J. Sladek, V. Sladek & M. Hrina Institute of Construction and Architecture, Slovak Academy of Sciences, Bratislava, Slovakia
|
|
- Roland Moore
- 5 years ago
- Views:
Transcription
1 Evaluation of fracture parameters for functionally gradient materials J. Sladek, V. Sladek & M. Hrina Institute of Construction and Architecture, Slovak Academy of Sciences, Bratislava, Slovakia Abstract Efficient numerical methods are presented to compute fracture parameters (stress intensity factor and T-term stress) for a cracked composite material body subjected to a stationary thermal load. The path independent integral representations for stress intensity factors and T-term stress in functionally gradient materials are derived. The thermomechanical fields along the integration path and in the enclosed domain are obtained by the boundary element method. The method is appropriate for computation of fracture parameters because the contour integral is evaluated over the path, which is far away from the crack tip. Numerical results are given for a thick-walled tube with edge cracks on the internal surface. 1 Introduction Cracking of structures due to thermal loading is an important phenomenon in many industrial applications, especially aerospace and nuclear engineering. Only composite materials can resist high temperatures effectively and simultaneously the thermal stresses could be relaxed significantly. Therefore, in recent years, the concept of so-called functionally gradient materials (FGM) has been introduced and applied to the development of structural component. In FGM the volume fractions are optimized to satisfy both operational and minimal stress requirements. In such materials, although the absence of sharp interfaces does largely reduce material property mismatch, cracks occur when they are subjected to external loading. Ceramics are brittle materials and microcracks play crucial roles in determining the strength and life of components under service conditions.
2 36 Damage and Fracture Mechanics VI Conventional fracture theories assume that the state of stress and strain in the vicinity of a crack tip is characterized by a single parameter. This single parameter, frequently called the first fracture parameter, can be the stress intensity factor (SEF) or J-integral. In literature there is a lot of original techniques for evaluation these fracture parameters. However, they are mostly restricted to homogeneous materials. Recently, some results have been published, where the SIF for a crack in the FGM is computed [1-3]. In all above mentioned papers the analytical methods have been used for solution of boundary value problems. In analytical methods the selection of a technique for evaluation of fracture parameters is not such significant as in a pure numerical approach. Namely, in numerical methods the most inaccurate quantities are obtained at the crack tip vicinity. Then, the results obtained directly from asymptotic expansion formulae are less accurate than those obtained from integral representations along a contour far away from the crack tip. Early theories of fracture mechanics assume that the stress and displacement states are controlled by the stress intensity factor. Significant quantitative changes could occur when the leading nonsingular terms of the crack tip series expansion are included. Recent numerical and experimental studies in elastostatics have attempted to describe fracture by using two parameters such that additional information could be gained. This second parameter is nonsingular corresponding to the uniform stress term in crack tip series expansion. It shall be referred to as the T-term stress [4]. Many possibilities exist for evaluation of the T-term [5-7]. Like to the SIF, all suggested methods are restricted to homogeneous bodies. The purpose of this paper is to present an accurate computational method for evaluation of the T-term stress in functionally gradient material bodies subjected to stationary thermal loading. The reciprocity theorem will be applied to derive a path independent integral for evaluating the T-term stresses. Suggested computational methods are applied to numerical examples for a thick-walled cracked tube. 2 Evaluation of thefirstfracture parameter In stationary thermoelasticity for non-homogeneous bodies the thermal and stress fields are described by the following governing equations [8] where 9 -g (1) 0 (2) is the temperature measured in the scale with its origin at the equilibrium state, %, and Q are body force vector and heat source, respectively and o-jj is stress tensor. The coefficient of heat conduction A is a continuous function of cartesian coordinates. A subscript preceded by a comma denotes a differentiation with respect to the corresponding cartesian coordinate. The strain tensor s^ is given by displacement gradients
3 Damage and Fracture Mechanics VI 37 In the presence of temperature gradient, the total strain tensor can be decomposed into its elastic part e^ and another one accounting for the free thermal expansion of medium. Thus, f,y=f*+a^0 (4) where a is the coefficient of thermal expansion and 5^ is the Kronecker delta. According to the Neumann hypothesis the stress tensor is related to the elastic strain in the usual way, viz 0",y=%,4 (5) where c^ is the tensor of material constants. For an isotropic continuum it is given by where // is the Lame constant (shear modulus) and v is Poisson's ratio. Note that the material is non-homogeneous in general, since the elasticity tensor and the coefficient of thermal expansion depend on position. The elastic strain energy density W = W(e^,x^ can be written in the following form Then, gradient of strain energy density is given as w *4,2 (6) "" &V*» l^l, ^^ UJexp, where "explicit" derivation of the strain energy density for non-homogeneous material and temperature gradient fields becomes ah " Y^K"^ "" ^'^^ ^ "" ^ ^y^^ (^) \ /exp/ Utilizing eqs. (3) and (5) the gradient of strain energy density can be rewritten into the form ^,m = k^w))-^.y^ +KJ^ (9) Then, from eqs. (9) and (2) it follows that W/m -^^wjly = ^^,m +K^Lpy (10) An integral form of equation (10) may be obtained upon application of the divergence theorem. If Q is a regular bounded region enclosed by a surface T whose unit outward normal vector is /*, it follows that
