Theory of the earth california institute of technology. Elastic constants study notes for mechanical engineering. It begins by defining the stress and strain tensors, and their connection via tensors of elastic constants or compliances. Strength of materials module 1 elastic constants e, k. The relation between single crystal elasticity and the effective elastic. Strength of materials module 1 elastic constants e. The two elastic constants are usually expressed as the youngs modulus e and the poissons ratio n. Pdf elastic properties play a key role in materials science and technology. That is, when the axial load w is elastic, the axial displacement, x, will fully recover upon unloading. The fundamental principles of composite material stiffness. Module 3 constitutive equations learning objectives understand basic stressstrain response of engineering materials.
Next the relationship between the elastic constants of individual crystallites to those of the polycrystal are developed by averaging according to micromechanical voigt, reuss, hill, kroner selfconsistent and texture. It results in the change in its original dimension. Most metallic materials are linear in their loaddisplacement relationship. The 12 elastic constants of wood three longitudinal modulus, three shear modulus, and six poissons ratio can be obtained using static tests but the methodology is expensive and laborious because it is necessary to use 6 specimens for one test. This observation is the basis for the definition of. The relationship between the elastic constants and the. There are three common elastic constants found in most reference books. This relationship was also illustrated with measurements. Mechanical deformation puts energy into a material.
Since only two are independent, there is an equation that, in theory, permits the calculation of the third when the other two have been measured experimentally. The external force applied on a specified area is known as stress, while the amount of deformation is called the strain. In continuum mechanics, the lame parameters also called the lame coefficients, lame constants or lame moduli are two materialdependent quantities denoted by. Elastic constants and their relationship lec5 youtube. However, the alternative elastic constants k bulk modulus.
It is the ratio between compressive stress and compressive strain or tensile stress and tensile strain. Most of these constants arise as constants of proportionality between stress and strain for various loading conditions. The relationship permits estimation of thermal expansion from only elastic constants second order and third order and atomic volume. The mathematical relationship between the elastic constants. Civil engineering mcqs elastic constants and their relationships %. Relationship between inplane load and strain relationship between out of plane bending moments and curvatures plate properties from the a, b and d matrices then analysis is equivalent to classical. Modulus of elasticity or youngs modulus e, bulk modulus k and modulus of rigidity or shear modulus m, c or. If a hydrostatic pressure, p, is applied to a volume of material, v, this will result in a slight reduction in volume, dv. Jul 14, 2017 this lecture includes following topics poissons ratio u youngs modulus or modulus of elasticity e shear modulus or modulus of rigidity c, g or n bulk modulus k relationship between. Different elastic constants and their relationships. Apparent elastic constants of cubic crystals and their pressure derivatives v. Stress, strain and hookes law lesson teachengineering. The solid is then characterised by an elastic modulus that relates strain to stress. However, the alternative elastic constants k bulk modulus slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising.
This assumption turns out to be an excellent predictor of the response of components which undergo small deformations. Mark well the various kinds of minerals, note their properties and their mode of origin. Elastic constants, relation between elastic constants. Elastic moduli of transversely isotropic materials the stressstrain relationship for a linearly elastic transversely isotropic material, based on the represen. Subject strength of materials topic module 1 elastic constants lecture 8 faculty venugopal sharma gate academy plus is an effort to initiate free online digital resources for the first. It is defined as the ratio of the linear stress to linear strain within elastic limit and denoted by e. However, while it is wellknown that liquids possess certain macroscopic elastic properties, their relationship with an analogous set of liquid phonons is far more tenuous. The ratio of the applied stresses to the strains generated will always be constant and is known as elastic constant. Elastic constants please find below pdf attachment. Methodology for the characterization of elastic constants of.
The distortion of the cube, is represented by the dotted lines. In such cases, we can use the relation between elastic constants to. For isotropic materials, only two independent elastic constants are needed for describing the stressstrain relationship, i. The elastic constants of a material can describe its response to an externally applied stress and provide useful information about its bonding character and structural stability.
