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C Bagni et al,1 Introduction, The origin of the finite element method FEM dates back to the mid twentieth century in the aerospace indus. try 13 57 where many relevant structures especially rockets can be considered as axisymmetric hence the. stress analysis of complex axisymmetric structures has always been of primary importance This encouraged. the development of axisymmetric finite elements in the first half of the 1960s 58 However even if axisym. metric finite elements have been originally developed for aerospace applications they have found a wide. application also in other fields mechanical industry transport and containment of fluids since many struc. tural components are characterised by an axisymmetric geometry such as pressure and containment vessels. pipes cooling towers tunnels and shafts, Speaking about aeronautical and mechanical components then fracture and fatigue problems are of great. interest and an accurate estimation of the stress state is essential for a reliable assessment Here gradient. elasticity theories come into play in fact as extensively discussed and proven in the literature see for. instance 10 15 17 19 20 42 46 they are able to overcome the deficiencies of classical elasticity in accurately. determining the stress fields in specific problems especially those for which it is necessary to account for micro. and nanoscale effects in the description of the overall macroscopic response In particular gradient enriched. theories are able to remove singularities from micro stress and micro strain fields in the neighbourhood. of sharp crack tips and in general they have a smoothing effect on the stresses in the presence of stress. concentrators such as notches and holes Moreover they can interpret size effects in a robust and effective. manner in agreement with experiments, The idea of enriching the equations of classical elasticity with higher order spatial gradients was proposed. by Cauchy in the early 1850s with the aim of studying more accurately the behaviour of discrete lattice. models 22 24 However this idea remained dormant until the beginning of the twentieth century when the. Cosserat brothers 26 proposed to enrich the kinematics of the three dimensional continuum equations with. three micro rotations and to include the couple stresses in the equations of motion. Nevertheless the most significant revival of gradient theories took place in the 1960s when several efforts. were spent in the extension of the Cosserat theory as well as couple stress theories In particular Toupin 56. proposed to improve the theory of classical elasticity by enriching the strain energy function with the full. first gradient of the strain Few years later Mindlin suggested to define both kinetic and deformation energy. density also in terms of micro strains 43 and enriched the theory of classical elasticity by including the second. gradient of the strain 44 Classical elasticity was further extended by Green and Rivlin 38 by including all. the gradients of the strain The aforementioned research works aiming to include a mathematically complete. set of higher order gradients into the classical elasticity equations resulted in rather complicated theories. Finally about 20 years later gradient elasticity theories saw a second significant renaissance in particular. through the work of Eringen 29 31 and Aifantis et al 1 10 46 leading to simpler theories containing only. the higher order terms needed to describe the phenomena of interest In particular in 1993 Ru and Aifantis 46. proposed a split operator allowing the solution of the original fourth order Aifantis theory as an un coupled. sequence of two systems of second order p d e as described in Sect 3 decreasing the continuity requirements. from C 1 to C 0 allowing a straightforward finite element implementation of the theory Furthermore as explained. in 17 the stress based formulation of the Ru Aifantis theory presents significant advantages if compared. to the displacement based and strain based formulations of the same theory These are the main reasons why. in this paper as well as in 19 the stress based Ru Aifantis theory has been chosen for the finite element. implementation, In the literature it is possible to find several applications of gradient elasticity to axisymmetric problems. In particular strain gradient theories such as the Mindlin theory have been successfully applied to study. different axisymmetric problems such as stress concentrations at spherical inclusions and cavities 21 25. circular holes 32 33 Boussinesq problem 35 36 Other researchers instead provided analytical solutions. to axisymmetric problems such as thick walled cylinder 34 annulus under internal and external pressure. and circular hole in infinite body 12 starting from the gradient elastic formulation proposed by Aifantis et al. Although gradient elasticity has been widely applied to axisymmetric problems leading to several analyt. ical solutions for simple benchmark problems the complex geometries of modern engineering components. make analytical solution of practical problems nearly impossible and therefore numerical solutions become. essential However as far as the authors are aware no complete finite element implementation of gradient. elasticity for axisymmetric problems has been proposed until now. Gradient enriched finite element methodology, In this paper a reduced gradient version of the gradient enriched finite element methodology presented.