4 38 Damage and Fracture Mechanics VI (11) Integral identity (11) is valid in a region where no field irregularities prevail. In a crack problem the stresses at the crack tip are singular and displacements are discontinuous on both crack surfaces. Therefore, a small region Q^ has to be excluded at the vicinity of a crack tip. This region is surrounded by F^ as shown in Fig. 1. Fig. 1. Integration paths and coordinate definitions. Both fields cr^ and %, are regular in the region Q-Q^. Contour F = FQ + F^ - T^ + r~ is a closed integration path in the anticlokwise direction. The radius s is considered to be very small and shrunk to zero in the limiting process. The crack surfaces F*,F~ are assumed to be free of tractions, tj = (Tijtij = 0, and crack is parallel to the x\ - axis. Then, eq. (11) can be written as -lim "o-o., (12) The left hand side of eq. (12) is equal to the definition of J - integral [9] for m = 1. Then, o-og (13) According to Sih et al. [10] the local character of thermal stresses at the crack tip vicinity is the same nature as problems with static mechanical stresses. Noda and Jin [11] analysed the asymptotic distribution of stresses for cracks in a nonhomogeneous body. It is found that the crack tip fields are the same as those in
5 Damage and Fracture Mechanics VI 39 the homogeneous material provided that the material properties of FGM are continuous. Finally, the stress intensity factors can be evaluated from the wellknown relation between Kj, K^ and the path independent J - integral [9]. The advantage of the presented method for computation of stress intensity factors through the J - integral technique consists in high accuracy because the contour integral is evaluated over the path far away from the crack tip. Inaccuracies of numerical calculations at the crack tip vicinity are avoided on this approach. Moreover, the discretization mesh is not required to be so fine as in a direct computation of SIF from the asymptotic expansion formulae. In such case results are sensitive to the distance of a selected point from the crack tip. 3 Evaluation of the second fracture parameter The T-term stress can be computed directly from the asymptotic expansion of stresses or displacements, if stress intensity factors and stress or displacement values at nodal points close to the crack tip (the validity of asymptotic expansions is assumed) are obtained from a numerical analysis. A drawback of this method is substantial dependence of results on the distance of a selected point from the crack tip. Therefore, it is required to suggest a method in which the T-term stress is expressed in terms of the solution at points far away from the crack tip. In the following an integral representation of the T-term stress will be found for a cracked body analysed in the framework of stationary thermoelasticity. Let { cr,y,%,,0 } and { a*-,w* } be two systems of fields (stresses, displacements and temperature) governed respectively by equations (l)-(5) and equations * = 0 (14) where c^ can be obtained from c..^ by replacing ju with //Q = const. In view of eqs (4), (5) and (15), one may write *^ - cjy^jin =J Iju-^-aOi nl_ i-^y (16) Having been interested in two-dimensional problems, one can rewrite the last identity, after some manipulations into the form
6 40 Damage and Fracture Mechanics VI where E 3-/ 2VK~\ for an isotropic material with E = 2//(l + v) and 13-4v for plane strain f v for plane strain 3-v i/=< v - for plane stress - for plane stress 1+v ^ U+v Applying the Gauss divergence theorem to eq. (17) and utilizing the governing equations (2) and (14), we get r with Q (18) J_ 2 _1-2F "'" The integral identity (18) is valid in a region, which is free of any irregularities. To avoid crack tip irregularities it is necessary to consider the region Q- O.^ bounded with the integration contour r - YQ + F* -F^ + F~ shown in Fig. 1. Then, eq. (18) can be written as -lim }F(0^^,^^,)dn (19) In the section 2 of this paper it is mentioned that the thermal stresses at the crack tip vicinity in material with a continuous non-homogeneity have the same singularity and the angular distributions as the mechanical stresses in a homogeneous material [12]. Let us try to find an appropriate auxiliary solution (u *,er *^ ), which enables us to obtain the integral expression of the T-term stress. In order to have a nonvanishing contribution of this term in the l.h.s of eq. (19), the auxiliary displacement w* must necessarily vary as r~'. Then, the traction vector /* is
7 Damage and Fracture Mechanics VI 41 proportional to r~*. Now, the products of auxiliary fields with the singular terms in the asymptotic expansions yield singular integrands on T^. These singular terms have to be cancelled out only due to the angular variation of the auxiliary fields. The asymptotic displacements and stresses can be split into two parts %*=<+%! and <?*/, =0-^+0-^, (20) where the superscript s denotes the terms which contain the stress intensity factors in the asymptotic expansion [12] and "a = -fki (1-1/2 )cosp - <^y(i +,/) sinp] (21) The auxiliary fields utilized in Kfouri's integral expression [5] of T-stresses have one order lower singularity than it is required for our auxiliary fields (u *, er*^ ). The first derivative of Kfouri's auxiliary fields (^, cr*/? ) with respect to coordinate component xi will satisfy above mentioned singularity of displacements and stresses. Substituting requirements on the %*=%,, and cr*=<7,y,i (22) are substituted into the l.h.s. of eq. (19), one can verify identity = 0 (23) Thus, the auxiliary fields given by eq. (22) satisfy requirements for cancellation of the singular terms. Their explicit expressions are given in [7]. On the other hand, the T-terms (u^, a^ ) give a finite contribution lim f /,"y,%i - o^%;fr = 19" (24) *-+Ur, ///o Using eqs. (20), (23) and (24), the integral representation of the T-term stress from eq. (19) becomes where (25) Unknown displacements, traction vector and temperature, required along the integration path and within of domain enclosed by the integration contour in eq. (25) can be received from a numerical or experimental analysis. In this paper the boundary element method is used.
