Alagappan Ponnalagu

Fatigue and damage are the least understood phenomena in the mechanics of solids. Recently, Alagappan et al. (“On a possible methodology for identifying the initiation of damage of a class of polymeric materials”, Proc R Soc Lond A Math Phys Eng Sci 2016; 472(2192): 20160231) hypothesized a criterion for the initiation of damage for a certain class of compressible polymeric solids, namely that damage will be initiated at the location where the derivative of the norm of the stress with respect to the stretch starts to decrease. This hypothesis led to results that were in keeping with the experimental work of Gent and Lindley(“Internal rupture of bonded rubber cylinders in tension. Proc. R. Soc. Lond. A 1959; 249, 195–205 :10.1098) and agrees qualitatively with the results of Penn (“Volume changes accompanying the extension of rubber”, Trans Soc Rheol 1970; 14(4): 509–517) on compressible polymeric solids. Alagappan et al. considered a body wherein there is a localized region in which the density is less than the rest of the solid. In this study, we show that the criterion articulated by Alagappan et al. is still applicable when bodies have multiple localized regions of lower density, thereby lending credence to the notion that the criterion might be reasonable for a large class of bodies with multiple inhomogeneities. As in the previous study, it is found that damage is not initiated at the location where the stresses are the largest but instead at the location where the densities tend to the lowest value. These locations of lower densities coincide with locations in which the deformation gradient is very large, suggesting large changes in the local volume, which is usually the precursor to phenomena such as the bursting of aneurysms.

In this paper we study the state of stress and strain in infinite elastic slabs of nonlinear viscoelastic solids containing elliptic holes subject to an uni-axial as well as a bi-axial state of stress. The geometry affords one to get some inkling concerning the states of stress and strain in bodies containing a crack by obtaining the limit of the solutions as the aspect ratio (in this case the ratio of the minor axis to the major axis) of the ellipse tends to zero. We consider two classes of non-linear viscoelastic bodies, the classical incompressible Kelvin–Voigt solid (Thomson in R Soc Lond 14:289–297, 1865; Voigt in Ann Phys 283(12):671–693, 1892) and a generalization of a compressible model due to Gent (Rubber Chem Technol 69(1):59–61, 1996). We also study for the sake of comparison the case of a nonlinear neo-Hookean elastic solid with an elliptic hole.

We study fatigue (weakness induced by cyclic loading) in a viscoelastic body described by a generalization of the Kelvin-Voigt constitutive relation, employing a novel damage initiation criterion developed by Alagappan et al. [13-15]. The main premise is that damage is a consequence of the inhomogeneity of the material which leads to some locations in the body being naturally weaker, say for instance due to the density being lower and the material moduli depending on the density and decreasing with density, leading ultimately to failure at that location. This approach has been used successfully for polymers, elastomers and concrete subject to monotonic loading. In this study, we consider the initiation of damage due to cyclic loading, which is referred to as fatigue. Since the body under consideration is viscoelastic, it dissipates energy in each cycle which leads to an increase in temperature. We shall not take the effect of the temperature of the material moduli, instead we assume that the material moduli depend on the density and the rate of dissipation. In the case of our specific study the shear modulus of the material depends on the density and dissipation (in the case of the constitutive relation considered the shear rate), and the structure of the shear modulus is such that it decreases with decrease in density and decreases with increase in dissipation (tantamount to the assumption that it decreases with increasing shear rate for the constitutive relation under consideration) leading to damage of the material. We find that after sufficient number of cycles, the body under consideration undergoes significant loss in load carrying capacity due to fatigue.

