The rational spirit in modern continuum mechanics 2005

THE RATIONAL SPIRIT IN MODERN CONTINUUM MECHANICS The Rational Spirit in Modern Continuum Mechanics Essays and Papers Dedicated to the Memory of Clifford Ambrose Truesdell III Edited by CHI-SING MAN University of Kentucky, Lexington, U.S.A and ROGER L FOSDICK University of Minnesota, Minneapolis, U.S.A Reprinted from Journal of Elasticity: The Physical and Mathematical Science of Solids, Vols 70, 71, 72 (2003) KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW eBook ISBN: Print ISBN: 1-4020-2308-1 1-4020-1828-2 ©2005 Springer Science + Business Media, Inc Print ©2004 Kluwer Academic Publishers Dordrecht All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Springer's eBookstore at: and the Springer Global Website Online at: http://ebooks.kluweronline.com http://www.springeronline.com Portrait by Joseph Sheppard Table of Contents Portrait by Joseph Sheppard v Foreword by Chi-Sing Man and Roger Fosdick xi Published Works of Clifford Ambrose Truesdell III xiii Serials Edited by Clifford Ambrose Truesdell III xli Eulogium by Roger Fosdick xliii Photograph: Bloomington, Indiana, 1959 xlv BERNARD D COLEMAN / Memories of Clifford Truesdell 1–13 ENRICO GIUSTI / Clifford Truesdell (1919–2000), Historian of Mathematics 15–22 WALTER NOLL / The Genesis of Truesdell’s Nonlinear Field Theories of Mechanics 23–30 JAMES SERRIN / An Appreciation of Clifford Truesdell 31–38 D SPEISER / Clifford A Truesdell’s Contributions to the Euler and the Bernoulli Edition Photograph: Baltimore, Maryland, 1978 STUART S ANTMAN / Invariant Dissipative Mechanisms for the Spatial Motion of Rods Suggested by Artificial Viscosity 39–53 55–64 MILLARD F BEATTY / An Average-Stretch Full-Network Model for Rubber Elasticity 65–86 MICHELE BUONSANTI and GIANNI ROYER-CARFAGNI / From 3-D Nonlinear Elasticity Theory to 1-D Bars with Nonconvex Energy 87–100 GIOVANNI BURATTI, YONGZHONG HUO and INGO MÜLLER / Eshelby Tensor as a Tensor of Free Enthalpy 101–112 SANDRO CAPARRINI and FRANCO PASTRONE / E Frola (1906–1962): An Attempt Towards an Axiomatic Theory of Elasticity 113–125 GIANFRANCO CAPRIZ and PAOLO MARIA MARIANO / Symmetries and Hamiltonian Formalism for Complex Materials 127–140 DONALD E CARLSON, ELIOT FRIED and DANIEL A TORTORELLI / Geometrically-based Consequences of Internal Constraints 141–149 vii viii YI-CHAO CHEN / Second Variation Condition and Quadratic Integral Inequalities with Higher Order Derivatives 151–167 ELENA CHERKAEV and ANDREJ CHERKAEV / Principal Compliance and Robust Optimal Design 169–196 JOHN C CRISCIONE / Rivlin’s Representation Formula is Ill-Conceived for the Determination of Response Functions via Biaxial Testing 197–215 CESARE DAVINI and ROBERTO PARONI / Generalized Hessian and External Approximations in Variational Problems of Second Order 217–242 F DELL’ISOLA, G SCIARRA and R.C BATRA / Static Deformations of a Linear Elastic Porous Body Filled with an Inviscid Fluid 243–264 GIANPIETRO DEL PIERO / A Class of Fit Regions and a Universe of Shapes for Continuum Mechanics 265–285 LUCA DESERI and DAVID R OWEN / Toward a Field Theory for Elastic Bodies Undergoing Disarrangements 287–326 MARCELO EPSTEIN and IOAN BUCATARU / Continuous Distributions of Dislocations in Bodies with Microstructure 327–344 ˙ MARCELO EPSTEIN and MAREK ELZANOWSKI / A Model of the Evolution of a Two-dimensional Defective Structure 345–355 J.L ERICKSEN / On the Theory of Rotation Twins in Crystal Multilattices 357–373 MAURO FABRIZIO and MURROUGH GOLDEN / Minimum Free Energies for Materials with Finite Memory 375–397 ROGER FOSDICK and LEV TRUSKINOVSKY / About Clapeyron’s Theorem in Linear Elasticity 399–426 M FOSS, W HRUSA and V.J MIZEL / The Lavrentiev Phenomenon in Nonlinear Elasticity 427–435 GIOVANNI P GALDI / Steady Flow of a Navier–Stokes Fluid around a Rotating Obstacle 437–467 TIMOTHY J HEALEY and ERROL L MONTES-PIZARRO / Global Bifurcation in Nonlinear Elasticity with an Application to Barrelling States