The Mathematical Theory of Elasticity covers plane stress and plane strain in the isotropic medium, holes and fillets of assignable shapes, approximate conformal mapping, reinforcement of holes, mixed boundary value problems, the third fundamental problem in two dimensions, eigensolutions for plane and axisymmetric states, anisotropic elasticity, thermal stress, elastic waves induced by thermal shock, three-dimensional contact problems, wave propagation, traveling loads and sources of disturbance, diffraction, and pulse propagation. The Mathematical Theory of Plasticity explores the theory of perfectly plastic solids, the theory of strain-hardening plastic solids, piecewise linear plasticity, minimum principles of plasticity, bending of a circular plate, and other problems.
The Mathematical Theory of Elasticity covers plane stress and plane strain in the isotropic medium, holes and fillets of assignable shapes, approximate conformal mapping, reinforcement of holes, mixed boundary value problems, the third fundamental problem in two dimensions, eigensolutions for plane and axisymmetric states, anisotropic elasticity, thermal stress, elastic waves induced by thermal shock, three-dimensional contact problems, wave propagation, traveling loads and sources of disturbance, diffraction, and pulse propagation. The Mathematical Theory of Plasticity explores the theory of perfectly plastic solids, the theory of strain-hardening plastic solids, piecewise linear plasticity, minimum principles of plasticity, bending of a circular plate, and other problems.
Elasticity and Plasticity: The Mathematical Theory of Elasticity and The Mathematical Theory of Plasticity
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Overview
The Mathematical Theory of Elasticity covers plane stress and plane strain in the isotropic medium, holes and fillets of assignable shapes, approximate conformal mapping, reinforcement of holes, mixed boundary value problems, the third fundamental problem in two dimensions, eigensolutions for plane and axisymmetric states, anisotropic elasticity, thermal stress, elastic waves induced by thermal shock, three-dimensional contact problems, wave propagation, traveling loads and sources of disturbance, diffraction, and pulse propagation. The Mathematical Theory of Plasticity explores the theory of perfectly plastic solids, the theory of strain-hardening plastic solids, piecewise linear plasticity, minimum principles of plasticity, bending of a circular plate, and other problems.
Product Details
| ISBN-13: | 9780486810478 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 03/17/2016 |
| Series: | Dover Books on Mathematics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 160 |
| File size: | 24 MB |
| Note: | This product may take a few minutes to download. |
About the Author
Engineer Philip Gibson Hodge (1920–2014) taught at several universities, including the Illinois Institute of Technology and the University of Minnesota. He wrote many books on plasticity, including Theory of Perfectly Plastic Solids with William Prager.
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The Mathematical Theory of Elasticity & The Mathematical Theory of Plasticity
By J. N. Goodier, P. G. Hodge Jr.
Dover Publications, Inc.
Copyright © 2016 J. N. GoodierAll rights reserved.
ISBN: 978-0-486-81047-8
CHAPTER 1
THE MATHEMATICAL THEORY OF ELASTICITY
1.Introduction. Scope and intention
An inquirer seeking a representative account of the theory of elasticity at the present time will not find it between any one pair of covers, nor in any one language. In 1927, the date of the last edition of Love's treatise, he might almost have done so, and Love's interpretation of "The Mathematical Theory of Elasticity" was wider than ours will be, for it included plates and shells. He will find that in the past 5 years a "one-foot shelf" of new books has appeared which, in company with the older books, provides an excellent comprehensive treatment of the subject, and a basis for the understanding and evaluation of the steady stream of research papers.
The proposed short list for the one-foot shelf is, in inverse chronological order: A. E. Green and W. Zerna, Theoretical Elasticity(1954); L. A. Galin, Contact Problems of the Theory of Elasticity(in Russian) (1953) ; N. I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity(3rd Russian edition, 1949, translated into English by J. R. M. Radok, 1953; a 4th Russian edition appeared in 1954); N. I. Muskhelishvili, Singular Integral Equations(2nd Russian edition, 1946, translated into English by J. R. M. Radok, 1953); G. N. Savin, Concentration of Stress around Holes(in Russian) (1951); S. G. Lekhnitzki,Theory of Elasticity of Anisotropic Bodies(in Russian) (1950).
The preponderance of Russian titles is striking, and these books are based almost entirely on recent Russian investigations. †Some 250 Russian papers on the subject, in the narrower sense adopted here, have been noticed in Mathematical Reviews and Applied Mechanics Reviews since 1940. Most of these, all the books of the short list above, and some others, have been available for this survey.
