**1. Introduction**

Applications of materials in the industry depend on their physical–chemical properties. Metal and semiconductor materials have had a big relevance for humanity development, facilitating the life and making possible technological advances still today. The properties and characteristics of metals are taken into account for the selection of materials according to the function we will perform.

Copper alloys are materials known for a wide range of applications. Currently more than 400 copper alloys are known, among which brass and bronze are distinguished. It is possible to vary the amounts of copper and zinc in these alloys to obtain a variety of brass with different properties [1, 2].

Brass is considered one of the most important copper alloys due to the amount of zinc that can be varied considerably, ranging between 5 and 45% by weight. It is possible to produce a wide variety of brass alloys with different technological properties for various commercial and industrial applications. Among the most commonly used brass in engineering areas is high-strength brass, suitable for heavy loads, with adequate properties to resist wear and corrosion. Among the main advantages of these materials are their mechanical properties by thermal treatment and their low cost [3–5].

In 2010 Ozgowiez et al. [6] analyzed the influence of recrystallization annealing temperature on the microstructure and mechanical properties of Cu 30% Zn brass, which was subjected to cold deformation with a variable tension in the rolling process. The mechanical test showed that the properties of the brass deteriorated and the properties of the plastic increased as the recrystallization temperature increased within the range of 400–650°C [7].

In 1967 Bailey studied the structure and strength of an alpha brass with 20%w Zn, 6%w Ni, and 1.5%w Al. Brass samples were subjected to solution heat treatment (SHT) at 800°C for 2 h followed by quenching with water. Subsequently, the samples were thermally treated for 2 h at 300, 400, 500, 600, and 700°C. The best mechanical properties were obtained at 500°C PHT temperature [8].

Infrared photothermal radiometry (PTR) is an optical technique used for the characterization of metallic materials such as brass, to determine the influence of the precipitation heat treatment (PHT) temperature on metallurgical microstructure, thermal properties, and microhardness. This technique is based in light absorption and consists of a laser beam incident on a sample, which will absorb the radiation and emit a thermal wave that is detected by an infrared sensor. The signals of thermal wave are translated as amplitude and phase parameters. Compared with other characterization techniques, this technique has no destructive properties and has the advantage of not having direct contact with the sample to be analyzed.

One of the most important aspects to consider in the PTR is the process of light absorption in the material to be analyzed. As trivial as this may sound, absorption very often turns out to be the most critical and cumbersome step in laser processing. An enormous amount of work has been dedicated to investigating laser absorption mechanisms under various circumstances, and a great deal can be learned from this work for the benefit of laser-material processing.

Absorption process can be thought of as secondary "source" of energy inside the material. While driven by the incident beam, it tends to develop its own dynamics and can behave in ways deviating from the laws or ordinary optics. It is this "secondary" source, rather than the beam emitted by the laser device, which determines what happens to the irradiated material [9].

Certain wavelengths of light can be selectively absorbed by a substance or a material according to its molecular structure. The absorption of light occurs when an incident photon promotes the transition of an electron from a state of lower to higher energy. Excited electrons eventually lose this gained energy, and by a spontaneous radiation process, they return to their initial state.

The radiation emitted by a molecule, or an atom, after it has absorbed energy to place itself in an excited state, is defined as luminescence, depending on the nature of the excited state. All spectrophotometric methods are based on two laws that combined which are known as Beer–Lambert law. This law states that the light absorbed by a semitransparent medium is independent of the intensity of incident light, and each successive layer of the medium absorbs an equal fraction of the light passing through it. This amount of light can be calculated by Eq. (1), where *I*0 is the incident light intensity, *I* is the transmitted light intensity, l is the length through which the light passes in the spectrophotometer cell, and k is the constant of the medium:

$$\log\left(\frac{I\_0}{I}\right) = kl \tag{1}$$

**59**

*Analysis of the Absorption Phenomenon through the Use of Finite Element Method*

set of nodes considering their adjacency relationships is called "mesh."

number of equations of said system is proportional to the number of nodes.

toward the exact solution of the system of equations.

nique applied to alpha brass are shown.

The FEM is widely used due to its generality and the ease of introducing complex calculation domains (in two or three dimensions). In addition, the method is easily adaptable to problems of heat transmission, fluid mechanics to calculate fields of velocities and pressures (computational fluid mechanics, CFD), or electromagnetic field. Given the practical impossibility of finding the analytical solution to these problems, often in engineering practice, numerical methods and, in particular, finite elements become the only practical alternative to calculation. One important property of the method is convergence; if finite element partitions are considered successively finer, the calculated numerical solution converges rapidly

For all the above, the FEM is an excellent method to consider the development of theoretical models for the analysis of the phenomenon of the absorption of radiation in materials. In this chapter, an analysis of absorption process as result of laser-material interactions using Beer–Lambert law is shown. In Section 2.1 a model absorption using COMSOL Multiphysics with finite element method in alpha brass is described. In Section 3 experimental results of photothermal radiometry tech-