of Cylindrical Columns 469–494 MOJIA HUANG and CHI-SING MAN / Constitutive Relation of Elastic Polycrystal with Quadratic Texture Dependence 495–524 MASARU IKEHATA and GEN NAKAMURA / Reconstruction Formula for Identifying Cracks 525–538 R.J KNOPS and PIERO VILLAGGIO / An Approximate Treatment of Blunt Body Impact 539–554 I-SHIH LIU / On the Transformation Property of the Deformation Gradient under a Change of Frame 555–562 ix KONSTANTIN A LURIE / Some New Advances in the Theory of Dynamic Materials 563–573 GERARD A MAUGIN / Pseudo-plasticity and Pseudo-inhomogeneity Effects in Materials Mechanics 575–597 A IAN MURDOCH / On the Microscopic Interpretation of Stress and Couple Stress 599–625 PABLO V NEGRÓN-MARRERO / The Hanging Rope of Minimum Elongation for a Nonlinear Stress–Strain Relation 627–649 MARIO PITTERI / On Certain Weak Phase Transformations in Multilattices 651–671 PAOLO PODIO-GUIDUGLI / A New Quasilinear Model for Plate Buckling 673–698 G RODNAY and R SEGEV / Cauchy’s Flux Theorem in Light of Geometric Integration Theory 699–719 U SARAVANAN and K.R RAJAGOPAL / A Comparison of the Response of Isotropic Inhomogeneous Elastic Cylindrical and Spherical Shells and Their Homogenized Counterparts 721–749 M ŠILHAVÝ / On SO(n)-Invariant Rank Convex Functions 751–762 ´ K WILMANSKI / On Thermodynamics of Nonlinear Poroelastic Materials 763–777 WAN-LEE YIN / Anisotropic Elasticity and Multi-Material Singularities 779–808 Foreword Through his voluminous and influential writings, editorial activities, organizational leadership, intellectual acumen, and strong sense of history, Clifford Ambrose Truesdell III (1919–2000) was the main architect for the renaissance of rational continuum mechanics since the middle of the twentieth century The present collection of 42 essays and research papers pays tribute to this man of mathematics, science, and natural philosophy as well as to his legacy The first five essays by B.D Coleman, E Giusti, W Noll, J Serrin, and D Speiser were texts of addresses given by their authors at the Meeting in memory of Clifford Truesdell, which was held in Pisa in November 2000 In these essays the reader will find personal reminiscences of Clifford Truesdell the man and of some of his activities as scientist, author, editor, historian of exact sciences, and principal founding member of the Society for Natural Philosophy The bulk of the collection comprises 37 research papers which bear witness to the Truesdellian legacy These papers cover a wide range of topics; what ties them together is the rational spirit Clifford Truesdell, in his address upon receipt of a Birkhoff Prize in 1978, put the essence of modern continuum mechanics succinctly as “conceptual analysis, analysis not in the sense of the technical term but in the root meaning: logical criticism, dissection, and creative scrutiny.” It is in celebration of this spirit and this essence that these research papers are dedicated to the memory of their bearer, driving force, and main promoter for half a century Most of these papers were presented at the Symposium on Recent Advances and New Directions in Mechanics, Continuum Thermodynamics, and Kinetic Theory – In Memory of Clifford A Truesdell III, held in Blacksburg, Virginia, in June 2002; parts of two papers were delivered at the meeting Remembering Clifford Truesdell, held in Turin in November 2002; and the rest was written especially for the present collection The portrait, a photo of which serves as the frontispiece of this collection, adorns the Clifford A Truesdell III Room of History of Science in the library of the Scuola Normale Superiore (Pisa, Italy), which was inaugurated in October 2003 and permanently houses Clifford Truesdell’s previously private collection of books, papers, and correspondence We are grateful to Mrs Charlotte Truesdell for helping us secure a digital file of this photo and for providing us with the list of published works of Clifford Truesdell C HI -S ING M AN University of