It is not intended to be an exhaustive and proportioned survey of all branches of the subject. The limitations of space, time, and competence would forbid that in any case. Its principal aim is to draw attention to those significant recent developments believed least known to readers whose first language is English. What is relatively well known or easily accessible in this sense is omitted or touched on only briefly and broadly. The bibliography includes only those books and papers actually discussed or cited.
2.Plane stress and plane strain in the isotropic medium
The major development of the present century in this branch of the subject has occurred chiefly in the work of Muskhelishvili and in the numerous investigations inspired by it. The importance and promise of the new methods and results were recognized and made available in English by I. S. Sokolnikoff some 15 years ago. They were concerned with the problems of prescribed boundary forces (the first fundamental problem) and of prescribed boundary displacements (the second fundamental problem). The later development has included the problem of mixed boundary conditions — the specified conditions involving both force and displacement, as when force is prescribed on one part of the boundary, displacement on the remainder — (the third fundamental problem). It is set forth, as of 1949, in Radok's translation of the book by Muskhelishvili [1].
While the new method for the third problem was emerging, there was extensive application of Muskhelishvili's methods for the first and second problems, predominantly in Russia. Most of this finds only mention in Muskhelishvili's own book, devoted as it is to method and comprehensive forms of solution rather than to detailed application. Here the book by Savin [1] is a valuable supplement, but it has not been translated. It provides complete solutions and many detailed evaluations for a great variety of problems of stress concentration at holes, several of which have been worked out independently in recent non-Russian papers. The book is not, however, limited to the plane problems of the homogeneous isotropic medium as dealt with in Muskhelishvili's book. It deals as extensively with stress concentration in the anisotropic medium, and also with the closely similar thin plate flexure problems, isotropic and (briefly) anisotropic. The bibliography of 135 items is naturally mainly Russian, but the relevant non-Russian literature appears to be fairly adequately represented, and some sections of the book are founded on it.
The basis for the isotropic problems is the now well-known Kolosov representation of stress (σx, σy, τxy) and displacement (u, v) in terms of two complex potentials z), ψ(z),
1. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
2. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
3. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where μ is the shear modulus, v Poisson's ratio, κ = 3 - 4v for plane strain and (3 -v)/(1 + v) for plane stress. These formulas for rectangular components lead very readily to components in curvilinear co-ordinates [xi], η, derived from a conformai mapping z= ωζ),ζ = [xi] + iη, which is usually to a unit circle in the [??]-plane of mathematical operation, from the physical region in the z-plane. The character of the parts of the complex potentials [??] (z), ψ(z) which are nonholomorphic (in the hole or in the material region), corresponding for instance to resultant force on a hole, or dislocation discontinuity at a cut, is made out in Muskhelishvili's book. The determination of the holomorphic parts remains. He showed that this may be effected from the boundary conditions by using the well-known Cauchy integral formula for analytic functions, although the application of this formula is not immediate and direct. Theorems derived from it, apparently for this purpose by Muskhelishvili himself, are required and are given with proofs in his book [1].
Savin [1] provides detailed treatment, in most cases with tables, curves, and charts, of an extensive set of particular problems of stress concentration at holes. The hole shapes are of four types: (1) the rectangle with rounded corners, (2) the triangle with rounded corners, (3) the ellipse, (4) the circle. These are taken with and without reinforcement, as disturbances of uniform normal or shearing stress, or simple distributions (bending, cantilever bending).
The actual shapes for which detailed results are given are selected, in types (1) and (2), by forming the Schwarz-Christoffel transformation for the exact rectangle or triangle, and developing it as a series, as, for instance, for the square:
(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Then the first two, three, or four terms only are retained, to give three shapes progressively closer to the square. Problems of holes of the ovaloid forms (c, m, n constants)
(5, 6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
have been solved (with evaluations) by Greenspan, Morkovin, and Green respectively. There is of course no necessity in Savin's choice of the coefficients as those of the SchwarzChristoffel expansion. Technical interest is likely to be focused, not on "ideal" holes with perfectly sharp corners and therefore usually infinite stress concentrations, but on corners rounded in a geometrically simple fashion, for which reliable stress estimates can be made. This question of choice of shape for evaluations such as those given by Savin is one of some importance, and will be raised again later.
The book gives a brief approximate treatment of a problem of considerable engineering interest which, as far as the writer has observed, has not yet appeared in the non-Russian literature. For a circular hole in a thin cylindrical shell, the effects of the curvature of the shell are represented in the stress-concentration factors (on longitudinal stress)
(7,8) k1 = 3 (1 + 0.43 ρ02/ah), k2 = 2.5 (1 + 2.3 ρ02/ah)
for fields of longitudinal tension, and of longitudinal and circumferential tensions as in a pressure cylinder with closed ends. Here ρ[0] is the radius of the hole, a the radius, and h the thickness of the shell. The source is Lourie [1, 2].
