Without recourse to constitutive assumptions, without knowledge of material properties, and without solving any of the conservation equations, we construct deformation ﬁelds for linear solids using Dynamic Mode Decomposition (DMD). Originally developed for analyzing experimental or computational ﬁeld vari- ables in ﬂuid mechanics, it takes as input vectors of ﬂow variables assembled as columns of a data matrix, and requires a remarkably few lines of code. It is a natural ﬁt for solid mechanics wherein surface displacements can be used for the analysis. Vectors of surface displacements are arrayed in columns to created a data matrix. Singular Value Decomposition of the time-shifted data matrix aﬀords selection of dominant modes (rank) in the deformation ﬁeld. The DMD algorithm operates on time-shifted data matrices, obtains a reduced-order model that can reconstruct the deformation ﬁeld, and also provides the domi- nant temporal and spatial modes of the deformation. In the case of linear elastic solids, DMD can be used to: reconstruct or predict the displacement states. In elastic-plastic solids, the transition from elastic to plastic results in the eigen- values of the low-rank data matrix going out of the unit circle (implying an unstable growth mode). Using a combination of ﬁnite element analyses and dis- placement measurements obtained using Digital Image Correlation (DIC), we make the case for using DMD for state-estimation and state prediction in elastic solids and identifying onset of plasticity in elastic-plastic solids. Keywords: Data-driven; Dynamic Mode Decomposition; Solid Mechanics; Reduced-Order Model Nomenclature 1 A Transition matrix to trasfer state ktok+ 1 2 Preprint submitted to European Journal of Mechanics / A Solids October 21, 2023 eA Low rank version A 3 Er Euclidean / Frobenius norm 4 f Vector of functions describing system dynamics 5 m Index of snapshot 6 n Index of measurement 7 r Rank of X 8 t Time 9 t,∆ttime and time interval 10 U,VUnitary matrices from SVD of X 11 eU,eVLow rank versions of U, andV 12 W Eigenvectors of eA 13 x State vector 14 X,X′Time-shifted data matrices 15 x , y xand ycoordinates 16 β β=m/n used in Gavish-Donoho rank truncation 17 xm Snapshot of mth measurement 18 Λ, λEigenvaluesof eA 19 µ System parameters 20 τ∗Gavish-Donoho threshold for rank truncation 21 Φ Spatial modes from DMD, size n×r 22 Ψ Temporal modes from DMD, size r×m 23 2 Abbreviations 24 CFD Computational Fluid Dynamics 25 DMD Dynamic Mode Decomposition 26 DIC Digital Image Correlation 27 POD Proper Orthogonal Decomposition 28 sgStrain gauge 29 SVD Singular Value Decomposition 30 vgVirtual gauge 31 1. Introduction 32 Measurements (or results of computations) of ﬁeld variables such as velocity, 33 temperature, and pressure in ﬂuids can be post-processed using the Dynamic 34 Mode Decomposition (DMD) algorithm to identify spatio-temporal coherent 35 features, and localized instabilities in the ﬂow. Originally developed by Schmid 36 (2010) and Rowley et al. (2009), and applied to ﬂow in channels and jets by 37 Rowley et al. (2009) and Schmid et al. (2011) has received wide attention and 38 within a decade of its introduction, it has found application in thermo-ﬂuids, 39 neuroscience, image-processing, and epidemiological studies; Brunton and Kutz 40 (2019). Monographs by Kutz et al. (2016), Brunton and Kutz (2019), and 41 the recent review by Schmid (2022) provide exposition, extensive bibliography, 42 and algorithms. It is a data-driven method that relies on data obtained through 43 computations, such as Computational Fluid Dynamics (CFD), or Eulerian mea- 44 surements. 