in 17 19 53 for plane stress strain and three dimensional problems has been developed for axisymmetric. problems with the aim of providing a comprehensive finite element methodology based on the Ru Aifantis. theory applicable to any kind of axisymmetric problem This includes even the non trivial case of non. axisymmetric loads where all the circumferential variables as well as their derivatives with respect to the. angular coordinate cannot be neglected while displacements and forces must be expressed in terms of. Fourier series Appropriate numerical integration schemes have also been established and comprehensive. convergence studies have been carried out allowing recommendations on optimal element size The proposed. methodology has also been used to analyse notched cylindrical bars subject to both static and fatigue loadings. showing that gradient elasticity could potentially be a very accurate and effective tool for the static and fatigue. assessment of notched components, After a brief description of the theoretical background of axisymmetric analysis and gradient elasticity the. ory provided respectively in Sects 2 and 3 in Sect 4 a possible C 0 continuous finite element implementation. of the Ru Aifantis theory with reduced gradient dependence is proposed for axisymmetric solids subject to. both axisymmetric and non axisymmetric loads In Sect 5 details about the best integration rules to adopt are. provided while in Sect 6 the proposed methodology is applied to solve the problem of an hollow cylinder. subject to internal pressure and the numerical results compared to the analytical solution proposed by Gao and. Park 34 In Sects 7 and 8 the convergence of the proposed methodology is studied for problems without. and with singularities respectively furthermore in Sect 8 recommendations on optimal element size are. provided Section 9 is dedicated to applications of the proposed methodology to different problems which. highlight the main advantages of the present gradient enriched methodology In particular these advantages. consist of the stress smoothing in the presence of stress risers removal of singularities from the stress fields. around the tips of sharp cracks which allows a more accurate static and fatigue assessment of components. characterised by stress concentrators and last but not least the ability to describe size effects. 2 Axisymmetric analysis,2 1 Axisymmetric loading, It is well known from the literature see e g 59 that in axisymmetric solids subject to axisymmetric loading. displacements strains and stresses are all independent of the circumferential coordinate of the cylindrical. coordinate system defined in Fig 1 This particularity enables the study of such a problem by simply considering. the generic plane section of the solid along its axis of revolution z shaded in Fig 1 subject to in plane loading. The displacement field is described by two components only since the circumferential component u 0. for axisymmetry reasons which are functions of the radial r and axial z coordinates only. u r z u r r z u z r z T 1, Fig 1 Axisymmetric solid cylindrical coordinate system and generic plane section of the solid to analyse. C Bagni et al, In what concerns the strains they can be collected in the strain vector. r z rr r z zz r z r z 2 r z r z T 2, where ignoring the spatial dependence for notational simplicity.
u r u z ur u z u r,rr zz 2 r z 3, Finally in the case of linear elastic materials the stress field is linked to the strain field through the following. constitutive relation,r z 0 0 0 1 2, where E and are the Young s modulus and Poisson s ratio respectively. 2 2 Non axisymmetric loading, In the case of an axisymmetric solid subject to non axisymmetric loads the circumferential component of the. displacements u must be taken into account in addition to the radial and axial components As a consequence. of this none of the strain and stress components can be considered null in particular the strain vector assumes. the following form ignoring the spatial dependence for notational simplicity. rr zz 2 r z 2 r 2 z T 5,where in addition to Eq 3 we have. 1 u r u u 1 u z,2 r u 2 z 6,and the stress vector is now defined as.
