Effects of microstructure on antiplane crack growth in couple-stress elastic materials

The problem of a propagating rectilinear crack in an elastic solid with microstructures subject to remote classical KIII field is investigated in the present work. The material behavior is described by the indeterminate theory of couple stress elasticity developed by Koiter. This constitutive model includes the characteristic lengths in bending and torsion and thus it is able to account for the underlying microstructure of the material as well as for the strong size effects arising at small scales and observed when the representative scale of the deformation field becomes comparable to the length scale of the microstructure, such as the grain size in a polycrystalline or granular aggregate. The stress and displacement fields near the tip of a Mode III propagting crack are thus expected to be strongly influenced by the microstructural characteristic lengths. The stationary full-field solution, already obtained [1] by using Fourier transforms and Wiener-Hopf technique, showed that ahead of the crack tip within a zone smaller than the characteristic length in torsion, the total shear stress and reduced tractions occur with the opposite sign with respect to the classical LEFM solution, due to the relative rotation of the microstructural particles currently at the crack tip. However, this zone was found to have limited physical relevance and to become vanishing small for a characteristic length in torsion of zero. In this limit case, the solution recovers the classical KIII field with square root stress singularity. Outside the zone where the total shear stress is negative, the full field solution exhibits a bounded maximum for the total shear stress ahead of the crack tip, whose magnitude was adopted as a measure of the critical stress level for crack advancing. The corresponding fracture criterion defines a critical stress intensity factor, which increases with the characteristic length in torsion. In the proposed research the previous analysis will be extended in order to consider the effects of crack speed and inertia terms on the stress and deformation fields, as well as on the stability of the crack propagation in the presence of microstructures

Effects of characteristic material lengths on ductile crack propagation.

The asymptotic fields near the tip of a crack steadily propagating in a ductile material under Mode III loading conditions areinvestigated by adopting an incremental version of the indeterminate theory of couple stress plasticity displaying linear strainhardening. The adopted constitutive model is able to account for the microstructure of the material by incorporating two distinctmaterial characteristic lengths. It can also capture the strong size effects arising at small scales, which results from theunderlying microstructures. The effects of microstructure on Mode III crack tip fields mainly consist in a substantial increase inthe singularities of the skew-symmetric stress and couple stress fields, which occurs also for small values of the strainhardening coefficient, whereas the symmetric stress field turns out to be non-singular according to the asymptotic crack tipfields for a stationary crack provided by the indeterminate theory of couple stress elasticity. The performed asymptotic analysisthus predicts a significant increase of the tractions level ahead of the crack-tip, due to the contribution of the rotation gradient

Evolution of multiple Martensite variants in a SMA thick-walled cylinder loaded by internal pressure

The stress and deformation fields in a SMA ring or a thick-walled cylinder loaded by internal pressure at constant temperature (over the start temperature of the martensitic transformation) are determined in closed form under plane stress loading conditions. The phenomenological SMA constitutive model incorporates the volume fractions of multi-variants Martensite, which are assumed to evolve linearly with the Tresca effective stress, according to the associative flow rule and the corner flow rule. Initially, the cylinder is everywhere in a state of Austenite. The application of an internal pressure then triggers the martensitic transformation starting from the inner radius of the cylinder wall and extending towards the outer radius. If the wall thickness is large enough, the tangential stress may vanish at the inner radius and correspondingly the stress state may reach a corner of the Tresca transformation condition, thus originating two different Martensite variants according to the corner transformation rule. The admissible phase partitions within the wall thickness originating during the loading process have been systematically investigated according to the ratio between the outer and inner radii. The results obtained here suggest that the loading process should be interrupted soon after the complete martensitic transformation is achieved at the inner radius of the cylinder to avoid permanent plastic deformations

Path-independent integrals around two circular holes in a thermoelastic medium

An analytic solution obtained by using bipolar coordinates is presented for thermal stresses in an infinite thermoelastic medium with two unequal circular holes kept at different temperatures. The Jk-integral vector and M- and L- integrals are derived for steady thermoelasticity and calculated on a closed contour encircling one or both holes

Thermal stress fields between two unequal circular holes in a ceramic medium

Thermal stresses play a significant role in a number of engineering problems ranging from the design of heat engines, nuclear plants and aircrafts to the enhancement of electronic devices and MEMS performance. In particular, the determination of stress concentrations due to thermal loadings is a main issue for the accurate design of many electronic devices, where a large number of conductive electric wires are embedded in a ceramic or Silicon matrix at a small distance from each other. In this case, the heat production due to the Joule effect may create high enough thermal stresses to cause cracking and rupture of the insulating ligament between the wires, thus reducing the performance of the device. Since cracks often initiate and propagate from the locations of stress concentration, such as holes and inclusions, then, an accurate evaluation of the stress concentration factor (SCF) in proximity of these defects is a prerequisite to assure the structural integrity of a number of ceramic components and to guarantee the proper functionality of many electronic devices. An analytic solution is presented here for thermal stresses in an infinite thermoelastic medium with two unequal circular cylindrical holes held at different temperatures, under steady-state heat flux. The most general representation for a biharmonic function in bipolar coordinates has been used. The stress field is decomposed in the sum of a particular stress field induced by the steady-state temperature distribution, which does not satisfy the conditions of vanishing tractions on the surfaces of the holes and vanishing remote stress field, and an auxiliary stress field required to satisfy these boundary conditions, which has been obtained for isothermal elasticity. The corresponding variations of the stress concentration factor, are determined in terms of the holes geometry and temperatures. Moreover, the Jk-integral vector and the M-integral are first generalized for steady state thermoelasticity and then calculated on a closed contour encircling one or both holes. Results are then presented for varying geometry of the holes