45 Snapshots of ﬁeld variables, constitute measurement vectors are obtained at 46 constant (or non-constant) intervals, to obtain data matrix. Singular Value De- 47 composition (SVD) on the data matrix is carried out and the dominant modes, 48 or rank, of the data are identiﬁed. Time-shifted data matrices are obtained 49 and used in the DMD algorithm to obtain a low-rank matrix, which is a signiﬁ- 50 cantly smaller than the larger (original) data matrix and whose eigenvalues and 51 eigenvectors describe the dynamics of the system. Accordingly, an excessively 52 large data matrix (such as those from CFD) is reduced in size owing to the 53 identiﬁcation of dominant modes. The DMD algorithm may be interpreted as 54 3 combining the advantages of the SVD for spatial dimensionality reduction, and 55 Fast Fourier Transform (FFT) for temporal frequency identiﬁcation; Brunton 56 and Kutz (2019). In this regard, DMD has advantages over Proper Orthogonal 57 Decomposition (POD), which has also been applied on snapshots of data, in 58 that DMD obtains eigenvalues that describe how the mode amplitudes evolve 59 with time; Kutz et al. (2016). 60 It bears emphasis, that it is a purely data-driven method and does not 61 require: computations of gradients, knowledge of material properties, solution 62 of the conservation equations, underlying physics (constitutive assumptions), or 63 knowledge of the system boundary conditions. 64 For linear systems, the DMD algorithm obtains the leading eigenvalues and 65 eigenvectors; and for periodic data, the decomposition is equivalent to a discrete 66 Fourier transform; Rowley et al. (2009). In passing, we note that DMD can be 67 implemented in a few lines of code; see, for instance, page 240 of Brunton and 68 Kutz (2019). 69 The DMD algorithm has been applied to systems that are linear, systems 70 that also display periodic responses, and quasi-periodic responses in non-linear 71 systems; Kutz (2013). The DMD operator can be connected to the Koopman 72 operator as discussed by Rowley et al. (2009). 73 Despite its popularity, DMD has not found wide use in solid mechanics. It 74 is a natural ﬁt to the discipline as displacement ﬁelds, from surfaces of solids, 75 which are routinely measured using Digital Image Correlation (DIC) or image- 76 processing methods, can be used to setup a data matrix. Saito and Kuno (2020) 77 obtained damping coeﬃcient and natural frequencies of a vibrating cantilever 78 using DMD. Das et al. (2020) used ﬁnite element computations, and obtained 79 snapshots of displacements and velocities to model the deformation of a pneu- 80 matically activated soft-robotic ﬁnger. Simha and Biglarbegian (2022), used 81 snapshots from ﬁnite element computations, and pointed out its power in de- 82 veloping reduced order models, and predicting states for elastic and non-linear 83 elastic materials. DMD models may fail to generalize beyond the training data, 84 or are susceptible to noise in the data, and to overcome these, Baddoo et al. 85 (2023) developed physics-informed DMD. They discuss self-adjoint operators 86 that are pervasive in mechanics. 87 In this paper, we further develop the application of the DMD to elastic, 88 and elastic-plastic solids. Data in this work are not noisy and we assume that 89 the underlying physics and physical properties are unknown . Section 3 presents 90 background on the algorithm, and the un-supervised rank truncation approach 91 4 advocated in this work. We use two-dimensional, ﬁnite element analyses to 92 make the case for applying DMD to surface displacement ﬁelds of elastic and 93 elastic-plastic solids; see Section 4. To support the theoretical analysis, DIC 94 measurements of displacement ﬁelds are used for DMD analyses, in Section 5. 95 Finally, we conclude with a Summary and Discussion in Section 6. 96 2. Notation 97 Boldface is used to denote vectors and matrices; Iis the identity matrix, †is 98 the Moore-Penrose inverse, ∗is the complex, conjugate transpose , superscript 99 −1is the inverse of a square matrix. Variable names and conventions are similar 100 to the treatment in Brunton and Kutz (2019). An overtilde is used to denote 101 matrices that are low-rank; those without the overtilde are full-rank. Frobenius 102 (or Euclidean) norm for matrix Xis denoted by ||X||2. 