rr 1 0 0 0,2 0 0 0 2 0,z 0 0 0 0 0 1 2, However it is still possible to solve this problem as a quasi bidimensional problem by expressing the load and. displacement components through Fourier series see e g 58 60. Gradient enriched finite element methodology, 3 Gradient elasticity theory and the Aifantis GradEla model. Gradient elasticity is a generic name given to a family of generalised continuum elasticity theories which. suggest to take into account the effects of the micro structure on the mechanical behaviour of a component. by including higher order strain or stress in the constitutive equation Such inclusion results to overcome. many of the limits of classical elasticity in describing particular phenomena amongst which are the removal. of singularities from strain and stress fields at crack tips the smoothing of these fields at stress concentrators. as well as the capture of size effects This ability is due to the introduction of internal length parameters the. phenomenological multipliers of the higher order gradient terms which in turn appear in the solution of. corresponding boundary value problems The magnitude of these internal length parameters proportional to. the size of the underlying micro structure of the material in relation to a characteristic specimen dimension. controls the component behaviour and allows for the descriptions of the aforementioned non standard effects. In 15 it is argued that Laplacian based gradient theories are the most versatile One of the most renowned. Laplacian based theories is the GradEla model developed by Aifantis et al 1 10 46 which represents a very. simple and effective approach characterised by just one additional parameter where the constitutive relations. are enriched by means of the Laplacian of the strain as follows. i j Ci jkl kl 2 kl mm 8, where i j and kl are respectively the stress and strain tensors Ci jkl is the constitutive tensor and is the. internal length scale The inclusion of the Laplacian is not arbitrary but it appears naturally when relating local. micro and average macro response and expanding the corresponding non local integral in a Taylor series. for appropriate forms of the influence kernel This leads to the following fourth order equilibrium equations. Ci jkl u k jl 2 u k jlmm bi 0 9, where u k is the displacement vector and bi are the body forces. Ru and Aifantis 46 developed a factorisation procedure which allows the solution of Eq 9 as a decoupled. sequence of two systems of second order partial differential equations p d e considerably simplifying the. finite element implementation of this model The first system of second order p d e to be solved is the standard. classical elasticity equilibrium,Ci jkl u ck jl bi 0 10.
from which it is possible to determine the field of the local displacements u ic that will be used as source term. in the following second set of second order p d e,u k 2 u k mm u ck 11. where u i represents the field of the non local or gradient affected displacements. Finally as shown in 17 39 40 and also discussed in 3 5 Eq 11 can be redefined in terms of stresses. i j 2 i j mm Ci jkl u ck l 12, where i j represents the field of the gradient enhanced stresses. 4 Implementation aspects, The Ru Aifantis theory described in Sect 3 has been already implemented in a unified finite element method. ology for both two and three dimensional problems 19 showing good convergence properties as well as. the ability to remove singularities from the stress field in the neighbourhood of stress risers While the afore. mentioned theory is based on the use of the full gradient of the stresses in this paper a simplified version of. the Ru Aifantis theory characterised by the use of a reduced gradient of the stresses is developed and imple. mented in a straightforward C 0 finite element framework for the analysis of axisymmetric solids subject to both. axisymmetric and non axisymmetric loads The choice of using a reduced gradient allows to obtain a simplified. yet reliable methodology more easily implementable in a finite element framework without renouncing to the. ability to remove singularities from the stress fields see 37 for another example of reduced gradient theory. C Bagni et al,4 1 Axisymmetric loads, The index notation used in Sect 3 is valid in Cartesian coordinates while axisymmetric problems are usually. addressed in cylindrical coordinates as in the following of the present paper Therefore in order to derive. the finite element equations for axisymmetric problems unambiguously we will now depart from the index. notation used in the previous section adopting a tensorial notation Furthermore two functionals one for each. of the two steps of the Ru Aifantis theory are defined as follows. W1 C d u b d,for the first step and,1 2 g gT g,S S d gT c d 14.