Kentucky Lexington ROGER F OSDICK University of Minnesota Minneapolis xi Published Works of Clifford Ambrose Truesdell III The year of publication is omitted from the entry unless it differs from the year under which the entry is listed Letters following a number indicate subsidiary separate publications, as follows: P A C L R RE T TC TE Preliminary report or preprint, Abstract, separately published or only published version, Condensed or extracted version, Lecture concerning part or all of the contents of main entry, Reprint, entire, Reprint of an extract, Translation, entire, Translation, condensed, Translation of an extract The list excludes some 600 reviews published between 1949 and 1971 in Mathematical Reviews, Applied Mechanics Reviews, Zentralblatt für Mathematik, Industrial Laboratories, and Mathematics of Computation but includes reviews published in other journals 1943 (Co-author P N EMÉNYI) A stress function for the membrane theory of shells of revolution, Proceedings of the National Academy of Sciences (U.S.A.) 29, 159–162 Other publication in 1943: No 3A1 1944 A LONZO C HURCH, Introduction to Mathematical Logic, Part I, Notes by C.A T RUESDELL, Annals of Mathematics Studies No 13, Princeton, University Press, vi + 118 pp Note by the editors: This list and the list on p 29 are slightly edited versions of those that we received from Mrs C Truesdell, to whom we are heartily grateful In our editorial work we have added a few entries, updated several items, and made a small number of other minor corrections To G.P Galdi, K Hutter, R.G Muncaster, F Pastrone, and D Speiser, we are beholden for their help in tracking down article titles and numbers of journal volumes In what follows, explanatory remarks set off by square brackets were made by Clifford Truesdell himself xiii xiv PUBLISHED WORKS OF C.A TRUESDELL 1945 The membrane theory of shells of revolution, Transactions of the American Mathematical Society 58, 96–166 3A1 The differential equations of the membrane theory of shells of revolution, Bulletin of the American Mathematical Society 49 (1943), 863– 864 3A2 The membrane theory of shells of revolution, Bulletin of the American Mathematical Society 51, 225 On a function which occurs in the theory of the structure of polymers, Annals of Mathematics 46, 144–157 −2m n , ∞ Generalizations of Euler’s summations of the series ∞ n=1 n n=0 (−) × (2n + 1)−2m−1 , etc., Annals of Mathematics 46, 194–195 Other publication in 1945: No 12A1 1946 (Co-author R.C P RIM) On Linearized Axially Symmetric Flow of a Compressible Fluid, U.S Naval Ordnance Laboratory Memorandum 8885, 16 December, pp On Behrbohm and Pinl’s linearization of the equation of two-dimensional steady polytropic flow of a compressible fluid, Proceedings of the National Academy of Sciences (U.S.A.) 32, 289–293 = U.S Naval Ordnance Laboratory Memorandum 8888, 18 December, pp 7A On Behrbohm and Pinl’s linearization of the two dimensional steady flow of a compressible adiabatic fluid, Bulletin of the American Mathematical Society 53 (1947), 59 Other publications in 1946: Nos 8A and 12A2 1947 On Sokolovsky’s “Momentless shells”, Transactions of the American Mathematical Society 61, 128–133 8A Same title, Bulletin of the American Mathematical Society 52 (1946), 240 (Co-author R.N S CHWARTZ) The Newtonian mechanics of continua, U.S Naval Ordnance Laboratory Memorandum 9223, 18 July, 25 pp 9A (Co-author R S CHWARTZ) On the Newtonian Mechanics of Continua, Bulletin of the American Mathematical Society 53, 1125 10 A note on the Poisson–Charlier functions, Annals of Mathematical Statistics 18, 450–454 11 Review of L Brand’s “Vector and Tensor Analysis”, Science 106, 623 Other publications in 1947: Nos 7A, 12A3, 13P, 14P, 16P, 16A, 23P, 48P 1948 12 An Essay toward a Unified Theory of Special Functions, based on the Functional Equation ∂F (z, α)/∂z = F (z, α + 1), Annals of Mathematics Studies No 18, Princeton, Princeton University Press, iv + 182 pp 794 W.