103 3. Background on DMD 104 A brief outline of DMD is presented here; general details may be found in 105 Brunton and Kutz (2019) and also details speciﬁc to mechanics in Simha and 106 Biglarbegian (2022). Partial diﬀerential equations, upon suitable discretization 107 can be reduced to a vector of dynamical equations as 108 dx dt=f(x, t;µ), where xis a state vector, tis time, fdenotes the dynamics, and µare system 109 parameters. The DMD algorithm assumes local linearity and replaces the above 110 with 111 dx dt=Ax, (1) which has the solution x(t) =x(0) exp(At), where x(0)is the initial condition. 112 Dynamic Mode Decomposition solves Eq. (1) by obtaining the snapshots of f(), 113 each of which is a column vector of size n×1and there are m+1measurements, 114 which obtains matrices XandX′as 115 X=2 64| | | | x1x2· · ·xm | | | |3 75 5 X′=2 64| | | | x2x3· · ·xm+1 | | | |3 75, (2) where the vertical lines in the matrix indicate that each vector is a stack of mea- 116 surements, and each of the vectors represents a state at a given time. Here, we 117 assume that the snapshots are sampled at constant intervals ∆t; non-constant 118 sampling of the snapshots is also possible; see Tu et al. (2014). In the case of 119 ﬂuids, the vectors comprise velocities/temperatures/pressure from ﬂow ﬁelds; in 120 the case of solids, surface displacements are a natural ﬁt. At its essence, DMD 121 obtains a best-ﬁt operator Athat advances the initial state to the next through 122 X′=AX , (3) whereAcan be interpreted as a least-squares/minimum-norm solution to X′= 123 AX. Using the Moore-Penrose inverse, denoted by †, leads to 124 A=X′X†. (4) For connections of Awith the Koopman operator, see Brunton and Kutz (2019). 125 Operator Acan be used to transition from state to state as 126 xk+1≈Axk. (5) Matrix Ais constructed so that the Euclidean norm ||xk+1−Axk||2is minimized 127 across all of the snapshots; Rowley et al. (2009). Equation (5) is the discrete 128 version of Eq. (1) sampled with ∆tandA= exp( A∆t).Using Eq. (5), noting 129 thatAcan be used in a recursive fashion, with a knowledge of the initial state 130 x1, terminal state xmis predicted using 131 X≈2 64| | | | x1Ax1· · ·Am−1x1 | | | |3 75. (6) The ﬁrst step in the DMD algorithm is to compute an SVD of Xas 132 X≈eUeΣeV∗, (7) 6 where eU∈Cn×r,eΣ∈Cr×r, and eV∈Cm×r, r≤mis the either the exact or 133 approximate rank of the data matrix X. Here, ∗denotes the complex conjugate 134 transpose. Columns of eUare the proper orthogonal modes of Xand satisfy 135 eUeU∗=eU∗eU=I. Likewise, eVeV∗=eV∗eV=I. The rmodes can be chosen 136 by identifying those deemed to have suﬃcient magnitude to contribute to the 137 dynamics; or they can be selected in an un-supervised manner as discussed next. 138 Taking the Moore-Penrose inverse of Eq. (7) yields X†=eVeΣ−1eU8, substi- 139 tuting in Eq. (4) gives 140 A=X′eVeΣ−1eU∗. (8) Similarity transformation is used to project Aonto the rPOD modes ( Schmid 141 (2010)) yields a lower dimensional matrix of size r×ras 142 eA=eU∗AeU=eU∗X′eVeΣ−1. (9) Matrix eU,can be used to reconstruct the full state xk, Brunton and Kutz (2019) 143 or Chapter 13 of Smith (2013). 144 Here we note that the full state xcan be reconstructed using the low-rank 145 state and eUas 146 xk=eUexk. (10) Eigenvectors and eigenvalues of eAcan be obtained through 147 eAW =WΛ, (11) where WandΛ(both size r×r)are the eigenvectors and eigenvalues of eA, 148 respectively. Schmid (2010) computes the eigenvectors of the high-dimensional 149 system using Φ = eUW ;alternatively, Tu et al. (2014) proposed the following 150 projection of size n×ras 151 Φ=X′eVeΣ−1W. (12) and derived the remarkable result that AΦ = ΦΛ , which shows that the 152 modes constructed using the reduced matrices are identical to the modes of the 153 high-dimensional matrix. 154 Noting that Eq. (9) implies A=eUeAeU∗, substituting in Eq. (5) and using 155 7 the properties of Uyields 156 xk+1=eUeAeU∗xk,and xk+2=eUeAeU∗xk+1=eUeA2eU∗xk. The above suggests the following recursive reconstruction using the low-rank 157 matrices as 158 xDMD m =2 64| | | | | x1eUeAeU∗x1eUeA2eU∗x1· · · eUeAm−1eU∗x1 | | | | |3 75. (13) Another method of reconstruction uses the eigenvalues and eigenvectors of 159 eAand further details may be found in Brunton and Kutz (2019). Listing of the 160 DMD pseudo-code is in Algorithm 1. 