For an isotropic, incompressible solid the strain energy can be written in terms of the first two invariants of the deformation or of the principal stretches (Holzapfel, 2000). The neo-Hookean model has one parameter and depends linearly on the first invariant. The form due to Mooney (1940), later generalised by Rivlin, adds the second invariant. Yeoh (1993) used a cubic polynomial in the first invariant only, the model of Ogden (1972) is a sum of powers of the principal stretches, and the models of Arruda and Boyce (1993) and Gent (1996) introduce a limiting chain extensibility to capture stiffening at large stretch. Flexible models fit one data set easily and can behave unphysically outside it. Ogden et al. (2004) demonstrated that least-squares fitting of such models is not unique, with several parameter sets giving nearly the same residual.

Digital image correlation tracks a random surface pattern between a reference image and deformed images and returns the displacement of small subsets of pixels. With one camera it measures in-plane motion of a flat specimen; with two calibrated cameras it measures shape and three-dimensional displacement, which removes the error caused by out-of-plane movement (Sutton et al., 2009; Pan et al., 2009). Strain is obtained by differentiating the displacement, an operation that amplifies noise and that depends on user settings: subset size, step and strain window. The good practices guide of the International Digital Image Correlation Society sets out how to choose and report them (Jones and Iadicola, 2018). Elastomers bring two special difficulties. The pattern must follow very large stretches without cracking or debonding, and thin specimens wrinkle or bulge if compressive stress develops anywhere.

Two families of method dominate. Finite element model updating, which goes back to Kavanagh and Clough (1971), simulates the test with trial parameters and adjusts them until computed displacements and forces match the measured ones. It is general, tolerates incomplete fields and uses the same software as the eventual design analysis, but each iteration requires a nonlinear solution. The virtual fields method applies the principle of virtual work directly to the measured strain field with chosen test functions and needs no forward simulation (Pierron and Grédiac, 2012); Promma et al. (2009) applied it to a hyperelastic model of rubber using heterogeneous tests. It is fast, but it requires strains over the whole region considered and assumes a through-thickness state, usually plane stress, in order to pass from surface measurements to volume integrals.

The specimen should produce a field rich enough to separate the parameters and should not fail in the arms before the centre has stretched appreciably. A cruciform with generous fillet radii, and where necessary slits along the arms to reduce their transverse stiffness, is a common starting point. Before any material is cut, the test is simulated with literature values for a comparable elastomer, and the simulated field is examined on a plot of the two principal stretches: points should spread between the uniaxial, planar and equibiaxial lines. At least three loading paths are planned, typically equibiaxial, one arm pair held while the other is extended, and an intermediate ratio. One path is set aside for validation and plays no part in fitting.

A stereo arrangement is preferred because thin elastomer sheets rarely stay flat. The speckle pattern is applied with a paint or powder that stretches with the surface; for very large stretch the pattern is best made from discrete particles, so that speckles separate without cracking. Feature size should span several pixels at the chosen magnification and remain resolvable when the surface area has multiplied. The cameras are calibrated with a target that fills the field of view, and a set of images of the unloaded specimen is recorded. Correlating those static images against each other gives the noise floor of displacement and strain for the actual set-up (Jones and Iadicola, 2018).

Filled elastomers soften during the first loading cycles to a given stretch, the effect reviewed by Mullins (1969), and virgin and cycled materials are in effect different materials. The procedure therefore fixes in advance which state is to be characterised. For the stabilised state, each specimen is cycled several times to the maximum stretch of the path before images are recorded, and a fresh specimen is used for each path, since softening depends on the deformation history. Tests are run slowly enough for rate effects to be small, which is verified by repeating one path at a different rate. Forces in all four arms are logged in step with the images.

Images are correlated with settings chosen from the noise study, and the reference image is updated incrementally where stretches are too large for a single reference. The outputs are displacement maps and, by differentiation, the in-plane deformation gradient, from which the principal stretches follow; the through-thickness stretch is obtained from incompressibility. Two checks are made. The fields should show the symmetries of the specimen, and departures indicate misalignment or slipping grips. The opposing arm forces should balance. Regions near grips and free edges, where correlation is poor, are masked, and the mask is recorded.

Candidate strain energy functions are chosen before fitting and kept few. A reasonable short list contains the neo-Hookean model as a baseline, a two-parameter form with second-invariant dependence, the Yeoh model, a low-order Ogden model and one limiting-extensibility model. The comparative study of Marckmann and Verron (2006) is a useful guide to how each behaves across deformation modes. For every candidate the admissible parameter range is written down, so that the optimiser cannot return a set that violates stability requirements within the tested stretches.

Either inverse route can be used, and using both on the same data is a strong cross-check. In finite element model updating, the measured boundary displacements at the edge of the region of interest are imposed on a model of that region, and the cost function combines the mismatch in interior displacements with the mismatch in arm forces, each scaled by its measurement uncertainty. Without the force term the parameters are determined only up to a common factor, because kinematics alone carry no stress scale. In the virtual fields method, virtual fields are chosen so that the measured forces enter the external virtual work, and the resulting equations are solved by least squares over all images. The optimisation is started from several well-separated initial guesses.

A low residual does not show that the parameters are known. Sensitivity of the cost function to each parameter is evaluated at the optimum, and strong correlation between parameters, or a flat direction in the cost surface, is reported as such. Confidence bounds can be estimated by repeating the identification on fields perturbed with noise at the measured level. The identified models are then used to predict the held-out loading path, and the predicted forces and fields are compared with those measured. If no candidate predicts it within the measurement uncertainty, the loop in Figure 1 is followed: the model list or the loading paths are revised.