On the Thermoelastic Response of a Gradient Elastic Half-Space Subjected to Thermal Shock on the Boundary
Abstract. The thermoelastic problem of a half-space subjected to thermal shock on its boundary is... more Abstract. The thermoelastic problem of a half-space subjected to thermal shock on its boundary is analysed for the case when the bulk is a microstructured solid, within the framework of the strain gradient theory of elasticity. Heat exchange through convection with a surrounding fluid, whose temperature suddenly increases by a specific amount, is imposed uniformly on the traction free surface of the half-space as a Robin type boundary condition for the temperature field. Both the weakly coupled problem of thermal stresses and the fully coupled thermoelastic response are studied. Classical Fourier heat transfer is assumed. The standard Galerkin finite element method is employed for the solution of the corresponding initial-boundary value problem which, due to the spatially uniform temperature of the free surface, is rendered one dimensional. Special finite elements are developed featuring quadratic Lagrange shape functions for the approximation of the temperature and Hermite polynomi...
Steady-state antiplane crack considering the flexoelectrics effect: surface waves and flexoelectric metamaterials
Archive of Applied Mechanics
The antiplane dynamic flexoelectric problem is stated as a dielectric solid that incorporates gra... more The antiplane dynamic flexoelectric problem is stated as a dielectric solid that incorporates gradients of electric polarization and flexoelectricity due to strain gradients. The work examines dielectric materials without piezoelectric coupling or nonlinear ferroelectric switching and considers the inverse flexoelectric effect. It is shown that the coupling of the mechanical with the electrical problem can be condensed in a single mechanical problem that falls in the area of dynamic couple stress elasticity. Moreover, static and steady state dynamic antiplane problems of flexoelectric and couple stress elastic materials can be modeled as anisotropic plates with a non-equal biaxial pre-stress. This analogy was materialized in a finite element code. In this work, we solved the steady-state problem of a semi-infinite antiplane crack located in the middle of an infinite flexoelectric material, with its crack-tip moving with constant velocity. The particular type of loading investigated serves to relate the present solutions with known results from classic elastodynamics. We investigated the influence of various parameters such as the shear wave velocity and two naturally emerging microstructural and micro-inertia lengths. In the context of flexoelectricity, the two lengths are due to the interplay of the elastic and the flexoelectric parameters. Furthermore, we investigated the subsonic and the supersonic steady state crack rupture and showed that the Mach cones depend on the microstructural as well as the micro-inertial lengths. An important finding of this work is the existence of surface waves of Bleustein–Gulyaev type that do not appear in classic elastodynamics, but have been found in piezoelectric materials. The case of dielectric metamaterials with negative electric susceptibility is examined for the first time. The results can be useful for other dispersive materials, provided we identify the pertinent microstructural and micro-inertial lengths in accord with the behavior of the material at high frequencies.
Uniformly moving antiplane crack in flexoelectric materials
European Journal of Mechanics - A/Solids
Abstract The antiplane dynamic flexoelectric problem is stated as a dielectric solid that incorpo... more Abstract The antiplane dynamic flexoelectric problem is stated as a dielectric solid that incorporates gradients of electric polarization and flexoelectricity due to strain gradients. It is shown that the coupling of the mechanical with the electrical problem can be condensed in a single mechanical problem that falls in the area of dynamic couple stress elasticity. Moreover, static and steady state dynamic antiplane problems of flexoelectric and couple stress elastic materials can be modeled as anisotropic plates with a non-equal biaxial pre-stress. This analogy was materialized in a finite element code. In this work we solved the steady-state problem of a semi-infinite antiplane crack located in the middle of an infinite flexoelectric material, with its crack-tip moving with constant velocity. The particular type of loading investigated serves to relate the present solutions with known results from classic elasto-dynamics. We investigated the influence of various parameters such as the shear wave velocity and two naturally emerging micro-structural and micro-inertia lengths. In the context of flexoelectricity, the two lengths are due to the interplay of the elastic and the flexoelectric parameters. We also investigated the connection of the electric boundary conditions with boundary conditions of the dynamic couple stress elasticity. Furthermore, we investigated the subsonic and the supersonic steady-state crack rupture and showed that the Mach cones depend on the micro-structural as well as the micro-inertial lengths. The results are important for all dielectrics such as ceramics, ice, perovskites and polymers that exhibit strong flexoelectric effects, often uncoupled from piezoelectricity (centrosymmetric materials). Moreover, the results can be useful for other dispersive materials, provided we identify the pertinent micro-structural and micro-inertial lengths in accord with the behaviour of the material at high frequencies.