161 3.1. Rank Truncation 162 Rank rcan be obtained by plotting the singular values of X, and choosing 163 dominant values that are deemed to contribute to the response. Alternatively, a 164 hard threshold proposed by Gavish and Donoho (2014) allows for unsupervised 165 selection of the rank of Xbased on its singular values. The selection being valid 166 when matrix size is large compared with the rank, and the signal-to-noise ratio 167 of the low-rank matrix stays constant. Gavish-Donoho propose the following 168 algorithm: 169 τ∗=f(β)median (diag(Σ)), where β=m/n is the ratio of number of snapshots to number of measurements, 170 diag stands for diagonal matrix, and median is the statistical measure. If the 171 signal-to-noise ratio is known, an analytical formula for f(β)is given by Gavish 172 and Donoho (2014); alternatively, they give an approximate function as 173 f(β)≈0.56β3−0.95β2+ 1.82β+ 1.43. Rank of Xcorresponds to the element of diag(Σ)that is closes to τ∗. 174 8 Algorithm 1 DMD algorithm; adapted from Kutz et al. (2016). 1.Set up time-shifted matrices XandX′. 2.Carry out SVD on X=UΣV∗, and choose the rank rfor truncation. 3.Obtain eU,eVandeΣusing r,U,V, andΣ. 4.Compute eA=eUΣeV∗. 5.Compute eigenvectors Wand eigenvalues ΛofeA. 4. DMD on Computational Data 175 We use the Abaqus ﬁnite element software to obtain snapshots of surface 176 displacements obtained from two-dimensional (2D) computations carried out in 177 thex−yplane, and apply the DMD algorithm on them. When the implicit 178 solver is used to solve static problems, time is a ﬁctitious parameter and serves as 179 a ﬁducial for the loading. Python scripts were used to output nodal coordinates 180 (x, y)and displacements (u, v)in 2D at constant intervals, and these serve as 181 inputs to the DMD algorithm. 182 4.1. Elastic Computations 183 Consider ﬁrst, the elastic case; a quarter ﬁnite element model of a plate 184 with a hole is shown in Fig. 1; xand ysymmetries were applied on the y= 0 185 and x= 0lines, respectively; and displacement-controlled loading was applied 186 on the right-hand edge of the plate. Constant strain, plane-stress triangular 187 elements (CPS3) were used for the mesh. Material was taken to be steel with 188 density, Young’s modulus, and Poisson’s ratio of 7800 kg/m3, 200 GPa, and 189 0.33, respectively. 190 A displacement of 0.05 mm was applied as a ramp over a period of 0.5 s. 191 Here, time in seconds is an arbitrary unit. Snapshots of displacements (u, v)for 192 every 0.01 seconds were obtained; that is, a total of m= 50 .The mesh had a 193 total of 1844 nodes. It follows then that, the entire displacement ﬁeld can be 194 represented by 3688×50numbers. Size of the snapshots and time-shifted data 195 matrices are: 196 xiis 3688 ×1 Xis 3688 ×49 X′is 3688 ×49. 9 0 20 x [mm]01020y [mm](a) x [mm]0 10 20 30 y [mm]0510152025Error [mm]2e-074e-076e-078e-07(b) m=50, r = 3 −1 0 1 Re(λ)−1.0−0.50.00.51.0Im(λ)−928.75−928.50Ψ(t) 392.5393.0 0.0 0.2 0.4 Time−4.125−4.100−4.075−0.050.00Φ −0.0250.000 0 10 Node #−0.0250.000Figure 1: Reconstruction of elastic deformation ﬁeld. (a) Quarter-model mesh. (b) Surface plot of error in mm, with all of the snapshots included. Bottom: Unit circle with eigenvalues ofeA, plots of ΨandΦ. We perform DMD using XandX′and obtain rank r,eU,eA; the initial state xi 197 is known by deﬁnition. We can obtain the displacement state of the plate for 198 snapshot musing the sequence in Eq. (13) and compare with the known xm 199 using the Frobenius norm 200 Er= xm−xDMD m 2. (14) We start with a data matrix of size n×m, and we attempt to reconstruct the 201 ﬁnal state with only eU,eA. That is data of size n×r+r×r. 202 Figure 1 presents the prediction error for each of the nodes in the ﬁnal state, 203 m= 50 and a rank r= 3.Here, with a knowledge of the initial vector x1and 204 3688×3 + 3×3pieces of data, we can reconstruct the deformation ﬁeld, which 205 is∼6%of all of the data. 206 The ﬁgure also shows the eigenvalues of the ˜Amatrix and all of them are 207 approximately unity and lie on the perimeter of the unit circles; vectors Φand 208 Ψare also plotted. In particular, Φrepresents spatial structures, and only the 209 10 ﬁrst twenty items (out of 3688) are shown and these represent xdisplacements. 