This gives the relationship between the position r of a point in the deformed solid and its position r before deformation. However, there are some materials which do not exhibit this linear relationship. Apr 27, 2015 there are three elastic constants are there. Using a lumped element model, the resonant frequency can temperature dependence of the elastic constants of doped silicon. Elastic constants and their relationship poisson s ratio which is the ratio of transverse to longitudinal strains only magnitude in tensile test specimen is yet another elastic constant. Elastic anisotropy and extreme poissons ratios in single.
In such cases, we can use the relation between elastic constants to understand the magnitude of deformation. Dec 17, 2016 civil engineering mcqs elastic constants and their relationships %. The bulk modulus the final elastic constant that is of interest to us is that of the bulk modulus. Different elastic constants and their relationships when an elastic body is subjected to stress, a proportionate amount of strain is produced. Jun 12, 2018 an elastic modulus also known as modulus of elasticity is a quantity that measures an object or substances resistance to being deformed elastically i. Journal of research of the national bureau of standards a. Elastic constants and their admissible values for incompressible and slightly compressible anisotropic materials. Consider a solid cube, subjected to a shear stress on the faces pq and rs and complimentary shear stress on faces qr and ps. The relationship between elasticwave velocities and density.
Again, note, that this relationship only holds if a pure shear is applied to a specimen. Stress produces a strain, but how much strain is produced depends on the solid itself. Petrus severinus 157 1 t he seismic properties of a material depend on compo sition, crystal structure, temperature, pressure and in some cases defect concentrations. In this chapter, we first introduce to the readers the basic constitutive equations for a general threedimensional anisotropic material with and without material symmetry, elastic constants and compliances and their relations to engineering constants, as well as transformation laws for elastic constants and compliances for both three and two. The elastic constants for the solid and its mass density. Modulus of elasticity or youngs modulus e, bulk modulus k and.
Elastic constants and their relationship strength of material. That is, when the axial load w is elastic, the axial displacement, x. So, here it is, you see, in this lecture we are going to, we are going to. Elastic constants youngs modulus, modulus of rigidity and. Mechanical properties and performance of materials samples of engineering materials are subjected to a wide variety of mechanical tests to measure their strength, elastic constants, and other material properties as well as their performance under a variety of actual use conditions and environments. Relation between elastic constants detailed explanation byjus. Nov 05, 2018 subject strength of materials topic module 1 elastic constants lecture 8 faculty venugopal sharma gate academy plus is an effort to initiate free online digital resources for the first. Other names are sometimes employed for one or both.
Determination of relation between elastic constants youtube. Temperature dependence of the elastic constants of doped silicon. The final elastic constant that is of interest to us is that of the bulk modulus. Pdf determination of relation between elastic constant and. Youngs modulus or modulus of elasticity e shear modulus or modulus of rigidity c, g or n bulk modulus k relationship between youngs. Pdf elastic constants and supersolidity in solid hcp 4he. Hookes law is a law of physics that states that the force f needed to extend or compress a spring by some distance x scales linearly with respect to that distancethat is, where k is a constant factor characteristic of the spring i. At small deformations these forces are proportional to the displacements of atoms.
Youngs modulus, bulk modulus, and rigidity modulus of any elastic solid are together called as elastic constants. Quantify the linear elastic stressstrain response in terms of tensorial quantities and in particular the fourthorder elasticity or sti ness tensor describing hookes law. The most difficult barrier when measuring these elastic constants is the ambiguity between the phase and group velocity in experimental measurements. A linear relation between sintered temperature and elastic constant in samples is observed. Elastic properties elastic constants elastic properties of solids are determined by interatomic forces acting on atoms when they are displaced from the equilibrium positions. If stress acts in three directions at a point it is called volumetric stress and produces volumetric strain. Their pressure derivatives appear in the relationship between the speeds of longitudinal and transverse waves in the body under hydrostatic and zero reference state. Modulus of elasticity the modulus of elasticity youngs modulus e is a material property, that describes its stiffness and is therefore one of the most important properties of solid materials. C 1 represents a rigidbody rotation about the origin and c 2 is a rigidbody displacement in the ydirection. In particular, the angular variation of the youngs modulus e and the rigidity shear modulus g are determined using general equations developed by. Elastic properties and determination of elastic constants. A universal tool for calculating elastic constants from first principles.
