness can be tuned through controlled surface tension. By reconciling surface and bulk contributions within a unified viscoelastic model, the new paper provides both conceptual clarity and experimental utility for the next generation of micromechanical characterization.