210 Node number 20 is on the y−symmetry plane on the right edge of the plate. 211 To understand what these vectors represent, consider the sum over the three 212 modes 213 Φ(20,0)Ψ(0.5,0) +Φ(20,1)Ψ(0.5,1) +Φ(20,2)Ψ(0.5,2) = 35 .0499998 , where the ﬁrst number in the parenthesis of Φis node number, and the second 214 number is the mode number; the ﬁrst number parenthesis of Ψis time and the 215 second is the mode number. Note that node 20 is on the xaxis and the right 216 edge of the the plate.The above sum diﬀers from the expected x−displacement of 217 35.05 mm by order of 10−7mm. Also note that Φdisplay a spike with value of 0, 218 these correspond to the nodes on the x−symmetry plane, which are restrained. 219 Alternatively, we use only few snapshots to predict the ﬁnal state, and the 220 reconstruction error depends on the number of snapshots. For instance, in Fig. 221 2(a), when m= 20 , r= 8 reconstruction yields error an order of magnitude 222 higher than the reconstruction using all of the snapshots. Using only 15 snap- 223 shots and r= 5yields errors of the order of 10−5mm. 224 Rank truncation using the Gavish and Donoho (2014) method yields 5modes 225 in this instance, and as shown in the plot of the eigenvalues of the ˜Amatrix, 226 can also be obtained by selecting the mode after which there is no signiﬁcant 227 change in the value of the eigenvalue. Figure 2 (e) shows the unit circle and 228 the location of the eigenvalues. As with the case with m= 49 , where we used 229 all of the snapshots, the three dominant eigenvalues are unity and on the unit 230 circle, however there are additional complex eigenvalues, which when matched 231 with the time-dependent amplitudes, which are initially oscillatory (of the order 232 of10−7mm) and decay to zero, do not contribute to the ﬁnal displacements. 233 4.2. Elastic-Plastic Computations 234 We assumed that the plate studied in the previous section is an elastic-plastic 235 material and apply DMD to the computed displacements. In addition to the 236 properties previously listed, yield strength was taken to be 100 MPa. In this 237 instance, we apply the DMD algorithm in an on-line sense. That is, acquire a 238 snapshot, augment the XandX′matrices and assess the resultant modes and 239 eigenvalues. Articles by Hemati et al. (2014); Nayak et al. (2023); Alfatlawi 240 et al. (2020); Barros et al. (2023) discuss other strategies and considerations 241 11 0102030010202e-064e-066e-06(a) m=20, r = 8 0102030010205e-061e-05(b) m=15, r = 5 −1 0 1 Re(λ)−101Im(λ)(d) m=15, r = 5 05101520 Modes−20−15−10−50log(λi/r ∑ iλi)(c) m=15, r = 5 −620.6−620.4(e) Ψ(t) 325.3325.4 2.502.751e−3 021e−7 0.0 0.1 Time [s]021e−7Figure 2: (a) and (b) reconstruction of elastic deformation ﬁeld with fewer snapshots. (c) Plot of normalized eigenvalues of the ˜Amatrix before truncation. (d) All eigenvalues are in the unit matrix. (e) Plots of the temporal modes.The ﬁrst three are dominant and the last two die out to zero. 12 when applying DMD for data that is acquired sequentially rather than as a 242 batch. 243 Figure 3 (a) shows the global load-displacement curve and the ﬁrst dotted 244 vertical line indicates that global response is elastic for snapshot m= 8, and 245 the second vertical line for m= 12 indicates that the global response is plas- 246 tic. Compare the linearity of the load-displacement curve with the dashed line, 247 plotted in the ﬁgure as a reference. 