8 42 Damage and Fracture Mechanics VI 4 Numerical results An infinitely long thick-walled tube with two radial cracks in opposite directions is considered. The cracks are situated on the internal surface of the tube. A permanent temperature gradient is prescribed with temperature T% = 30 C on the internal surface (radius R% = 8cm) and T: = 200 C on the external surface (% = 10cm). The shear modulus is assumed to be dependent on temperature linearly where 0 = 7-7,, ju^ = 310^ MPa and Poisson's ratio is a constant, y =0.3. The coefficient of thermal expansion a is assumed to be independent on temperature, a = a^ = ~"(deg~^). Three various relaxation parameters #1 are considered in the numerical analysis, B, =15, 40 and 63MPa/deg. The functionally gradient material with prescribed material behaviour is fabricated in such a way that the volume fractions of ceramics and metals are varied continuously in a predetermined composition profile. For material #3 (#1 = 63MPa/deg) the ratio of shear moduli on internal and external surface of tube is given as //, / ju^ = at the prescribed temperature field. Plane strain conditions are considered in the analysis. Because of the symmetry of the problem only a quarter of the cross section is discretized. The boundary contour is divided into 34 conforming elements with a quadratic approximation. The domain Q is discretized by 30 quadratic quadrilateral cells. The crack length is denoted by a. The normalized stress intensity factor is defined as Variations of fj with the crack length ratio a/\v are given in Fig. 2 for homogeneous and non-homogeneous materials. One can see that the character of all the curves is practically the same. Small differences are observed only for larger sizes of cracks, since at the vicinity of external surface of tube a substantial influence of the parameter B\ on the hoop stresses has been observed for a thickwalled tube without a crack. The normalized T-term stress is frequently called in literature as the biaxiality parameter. It is denoted by B = T/<JQ, where the stress OQ = {p'^j/naf EC, = E^/(l-v^) for plane strain conditions and E^ = 2//Q (1 + v).the dependence of the biaxial parameter B on the crack length ratio a/\v is given in Fig. 3. The similar dependence can be observed also for a single edge cracked pure bending specimen. In a pure bending specimen it is a linear variation of normal stresses over the cross section, similarly to a thick-walled tube under a thermal gradient loading, where normal stresses are almost linear. with
9 Damage and Fracture Mechanics VI 43 * homogeneous material non-homog. material #1 * non-homog. material #2 * non-homog. material #3 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 a/w Fig. 2. Variations of the SIF with the length of crack in a thick-walled tube 0,8 0,6 0,4 non-homog. material #3 homogeneous material CQ 0,2 0,0 0,70-0,65-0,60-0,55-0,50-0,45-0,35- -0,2- -0,4- -0,6- -0,8- r i i OJ 0,2 0,3 0, , ,6, -, 0,7 P- 0,8 a/w Fig. 3. Values of the biaxiality parameter B for a cracked thick-walled tube Then, the similarity of biaxial parameter variations for both cases is not surprising. One can see in Fig. 3 a moderate dependence of B on the relaxation parameter B,, which governs the linearity of the Young modulus on the thickness of tube.
10 44 Damage and Fracture Mechanics VI 5 Conclusions This paper presents the numerical methods for evaluation of stress intensity factors and T-term stresses for crack problems in functionally gradient materials under a stationary thermal loading. The path independent integral representations for both fracture parameters are derived. The present integral methods are numerically more expedient than those based on the direct computation of fracture parameters from the asymptotic expansion of the stresses and/or displacements. The integral approach is well suited for elastic boundary element type analyses. An advanced BEM formulation is employed for stationary thermoelasticity crack problems in a functionally gradient material where material parameters are varied continuously. References [1] Noda, N. & Jin, Z.H. Thermal stress intensity factors for a crack in a functionally gradient material. Int. J. Solids Structures, 30, pp , [2] Erdogan, F. & Wu, B. Analysis of FGM specimens for fracture toughness testing. Trans. American Ceramic Soc. Functionally Gradient Materials, 34, pp.39-46, [3] Nemat-Alla, M. & Noda, N. Thermal stress intensity factor for functionally gradient half space with an edge crack under thermal load. Archive Appl. Mech., 66, pp , [4] Rice, J.R. Limitations to the small scale yielding approximation for crack tip plasticity. J. Mech. Phys. Solids, 22, pp , [5] Kfouri, A.P. Some evaluations of the elastic T-term using Eshelby's method. Int. J. Fracture, 30, pp , [6] Sherry, A.H., France, C.C. & Goldthorpe, M.R. Compendium of T-stress solutions for two and three dimensional cracked geometries. Fatigue Fract. Engn. Mater. Struct, 18, pp , [7] Sladek, J., Sladek, V. & Fedelinski, P. Contour integrals for mixed-mode crack analysis; effect of nonsingular terms. Theoret. Appl. Fracture Mech., 27, pp , [8] Nowacki, W. Dynamic Problems of Thermoelasticity, PWN, Warsaw, [9] Kishimoto, K., Aoki, S. & Sakata, M. On the path independent integral - J. J. Engn. Fracture Mech., 13, pp , [10] Sih, G.C. On the singular character of thermal stresses near a crack tip. J. ^X Afgc/,., 29, pp , [11] Noda, N. & Jin, Z.H. Crack tip singularity fields in non-homogeneous body under thermal stress fields. JSMEInt. Jour. ser. A, 38, pp , [12] Williams, M.L. On the stress distribution at the base of stationary crack. J. Appl. Mech., 24, pp , 1957.
Critical applied stresses for a crack initiation from a sharp V-notch
Focussed on: Fracture and Structural Integrity related Issues Critical applied stresses for a crack initiation from a sharp V-notch L. Náhlík, P. Hutař Institute of Physics of Materials, Academy of Sciences
More informationThermal load-induced notch stress intensity factors derived from averaged strain energy density
Available online at www.sciencedirect.com Draft ScienceDirect Draft Draft Structural Integrity Procedia 00 (2016) 000 000 www.elsevier.com/locate/procedia 21st European Conference on Fracture, ECF21, 20-24
More informationStress intensity factor analysis for an interface crack between dissimilar isotropic materials
Stress intensity factor analysis for an interface crack between dissimilar isotropic materials under thermal stress T. Ikeda* & C. T. Sun* I Chemical Engineering Group, Department of Materials Process
More informationCRACK INITIATION CRITERIA FOR SINGULAR STRESS CONCENTRATIONS Part I: A Universal Assessment of Singular Stress Concentrations
