Elasticity (eBook)

Theory, Applications, and Numerics

Martin H. Sadd (Autor)

eBook Download: PDF | EPUB

2014 | 3. Auflage
600 Seiten
Elsevier Science (Verlag)
978-0-12-410432-7 (ISBN)

Elasticity: Theory, Applications, and Numerics, Third Edition, continues its market-leading tradition of concisely presenting and developing the linear theory of elasticity, moving from solution methodologies, formulations, and strategies into applications of contemporary interest, such as fracture mechanics, anisotropic and composite materials, micromechanics, nonhomogeneous graded materials, and computational methods. Developed for a one- or two-semester graduate elasticity course, this new edition has been revised with new worked examples and exercises, and new or expanded coverage of areas such as spherical anisotropy, stress contours, isochromatics, isoclinics, and stress trajectories. Using MATLAB software, numerical activities in the text are integrated with analytical problem solutions. These numerics aid in particular calculations, graphically present stress and displacement solutions to problems of interest, and conduct simple finite element calculations, enabling comparisons with previously studied analytical solutions. Online ancillary support materials for instructors include a solutions manual, image bank, and a set of PowerPoint lecture slides. - Thorough yet concise introduction to linear elasticity theory and applications - Only text providing detailed solutions to problems of nonhomogeneous/graded materials - New material on stress contours/lines, contact stresses, curvilinear anisotropy applications - Further and new integration of MATLAB software - Addition of many new exercises - Comparison of elasticity solutions with elementary theory, experimental data, and numerical simulations - Online solutions manual and downloadable MATLAB code

Martin H. Sadd is Professor Emeritus of Mechanical Engineering and Applied Mechanics at the University of Rhode Island. He received his Ph.D. in mechanics from the Illinois Institute of Technology and began his academic career at Mississippi State University. In 1979 he joined the faculty at Rhode Island and served as department chair from 1991 to 2000. Professor Sadd's teaching background is in the area of solid mechanics with emphasis in elasticity, continuum mechanics, wave propagation, and computational methods. He has taught elasticity at two academic institutions, in several industries, and at a government laboratory. Professor Sadd's research has been in computational modeling of materials under static and dynamic loading conditions using finite, boundary, and discrete element methods. Much of his work has involved micromechanical modeling of geomaterials including granular soil, rock, and concretes. He has authored more than 75 publications and has given numerous presentations at national and international meetings.

14.6. Torsion problem

We now wish to re-examine the torsion of elastic cylinders for the case where the material is nonhomogeneous. The basic formulation and particular solutions were given in Chapter 9 for the homogeneous case and in Chapter 11 for anisotropic materials. Although a vast amount of work has been devoted to these problems, only a few studies have investigated the corresponding inhomogeneous case. Early work on the torsion of nonhomogeneous cylinders includes Lekhnitskii (1981); later studies were done by Rooney and Ferrari (1995) and Horgan and Chan (1999c). As expected, most closed-form analytical solutions for the inhomogeneous problem are limited to cylinders of revolution, normally with circular cross-sections.

Following the work of Horgan and Chan (1999c), we consider the torsion of a right circular cylinder of radius a, as shown in Figure 14.24. The cylindrical body is assumed to be isotropic, but with graded shear modulus that is a function only of the radial coordinate μ = μ(r). The usual boundary conditions require zero tractions on the lateral boundary S and a resultant pure torque loading T over each end section R.

FIGURE 14.24 Torsion of a Nonhomogeneous Circular Cylinder.

The beginning formulation steps remain the same as presented previously, and thus the displacements, strains, and stresses are the same as given in Chapter 9

(14.6.1)

(14.6.2)

(14.6.3)

It again becomes useful to introduce the Prandtl stress function, ϕ = ϕ(x,y)

(14.6.4)

so that the equilibrium equations are satisfied identically. We can again generate the compatibility relation among the two nonzero stress components by differentiating and combining relations (14.6.3)2,3 to eliminate the displacement terms. Substituting (14.6.4) into that result gives the governing relation in terms of the stress function

(14.6.5)

where the shear modulus μ must now be left inside the derivative operations because the material is inhomogeneous. Recall that, for the homogeneous case, relation (14.6.5) reduced to the Poisson equation ▿2ϕ = 2μα.

Incorporation of the boundary condition that tractions vanish on the lateral surface S leads to identical steps as given previously by equations (9.3.10)–(9.3.12), thus leading to the fact that the stress function must be a constant on all cross-section boundaries

(14.6.6)

For simply connected sections, the constant may again be chosen as 0. Invoking the resultant force conditions on the cylinder end planes (domain R), as given by relations (9.3.14), again yields

(14.6.7)

and the torsional rigidity J can again be defined by J = T/α.