for the second where uc and c are respectively the local displacement and local strain vectors g is the. non local stress vector C is the constitutive matrix defined in Eq 4 S C 1 is the compliance matrix b. is the vector of the body forces in the domain t is the vector of the prescribed traction on the free portion. n of the boundary, Imposing the stationarity of the first functional that is W1 0 the usual global system of standard. elasticity equations is obtained,Kdc f 0 15, where d is the vector of the nodal local displacements K Bu CBu d is the stiffness matrix and. n the force vector, Repeating the same procedure for the second functional W2 leads to. 2 N N N T N,N T Bu d dc 0 16, where sg is the vector of the nodal non local stresses and N is just an expanded version of the shape function. matrix Nu in order to accommodate all four non local stress components so that g N sg. At this point solving Eq 15 for dc and using the result as source term in Eq 16 the nodal values of. the gradient enriched stresses sg can be easily calculated. 4 2 Non axisymmetric loads, The finite element equations for the case of axisymmetric solids subject to non axisymmetric loads can be.
determined by following the same process described in Sect 4 1. Regarding the first step of the Ru Aifantis theory the functional defined by Eq 13 can be considered by. taking into account that now the constitutive matrix is C defined in Eq 7 and the displacements including. now also the circumferential component are described by means of Fourier series as. uc sh Nu dc s,h ah Nu dc a,cos h 0 0 sin h 0 0,sh 0 cos h 0 and ah 0 sin h 0 18. 0 0 sin h 0 0 cos h, where h is the order of the harmonic while the superscripts s and a indicate respectively the symmetric and. anti symmetric components of the displacements and of the loads with respect to the 0 axis. Gradient enriched finite element methodology,The local strains are defined as follows. and substituting Eq 17 into 19,Bsu h dc s,h Bau h dc a. where Bsu h L sh Nu and Bau h L ah Nu are the two strain displacement matrices for symmetric and. antisymmetric displacements loads respectively, After these considerations the first step of the Ru Aifantis theory still consists in Eq 15 with the.
difference that now also the nodal forces are expressed in terms of Fourier series as. fr h cos h fr h sin h,s cos h a sin h,f fz f f 21,s sin h f a cos h. and the stiffness matrix is defined as, where khl is the stiffness contribution of the hth and lth harmonics given by. khl Bu h C Bu l dV,C Bu l Bu l dV,BsT C Bs BsT C Ba. u h u l u h u l,u h C Bu l Bu h C Bu l, Considering that dV r dr dzd r d dA it can be easily demonstrated that. u h C Bu l r d dA u h C Bu l r d dA 0, which means that the symmetric and anti symmetric terms are decoupled as it should be given the orthogo.
nality of the Fourier series Furthermore it can also be proved that. u h C Bu l r d dA u h C Bu l r d dA 0 when h l,C Bagni et al. Hence the off diagonal terms of the global stiffness matrix K are all null and each harmonic can be treated. separately allowing application of the principle of superposition to determine the final result. In what concerns the second step of the Ru Aifantis theory considering for simplicity only the symmetric. components defining the following functional,W2 s g s T g s. S s h h s h h dV,s h h s h h S,s g s T c s,where S C 1 while s h is defined as see e g 58. cos h 0 0 0 0 0,0 0 cos h 0 0 0,0 0 cos h 0 0,0 0 0 0 sin h 0. 0 0 0 0 0 sin h,2 that is W, and imposing the stationarity of W 2 0 the following solving system of equations is obtained.
