Abstract
In much of this text we shall be concerned with the study of elastic bodies. Accordingly, we shall now present a brief treatment of the theory of elasticity. In developing the theory we shall set forth many concepts that are needed for understanding the variational techniques soon to be presented.
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Notes
- 1.
Note that normal stress actually follows this very same convention.
- 2.
We are thus tacitly assuming that the quantities in Eq. 1.1 vary in a continuous manner.
- 3.
We have only considered body force distributions here. If one assumes body-couple distributions that may occur as a result possibly of magnetic or electric fields on certain kinds of dielectric and magnetic materials, we will have in Eq. (1.8) the additional integral \({x\prime_1},\,{x\prime_2},\,{x\prime_3}\) M dv where M is the couple-moment vector per unit volume. The result is that Eq. (1.8) becomes ε ijk τ kj + M i = 0. The stress tensor is now no longer a symmetric tensor. Such cases are beyond the scope of this text.
- 4.
a 1i represents the set of direction cosines between the \({x\prime_1}\) axis and the x 1, x 2 and x 3 axes. It is therefore a unit vector in the \({\tau \prime_{31}} = T_i^{(3\prime)}{a_{1\,\,i}} = \left( {{\tau _{ij}}{a_{3\,j}}} \right){a_{1\,i}} = {a_{3\,\,j}}{a_{1\,\,i}}{\tau _{ij}} = {a_{3\,\,j}}{a_{1\,\,i}}{\tau _{ji}}\) direction.
- 5.
We could also have arrived at this equation by using one of the quotient rules of Appendix I in conjunction with Cauchy’s formula
$$T_i^{(\nu )}$$since v j is an arbitrary vector (except for length) and \(T_i^{(\nu )} = {\tau _{ij}}{\nu _j}\) is a vector.
- 6.
This assumes that ∂j/∂x i , i = 1,2,3 are all of the same order of magnitude. Thus consider
$${{\partial {u_2}} \over {\partial {x_1}}}{{\partial J} \over {\partial {x_2}}}\,{\rm{and}}\,{{\partial {u_3}} \over {\partial {x_1}}}{{\partial J} \over {\partial {x_3}}}$$It is clear we can neglect ∂u 1/∂x 1 compared to unity. To neglect the last two terms means that
$${\partial \over {\partial {\xi _i}}} = {\partial \over {\partial {x_i}}}$$must be small compared to ∂J/∂x 1. For this to be true the ∂J/∂x i must be of the same order of magnitude.
- 7.
You will recall that in developing the equations of equilibrium we employed only a single reference, and the tacit assumption taken was that the geometry employed for the equation was the deformed geometry. For infinitesimal deformation we may use the undeformed geometry rather than the deformed geometry for expressing equations of equilibrium. We shall discuss this question and other related questions in Chap. 8
- 8.
A simply-connected region is one for which each and every curve can be shrunk to a point without cutting a boundary.
- 9.
From our discussion thus far, we can say that the Helmholtz function represents the energy that can be converted to mechanical work in a reversible isothermal process. This is often called free energy. This is one reason for its importance in thermodynamics.
- 10.
A positive definite function has the property of never being less than zero for the range of the variables involved and is zero only when the variables are zero. This means U 0 = 0 at a point when the strains are zero at the point.
- 11.
- 12.
A homogeneous material has the same composition throughout.
- 13.
G is the shear modulus and is often represented by the letter μ in the literature.
- 14.
Having the same rigid-body resultant.
- 15.
Goodier, J. N., “A General Proof of St. Venant’s Principle,” Philosophy Magazine, 7, No. 23, 637 (1937). Hoff, N. J., “The Applicability of Saint Venant’s Principle to Airplane Structures,” J. of Aero. Sciences, 12, 455 (1945). Fung, Y. C., “Foundation of Solid Mechanics,” Prentice-Hall Inc., 1965, p. 300.
- 16.
Hooke’s law for plane stress giving stress in terms of strain will be used later in the text and can be found from Eq. (1.117) by straightforward algebraic steps to be:
$$\begin{array}{lll} \varepsilon _{xx} = \frac{1}{E}\left( {\tau _{xx} - \nu \tau _{yy} } \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\gamma _{xy} = \frac{1}{G}\tau _{xy} \cr \varepsilon _{yy} = \frac{1}{E}\left( {\tau _{yy} - \nu \tau _{xx} } \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\gamma _{xz} = 0 \cr \varepsilon _{zz} = - \frac{\nu }{E}\left( {\tau _{xx} - \tau _{yy} } \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\gamma _{yz} = 0 \end{array} $$(1.118) - 17.
One can show that the error incurred using Eq. (1.119) to determine Φ leads to results that are very close to the correct result for thin plates. (Timoshenko and Goodier, “Theory of Elasticity,” McGraw-Hill Book Co.)
- 18.
One can show that when the bending moment varies linearly with x the flexure formula is exact. See: Borg, S. R., “Matrix-Tensor Methods in Continuum Mechanics,” Section 5.5. D. Van Nostrand Co., Inc., 1962.
- 19.
It is to be pointed out that if we fix the beam in another manner at the end (for example we can reasonably assume that ∂u/∂y = 0 at 0 which is a case you will be asked to investigate) there results a different deflection curve for the centerline. Thus we see that, whereas St. Venant’s principle assumes that the stresses will not be affected away from the support for such changes, this does not hold for the deflection curve.
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Dym, C.L., Shames, I.H. (2013). Theory of Linear Elasticity. In: Solid Mechanics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6034-3_1
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