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Just a few short minutes a day to blend one up and enjoy So do yourself a favor shell be chomping forThere are 3 options so you can customize whats right for YOU. Each comes with an instantly downloadable version that youll get access to immediately when you order today. The Bronze Version is the digital only you can read the PDF ebooks on your laptop, your tablet, or your phone. Constitutive laws. Linear Elasticity. Chapter 3. Constitutive. Models  Relations between Stress and Strain. Linear elastic material behavior. You. are probably familiar with the behavior of a linear elastic material from. Isotropic, linear elastic. If. you conduct a uniaxial tensile test on almost any material, and keep the. The specimen deforms reversibly  If you remove the loads, the solid returns. The strain in the specimen depends only on. For most materials, the stress is a linear. Because the strains are small, this is true. Cauchy stress or nominal stress, and is. Lagrange strain or infinitesimal. For most, but not all, materials, the. Thus, if you cut a tensile specimen out of a block of material, as. Such materials are. If you heat a specimen of the material. If the material is isotropic. Stressstrain. relations for isotropic, linear elastic materials. Youngs Modulus, Poissons. Thermal Expansion Coefficient. Before writing down stressstrain. Because the model only works for. Deformation is characterized using the. Section 2. 1. 7.   This is convenient for calculations, but. All stress measures are taken to be. We can use the Cauchy stress  as the stress measure. You probably already know the stressstrain. They are repeated below for convenience. Here, E. Youngs modulus and Poissons ratio,  is the coefficient of thermal expansion, and. The remaining relations can be. The inverse relationship can be expressed as HEALTH. WARNING Note the factor of 2 in the strain vector. Most texts, and most FEM codes use this. In. addition, shear strains and stresses are often listed in a different order in. For. isotropic materials this makes no difference, but you need to be careful when. We can write this expression in a much more convenient form using. Verify for yourself. The. inverse relation is  The stress strain relations are often expressed using the elastic modulus tensor  or the elastic. In terms. of elastic constants,  and  are 3. Reduced stress strain equations. For plane strain. For a. plane strain deformation. The stress strain laws are therefore   In index. Greek subscripts  can have values 1 or 2. For a. Representative values for density, and elastic constants of. Most. of the data in the table below were taken from the excellent introductory. Engineering Materials, by M. F. Ashby and D. R. H. Jones, Pergamon Press. The remainder are from random web pagesNote the units values of E are given in. G stands for Giga, and is short for. The units for density are in   thats Mega grams. One mega gram is 1. Material. Mass density Youngs Modulus Poisson Ratio Expansion coeft  Tungsten Carbide. Silicon Carbide. 2. Tungsten. 13. 4. 41. Alumina. 3. 9. 39. Titanium Carbide. Silicon Nitride. 3. Nickel. 8. 9. 21. CFRP1. 5 1. 6. 70  2. Iron. 7. 9. 19. 60. Low alloy steels. Stainless steel. 7. Mild steel. 7. 8. Copper. 8. 9. 12. Titanium. 4. 5. 11. Silicon. 2. 5 3. Silica glass. Aluminum alloys. Concrete. GFRP1. 4 2. 2. 7 4. Wood, parallel grain. Polyimides. 1. 4. Nylon. 1. 1  1. 2. PMMA1. 2. 3. 4. 0. Polycarbonate. 1. Natural Rubbers. 0. PVC1. 3 1. 6. 0. Other Elastic Constants  bulk, shear and Lame modulus. Youngs modulus and Poissons ratio are the most common. For example, we define the shear modulus,  bulk. Lame modulus of an. A nice table relating. Enjoy 3. 2. 6 Physical. Interpretation of elastic constants for isotropic solids. It is important to have a feel for the physical. E. modulus. E is the slope of the stressstrain. It has. dimensions of stress   and is usually large  for steel,. You can think of E as a measure of. The larger the value of E, the stiffer the solid. For. E 0.  Poissons ratio  is the ratio of lateral to longitudinal. It is dimensionless and typically ranges. For. a stable material,. It is a measure of the compressibility of the solid. If. the solid is incompressible  its volume remains constant, no matter how. If. then stretching a specimen causes no lateral contraction. Some bizarre materials have     if. Thermal. expansion coefficient quantifies. It has dimensions of degrees. Kelvin 1 and is usually very small. For steel,  The bulk. It has a large value usually bigger than E. The shear. Its value is usually somewhat smaller than E. Strain Energy Density for. Isotropic Solids. Note the following. If you. deform a block of material, you do work on it or, in some cases, it may do. In an. elastic material, the work done during loading is stored as recoverable. If you. unload the material, the specimen does work on you, and when it reaches its. The. work done to deform a specimen depends only on the state of strain at the end. It is independent of the. Based. on these observations, we define the strain. To. write down an expression for the strain energy density, it is convenient to. Work. is done on the specimen only during mechanical loading. It is straightforward to show that the. You can also re write. Observe that 3. 2. Stress strain relation for a general anisotropic linear elastic material  the elastic stiffness and compliance tensors. The. simple isotropic model described in the preceding section is unable to. This is. because some materials do have a characteristic orientation. For example, in a block of wood, the grain. The block will be stiffer if it is loaded. The same observation applies to fiber. Generally, single crystal specimens of a. Even. polycrystalline metals may be anisotropic, because a preferred texture may. A. more general stressstrain. The most general linear stressstrain. Here. is a fourth order tensor horrors, known. The stress strain relation is invertible where. At. first sight it appears that the stiffness tensor has 8. Imagine having to measure and keep track of. Fortunately,  must have the following symmetries This. The compliance tensor has the. To see the origin of the symmetries of. The. stress tensor is symmetric, which is only possible if   If a. The. previous two symmetries imply. To see that. note that by definition and. Combining these, Now, note that so that These symmetries allow us. The inverse has the form where. To. satisfy Drucker stability, the eigenvalues of the elastic stiffness and. HEALTH WARNING The shear strain and shear stress components are not. The conventions used here. But many. sources use other conventions. Be. Physical Interpretation of the Anisotropic Elastic Constants. It. is easiest to interpret. Imagine applying a uniaxial stress, say. In. general, this would induce both extensional and shear deformation in the. The strain induced by  the uniaxial stress would be All the constants have dimensions. The constant  looks like a uniaxial compliance, like  , while the ratios  are generalized versions of Poissons ratio. The shear terms are new  in an isotropic material, no shear strain is. Strain energy density for. The strain energy density of an anisotropic material is 3. Basis change formulas for anisotropic. The. material constants  or  for a particular material are usually. When. solving problems involving anisotropic materials it is frequently necessary. Since  is a fourth rank tensor, the basis change. We define the usual transformation tensor  with components. This. is an orthogonal matrix satisfying. In practice, the matrix can be computed in terms of the angles between the. It is straightforward to show that stress, strain, thermal. The basis change formula. K is computed as and the modulo function. Although. these expressions look cumbersome they are quite convenient for computer. The basis change for the.