Savin's accounts of the strip with a centrally placed circular hole, the circular hole in the semi-infinite plate, and two holes intersecting or nonintersecting, are in terms of Fourier methods and real variables, and in the main are drawn from the work of Howland, Mindlin, and Ling. A recent addition to calculated results of this kind is given by Isida [1], for a strip with an eccentric circular hole, under bending and under tension, and is complemented by photoelastic confirmation. References of the period 1940–1950 may be found in an earlier survey. A paper on the strip with a semicircular notch in each edge by Ling [1] constructs a sequence of stress functions each member of which is an infinite series (here of Fourier type). Superposition of such members, or their construction by satisfying first one boundary, then another, is of course well known, and is exemplified by numerous papers on combinations of circular and straight boundaries (as Howland's ). But Ling effects an improvement in these sequences, under the term "promotion of rank." The earlier Fourier harmonics of each member of the sequence are removed by suitable subtraction of preceding members, and in the new sequence each member starts at a Fourier harmonic of higher order than its predecessor.
In general, problems of regions of more than twofold connectivity have not yielded to the complex variable methods, since they are not amenable to conformai transformation within or outside the unit circle. Muskhelishvili [1] (Radok's translation, p. 397) touches on reductions to Fredholm integral equations by Sherman and Mikhlin. In later papers Sherman [1, 2, 3] devotes lengthy analyses in the complex variable to the heavy vertical plate (putting the matter in plane stress terms) with two circular holes, or with two elliptical holes. A free straight horizontal upper edge is contemplated, but the problems really concern the infinite plate, the holes being taken too small and too far down to be influenced by the free edge. The circular hole cases seem suitable for the general Fourier form of solution in bipolar co-ordinates constructed by Jeffery in 1921, and used for numerous particular problems since. On the whole it appears that the multiply-connected regions stand in the way of a general unification of the two-dimensional theory in terms of complex function theory.
Gravitational stress affected by noncircular holes or tunnels is not considered in Savin's book. Yu [1] has derived expressions for the complex potentials and the stress at the hole for the ovaloid holes of the transformation (5). He has also given the forms taken in a Muskhelishvili representation by the solutions for the heavy circular annulus supported at a point, and the heavy disk supported by concentrated forces. Savin does include the annulus with diametrically opposite forces (the "proving-ring problem") and gives some Russian evaluations.
3.Holes and fillets of assignable shapes. Approximate conformai mapping
The ovaloid holes given by the transformations (4), (5), and (6) have been extensively studied and their stress concentrations evaluated, because they are mathematically convenient and give shapes approaching rectangles, rhombi, and triangles with rounded corners. The practical estimation of such studies will depend on whether they can be used with confidence for actual holes, which commonly will have straight sides and circular arc roundings at the corners. The shape constants m, n of the transformation (5) can be adjusted to give, for a rectangle, the required ratio of sides, and the ratio of the minimum radius of curvature at the corners to a side. But the rounding off is not of circular arc form, and the sides are not quite straight. In general, the attraction of mathematically convenient exact conformai mappings tends to make the development of the two-dimensional theory resemble that of the Saint-Venant theory of torsion, in which a large accumulation of results is available for mathematically simple but otherwise very improbable shapes.
Greater practical interest would be engaged by a method which could cope directly with an actual given shape. Such a method is offered by Kikukawa [1, 2, 3], and its possibilities demonstrated by complete calculations for a number of striking examples of stress concentration in the stretched plate with (a) a hole of rhombic form, with circular arc fillets at the corners; (b) a double notch, each notch having straight parallel sides joined to U-shape by a semicircle; (c) circular quadrant fillets at a change from finite to infinite width. The results for case (b) agree very well with those of Neuber for the double hyperbolic notch. The results for case (a) show, if they are valid, that the ovaloid approximation to the shape attainable by a mapping such as (5) is not adequate. No previous theoretical evaluation of case (c) is known to the writer, although there are photoelastic measurements.
Kikukawa's method proceeds from an initial mapping z= z0(ζ) in simple definite form, for instance, the mapping (5) for case (a) above. This is amended to bring the hole shape sufficiently close to the prescribed form. The method of Theodorsen and Garrick, devised for airfoil shapes, is not found suitable.
