Siavouche Nemat-Nasser

Interacting micro-cracks near the tip in the process zone of a macro-crack

Abstract F or a linearly elastic and isotropic solid containing two or more cracks, cavities and other interacting defects of complex geometries, a method called “the method of pseudo-tractions” has been recently proposed by H ori and N emat -N asser (1983, 1985a), which can effectively solve two-dimensional problems of this kind, when cracks or cavities with sharp corners are suitably far apart. The method, however, breaks down when a crack or cavity is situated very close to the tip of another crack, which is the case when the process zone at the tip of a crack contains many micro-cracks. In this work, modifications of the method of pseudo-tractions are introduced, which will permit effective calculation of the stress intensity factors when a large crack interacts with small cracks which are situated very close to its tip in its process zone. Explicit asymptotic expressions are obtained for the stress intensity factors of the macro-crack, as well as those of the micro-cracks. It is shown that the presence of the microcracks in the process zone of a macro-crack may induce out-of-plane crack growth even under far-field hydrostatic tension. Several illustrative examples are worked out, including two collinear cracks for which an exact solution exists, arriving at an excellent correlation.

Toughening by partial or full bridging of cracks in ceramics and fiber reinforced composites

Abstract A complete solution is given for a fully or partially bridged straight crack in transversely isotropic elastic materials which may correspond to unidirectionally fiber-reinforced ceramics or other brittle composites. The stiffness of the bridging materials may have an arbitrary variation along the crack, representing partially failed fibers or ligaments. The crack may have any orientation with respect to the axis of the material symmetry. The solution is explicit in terms of the Chebychev polynomials when the bridging-forces are linearly dependent on the crack-opening-displacement. In addition, uniformly valid asymptotic solutions are developed for fully or partially bridging cracks. For the case when the crack is short relative to a length scale which depends on the material properties, the method yields a complete asymptotic solution when the bridging forces are linearly or non-linearly dependent on the crack-opening-displacement (a square-root dependence, corresponding to continuous fibers, is used for illustration). For the case of long cracks, the proposed asymptotic is effective, but the results are not presented in this work. The mechanism of crack kinking is studied for an oblique partially or fully bridged, or unbridged crack in a macroscopically transversely isotropic elastic solid. The crack is assumed to grow in the matrix material (containing unbroken strong fibers) under local driving forces which are calculated on the basis of the overall anisotropic material response. The results of various fracture criteria are studied. It is illustrated that, under far-field tensile forces normal to the crack, the criterion of the maximum opening mode stress intensity factor in the homogenized anisotropic solid (i.e., the orientation for which the strength of the singularity associated with the tensile hoop stress is maximum) produces results which suggest crack growth more or less parallel to the fibers, whereas the results based on the maximum Mode I stress intensity factor in the isotropic matrix material and/or on the local symmetry criterion (again, for the isotropic matrix) predict crack extension more or less normal to the reinforcing fibers.

Void collapse and void growth in crystalline solids

The problem of large deformation and eventual collapse (or growth to a final shape, depending on the microstructure and loading) of voids in a crystalline solid which undergoes plastic flow by slip on crystallographic planes is considered and solved analytically for plane problems, using certain reasonable simplifying assumptions. It is shown that because of local anisotropic plastic flow, an initially circular (in two dimensions or spherical in three dimensions) void quickly deforms into a noncircular (or nonspherical) shape, even under all‐around uniform far‐field pressure or tension. Using an incremental solution, the shape of the void at each loading stage is approximated by an equivalent ellipse, and this procedure is continued until either the void collapses into a crack (which occurs always under compression and in special cases even under tension) or attains a constant aspect ratio (under tension only). The residual stresses in the vicinity of the collapsing void are calculated and used to establi...

Asymptotic solution of a class of strongly singular integral equations

A class of strongly singular, nonlinear integral equations is considered and sufficient conditions for the uniqueness of the solution and, in the linear and homogeneous case, the nonpositiveness of the associated eigenvalues are obtained. When the singular integral is proportional to a small perturbation parameter, a procedure for the construction of the uniformly valid leading term of the asymptotic solutions for both linear and nonlinear equations is presented. The results are illustrated by means of an example that emerges in the analysis of microcrack growth in reinforced ceramics.

Compression-induced high strain rate void collapse, tensile cracking, and recrystallization in ductile single and polycrystals

Abstract A series of dynamic compression experiments were performed on single-crystal and polycrystal copper specimens, as well as on mild steel (AISI 1018 steel) and pure iron, containing pre-existing cavities, using the split Hopkinson compression bar. It is found that void collapse occurring under purely compressive axial loading leads to tensile cracking (presumably upon unloading). Microcracks are nucleated at intersections of slip planes, growing along slip lines as well as in the general direction normal to the applied compression. The resistance to void collapse in loading and the extent of fracturing in unloading increase with increasing overall nominal rate of compressive loading. For pure copper single crystals, recrystallization occurs when the strains and strain rates are sufficiently high. Tensile cracks are then observed to grow into the newly formed crystal grains.

