Elastic moduli: Overview and characterization methods

Technical Review ITC-ME/ATCP Elastic moduli: Overview and characterization methods ATCP Physical Engineering www.atcp-ndt.com Authors: Leiliane Cristina Cossolino (Cossolino LC) and Antônio Henrique Alves Pereira (Pereira AHA) Published online on 21/10/2010

INDEX 1. INTRODUCTION... 3 1.1 The Elastic Modules... 3 1.2 Elastic Modules for isotropic materials... 4 1.3 Applications and importance... 6 2. CHARACTERIZATION METHODS... 8 2.1 Quasi-static methods (destructive testing)... 8 2.1.1 Introduction... 8 2.1.2 Experimental method... 8 2.1.3 Calculations... 9 2.2 DYNAMIC METHODS (NON-DESTRUCTIVE)... 10 2.2.1 Introduction... 10 2.2.2 History... 10 2.2.3 Vibration modes... 11 2.2.4 Natural vibration frequencies... 11 2.3 Experimental methods... 14 2.3.1 Impulse excitation... 14 2.3.2 Frequency Sweep... 17 2.3.3 Calculations... 19 2.4 Ultrasound methods (pulse-echo)... 21 2.4.1 Pulse generation and reception in the ultrasonic device... 22 2.4.2 Calculations... 23 3. COMPARATIVE ANALYSIS... 23 4. CONCLUSIONS... 24 5. APPENDIX A... 25 6. REFERENCES... 27 2 Visit our site: http://www.atcp-ndt.com/products/materials-characterization/sonelastic.html

1. INTRODUCTION This article provides an overview of the meaning of the elastic moduli and the relevance of these properties to the materials science and engineering. It also outlines the different characterization methods, destructive and nondestructive, and discusses the strengths and weaknesses of each method and its variations. The purpose of this article is to provide the engineer or researcher, the basic information about the elastic modules, with particular focus on methods of measurement in order to facilitate the characterization and use of knowledge of these properties, which are fundamental to all classes of materials. Many materials are usually subjected to forces or loads, such as the aluminum alloy from which the wing of an aircraft is constructed or the steel employed in the shaft of an automobile. In these situations knowing the characteristics of the material, among these, the elastic modules, is necessary, so as to design the parts and devices such that, any resulting deformation is foreseeable and not excessive to the point of plastic deformation, accelerated fatigue or even fracture occurs. The mechanical behavior of a specific material depends largely on its response (or deformation) to the load or force that is submitted. The property that correlates the elastic deformation with the stress is the elastic moduli, which have different definitions depending on the kind of applied stress. In addition to the elastic modules, other properties are equally important, such as mechanical strength and ductility, in the case of metallic materials. Mechanical properties measurement is achieved by carefully programmed laboratory experiments in accordance with conditions governed by rules. In the specific case of the elastic moduli, the employed methods can be dynamics, by vibrations with small amplitudes of deformation, or statics, submitting the specimen to a known voltage and simultaneously measuring the induced deformation. These assays can be conducted either at room temperature or at high temperatures, with or without a controlled atmosphere. Knowledge of the elastic moduli is the attention focus of various professionals (for example, material producers and consumers, research organizations, government agencies), with different needs and applications. Consequently it is necessary to have consistency in the way in which the tests are conducted and in the interpretation of the results. This consistency is achieved through the use of standard assay techniques. The establishment and publication of standard norms are often coordinated by professional societies. In Brazil, ABNT (Brazilian Technical Standards Association) and the United States ASTM (American Society for Testing and Materials), among others, are responsible for standardization of materials test. Its publications are updated annually and a series of standards is related to the determination of elastic moduli. Materials are often selected for structural applications due to its desirable combinations of mechanical properties, such as stiffness (elastic modulus), mechanical strength, durability and economy of energy resources. The role of structural engineers is to determine the distribution of tensions and stresses in the materials that are subject to well-defined loads. This can be accomplished by experimental techniques and/or by theoretical and mathematical analysis of the stress which depends on knowledge of elastic moduli. Materials engineers and metallurgical engineers, on the other hand, are concerned with the production and manufacture of materials to meet service requirements as provided by these stress analysis. This necessarily involves an understanding of the relationship between the microstructure (i.e., the internal characteristics) of the material and its mechanical properties, which are directly correlated with the elastic modules, allowing the characterization of these properties also in quality control. 1.1 The Elastic Modules Elastic moduli are key parameters for engineering and materials application, since they are connected to the description of various other mechanical properties such as yield stress, the tensile strength, the critical temperature variation for cracks propagation under the action of heat shock, etc. They are intrinsic properties of materials that describe the relationship between stress and strain in the elastic range and depend on the chemical composition, microstructure and defects such as, for example, pores and cracks. 3 Visit our site: http://www.atcp-ndt.com/products/materials-characterization/sonelastic.html