Uniformly moving screw dislocation in flexoelectric materials
European Journal of Mechanics - A/Solids
Abstract The anti-plane dynamic flexoelectric problem is stated as a dielectric solid that incorp... more Abstract The anti-plane dynamic flexoelectric problem is stated as a dielectric solid that incorporates gradients of electric polarization and flexoelectricity due to strain gradients. It is shown that the coupling of the mechanical with the electrical problem can be condensed in a single mechanical problem that falls within the area of dynamic couple stress elasticity. Moreover, static and steady state dynamic anti-plane problems of flexoelectric or couple stress elastic materials can be modeled as anisotropic plates with a non-equal biaxial pre-stress. This analogy is materialized in a finite element code. Screw dislocations play prominent role in crystal growth, plasticity, development of thin epitaxial films, micro-components and opto-mechanical devices and are important for our modern understanding of seismology. In general, screw dislocations are treated in the context of anti-plane elasticity. In this work we solve the steady-state problem of a screw dislocation in flexoelectric materials, moving with constant velocity. We investigate the influence of various parameters such as the shear wave velocity and two naturally emerging micro-structural and micro-inertia lengths. In the context of flexoelectricity, the two lengths are attributed to the interplay of the elastic and the flexoelectric parameters. We also investigate the connection of the electric boundary conditions with the boundary conditions of the dynamic couple stress elasticity. Furthermore, we investigate the subsonic and the supersonic steady state dislocation motion and find that the Mach cones depend on the micro-structural as well as the micro-inertial lengths. The presented results are important for all crystalline dielectrics such as ceramics, ice and perovskites that exhibit strong flexoelectric effect, often uncoupled from piezoelectricity.
Anti-plane loading of microstructured materials in the context of couple stress theory of elasticity: half-planes and layers
Archive of Applied Mechanics
The problem of calculating the displacements and stresses in a layered system often arises in eng... more The problem of calculating the displacements and stresses in a layered system often arises in engineering analysis and design, ranging from the field of mechanical engineering, to the field of materials science and soil mechanics. The present work focuses on the anti-plane response of half-planes and layers of finite thickness bonded on rigid substrates, under a point load, in the context of couple stress elasticity. The theory of couple stress elasticity is used to model the material microstructure and incorporate the size effects into the macroscopic response. This problem in plane strain configuration is referred to as Burmister’s problem. The purpose is to derive the pertinent Green’s functions that can be effectively used for the formulation of anti-plane contact problems in the context of couple stress elasticity. Full-field solutions regarding the out-of-plane displacements, the strains and the equivalent stress are presented, and of special importance is the behavior of the new solutions near to the point of application of the force where pathological singularities and discontinuities exist in the classical solutions.
In the present paper we explore the response of a half-plane indented by a tilted flat punch with... more In the present paper we explore the response of a half-plane indented by a tilted flat punch with sharp corners in the context of couple-stress elasticity theory. Contact conditions arise in a number of modern engineering applications ranging from structural and geotechnical engineering to micro and nanotechnology. As the contact scales reduce progressively the effects of the microstructure upon the macroscopic material response cannot be ignored. The generalized continuum theory of couple-stress elasticity introduces characteristic material lengths in order to describe the pertinent scale effects that emerge from the underlying material microstructure. The problem under investigation is interesting for three reasons: Firstly, the indentor's geometry is simple so that benchmark results may be extracted. Secondly, important deterioration of the macroscopic results may emerge in the case that a tilting moment is applied on the indentor inadvertently or in the case that the flat punch itself is not self-aligning so that asymmetrical contact pressure distributions arise on the contact faces. Thirdly, the voluntary application of a tilting moment on the flat punch during an experiment gives rise to potential capabilities of the flat punch for the determination of the material microstructural characteristic lengths. The solution methodology is based on singular integral equations which have resulted from a treatment of the mixed boundary value problems via integral transforms and generalized functions. The results show significant departure from the predictions of classical elasticity revealing that valuable information may be deducted from the indentation of a tilted punch of a microstructured solid.