-L YIN Figure Deformed 3-layer model under shear Root” command in conjunction with contour-plotting However, the argument principle provides crucial help for ascertaining the number of roots (and the multiplicity of repeated roots) within any closed curve Each real or complex root λ determines a wedge eigensolution whose analytical expression in the kth sector is given by equations (4.2)–(4.6) (v) The wedge eigensolutions associated with the successive wedge eigenvalues are linearly combined to form a (truncated) eigenseries The coefficients of combination are determined by collocation along the path Generally there are more data points than coefficients, and a least-square error criterion is used to best fit the data of stress potentials on the collocation path The composite structural model to be studied has three layers of equal thickness cm The middle layer has the length 12 cm in the x-direction while the top and bottom layers are 14 cm long, as shown in Figure in its deformed state under a shear loading All three layers are made of the same unidirectional graphite-epoxy composite whose homogenized anisotropic elastic properties are characterized by the following elastic constant: E2 = E3 = 10.3 GPa, v12 = v13 = v23 = 0.28, E1 = 181 GPa, (5.1) G23 = 4.023 GPa G12 = G13 = 7.17 GPa, The fiber axis in the bottom, middle and top layers are oriented at angles 30◦ , 0◦ and −60◦ , respectively, with respect to the z-direction The middle layer has a crack in a 45◦ inclined plane that runs through the entire thickness of the layer, and also through the entire width in the z-direction This is a matrix crack since the crack plane is parallel to the fibers Since the stress solutions near the singularities will be examined at various scale lengths including those that are much smaller than fiber diameters, the elasticity solutions to be obtained are not so much appropriate to the three-layer composite model as to the layerwise homogenized model That is, strictly speaking, our analy- 795 MULTI-MATERIAL SINGULARITIES sis and solutions concern the layerwise homogenized model, not the composite model But the three layers will still be designated as 30◦ , 0◦ and −60◦ layers The lower surface of the model is fixed and the upper surface is moved rigidly in the negative x-direction through a distance 1/100 cm Plane strain condition εz = is maintained for the model The material eigenvalues of the three layers are all purely imaginary: −60◦ layer 0◦ layer 30◦ layer ±0.871351 i, ±i, ±0.959036 i, ±1.25958 i, ±i, ±1.093396 i, ±4.26833, ±i, ±2.596064 i (5.2) The 0◦ middle layer is transversely isotropic, and therefore degenerate (but not extradegenerate) It has two zeroth-order material eigenvectors and one first-order eigenvector There are six multimaterial singularities in this model The singularities at the two ends of the inclined crack are associated with trisectors wedges At points B and C, one has the well-known free-edge singularities, which will not be studied in this work The bisector wedges at A and B are similar but not identical to those at D and C, respectively, due to the different orientations of the top and bottom layers For the two bisector wedges at the reentrant corners A and D, and for the two trisector wedges at the ends of the inclined crack, the wedge eigenvalues are shown in Table I For these four wedges, all singular eigenvalues are real Hence the elasticity solutions not show oscillatory behavior in the immediate vicinities of the singularities In addition, as one approaches the vertex of a wedge, the elasticity solution approaches the asymptotic limit determined by the real eigenvector associated with the dominant real eigenvalue Both the radial and angular dependence of this real asymptotic solution are determined by the geometry and material of the wedge sectors and the edge boundary conditions except for a real amplitude factor which is determined by remote loading Thus, changes in remote loading can only affect this amplitude factor but cannot