1.2 Elastic Modules for isotropic materials In the case of isotropic materials, in which the properties are not dependent on which directions are measured, the modules are: a) Elasticity modulus or Young's modulus (uniaxial), E or Y; is a magnitude that is proportional to the stiffness of a material when it is subjected to an external traction or compression. Basically it is the ratio of applied stress and strain experienced by the body, when the behavior is linear, as shown in the following equation: Where: E = Elasticity modulus or Young's modulus (Pa), σ = Applied stress (Pa), ε = Longitudinal elastic deformation of the specimen (dimensionless). Imagine that we have a rubber and a metal, and we apply the same stress to both, we can verify a far greater deformation in the rubber than in metal. This shows that the Young's modulus of the metal is higher than rubber and therefore, it is necessary to apply a greater tension to bring about the deformation observed in the rubber (if possible!) (Figure 1). (1) Figure 1: An illustration of the differences in the values of Young's modules of two different materials. b) Shear modulus, G, é is defined for the shear stress; Where: G = Shear modulus (Pa), τ = Shear stress (Pa), γ = Elastic shear deformation of the specimen (dimensionless). The shear stress is related to parallel force applied to the surface, in order to cause the sliding of the parallel planes relative to each other (Figure 2). In this case, the shear strain γ, can be calculated by the tangent of the θ angle. (2) 4 Visit our site: http://www.atcp-ndt.com/products/materials-characterization/sonelastic.html

Figure 2: Representation of shear stress (γ) 1. c) The bulk modulus, K, describes volumetric elasticity, or tendency of an object to deform in all directions when uniformly loaded in all directions (hydrostatic). This module is defined as the relation between the volumetric tension and the volumetric deformation, and is the inverse of compressibility. (3) Where: K = Volumetric module (Pa), P = Pressure (Pa), V /V 0 = Volumetric deformation (dimensionless). d) Poisson's ratio, measures the deflection transverse (in relation to the longitudinal direction of load application) of a homogeneous and isotropic material. Poisson s ratio is not the relation between stress and strain, but between orthogonal deformations. 1,2 (4) Where: µ = Poisson's ratio (dimensionless), ε x = Deformation in the x direction, which is transverse ε y = Deformation in the y direction, which is transverse ε z = Deformation in the z direction, which is longitudinal ε x, ε y e ε z are also dimensionless quantities, since they are deformations. The negative sign in equation 4 is adopted because the transverse and longitudinal deformations have opposite signs. Conventional materials contract transversely when stretched longitudinally, and transversely shrink when compressed longitudinally. The transverse contraction in response to the longitudinal extent due to a tensile mechanical stress corresponds to a positive Poisson's ratio. When you stretch a rubber, for example, you'll notice that it will contract in the direction perpendicular to the one you initially stretched. On the other hand, when the material has a negative Poisson's ratio (which is a rather special case) it expands transversally when pulled. Materials that have a negative Poisson's ratio are named auxetics and also known as anti-rubbers. 3 For an isotropic material, the shear modulus, Young s modulus and Poisson s ratio are related by the expression: (5) and the Young s modulus, the volumetric modulus and Poisson s ratio, by the expression: 3 12 (6) 5 Visit our site: http://www.atcp-ndt.com/products/materials-characterization/sonelastic.html

For most metals that have a Poisson's ratio of 0.25, G is equivalent to approximately 0.4E, thus, if the value of a module is known, the other can be estimated. Many materials are elastically anisotropic, i.e., the elastic behavior (e.g. the magnitude of E) varies according to the crystallographic direction (see Appendix A). For these materials, the elastic properties are only completely characterized by specifying different elastic constants, their number depending on the structural characteristics of the crystal. Even for isotropic materials, at least two constants must be given for a comprehensive characterization of the elastic properties. Since the grain orientation is random in most non-textured polycrystalline material, these can be considered isotropic. Inorganic glasses are also isotropic. The table below gives some examples of materials with their respective values of Young's modulus, E, shear modulus, G, and Poisson's ratio, µ. Table 1: Materials properties. Class Material Young s Modulus (GPa) Shear Modulus (GPa) Poisson s Ratio Metals Polymers Ceramics Steel 207 83 0,30 Copper 110 46 0,34 Brass 97 37 0,34 Aluminum 69 25 0,33 Tin 44,3 18 0,33 Lead 13,5 5,6 0,44 Polyethylene (PET) 2,46-4,14 - - Polyvinyl chloride (PVC) 2,41-4,14-0,38 Epoxy 2,41 - - Polycarbonate (PC) 2,38-0,36 Polyester 2,04-4,41 - - Nylon 1,59-3,79-0,39 Diamond (Synthetic) 800-925 - 0,20 Aluminum oxide (99.9% purity) 380-0,22 Silicon nitride (Sintered) 304-0,28 Silica glass, lime and soda 69-0,23 Concrete 25,4-36,6 a - 0,20 Graphite (extruded) 11 - - a Secant modulus of elasticity taken in 25% of the limit of tensile strength. Source: CALLISTER, Jr., W.D. Materials Science and Engineering. 7 º ed. New York: John Wiley & Sons, Inc, 2007. 1.3 Applications and importance Determination of elastic properties is important for the description of other mechanical properties, such as the following examples: a) Shear stress of metals, τ e, based on the theory of dislocations. 4 The critical resolved shear stress, τ e, of metals is directly proportional to the shear modulus and can be correlated to macroscopic yield stress (Equation 7). (7) Wherein G is the shear modulus, b is the Burgers vector l is the mean free length of the dislocations lines. 6 Visit our site: http://www.atcp-ndt.com/products/materials-characterization/sonelastic.html