Mechanics of void growth and void collapse in crystals

Abstract The mechanics of void deformation in single crystals is studied in a fully three-dimensional setting, taking into account all twelve slip systems for fcc and bcc crystals. The increments of the local field variables are calculated analytically using Eshelbys approach for the three-dimensional inclusion problem. The collapse or growth of voids in three dimensions is investigated when a rate-dependent elastoplastic material is subjected to compression and tension at a high rate. The deformation of an initially spherical cavity is calculated incrementally, and, it is shown that, even under all-around uniform loading, a void deforms into a complicated shape which is defined by the structure and symmetry of the slip systems. An equivalent ellipsoid is used to approximate the deformed void shape at each incremental step, and the procedure is continued until the equivalent ellipsoid collapses into a crack or a needle, or it expands or shrinks in a self-similar manner. Several numerical examples are presented, and the numerical results are compared with the experimental observations, obtaining good correlation. The significant effects of loading rate on the material response are illustrated. It is shown that the material becomes stronger at higher loading rates. The three-dimensional final void geometry under various loading conditions is studied. The difference between void deformation mechanism under tension and compression is illustrated. The possible overall failure mechanism caused by collapse and/or growth of pre-existing cavities is discussed. From the comparison of the corresponding results with those of the two-dimensional case it is shown that the double-slip system in two dimensions can be used effectively to simulate the three-dimensional problem.

Electromechanical response of ionic polymer metal composites

An ionic polymer-metal composite (IPMC) consisting of a thin Nafion sheet, platinum plated on both faces, undergoes large bending motion when an electric field is applied across its thickness. Conversely, a voltage is produced across its faces when it is suddenly bent. A micromechanical model is developed which accounts for the coupled ion transport, electric field, and elastic deformation to predict the response of the IPMC, qualitatively and quantitatively. First the basic 3D coupled field equations are presented, and then the results are applied to predict the response of a thin sheet of an IPMC. Central to our theory is the recognition that the interaction between an imbalanced charge density and the backbone polymer can be presented by an eigenstress field. The constitutive parameter connecting the eigenstress to the charge density is calculated directly using a single microstructural model for Nafion. The results are applied to predict the response of samples of IPMC, and good correlation with experimental data is obtained. Experiments show that the voltage induced by a sudden imposition of a curvature, it two orders of magnitude less than that required to produce the same curvature. The theory accurately predicts this result. The theory also shows the relative effects of different counter ions, e.g., sodium versus lithium, on the response of the composite to an applied voltage or a curvature.

Tailoring actuation of ionic polymer-metal composites through cation combination

An ionic polymer-metal composite (IPMC) consisting of a thin perfluorinated ionomer (usually, Nafion® or Flemion®) strip, platinum and/or gold plated on both faces, undergoes large bending motion when a small electric field is applied across its thickness. When the same membrane is suddenly bent, a small electric potential of the order of millivolts is produced across its surfaces. This actuation and sensing response depends on the structure of the ionomer, the morphology of the metal electrodes, the nature of cations, and the level of hydration. IPMCs in alkali-metal cation form under direct current (DC) show a fast motion towards the anode, followed by a slow relaxation. For Nafion-based IPMCs, this slow relaxation is towards the cathode, whereas for Flemion-based IPMCs, the slow relaxation continues the initial fast motion towards the anode. In contrast, the actuation of both Nafion- and Flemion-based IPMCs in tetrabutylammonium (TBA+) cation form consists of a continuous slow motion towards the anode. We have discovered that when an IPMC is neutralized by combined Na+ and TBA+ cations to produce a suitable Na-TBA-form membrane, different actuation behavior results. The proportion of the cations can be tailored to obtain a desired actuation response, e.g., to control the duration, speed, and the maximum amplitude of the initial motion towards the anode, or the magnitude and the speed of the subsequent relaxation. A series of cation combination tests on both Nafion- and Flemion-based IPMCs are carried out. Various essential physical properties of the IPMCs in various cation compositions are measured and compared. A summary of these results is presented.

Dynamic response of crystalline solids with microcavities

Based on an approximate method by the authors for calculating void deformation in crystalline solids, the global response of a small continuum material element which contains microcavities is studied. A rate‐dependent power‐law plastic flow by double‐slip is assumed to govern the local inelastic deformation. The local field variables are analytically calculated in an incremental manner. The average stress and strain are then computed by the integration of the local stress and strain over the continuum element. These average variables are used to describe the overall response of the material element under high loading rates. Several illustrative examples are given. It is shown that the global response of the material is significantly affected by the loading rate: the material response becomes tougher as the loading rate increases, but once the entire matrix becomes plastic, a strong ductility develops. It is observed that the large overall plastic deformation of crystalline solids stems not only from a uni...

Micromechanical analysis of ionic clustering in Nafion perfluorinated membrane

The cluster morphology in a water-swollen Nafion perfluorinated membrane is studied using a micromechanics approach. The cluster size is determined from the minimization of the free energy as a function of the equivalent-weight of Nafion, the volume fraction of water, and the temperature, taking into account the electrostatic dipole interaction energy, the elastic polymer chain reorganization energy, and the cluster surface energy, leading to results which are in accord with experimental observations. By minimizing the sum of (1) the electro- elastic interaction energy between an ionic cluster and the fluorocarbon matrix, and (2) the cluster surface energy, it is concluded that the effective cluster shape is spherical in the absence of an electric field, and an oblate spheroid when an electric field is applied. The effect of cluster morphology on the effective electro-elastic moduli and the effective ionic conductivity is then studied by a micromechanical multi-inclusion model. The result seems to describe the available empirical relation when a spherical cluster shape is assumed. It correctly predicts the insulator-to-conductor transition which occurs in Nafion, as the water volume fraction is increased.