change the ratios of stress components of the asymptotic solution This is in stark contrast with the case of complex conjugate dominant singularities where the stress ratios of the asymptotic solution (for example, the stress-intensity factors of interface cracks) generally depend on remote loading The distribution of eigenvalues in the complex plane has a similar pattern for the two corner wedges, and also for the two trisector wedges, despite large difference in the axial stiffness of the −60◦ and 30◦ layers (110.693 GPa and 24.690 GPa, respectively) In general, the wedge eigenvalues are strongly affected by the edge conditions and the wedge geometry (including the number of sectors and sector angles), but are less sensitive to moderate changes in the elastic constants of the sectors The two bisector wedges have simple integer eigenvalues 1, 2, 4, 6, etc For the trisector wedges, every positive integer is a triple eigenvalue However, one of the three eigenvectors associated with λ = is a rigid rotation mode that contributes no stress It is interesting to notice that the dominant singular eigenvalues [0/30] at D 0.5585964 0.7150752 0.9339855 1.2822588 1.6422164 ± 0.2330850 i 2.2631785 ± 0.3099347 i 2.7187519 2.9863063 ± 0.2948049 i 3.2811556 3.7095433 ± 0.4676336 i 4.2072597 ± 0.4663597 i 4.7189494 4.9936378 ± 0.3531516 i 5.2810307 5.8077471 ± 0.6126472 i 6.1164562 ± 0.5507335 i 6.7189994 6.9967129 ± 0.3782984 i 7.2809934 7.9133476 ± 0.8085709 i [−60/0] at A 0.5924654 0.6760317 0.9679459 1.3290965 1.6881490 ± 0.2455260 i 2.1615945 ± 0.2719786 i 2.6696804 2.9973732 ± 0.1681712 i 3.3304593 3.8604587 ± 0.7342584 i 4.0072164 ± 0.2662313 i 4.6694182 4.9992643 ± 0.1843503 i 5.3306112 5.8853940 ± 1.0211285 i 6.0013338 ± 0.2422297 i 6.6693539 6.9997085 ± 0.1893604 i 7.3306568 7.9007901 ± 1.2123062 i 0.4891194 0.5056793 0.6130216 1, 1, 1.4577165 1.5280594 ± 0.0266804 i 2, 2, 2.4480841 ± 2.4652695 2.5869116 3, 3, 3.5396360 ± 0.0972610 i 3.7955437 4, 4, 4.2567490 ± 0.1718115 i 4.4462049 5, 5, 5.3228318 ± 0.2168114 i 5.4462668 6, 6, 6.5306451 6.5936590 ± 0.3364298 i 7, 7, 7.5448581 7.8660481 ± 1.1948926 i 7.9450938 ± 0.3830909 i 8, 8, Lower end of crack Upper end of crack 4.4694496 5, 5, 5.2557896 ± 0.4775619 i 5.4666851 6, 6, 6.5271205 6.5893884 ± 0.5830187 i 7, 7, 7.5344069 7.7590945 ± 1.9577296 i 7.9333809 ± 0.6565767 i 8, 8, 0.4732559 0.5271834 0.6922293 1, 1, 1.3592020 1.4947400 ± 0.0418597 i 2, 2, 2.4973147 ± 0.1793125 i 2.5318482 3, 3, 3.5437306 3.8255711 ± 0.9599962 i, 3.8721508 ± 0.3127019 i 4, 4, Table I Engenvalues of four multimaterial wedges 796 W.-L YIN 797 MULTI-MATERIAL SINGULARITIES of the two trisector wedges are smaller than 0.5 (λ = 0.4891194 and 0.4732559, respectively, for the wedge at the lower and upper end of the crack) That is, the strengths of the singularities exceed that of an interface crack For the bisector wedge at A, an elasticity solution is obtained by the preceding procedure, using the traction data on a circle of radius r0 = cm generated by a finite element analysis using over 2000 triangular elements Twenty-two wedge eigensolutions are combined, including all eigensolutions with λ ≤ The resulting interfacial stresses between 0◦ and −60◦ layers are shown in Figure The leading term of the eigenseries contributes the dominant singular stress field of the order r λ−1 , where λ = 0.5924654 is the first eigenvalue When this asymptotic stress field is multiplied by the factor r 1−λ , the result is independent of r, and is a function of θ only The values of its components on the upper and lower sides of the interface θ = may be taken as the generalized stress intensity factors Sij+ and Sij− (In fact Sij+ and Sij− determine each other algebraically due to the three continuity conditions of tractions and another three continuity conditions of tangential strains): + − = Syy = 5.89400, Syy + Sxx = 20.8647, − = 4.12336, Sxx + − Sxy = Sxy = −2.41276, + Szz = 6.84576, − Szz = 2.77966, + − Syy = Syy = −1.27436, + Sxz = −8.23518, − Sxz = 0.94912, (5.3) where the unit of the stress is MPa Figure Interfacial stresses of the bisector wedge 798 W.