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Chapter 8 Large Strains

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Chapter 8 Large Strains Chapter 8 Large Strains Introduction Most geological deformation, whether distorted fossils or fold and thrust belt shortening, accrues over a long period of time and can no longer be analyzed with the assumptions of infinitesimal strain. Fortunately, these large, or finite strains have the same starting point that infinitesimal strain does: the deformation and dis- placement gradient tensors. However, we must clearly distinguish between gradi- ents in position or displacement with respect to the initial (material) or to the final (spatial) state and several assumptions from the last Chapter — small angles, addi- tion of successive phases or steps in the deformation — no longer hold. Finite strain can get complicated very quickly with many different tensors to worry about. Most of our emphasis here will be on the practical measurement of finite strain rather than the details of the theory but we do have to review a few basic concepts first, so that we can appreciate the differences between finite and infinitesimal strain. Some of these differences have a profound impact on how we analyze de- formation. CHAPTER 8 FINITE STRAIN Comparison to Infinitesimal Strain A Plethora of Finite Strain Tensors There are lots of finite strain tensors and they come in pairs: one referenced to the initial state and the other referenced to the final state. The derivation of these tensors is usually based on Figure 7.3 and is tedious but straightforward; we will skip the derivation here but you can see it in Allmendinger et al. (2012) or any good continuum mechanics text. The first tensor is the Lagrangian strain tensor: 1 ⎡ ∂ui ∂u j ∂uk ∂uk ⎤ 1 " Lij = ⎢ + + ⎥ = ⎣⎡Eij + E ji + EkiEkj ⎦⎤ (8.1) 2 ⎣∂ X j ∂ Xi ∂ Xi ∂ X j ⎦ 2 where Eij is the displacement gradient tensor from the last Chapter. Recall that the infinitesimal strain tensor, ε, is (from Eqn. 7.20): 1 "ε = E + E (8.2) ij 2 ( ij ji ) This highlights the first distinction between infinitesimal and finite strain: the for- mer ignores the higher order term, " EkiEkj . Expansion of the different terms in Equation (8.1) follows the usual summation convention rules. For example, to ex- pand for L11, we write: 1 1 2 2 2 " L11 = (E11 + E11 + E11E11 + E21E21 + E31E31 ) = E11 + E11 + E21 + E31 (8.3) 2 2 ( ) 1 In contrast, "ε11 = (E11 + E11 ) = E11 2 Just for practice, let’s also expand for L13: 1 1 " L13 = (E13 + E31 ) + (E11E13 + E21E23 + E31E33 ) (8.4) 2 2 MODERN STRUCTURAL PRACTICE "152 R. W. ALLMENDINGER © 2015-16 CHAPTER 8 FINITE STRAIN As before, the higher order non-linear term on the right hand side of the equation is ignored in infinitesimal strain. The Eulerian finite strain tensor, " Lij , is the same as the Lagrangian strain tensor but referenced to the final state: 1 ⎡ ∂ui ∂u j ∂uk ∂uk ⎤ 1 " Lij = ⎢ + + ⎥ = ⎣⎡eij + eji + ekiekj ⎦⎤ (8.5) 2 ⎣∂ x j ∂ xi ∂ xi ∂ x j ⎦ 2 where eij is the displacement gradient tensor referenced to the final state. Alternatively, we can start with the deformation gradient tensor, Dij (Eqns. 7.8 and 7.13) and derive the Green deformation tensor: ∂ xk ∂ xk " Cij = = Dki Dkj (8.6) ∂ Xi ∂ X j referenced to the initial or material state. The equation (8.6) expansion for C11 and C13 (as before) is: 2 2 2 "C11 = D11D11 + D21D21 + D31D31 = D11 + D21 + D31 (8.7a) "C13 = D11D13 + D21D23 + D31D33 (8.7b) And, finally, the Cauchy deformation tensor, "Cij , is similar to Equation (8.6), but referenced to the final or spatial state: ∂ Xk ∂ Xk "Cij = = dkidkj (8.8) ∂ xi ∂ x j where dij is the deformation gradient tensor referenced to the final state. For a vec- tor parallel to the X1 axis, the quadratic elongation, λ, is equal to the square of the stretch, S, is equal to C11. 2 " λ(1) = S(1) = C11 = 1+ 2L11 (8.9) And, a line parallel to the x1 axis in the final state: MODERN STRUCTURAL PRACTICE "153 R. W. ALLMENDINGER © 2015-16 CHAPTER 8 FINITE STRAIN 1 1 " = 2 = C11 = 1− 2L11 λ(1) S(1) These tensors are not all independent of each other. The relations between them are: 1 " L = C −δ (8.9) ij 2 ( ij ij ) And 1 " L = δ − C (8.10) ij 2 ( ij ij ) Where "δ ij is the Kronecker delta that represents the components of the identity matrix. All of these tensors are symmetric. Thus " Lij and Cij have the same principal axes, Likewise, " Lij and Cij also have the same principal axes, which are different than those for " Lij and Cij . The difference in orientation between the principal axes of " Lij and " Lij is the amount