Plastic or permanent deformation of crystalline substances occurs primarily by movement of the crystal imperfections, called dislocations. b) The fracture tension (tensile) of brittle materials, σ f, by the Griffith theory. 1,5,6 According to Griffith theory, the critical tensile stress, σ f, required for catastrophic fracture (catastrophic crack propagation) in a brittle material is proportional to the square root of Young's modulus (E) which is given by the equation below: (8) where γ S is the energy of the specific surface and is half the length of the internal defect (straight through-elliptical). During the crack propagation in a material, a release of energy occurs known as elastic strain energy; a part of this energy is stored in that material as it is elastically deformed. There are variations of Equation 8 for materials susceptible to plastic deformations and highly ductile. c) The critical temperature variation, T c, for crack propagation of a material under the action of heat shock, according to Hasselman theory. 7 The critical temperature variation, T c, or thermal shock resistance is inversely proportional to the square root of the Young's modulus (E). Variations above T c cause thermal shock damage, which is the degradation of mechanical properties of a material induced by thermal nature stresses. The degradation is caused by microstructural changes, especially the emergence and propagation of cracks and micro-cracks. According to the unified theory of Hasselman 7, evaluation of the thermal shock resistance of a material can be accomplished in two ways: first, by the ability of the material to prevent crack initiation, and second, by the ability of the material to minimize the propagation of these cracks when its occurrence is inevitable. With this theory, it is possible to estimate the variation of critical temperature ( T c to propagate long or short cracks 7. For short cracks, T c is given by the following equation: γ µ µ (9) where E 0 is the Young's modulus of the material without cracks, γ wof is the total energy of fracture, α, the linear thermal expansion coefficient, µ the Poisson ratio, and l the length of the crack. 7 For long cracks, is given T c by the following equation: γ µ (10) where N is the number of cracks, all of length l, per volume unit. 7 The importance of characterization of the elastic moduli in the context of materials science and engineering can be seen from these examples. More specifically, measurement of the elastic modulus is used in: - Finite element simulations of mechanical stresses and strains. - Quality control: the elastic moduli are sensitive to processing and microstructure. - Evaluation of damage caused by physical and chemical degradation of the material. - Evaluation of thermal shock damage. - Evaluation of damage per load in construction concrete. 7 Visit our site: http://www.atcp-ndt.com/products/materials-characterization/sonelastic.html

2. CHARACTERIZATION METHODS The elastic modulus may be characterized by quasi-static methods, dynamic methods, or by ultrasound. The quasi-static or isothermal methods are based on usually destructive mechanical tests, and the dynamic or adiabatic, in non-destructive resonance techniques. The values determined by dynamic methods are higher than those determined by static at a typical rate of 0.5% for metals. For other materials, the difference can be large depending on the inelastic effects. 2.1 Quasi-static methods (destructive testing) 2.1.1 Introduction In quasi-static methods destructive mechanical tests are usually performed in which the specimens are usually destroyed after the completion thereof. They consist of applying a load slowly while monitoring the induced deformation. First, we will introduce the concept of stress-strain curve. The stress corresponds to a force or load, per area unit, applied to a material, and the strain is the change in dimensions, per unit of the original dimension. In the case of static methods, the load that can be static or change relatively slowly over time, it is uniformly applied over a straight section or body surface, and the deformation is measured and related to an elastic modulus which may be Young's or Shear modulus depending on the type of test. There are three main ways according to which a load may be applied: tension and compression for determining the Young's and shear modulus or torsionally for shear modulus, the tensile tests being most common. In those tests the specimen is deformed when a tensile load is applied uniaxially, along the longer dimension of the specimen. These tests are commonly carried out on metals, at room temperature, for the ease of holding the specimen in the accessories of the testing machine. For the dynamic methods (which will be seen in detail later) elastic moduli are determined from the natural frequency of vibration (resonance) of the specimen with minimum amplitudes of vibration (deformation). Figure 3 shows the ways in which loads may be applied in a test using the quasi-static method. Figure 3: (a) Schematic illustration of how an elongation and tensile load produces a positive voltage. The dashed lines represent the shape before the deformation and the solid lines after the deformation. (b) An illustration of how a compressive force produces contraction and negative stress. (c) Representation of the torsional deformation (i.e., angle of twist φ) produced by an applied torque τ. 1 2.1.2 Experimental method In the quasi-static methods, also called isothermals, the specimen is subjected to mechanical stress (σ), which, as mentioned above, can be traction, compression or shear, and the deformations undergone by the body due to the applied stress, are measured. In Figure 4 we can visualize a test of this type, in which the specimen is subjected to a tensile force and responds with an elongation. 8 Visit our site: http://www.atcp-ndt.com/products/materials-characterization/sonelastic.html

Figure 4: Schematic representation of a mechanical testing machine used to conduct tensile tests and obtain a stress-strain curve, after force in stress and elongation into deformation. The body is elongated by mobile bar. A load cell and a strain gauge measures, respectively, the magnitude of the applied load and elongation. 1 2.1.3 Calculations Deformation (ε) of a body depends on the stress magnitude (σ), which it is subjected. For most crystalline materials, when applied relatively low stress levels, stress and deformation are proportional to each other, this proportionality being represented by the Hooke 1 Law and can be represented by the equation: (11) where: E = Elasticity Modulus or Young's modulus (Pa), σ = Applied Stress (Pa), ε = Longitudinal deformation of the specimen (dimensionless). Hooke's law is valid for the linear elastic regime. In this region, the Young's modulus can be obtained by the slope of the stress-strain graph, as shown in Figure 5. Figure 5: Stress-strain curve in linear elastic regime. 8 There are some materials (e.g., gray cast iron and concrete) to which this initial elastic portion of the stressstrain curve is not linear and therefore cannot be determined the Young's modulus as described above. For this nonlinear behavior, both the tangent modulus and the secant modulus are commonly used. Tangent modulus is taken as the slope of the stress-strain curve at a particular stress level, whereas the secant modulus is the slope of a secant line from the origin to a given point on the curve σ-ε. The determination of these modules is illustrated in Figure 6. 9 Visit our site: http://www.atcp-ndt.com/products/materials-characterization/sonelastic.html