**2. Analysis of laser-material interactions using Beer–Lambert law**

supplied by it will be absorbed by the material. This process can be described through Beer–Lambert law. If we consider monochromatic and collimated light,

When a light beam incident upon a semitransparent material, part of the energy

Calculations are carried out on this mesh of points called nodes, which serve as the basis for the discretization of the domain in finite elements. The creation of the mesh is done in a stage prior to the calculations called pre/process and usually with special programs called mesh generators. According to these adjacency or connectivity relations, the value of a set of unknown variables defined in each node and called degrees of freedom is related. The set of relations between the values of a given variable between the nodes can be written as a system of linear (or linearized) equations. The matrix of the said system of equations is called the system's stiffness matrix. The

properties of the materials and parameters that describe the process system, such as the direction or directions of the incidence of light and wavelength, among others. Finite element method (FEM) is one of the most practical methods for solving engineering problems. This general numerical method is used to approximate solutions of partial differential equations very complicated in the resolution of problems that involve a high degree of mathematical and physical–mathematical performance. FEM is applied to character geometries and multiphysical behavior, both of material properties and of problems where it is generally not possible to obtain any analytical solution directly with the use of mathematical expressions. Some of the areas that use the finite element method to solve problems are heat transfer, fluid flow, mass transport, distribution of electromagnetic fields, and structural analysis, among others. FEM allows obtaining an approximate numerical solution on a body, structure, or domain (continuous medium). On this domain certain differential equations are defined in an integral form that characterizes the physical behavior of the problem, dividing it into a high number of non-subdomains intersecting each other called "finite elements." This set of finite elements forms a partition of the domain also called discretization. Within each element there are a series of representative points called nodes. Two nodes are adjacent if they belong to the same finite element; in addition, a node on the edge of a finite element can belong to several elements. The

*DOI: http://dx.doi.org/10.5772/intechopen.86924*

In the absorption spectrometry, the comparison of incident light intensity before and after the interaction with a sample can be carried out nowadays, by means of different software. In the development of theoretical models, it is possible to include

#### *Analysis of the Absorption Phenomenon through the Use of Finite Element Method DOI: http://dx.doi.org/10.5772/intechopen.86924*

properties of the materials and parameters that describe the process system, such as the direction or directions of the incidence of light and wavelength, among others.

Finite element method (FEM) is one of the most practical methods for solving engineering problems. This general numerical method is used to approximate solutions of partial differential equations very complicated in the resolution of problems that involve a high degree of mathematical and physical–mathematical performance. FEM is applied to character geometries and multiphysical behavior, both of material properties and of problems where it is generally not possible to obtain any analytical solution directly with the use of mathematical expressions. Some of the areas that use the finite element method to solve problems are heat transfer, fluid flow, mass transport, distribution of electromagnetic fields, and structural analysis, among others.

FEM allows obtaining an approximate numerical solution on a body, structure, or domain (continuous medium). On this domain certain differential equations are defined in an integral form that characterizes the physical behavior of the problem, dividing it into a high number of non-subdomains intersecting each other called "finite elements." This set of finite elements forms a partition of the domain also called discretization. Within each element there are a series of representative points called nodes. Two nodes are adjacent if they belong to the same finite element; in addition, a node on the edge of a finite element can belong to several elements. The set of nodes considering their adjacency relationships is called "mesh."

Calculations are carried out on this mesh of points called nodes, which serve as the basis for the discretization of the domain in finite elements. The creation of the mesh is done in a stage prior to the calculations called pre/process and usually with special programs called mesh generators. According to these adjacency or connectivity relations, the value of a set of unknown variables defined in each node and called degrees of freedom is related. The set of relations between the values of a given variable between the nodes can be written as a system of linear (or linearized) equations. The matrix of the said system of equations is called the system's stiffness matrix. The number of equations of said system is proportional to the number of nodes.

The FEM is widely used due to its generality and the ease of introducing complex calculation domains (in two or three dimensions). In addition, the method is easily adaptable to problems of heat transmission, fluid mechanics to calculate fields of velocities and pressures (computational fluid mechanics, CFD), or electromagnetic field. Given the practical impossibility of finding the analytical solution to these problems, often in engineering practice, numerical methods and, in particular, finite elements become the only practical alternative to calculation. One important property of the method is convergence; if finite element partitions are considered successively finer, the calculated numerical solution converges rapidly toward the exact solution of the system of equations.

For all the above, the FEM is an excellent method to consider the development of theoretical models for the analysis of the phenomenon of the absorption of radiation in materials. In this chapter, an analysis of absorption process as result of laser-material interactions using Beer–Lambert law is shown. In Section 2.1 a model absorption using COMSOL Multiphysics with finite element method in alpha brass is described. In Section 3 experimental results of photothermal radiometry technique applied to alpha brass are shown.