N T sT s N N T sT s N,2 N T N dV sh N T sT s,h CBu h dV dc s. which can be easily numerically solved for the non local stress sh The same system must be solved also. for the anti symmetric components if needed Finally once the solutions for all the necessary harmonics are. calculated the global solution can be determined through superposition of the effects. 5 Numerical integration, For the numerical solution of Eqs 15 16 and 28 the Gauss quadrature rule has been used As summarised. in Table 1 in what concerns Eq 15 first step since it coincides with the p d e of classical elasticity the stan. dard number of integration points has been considered in particular the bi quadratic serendipity quadrilaterals. have been under integrated, For the numerical solution of Eqs 16 second step with axisymmetric loads and 28 second step with. non axisymmetric loads instead higher integration rules are needed except for the bi linear quadrilateral. elements if 0 as reported in Table 1 On the other hand if 0 the quadrature rules used for the. first step can be adopted for all the elements without any rank deficiency problem similar to the 2D plane. strain stress and 3D problems as described in 19,Gradient enriched finite element methodology. Table 1 Number of Gauss points used in the first step of the Ru Aifantis theory and formally required in the second step when. Elements Order Gauss points,1st step Triangles Linear 1.
Quadratic 3,Quadrilaterals Bi linear 2 2,Bi quadratic 2 2. 2nd step Triangles Linear 3,Quadratic 6 degree of precision 4. Quadrilaterals Bi linear 2 2,Bi quadratic 3 3, 6 Internally pressurised hollow cylinder comparison with Gao and Park 2007. The analysed problem consists of a thick walled hollow cylinder of inner radius a 0 5 m outer radius. b 1 5 m and length L 8 m subject to an internal pressure pi 10 MPa Fig 2 The material is. characterised by a Young s modulus E 1000 MPa Poisson s ratio 0 25 and the length scale has. been set as 0 1 m Concerning the boundary conditions for the first step Eq 15 the cylinder is simply. supported in the axial direction at both ends Fig 2 while for the second step Eq 16 homogeneous natural. boundary conditions are taken throughout since as also described in 15 this choice is the most widely. accepted amongst the scientific community when dealing with gradient elasticity. For symmetry of the cylinder with respect to a plane normal to the z axis only half of the cross section. has been modelled using 16 64 bi linear and bi quadratic quadrilateral elements and the double of linear. and quadratic triangular elements note that for triangular elements the meshes are obtained by subdividing. each quadrilateral in two triangles Since the displacements determined according to the proposed methodol. ogy correspond to the classical elastic displacements the numerical solution for the displacements has been. compared with the following analytical solution of classical elasticity known from the literature 54. pi a 2 r 1 b2,ur 1 2 1 29,E b2 a 2 r2, Regarding the stresses instead the gradient enriched stress fields numerically obtained by applying the devel. oped methodology have been compared to the analytical solutions proposed by Gao and Park 34. Fig 2 Benchmark problem geometry and boundary conditions. C Bagni et al,Classical elasticity,7 Proposed methodology 2.
6 Gao and Park 2007,Proposed methodology,0 5 0 7 0 9 1 1 1 3 1 5 0 5 0 7 0 9 1 1 1 3 1 5. Gao and Park 2007 Gao and Park 2007,10 Proposed methodology Proposed methodology. 0 5 0 7 0 9 1 1 1 3 1 5 0 5 0 7 0 9 1 1 1 3 1 5, Fig 3 Comparison between the numerical solutions obtained with bi quadratic quadrilateral elements and their correspondent. analytical counterparts, In Fig 3 the displacement and stress fields obtained by using bi quadratic quadrilateral elements are. compared with the relative analytical solutions Similar results have also been obtained with all the other. implemented elements, From Fig 3 it can be observed that the numerical estimation of the radial displacements u r perfectly matches.