Figure 6: Nonlinear stress-strain curve. 8 The higher the Young's modulus the more rigid the material and lower the elastic deformation resulting from a given stress. The error in measurements by this method is around 15% and is dependent on the elastic moduli of measurement accessories, the stiffness of the force frame of the machine itself, the rounding in the calculations, accuracy of the equipment and on the person who performs the test 9. More modern methods and non-destructive (next topic) are preferred and can be used with relative simplicity and precision. 2.2 DYNAMIC METHODS (NON-DESTRUCTIVE) 2.2.1 Introduction The dynamic methods allow getting both quantitative (elastic modules) and qualitative information about the integrity of a mechanical component, beyond the control of properties such as phase changes, for example. The specimen is not discarded after the test and can be used in its function normally or tested many times. For example, in the study of thermal shock damage, it is possible to assess the development of mechanical strength by measuring the Young's modulus simultaneously with the application of successive cycles of thermal shock. If another technique was used, the measurement of modulus of rupture, for example, a sample would be required for each measurement. The dynamic methods are standardized and widely used, especially for brittle composite materials. 2.2.2 History The first practical implementation of the dynamic method was developed in 1937 10 by Forster. In this method, a bar of rectangular cross-section of the material to be characterized is hung by two wires, one of them connected to an actuator and the other to a sensor. The resonance frequencies are determined with an excitation sine wave of variable frequency associated with the observation of the peak response of the sample captured by the sensor. This method also is mainly used for ultra-high temperature in a controlled atmosphere. Other methods have emerged around the same time, however, were not practical for the need for application of high voltage and the setting of metallic parts in the specimen. 11 The mathematical bases for the accurate calculations of the dynamic elastic moduli were developed between the 1940s and 1960s. In 1945 Pickett 12 presented the equations for the calculation of the elastic modulus and Poisson's ratio from the fundamental modes of vibration. The equations proposed by Pickett rely on empirical correction factors for bars and cylinders with a low aspect ratio. In 1960 Kaneko 13 introduced a refinement to Pickett s equation generalizing them to vibration modes of any order, not only for the fundamental modes. In the 1960s and 1970s was developed the impulse excitation method and Grindosonic14 equipment were developed, which popularized the characterization of the dynamic elastic modules and extended the method to the field of quality control and inspection. In this method, from certain mechanical boundary conditions, the specimen is excited in a particular mode of vibration by a "stroke". The equipment captures this vibration with a piezoelectric sensor or microphone and tells the user its resonance frequency, from which the modules are calculated. 10 Visit our site: http://www.atcp-ndt.com/products/materials-characterization/sonelastic.html

The Grindosonic is marketed with the same features so far, but in the 1990s automated measuring systems were developed and are currently being improved for the characterization of elastic modules of refractory materials as a function of time and temperature. 13,15 These systems are computer based and have several advantages compared to the traditional Grindosonic, particularly in frequency discrimination, such as Sonelastic (http://www.atcpndt.com/products/materials-characterization/sonelastic.html), which in addition to the fundamental frequency also lists the harmonic frequencies present and their respective damping. Given the assumed importance, the procedures for non-destructive characterization by dynamic methods were standardized 16,17, and made efforts between laboratories for harmonization 18 and preparation of studies and good practice characterization guides. 1,9 The dynamic methods are subdivided into: - Impulse Excitation Technique - Frequency Sweep 2.2.3 Vibration modes The principle of adiabatic or dynamic method is to calculate the elastic modules from the natural frequencies of vibration of the specimen and its geometrical parameters 12,13. These frequencies, together with the dimensions and weight, have a univocal relationship with the elastic modulus. The dynamic methods have the advantage of using small samples and being rapid and non-destructive (which enables control of material properties and their variation), as we mentioned earlier, besides the application of low loads. Basically there are three modes of vibration used, for example, in prismatic test bodies: longitudinal or transverse, flexural and torsional. The first two allows the calculation of Young's modulus, and the latter allows the determination of the shear modulus and Poisson's ratio. From the experimental point of view, the method can be separated into two parts: the first consists of the excitation, acquisition and detection of resonance frequencies, and second, the use of mathematical relationships and computational procedures, to obtain the elastic modulus from resonant frequencies. 2.2.4 Natural vibration frequencies The basic model of vibration of a simple oscillating system consists of a body with a given mass, a spring having a mass negligible compared to the mass of the body and a damper as shown in Figure 7. The force-elongation ratio (or contraction) experienced by the mass-spring assembly is considered linear and can be described by Hooke's law: F = kx, where k is the spring stiffness and x is the displacement of the position of static equilibrium suffered by mass. The damping is described by a force proportional to the speed, i.e. F = cv, where v is the speed of displacement and c is the damping coefficient. Figure 7: Model of a damped harmonic oscillator. Model of a damped harmonic oscillator: + +0 (12) 11 Visit our site: http://www.atcp-ndt.com/products/materials-characterization/sonelastic.html

