Plane Elasticity Theory

Galin, L. A.

doi:10.1007/978-1-4020-9043-1_2

Cited by 22 publications

(49 citation statements)

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“…In the further limit c-> 0, F{E) -*• £ and K(c) -> \n, so (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21) in the adhesive region 0 < x < c. Fig. 3 shows computed curves of qj/ip for the case v -0 for three values of c, for comparison with this expression.…”

Section: -2 a Limiting Solutionmentioning

confidence: 97%

“…Since \q/p\ is at most equal to fi, the term in square brackets in equation (2-11) is of order/ty times the normal stress term,.and a first approximation 254 D. A. SPENCE that throws some light on the physics can be found by omitting the term, when (2)(3)(4)(5)(6)(7)(8)(9)(10)(11) has the normalized solution…”

Section: -2 a Limiting Solutionmentioning

confidence: 99%

“…where tan?ra = /iy (0 < a < which are to be solved subject to the requirement that r vanishes on (c, 1) and is bounded and non-negative on (0, c) in accordance with (2)(3)(4)(5)(6)(7). A singularity in p at x = 1 can however be admitted in the solution.…”

Section: Reduction Of the Dual Problem To A Fredholm Equationmentioning

confidence: 99%

“…as c-»0, so that the width of the adhesive region is given, according to (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18), by as y as c = for fixed y. In the further limit c-> 0, F{E) -*• £ and K(c) -> \n, so (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21) in the adhesive region 0 < x < c. Fig.…”

Section: -2 a Limiting Solutionmentioning

confidence: 99%

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An eigenvalue problem for elastic contact with finite friction

Spence¹

1973

Math. Proc. Camb. Phil. Soc.

108

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The two-dimensional indentation of an elastic half space by a rigid punch under a slowly applied normal load is considered, for the case in which there is a finite coefficient of friction (i between the surfaces. The contact area is then divided into an inner adhesive region -c < x < c in which the surface displacements are known, surrounded by regions c < \x\ < 1 in which the friction is limiting and the lateral displacement (which must increase in proportion to the overall load) is not known in advance. The problem is formulated in terms of a coupled pair of singular integral equations for the normal and shear stresses cr and r at the surface; these are combined to give a single homogeneous Fredholm equation with positive kernel for a quantity proportional to the difference T -/t > 0, gives the adhesive boundary c in terms of ji and Poisson's ratio v. A similarity transformation shows that c has the same value for both flat-faced and power law punches.1. Introduction. The problem posed by the indentation of an elastic half space by a rigid punch has a straightforward solution in the case of frictionless contact, for which a boundary condition of zero shear stress beneath the punch applies. As a step towards the physical situation in which the shear stress is controlled by friction, solutions have also been found in a number of cases for the opposite limit of fully adhesive contact, in which no relative slip between the indentor and the half-space is allowed once contact has been established. The simplest such case is the symmetrical two-dimensional indentation of a half space by a flat punch over the interval -1 < x < 1 considered by Muskhelishvili ( (7), pp. 475-7). The solution of the linear elastic equations in this case, although physically realistic near the centre of the punch, is not so near the edges, where the ratio of shear r to normal stress cr at the surface, given by

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Section: -2 a Limiting Solutionmentioning

confidence: 97%

Section: -2 a Limiting Solutionmentioning

confidence: 99%

Section: Reduction Of the Dual Problem To A Fredholm Equationmentioning

confidence: 99%

Section: -2 a Limiting Solutionmentioning

confidence: 99%

See 2 more Smart Citations

An eigenvalue problem for elastic contact with finite friction

Spence¹

1973

Math. Proc. Camb. Phil. Soc.

