Abstract

In this paper we investigated if temperature effects the propagation and growth of cracks in a wine glass tuning nearest to a sound- machine. We based upon the laws of classic and fracture mechanics and we showed that the wine glass breaks when initially its temperature overcomes the temperature of the room and at continuity starts decreasing before equalizing.

1. Introduction

All material bodies: solids, liquids and gasses are vibrating. The wine glass is vibrating nearest to 5.5Hz. ^{1, 2} and its acoustics has been considered by many researchers ^{3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}.

The purpose of this paper isto investigate if temperature plays any role to propagation and growth of cracks in a wine glass, tuning nearest to a sound machine.

For that reason we will base upon the laws of classic mechanics: balance of mass, momentum and energy.

2. The Problem and Its Physical Approximation

i) The mathematical modeling

We assume that at , the wine glass had a density of mass , a temperature and an internal energy per unit volume . Also its displacement and velocity were and respectively. Finally the room had a temperature and the sound machine indicated in Figure 1 was non-activated.

At the sound machine starts vibrating tuning nearest the wine-glass. Therefore the wine glass is under a constraint resonated vibration without damping, due to the sound wave of sound machine described by the following differential equation:

(1)

where , m and k are respectively: the horizontal displacement, the mass and the stiffness constant of the wine glass, while is the amplitude of vibration of sound machine. We assume that the rest displacements vanish.

The above equation is written as:

(2)

where is the radial frequency of the wine glass in units of rad/sec and . The general solution of the above is:

(3)

Assuming that and the above reduces to the form:

(4)

Therefore the velocity of material particles of wine glass is:

(5)

while its acceleration is:

(6)

The balance of mass of wine glass is:

(7)

where is its density. Substituting (5) into (7) it follows:

(8)

The dynamics stress - equation for the wine glass are:

We assume that:

Inserting the above into (9)_1-2-3 it is possible to obtain:

where A(t) is an unknown function, while B, C, D and E are unknown constants.

The balance of energy is:

(12)

where , are respectively: the thermal flux and the energy transferred by the sound wave and added to wine glass given by:

where M is a vibration constant of sound wave. Finally is the energy supply due to thermal reasons added /subtracted to /from the glass, given by:

(14)

where K and are respectively a thermal constant and the difference of temperature between wine glass and room . The temperature of room assume to be constant.

If initially then and therefore vanishes.

If then the energy supply W₂ should be written as:

because the temperature of the wine glass increases for and tends to be equal with that of room at a particular time moment . At this case the last is heating, that is an amount of thermal energy is added to it for .

If then the energy supply W₂ should be written as:

because the temperature of wine glass decreases for and tends to be equal with that of room at a particular time moment . At this case the last is cooling, that is it looses an amount of thermal energy for .

Then (12) because of (4), (5), (11),(13) and (14) concludes to:

(17)

The above satisfies the initial condition:

(18)

ii) The solution of the problem:

1) If , then (17) reduces to:

(19)

From the above it is possible to conclude that E(t)=de(t)/dt is an increasing function, that is _. Consequently the internal energy of wine glass which means that the last is under an elastic deformation.

2) If _, then (17) due to (15)_1-2 is written as:

From the above it is possible to conclude that is an increasing function in both branches, that is _. for and the same goes as at previous case.

3) If , then (17) due to (16) _1-2 is written as:

If 0<t≤ (KΔθ²(t)/Mf_o²)^1/2 then (21)₁ takes the form:

(22)

From the above it is possible to conclude that is a decreasing function for , that is _. Consequently the internal energy of wine glass for _. Accordingly to Griffith's energy criterion ^{18, 19} macroscopically propagation and growth of crack will be arised which microscopically due to the breaking of the chemical connections between atoms. Therefore at time that satisfies the condition the wine glass will be fractured.

3. Discussion and Conclusion

Our model is verified by experimental studies. Particularly it has been shown that a sound machine resonated nearest to wine glass can break it ^{1, 2, 9, 17, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30}.

From the above we result that temperature effects the propagation and growth of cracks in a wine -glass tuning nearest to a sound machine. The last happens when temperature of wine glass overcomes the temperature of room and starts decreasing before equalizing.

References