Engineering MECHANICS, Vol. 14, 2007, No. 6, p. 399 408 399 CRACK INITIATION CRITERIA FOR SINGULAR STRESS CONCENTRATIONS Part I: A Universal Assessment of Singular Stress Concentrations Zdeněk Knésl, Jan
More informationFRACTURE MECHANICS FOR MEMBRANES
FRACTURE MECHANICS FOR MEMBRANES Chong Li, Rogelio Espinosa and Per Ståhle Solid Mechanics, Malmö University SE 205 06 Malmö, Sweden chong.li@ts.mah.se Abstract During fracture of membranes loading often
More informationFracture Mechanics, Damage and Fatigue Linear Elastic Fracture Mechanics - Energetic Approach
University of Liège Aerospace & Mechanical Engineering Fracture Mechanics, Damage and Fatigue Linear Elastic Fracture Mechanics - Energetic Approach Ludovic Noels Computational & Multiscale Mechanics of
More informationLinear Elastic Fracture Mechanics
Measure what is measurable, and make measurable what is not so. - Galileo GALILEI Linear Elastic Fracture Mechanics Krishnaswamy Ravi-Chandar Lecture presented at the University of Pierre and Marie Curie
More informationA simple plane-strain solution for functionally graded multilayered isotropic cylinders
Structural Engineering and Mechanics, Vol. 24, o. 6 (2006) 000-000 1 A simple plane-strain solution for functionally graded multilayered isotropic cylinders E. Pan Department of Civil Engineering, The
More informationTHERMOELASTIC ANALYSIS OF THICK-WALLED FINITE-LENGTH CYLINDERS OF FUNCTIONALLY GRADED MATERIALS
Journal of Thermal Stresses, 28: 391 408, 2005 Copyright # Taylor & Francis Inc. ISSN: 0149-5739 print/1521-074x online DOI: 10.1080/01495730590916623 THERMOELASTIC ANALYSIS OF THICK-WALLED FINITE-LENGTH
More informationDepartment of Structural, Faculty of Civil Engineering, Architecture and Urban Design, State University of Campinas, Brazil
Blucher Mechanical Engineering Proceedings May 2014, vol. 1, num. 1 www.proceedings.blucher.com.br/evento/10wccm A SIMPLIFIED FORMULATION FOR STRESS AND TRACTION BOUNDARY IN- TEGRAL EQUATIONS USING THE
More informationTransactions on Engineering Sciences vol 6, 1994 WIT Press, ISSN
The treatment of crack propagation in inhomogeneous materials using the boundary element method A. Boussekine," L. Ulmet," S. Caperaa* " Laboratoire de Genie Civil, Universite de Limoges, 19300 Egletons,
More informationOn the uniformity of stresses inside an inhomogeneity of arbitrary shape
IMA Journal of Applied Mathematics 23) 68, 299 311 On the uniformity of stresses inside an inhomogeneity of arbitrary shape Y. A. ANTIPOV Department of Mathematics, Louisiana State University, Baton ouge,
More informationA modified quarter point element for fracture analysis of cracks
ndian Journal of Engineering & Materials Sciences Vol. 14, February 007, pp. 31-38 A modified quarter point element for fracture analysis of cracks Sayantan Paul & B N Rao* Structural Engineering Division,
More informationMaterials and Structures
Journal of Mechanics of Materials and Structures BRITTLE FRACTURE BEYOND THE STRESS INTENSITY FACTOR C. T. Sun and Haiyang Qian Volume 4, Nº 4 April 2009 mathematical sciences publishers JOURNAL OF MECHANICS
More information17th European Conference on Fracture 2-5 September,2008, Brno, Czech Republic. Thermal Fracture of a FGM/Homogeneous Bimaterial with Defects
-5 September,8, Brno, Czech Republic Thermal Fracture of a FGM/Homogeneous Bimaterial with Defects Vera Petrova, a, Siegfried Schmauder,b Voronezh State University, University Sq., Voronezh 3946, Russia
More informationCrack Tip Plastic Zone under Mode I Loading and the Non-singular T zz -stress
Crack Tip Plastic Zone under Mode Loading and the Non-singular T -stress Yu.G. Matvienko Mechanical Engineering Research nstitute of the Russian Academy of Sciences Email: ygmatvienko@gmail.com Abstract:
More informationFundamentals of Linear Elasticity
Fundamentals of Linear Elasticity Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research of the Polish Academy
More informationDEVELOPMENT OF TEST GUIDANCE FOR COMPACT TENSION FRACTURE TOUGHNESS SPECIMENS CONTAINING NOTCHES INSTEAD OF FATIGUE PRE-CRACKS
Transactions, SMiRT-23 Division II, Paper ID 287 Fracture Mechanics and Structural Integrity DEVELOPMENT OF TEST GUIDANCE FOR COMPACT TENSION FRACTURE TOUGHNESS SPECIMENS CONTAINING NOTCHES INSTEAD OF
More informationA truly meshless Galerkin method based on a moving least squares quadrature
A truly meshless Galerkin method based on a moving least squares quadrature Marc Duflot, Hung Nguyen-Dang Abstract A new body integration technique is presented and applied to the evaluation of the stiffness
More informationModule 7: Micromechanics Lecture 34: Self Consistent, Mori -Tanaka and Halpin -Tsai Models. Introduction. The Lecture Contains. Self Consistent Method
Introduction In this lecture we will introduce some more micromechanical methods to predict the effective properties of the composite. Here we will introduce expressions for the effective properties without
More informationCRACK-TIP DRIVING FORCE The model evaluates the eect of inhomogeneities by nding the dierence between the J-integral on two contours - one close to th
ICF 100244OR Inhomogeneity eects on crack growth N. K. Simha 1,F.D.Fischer 2 &O.Kolednik 3 1 Department ofmechanical Engineering, University of Miami, P.O. Box 248294, Coral Gables, FL 33124-0624, USA
More informationPROPAGATION OF CURVED CRACKS IN HOMOGENEOUS AND GRADED MATERIALS
PROPAGATION OF CURVED CRACKS IN HOMOGENEOUS AND GRADED MATERIALS Abstract Matthew T. Tilbrook, Robert J. Moon and Mark Hoffman School of Materials Science and Engineering University of New South Wales,
More informationOn the Path-Dependence of the J-Integral Near a Stationary Crack in an Elastic-Plastic Material
Dorinamaria Carka Chad M. Landis e-mail: landis@mail.utexas.edu Department of Aerospace Engineering and Engineering Mechanics, University of Texas at Austin, 10 East 4th Street, C0600 Austin, TX 7871-035
More informationFailure modes of glass panels subjected to soft missile impact
Failure modes of glass panels subjected to soft missile impact L. R. Dharani & J. Yu Dept. of Mech. and Aerospace Engineering and Engineering Mechanics, University of Missouri-Rolla, U.S.A. Abstract Damage
More informationNONLINEAR LOCAL BENDING RESPONSE AND BULGING FACTORS FOR LONGITUDINAL AND CIRCUMFERENTIAL CRACKS IN PRESSURIZED CYLINDRICAL SHELLS
NONINEAR OA BENDING RESPONSE AND BUGING FATORS FOR ONGITUDINA AND IRUMFERENTIA RAKS IN PRESSURIZED YINDRIA SHES Richard D. Young, * heryl A. Rose, * and James H. Starnes, Jr. NASA angley Research enter
More informationTwo semi-infinite interfacial cracks between two bonded dissimilar elastic strips
