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2010 | OriginalPaper | Buchkapitel

5. Thermoelasticity

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

Erschienen in: Theory ofElasticity

Verlag: Springer Berlin Heidelberg

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Abstract

Let us consider now the problem of simulating mathematically the behavior of a solid body submitted simultaneously to external loads and thermal load. In this analysis the Thermodynamics collaborate with the Mechanics holding a greater role than the one played in the ancient Theory of elasticity. In fact a new type of problem comes into play, that of the Transmission of the heat in the solid bodies.

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Fußnoten
1
T is also called thermodynamic temperature. The scale of the absolute temperatures was established by Lord Kelvin. He assigned to the water and the ice, under conditions of thermodynamic equilibrium and at pressure of 1 bar, the value T = 273.16 °K.
 
2
As a rule, the fundamental relationship is gotten hypothesizing a model, that is conjecturing an expression of it and then going to verify its field of validity.
 
3
In (5.1.4) R 0 denotes an universal constant for all the gases, equal to R 0 = 8,316.6 Nm/°K = 1.986 Cal/°K. We also sometimes put \(R = R_0 /M\) in (5.1.4). The (5.1.4) summarizes the laws of Boyle (pV=const.), of Gay–Lussac (\(V=V_0\left(1+\frac{T}{272.24}\right),p=p_0\left(1+\frac{T}{272.24}\right)\)) and of Avogadro (equal volumes of any gas, in equal conditions of temperature and pressure, contain the same number of molecules).
 
4
This result is drawn in a simple way employing the kinetic theory.
 
5
For a perfect gas it results A = 1.
 
6
We call unworking a wall that doesn't allow exchanges of work.
 
7
We denote with \(t\in[0,+\infty[\) the initial instant and with t f the final instant, that is a real number \(t_f > t_i\) or the symbol +∞.
 
8
A system is said isolated if the flow of any greatness through the frontier Σ(t) of the volume V(t) occupied by the system at the instant t is zero.
 
9
In this simplified formulation there is an incongruity, because under conditions of thermodynamic equilibrium the flows and the productions are zero. However it exists the exact formulation, that preliminarily gives all the concepts of the Thermodynamics of equilibrium, where this incongruity doesn't subsist.
 
10
A greatness associated to a point is said scalar or tensor of order 0 if is individualized by a real number; vector or tensor of order 1 if is individualized by an ordered triplet of real numbers; tensor or tensor of order 2 if is individualized by an ordered sextuple of real numbers or by two vectors. It benefits to underline that to a vector a direction is associated; to a tensor two directions are associated or, if we prefer, a plane and a direction.
 
11
This because the flow of work is provoked by a cause that we call force.
 
12
Let \({\textit {\textbf u}}=(u_x,u_y,u_z),v=(v_x,v_y,v_z)\) be two any vectors of ℜ3. We call tensor product of u and v the tensor
$$ \overline{ {\textit {\textbf u}} \cdot {\textit {\textbf v}} } = \left[ {\begin{array}{*{20}c} {u_x v_x } & {u_x v_y } & {u_x v_z } \\ {u_y v_x } & {u_y v_y } & {u_y v_z } \\ {u_z v_x } & {u_z v_y } & {u_z v_z } \\ \end{array}} \right]\,.$$
Let
$$\breve{\boldsymbol \tau}=\left[ {\begin{array}{*{20}c} {\tau _{xx} } & {\tau _{xy} } & {\tau _{xz} } \\ {\tau _{yx} } & {\tau _{yy} } & {\tau _{yz} } \\ {\tau _{zx} } & {\tau _{zy} } & {\tau _{zz} } \\ \end{array}} \right], {\textit {\textbf n}} = \left[ {\begin{array}{*{20}c} {n_x } \\ {n_y } \\ {n_z } \\ \end{array}}\right]$$
be two any tensor and vector of ℜ3. We call scalar product of \(\breve{\boldsymbol \tau}\) and n the vector
$$\breve{\boldsymbol \tau} \times {\textit {\textbf n}} = \left[ {\begin{array}{*{20}c} {(\breve{\boldsymbol \tau} \times {\textit {\textbf {n}}})_x}\\{(\breve{\boldsymbol \tau} \times {\textit {\textbf n}})_y}\\{(\breve{\boldsymbol \tau} \times {\textit {\textbf n}})_z}\\ \end{array}} \right] = \left[ {\begin{array}{*{20}c} {\tau _{xx} n_x + \tau _{xy} n_y + \tau _{xz} n_z } \\ {\tau _{yx} n_x + \tau _{yy} n_z + \tau _{yz} n_y } \\ {\tau _{zx} n_x + \tau _{zy} n_y + \tau _{zz} n_z } \\ \end{array}} \right]\,.$$
Finally we denote with \({\textit {\textbf {x,\,y,\,z}}}\) the unit vectors of the reference frame and consider any tensor \(\breve{\boldsymbol \tau}\) of ℜ3 and any regular surface Σ of ℜ3. We call flow of the tensor \(\breve{\boldsymbol \tau}\) through the vector
$$ \boldsymbol \Phi_{\breve{\boldsymbol \tau}\times {\textit {\textbf n}}}=\int_{\Sigma}(\breve{\boldsymbol \tau}\times {\textit {\textbf n}})d\sigma=\left(\int_{\Sigma}(\breve{\boldsymbol \tau}\times {\textit {\textbf {n}}})_x d\sigma\right){\textit {\textbf x}}+\left(\int_{\Sigma}(\breve{\boldsymbol \tau}\times {\textit {\textbf{n}}})_y d\sigma\right){\textit {\textbf y}}+\left(\int_{\Sigma}(\breve{\boldsymbol \tau}\times {\textit {\textbf {n}}})_z\ d\sigma\right){\textit {\textbf z}}\,.$$
 
13
Applying the same reasoning to an elementary parallelepiped with faces parallel to the coordinated planes and containing the generic point \((x,y,z)\in V,\) we obtain three scalar differential equations (found at first by Stokes-Navier) to satisfy in V.
 