or + +0 (13) where represents the acceleration, the speed and x the displacement. Rearranging the equation (12) and assuming: and ζ we have: + 2ζ + 0 (14) where ω 0 is called natural frequency of vibration (or resonance frequency) and ζ is the rate of damping or simply the damping. Longitudinal vibration of bars Longitudinal waves are those in which vibration occurs in the same direction of wave movement. The longitudinal vibration of bars of square or circular cross section is recommended as the most accurate method for determining the Young's modulus. However, if the section is too far from the square geometry, the errors involved in the calculation of the elastic properties may be great. In this case, the use of longitudinal vibration is not recommended and the Young's modulus can be obtained more precisely by the flexural vibration. Further details of longitudinal vibration of bars can be obtained from the reference number 9. Flexural vibration of bars Transverse or flexural mode of vibration is the most complex of the three modes in consideration in relation to the way the resonance frequency is affected, not only by the length and cross section, but the ratio between the two 19. In the case of thin bars, it is easier to excite flexural vibrations than the longitudinal vibration. Therefore, the flexural vibration is highly recommended for the determination of the Young's modulus of thin bars. There are a number of nodes (point of zero amplitude, destructive interference) and anti-nodes or bellies (maximum amplitude, constructive interference) along the length of a bar supported freely. At the lowest resonance frequency or fundamental frequency (the fundamental mode) the nodal points are located at 0.224 L from each end, with the anti-nodes in the center and at each end. Figure 8 illustrates the points of nodes and anti-nodes for flexural vibration of a bar. n = 1 n = 2 Figure 8: (a) Illustration of the location of the nodes and anti-nodes for a standing wave on a string with free ends. (b) Flexural mode of vibration for a bar of rectangular section, with the nodal lines indicated. 9 12 Visit our site: http://www.atcp-ndt.com/products/materials-characterization/sonelastic.html

In Figure 8 we have shown nodal lines for the different vibration modes (n = 1, 2, 3...). Note, for example, that for n = 1, these nodal lines, or a minimum amplitude, are at the ends (0.224 L to each side), which means that we have a maximum amplitude at the center of the bar and their ends (as can be best seen in Figure 9). Figure 9: Rectangular bar excited to capture flexural frequencies. 16 Thus, we found that the fundamental flexural mode occurs during an impact when the bar is supported at its nodal points and the same is applied in place of the belly (center) (Figure 9). Torsional vibration of bars The torsional vibration mode allows calculating the shear modulus, and then the Poisson's ratio. The location of the nodes can be seen in Figure 10. Figure 10: Torsional vibration mode for a bar of rectangular section, with the indicated nodal lines. 9 We can note then that to obtain the torsional frequencies the impact must be outside the center of the bar, where we have a nodal point. Figure 11 provides the scheme where the excitation and reception of such vibration should be. Figure 11: Bar of rectangular section excited to capture torsional frequencies. 16 13 Visit our site: http://www.atcp-ndt.com/products/materials-characterization/sonelastic.html

The following figure illustrates the positions of the nodes of vibration for the three types of resonance: longitudinal, flexural and torsional, presented, as well as the direction of movement. It can be observed in Figure 12 that the node for the fundamental resonant in the longitudinal and torsional modes is located 1/2L, while for the fundamental resonance in flexural or transversal modes is at 0.224 from the ends of the bar, being L the length of the bar. Figure 12: Position of the nodes of vibration for longitudinal, transverse or flexural and torsional resonance modes. 8 2.3 Experimental methods 2.3.1 Impulse excitation In the impulse excitation technique, the specimen receives a short duration impact and responds with vibrations at its natural frequencies of vibration 18 under the imposed boundary conditions. Figure 13 shows a basic scheme of the positioning of the sample for measuring the torsional and flexional resonant frequencies for this method. An impulse device is an equipment that applies an impact on the specimen to generate mechanical vibrations, without damaging it, and the transducer captures the acoustic response and transforms it into an electrical signal so that we can read the resonance frequencies 14 Visit our site: http://www.atcp-ndt.com/products/materials-characterization/sonelastic.html

Figure 13: Basic schema of the positioning of the sample for measuring the resonant frequencies using the impulse excitation technique. 16 Note that the sample should be supported in the position of the nodes of the fundamental resonance (located at 0.224*L of each end) and the knock must be given in place of higher amplitude, as seen previously (Figures 9 and 11), for excitation of flexural and torsional frequencies, respectively. The frequencies are related to elastic modules by mathematical methods as described on the item Calculations, further on. A system to obtain the natural frequencies by the impulse excitation technique (see Figures 14-17) is basically composed by the following parts, in accordance with ASTM E1876-07: 16 - Impulse device, - Receiver transducer that converts the emerging mechanical vibration into an electrical signal, - Electronic system: amplifier, frequencimetro, - Support System. Support for bars and cylinders Precision support for bars Support for plates and discs Basic support for bars Figure 14: Examples of support for various specimen geometries manufactured by ATCP. 15 Visit our site: http://www.atcp-ndt.com/products/materials-characterization/sonelastic.html

Automatic electromagnetic impulse device - Enables automatic measurement as a function of time - Adjustment of time and voltage pulse - Manual or via PC (USB) - Controlled and reproducible excitation Figure 15: Example of automatic electromagnetic pulser manufactured by ATCP. Figure 16: Equipment developed by ATCP Physical Engineering (Sonelastic ) for measuring the elastic modules based on the impulse excitation technique. Figure 17: Equipment developed by ATCP Physical Engineering, (Sonelastic ), to determine the elastic modules based in the impulse excitation technique. In this configuration the software is embedded in hardware. 16 Visit our site: http://www.atcp-ndt.com/products/materials-characterization/sonelastic.html

