Basic Equations of Elasticity


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1 A Basic Equations of Elasticity A.1 STRESS The state of stress at any point in a loaded bo is defined completely in terms of the nine components of stress: σ xx,σ yy,σ zz,σ xy,σ yx,σ yz,σ zy,σ zx,andσ xz, where the first three are the normal components and the latter six are the components of shear stress. The equations of internal equilibrium in terms of the nine components of stress can be derived by considering the equilibrium of moments and forces acting on the elemental volume shown in Fig. A.1. The equilibrium of moments about the x, y, an axes, assuming that there are no bo moments, leads to the relations σ yx = σ xy, σ zy = σ yz, σ xz = σ zx A.1) Equations A.1) show that the state of stress at any point can be defined completely by the six components σ xx,σ yy,σ zz,σ xy,σ yz,andσ zx. A.2 STRAIN DISPLACEMENT RELATIONS The deformed shape of an elastic bo under any given system of loads can be described completely by the three components of displacement u, v, and w parallel to the directions x, y, an, respectively. In general, each of these components u, v, andw is a function of the coordinates x, y, an. The strains and rotations induced in the bo can be expressed in terms of the displacements u, v, andw. We shall assume the deformations to be small in this work. To derive the expressions for the normal strain components ε xx and ε yy and the shear strain component ε xy, consider a small rectangular element OACB whose sides of lengths and ) lie parallel to the coordinate axes before deformation. When the bo undergoes deformation under the action of external load and temperature distribution, the element OACB also deforms to the shape O A C B, as shown in Fig. A.2. We can observe that the element OACB has two basic types of deformation, one of change in length and the other of angular distortion. Since the normal strain is defined as change in length divided by original length, the strain components ε xx and ε yy can be found as change in length of the fiber OA which lies in the x direction before deformation ε xx = original length of the fiber = { + [u + u/)] u} = u A.2) 700 Vibration of Continuous Systems. Singiresu S. Rao 2007 John Wiley & Sons, Inc. ISBN:
2 A.2 Strain Displacement Relations 701 z s zz + s zz s yy s yz s yx s zx + s zx 0 s xy s xz + s xz s xy + s xy s zy + s zy s xz s zx s xx s yz + s yz s yx + s yx s yy + s yy y s xx + s xx s zy s zz x Figure A.1 Stresses on an element of size. u + u B C B v + v C q 2 A y O v u O A q 1 u + u v + v x Figure A.2 Deformation of an element.
3 702 Basic Equations of Elasticity change in length of the fiber OB which lies in the y direction before deformation ε yy = original length of the fiber OB = { + [v + v/)] v} = v A.3) The shear strain is defined as the decrease in the right angle between fibers OA and OB, which were at right angles to each other before deformation. Thus, the expression for the shear strain ε xy can be obtained as [v + v/)] v ε xy = θ 1 + θ 2 + [u + u/)] u + [u + u/)] u + [v + v/)] v A.4) If the displacements are assumed to be small, ε xy canbeexpressedas ε xy = u + v A.5) The expressions for the remaining normal strain component ε zz and shear strain components ε yz and ε zx can be derived in a similar manner as ε zz = w A.6) ε yz = w + v A.7) ε zx = u + w A.8) A.3 ROTATIONS Consider the rotation of a rectangular element of sides and as a rigid bo by a small angle, as shown in Fig. A.3. Noting that A D and C E denote the displacements of A and C along the y and x axes, the rotation angle α canbeexpressedas α = v = u A.9) Of course, the strain in the element will be zero during rigidbo movement. If both rigidbo displacements and deformation or strain occur, the quantity ω z = 1 v 2 u ) A.10) can be seen to represent the average of angular displacement of and the angular displacement of, and is called rotation about the z axis. Thus, the rotations of an elemental bo about the x, y, an axes can be expressed as ω x = 1 w 2 v ) A.11) ω y = 1 u 2 w ) A.12) ω z = 1 v 2 u ) A.13)
