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2010 | OriginalPaper | Buchkapitel

1. The Three-Dimensional Problem

verfasst von : Prof. Dr.-Ing. Aldo Maceri

Erschienen in: Theory ofElasticity

Verlag: Springer Berlin Heidelberg

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Abstract

Let the solid body C, elastic or not, occupies with continuity some region V of the three-dimensional space. Let us employ a Cartesian orthogonal reference frame O, x, y, z and denote with x, y, z the unit vectors of the coordinate axes x, y, z (Fig. 1.1.1).

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Nächstes Kapitel The Problem of Saint Venant

This section of the Mathematical theory of elasticity (formulation of the problem of the elastic equilibrium in classical terms) involves delicate matters of the modern Mathematical analysis whose study is still incomplete.

We denote with dist(A,B) the distance between any two points A, B.

The definition of elongation according the direction r (at the point P) in use in the Mathematical theory of the elasticity is $\varepsilon _r = \lim _{Q \rightarrow F} \frac{{dist(P',\,Q') - dist(P,\,Q)}}{{dist(P,\,Q)}}$.

In fact, if a,b are real numbers such that $0 < a < 1$ and $0 < b < 1$, then we have $0 < ab < a$, $0 < ab < b$.

Precisely, if α, β, γ are real numbers with modulus very smaller than 1, then $u = \gamma \gamma + \beta \,z$, $v = - \gamma x + \alpha \,z$, $- \beta x - \alpha y$ are, with good approximation, the components of the displacement of an any point (x, y, z) of a rigid body because of three anticlockwise rotations around axes passing through (0, 0, 0). The first is of intensity (infinitesimal) α and around the axis parallel to x, the second is of intensity (infinitesimal) β and around the axis parallel to y, the third is of intensity (infinitesimal) γ and around the axis parallel to z.

If α is very small, we have with good approximation $\alpha = sin\alpha = tg\alpha$, $cos\alpha = 1$ (Fig. 1.1.12).

If x ^′ belongs to a neighborhood of x and f is a real function of a real variable, differentiable in x, it results with good approximation

$$f(x') \cong f(x) + \left[ {\frac{{df}}{{dx}}(x)} \right](x' - x),$$

The necessity is obvious. The sufficiency involves delicate matters of the modern Mathematical analysis whose study is still incomplete.

We call transducer a device sensitive to the greatness to measure and that, giving an electric signal or another indication (as for instance the position of the needle in a dial), allows to measure her.

A mechanical comparator was quietly able to appreciate the thousandth one of millimeter.

Sir Charles Wheatstone, Gloucester 1802 – Paris 1875.

In the liquids the model is such that the stress vector t _n acting on α _n has as line of action the orthogonal n to α _n, is of compression and of constant intensity for every n (hydrostatic pressure).

Augustin Louis Cauchy, Paris 1789 – Sceaux 1857. It is one of the greatest mathematicians of all time.

That is volumetric loads, superficial loads acting on S_p and constraint reactions acting on S_u.

In the three-dimensional space we call equivalent to zero (or balanced) a system of forces such that its resultant has zero intensity and its moment with respect to any axis of the space has zero intensity.

We denote with meas_n(Δ) the n-dimensional measure of Δ.

Christian Otto Mohr, Wesselburen 1835 – Dresden 1918.

In Physics we call work the scalar product of a force and of the displacement of its point of application.

Karl Friedrich Gauss, Brunswick 1777 – Gottingen 1855. He is one of the greatest mathematicians that has ever had the humanity. The Gauss’s formulas transform the surface integral in volume integral. Precisely, if f it is a function defined in V , if f and the boundary S of V are regular and if $n = \left( {n_x ,\,n_y ,\,n_z } \right)$ is the normal one to S, it results (Fig. 1.3.2) that the surface integral on S of f n _x [resp. f n _y] [resp. f n _z] is equal to the volumetric integral on V of the x [resp. y] [resp. z] partial derivative of f.

This is an ancient type of test machine. The load was increased manually adding small weights (one to the time) on the dish of the test machine.

Robert Hooke, Wight 1635 – London 1703.

Louis Marie Henri Navier, Dijon 1785 – Paris 1836.

Thomas Young, Milverton 1773 – London 1829.

Siméon Denis Poisson, Pithiviers 1781 – Paris 1840.

Gabriel Lamé, Tours 1795 – Paris 1870.

The result 0 < ν < 1/2 is notably important in the Mathematical theory of plates.

