The Effect Of Surface Elasticity On Nano-/Microscale Mode Ⅲ Crack Tip Field

The integrity,reliability,and stability behaviors of nanomaterials and nanostructures have consistently been regarded as fundamental topics attracting the attention of a large number of researchers.Fabrication processes and mechanisms governing crystal growth inevitably lead to the appearance of defects in manufactured materials and structures,for example,dislocations,grain boundaries,cracks and voids.Consequently,the elastic analysis of flawed micro-nano materials and structures is particularly important in gaining an understanding of failure mechanisms.However,the mechanical properties of materials are significantly influenced by their microstructure which contributes increasingly to overall deformation as the material’s constitutive microstructural elements decrease in size.This is true,in particular,for micro-nano devices whose properties differ significantly from those of their macro(large-)scale counterparts,for example,bulk materials.These‘microstructural contributions’ to deformation are well-known to have a significant impact on the static and dynamic mechanical properties of structures.In attempting to enhance classical elasticity theory to accommodate the contribution of small-scale effects,the incorporation of surface elasticity on the bounding surfaces of structures has been shown to be quite successful in capturing this size-dependent mechanical behavior of micro-/nanoscale structures.Accordingly,the study of fracture at small length scales utilizing models which incorporate surface energetic boundaries has received increasing attention in the literature.However,the analysis of many practical problems in this area remains absent from the literature.This can be attributed to the complicated nature of the ensuing mathematical models which often lead to extremely complicated initialboundary value problems,for example,when the influence of surface elasticity is incorporated in fracture problems involving finite domains,functionally graded materials and interface cracks.The following questions arise naturally not just because of their scientific interest but perhaps more importantly as a result of their practical importance.What are the significant differences between micro-nano scale fracture problems and large-scale fracture problems when the corresponding solids are subjected to impact load? What effect does it have on the three-dimensional fracture of the surface layer and the plastic field at a typical crack tip in a fractured material? How do we analyze the corresponding fracture phenomenon?The solution of these problems has important guiding significance for reasonable safety assessment and optimal design of micro-nano electromechanical systems.Accordingly,the present thesis focuses on the fracture of materials and mechanisms considered at micro/nano scales.The influence of surface elasticity on a crack tip field is discussed using classical elasticity theory enhanced by the incorporation of surface elasticity.Using continuum-based models,we present effective methods to solve the above-mentioned complex boundary value problems.Specifically:(1)A mode-Ⅲ(anti-plane)static crack model considering surface effect is proposed.A singular integro-differential equation for the nonclassical mixed boundary value problem combining classical elasticity with surface elasticity is established.The problem of mode Ⅲ anti-plane crack and interface crack embedded in finite thickness homogeneous isotropic elastic layer and bi-material layer is studied.The classical elasticity incorporating surface elasticity is employed to reduce a nonclassical mixed boundary value problem,where the layer interior obeys the traditional constitutive relation and the surfaces of the layer and the crack are dominated by the surface constitutive relation.The influences of surface elasticity on the elastic field and stress intensity factor are examined.It is shown that surface elasticity decreases the bulk stress and its intensity factor near the crack tips for positive surface shear modulus and gives rise to an opposite trend for a negative surface shear modulus(2)A mode-Ⅲ(anti-plane)static crack model in functionally graded materials considering the surface effect is proposed,and the hypersingular integro-differential equation about the out-of-plane displacement of the crack is mainly discussed and established.Next,we study the influence of surface elasticity on the stress intensity factor of an anti-plane shear crack embedded in an elastic strip made of functionally graded materials.Surface elasticity is applied on the strip surfaces and crack faces,and classical elasticity is invoked for the strip interior.The associated problem is converted to a hypersingular integro-differential equation for the out-of-plane displacement on the crack faces using via the use of Fourier transforms and then to a singular integral differential equation with Cauchy kernel.Stress intensity factors at the crack tips and the out-of-plane displacement on the crack faces are calculated numerically.It is found that surface elasticity and the corresponding gradient index strongly alter the bulk stress and its intensity factors near the crack tips.(3)A mode-Ⅲ(anti-plane)dynamic crack in nano-scale is proposed,and the hypersingular integro-differential equation of dynamic fracture in nano-scale is established.The transient response of linear elastic solid with micro-cracks is analyzed when the solid is subjected to antiplane impact load.As noted above and pervasive throughout the thesis,we account for the nanoscale by incorporating surface elasticity into the model of deformation.The problem is formulated as a nonclassical mixed initialboundary value problem.Most significant is the fact that both surface stress and surface mass inertia are included in the boundary condition on the crack surface.The Gaver-Stehfest algorithm is then applied to perform a numerical inversion of the Laplace transform from which we obtain dynamic stress intensity factors in the time domain.Numerical results characterizing the transient response of the crack-tip field are presented with particular emphasis on the influence of surface elasticity on the dynamic fracture parameters.(4)A penny-shaped torsional crack model considering the surface effect in nanometer size is proposed,and the hypersingular integral equation of the nonstandard mixed boundary value problem for elastoplastic problems is established.The influence of surface elasticity on the elastoplastic deformation of penny-shaped torsional crack is studied.We consider the torsional deformation of an infinite three-dimensional isotropic elastic solid weakened by a penny-shaped crack whose boundary is enhanced by the incorporation of surface elasticity.Furthermore,for completeness and a more comprehensive mathematical model,we extend our ideas to include the presence of a Dugdale-type plastic zone around the crack front.The ensuing elastoplastic problem gives rise to a non-standard mixed boundary value problem which is then reduced to a hypersingular integral equation upon application of the Hankel transform.The hypersingular integral equation is subsequently solved numerically.Our results demonstrate the effect of surface elasticity on the width of the plastic zone thus providing a significant shielding effect while increasing the load-bearing capacity of the solid in the vicinity of the defect.Our conclusions will be of particular interest when the presence of surface elasticity along a crack boundary is used in enhanced continuum-based models to study fracture at micro-and nano-scales,for example,in the prediction of fracture and failure in micro-and nano-structures.