-L YIN The results of Figure can only be deciphered for the range 10−2 r¯ 1, where r¯ ≡ r/r0 In Figures 3(a)–(c), the elasticity solutions of the interfacial stresses are normalized through multiplication by the factor r¯ 1−λ , and plotted in solid curves as functions of log10 (¯r ) It is seen that the normalized σy and τyz approach the asymptotic solution very slowly (Figures 3(a) and (c)) as compared to τxy Significant discrepancies are found even as r decreases to the subatomic scale and beyond (notice that since r0 = cm, r = 10−9 m corresponds to log10 (¯r ) = −7) This is due to the closeness of the first two eigenvalues A two-term approximate solution obtained by discarding all except the first two eigensolutions in the 22-term elasticity solution yields interfacial stresses that are shown as broken curves in Figures 3(a)–(c) These results show excellent agreement with the 22-term solution except for the domain r¯ > 10−3 The tangential stresses σx , σz and τxz are discontinuous across the interface Their normalized values on the upper and lower sides of the interface are shown in Figures 4(a) and (b) with r¯ also plotted in the logarithmic scale The preceding solution is compared with five additional elasticity solutions using truncated eigenseries of various lengths Each solution includes all terms associated with wedge eigenvalues that have real parts not greater than N, where N is taken successively to be 1, 2, 3, and 10 (N = corresponds to the pre- (a) Figure (a) σy of 22-, 2- and 1-term series; (b) τxy of 22-, 2- and 1-term series; (c) τyz of 22-, 2- and 1-term series MULTI-MATERIAL SINGULARITIES (b) (c) Figure (Continued.) 799 800 W.-L YIN (a) (b) Figure Tangential stresses (a) on the upper side (b) on the lower side of the interface MULTI-MATERIAL SINGULARITIES 801 Figure Syy of various solutions, bisector wedge ceding 22-terms solution) The generalized stress intensity factors Syy for all six solutions are shown in Figure For each solution, other components of Sij+ and Sij− are determined by the same stress ratios of the dominant singularity as given in equation (5.3) Except for the two solutions with the smallest number of terms, the results of the other solutions are in close agreement The eigenseries converge rapidly and, for this wedge, an accurate solution requires only eigensolutions with λ < For the trisector wedge at the lower end of the inclined crack, the traction data on r ≡ r0 = is obtained from the same finte-element analysis A 29-term eigenseries including all eigensolutions with Re[λ] is used in collocation The resulting interfacial stresses on the interface to the left of the singularity are shown in Figure The results are normalized with respect to r¯ −0.510881, and plotted versus log10 (¯r ) in Figures 7(a)–(c), where r¯ covers a much wider range from 10−50 to The corresponding asymptotic solutions are shown as dashed horizontal lines Any lingering faith that the asymptotic solution generally provides useful and realistic parameters for characterizing the criticality of stress singularity and for predicting failure initiation must be dispelled by the behavior of the interfacial peeling stress as shown in Figure 7(a) The peeling stress of the elasticity solution is nearly ten times greater than the asymptotic solution at r¯ = 10−5 , and about eight times greater at r¯ = 10−10 It is more than three times bigger even at r¯ = 10−50 On the other hand, τxy approaches the asymptotic solution much faster, as shown in Figure 7(b) Therefore, the stress ratios of the elasticity solution are very different from the generalized stress intensity factors The asymptotic solution has neither a physical relation nor a mathematical semblance to the stress field in the wedge at any physically meaningful length scale, because the region of dominance of the asymptotic solution has an extraordinarily small size of the order much smaller 802 W.