248 Error plots in Fig. 3 tell a diﬀerent story. The reconstruction error for state 249 m= 7is of the order of 10−6mm, which suggests, based on the reconstruction 250 errors in the previous section, that the local response is elastic. For state m= 8, 251 the reconstruction error blows up to the order of 100 mm, which suggests that 252 the DMD algorithm has broken down, and presumably this is is owing to local 253 plasticity, which is in contrast to the observation in the load-displacement curve 254 that indicates global linearity for m= 8. Even if we double the number of 255 retained modes to r= 6,the error is not impacted. 256 This is a key result and indicates the global load-displacement curve displays 257 the onset of non-linear behavior in the structure, when it ceases to be a straight 258 line; the DMD algorithm, on the other hand, identiﬁes the onset of local plas- 259 ticity, especially at the north pole of the hole. We make further case for 260 this in the next ﬁgure. 261 With the acquisition of more snapshots, the reconstruction error improves 262 to approximately 10−3mm and is shown in Figure 3 (b) for a state m= 15 . As 263 plasticity spreads in the plate with further increase of global displacement, the 264 gradients in displacement are informed by the additionally acquired snapshots. 265 However, owing to the non-linear material response, the reconstruction error 266 does not go below the value of 10−3mm. 267 The Abaqus post-processor outputs a binary (0 or 1) ﬁeld variable called 268 yield ﬂag whose level indicates whether the von-Mises stress has exceeded the 269 yield strength of the material. Figure 4 (a) presents a close up view of the 270 perimeter of the hole in the plate and superimposed is the contours of the yield 271 ﬂag for m= 7there is incipient yield and because of nodal averaging, the value 272 of the ﬂag is below unity around the perimeter of the hole. Subsequently, when 273 m= 8, contours of the yield ﬂag around the north-pole of the hole shown in Fig. 274 4 (b) indicates plasticity around the north-pole of the hole and it corroborates 275 the result obtained using the DMD algorithm. 276 The more striking result is in Fig. 4 (c) wherein the eigenvalue of the the ˜A 277 matrix (size r×r)is outside the unit circle, in the right side of the imaginary 278 13 0102030010201e-062e-063e-06(b) Error [mm] m = 7, r = 2 0102030010202e+024e+02(c) m =8, r = 3 0102030010201e-032e-033e-03(d) m =12, r = 50.000 0.005 0.010 0.015 Displacement [mm]0500100015002000Load [N]m= 8 m= 12(a) 0.000 0.050 0.100 0.150Time [s]Figure 3: (a) Global load-displacement plot. Vertical lines correspond to elastic behavior form= 8,and onset of global plastic response for m= 12 . Dashed line shows departure from linearity in the force-displacement curve. (b) Error surface plot prior to onset of local plasticity for m= 7. (c) Error surface plot after onset of local plasticity - corresponds to ﬁrst vertical dashed line in panel, m= 8.(a). (d) Error surface plot at 0.012 mm of global displacement; m= 12 . 14 −2 0 2 Re(λ)−2−10123Im(λ)(c) m = 8, r = 3 15 180 0.00 0.02 0.04 0.06 0.08 Time [s]−2000200400600800Ψ(d) Φ(t) m= 80 2 4 6 80246(a) Yield Flag m = 7 0.00.7 0 2 4 6 80246(b) Yield Flag m = 8 01Figure 4: (a) Zoomed in contour plot of yield ﬂag for state m= 7; no signiﬁcant plasticity yet. (b) Contour plot of yield ﬂag from m= 8; signiﬁcant plasticity around the north pole of the hole. (d) Unit circle showing the three eigenvalues. Two are close to unity on the real axis. The third is out of the unit circle (at approximately 17.5) and shown as a ﬁlled in square in the inset. (d) Plot of modes, showing the dominant mode displaying a signiﬁcant increase in magnitude with the onset of plasticity. 15 plane, indicating instability in the sense of classical control theory. Correspond- 279 ing mode shows a dramatic rise with time; Fig. 4 (d). In what follows, we use 280 Digital Image Correlation on elastic-plastic solids to assess whether the latter 281 result can be reproduced when DMD is applied to experimental measurements. 