2.3.2 Frequency Sweep The principle of the sweep method consists of frequency stimulation of the specimen with variable frequency to find the resonant frequencies of vibration of the sample. After that, by means of mathematical relationships, it is easy to determine the elastic modules from the resonant frequencies. In this kind of measurement, called bar resonance, the sample is suspended by wires that also can be used to excite and detect the vibration. The size and shape of the sample and the types of excitation of vibration shall comply with the established mathematical solutions and for this, the bar of square and circular cross section are the most common, having longitudinal, flexural and torsional excitation. The sample size generally depends on the material to be tested. The dimensions should be in a range of values, so that the resonant frequencies are within the limits that the equipment can measure (mainly the transducers). The most delicate part of the system is the coupling between the transducer (excitation and reception) and the sample. Ideally, the coupling should not interfere with the natural frequencies of vibration of the sample, and for this the coupling must not impose inertia to the system, i.e. add significant mass which can affect the frequency of the normal modes of vibration. This should be done in a controlled manner, in order to generate evaluable impact. There are devices in which the sample is suspended by wires and those in which it rests on the nodes of vibration, for example, the Scanelastic (http://www.atcp.com.br/pt/produtos/caracterizacaomateriais/scanelastic.html). Figure 18 shows a coupling system sample-transducer made by wires, for the measurement of flexural and torsional frequencies. In the case of samples suspended by wires, they must not be supported in its nodal position (0.224 L of each end of the bar) and must be first supported at 0.1 L of the bar ends, and subsequently adjusted experimentally to maximize the vibrational deflection. 20 Figure 18: Positioning of the sample for measuring torsional and flexural resonant frequencies using the bar resonance method. 8 In the coupling case where there is direct contact between the coupling rods of the transducers and the sample, they must allow the sample to oscillate without significant restrictions on the desired mode of vibration. Figure 19 shows an example of such a coupling for flexural and torsional modes of vibration. In this case the sample is supported in the position of the nodes of the fundamental resonant: 0.224 L and 0.5 L of each end for the flexural and torsional vibrations modes, respectively. 17 Visit our site: http://www.atcp-ndt.com/products/materials-characterization/sonelastic.html

Figure 19: Positioning of the sample for measuring torsional and flexional resonant frequencies, using the scanning method with the sample resting on a support. 8 The components used in the dynamic method for resonating bar, according to the ASTM E1875-00 are: 17 - Signal generator, - Exciter transducer which converts the electric signal into mechanical vibration, - Supporter, - Receiving transducer that converts mechanical energy of vibration into an electrical signal, - Frequency meter, - Oscilloscope. Figure 20 shows an ATCP Physical Engineering equipment, the Scanelastic, which uses the resonance technique for bars. Figure 20: Scanelastic equipment, developed by ATCP Physical Engineering, for measuring the elastic moduli by the method of scanning frequencies (bars resonance). http://www.atcp.com.br/pt/produtos/caracterizacaomateriais/scanelastic.html In the following paragraphs we will show the calculations used to relate the flexural and/or torsional frequencies with the elastic modulus. It is important to mention that most modern equipment uses advanced software to perform these types of calculations, simplifying measurement and data processing. 18 Visit our site: http://www.atcp-ndt.com/products/materials-characterization/sonelastic.html

2.3.3 Calculations There are different mathematical models for each type of excitation applied to the specimen. Some of these models are described below, according to the ASTM E-1876 16 and E-1875 17 : Young s Module (flexural vibration) Rectangular section bar excited in flexion: 16 0,9465 (15) wherein m is the mass of the bar, L the length, b the height and t the width, f f is the flexural fundamental resonance frequency and of flexural and T 1 is a correction factor for the fundamental flexural mode given by: 1+6,585 (1+0,0752+0,8109 ) 0,868 [,,, ( ) ],, (,, )( (16) ) µ is the Poisson ratio. a) if L/t 20, the correction factor T 1 in Equation 16, can be simplified to the following equation: =[1,000+6,585 ] (17) and E can be calculated directly by Equation 15. b) if L/t < 20 and Poisson's ratio is known, then T 1 can be calculated directly from Equation 16. c) if L/t < 20 and Poisson's ratio is not known, then an iterative process must be used for calculating the Poisson ratio, based on the experimental values for Young's and shear modules. The iterative process is schematically shown in Figure 21. Its steps are as follows: -Determine the resonant frequencies of the fundamental flexural mode (f f ) and torsional (f t ) of the specimens. The shear modulus is calculated by Equations 21 and 22 (G) from the fundamental torsional resonance frequency and the size and weight of the sample. - Using equations 15 and 16 we can obtain the Young's modulus (E) from the main flexural frequency, size, mass and an initial value of the Poisson's ratio (µ 0 ). Specimen of circular cross section with a flexural excitement: = 1,6067 ( )( ) (18) where D is the diameter, f f is the fundamental flexural resonance frequency and T 1 ' is a correction factor for the fundamental flexural mode, given by: =1+4,939 (1+0,0752+0,8109 ) 0,4883 19 Visit our site: http://www.atcp-ndt.com/products/materials-characterization/sonelastic.html