We extend the methodology introduced for the initiation of damage within the context of a class of elastic solids to a class of viscoelastic solids (Alagappan et al. 2016 Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci. 472, 20160231. (doi:10.1098/rspa.2016.0231)). In a departure from studies on damage that consider the body to be homogeneous, with initiation of damage being decided by parameters that are based on a quantity such as the strain, that requires information concerning a special reference configuration, or the use of ad hoc parameters that have no physically meaningful origins, in this study we use a physically relevant parameter that is completely determined in the current deformed state of the body to predict the initiation of damage. Damage is initiated due to the inhomogeneity of the body wherein certain regions in the body are unable to withstand the stresses, strains, etc. The specific inhomogeneity that is considered is the variation of the density in the body. We consider damage within the context of the deformation of two representative viscoelastic solids, a generalization of a model proposed by Gent (1996 Rubber Chemistry and Technology 69, 59-61. (doi:10.5254/1.3538357)) for polymeric solids and a generalization of the Kelvin-Voigt model. We find that the criterion leads to results that are in keeping with the experiments of Gent & Lindley (1959 Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci. 249, 195-205. (doi:10.1098/rspa.1959.0016)).

In this paper, we provide a possible methodology for identifying the initiation of damage in a class of polymeric solids. Unlike most approaches to damage that introduce a damage parameter, which might be a scalar, vector or tensor, that depends on the stress or strain (that requires knowledge of an appropriate reference configuration in which the body was stress free and/or without any strain), we exploit knowledge of the fact that damage is invariably a consequence of the inhomogeneity of the body that makes the body locally 'weak' and the fact that the material properties of a body invariably depend on the density, among other variables that can be defined in the current configuration, of the body. This allows us to use density, for a class of polymeric materials, as a means to identify incipient damage in the body. The calculations that are carried out for the biaxial stretch of an inhomogeneous multi-network polymeric solid bears out the appropriateness of the thesis that the density of the body can be used to forecast the occurrence of damage, with the predictions of the theory agreeing well with experimental results. The study also suggests a meaningful damage criterion for the class of bodies being considered.

We study the initiation of damage in a polymeric body in which there is a line defect due to the formation of a "weld line" that occurs when two polymer streams join together and then solidify. We show that damage initiates in the region of weakness, namely the "weld line" based on a criterion for damage that was developed earlier. We also show that if there are other stress concentrators also additionally present, such as a hole, then there is a competition between the stresses induced due to the weakness and the stress as a consequence of the stress concentrator (in this instance a hole). This study adds more credence to the criterion for the initiation of damage that is based completely on knowledge of information at the current configuration of the body, that is, the criterion for damage is not based on the value of quantities that also need information based on a reference configuration such as the stress or strain.

Damage in concrete has been modelled using various approaches such as fracture mechanics, continuum damage mechanics and failure envelope theories. This study proposes a new approach to model the initiation of damage in concrete that addresses some limitations associated with the existing approaches. The proposed approach defines damage in terms of changes in the density of the material at the microscopic level, where such changes are induced by mechanical loading. The suggested approach is used to simulate the response of 2D concrete bodies to uni-axial tension and uni-axial compression. The simulation results indicate that the proposed model, by means of a single constitutive function, is able to correctly predict failure patterns and aptly capture the damage mechanisms under both uni-axial tension and uni-axial compression loadings using only the information related to the microstructure, the density field and the stiffness field. As a continuation, in Part II, the ability of the D3-M approach to model fully coupled chemo-mechanical damage in concrete using a single constitutive equation will be demonstrated.

We present a mathematical model to study the dissipation of energy in a composite helmet that is subjected to blast loading. In addition to assuming a constitutive relation for the material of the helmet, we also assume models for the skull, and brain, all of them being treated as homogeneous isotropic bodies. A rate type model with non-linear dissipative response characteristics is assumed within the context of a thermodynamic framework for the helmet, as well as the skull, and brain. We study the effect of orientation, the region of impact, the dissipative materials used in the helmet, and the interface condition between the helmet and head assembly. We present a numerical simulation corresponding to different dissipative mechanisms that lead to multiple wave transmissions and reflections that could be detrimental to the brain, highlighting the necessity to pick the materials for the helmet with great care.