-L YIN Figure Interfacial stresses on the left interface, trisector wedge (a) Figure (a) σy , (b) τxy , (c) τyz of 29-, 3- and 1-term series MULTI-MATERIAL SINGULARITIES (b) (c) Figure (Continued.) 803 804 W.-L YIN Figure Syy of various solutions, left interface of the trisector wedge than 10−50 Outside this minute region, the next two eigensolutions contribute significantly, as shown by the dashed curves in Figures 7(a)–(c), which combine the results of the first three terms of the 29-term solution The 3-term solution agrees closely with the full 29-term solution in the region r¯ < 10−5 Notice that the trisector wedge has a series of clusters of three closely spaced eigenvalues on or near the real axis, as seen in Table I On the interface to the right of the singularity, the interfacial streses are found to be significantly smaller and the plots are not shown The stresses σy and τxy approach the asymptotic solution more rapidly compared to the left interface, but τyz still has a very slow rate of approach The generalized stress intensity factors are given by the asymptotic stresses on the left and right interfaces multiplied by r¯ 0.510881 One has (all results in the unit MPa) On the left interface: Syy = 0.229723, On the right interface: Syy = 0.307952, Sxy = −0.577359, Sxy = 0.510823, Syz = 0.888552, Syz = −0.323603 In the interior segment of the interface at sufficiently large distances away from the bisector and trisector singularities, the stresses σy , τxy and τyz reach constant values 0, −7.85475 MPa and 0, respectively These limiting values are given by the layerwise constant stresses in an otherwise identical model without the inclined crack and with infinite length in the axial direction In the opposite limit of extremely small r, the stress τxy on the left and right interface approaches negative and positive asymptotic results, respectively, whose ratio is independent of loading (since the dominant singular value is real) Besides the 29-term solution, five additional solutions using all eigensolutions with λ N are obtained, where N assumes the values 1, 2, 3, and For the stress intensity factor Syy on the right interface, the results of the various solutions MULTI-MATERIAL SINGULARITIES 805 Figure Interface stresses of the bisector wedge in a realistic range of r are compared in Figure The 11-term solution (N = 2) yields relatively poor results, with an error of more than 20% compared to the 29-term solution The last four solutions, all including the eigensolutions with λ 3, are in excellent agreement On different interfaces the elasticity solutions show different stress ratios These ratios also vary greatly with r It is only meaningful to make comparison in a physically relevant range of r¯ , and comparisons of the interfacial stresses for different wedges require normalization with respect to a common power, which is conveniently taken to be r¯ −1/2 Figures and 10 show the results of the bisector wedge, and of the left interface of the trisector wedge, both normalized with relog10 (¯r ) (corresponding to spect to r¯ −1/2 and plotted over the range −7 r 0.01 m) Within this range, the stresses on the left interface of the 10−9 m trisector wedge (Figure 10) are much higher than those on the right interface (not shown) But the generalized stress intensity factors of the right interface exceed those on the left by more than one third The relative intensity of the peeling stress of the two wedges as shown in Figures and 10 appears to depend on the length scale and, therefore, may require consideration of damage mechanism and microstructure In the literature, so much attention has been directed to the order of stress singularities that it is almost taken for granted that a stronger order is invariably more threatening In the present case, the trisector wedge has a stronger order with 806 W.