We call constraint’s reactions the superficial forces, applied in the points of S _u, that the constraints act on the body C.

Eugenio Beltrami, Cremona 1835 – Roma 1900.

Gustave Robert Kirchhoff, Kònigsberg 1824 – Berlin 1887.

Sergei Sobolev , St. Petersburg 1908 – St. Petersburg 1989. This mathematician gave a very big contribution to the Functional analysis.

By using the Functional analysis.

The surface of separation is function of the time t. We call impermeable [resp. adiabatic] [resp. unergodic] a surface of separation that blocks the flow (that is the exchange) of mass [resp. heat] [resp. work] between system and environment.

The temperature is preliminarily defined by experimental way introducing a reproducible scale, that we call thermometer.

If it does not depart from a state of thermodynamic equilibrium or if it does not reach a state of thermodynamic equilibrium, the evolution is called process.

It does not make sense to speak of heat contained in a body. It can only be talked of heat (or thermal energy) exchanged among two bodies, that is between environment and system. This exchange can happen for conduction or for radiation. The heat can be transmitted for conduction from the environment to the system, or vice versa, when and only when there is a difference of temperature between the environment and the system. In Statistic thermodynamics it is seen that the heat transmitted for conduction is an exchange of mechanical energy at microscopic state.

It does not make sense to speak of work contained in a body. It can only be talked of work executed by a body on another (as exchange of mechanical energy between environment and system) and of work executed by a field of forces on a body.

The laws of the Physics are conjectures confirmed by all the known experimental data.

That is a function that only depends by the variables of state.

We denote with ΔE the variation of the energy of the system during the transformation, that is $\Delta E = E\left( {t_f } \right) - E\left( {t_i } \right)$. If Σ is permeable we must add to first member of (1.6.1) the exchange of chemical energy associate to the exchange of mass. It is obvious that the [1.6.1] furnishes the energy of the system for less than a constant. So to E(t _i) we can assign any value.

We denote with ρ [resp.v] the density [resp. velocity].

The internal energy is a microscopic energy of the system. Precisely, in Statistic thermodynamics we see that internal energy is constituted by all the forms of energy existing at microscopic level. They are the kinetic energy of translation, of rotation and of vibration of the elementary particles and the energy of position of the elementary particles, which are reciprocally attracted with force that depends on the mutual distance.

We notice that the internal energy cannot be separate in a mechanical part and in a thermal part.

To a such transformation the first principle of the Thermodynamics does not oppose it.

Q₁ constitutes a loss of energy.

Rudolf Julius Emmanuel Clausius, Kòslìn 1822 – Bonn 1888.

Unlike the internal energy, it is not possible to attribute to the entropy a physical meaning.

In the Thermoelasticity and in the Plasticity transformations with Q # 0 are studied.

In the Plasticity and in the Thermoelasticity the transformations are irreversible.

The forces of inertia are of primary importance in the Dynamics of structures.

Benoit Paul Emile Clapeyron, Paris 1799 – Paris 1864.

The mutual work is also called dragging work.

Enrico Betti, Pistoia 1823 – Soiana 1892.

Alberto Carlo Castigliano, Asti 1847 – Milano 1884.

Up to 1980 approximately, to appraise the structural safety in the respects of the collapse the Method of the admissible stress has only been employed. Since then, the Method of the ultimate limit states has been used. The second of such methods allows a more precise calculation. With it the threshold not to be overcome is the collapse. However the first of such methods allows a much more prudential calculation. With it the threshold not to be overcome is the yelding (i.e. practically the point B of Fig. 1.4.4). As a consequence, if because of exceptional events the threshold of crisis is reached

if the calculation has been made with the second of such methods the body collapses,
if the calculation has been made with the first of such methods the body still has the ability of resistance of the whole plastic field, so that it absorbs in safety the exceptional load suffering plastic deformations.

Therefore, if the structure is desired to be operational for very long time, the Method of the admissible stress still today constitutes a profit tool for the structural design.

Henry Edouard Tresca, Dunkerque 1814 – Paris 1885.

Richard von Mises, Leopoli 1883 – Boston 1953.

Titel: The Three-Dimensional Problem
verfasst von: Prof. Dr.-Ing. Aldo Maceri
Verlag: Springer Berlin Heidelberg
Buch: Theory ofElasticity
Print ISBN: 978-3-642-11391-8

Electronic ISBN: 978-3-642-11392-5

Copyright-Jahr: 2010
DOI: https://doi.org/10.1007/978-3-642-11392-5_1

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