of rotation during the deformation. Multiple Deformations In infinitesimal strain, the displacement gradient tensors are simply added together to sum the total deformation as we saw in the case of the moment tensor for earthquakes or faults in the previous chapter (Eqn. 7.37): " total E = ∑ n E = 1 E + 2 E +…+ n E (8.11) Let’s see how the finite strain version compares. Using the deformation gradient tensor for the first deformation, we get: " 1dx = 1DdX (8.12) The final state of the first deformation is the initial state of the second deformation — that is, " 2dX = 1dx — so for the second deformation, we can write: MODERN STRUCTURAL PRACTICE "154 R. W. ALLMENDINGER © 2015-16 CHAPTER 8 FINITE STRAIN " 2dx = 2 D 2dX = 2 D( 1DdX) = 2 D1DdX (8.13) We know that E = D – I, so the expansion of the 2D1D term in Equation (8.13) is: " 2 D1D = ( 2 E + I)( 1E + I) = 1E + 2 E + ( 2 E1E) + I (8.14) So, superposing deformations really has a higher order term, "( 2 E1E), that is ig- nored in the infinitesimal strain summing of displacement gradient tensors. This term is a matrix multiplication which is non-commutative and thus, because " 2 E1E ≠ 1E 2 E , the order in which the strain and/or rotation occurs makes a difference in the final result. Figure 8.1 illustrates this important principle for a stretch and a rotation. (a) (b) Figure 8.1 — The order in which strains and rotations occur make a difference! (a) A horizontal stretch of 2 followed by a rotation of 45°. (b) a rotation of 45° followed by a horizontal stretch of 2. In both cases, the initial square is shown in pink. Mohr’s Circle for Finite Strain in the Deformed State The Mohr Circle construction for the Cauchy deformation tensor (Eqn. 8.8) is particularly useful in practical strain analysis. This construction is derived, just like all Mohr’s Circles, by rotating the coordinate system about a principal axis and you can see the derivation in Allmendinger et al. (2012). The equations are given by: MODERN STRUCTURAL PRACTICE "155 R. W. ALLMENDINGER © 2015-16 CHAPTER 8 FINITE STRAIN γ′= γ/λ a ψmax a′ a′ 2θ′LNFE S = √λ 3 3 ψ max 2θ′a θ′a S = √λ λ′ 1 2 3 λ′ θ′b 1 1 1 3 2θ′b b′ (a) (b) b b′ Figure 8.2 — (a) the finite strain ellipse (red) and the initial circle with radius of one (blue). Two lines are shown in the undeformed (a and b) and deformed (a′ and b′) state. Because the initial length is 1, the major and minor axes of the ellipse are S1 and S3, respectively. Line a′ is drawn in the orientation of the line that experiences the maximum shear strain in the body. (b) The Mohr’s Circle for finite strain in the deformed state. Lines a′ and b′ are plotted in their correct positions. Note that the angular shear is measured by drawing a line for the origin of the plot to the point on the circle, with the maximum angular shear determined by the line from the origin that is tangent to the circle (line a′). The orientation of a line of no finite elongation (LNFE), which has a λ′ = 1, is also shown. (C1 + C3 ) (C1 − C3 ) (C1 − C3 ) " C ′ = + cos2θ and "C ′ = sin2θ (8.15) 11 2 2 13 2 You will more commonly see these written as: (λ1′ + λ3′) (λ1′ − λ3′) γ (λ′1 − λ′3 ) " λ′ = + cos2θ and "γ ′ = = sin2θ (8.16) 2 2 λ 2 1 1 where " λ′ = = λ S2 Figure 8.2 shows a finite strain ellipse on the left and the corresponding Mohr’s Circle construction on the right. This Mohr’s Circle construction is very useful, particularly where the deformation is in plane strain. MODERN STRUCTURAL PRACTICE "156 R. W. ALLMENDINGER © 2015-16 CHAPTER 8 FINITE STRAIN Progressive Strain In Figure 8.2a, you see the finite strain ellipse (in light red) superimposed on the initial circle (in light blue). The points of intersection between the ellipse and circle define lines that are currently at the same length as they were when they started out; we call these lines of no finite elongation (LNFE) and they are sym- metric with respect to the principal axes of strain. You can imagine making the same sort of diagram for the infinitesimal strain ellipse, which would be only very (a) (b) LNFE 1a LNIE 3 LNFE 1a 3 LNIE, LNFE LNFE LNIE LNIE (c) (d) 3: initially lengthened, will continue to lengthen Extension (+) 2: initially shortened but now longer than initial Time 1b: shorter than initial but will lengthen in next step Extension (–) 1a: will shorten in next increment Figure 8.3 — Illustration of progressive strain. (a) A pure shear strain path where the prin- cipal axes of finite strain and incremental infinitesimal strain remain parallel throughout the deformation.
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