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1.1 Stress, Strain, and Displacement ! Wave Equation 32 geophysics 130: introduction to seismology displacement 1.1 Stress, strain, and displacement wave equation geometric law ! From the relationship between stress, strain, and displacement, we can derive a 3D elastic wave equation. Figure 1.1 shows relationships equation of motion constitutive law between each pair of parameters. In this section, I will show each stress strain term in Figure 1.1. Figure 1.1: Relationship of each parame- ter. 1.1.1 Displacement Displacement, characterizes vibrations, is distance of a particle from its position of equilibrium: u1(x, t) u(x, t)=0 u2(x, t) 1 .(1.1) u (x, t) x3 B 3 C @ A x2 x1 s33 s32 1.1.2 Stress s31 s22 s13 s 21 s s23 12 s11 s11 Stress characterizes forces applied to a material: s23 s12 s21 s 13 s22 s31 s32 s11 s12 s13 (x1, x2, x3) (x1 + dx1, x2, x3) s33 s = s = s s s ,(1.2) ij 0 21 22 23 1 Figure 1.2: Stresses. s s s B 31 32 33 C @ A which is a tensor, and the first subscript indicates the surface applied and the second the direction (Figure 1.2). u(x + dx) x + dx 1.1.3 Strain u(x) x Strain characterizes deformations under stress. If stresses are applied = to a material that is not perfectly rigid, points within it move with respect to each other, and deformation results. parallel translation Let us consider an elastic material which moves u(x) (Figure 1.3). When the original location of the material is x, the displacement of a nearby point originally at x + dx can be written as + rotation ∂ui(x) ui(x + dx) ui(x)+ dxj = ui(x) + dui , ⇡ ∂xj parallel translation rotation+de f ormation + | {z } |{z} (1.3) deformation Figure 1.3: Displacement includes parallel translation, rotation, and deformation (strain). basic physics for seismology 33 Therefore, in the first-order assumption, ∂ui(x) dui = dxj ∂xj 1 ∂ui ∂uj 1 ∂ui ∂uj = + dxj + dxj 2 ∂xj ∂xi ! 2 ∂xj − ∂xi ! 1 1 = (u + u )dx + ( u dx) 2 i,j j,i j 2 r⇥ ⇥ i =(eij + wij)dxj,(1.4) where wij is a rotational translation term (diagonal term is zero, w = w ). Then e = e is the strain tensor, which contains the ij − ji ij spatial derivatives of the displacement field. With the definition of eij, the tensor is symmetric and has 6 independent components. u1,1 1/2(u1,2 + u2,1) 1/2(u1,3 + u3,1) eij = 0 1/2(u2,1 + u1,2) u2,2 1/2(u2,3 + u3,2) 1 (1.5) 1/2(u + u ) 1/2(u + u ) u B 3,1 1,3 3,2 2,3 3,3 C @ A If the diagonal terms of eij are zero, we do not have volume changes. The volume increase, dilatation, is given by the sum of the extensions in the xi directions: ∂u1 ∂u2 ∂u3 eii = + + = tr(e)= u = q (1.6) ∂x1 ∂x2 ∂x3 r· This dilatation gives the change in volume per unit volume associ- ated with the deformation. ∂ui/∂xi mentions displacement of the xi direction changes along the direction of xi. ∂u ∂u ∂u ∂u ∂u ∂u 1 + 1 dx 1 + 2 dx 1 + 3 dx 1 + 1 + 2 + 3 dx dx dx =(1 + q)V = V + DV, ∂x 1 ∂x 2 ∂x 3 ⇡ ∂x ∂x ∂x 1 2 3 ✓ 1 ◆ ✓ 2 ◆ ✓ 3 ◆ ✓ 1 2 3 ◆ (1.7) where q = DV/V. 1.1.4 Geometric law Relationship between displacement and strain, which represents geometric properties (deformation). As we have already found in equation 1.4, 1 e = u +( u)T (1.8) 2 r r ⇣ ⌘ 1.1.5 Equation of motion Relationship between displacement and stress, which represents dynamic properties (motion). 34 geophysics 130: introduction to seismology We write Newton’s second law in terms of body forces and stresses. When I consider the stresses in the x2 direction (the red arrows in Figure 1.2), s (x + dx ˆn ) s (x) dx dx { 12 1 1 − 12 } 2 3 + s (x + dx ˆn ) s (x) dx dx { 22 2 2 − 22 } 1 3 + s (x + dx ˆn ) s (x) dx dx { 32 3 3 − 32 } 1 2 ∂2u + f dV = r 2 dV (1.9) 2 ∂t2 where dV = dx1dx2dx3. With a Taylor expansion, ∂s ∂s ∂s ∂2u 12 + 22 + 32 dV + f dV = r 2 dV (1.10) ∂x ∂x ∂x 2 ∂t2 ✓ 1 2 3 ◆ We also have similar equations for x1 and x2 directions, and by using the summation convention, ∂2u (x, t) s (x, t) + f (x, t) = r i ij,j i ∂t2 sur f ace f orces body f orces | {z } | {zs +}f = ru¨ .