The Journal of Strain Analysis for Engineering Design, 2015
In this study, we derive general solutions for two-dimensional plane strain contact problems with... more In this study, we derive general solutions for two-dimensional plane strain contact problems within the framework of the generalized continuum theory of couple-stress elasticity. This theory introduces characteristic material lengths and is able to capture the associated scale effects that emerge from the material microstructure which are often observed in indentation tests used for the material characterization. The contact problems are formulated in terms of singular integral equations using a Green’s function approach. The pertinent Green’s function obtained through the use of integral transforms corresponds to the solution of the two-dimensional Flamant–Boussinesq half-plane problem in couple-stress elasticity. The results show a strong dependence on the microstructural characteristics of the material when this becomes comparable to the characteristic dimension of the problem, which in the case of an indentation test is the contact length/area.
Contact Mechanics in the Framework of Couple Stress Elasticity
The purpose of this work is to present general solutions for two-dimensional (2D) plane-strain co... more The purpose of this work is to present general solutions for two-dimensional (2D) plane-strain contact problems within the framework of the generalized continuum theory of couple-stress elasticity. This theory is able to capture the scale effects, which are often observed in indentation problems with contact lengths comparable to the material microstructure. To this end, we formulate a number of basic contact problems in terms of singular integral equations using the pertinent Green’s function that corresponds to the solution of the analogue of the Flamant-Boussinesq problem of a half-space in couple-stress elasticity. In addition, we also provide results concerning the more complex traction boundary-value problem involving a deformable layer (again within couple-stress elasticity) of finite thickness superposed on a rigid half-space. We show that the contact behavior of materials with couple-stress effects depends strongly upon their microstructural characteristics, especially when...
Journal of Engineering Materials and Technology, 2010
Functionally graded materials (FGMs) are composite materials that exhibit a microstructure that v... more Functionally graded materials (FGMs) are composite materials that exhibit a microstructure that varies locally in order to achieve a specific type of local material properties distribution. In recent years, FGMs appear to be more interesting in engineering application since they present an enhanced performance against deformation, fracture, and fatigue. The purpose of the present work is to present evidence of the excellent strength properties of a new graded composite that is inspired by the human teeth. The outer surface of the teeth exhibits high surface strength while it is brittle and wear resistant, whereas the inner part is softer and flexible. The specific variation in Young's modulus along the thickness of the presented composite is of particular interest in our case. The present work presents a finite element analysis and an experimental verification of an actual composite with elastic modulus that follows approximately the theoretical distribution observed in the teeth.
Tanner's simple model of crystallization for power-law fluids extended
ABSTRACT Tanner et al. (Rheol. Acta, 48, 2009, 499-507) presented a simple model for power-law fl... more ABSTRACT Tanner et al. (Rheol. Acta, 48, 2009, 499-507) presented a simple model for power-law fluids in which it was possible to derive semi-analytical solutions based on some key simplifying assumptions. These include shear flows in tubes and channels, a 'step function' or 'amorphous-frozen' model of the viscosity changes due to crystallization, and a power-law index of 1/3 valid for a crystallizing poly(butene-1) polymer for which experiments were available. Their work compared favorably with experimental data for the onset of crystallization times. In the present work, we have repeated and verified the Tanner model and extended it to power-law indices from 1 (Newtonian behavior) down to 0 (extreme shear thinning) in order to study the effect of the different problem parameters and place a set of results that will act as reference for future and more detailed computational calculations through the Finite Element Method.
The normal impact of thermal barrier coatings by spherical particles is analysed by the finite el... more The normal impact of thermal barrier coatings by spherical particles is analysed by the finite element method in order to determine the transient stress state and potential for erosion. During the impact event, the deformation response can progress from elasto-dynamic to quasi-static elastic and then plastic. The precise sequence depends upon the size and velocity of the impacting particle, and upon the geometric and mechanical properties of the coating. Each deformation mechanism involves transient tensile stresses at characteristic depths within the coating, and these can lead to fracture of the columns from pre-existing flaws. Analytical solutions are presented that describe the transient stress state within the coating for each deformation mechanism; the analytical models are validated by independent finite element simulations in order to evaluate independent parameters. The analytical framework is used to construct deformation mechanism maps for the erosion of the thermal barrier coatings. It is concluded that erosion is a consequence of a set of potentially active deformation mechanisms.