Apparent elastic constants of cubic crystals and their. Students are introduced to hookes law as well as stressstrain relationships. Mechanical properties of metals western university. The offdiagonal elastic constant depends on a wavevelocity measured along a nonsymmetry axis. In light of the universitys newly released policy regarding covid19 recommending the community minimize social contact we encourage students to utilize virtual options when seeking support or information from the earth systems program. Foams can be anisotropic, depending on their cell structure. Elastic constants and their relationship strength of material different types of stresses and their corresponding strains within elastic limit are related which are referred to as elastic constants. Experimental values for the elastic constants of a. Stewart abstract when information about density 0 is unavailable, it is often estimated from pwave velocity vp using an empirical relationship such as gardners equation. Stress nine quantities are required to define the state of stress at a point.
When a body is subjected to three mutually perpendicular stresses of same. The relationship between elastic wave velocities and density in sedimentary rocks. Relation between elastic constants detailed explanation. Different types of stresses and their corresponding strains within elastic limit are related which are referred to as elastic constants. For this case, there are only two independent elastic constants. In the solid state the elastic properties of a substance are related quite simply and directly to its phonon spectrum. Apparent elastic constants and their pressure derivatives theapparentsecondorderelasticconstantsplay an important role in the calculation of the thirdorder elastic constants. Engineering elastic constants there are three purposes to this block of lectures. Nov 01, 2016 elastic constants youngs modulus, modulus of rigidity and bulk modulus november 1, 2016 october 27, 2017 pankaj mishra 1 comment engineering mechanics, mechanics of solids, strength of material, strength of materials. This cyberspace method of construction is thought to be more accurate than the conventional mechanical testing procedures for evaluating the elastic. The position of the curve is determined by the constants of integration.
The relationships between the elastic moduli and compliances of transversely isotropic and orthotropic materials, which correspond to different appealing sets of linearly independent fourthorder base tensors used to cast the elastic moduli and compliances tensors, are derived by performing explicit inversions of the involved fourthorder tensors. Elastic constants and their related properties were obtained by using the irelast package. Derive the relationship between the elastic constants, i. When an elastic body is subjected to stress, a proportionate amount of strain is produced. Thermal expansion and elastic constants 1 hassel ledbetter 2 we give a simple, useful relationship between thermal expansion, avv, and elastic constants.
By definition, an orthotropic material has at least 2 orthogonal planes of symmetry, where material properties are independent of direction within each plane. Different elastic constants and their relationships the constructor. Just as there is a general relationship between stress and strain. Deviations from known elastic relations are reported near the density where the supersolid fraction is maximum. To complete our quick journey through continuum mechanics, to provide you with a continuum version of a constitutive law at least for linear elastic materials spq e. However, the alternative elastic constants k bulk modulus and or g shear modulus can also be used. Although only 2 independent elastic constants are required to completely the properties of an isotropic, elastic medium, there are many different constants from which to choose. Calculator for exploring relations among the elastic constants. Pdf theory of elastic constants of cubic transition. Measurements of elastic constants in anisotropic media.
Relationships between elastic constants from fung, a first course in continuum mechanics although only 2 independent elastic constants are required to completely the properties of an isotropic, elastic medium, there are many different constants from which to choose. If one considers wood to be an orthotropic material, this means knowing 12 elastic constants. The database of elastic constants of 141 human cancellous bone specimens employed here was constructed by imaging real specimens and then computationally determining their elastic constants. This calculator computes the interrelations among the 5 commonly used elastic constants. The relationship between elastic anisotropy and extreme poissons ratio behaviour either positive or negative in singlecrystalline materials has been investigated using experimentally determined singlecrystal elastic constants for a wide range of solid materials. Hence, the computation of the constants is equivalent to adjusting the position of the elastic. This lecture includes following topics poissons ratio u youngs modulus or modulus of elasticity e shear modulus or modulus of rigidity c, g or n bulk modulus k relationship between. An elastic modulus also known as modulus of elasticity is a quantity that measures an object or substances resistance to being deformed elastically i. Apr 24, 2018 the two elastic constants are usually expressed as the youngs modulus e and the poissons ratio n. Elasticity constant an overview sciencedirect topics. Thomas young 17731829 defined his elastic modulus e under uniaxial stress see fig. Elastic constant represents the elastic behaviour of objects.
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