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Faculty Publications from the Department of Engineering Mechanics Mechanical & Materials Engineering, Department of 9-2003
More informationSize effect in the strength of concrete structures
Sādhanā Vol. 27 Part 4 August 2002 pp. 449 459. Printed in India Size effect in the strength of concrete structures B L KARIHALOO and Q Z XIAO Division of Civil Engineering School of Engineering Cardiff
More informationInfluence of impact velocity on transition time for V-notched Charpy specimen*
[ 溶接学会論文集第 35 巻第 2 号 p. 80s-84s (2017)] Influence of impact velocity on transition time for V-notched Charpy specimen* by Yasuhito Takashima** and Fumiyoshi Minami** This study investigated the influence
More informationMechanics PhD Preliminary Spring 2017
Mechanics PhD Preliminary Spring 2017 1. (10 points) Consider a body Ω that is assembled by gluing together two separate bodies along a flat interface. The normal vector to the interface is given by n
More informationEFFECTS OF THERMAL STRESSES AND BOUNDARY CONDITIONS ON THE RESPONSE OF A RECTANGULAR ELASTIC BODY MADE OF FGM
Proceedings of the International Conference on Mechanical Engineering 2007 (ICME2007) 29-31 December 2007, Dhaka, Bangladesh ICME2007-AM-76 EFFECTS OF THERMAL STRESSES AND BOUNDARY CONDITIONS ON THE RESPONSE
More informationStresses and Displacements in Functionally Graded Materials of Semi-Infinite Extent Induced by Rectangular Loadings
Materials 2012, 5, 210-226; doi:10.3390/ma5020210 Article OPEN ACCESS materials ISSN 1996-1944 www.mdpi.com/journal/materials Stresses and Displacements in Functionally Graded Materials of Semi-Infinite
More informationConsistent Formulations of the Interaction Integral Method for Fracture of Functionally Graded Materials
Jeong-Ho Kim Glaucio H. Paulino 2 e-mail: paulino@uiuc.edu Department of Civil and Environmental Engineering, Newmark Laboratory, The University of Illinois at Urbana-Champaign, 205 North Mathews Avenue,
More informationElastic Crack Interaction Limit of Two Interacting Edge Cracks in Finite Body
Elastic Crack Interaction Limit of Two Interacting Edge Cracks in Finite Body R. Daud, M.A. Rojan Division of Applied Mechanics, School of Mechatronic Engineering, Pauh Putra Campus, Universiti Malaysia
More informationCOMPUTATIONAL ELASTICITY
COMPUTATIONAL ELASTICITY Theory of Elasticity and Finite and Boundary Element Methods Mohammed Ameen Alpha Science International Ltd. Harrow, U.K. Contents Preface Notation vii xi PART A: THEORETICAL ELASTICITY
More informationExamination in Damage Mechanics and Life Analysis (TMHL61) LiTH Part 1
Part 1 1. (1p) Define the Kronecker delta and explain its use. The Kronecker delta δ ij is defined as δ ij = 0 if i j 1 if i = j and it is used in tensor equations to include (δ ij = 1) or "sort out" (δ
More informationIntegral equations for crack systems in a slightly heterogeneous elastic medium
Boundary Elements and Other Mesh Reduction Methods XXXII 65 Integral equations for crack systems in a slightly heterogeneous elastic medium A. N. Galybin & S. M. Aizikovich Wessex Institute of Technology,
More informationINFLUENCE OF TEMPERATURE ON BEHAVIOR OF THE INTERFACIAL CRACK BETWEEN THE TWO LAYERS
Djoković, J. M., et.al.: Influence of Temperature on Behavior of the Interfacial THERMAL SCIENCE: Year 010, Vol. 14, Suppl., pp. S59-S68 S59 INFLUENCE OF TEMPERATURE ON BEHAVIOR OF THE INTERFACIAL CRACK
More informationMixed-Mode Crack Propagation in Functionally Graded Materials
Materials Science Forum Vols. 492-493 (25) pp. 49-414 online at http://www.scientific.net 25 Trans Tech Publications, Switzerland Mixed-Mode Crack Propagation in Functionally Graded Materials Jeong-Ho
More informationTransactions on Modelling and Simulation vol 9, 1995 WIT Press, ISSN X
A path-independent integral for the calculation of stress intensity factors in three-dimensional bodies C.M. Bainbridge," M.H. Aliabadi," D.P. Rooke* "Wessex Institute of Technology, Ashurst Lodge, Ashurst,
More informationDAMPING OF GENERALIZED THERMO ELASTIC WAVES IN A HOMOGENEOUS ISOTROPIC PLATE
Materials Physics and Mechanics 4 () 64-73 Received: April 9 DAMPING OF GENERALIZED THERMO ELASTIC WAVES IN A HOMOGENEOUS ISOTROPIC PLATE R. Selvamani * P. Ponnusamy Department of Mathematics Karunya University
More informationTransactions on Engineering Sciences vol 13, 1996 WIT Press, ISSN
Computational debonding analysis of a thermally stressed fiber/matrix composite cylinder F.-G. Buchholz," O. Koca* "Institute ofapplied Mechanics, University ofpaderborn, D-33098 Paderborn, Germany ^Institute
More informationShock wave speed and stress-strain relation of aluminium honeycombs under dynamic compression
EPJ Web of Conferences 83, 7 (8) DYMAT 8 https://doi.org/.5/epjconf/8837 Shock wave speed and stress-strain relation of aluminium honeycombs under dynamic compression Peng Wang,*, Jun Zhang, Haiying Huang,
More informationFinite element analysis of longitudinal debonding between fibre and matrix interface
Indian Journal of Engineering & Materials Sciences Vol. 11, February 2004, pp. 43-48 Finite element analysis of longitudinal debonding between fibre and matrix interface K Aslantaş & S Taşgetiren Department
More informationEXPERIMENTAL STUDY OF THE OUT-OF-PLANE DISPLACEMENT FIELDS FOR DIFFERENT CRACK PROPAGATION VELOVITIES
EXPERIMENTAL STUDY OF THE OUT-OF-PLANE DISPLACEMENT FIELDS FOR DIFFERENT CRACK PROPAGATION VELOVITIES S. Hédan, V. Valle and M. Cottron Laboratoire de Mécanique des Solides, UMR 6610 Université de Poitiers
More informationThree-dimensional thermoelastic deformations of a functionally graded elliptic plate
JCOMB Composites: Part B () 9 6 www.elsevier.com/locate/compositesb Three-dimensional thermoelastic deformations of a functionally graded elliptic plate Z.-Q. Cheng a, R.C. Batra b, * a Department of Modern
More informationChapter 7. Highlights:
Chapter 7 Highlights: 1. Understand the basic concepts of engineering stress and strain, yield strength, tensile strength, Young's(elastic) modulus, ductility, toughness, resilience, true stress and true
More informationVERIFICATION OF BRITTLE FRACTURE CRITERIA FOR BIMATERIAL STRUCTURES
VERIFICATION OF BRITTLE FRACTURE CRITERIA FOR BIMATERIAL STRUCTURES Grzegorz MIECZKOWSKI *, Krzysztof MOLSKI * * Faculty of Mechanical Engineering, Białystok University of Technology, ul. Wiejska 45C,
More informationPrediction of geometric dimensions for cold forgings using the finite element method
Journal of Materials Processing Technology 189 (2007) 459 465 Prediction of geometric dimensions for cold forgings using the finite element method B.Y. Jun a, S.M. Kang b, M.C. Lee c, R.H. Park b, M.S.