the correspondent analytical counterpart Regarding the stress components instead the numerical solutions. obtained by applying the proposed methodology show some differences when compared with the analytical. solutions proposed by Gao and Park 34 In particular it is possible to observe that both the numerical and. zz stress profiles present qualitative and quantitative differences if compared to their analytical counterparts. This is due to the fact that while the methodology presented in this paper is stress based the analytical. solutions proposed by Gao and Park 34 are obtained through a displacement based formulation and therefore. the stress components are calculated following different procedures Furthermore the displacement based. formulation used by Gao and Park 34 requires just two higher order boundary conditions in particular they. set homogeneous natural boundary conditions of the radial stress at the inner and outer surfaces while the stress. based methodology proposed in this paper requires more higher order boundary conditions homogeneous. natural boundary conditions of all the stress components at the inner and outer surfaces have been chosen. Therefore the aforementioned differences are also due to a different choice of the boundary conditions Finally. it can be easily explained why zz obtained by applying the proposed methodology should be constant To do. so let us consider first the exact stress solutions of classical elasticity known from the literature 54. pi a 2 b2 pi a 2 b2 pi a 2,rr 1 2 1 zz 2 30,2 r2 b2 a2. In particular it can be observed that zz is constant along the radial coordinate and therefore due to the way. the proposed methodology post processes the stress fields the longitudinal stress component is not affected. by any gradient enrichment leading to zz zz const, Only quantitative differences can be seen instead in the radial stress component due to the fact that while. in the proposed methodology the stress components are uncoupled in the analytical solution proposed by Gao. and Park 34 the stress components are coupled,Gradient enriched finite element methodology. Furthermore the analytical solution proposed by Gao and Park 34 is obtained through a full gradient. methodology while as already explained in Sect 4 the methodology proposed in the present paper is based. on the use of a reduced gradient, Due to the aforementioned differences although justified the analytical solution proposed by Gao and. Park cannot be used as benchmark solution for the convergence study performed in the following section where. the reference solution will be approximated through Richardson extrapolation 45. 7 Convergence study, The convergence behaviour of the implemented elements in the case of both axisymmetric and non.
axisymmetric loads has been studied For this purpose the L2 norm error defined as. where e and c are respectively the extrapolated and calculated values of the stresses has been plotted against. the number of degrees of freedom nDoF From now on the reference solutions have been approximated. through Richardson extrapolation 45, As already explained in 19 the stresses are determined as primary variable by solving the second step of. the Ru Aifantis theory either Eq 16 or 28 and not as secondary variables like in standard finite element. methodologies based on classical elasticity This means that although the error conserves its proportionality. with the nDoF the theoretical convergence rate for the stresses is different from the one determined in classical. elasticity In particular Helmholtz equations such as Eqs 16 and 28 are characterised by the following. proportionality 41,e O n DoF 2 O n DoF p 32,where n is the polynomial order. Equation 32 clearly shows that the e 2 nDoF curve in a bi logarithmic system of axis is represented. by a straight line whose slope represents the convergence rate p of the numerical solution to the extrapolated. solution In particular the theoretical convergence rate is p 1 for linear elements and p 1 5 for. quadratic elements, In what concerns the case of axisymmetric load the thick walled hollow cylinder problem presented in. Sect 5 has been considered The domain has been modelled by using all the implemented elements starting. from a 4 16 mesh and refining the mesh by doubling the number of elements in both directions 4 times up. to a 64 256 mesh The length parameter has been set as 0 1 m. Regarding the case of axisymmetric solids subject to non axisymmetric loads two different problems have. been analysed The first one consists of a cylindrical bar of radius R 1 m and length L 8 m subject to a. bending moment M 106 Nm Fig 4 left while the second one consists of the same cylinder subject now. to a torsional moment T 106 Nm Fig 4 right In both cases Young s modulus E 1000 MPa Poisson s. ratio 0 25 and the length scale has been set as 0 1 m. For symmetry of the cylinder with respect to a plane normal to the z axis in both cases only half of the. cross section has been modelled Fig 4 Also in this case all the implemented elements have been used to. discretise the domain starting from a 4 16 mesh and performing the same mesh refinement described in the. case of axisymmetric loads