Hamiltonian/Stroh formalism for anisotropic media with microstructure

Moving from variational principles, we develop the Hamiltonian formalism for generally anisotropic microstructured materials, in an attempt to extend the celebrated Stroh formulation. Microstructure is expressed through the indeterminate (or Mindlin-Tiersten's) theory of couple stress elasticity. The resulting canonical formalism appears in the form of a Differential Algebraic system of Equations (DAE), which is then recast in purely differential form. This structure is due to the internal constraint that relates the micro- to the macro- rotation. The special situations of plain and anti-plane deformations are also developed and they both lead to a 7-dimensional coupled linear system of differential equations. In particular, the antiplane problem shows remarkable similarity with the theory of anisotropic plates, with which it shares the Lagrangian. Yet, unlike for plates, a classical Stroh formulation cannot be obtained, owing to the difference in the constitutive assumptions. Nonetheless, the canonical formalism brings new insight into the problem's structure and highlights important symmetry properties

A piezoelectric based energy harvester with dynamic magnification: modelling, design and experimental assessment

This work presents a simple and innovative piezoelectric energy harvester, inspired by fractal geometry and intrinsically including dynamic magnification. Energy harvesting from ambient vibrations exploiting piezoelectric materials is an efficient solution for the development of self-sustainable electronic nodes. After an initial design step, the present work investigates the eigenfrequencies of the proposed harvester, both through a simple free vibration analysis model and through a computational modal analysis. The experimental validation performed on a prototype, confirms the accurate frequency response predicted by these models with five eigenfrequencies below 100 Hz. Despite the harvester has piezoelectric transducers only on a symmetric half of the top surface of the lamina, the rate of energy conversion is significant for all the investigated eigenfrequencies. Moreover, by adding a small ballast mass on the structure, it is possible to excite specific eigenfrequencies and thus improving the energy conversion

Propagation of cracks and dislocations in 2D quasicrystals

A closed-form solution is provided for the stress, strain and velocity fields due to a planar crack steadily propagating in an elastic quasicrystal with fivefold symmetry at speed lower than the bulk wave-speeds. The cases of a semi-infinite rectilinear crack and a Griffith crack which propagates maintaining a constant length, according to the Yoffe model, are considered. Crack face loading and remote loading conditions are taken into consideration. The dynamic theory of quasicrystal with inertia forces, but neglecting dissipative phonon activity, is assumed to govern the motion of the medium. The phonon and phason stress fields turn out to be square-root singular at crack tip. The energy release rate is positive for subsonic and subRayleigh crack propagation

Finite Thin Cover on an Orthotropic Elastic Half Plane

The present work deals with the mechanical behaviour of thin films bonded to a homogeneous elastic orthotropic half plane under plain strain condition and infinitesimal strain. Both the film and semi-infinite substrate display linear elastic orthotropic behaviour. By assuming perfect adhesion between film and half plane together with membrane behaviour of the film, the compatibility condition between the coating and substrate leads to a singular integral equation with Cauchy kernel. Such an equation is straightforwardly solved by expanding the unknown interfacial stress in series of Chebyshev polynomials displaying square-root singularity at the film edges. This approach allows handling the singular behaviour of the shear stress and, in turn, reducing the problem to a linear algebraic system of infinite terms. Results are found for two loading cases, with particular reference to concentrated axial forces acting at the edges of the film. The corresponding mode II stress intensity factor has been assessed, thus providing the stress concentrations at both ends of the covering. Possible applications of the results here obtained range from MEMS, NEMS, and solar Silicon cell for energy harvesting to welded joint and building foundation

Flexural edge waves generated by steady-state propagation of a loaded rectilinear crack in an elastically supported thin plate

The problem of a rectilinear crack propagating at constant speed in an elastically supported thin plate and acted upon by an equally moving load is considered. The full-field solution is obtained and the spotlight is set on flexural edge wave generation. Below the critical speed for the appearance of travelling waves, a threshold speed is met which marks the transformation of decaying edge waves into edge waves propagating along the crack and dying away from it. Yet, besides these, and for any propagation speed, a pair of localized edge waves, which rapidly decay behind the crack tip, is also shown to exist. These waves are characterized by a novel dispersion relation and fade off from the crack line in an oscillatory manner, whence they play an important role in the far field behaviour. Dynamic stress intensity factors are obtained and, for speed close to the critical speed, they show a resonant behaviour which expresses the most efficient way to channel external work into the crack. Indeed, this behaviour is justified through energy considerations regarding the work of the applied load and the energy release rate. Results might be useful in a wide array of applications, ranging from fracturing and machining to acoustic emission and defect detection