More informationFRACTURE MECHANICS OF COMPOSITES WITH RESIDUAL STRESSES, TRACTION-LOADED CRACKS, AND IMPERFECT INTERFACES
Proc. 2 nd ESIS TC4 Conference on Polymers and Composites, in press, 1999 Author prepared reprint FRACTURE MECHANICS OF COMPOSITES WITH RESIDUAL STRESSES, TRACTION-LOADED CRACKS, AND IMPERFECT INTERFACES
More informationIntroduction to fracture mechanics
Introduction to fracture mechanics Prof. Dr. Eleni Chatzi Dr. Giuseppe Abbiati, Dr. Konstantinos Agathos Lecture 6-9 November, 2017 Institute of Structural Engineering, ETH Zu rich November 9, 2017 Institute
More informationASSESSMENT OF DYNAMICALLY LOADED CRACKS IN FILLETS
ASSESSMENT OF DNAMICALL LOADED CRACKS IN FILLETS Uwe Zencker, Linan Qiao, Bernhard Droste Federal Institute for Materials Research and Testing (BAM) 12200 Berlin, Germany e-mail: zencker@web.de Abstract
More informationMECHANICS OF MATERIALS. EQUATIONS AND THEOREMS
1 MECHANICS OF MATERIALS. EQUATIONS AND THEOREMS Version 2011-01-14 Stress tensor Definition of traction vector (1) Cauchy theorem (2) Equilibrium (3) Invariants (4) (5) (6) or, written in terms of principal
More informationElastic-Plastic Fracture Mechanics. Professor S. Suresh
Elastic-Plastic Fracture Mechanics Professor S. Suresh Elastic Plastic Fracture Previously, we have analyzed problems in which the plastic zone was small compared to the specimen dimensions (small scale
More informationAlternative numerical method in continuum mechanics COMPUTATIONAL MULTISCALE. University of Liège Aerospace & Mechanical Engineering
University of Liège Aerospace & Mechanical Engineering Alternative numerical method in continuum mechanics COMPUTATIONAL MULTISCALE Van Dung NGUYEN Innocent NIYONZIMA Aerospace & Mechanical engineering
More informationQuasi Static Thermal Stresses in A Limiting Thick Circular Plate with Internal Heat Generation Due To Axisymmetric Heat Supply
International Journal of Mathematics and Statistics Invention (IJMSI) E-ISSN: 2321 4767 P-ISSN: 2321-4759 Volume 1 Issue 2 ǁ December. 2013ǁ PP-56-63 Quasi Static Thermal Stresses in A Limiting Thick Circular
More informationModel-independent approaches for the XFEM in fracture mechanics
Model-independent approaches for the XFEM in fracture mechanics Safdar Abbas 1 Alaskar Alizada 2 and Thomas-Peter Fries 2 1 Aachen Institute for Computational Engineering Science (AICES), RWTH Aachen University,
More informationFinite Element Computations of Complex Stress Intensity Factor Magnitude of Interfacial Crack in Bi-Material Composites
16 th International Conference on AEROSPACE SCIENCES & AVIATION TECHNOLOGY, ASAT - 16 May 26-28, 215, E-Mail: asat@mtc.edu.eg Military Technical College, Kobry Elkobbah, Cairo, Egypt Tel : +(22) 2425292
More informationDetermination of Stress Intensity Factor for a Crack Emanating From a Rivet Hole and Approaching Another in Curved Sheet
International OPEN ACCESS Journal Of Modern Engineering Research (IJMER) Determination of Stress Intensity Factor for a Crack Emanating From a Rivet Hole and Approaching Another in Curved Sheet Raghavendra.
More informationEffective stress assessment at rectangular rounded lateral notches
Focussed on characterization of crack tip fields Effective stress assessment at rectangular rounded lateral notches Enrico Maggiolini, Roberto Tovo, Paolo Livieri University of Ferrara Enrico.maggiolini@unife.it,
More information202 Index. failure, 26 field equation, 122 force, 1
Index acceleration, 12, 161 admissible function, 155 admissible stress, 32 Airy's stress function, 122, 124 d'alembert's principle, 165, 167, 177 amplitude, 171 analogy, 76 anisotropic material, 20 aperiodic
More informationCellular solid structures with unbounded thermal expansion. Roderic Lakes. Journal of Materials Science Letters, 15, (1996).