108

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“…On each sample 2 series of statistical nanoindentation tests were carried out that differ in the loading protocol: the first series assesses the microstructure from indentation hardness and indentation stiffness results; obtained by a trapezoidal load history with a maximum load of P max ϭ 2 mN, applied in 10 s, kept constant over 5 s and unloaded in 10 s. Following fast loading, the 5-s dwelling time is short enough to ensure that the indentation hardness, H ϭ P max /A c (with A c the projected area of contact between the indenter probe and the indented surface), is representative of the strength content (19,23,24). In turn, the unloading is fast enough so that the indentation modulus M, determined from the unloading slope S ϭ (dP/ dh) hϭh max at the end of the holding phase truly relates to the elasticity content of the indented material (25), namely the indentation modulus M ϭ E/(1 Ϫ 2 ) ϭ S/(2a U ), with E the Young's modulus, the Poisson's ratio, and a U ϭ ͌ A c / the radius of contact between the indenter probe and the indented surface upon unloading (21,22,26). The load protocol of the second series differs from the first in a 180-s long dwelling period, which allows the assessment of the contact creep compliance rate, L (t) Х 2a U ḣh/P max , and the creep compliance, L(t) ϭ 2a U ⌬h(t)/P max ϩ const, from the time-dependent indentation depth rate ḣ(t) and the change in indentation depth ⌬h(t) ϭ h(t) Ϫ h 0 in excess of the indentation depth h 0 Ϸ 200 nm Ϯ 45 nm recorded during the 10-s loading to P max ϭ 2 mN (Fig.…”

mentioning

confidence: 99%

Nanogranular origin of concrete creep

Vandamme

Ulm

2009

Proc. Natl. Acad. Sci. U.S.A.

379

192

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Concrete, the solid that forms at room temperature from mixing Portland cement with water, sand, and aggregates, suffers from time-dependent deformation under load. This creep occurs at a rate that deteriorates the durability and truncates the lifespan of concrete structures. However, despite decades of research, the origin of concrete creep remains unknown. Here, we measure the in situ creep behavior of calcium-silicate-hydrates (C-S-H), the nano-meter sized particles that form the fundamental building block of Portland cement concrete. We show that C-S-H exhibits a logarithmic creep that depends only on the packing of 3 structurally distinct but compositionally similar C-S-H forms: low density, high density, ultra-high density. We demonstrate that the creep rate (Ϸ1/t) is likely due to the rearrangement of nanoscale particles around limit packing densities following the free-volume dynamics theory of granular physics. These findings could lead to a new basis for nanoengineering concrete materials and structures with minimal creep rates monitored by packing density distributions of nanoscale particles, and predicted by nanoscale creep measurements in some minute time, which are as exact as macroscopic creep tests carried out over years.C oncrete is the most-used construction material on earth. The annual worldwide production stands at 20 billion tons and increases per annum by 5%. However, the fundamental causes of concrete creep are still an enigma, and have deceived many decoding attempts from both experimental (1-3) and theoretical sides (4-8). In the United States alone, concrete creep is partly responsible for an estimated 78.8 billion dollars required annually for highway and bridge maintenance. Although it is generally agreed that the complex creep behavior of concrete materials is largely related to the viscoelastic response of the primary hydration product and binding phase of hardened Portland cement paste, the calcium-silicate-hydrate (C-S-H), the creep properties of C-S-H have never been measured directly. C-S-H precipitates when cement and water are mixed, as clusters of nanoscale colloidal particles (9, 10) that cannot be recapitulated ex situ in bulk form suitable for macroscopic testing. Over decades, therefore, concrete creep properties have been probed on the composite scale of mortar and concrete (11), with the conclusion that there are 2 distinct creep phenomena at play ( Fig. 1 A and B): a short-term volumetric creep and a long-term creep associated with shear deformation (3,7,12,13), with a creep rate evolving as a power function t Ϫn of exponent n between 0.9 and 1 (14). The assessment of this long-term creep is most critical for the durability of concrete structures, and requires, for a specific concrete composition and structural application, years of expensive macroscopic testing (11,14). After more than 40 years of research (4-8), basic questions persist regarding the physical origin of this logarithmic creep and its link with microstructure and composition.In this study, we investiga...

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The investigation of singular integro-differential equations relating to adhesive contact problems of the theory of viscoelasticity

Shavlakadze

Odishelidze

Criado-Aldeanueva

2021

Z. Angew. Math. Phys.

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The exact and approximate solutions of singular integro-differential equations relating to the problems of interaction of an elastic thin finite or infinite non-homogeneous patch with a plate are considered, provided that the materials of plate and patch possess the creep property. Using the method of orthogonal polynomials the problem is reduced to the infinite system of Volterra integral equations, and using the method of integral transformations this problem is reduced to the different boundary value problems of the theory of analytic functions. An asymptotic analysis is also performed.

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Plane Elasticity Theory

Cited by 22 publications

References 0 publications

An eigenvalue problem for elastic contact with finite friction

An eigenvalue problem for elastic contact with finite friction

Nanogranular origin of concrete creep

The investigation of singular integro-differential equations relating to adhesive contact problems of the theory of viscoelasticity

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