14
Usually we can denote \(\tau _{xx}\) [resp. \(\tau _{yy}\)] [resp. \(\tau _{zz}\)] also with the symbol \(\sigma _x\) [resp. \(\sigma _y\)] [resp. \(\sigma _z\)].
 
15
However more than an elastic interpretation of the number of Mach, this is an interpretation of the velocity of the sound.
 
16
The enthalpy H = U + p V is a thermodynamic potential.
 
17
This is the most arduous assignment in the analysis of a fluid dynamics problem, because a wrong choice leads as a rule to wrong results.
 
18
For the air \(\gamma \)=1.4.
 
19
We denote with the index 1 [resp. 2] the greatness before [resp. after] the shock wave.
 
20
For instance a duct submitted to the solar energy. Another class of problems reenters in this scheme, the rooms of combustion. They are ducts in which a chemical reaction happens that leaves, in first approximation, the composition of the fluid unchanged and generates heat.
 
21
If to the duct a flow of heat J q is administered, it is \(Q = \int_{\Sigma} {J_q}d \Sigma\). If in the duct a reaction happens that generates a quantity q of energy for unity of time and for unity of mass, it is \(Q = \int_M q dM\).
 
22
The wave of detonation is a surface of discontinuity through which the conversion of a form of energy in another happens. It, unlike the shock wave, doesn't leave the total energetic level unchanged because some chemical reactions happen due to the collisions of the molecules in the inside of the wave.
 
23
This happens if the number of elementary particles that constitute the fluid is very big.
 
24
In the case of motion in a duct we will choose as L r the diameter of the duct.
 
25
The product \(R_e P_r\) is called number of Peclet.
 
26
A gas perfect slips on the wall.
 
27
It is the case of two fluid dynamics fillets of current.
 
28
As the slab is parallel to the fluid current they are absent forces of pressure and for this the unique resistance to the motion is due to the presence of the limit layer. If the slab is inclined with respect to the current (Fig. 5.2.19) the resistance is also due to the forces of pressure. In such case the current external to the limit layer is meaningfully different from that unmolested. This is not verified if the slab is parallel to the current.
 
29
The dynamic pressure coincides with the kinetic energy for unity of volume.
 
30
The motion related to values of \(R_{e}\) insides to the interval can be considered as an unstable regime. If it is laminar, a small perturbation is enough to make it turbulent and vice versa. In nature there are a lot of small perturbations so in the aforesaid interval the motion continually passes from laminar to turbulent and vice versa.
 
31
As already specified on the interval of the values of \(R_{ecr}\), in reality the transition from the laminar regime to turbulent gradually happens in a neighborhood of \(x_{cr}\).
 
32
In the (5.1.11), that Fourier get by experimental way, k is called constant of thermal conductibility of the material and is measured in kcal/(cm °C s).
 
33
Q is also called thermal power.
 
34
L is also called mechanical power.
 
35
For the iron \(\alpha = 12 \cdot 10^{-6\,\circ} \textrm{C}^{-1}\). For instance a track long 18 m lengthens from the winter to the summer \((\Delta T=40\,^{\circ}\textrm{C})\) of \(\Delta l = al \Delta T=12\cdot10^{-6}\cdot 40\cdot18=0.0086\, \textrm{m}=8.6\,\textrm{mm}\).
 
36
In his fundamental paper back in 1837.
 
37
The (5.3.18), (5.3.19), and (5.3.20) are obtained from the (5.3.4) employing the (5.3.12) and (5.3.1).
 
38
If two particles have different temperatures, the heat moves from the greater temperature to the smaller temperature by one of the three following ways: conduction, convection, radiation. The first way is if the particles belong to solids in contact; the second if the particles are fluid; the third one if the particles are separated from the void and at least one of the temperatures is elevated.
 
39
If the solid in examination is immersed in a fluid the phenomenon of the convection intervenes. In such case in rigor the problem is rather complex, because the fluid dynamics problem and that of the thermoelastic solid should be treated as an unique problem.
 
40
This is the phenomenon of radiation. In the law of Stefan-Boltzmann , and depend from the relative orientation of the surfaces to which the two particles belong, from their distance and from the properties of absorption and reflection of the surfaces. In this phenomenon the transmission of the heat is meaningful only for high absolute temperatures (T ≥ 400°F).
 
41
This problem intervenes in the design of a gas turbine.
 
42
The determination of the thermal stresses in the wall of a cylinder is of great practical importance in the design of the endothermic motors.
 
43
Such condition is exact in the indefinitely long cylinder, where every plane of normal z is of symmetry.
 
44
This is the thermal load of greater technical interest for the deflected beams.
 
Metadaten
Titel
Thermoelasticity
verfasst von
Prof. Dr.-Ing. Aldo Maceri
Copyright-Jahr
2010
Verlag
Springer Berlin Heidelberg
DOI
https://doi.org/10.1007/978-3-642-11392-5_5

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