-L YIN Figure 10 Trisector wedge, left interface stresses over a realistic range of r the lowest eigenvalue 0.489119, versus 0.592465 for the bisector wedge But this difference may have very little to with the severity of the interfacial stresses in the two wedges over the physically relevant range of scales Results in Figure for the bisector wedge are more severe than the right interface of the trisector wedge Yet for exceedingly small length scales, the mathematical solution of the stresses on the latter interface will grow faster and exceed in magnitude the stresses in the bisector wedge Ultimately, if interface failure prediction is to be based on the elasticity solution, one must restrict attention to a physically relevant range of r, and ignore the mathematical solution beyond that range That is, one must formulate and apply failure criteria to the relevant analysis results such as shown in Figures and 10 Notice that these figures require accurate elasticity analysis as presented in this work In order to obtain results at a length scale r¯ = 10−N directly, one almost needs a conventional finite-element analysis with close to 102N elements, unless a substructuring method is used Solutions of the preceding two examples lead to the following observations: (1) The two-step substructure approach, in which a conventional finite-element analysis is used to generate the traction boundary conditions on a path encircling a singularity, and an elasticity solution of the multimaterial wedge is obtained by combining eigensolutions, provides a reliable, highly efficient and accurate method for the analysis of singularities in heterogeneous structures MULTI-MATERIAL SINGULARITIES 807 (2) The eigenseries converge rapidly In the two examples, the various elasticity solutions that include all eigensolutions with λ are in excellent agreement However, a solution that excludes some terms with smaller λ may incur very significant error (3) When the lowest singular eigenvalue is real, the elasticity solution approaches the dominant singular solution asymptotically as r → but the approach may be very slow As r decreases to subatomic size and even to 10−50 r0 in the present problems, the elasticity solution for the interfacial stress may still be significantly different from the asymptotic solution For the intervening range of r, the ratios of the stress components of the elasticity solution vary widely with r, and they can be very different from the ratios of the generalized stress intensity factors (4) Hence the asymptotic solution (including both the order of singularity and the generalized stress intensity factors) cannot be used generally to characterize the criticality of stress singularity, and to be used as the main basis for the prediction of failure initiation Interface failure may be affected by stress contributions from the second and third eigensolutions, and failure criteria must be based on relevant size scales as determined by the model geometry and microstructure (fiber diameters, etc.), since the mathematical results of the stress level in a minute region of subatomic size have no relevance to physical processes including failure initiation References W.-L Yin, A general analysis methodology for singularities in composite structures In: Proc AIAA/ASME/ASCE/AHS/ASC 38th SDM Conference, Kissimere, FL, 7–10 April 1997, pp 2238–2246 W.-L Yin, Mixed mode stress singularities in anisotropic composites In: Y.D.S Rajapakse and G.A Kardomateas (eds), Thick Composites for Load Bearing Structures, AMD 235 ASME, New York (1999) pp 33–45 W.-L Yin, K.C Jane and C.-C Lin, Singular solutions of multimaterial wedges under thermomechanical loading In: G J Simitses (ed.), Analysis and Design Issues for Modern Aerospace Vehicles – 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Rational Spirit in Modern Continuum Mechanics, ... Rational Mechanics and Analysis The footnote urges the rational student “to cleave the stinging fog of pseudo-philosophical mysticism” hiding the mathematics behind a certain formulation of the Second