Multiscale Failure Modeling: From Atomic Bonds to Hyperelasticity with Softening

Abstract

Separation of two particles is characterized by a magnitude of the bond energy that limits the accumulated energy of the particle interaction. In the case of a solid comprised of many particles there exist a magnitude of the average bond energy that limits the energy that can be accumulated in a small material volume. The average bond energy can be calculated if the statistical distribution of the bond density is known for a particular material. Alternatively, the average bond energy can be determined in macroscopic experiments if the energy limiter is introduced in a material constitutive model. Traditional continuum models of materials do not have energy limiters and, consequently, allow for the unlimited accumulation of the strain energy. The latter is unphysical, of course, because no material can sustain large enough strains without failure. The average bond energy limits the strain energy and controls material softening, which indicates failure. Thus, by limiting the strain energy we include a description of material failure in the constitutive model. Generally, elasticity including energy limiters can be called softening hyperelasticity because it can describe material failure via softening. We illustrate the capability of softening hyperelasticity in examples of brittle fracture and arterial failure.

First, we analyze the overall strength of arteries under the blood pressure. For this purpose we enhance various arterial models with the energy limiters. The models vary from the phenomenological Fung-type theory to the microstructural theories regarding the arterial wall as a bi-layer fiber-reinforced composite. Based on the simulation results we find, firstly, that residual stresses accumulated during artery growth can significantly delay the onset of arterial rupture like the pre-existing compression in the pre-stressed concrete delays the crack opening. Secondly, we find that the media layer is the main load-bearing layer of the artery. And, thirdly, we find that the strength of the collagen fibers dominates the media strength.

Second, we numerically simulate tension of a thin plate with a preexisting central crack within a softening hyperelasticity framework and we find that the critical load essentially depends on the crack sharpness: the sharper is the crack the lower is the critical load. The latter also means that the fracture toughness of brittle materials cannot be calibrated in experiments uniquely. Such a conclusion qualitatively corresponds to the results of the experimental tests on the calibration of the fracture toughness of ceramics, for example. The practical implication of our results is a recommendation to calibrate toughness in experiments where the size of the notch tip is comparable with a characteristic length of the material microstructure, e.g. grain size, atomic distance etc. In other words, toughness can be calibrated only under conditions where the hypothesis of continuum fails.