Concerning the boundary conditions in both problems for the first step Eq 15. with K defined by Eq 22 homogeneous essential boundary conditions have been imposed so that u z 0. bending problem and u z u 0 torsional problem along the axis of symmetry Fig 4 while for the. second step Eq 28 homogeneous natural boundary conditions have been chosen as in the previous case. In Fig 5 the L2 norm error is plotted against the nDoF in a bi logarithmic system for the three analysed. problems from which it is possible to appreciate that all the implemented elements produce numerical solutions. characterised by convergence rates in good agreement with the theoretical predictions when applied to any. kind of axisymmetric problem,C Bagni et al, Fig 4 Cylindrical bars under pure bending left and pure torsion right geometry and boundary conditions. Fig 5 Stress error versus nDoF for internal pressure in hollow cylinder left and plain cylindrical bars subject to pure bending. centre and pure torsion right The slope of the lines represents the convergence rate. 8 Convergence in the presence of singularities and recommendations on optimum element size. As is well known 1 3 5 10 46 one of the most important abilities of gradient elasticity is the removal of. singularities from the strain and stress fields such as those obtained at the tips of sharp cracks when classical. elasticity is used See also the analyses in 15 17 19 as well as a more recent discussion in 6 7 Since the. presence of cracks reduces drastically the convergence rate of standard finite element methodologies very. fine meshes are required in the neighbourhood of the stress riser to ensure that convergence of the solution is. reached with significant increasing in computational cost Hence the analysis of the convergence behaviour of. the proposed gradient enriched finite element methodology is of great interest Furthermore an accurate error. estimation has allowed the formulation of recommendations on optimal element size also for axisymmetric. finite elements that together with the recommendations provided in 19 for plane and three dimensional finite. elements constitutes a complete and useful meshing guideline for gradient enriched finite elements. To analyse the aforementioned aspects three different problems Fig 6 consisting of a cylindrical bar. characterised by the presence of a circumferential crack and subject respectively to axial load F 106 N. axisymmetric bending moment M 106 Nm and torque T 106 Nm non axisymmetric loads have. been studied The cylinder has a radius R 1 m a length L 8 m and the crack is 0 25 m deep Material. properties boundary conditions and meshes are taken as in Sect 7. As in Sect 7 the convergence rate is determined as the slope of the straight line obtained by plotting in a. bi logarithmic graph the L2 norm error defined by Eq 31 versus the nDoF Fig 7 From the literature 59. it is known that in the presence of singularities the error on classical stresses is proportional to the nDoF as. e O nDoF min n 2 O nDoF p 33, where 0 5 for a nearly closed crack which leads to a theoretical convergence rate p 0 25 for both.
linear and quadratic elements,Gradient enriched finite element methodology. Fig 6 Cracked cylindrical bars under uniaxial tensile load left pure bending centre and pure torsion right. Fig 7 Stress error versus nDoF for cracked cylindrical bars subject to uniaxial tensile load left pure bending centre and pure. torsion right The slope of the lines represents the convergence rate. From Fig 7 it is possible to observe that when singularities are involved both linear and quadratic. elements show approximatively the same convergence rate in accordance with Eq 33 Furthermore the. proposed gradient enriched methodology produces a significant improvement in terms of convergence rate in. particular the solutions of the three analysed problems are all characterised by a convergence rate almost three. times higher than the correspondent theoretical value typical of standard finite element methodology based. on classical elasticity As already explained in 19 this improvement is due to two main factors. removal of singularities from the stress fields, gradient enriched stresses are primary variables instead of secondary variables as in standard classical. elasticity based finite element methodologies, Analysing then the error affecting the numerical solutions determined through Eq 31 it is possible to. provide recommendations on optimal element size Table 2 summarises the ratios between the element size. and the length scale that ensure an error of about 5 or lower The recommendations summarised in Table 2. highlight the fact that using the proposed methodology even in the presence of singularities it is possible to. obtain accurate solutions using a fairly coarse mesh reducing consistently the computational efforts. 9 Applications,9 1 Removal of singularities, As already mentioned in Sect 1 one of the features of the proposed methodology is the ability to remove. singularities from the stress field such as those emerging near the tip of sharp cracks This ability can be easily. C Bagni et al, Table 2 Recommendations on optimal element size to guarantee an error of 5 or lower.