Thermal barrier coatings with a columnar microstructure are prone to erosion damage by a mechanis... more Thermal barrier coatings with a columnar microstructure are prone to erosion damage by a mechanism of surface cracking upon impact by small foreign particles. In order to explore this erosion mechanism, the elastic indentation response and the elasticplastic indentation responses of a columnar thermal barrier coating to a spherical indenter are determined by the finite element method and by analytical models. It is shown that the indentation response is intermediate between that of a homogeneous half-space and that given by an elastic-plastic mattress model (with the columns behaving as independent non-linear springs). The sensitivity of the indentation behaviour to geometry and to the material parameters is explored: the diameter of the columns, the gap width between columns, the coefficient of Coulomb friction between columns and the layer height of the thermal barrier coating. The calculations reveal that the level of induced tensile stress is sufficient to lead to cracking of the columns at a depth of about the column radius. It is also demonstrated that the underlying soft bond coat can undergo plastic indentation when the coating comprises parallel columns, but this is less likely for the more realistic case of a random arrangement of tapered columns.
Numerical simulations have been undertaken for the creeping pressure-driven flow of a Bingham pla... more Numerical simulations have been undertaken for the creeping pressure-driven flow of a Bingham plastic past a cylinder kept between parallel plates. Different gap/cylinder diameter ratios have been studied ranging from 2:1 to 50:1. The Bingham constitutive equation is used with an appropriate modification proposed by Papanastasiou, which applies everywhere in the flow field in both the yielded and practically unyielded regions. The emphasis is on determining the extent and shape of yielded/unyielded regions along with the drag coefficient for a wide range of Bingham numbers. The present results extend previous simulations for creeping flow of a cylinder in an infinite medium and provide calculations of the drag coefficient around a cylinder in the case of wall effects.
Numerical simulations have been undertaken for the creeping pressure-driven flow of a Bingham pla... more Numerical simulations have been undertaken for the creeping pressure-driven flow of a Bingham plastic past a cylinder kept between parallel plates. Different gap/cylinder diameter ratios have been studied ranging from 2:1 to 50:1. The Bingham constitutive equation is used with an appropriate modification proposed by Papanastasiou, which applies everywhere in the flow field in both the yielded and practically unyielded regions. The emphasis is on determining the extent and shape of yielded/unyielded regions along with the drag coefficient for a wide range of Bingham numbers. The present results extend previous simulations for creeping flow of a cylinder in an infinite medium and provide calculations of the drag coefficient around a cylinder in the case of wall effects.
The present work is concerned with the benchmark problem of flow in a lid-driven square cavity. T... more The present work is concerned with the benchmark problem of flow in a lid-driven square cavity. The geometry offers a typically simple two-dimensional flow, for which accurate solutions exist for Newtonian fluids [1], while for non-Newtonian viscoelastic fluids numerical solutions have also appeared recently . An important class of non-Newtonian materials exhibits a yield stress, which must be exceeded before significant deformation can occur. The models presented for such so-called viscoplastic materials include the Bingham, Herschel-Bulkley and Casson . Several researchers have studied these materials in non-trivial flows (see ). For the benchmark problem at hand and to the authors' best knowledge, the only numerical viscoplastic work was done by Bercovier and Engelman [9] with a regularized Bingham model. These authors included inertia (Re = 1) and studied four values of the yield stress (2.5, 5, 7.5, 10) showing in an elementary way the yielded/unyielded regions. At that early time, only 100 (10 × 10) elements were used for the computations.
The capillary flow of a commercial low-density polyethylene (LDPE) melt was studied both experime... more The capillary flow of a commercial low-density polyethylene (LDPE) melt was studied both experimentally and numerically. The excess pressure drop due to entry (Bagley correction), the compressibility, the effect of pressure on viscosity, and the possible slip effects on the capillary data analysis have been examined. Using a series of capillary dies having different diameters, D, and length-to-diameter L/D ratios, a full rheological characterization has been carried out, and the experimental data have been fitted both with a viscous model (Carreau-Yasuda) and a viscoelastic one (the Kaye-Bernstein, Kearsley, Zapas/Papanastasiou, Scriven, Macosko, or K-BKZ/PSM model). Particular emphasis has been given on the pressure-dependence of viscosity, with a pressure-dependent coefficient b p . For the viscous model, the viscosity is a function of both temperature and pressure. For the viscoelastic K-BKZ model, the time-temperature shifting concept has been used for the non-isothermal calculations, while the time-pressure shifting concept has been used to shift the relaxation moduli for the pressure-dependence effect. It was found that only the viscoelastic simulations were capable of reproducing the experimental data well, while any viscous modeling always underestimates the pressures, especially at the higher apparent shear rates and L/D ratios. POLYM. ENG. SCI., 52:649-662, 2012. ª
On the Thermoelastic Response of a Gradient Elastic Half-Space Subjected to Thermal Shock on the Boundary
Abstract. The thermoelastic problem of a half-space subjected to thermal shock on its boundary is... more Abstract. The thermoelastic problem of a half-space subjected to thermal shock on its boundary is analysed for the case when the bulk is a microstructured solid, within the framework of the strain gradient theory of elasticity. Heat exchange through convection with a surrounding fluid, whose temperature suddenly increases by a specific amount, is imposed uniformly on the traction free surface of the half-space as a Robin type boundary condition for the temperature field. Both the weakly coupled problem of thermal stresses and the fully coupled thermoelastic response are studied. Classical Fourier heat transfer is assumed. The standard Galerkin finite element method is employed for the solution of the corresponding initial-boundary value problem which, due to the spatially uniform temperature of the free surface, is rendered one dimensional. Special finite elements are developed featuring quadratic Lagrange shape functions for the approximation of the temperature and Hermite polynomi...
