Gradient-elastic stress analysis near cylindrical holes in a plane under bi-axial tension fields

Abstract This paper is devoted to a gradient-elastic stress analysis of an infinite plate weakened by a cylindrical hole and subjected to two perpendicular and independent uni-axial tensions at infinity. The problem setting can be considered as an extension and generalization of the well-known Kirsch problem of the classical elasticity theory which is here extended with respect to the external loadings and generalized with respect to the continuum framework. A closed-form solution in terms of displacements is derived for the problem within the strain gradient elasticity theory on plane stress/strain assumptions. The main characters of the total and Cauchy stress fields are analyzed near the circumference of the hole for different combinations of bi-axial tensions and for different parameter values. For the original Kirsch problem concerning a uni-axially stretched plate, the analytical solution fields for stresses and strains are compared to numerical results. These results are shown to be in a full agreement with each other and, in particular, they reveal a set of new qualitative findings about the scale-dependence of the stresses and strains provided by the gradient theory, not common to the classical theory. Based on these findings, we finally consider the physicalness of the concepts total and Cauchy stress appearing in the strain gradient model.

[1]  G. Lamé,et al.  Leçons Sur la Théorie Mathématique de L'élasticité des Corps Solides , 2009 .

[2]  E. Cosserat,et al.  Théorie des Corps déformables , 1909, Nature.

[3]  N. Eshel,et al.  Axi-symmetric problems in elastic materials of grade two , 1975 .

[4]  M. Lazar,et al.  Dislocations in second strain gradient elasticity , 2006 .

[5]  J. D. Eshelby.  The determination of the elastic field of an ellipsoidal inclusion, and related problems , 1957, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[6]  R. Peterson,et al.  Stress Concentration Factors , 1974 .

[7]  S. Deng,et al.  Wedge and Twist Disclinations in Second Strain Gradient Elasticity , 2007 .

[8]  E. Aifantis,et al.  Numerical modeling of size effects with gradient elasticity - Formulation, meshless discretization and examples , 2002 .

[9]  E. Aifantis,et al.  Eshelby’s inclusion problem in the gradient theory of elasticity: Applications to composite materials , 2011 .

[10]  N. Aravas.  Plane-Strain Problems for a Class of Gradient Elasticity Models—A Stress Function Approach , 2011 .

[11]  S. Markolefas,et al.  Theoretical analysis of a class of mixed, C0 continuity formulations for general dipolar Gradient Elasticity boundary value problems , 2007 .

[12]  H. Shodja,et al.  Ab initio calculations of characteristic lengths of crystalline materials in first strain gradient elasticity , 2013 .

[13]  J. Bleustein.  A note on the boundary conditions of toupin's strain-gradient theory , 1967 .

[14]  M. Lazar.  The fundamentals of non-singular dislocations in the theory of gradient elasticity: Dislocation loops and straight dislocations , 2012, 1209.1997.

[15]  T. Cook,et al.  Strain-gradient effects around spherical inclusions and cavities , 1966 .

[16]  Y. S. Zhang,et al.  Size dependence of Young's modulus in ZnO nanowires. , 2006, Physical review letters.

[17]  E. Aifantis,et al.  Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results , 2011 .

[18]  P. Sharma,et al.  Inclusions and inhomogeneities in strain gradient elasticity with couple stresses and related problems , 2005 .

[19]  Paul Steinmann,et al.  Isogeometric analysis of 2D gradient elasticity , 2011 .

[20]  B. Guzina,et al.  A physical perspective of the length scales in gradient elasticity through the prism of wave dispersion , 2013 .

[21]  Francesco dell’Isola,et al.  Geometrically nonlinear higher-gradient elasticity with energetic boundaries , 2013 .

[22]  Joel Goldstein,et al.  Measurement of the ratios of branching fractions B(Bs0 → Ds- π+)/B(B0 → D-π+) and B(B+ → anti-D0π+)/B(B0 → D-π+) , 2007 .

[23]  A. Weck,et al.  Plane stress local failure criterion for polycarbonate containing laser drilled microvoids , 2013 .

[24]  Antti H. Niemi,et al.  Variational formulation and isogeometric analysis for fourth-order boundary value problems of gradient-elastic bar and plane strain/stress problems , 2016 .

[25]  H. Georgiadis.  The Mode III Crack Problem in Microstructured Solids Governed by Dipolar Gradient Elasticity: Static and Dynamic Analysis , 2003 .