Elements Order Element size,Triangles Linear 1,Quadratic 5 2. Quadrilaterals Bi linear 3 2,Bi quadratic 5 2, shown by considering the problems presented in Sect 8 and comparing the stress profiles obtained by applying. the proposed methodology with those produced by classical elasticity In Fig 8 in fact it can be seen that while. the classical elastic stress fields are characterised by an unbounded peak in correspondence with the crack. tip the gradient enriched ones converge to a unique finite solution upon mesh refinement the stress profiles. presented in Fig 8 were obtained by using bi quadratic quadrilateral elements similar results are produced by. all the other implemented elements,9 2 Static and fatigue assessment of notched bars. The aforementioned ability to remove singularities or more in general the stress smoothing in correspon. dence with stress concentrators leads to more physically realistic results and therefore to more accurate static. and fatigue assessments of components presenting stress risers In particular it allows the static and fatigue. assessments by considering the values of the relevant stresses directly at the crack notch tip and not into the. analysed body as it happens in most of the existing assessment procedures with evident simplifications in the. assessment process, In 16 48 similarities and differences between gradient elasticity and the Theory of Critical Distances. TCD for an in depth description of this theory see 52 are investigated showing the advantages of these. two theories in the static and fatigue assessment of cracked components Furthermore in 48 a relation linking. the two length parameters characterising the aforementioned theories has been proposed namely. where L is the characteristic length of the TCD defined as in 52. The proposed methodology has been applied to notched cylindrical bars Fig 9 subject to both static and. fully reversed cyclic loadings load ratio R 1 in order to show the accuracy of the proposed methodology. with defined through Eq 34 in the determination of both static and fatigue strength of notched components. for more results see also 20, The accuracy of the proposed methodology has been tested against a wide range of materials both brittle.
and ductile and notch root radii, All the relevant data about the analysed problems are summarised in Tables 3 and 4 where 0 is the inherent. material strength th is the ultimate tensile nominal stress 0 is the plain fatigue limit stress range and. th is the range of the nominal fatigue strength all the nominal stresses are referred to the gross section of. the specimens, The accuracy of the proposed methodology in estimating both the static strength and the fatigue limit of. notched cylindrical bars has been verified by defining the following errors. eff 0 eff 0,error and error 35, for static and fatigue problems respectively where eff and eff are respectively the stress for static. problems and the stress range for fatigue problems numerically obtained at the notch tip. In Fig 10 the aforementioned errors are plotted for the different analysed materials from which it is. possible to appreciate the accuracy of the proposed methodology with errors mainly ranging between 10. and 30 dashed lines Considering the usual error band 20 it is possible to observe that the present. methodology produces results with errors falling into a band of the same size but 10 more conservative. Gradient enriched finite element methodology,4x16 elements 4x16 elements. 15 8x32 elements 15 8x32 elements,16x64 elements 16x64 elements.
32x128 elements 32x128elements,10 64x256 elements 64x256 elements. 0 0 2 0 4 0 6 0 8 1 0 0 2 0 4 0 6 0 8 1,7 4x16 elements 4x16 elements. 8x32 elements 6 8x32 elements,6 16x64 elements,16x64 elements. 5 32x128 elements 32x128 elements,64x256 elements zz Pa 4 64x256 elements. 0 0 2 0 4 0 6 0 8 1 0 0 2 0 4 0 6 0 8 1,4x16 elements 4x16 elements.
3 8x32 elements 3 8x32 elements,16x64 elements 16x64 elements. 32x128 elements 32x128 elements,2 64x256 elements 2 64x256 elements. 0 0 2 0 4 0 6 0 8 1 0 0 2 0 4 0 6 0 8 1, Fig 8 Stress profiles obtained by applying classical left and gradient right elasticity for uniaxial tensile loading top pure. bending centre and pure torsion bottom,Fig 9 Geometry of the cylindrical notched bars.

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