[,,,,,,, (19) ] a) if L/D 20, the correction factor T 1 can be simplified to: [1,000+4,939( ) ] (20) b) if L/D < 20 and Poisson's ratio is known, then T 1 may be calculated directly from Equation 19. c) if L/D < 20 and Poisson's ratio is not known, then the iterative process (shown in Figure 21 and explained later) should be used. Always remember to use the equations for the specimen of circular cross section: Equations 18, 19 and 23. Shear modulus (torsional vibration) Exciting a rectangular cross section bar in torsion: = (21) wherein G is the shear modulus, f t the fundamental torsional resonance frequency and R is a factor dependent on the ratio between width and height of the sample 12 given by: =,, For a torsionally excited cylindrical specimen: 1+, 0,060 1) (22) = 16 ( ) (23) Regardless of the method and of isotropic condition of the material, the Poisson ratio is related to E and G, as is given by the following equation: = 1 (24) where µ is the Poisson ratio, E is the Young's modulus and G is the shear modulus. The values of shear modulus and Young's modulus calculated in the two previous items are substituted into Equation (24) for the Poisson ratio. A new value for the Poisson ratio (µ x ) is generated to start iteration. The steps are repeated until a non-significant difference (2% or less) is observed between the current value and last value computed for the Poisson ratio. 20 Visit our site: http://www.atcp-ndt.com/products/materials-characterization/sonelastic.html

Figure 21: Iterative process for the simultaneous determination of E, G and µ. 16 2.4 Ultrasound methods (pulse-echo) The ultrasonic methods, called pulse-echo, are also non-destructive and consist of sending ultrasonic waves (high frequency mechanical waves above 20 khz, inaudible for humans) and capturing its echo. In this method, an ultrasonic pulse is emitted by an electro-acoustic transducer. The energy of this pulse is partially or totally reflected back to the transducer upon encountering a reflective surface. The value of the speed of sound is then calculated, to traverse this distance and then the Young's modulus. The same principle is used in the sonar, which is an auxiliary of maritime navigation. This device emits ultrasonic waves by a device placed on the ships and coupled with a sound receiver. The sound propagates in water, and is reflected on the ocean floor or objects (shoal, for example), returns and is captured by the receiver which computes times variations between emission and reception of sound. Some animals such as dolphins and bats also emit ultrasonic waves for locomotion and prey capture, is what is called echolocation, or "biosonar." When the ultrasonic wave hits an object, it is reflected and captured by the animal, then he can estimate the distance to the prey, if there is any object in the path, etc. Figure 22 illustrates the scheme of emission and reception of ultrasonic waves for a bat, which is analogous to the sonar and other equipment with the same principle. Figure 22: Echo-location mechanism of a bat. (Fonte: http://en.wikipedia.org/wiki/animal_echolocation). 21 Visit our site: http://www.atcp-ndt.com/products/materials-characterization/sonelastic.html

The ultrasonic method is widely used mainly because of its simplicity. But a large uncertainty surrounds the importance of Poisson's ratio in this case and of the impossibility to calculate it, because we only measured the longitudinal velocity of sound and the traverse speed would be necessary to know also. Thus, its value must be estimated, resulting in measurement errors, proportional to the dispersion between the true value of Poisson's ratio and the estimated. Furthermore, it is necessary that the contact surface between the sample and the transducer is filled with a "coupling agent" who allows the transfer of energy, but eliminates the air between them. 2.4.1 Pulse generation and reception in the ultrasonic device In Figure 23 the device produces an ultrasonic pulse (1) through the crystal. It propagates through the sample, and in this moment the circuitry of the device starts counting the time. 21 Figure 23: Propagation scheme of the ultrasonic. 21 By focusing on an interface, i.e., in the discontinuity at the distance "S", is the reflection of the wave (2) that is detected by the crystal, causing an electrical signal that is interpreted and amplified by the device and represented by the echo reflection (3) on the screen of the ultrasound apparatus (see Figure 24). The position of the echo on the screen is proportional to the measured return time of the signal and also to the path followed by the sound (S) to the discontinuity in the sample. 21 Figure 24: Echo capture mechanism. 21 Components used in the ultrasonic method, according to ASTM C597 09: 22 - Pulse generator, - Amplifier, - Transducer which turns the electronic pulse in mechanical vibrations, - Time acquisition circuit, - Reference bar in which the transmission time of a longitudinal wave is known, - Cable connectors, - Coupling agent to ensure efficient energy transfer between the sample and the transducers and remove the air between them. 22 Visit our site: http://www.atcp-ndt.com/products/materials-characterization/sonelastic.html

2.4.2 Calculations The following calculations are made according to ASTM C597. 22 Pulse speed is given by: (25) wherein L is the distance between the transducers and t the pulse delay time to traverse the space between them. From the speed of the pulse we can calculate the Young's modulus E: (26) where ρ is the density of the material and µ the Poisson ratio. 3. COMPARATIVE ANALYSIS Given the importance of knowledge of the elastic modules, even in the determination of other essential mechanical properties for projects in the materials context, quality control and various branches of engineering, we will introduce and explain several concepts in order to show the principal measurement techniques of these constants, as well as facilitate its use and interpretation of results. Table 2 shows a comparison between the main characteristics of the methods presented in this article. Table 2: Comparative analysis of quasi-static methods and ultrasound Destructive Testing Measurement Uncertainty Time Characterizable Measurement Samples Elastic Constants Measured as a function of Temperature Quasi-static methods Yes 15 % or more ** principally metals focus on E difficult Dinamic methods Ultrasonic methods No < 2 % seconds any solid material No up to 15%* seconds any solid material E, G and µ (simultaneously) E (with µ estimated and not measured) easy difficult * Dependent on the value of estimated Poisson's ratio ** Depends on the equipment and on who performs the test Thanks to Table 2 we can note some of the main advantages and disadvantages of each method and choose the one that best suits our needs and working conditions. - Non-destructive testing: is greatly appreciated because they allow control of material properties such as phase changes. In addition many materials, when subjected to quality control, need to return intact to their workplaces after inspection. In the engineering industry, particularly in aeronautics, it is very common to need to inspect parts during their lifetime. In such cases, you cannot break or impair the part or component to be tested, because it should be relocated in the system. These tests allow obtaining both quantitative and qualitative information about the integrity of a mechanical component, thus allowing the professional in charge to secure its replacement before the component fails in operation. They are widely used in maintenance and inspection of machinery and engines and, depending on the test to be applied, can provide low operating costs, practicality and speed of test. 23 Visit our site: http://www.atcp-ndt.com/products/materials-characterization/sonelastic.html