1 Cellular solid structures with unbounded thermal expansion Roderic Lakes Journal of Materials Science Letters, 15, 475-477 (1996). Abstract Material microstructures are presented which can exhibit coefficients
More informationA fracture parameter for welded structures with residual stresses
Computational Mechanics 22 (1998) 281±288 Ó Springer-Verlag 1998 A fracture parameter for welded structures with residual stresses Y.-C. Hou, J. Pan Abstract In this paper, a nite element procedure to
More informationARTICLE A-8000 STRESSES IN PERFORATED FLAT PLATES
ARTICLE A-8000 STRESSES IN PERFORATED FLAT PLATES Delete endnote 18, which says "Express metric values in exponential form" A-8100 INTRODUCTION A-8110 SCOPE (a) This Article contains a method of analysis
More informationElastic behaviour of an edge dislocation near a sharp crack emanating from a semi-elliptical blunt crack
Chin. Phys. B Vol. 19, No. 1 010 01610 Elastic behaviour of an edge dislocation near a sharp crack emanating from a semi-elliptical blunt crack Fang Qi-Hong 方棋洪, Song Hao-Peng 宋豪鹏, and Liu You-Wen 刘又文
More informationROTATING RING. Volume of small element = Rdθbt if weight density of ring = ρ weight of small element = ρrbtdθ. Figure 1 Rotating ring
ROTATIONAL STRESSES INTRODUCTION High centrifugal forces are developed in machine components rotating at a high angular speed of the order of 100 to 500 revolutions per second (rps). High centrifugal force
More informationExample-3. Title. Description. Cylindrical Hole in an Infinite Mohr-Coulomb Medium
Example-3 Title Cylindrical Hole in an Infinite Mohr-Coulomb Medium Description The problem concerns the determination of stresses and displacements for the case of a cylindrical hole in an infinite elasto-plastic
More informationA Piezoelectric Screw Dislocation Interacting with an Elliptical Piezoelectric Inhomogeneity Containing a Confocal Elliptical Rigid Core
Commun. Theor. Phys. 56 774 778 Vol. 56, No. 4, October 5, A Piezoelectric Screw Dislocation Interacting with an Elliptical Piezoelectric Inhomogeneity Containing a Confocal Elliptical Rigid Core JIANG
More informationME 2570 MECHANICS OF MATERIALS
ME 2570 MECHANICS OF MATERIALS Chapter III. Mechanical Properties of Materials 1 Tension and Compression Test The strength of a material depends on its ability to sustain a load without undue deformation
More informationCALCULATION OF FRACTURE MECHANICS PARAMETERS FOR AN ARBITRARY THREE-DIMENSIONAL CRACK, BY THE EQUIVALENT DOMAIN INTEGRAL METHOD 1
CALCULATION OF FRACTURE MECHANICS PARAMETERS FOR AN ARBITRARY THREE-DIMENSIONAL CRACK, BY THE EQUIVALENT DOMAIN INTEGRAL METHOD 1 G. P. NIKISHKOV 2 and S. N. ATLURI 3 Center for the Advancement of Computational
More informationScienceDirect. Hamiltonian approach to piezoelectric fracture
Available online at www.sciencedirect.com ScienceDirect Procedia Materials Science 3 ( 014 ) 318 34 0th European Conference on Fracture (ECF0) Hamiltonian approach to piezoelectric fracture J.M. Nianga
More informationHomework Problems. ( σ 11 + σ 22 ) 2. cos (θ /2), ( σ θθ σ rr ) 2. ( σ 22 σ 11 ) 2
Engineering Sciences 47: Fracture Mechanics J. R. Rice, 1991 Homework Problems 1) Assuming that the stress field near a crack tip in a linear elastic solid is singular in the form σ ij = rλ Σ ij (θ), it
More informationULTRASONIC REFLECTION BY A PLANAR DISTRIBUTION OF SURFACE BREAKING CRACKS
ULTRASONIC REFLECTION BY A PLANAR DISTRIBUTION OF SURFACE BREAKING CRACKS A. S. Cheng Center for QEFP, Northwestern University Evanston, IL 60208-3020 INTRODUCTION A number of researchers have demonstrated
More information3D Finite Element analysis of stud anchors with large head and embedment depth
3D Finite Element analysis of stud anchors with large head and embedment depth G. Periškić, J. Ožbolt & R. Eligehausen Institute for Construction Materials, University of Stuttgart, Stuttgart, Germany
More informationStress intensity factors under combined tension and torsion loadings
Indian Journal of Engineering & Materials Sciences Vol. 19, February 01, pp. 5-16 Stress intensity factors under combined tension and torsion loadings A E Ismail a *, A Ariffin b, S Abdullah b & M J Ghazali
More information3 2 6 Solve the initial value problem u ( t) 3. a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1
Math Problem a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1 3 6 Solve the initial value problem u ( t) = Au( t) with u (0) =. 3 1 u 1 =, u 1 3 = b- True or false and why 1. if A is
More informationHeterogeneous structures studied by interphase elasto-damaging model.