4 A.4 Stress Strain Relations 703 y B C C E A B a a O D A x Figure A.3 Rotation of an element. A.4 STRESS STRAIN RELATIONS The stress strain relations, also known as the constitutive relations, of an anisotropic elastic material are given by the generalized Hooke s law, based on the experimental observation that strains are linearly related to the applied load within the elastic limit. The six components of stress at any point are related to the six components of strain linearly as σ xx σ yy σ zz σ yz σ zx σ xy = C 11 C 12 C 13 C 16 C 21 C 22 C 23 C 26 C 31 C 32 C 33 C 36 C 61 C 62 C 63 C 66 ε xx ε yy ε zz ε yz ε zx ε xy A.14) where the C ij denote one form of elastic constants of the particular material. Equation A.14) has 36 elastic constants. However, for real materials, the condition for the elastic energy to be a singlevalued function of the strain requires the constants C ij to be symmetric; that is, C ij = C ji. Thus, there are only 21 different elastic constants in Eq. A.14) for an anisotropic material. For an isotropic material, the elastic constants are invariant, that is, independent of the orientation of the x, y, an axes. This reduces to two the number of independent elastic constants in Eq. A.14). The two independent elastic constants, called Lamé s elastic constants, are commonly denoted as λ and µ. The Lamè constants are related
5 704 Basic Equations of Elasticity to C ij as follows: C 11 = C 22 = C 33 = λ + 2µ C 12 = C 21 = C 31 = C 13 = C 32 = C 23 = λ C 44 = C 55 = C 66 = µ all other C ij = 0 Equation A.14) can be rewritten for an elastic isotropic material as A.15) σ xx = λ + 2µε xx σ yy = λ + 2µε yy σ zz = λ + 2µε zz σ yz = µε yz A.16) σ zx = µε zx σ xy = µε xy where = ε xx + ε yy + ε zz A.17) denotes the dilatation of the bo and denotes the change in the volume per unit volume of the material. Lamé s constants λ and µ are related to Young s modulus E, shear modulus G, bulk modulus K, and Poisson s ratio ν as follows: µ3λ + 2µ) E = λ + µ A.18) G = µ A.19) K = λ µ A.20) or λ = µ = ν = λ 2λ + µ) νe 1 + ν)1 2ν) A.21) A.22) E 21 + ν) = G A.23) A.5 EQUATIONS OF MOTION IN TERMS OF STRESSES Due to the applied loads which may be namic), stresses will develop inside an elastic bo. If we consider an element of material inside the bo, it must be in namic equilibrium due to the internal stresses developed. This leads to the equations of motion of a typical element of the bo. The sum of all forces acting on the element shown in Fig. A.1 in the x direction is given by Fx = σ xx + σ ) xx σ xx + + σ zx + σ zx ) σ zx σ xy + σ ) xy σ xy = σ xx + σ xy + σ zx A.24)
6 A.6 Equations of Motion in Terms of Displacements 705 According to Newton s second law of motion, the net force acting in the x direction must be equal to mass times acceleration in the x direction: Fx = ρ 2 u A.25) where ρ is the density, u is the displacement, and 2 u/ is the acceleration parallel to the x axis. Equations A.24) and A.25) lead to the equation of motion in the x direction. A similar procedure can be used for the y and z directions. The final equations of motion can be expressed as σ xx σ xy σ zx + σ xy + σ yy + σ yz + σ zx + σ yz + σ zz = ρ 2 u = ρ 2 v = ρ 2 w A.26) A.27) A.28) where u, v, andw denote the components of displacement parallel to the x, y, an axes, respectively. Note that the equations of motion are independent of the stress strain relations or the type of material. A.6 EQUATIONS OF MOTION IN TERMS OF DISPLACEMENTS Using Eqs. A.16), the equation of motion, Eq. A.26), can be expressed as λ + 2µε xx) + µ ε xy) + µε xz) = ρ 2 u A.29) Using the strain displacement relations given by Eqs. A.2), A.4), and A.8), Eq. A.29) can be written as λ + 2 µ u ) + which can be rewritten as [ v µ + u )] + [ w µ + u )] = ρ 2 u A.30) λ + µ) + µ 2 u = ρ 2 u A.31) where is the dilatation and 2 is the Laplacian operator: 2 = A.32)
7 706 Basic Equations of Elasticity Using a similar procedure, the other two equations of motion, Eqs. A.27) and A.28), can be expressed as λ + µ) + µ 2 v = ρ 2 v λ + µ) + µ 2 w = ρ 2 w A.33) A.34) The equations of motion, Eqs. A.31), A.33), and A.34), govern the propagation of waves as well as the vibratory motion in elastic bodies.
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