Steady-state antiplane crack considering the flexoelectrics effect: surface waves and flexoelectric metamaterials
Archive of Applied Mechanics
The antiplane dynamic flexoelectric problem is stated as a dielectric solid that incorporates gra... more The antiplane dynamic flexoelectric problem is stated as a dielectric solid that incorporates gradients of electric polarization and flexoelectricity due to strain gradients. The work examines dielectric materials without piezoelectric coupling or nonlinear ferroelectric switching and considers the inverse flexoelectric effect. It is shown that the coupling of the mechanical with the electrical problem can be condensed in a single mechanical problem that falls in the area of dynamic couple stress elasticity. Moreover, static and steady state dynamic antiplane problems of flexoelectric and couple stress elastic materials can be modeled as anisotropic plates with a non-equal biaxial pre-stress. This analogy was materialized in a finite element code. In this work, we solved the steady-state problem of a semi-infinite antiplane crack located in the middle of an infinite flexoelectric material, with its crack-tip moving with constant velocity. The particular type of loading investigated serves to relate the present solutions with known results from classic elastodynamics. We investigated the influence of various parameters such as the shear wave velocity and two naturally emerging microstructural and micro-inertia lengths. In the context of flexoelectricity, the two lengths are due to the interplay of the elastic and the flexoelectric parameters. Furthermore, we investigated the subsonic and the supersonic steady state crack rupture and showed that the Mach cones depend on the microstructural as well as the micro-inertial lengths. An important finding of this work is the existence of surface waves of Bleustein–Gulyaev type that do not appear in classic elastodynamics, but have been found in piezoelectric materials. The case of dielectric metamaterials with negative electric susceptibility is examined for the first time. The results can be useful for other dispersive materials, provided we identify the pertinent microstructural and micro-inertial lengths in accord with the behavior of the material at high frequencies.
Uniformly moving antiplane crack in flexoelectric materials
European Journal of Mechanics - A/Solids
Abstract The antiplane dynamic flexoelectric problem is stated as a dielectric solid that incorpo... more Abstract The antiplane dynamic flexoelectric problem is stated as a dielectric solid that incorporates gradients of electric polarization and flexoelectricity due to strain gradients. It is shown that the coupling of the mechanical with the electrical problem can be condensed in a single mechanical problem that falls in the area of dynamic couple stress elasticity. Moreover, static and steady state dynamic antiplane problems of flexoelectric and couple stress elastic materials can be modeled as anisotropic plates with a non-equal biaxial pre-stress. This analogy was materialized in a finite element code. In this work we solved the steady-state problem of a semi-infinite antiplane crack located in the middle of an infinite flexoelectric material, with its crack-tip moving with constant velocity. The particular type of loading investigated serves to relate the present solutions with known results from classic elasto-dynamics. We investigated the influence of various parameters such as the shear wave velocity and two naturally emerging micro-structural and micro-inertia lengths. In the context of flexoelectricity, the two lengths are due to the interplay of the elastic and the flexoelectric parameters. We also investigated the connection of the electric boundary conditions with boundary conditions of the dynamic couple stress elasticity. Furthermore, we investigated the subsonic and the supersonic steady-state crack rupture and showed that the Mach cones depend on the micro-structural as well as the micro-inertial lengths. The results are important for all dielectrics such as ceramics, ice, perovskites and polymers that exhibit strong flexoelectric effects, often uncoupled from piezoelectricity (centrosymmetric materials). Moreover, the results can be useful for other dispersive materials, provided we identify the pertinent micro-structural and micro-inertial lengths in accord with the behaviour of the material at high frequencies.