Because we wish to use a simple radial shear modulus variation, μ = μ(r), it is more convenient to use a polar coordinate formulation. For the circular cylinder under study, the problem reduces to an axisymmetric formulation independent of the angular coordinate and the warping displacement vanishes. Under these conditions the displacements, strains, and stresses reduce to

(14.6.8)

Relations (14.6.4) and (14.6.5) for the stress function formulation then reduce to a system in terms of only the radial coordinate r

(14.6.9)

with boundary condition ϕ(a) = 0.

The governing differential equation (14.6.9) can be easily integrated to give the general solution

(14.6.10)

where C1 and C2 are arbitrary constants. We require that the solution for ϕ remain bounded as r → 0, thus implying that each integral term in (14.6.10) be finite at the origin. Restricting ourselves to the plausible case where the shear modulus is expected to be nonzero but bounded at the origin, , while the second integral, , is singular. Based on these arguments, C1 must be set to 0. Finally, the boundary condition ϕ(a) = 0 determines the final constant C2 and produces the general solution

(14.6.11)

With this result, the shear stress and torsional rigidity then become

(14.6.12)

To explore the effects of inhomogeneity, let us consider some specific gradations in shear modulus. Following some of the examples discussed by Horgan and Chan (1999c), we consider two cases of the following form

(14.6.13)

where n ≥ 0 and μo > 0 and m are material constants. Note that, for either example, as n → 0 we recover the homogeneous case μ(r) = μo. Also, as r → 0, μ → μo, and so these material examples all have finite shear modulus at r = 0.

Plots of these shear modulus gradations are shown in Figure 14.25 for various cases of material parameter m with n = 1. For the model given by (14.6.13)1, three cases are shown. The m = 1 case corresponds to a linearly increasing shear modulus from the central axis of the shaft, while m = −1 or −3 gives a nonlinear decreasing gradation in material stiffness. The figure also shows the modulus variation for the exponential graded model given by (14.6.13)2 for the case n = 1. All gradation forms (14.6.13) allow simple solutions to be generated for the stress function, shear stress, and torsional rigidity.

FIGURE 14.25 Shear Modulus Behavior for Torsion Problems (n = 1).

Solutions for the gradation model given by (14.6.13)1 are found to be

(14.6.14)

(14.6.15)

Note that the solution for the stress function requires integration through relation (14.6.11), and thus closed-form solutions can only be determined for integer and other special values of the parameter m. From relation (14.6.15), it can be shown that if m ≥ −1, the maximum shear stress always occurs at the boundary r = a. Recall that this result was found to be true in general for all homogeneous cylinders of any cross-section geometry (see Exercise 9.5). However, for the inhomogeneous case when m < −1, the situation changes and the location of maximum shear stress can occur in the cylinder’s interior.

Horgan and Chan (1999c) have shown that for the case with n > 0, the choice of m < −1 − 1/n produces a maximum shear stress τθz at r = −a/n(1 + m). These results imply that modulus gradation can be adjusted to allow control of the location of (τθz)max. Dimensionless shear stress distributions for model (14.6.13)1 are shown in Figure 14.26 for various cases of material parameters m and n. As expected, higher stresses occur for a gradation with increasing shear modulus. For the homogeneous case, the shear stress distribution will be linear, as predicted from both elasticity theory and mechanics of materials. For the nonhomogeneous cases with n = 1 and m = ±1, it is noted that the maximum shear stress occurs on the boundary of the shaft. However, for the case shown with n = 1 and m = −3, the maximum stress occurs interior at r = a/2 according to our previous discussion.

Considering next the solutions for the gradation model (14.6.13)2, relations (14.6.11) and (14.6.12) give

(14.6.16)

It can easily be shown that if n ≤ 1 the maximum shear stress will exist on the outer boundary, while if n > 1 the maximum moves to an interior location within the shaft. Nondimensional shear stress distributions for this exponential gradation are shown in Figure 14.27 for several values of the parameter n. As observed in the previous model, for the case with decreasing radial gradation (n > 0), the shear stress will always be less than the corresponding homogeneous distribution. With n < 0, we have an increasing radial gradation that results in stresses larger than the homogeneous values.

The discussion of the location of maximum shear stress can...

Erscheint lt. Verlag	22.1.2014
Sprache	englisch
Themenwelt	Naturwissenschaften ► Physik / Astronomie ► Mechanik
	Technik ► Bauwesen
	Technik ► Fahrzeugbau / Schiffbau
	Technik ► Luft- / Raumfahrttechnik
	Technik ► Maschinenbau
ISBN-10	0-12-410432-0 / 0124104320
ISBN-13	978-0-12-410432-7 / 9780124104327

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