[26]  Gérard A. Maugin,et al.  Generalized Continuum Mechanics: What Do We Mean by That? , 2010 .

[27]  On dislocations in a special class of generalized elasticity , 2005, cond-mat/0504291.

[28]  Ioannis Vardoulakis,et al.  Two finite element discretizations for gradient elasticity , 2009 .

[29]  S. K. Park,et al.  Variational formulation of a simplified strain gradient elasticity theory and its application to a pressurized thick-walled cylinder problem , 2007 .

[30]  P. Sharmaa,et al.  A novel atomistic approach to determine strain-gradient elasticity constants : Tabulation and comparison for various metals , semiconductors , silica , polymers and the ( Ir ) relevance for nanotechnologies , 2007 .

[31]  M. Lazar,et al.  The Eshelby stress tensor, angular momentum tensor and dilatation flux in gradient elasticity , 2007 .

[32]  M. Mousavi.  Dislocation-based fracture mechanics within nonlocal and gradient elasticity of bi-Helmholtz type—Part II , 2016 .

[33]  C. Polizzotto.  A gradient elasticity theory for second-grade materials and higher order inertia , 2012 .

[34]  Xin-Lin Gao,et al.  Strain gradient solution for a finite-domain Eshelby-type plane strain inclusion problem and Eshelby's tensor for a cylindrical inclusion in a finite elastic matrix , 2011 .

[35]  Xin-Lin Gao,et al.  Solution of Eshelby's inclusion problem with a bounded domain and Eshelby's tensor for a spherical inclusion in a finite spherical matrix based on a simplified strain gradient elasticity theory , 2010 .

[36]  S. Lurie,et al.  Exact solution of Eshelby–Christensen problem in gradient elasticity for composites with spherical inclusions , 2016 .

[37]  Fan Yang,et al.  Experiments and theory in strain gradient elasticity , 2003 .

[38]  N. Aravas,et al.  Mixed finite element formulations of strain-gradient elasticity problems , 2002 .

[39]  Gérard A. Maugin,et al.  A Historical Perspective of Generalized Continuum Mechanics , 2010 .

[40]  Theodoros Triantafyllidis,et al.  Torsional vibrations of a column of fine-grained material: A gradient elastic approach , 2015 .

[41]  Julien Réthoré,et al.  Gradient-elasticity for honeycomb materials: Validation and identification from full-field measurements , 2015 .

[42]  X.-L. Gao,et al.  Green’s function and Eshelby’s tensor based on a simplified strain gradient elasticity theory , 2009 .

[43]  E. Aifantis,et al.  On Some Aspects in the Special Theory of Gradient Elasticity , 1997 .

[44]  R. D. Mindlin.  Micro-structure in linear elasticity , 1964 .

[45]  N. Fleck,et al.  FINITE ELEMENTS FOR MATERIALS WITH STRAIN GRADIENT EFFECTS , 1999 .

[46]  R. D. Mindlin.  Second gradient of strain and surface-tension in linear elasticity , 1965 .

[47]  X.-L. Gao,et al.  Strain gradient solution for Eshelby’s ellipsoidal inclusion problem , 2010, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[48]  Viacheslav Balobanov,et al.  Isogeometric Analysis of Gradient-Elastic 1D and 2D Problems , 2016 .

[49]  N. Eshel,et al.  Effects of strain-gradient on the stress-concentration at a cylindrical hole in a field of uniaxial tension , 1970 .

[50]  G. E. Exadaktylos,et al.  Gradient elasticity with surface energy: Mode-I crack problem , 1998 .

[51]  R. D. Mindlin,et al.  On first strain-gradient theories in linear elasticity , 1968 .

[52]  Antti H. Niemi,et al.  Differential cubature method for gradient-elastic Kirchhoff plates , 2015 .

[53]  Sergei Khakalo,et al.  Isogeometric analysis of higher-order gradient elasticity by user elements of a commercial finite element software , 2017, Comput. Aided Des..

[54]  Strain gradient solution for the Eshelby-type polyhedral inclusion problem , 2012 .

[55]  Wolfgang H. Müller,et al.  Applications of Strain Gradient Theories to the Size Effect in Submicro-Structures incl. Experimental Analysis of Elastic Material Parameters , 2022 .

[56]  R. Toupin.  Elastic materials with couple-stresses , 1962 .