- Uncertainty in measurement: is an important parameter to be considered, since for quality control tests, the measurement accuracy is crucial for the reliability of results and for the statement of good working condition. - Measurement time: taking into account the need for characterizing large numbers of parts by the industrial area, acquisition time becomes a factor indispensable for speeding measures. - Characterizable samples: industries often need equipment which characterize a wide range of samples and for that, give preference to a device that meets the growing demand, always seeking a good cost-benefit relationship. - Determination of elastic constants: due to the importance of the elastic constants in the determination of other mechanical properties, knowledge of the elastic modulus, E, shear modulus, G, and Poisson's ratio, µ, simultaneously, are preferred. - Measured as a function of temperature: this type of measurement allows the verification of changes in elastic properties of materials when subjected to heat treatments. You can follow the appearance of new phases, softening of the glassy phase, propagation of cracks or damages, in general. Many materials works under considerable variations of energy and therefore changes in their properties are important questions to be addressed. The equipment that makes this type of measurement should be of easy driving and withstand considerably high temperatures. All the above features, as well as ease of handling, cost, convenience and confidence in the results are determining characteristics for choosing one method or another and getting a system to measure the elastic modules. 4. CONCLUSIONS In this article an overview of what elastic modules are, its importance and ways to measure them was present. A comparison was also made between different measurement techniques, pointing out their advantages and disadvantages. Thus, we can infer that: - The elastic modules are important characteristics of the materials since they allow to design and ensure the proper functioning of parts that are under the influence of some kind of tension, as well as determining other essential mechanical properties; - The quasi-static methods are destructive, which makes them disadvantageous, since many of the tested materials should return to normal service after testing. Moreover, the imprecision of the measures is a factor that must be taken into consideration; - The dynamic methods, which are non-destructive, present more accurate results than those achieved by the quasi-static method, besides allowing repetitive research in the material, a greater acquisition speed and the possibility of make tests in function of temperature; - The ultrasonic method is widely used, mainly due to their relative simplicity, but a large uncertainty in the measurement arises from the impossibility of measuring the Poisson's ratio and the need of its value in the calculations; And finally, we can say that the identification of the characteristics of each method (advantages and disadvantages) allows a prudent choice that best fits your needs and availability, whether financial, technical or practical. 24 Visit our site: http://www.atcp-ndt.com/products/materials-characterization/sonelastic.html

5. APPENDIX A Elastic moduli for anisotropic materials The physical properties of single crystals of some substances depend on the crystallographic direction in which measurements are taken. For example, the elasticity modulus, electrical conductivity and refractive index can have different values in the directions [100] and [111]. This dependence on the crystallographic direction of the properties is known as anisotropy, and is associated with the difference of atomic or ionic spacing between the different directions. The anisotropy increases with the decrease in structural symmetry triclinics structures are normally strongly anisotropic. 1 The waves, for example, which propagate in an anisotropic medium, differ in many aspects from those that propagate in isotropic media. In the anisotropic medium we can find three different types of plane waves propagating, namely the quasi-compressional (qp), which is the fastest, and two other so-called quasi-shear (QS1 and QS2). These are spread with different speeds, which cause the occurrence of the partition of the waves S. The velocity vector (perpendicular to the wavefront) has directions different from that corresponding to the group velocity (which is the direction of propagation of elastic energy, i.e., the direction of the ray). This means that the rays in anisotropic media are not perpendicular to the wavefront. The particle motion (displacement vector) in the wave qp is different from the direction of the group velocity and phase vectors. Likewise, for qs waves, the displacement vector is not perpendicular to the ray or normal to the wavefront. Cases may occur where the propagation velocity of the P wave is less than the speed of the S wave propagating in the same direction. 23 In the study of states of stress and strain in an anisotropic solid produced by external loads, the following hypotheses should be considered: 24 - Tensions in any plane of the solid and in its surface are forces per area unit; - Small deformations; - The material follows Hooke's Law (linear elasticity); - Initial stress unrelated to the external load is negligible (for example: thermal stress). Thus, one can approximate the theory of elastic anisotropy to the classical theory of homogeneous and inhomogeneous linear elasticity. Thus, problems of dynamic, instability, vibrations, large deformations, as well as non-elastic anisotropy are not considered. The reference system is a cylinder r, θ, z and the stress tensor is given by: [ ] (1) In Cartesian coordinates: [ ] (2) The projected displacement of a point, generally, is written by: u r, u θ, u z in cylindrical coordinates. The deformation tensor is given by: 25 Visit our site: http://www.atcp-ndt.com/products/materials-characterization/sonelastic.html