Uniformly moving screw dislocation in flexoelectric materials
European Journal of Mechanics - A/Solids
Abstract The anti-plane dynamic flexoelectric problem is stated as a dielectric solid that incorp... more Abstract The anti-plane dynamic flexoelectric problem is stated as a dielectric solid that incorporates gradients of electric polarization and flexoelectricity due to strain gradients. It is shown that the coupling of the mechanical with the electrical problem can be condensed in a single mechanical problem that falls within the area of dynamic couple stress elasticity. Moreover, static and steady state dynamic anti-plane problems of flexoelectric or couple stress elastic materials can be modeled as anisotropic plates with a non-equal biaxial pre-stress. This analogy is materialized in a finite element code. Screw dislocations play prominent role in crystal growth, plasticity, development of thin epitaxial films, micro-components and opto-mechanical devices and are important for our modern understanding of seismology. In general, screw dislocations are treated in the context of anti-plane elasticity. In this work we solve the steady-state problem of a screw dislocation in flexoelectric materials, moving with constant velocity. We investigate the influence of various parameters such as the shear wave velocity and two naturally emerging micro-structural and micro-inertia lengths. In the context of flexoelectricity, the two lengths are attributed to the interplay of the elastic and the flexoelectric parameters. We also investigate the connection of the electric boundary conditions with the boundary conditions of the dynamic couple stress elasticity. Furthermore, we investigate the subsonic and the supersonic steady state dislocation motion and find that the Mach cones depend on the micro-structural as well as the micro-inertial lengths. The presented results are important for all crystalline dielectrics such as ceramics, ice and perovskites that exhibit strong flexoelectric effect, often uncoupled from piezoelectricity.
Anti-plane loading of microstructured materials in the context of couple stress theory of elasticity: half-planes and layers
Archive of Applied Mechanics
The problem of calculating the displacements and stresses in a layered system often arises in eng... more The problem of calculating the displacements and stresses in a layered system often arises in engineering analysis and design, ranging from the field of mechanical engineering, to the field of materials science and soil mechanics. The present work focuses on the anti-plane response of half-planes and layers of finite thickness bonded on rigid substrates, under a point load, in the context of couple stress elasticity. The theory of couple stress elasticity is used to model the material microstructure and incorporate the size effects into the macroscopic response. This problem in plane strain configuration is referred to as Burmister’s problem. The purpose is to derive the pertinent Green’s functions that can be effectively used for the formulation of anti-plane contact problems in the context of couple stress elasticity. Full-field solutions regarding the out-of-plane displacements, the strains and the equivalent stress are presented, and of special importance is the behavior of the new solutions near to the point of application of the force where pathological singularities and discontinuities exist in the classical solutions.
In the present paper we explore the response of a half-plane indented by a tilted flat punch with... more In the present paper we explore the response of a half-plane indented by a tilted flat punch with sharp corners in the context of couple-stress elasticity theory. Contact conditions arise in a number of modern engineering applications ranging from structural and geotechnical engineering to micro and nanotechnology. As the contact scales reduce progressively the effects of the microstructure upon the macroscopic material response cannot be ignored. The generalized continuum theory of couple-stress elasticity introduces characteristic material lengths in order to describe the pertinent scale effects that emerge from the underlying material microstructure. The problem under investigation is interesting for three reasons: Firstly, the indentor's geometry is simple so that benchmark results may be extracted. Secondly, important deterioration of the macroscopic results may emerge in the case that a tilting moment is applied on the indentor inadvertently or in the case that the flat punch itself is not self-aligning so that asymmetrical contact pressure distributions arise on the contact faces. Thirdly, the voluntary application of a tilting moment on the flat punch during an experiment gives rise to potential capabilities of the flat punch for the determination of the material microstructural characteristic lengths. The solution methodology is based on singular integral equations which have resulted from a treatment of the mixed boundary value problems via integral transforms and generalized functions. The results show significant departure from the predictions of classical elasticity revealing that valuable information may be deducted from the indentation of a tilted punch of a microstructured solid.