The initial mapping z = z0(ζ) gives a curve C0 (e.g., ovaloid) in the z-plane somewhat different from the given curve C]e.g., the hole of case (a) above], as shown in Fig.1b. The point Q0 on C0 corresponds to a point ζ = eiτ on the unit circle of the [??]-plane (Fig. 1a). The point Q is the unknown, but neighboring, point on C derived from the desired, but of course unknown, mapping function z(ζ). Writing
z(ζ) - z0(ζ) = Δz(ζ)
the vector Q0Q is Δz(ζ) at ζ = eiτ If Δn(τ) and Δt(τ) are the components of Q0Q along the normal and tangent to C0 at Q0, we have (Fig. 1c)
(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where y, the angle between the normal and the x-axis, is given by
(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
From (9) and (10)
(11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The domain bounded by the single curve C in the z-plane can be mapped into the interior of the unit circle T in the ζ-plane by z = z(ζ), where z(ζ) is holomorphic in T and z'(ζ) ≠ 0 in T. The mapping is unique under the normalization conditions z(0) = a, and arg z'(0) = 0 Both z(ζ) and z(ζ) are chosen accordingly.
The function on the left of (11) is holomorphic in T because
(i) Δz(ζ) is holomorphic in T, and Δz(0) = 0 by normalization.
(ii)z0'(ζ) is holomorphic in T.
There is accordingly a Maclaurin series
(12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where Δmp = Δmp' + i Δmp" (p= 1, 2, ...), but Δm0 is real because Δz'(0) and z0'(0) are real. It follows from (11) that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
and so
(13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Now if Δn(τ) were known, the coefficients of the series (12) could be evaluated by these formulas (13). In fact, we have only the two curves C0 and C, and do not know the point Q which, by the desired mapping, corresponds to eiτ on the unit circle |ζ| = 1. But Δn(τ), being the projection of Q0Q on the normal to C0 at Q0, is nearly equal to the distance [??]n(τ) from C0 to C along the normal. This is known, and it is used in place of Δn(τ) to evaluate approximately the coefficients (13) of the series (12). The complex potentials of (1), (2), and (3) are found by a simple expansion procedure.
For the hole of case (a) above, Kikukawa finds that two terms of (12) make a substantial difference to the stress calculated for the initial ovaloid, but the third term brings little further change. If these results are confirmed, the method will add much to the effectiveness of the two-dimensional theory, as well as to plate flexure theory. These biharmonic problems are much less suitable for numerical methods, as relaxation, than are the potential problems of torsion or flexure.
4.Reinforcement of holes
The artificiality of the (in the exact sense) mathematically amenable forms is no less troublesome in the problem of the hole reinforced by a ring. The circular hole and ring offer no difficulty, and have been well investigated. References may be found in the recent paper by Heller [1] and in the writer's earlier survey (footnote, p. 7). Problems of nests of circular rings in holes (a hole with a ring welded inside it, then another, and another, etc.), the rings being of different materials, appear in several Russian papers. The usual fields of tension, bending, and shear prevail at infinity. Each ring as well as the plate is treated by plane stress theory. References, and evaluations for a single ring of different material (or equivalently under plane stress assumptions, of a different thickness), are given in Ch. 5 of Savin [1]. Being for specialized values, the evaluations do not cover a representative range of conditions, and are merely illustrative. The other investigations have proceeded independently, most of them treating the ring as a curved bar (the "bead" reinforcement). Heller [1], in an analysis of this kind for a reinforced circular hole centrally placed in a plate carrying bending and shear stress, finds as Reissner and Morduchov [1] had suggested, that the reinforcement does its work in tension or compression rather than bending.
(Continues...)
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Table of Contents
This volume comprises two classic essays on the mathematical theories of elasticity and plasticity by authorities in this area of engineering science. Undergraduate and graduate students in engineering as well as professional engineers will find these works excellent texts and references.
The Mathematical Theory of Elasticity covers plane stress and plane strain in the isotropic medium, holes and fillets of assignable shapes, approximate conformal mapping, reinforcement of holes, mixed boundary value problems, the third fundamental problem in two dimensions, eigensolutions for plane and axisymmetric states, anisotropic elasticity, thermal stress, elastic waves induced by thermal shock, three-dimensional contact problems, wave propagation, traveling loads and sources of disturbance, diffraction, and pulse propagation. The Mathematical Theory of Plasticity explores the theory of perfectly plastic solids, the theory of strain-hardening plastic solids, piecewise linear plasticity, minimum principles of plasticity, bending of a circular plate, and other problems.
Dover (2016) republication of the edition originally published by John Wiley & Sons, New York, 1958.
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