A Non-Linear Spring Model for Solid Material

I. Introduction
The properties of a solid material as yield strength, modulus and linear coefficient of thermal expansion CTE can be shown to have an intimate relation. We will assume the material to be characterized well by a non-linear spring. Such model description has been successfully applied for the derivation of vibrational energy levels in diatomic molecules by Philip Morse [1]. The energy density V(ϵ), where ϵ is the strain in the system is given by:

where E is the modulus of the material, the ratio between strain and stress and β is the parameter that determines the non-linearity of the system. In the limit of small strain equation (1) reduces to the familiar equation for the energy density in a solid

II. Energy density diagram
The fundamental shape of a curve describing is shown in fig. 1.

III. Pull force graph
The yield stress Y is determined by the point where the pull force needed for elongation is maximum. From the equation given in eq. (1) one can derive the expression for the pull force FY(ϵ) = dV(ϵ)/dϵ:

A schematic view of such diagram showing the elongation versus force FY(ϵ) is given in fig.2.

IV. Yield Strength
The yield strength or maximum force is found through double differentiation of equation (1) and substituting the solution into equation (2). The second derivative is given as:

It can be shown that the value of ϵ, for which eq. (3) equals zero, ϵmax is given as:

Substitution into eq. (2) yields the value for the yield strength YS:

V. Coefficient of Thermal Expansion
Dulong and Petit [2] have found that the molar heat capacity of solid crystalline metals are all about equal and have the value of 3 R, where R is the gas constant with value 8.314 J/Mole/K. Einstein used this to state that in a Mole of such materials there are 3 N  oscillators, where N is Avogadro’s number. In the high temperature limit an oscillator has energy content k T, where k is Boltzmann’s constant and T is the temperature. From eq. (1) the energy diagram per atom can be shown to be

where N is the number of atoms per unit volume. If we let the kinetic energy be W, the turning points ϵ1 and ϵ2 in the potential well are given as

The average ϵavg of the turning points ϵ1 and ϵ2 is given as

As the argument of the natural logarithm is close to 1, ϵavg can be expanded as a series in W of which we only show two terms

The thermal coefficient of expansion CTE is the temperature derivative of the average thermal strain. After averaging ϵavg using Boltzmann statistics and subsequently differentiating with respect to temperature T and inserting the values for the number density N we obtain finally for the expansion coefficient CTE

Where R is the gas constant R= N k, where N  is Avogadro’s number and k is Boltzmann’s constant. M is the molar mass and ρ is specific density of the material. From this simple theory it is clear that the expansion coefficient is not a constant but is a temperature dependent function.

VI. References
[1] Morse, P. M. “Diatomic molecules according to the wave mechanics.
II. Vibrational levels”. Phys. Rev. 34. pp. 57–64. (1929)
[2] Einstein Albert “Theorie der Strahlung und die Theorie der Spezifische Wärme”, Annalen der Physik. 4. 22: 180-190, (1906)

VII. Authors
N. van Veen and A. Kodentsov

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