The Journal of Strain Analysis for Engineering Design, 2015
In this study, we derive general solutions for two-dimensional plane strain contact problems with... more In this study, we derive general solutions for two-dimensional plane strain contact problems within the framework of the generalized continuum theory of couple-stress elasticity. This theory introduces characteristic material lengths and is able to capture the associated scale effects that emerge from the material microstructure which are often observed in indentation tests used for the material characterization. The contact problems are formulated in terms of singular integral equations using a Green’s function approach. The pertinent Green’s function obtained through the use of integral transforms corresponds to the solution of the two-dimensional Flamant–Boussinesq half-plane problem in couple-stress elasticity. The results show a strong dependence on the microstructural characteristics of the material when this becomes comparable to the characteristic dimension of the problem, which in the case of an indentation test is the contact length/area.
Contact Mechanics in the Framework of Couple Stress Elasticity
The purpose of this work is to present general solutions for two-dimensional (2D) plane-strain co... more The purpose of this work is to present general solutions for two-dimensional (2D) plane-strain contact problems within the framework of the generalized continuum theory of couple-stress elasticity. This theory is able to capture the scale effects, which are often observed in indentation problems with contact lengths comparable to the material microstructure. To this end, we formulate a number of basic contact problems in terms of singular integral equations using the pertinent Green’s function that corresponds to the solution of the analogue of the Flamant-Boussinesq problem of a half-space in couple-stress elasticity. In addition, we also provide results concerning the more complex traction boundary-value problem involving a deformable layer (again within couple-stress elasticity) of finite thickness superposed on a rigid half-space. We show that the contact behavior of materials with couple-stress effects depends strongly upon their microstructural characteristics, especially when...
Journal of Engineering Materials and Technology, 2010
Functionally graded materials (FGMs) are composite materials that exhibit a microstructure that v... more Functionally graded materials (FGMs) are composite materials that exhibit a microstructure that varies locally in order to achieve a specific type of local material properties distribution. In recent years, FGMs appear to be more interesting in engineering application since they present an enhanced performance against deformation, fracture, and fatigue. The purpose of the present work is to present evidence of the excellent strength properties of a new graded composite that is inspired by the human teeth. The outer surface of the teeth exhibits high surface strength while it is brittle and wear resistant, whereas the inner part is softer and flexible. The specific variation in Young's modulus along the thickness of the presented composite is of particular interest in our case. The present work presents a finite element analysis and an experimental verification of an actual composite with elastic modulus that follows approximately the theoretical distribution observed in the teeth.
Tanner's simple model of crystallization for power-law fluids extended
ABSTRACT Tanner et al. (Rheol. Acta, 48, 2009, 499-507) presented a simple model for power-law fl... more ABSTRACT Tanner et al. (Rheol. Acta, 48, 2009, 499-507) presented a simple model for power-law fluids in which it was possible to derive semi-analytical solutions based on some key simplifying assumptions. These include shear flows in tubes and channels, a 'step function' or 'amorphous-frozen' model of the viscosity changes due to crystallization, and a power-law index of 1/3 valid for a crystallizing poly(butene-1) polymer for which experiments were available. Their work compared favorably with experimental data for the onset of crystallization times. In the present work, we have repeated and verified the Tanner model and extended it to power-law indices from 1 (Newtonian behavior) down to 0 (extreme shear thinning) in order to study the effect of the different problem parameters and place a set of results that will act as reference for future and more detailed computational calculations through the Finite Element Method.
The normal impact of thermal barrier coatings by spherical particles is analysed by the finite el... more The normal impact of thermal barrier coatings by spherical particles is analysed by the finite element method in order to determine the transient stress state and potential for erosion. During the impact event, the deformation response can progress from elasto-dynamic to quasi-static elastic and then plastic. The precise sequence depends upon the size and velocity of the impacting particle, and upon the geometric and mechanical properties of the coating. Each deformation mechanism involves transient tensile stresses at characteristic depths within the coating, and these can lead to fracture of the columns from pre-existing flaws. Analytical solutions are presented that describe the transient stress state within the coating for each deformation mechanism; the analytical models are validated by independent finite element simulations in order to evaluate independent parameters. The analytical framework is used to construct deformation mechanism maps for the erosion of the thermal barrier coatings. It is concluded that erosion is a consequence of a set of potentially active deformation mechanisms.
