On Incompressibility Constraint and Crack Direction in Soft Solids

Most soft materials resist volumetric changes much more than shape distortions. This experimental observation led to the introduction of the incompressibility constraint in the constitutive description of soft materials. The incompressibility constraint provides analytical solutions for problems which, otherwise, could be solved numerically only. However, in the present work, we show that the enforcement of the incompressibility constraint in the analysis of the failure of soft materials can lead to somewhat nonphysical results. We use hyperelasticity with energy limiters to describe the material failure, which starts via the violation of the condition of strong ellipticity. This mathematical condition physically means inability of the material to propagate superimposed waves because cracks nucleate perpendicular to the direction of a possible wave propagation. By enforcing the incompressibility constraint, we sort out longitudinal waves, and consequently, we can miss cracks perpendicular to longitudinal waves. In the present work, we show that such scenario, indeed, occurs in the problems of uniaxial tension and pure shear of natural rubber. We also find that the suppression of longitudinal waves via the incompressibility constraint does not affect the consideration of the material failure in equibiaxial tension and the practically relevant problem of the failure of rubber bearings under combined shear and compression.

On Cavitation in Rubberlike Materials

Microscopic voids can irreversibly grow into the macroscopic ones under hydrostatic tension. To explain this phenomenon, it was suggested in the literature to use the asymptotic value of the hydrostatic tension in the plateau yieldlike region on the stress–stretch curve obtained for the neo-Hookean model. Such an explanation has two limitations: (a) it relies on analysis of only one material model and (b) the hyperelasticity theory is used for the explanation of the failure phenomenon. In view of the mentioned limitations, the objective of the present note is twofold. First, we simulate the cavity expansion in rubber by using various experimentally calibrated hyperelastic models in order to check whether the stress–stretch curves have the plateau yieldlike regions independently of the constitutive law. Second, we repeat simulations via these same models enhanced with a failure description. We find (and that was not reported in the literature) that the process of cavity expansion simulated via hyperelastic constitutive models exhibiting stiffening, due to unfolding of long molecules, is completely stable and there are no plateau yieldlike regions on the stress–stretch curves associated with cavitation. In addition, we find that the instability in the form of yielding observed in experiments does appear in all simulations when the constitutive laws incorporate failure description with energy limiters.

REVIEW OF THE ENERGY LIMITERS APPROACH TO MODELING FAILURE OF RUBBER

ABSTRACT Nonlinear theories of elasticity describe rubber deformation but not failure; however, in reality, rubbers do fail. In the present work, we review a new approach of energy limiters that allows for unifying hyperelasticity theories with failure descriptions, and we discuss results of this unification. First, we introduce the energy limiter concept, which allows the enforcement of failure descriptions in elasticity theories. The limiter provides the saturation value for the strain energy, hence indicating the maximal energy that may be stored and dissipated by an infinitesimal material volume. The limiter is a material constant that can be calibrated via macroscopic experiments. Second, we illustrate the new approach with examples in which failure initiation is predicted but its propagation is not tracked. Examples include the problems of crack initiation, cavity instability, and rupture of inflating membranes. In addition, the traditional strength-of-materials criteria are reassessed. Third, the theory is used for three-dimensional explicit finite element simulations of a high-velocity penetration of a stiff elastic body into a rubber plate. These simulations show that a high-velocity penetration of a flat projectile leads to a diffused nonlocal failure, which does not trigger the mesh sensitivity. To the contrary, a low-velocity penetration of a sharp projectile leads to a highly localized cracklike failure, which does trigger the mesh sensitivity. Calculation of the characteristic length of failure localization allows for setting the mesh size that provides regularization of the simulations. The fact that the calculation is based on results of solely macroscopic experiments is noteworthy.

CAVITATION INSTABILITY IN RUBBER

Rubber materials and structures can fracture because tensile deformation and growth of small pre-existing voids become unstable, leading to failure localization and crack propagation. Thus, it is important to predict the onset of static instability of the growing voids. We consider two typical cases of interest: the instability of 3D voids under the remote hydrostatic tension in the bulk and the instability of 2D voids under the remote biaxial tension in the membrane. For the purpose of analysis, we use constitutive models of natural and styrene-butadiene rubbers with the failure description enforced by energy limiters. The limiters provide the saturation value for the strain energy which indicates the maximum energy that can be stored and dissipated by an infinitesimal material volume. We find that the unstable growth of a 3D bulk void can start when the remote hydrostatic tension reaches the value of ~2 ÷ 3 MPa and the unstable growth of a 2D membrane void can start when the remote biaxial tension reaches the value of ~50 ÷ 60 MPa.