Heterogeneous structures studied by interphase elasto-damaging model. Giuseppe Fileccia Scimemi 1, Giuseppe Giambanco 1, Antonino Spada 1 1 Department of Civil, Environmental and Aerospace Engineering,
More informationA short review of continuum mechanics
A short review of continuum mechanics Professor Anette M. Karlsson, Department of Mechanical ngineering, UD September, 006 This is a short and arbitrary review of continuum mechanics. Most of this material
More informationFrontiers of Fracture Mechanics. Adhesion and Interfacial Fracture Contact Damage
Frontiers of Fracture Mechanics Adhesion and Interfacial Fracture Contact Damage Biology, Medicine & Dentistry The Next Frontiers For Mechanics One of the current challenges in materials & mechanics is
More informationThe Effects of Transverse Shear on the Delamination of Edge-Notch Flexure and 3-Point Bend Geometries
The Effects of Transverse Shear on the Delamination of Edge-Notch Flexure and 3-Point Bend Geometries M. D. Thouless Department of Mechanical Engineering Department of Materials Science & Engineering University
More informationTransactions on Modelling and Simulation vol 9, 1995 WIT Press, ISSN X
Elastic-plastic model of crack growth under fatigue using the boundary element method M. Scibetta, O. Pensis LTAS Fracture Mechanics, University ofliege, B-4000 Liege, Belgium Abstract Life of mechanic
More informationApplication of a non-local failure criterion to a crack in heterogeneous media S. Bavaglia*, S.E. Mikhailov*
Application of a non-local failure criterion to a crack in heterogeneous media S. Bavaglia*, S.E. Mikhailov* University of Perugia, Italy Email: mic@unipg.it ^Wessex Institute of Technology, Ashurst Lodge,
More informationS.P. Timoshenko Institute of Mechanics, National Academy of Sciences, Department of Fracture Mechanics, Kyiv, Ukraine
CALCULATION OF THE PREFRACTURE ZONE AT THE CRACK TIP ON THE INTERFACE OF MEDIA A. Kaminsky a L. Kipnis b M. Dudik b G. Khazin b and A. Bykovtscev c a S.P. Timoshenko Institute of Mechanics National Academy
More informationThe Elastic-Viscoelastic Correspondence Principle for Functionally Graded Materials, Revisited
S. Mukherjee ASME Fellow Department of Theoretical and Applied Mechanics, Cornell University, Kimball Hall, Ithaca, NY 14853 e-mail: sm85@cornell.edu Glaucio H. Paulino 1 Mem. ASME Department of Civil
More informationFast multipole boundary element method for the analysis of plates with many holes
Arch. Mech., 59, 4 5, pp. 385 401, Warszawa 2007 Fast multipole boundary element method for the analysis of plates with many holes J. PTASZNY, P. FEDELIŃSKI Department of Strength of Materials and Computational
More informationTensile stress strain curves for different materials. Shows in figure below
Tensile stress strain curves for different materials. Shows in figure below Furthermore, the modulus of elasticity of several materials effected by increasing temperature, as is shown in Figure Asst. Lecturer
More informationDiscrete Analysis for Plate Bending Problems by Using Hybrid-type Penalty Method
131 Bulletin of Research Center for Computing and Multimedia Studies, Hosei University, 21 (2008) Published online (http://hdl.handle.net/10114/1532) Discrete Analysis for Plate Bending Problems by Using
More informationAN EFFECTIVE SOLUTION OF THE COMPOSITE (FGM S) BEAM STRUCTURES
Engineering MECHANICS, Vol. 15, 2008, No. 2, p. 115 132 115 AN EFFECTIVE SOLUTION OF THE COMPOSITE (FGM S) BEAM STRUCTURES Justín Murín, Vladimír Kutiš* The additive mixture rules have been extended for
More informationModelling the nonlinear shear stress-strain response of glass fibrereinforced composites. Part II: Model development and finite element simulations
Modelling the nonlinear shear stress-strain response of glass fibrereinforced composites. Part II: Model development and finite element simulations W. Van Paepegem *, I. De Baere and J. Degrieck Ghent
More informationNumerical analyses of cement-based piezoelectric smart composites
Numerical analyses of cement-based piezoelectric smart composites *Jan Sladek 1, Pavol Novak 2, Peter L. Bishay 3, and Vladimir Sladek 1 1 Institute of Construction and Architecture, Slovak Academy of
More informationSupplementary Figures
Fracture Strength (GPa) Supplementary Figures a b 10 R=0.88 mm 1 0.1 Gordon et al Zhu et al Tang et al im et al 5 7 6 4 This work 5 50 500 Si Nanowire Diameter (nm) Supplementary Figure 1: (a) TEM image
More informationAdvanced Structural Analysis EGF Cylinders Under Pressure
Advanced Structural Analysis EGF316 4. Cylinders Under Pressure 4.1 Introduction When a cylinder is subjected to pressure, three mutually perpendicular principal stresses will be set up within the walls
More informationPROPAGATION OF A MODE-I CRACK UNDER THE IRWIN AND KHRISTIANOVICH BARENBLATT CRITERIA
Materials Science, Vol. 39, No. 3, 3 PROPAGATION OF A MODE-I CRACK UNDER THE IRWIN AND KHRISTIANOVICH BARENBLATT CRITERIA I. I. Argatov, M. Bach, and V. A. Kovtunenko UDC 539.3 We study the problem of
More informationTransactions on Engineering Sciences vol 6, 1994 WIT Press, ISSN
A computational method for the analysis of viscoelastic structures containing defects G. Ghazlan," C. Petit," S. Caperaa* " Civil Engineering Laboratory, University of Limoges, 19300 Egletons, France &
More informationExercise: concepts from chapter 8
Reading: Fundamentals of Structural Geology, Ch 8 1) The following exercises explore elementary concepts associated with a linear elastic material that is isotropic and homogeneous with respect to elastic
More informationChapter 3. Load and Stress Analysis
Chapter 3 Load and Stress Analysis 2 Shear Force and Bending Moments in Beams Internal shear force V & bending moment M must ensure equilibrium Fig. 3 2 Sign Conventions for Bending and Shear Fig. 3 3
More informationInternational Journal of Pure and Applied Mathematics Volume 58 No ,
International Journal of Pure and Applied Mathematics Volume 58 No. 2 2010, 195-208 A NOTE ON THE LINEARIZED FINITE THEORY OF ELASTICITY Maria Luisa Tonon Department of Mathematics University of Turin
More informationEngineering Sciences 241 Advanced Elasticity, Spring Distributed Thursday 8 February.
Engineering Sciences 241 Advanced Elasticity, Spring 2001 J. R. Rice Homework Problems / Class Notes Mechanics of finite deformation (list of references at end) Distributed Thursday 8 February. Problems
More informationIntroduction to the J-integral
Introduction to the J-integral Instructor: Ramsharan Rangarajan February 24, 2016 The purpose of this lecture is to briefly introduce the J-integral, which is widely used in fracture mechanics. To the
More informationFinite Element Method in Geotechnical Engineering
Finite Element Method in Geotechnical Engineering Short Course on + Dynamics Boulder, Colorado January 5-8, 2004 Stein Sture Professor of Civil Engineering University of Colorado at Boulder Contents Steps
More informationEnd forming of thin-walled tubes
Journal of Materials Processing Technology 177 (2006) 183 187 End forming of thin-walled tubes M.L. Alves a, B.P.P. Almeida b, P.A.R. Rosa b, P.A.F. Martins b, a Escola Superior de Tecnologia e Gestão
More information