(1.11) r· This is the equation of motion, which is satisfied everywhere in a continuous medium. When the right-hand side in equation 1.11 is zero, we have the equation of equilibrium, s (x, t)= f (x, t),(1.12) ij,j − i and if no body forces are applied, we have the homogeneous equa- tion of motion ∂2u (x, t) s (x, t)=r i .(1.13) ij,j ∂t2 1.1.6 Constitutive equations Relationship between stress and strain, which represents material properties (strength, stiffness). Here, we consider the material has a linear relationship between stress and strain (linear elastic). Linear elasticity is valid for the short time scale involved in the propagation of seismic waves. Based on Hooke’s law, the relationship between stress and strain is sij = cijklekl s = c e,(1.14) where constant cijkl is the elastic moduli, which describes the proper- ties of the material. basic physics for seismology 35 Not all components of cijkl are independent. Because stress and strain tensors are symmetric and thermodynamic consideration; cijkl = cjikl = cijlk = cklij.(1.15) Therefore, we have 21 independent components in cijkl. With Voigt Strain energy is defined by recipe, we change the subscripts with 1 W = sijeijdV 2 Z 11 1, 22 2, 33 3, 23 4, 13 5, 12 6, 1 ! ! ! ! ! ! = cijkleijekldV, 2 Z and we can write the elastic moduli as c (i, j = 1, 2, ,6). With ij ··· Therefore, cijkl = cklij. these 21 components, we can describe general anisotropic media. 1.1.7 Wave equation (general anisotropic media) geometric law e = 1 u +( u)T (eq 2 r r Wave equation describes vibrations (u) at each space (x) and time (t) 1.8) under material properties (c, r); • small perturbation f (u, x, t, r, c)=F.(1.16) equation of motion s = ru¨ (eq 1.11) r· • small perturbation In homogeneous case (F = 0), • continuous material f (u, x, t, r, c)=0. (1.17) constitutive law s = c e (eq 1.14) We eliminate s and e by plugging in equations 1.8, 1.11, and 1.14. • small perturbation • continuous material 1 c u +( u)T = ru¨ (1.18) • elastic material r· 2 r r ⇢ ✓ h i◆ This is a general wave equation for anisotropic elastic media. 1.1.8 Elastic moduli in isotropic media On a large scale (compared with wave length), the earth has approxi- mately the same physical properties regardless of orientation, which is called isotropic. In the isotropic case, cijkl has only two indepen- dent components. One pair of the components are called the Lamé constants l and µ, which are defined as cijkl = ldijdkl + µ(dikdjl + dildjk).(1.19) µ is called the shear modulus, but l does not have clear physical explanation. By using the Voigt recipe, equation 1.18 can be written with a matrix form; l + 2µl l000 0 ll+ 2µl0001 lll+ 2µ 000 c = B C (1.20) ij B µ C B 000 00C B C B 0000µ 0 C B C B 00000µ C B C @ A 36 geophysics 130: introduction to seismology In the isotropic media, equation 1.14 becomes sij = lekkdij + 2µeij = lqdij + 2µeij s = ltr(e)I + 2µe (1.21) where q is the dilatation. There are other elastic moduli, which are related to the Lamé constants, such as bulk modulus (K), Poisson’s ratio (n), and Young’s modulus (E) (Table 1.1.8). Table 1.1: Elastic moduli (l, µ)(l, n)(K, l)(E, µ)(K, µ)(E, n)(µ, n)(K, n)(K, E) 2 l(1+n) Eµ E 2µ(1+n) K l + 3 µ 3n 3(3µ E) 3(1 2n) 3(1 2n) l l − 3K 2µ − − 3 E − K E n 2(l+µ) 3K l 2µ 2(3K+µ) 6−K µ(3l+2µ) l(1+n)(1 2n) 9K(K− l) 9Kµ E l+µ n − 3K −l 3K+2µ 2µ(1 + n) 3K(1 2n) − − l µ 1.1.9 Wave equation in isotropic media geometric law e = 1 u +( u)T (eq 2 r r Using equation 1.21 instead of equation 1.14, we can derive the wave 1.8) equation in an isotropic medium. equation of motion s = ru¨ (eq 1.11) r· From equations 1.8, 1.11, and 1.21, the isotropic wave equation is constitutive law s = ltr(e)I + 2µe (eq 1.21) ru¨ =(l + 2µ) ( u) µ u,(1.22) r r· − r⇥r⇥ with an assumption of slowly-varying material ( l 0 and µ r ⇡ r ⇡ 0). u volumetric deformation r· u shearing deformation r⇥ 24 geophysics 130: introduction to seismology 2.3.11 Principal stresses For any stress tensor, we can always find a direction of nˆ that defines the plane of no shear stresses.
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