Thermal barrier coatings with a columnar microstructure are prone to erosion damage by a mechanis... more Thermal barrier coatings with a columnar microstructure are prone to erosion damage by a mechanism of surface cracking upon impact by small foreign particles. In order to explore this erosion mechanism, the elastic indentation response and the elasticplastic indentation responses of a columnar thermal barrier coating to a spherical indenter are determined by the finite element method and by analytical models. It is shown that the indentation response is intermediate between that of a homogeneous half-space and that given by an elastic-plastic mattress model (with the columns behaving as independent non-linear springs). The sensitivity of the indentation behaviour to geometry and to the material parameters is explored: the diameter of the columns, the gap width between columns, the coefficient of Coulomb friction between columns and the layer height of the thermal barrier coating. The calculations reveal that the level of induced tensile stress is sufficient to lead to cracking of the columns at a depth of about the column radius. It is also demonstrated that the underlying soft bond coat can undergo plastic indentation when the coating comprises parallel columns, but this is less likely for the more realistic case of a random arrangement of tapered columns.
Numerical simulations have been undertaken for the creeping pressure-driven flow of a Bingham pla... more Numerical simulations have been undertaken for the creeping pressure-driven flow of a Bingham plastic past a cylinder kept between parallel plates. Different gap/cylinder diameter ratios have been studied ranging from 2:1 to 50:1. The Bingham constitutive equation is used with an appropriate modification proposed by Papanastasiou, which applies everywhere in the flow field in both the yielded and practically unyielded regions. The emphasis is on determining the extent and shape of yielded/unyielded regions along with the drag coefficient for a wide range of Bingham numbers. The present results extend previous simulations for creeping flow of a cylinder in an infinite medium and provide calculations of the drag coefficient around a cylinder in the case of wall effects.
Numerical simulations have been undertaken for the creeping pressure-driven flow of a Bingham pla... more Numerical simulations have been undertaken for the creeping pressure-driven flow of a Bingham plastic past a cylinder kept between parallel plates. Different gap/cylinder diameter ratios have been studied ranging from 2:1 to 50:1. The Bingham constitutive equation is used with an appropriate modification proposed by Papanastasiou, which applies everywhere in the flow field in both the yielded and practically unyielded regions. The emphasis is on determining the extent and shape of yielded/unyielded regions along with the drag coefficient for a wide range of Bingham numbers. The present results extend previous simulations for creeping flow of a cylinder in an infinite medium and provide calculations of the drag coefficient around a cylinder in the case of wall effects.
The present work is concerned with the benchmark problem of flow in a lid-driven square cavity. T... more The present work is concerned with the benchmark problem of flow in a lid-driven square cavity. The geometry offers a typically simple two-dimensional flow, for which accurate solutions exist for Newtonian fluids [1], while for non-Newtonian viscoelastic fluids numerical solutions have also appeared recently . An important class of non-Newtonian materials exhibits a yield stress, which must be exceeded before significant deformation can occur. The models presented for such so-called viscoplastic materials include the Bingham, Herschel-Bulkley and Casson . Several researchers have studied these materials in non-trivial flows (see ). For the benchmark problem at hand and to the authors' best knowledge, the only numerical viscoplastic work was done by Bercovier and Engelman [9] with a regularized Bingham model. These authors included inertia (Re = 1) and studied four values of the yield stress (2.5, 5, 7.5, 10) showing in an elementary way the yielded/unyielded regions. At that early time, only 100 (10 × 10) elements were used for the computations.
The capillary flow of a commercial low-density polyethylene (LDPE) melt was studied both experime... more The capillary flow of a commercial low-density polyethylene (LDPE) melt was studied both experimentally and numerically. The excess pressure drop due to entry (Bagley correction), the compressibility, the effect of pressure on viscosity, and the possible slip effects on the capillary data analysis have been examined. Using a series of capillary dies having different diameters, D, and length-to-diameter L/D ratios, a full rheological characterization has been carried out, and the experimental data have been fitted both with a viscous model (Carreau-Yasuda) and a viscoelastic one (the Kaye-Bernstein, Kearsley, Zapas/Papanastasiou, Scriven, Macosko, or K-BKZ/PSM model). Particular emphasis has been given on the pressure-dependence of viscosity, with a pressure-dependent coefficient b p . For the viscous model, the viscosity is a function of both temperature and pressure. For the viscoelastic K-BKZ model, the time-temperature shifting concept has been used for the non-isothermal calculations, while the time-pressure shifting concept has been used to shift the relaxation moduli for the pressure-dependence effect. It was found that only the viscoelastic simulations were capable of reproducing the experimental data well, while any viscous modeling always underestimates the pressures, especially at the higher apparent shear rates and L/D ratios. POLYM. ENG. SCI., 52:649-662, 2012. ª
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Papers by Thanasis Zisis