282 5. DMD on Experimental Data 283 5.1. Materials and Methods 284 Standard ASTM E8 tensile specimens of cold-rolled 1018 steel and aluminum 285 Al6061 plates with a hole in them were axially loaded in a 50-kN Instron frame. 286 The tensile specimen was 18.36-mm wide, 3.2-mm thick, and had a gauge sec- 287 tion of 50 mm. The Aluminum plate was 25-mm wide, 3-mm thick, 200-mm 288 long, and had a hole in the center of 5-mm radius. For both of the specimen 289 geometries, three tests were conducted to ensure repeatability. It bears empha- 290 sis that knowledge of the material properties and specimen geometries are not 291 required for the DMD algorithm. 292 Figure 5 presents photos of the experimental setup for the two-dimensional 293 DIC. Speckle patterns were applied by spraying black and white paint on one 294 side and a CEA-06-125UN-350 Micro Measurements resistance strain gauge was 295 bonded to the other side of the specimen. In the tensile specimen, the gauge 296 was bonded in the gauge section, and in the structure with the hole, the gauge 297 was bonded in the region where the deformation would be elastic. 298 A standard Digital Single Lens Reﬂex camera (Canon, EOS Rebel T7 with 299 24-megapixel sensor) in conjunction with a Sigma 18-300 mm F33.5-6.3 Macro 300 lens was used to record video of the deformation of the specimen’s surface. Start 301 of the loading and video recording were synchronized and video was captured 302 a framing rate of 30 frames per second. The strain gauge was bonded using 303 standard procedure, and is used with a Tacuna Systems (EMBSGB200) load- 304 cell ampliﬁer to measure strain. 305 The video was recorded as an mp4 ﬁle and images were sampled with a 306 ﬁxed frequency (skipping 2 frames). These images were then input to a Digital 307 Image Correlation engine (DICe) developed by Turner (2015). The latter is an 308 open-source DIC analysis software. We used it in two modes; ﬁrst by specifying 309 a window and tracking subsets of the order of 15 pixels. This resulted in coordi- 310 nates of approximately 1400 points being tracked. That is, 1400 displacements 311 (u, v)pairs per snapshot. As we know the number of frames skipped in the video 312 ﬁle and the framing rate, the time stamp of each snap shot can be obtained. 313 16 hole radius 5 mm 25 mm25 mm Figure 5: Photos of the experimental setup. Left: DIC side of dog-bone specimen in grips of tensile frame. Right top: strain gauge installed opposite to the DIC side of the specimen. Bottom: Regions of Interest used in the plate-with-hole aluminum 6061-specimen, and window used in analysis of tensile 1018-steel specimen. Alternatively, we speciﬁed pairs of regions of interest each of the order of 20 314 pixels and used the average coordinates of the center to compute true strains. 315 We compare the latter strains with the strain measurement from the bonded 316 resistance strain gauge to validate our DIC setup. 317 5.2. DMD on Tensile Specimen 318 We ﬁrst address the results of the analysis on the tensile specimens; see Fig. 319 6. Graphs in the top row show the results of the strain history (in the gauge 320 section of the specimen) from the bonded resistance strain gauge and virtual 321 strain gauges. Good agreement between the resistance gauge and the virtual 322 gauges from DIC, validates our optical and post-processing workﬂow. Three 323 vertical dotted lines are shown at approximately 18, 21 and 31 s, respectively. 324 Axial load history is also shown in the graph through a second y−axis. A 325 dotted reference line is also shown, which identiﬁes the onset of global non- 326 linearity . 327 The ﬁrst dotted vertical line at 18 s is in the elastic portion of the global 328 response, the second vertical line at 21 s is at the start of global non-linear 329 response, and the third vertical line at 31 s is when the load is almost constant. 330 17 0 5 10 15 20 25 30 35 Time [s]0.00.51.0Strain1e−2 vg1 vg2 sg 020 Load [kN]Loadref. line 0 10 Time [s]0.00.51.0Ψ/m ax(Ψ)56 Modes 0.95 1.00 Re(λ)−0.20.00.2Im(λ)0 10 200.00.51.064 Modes 0.95 1.00−0.20.00.20 200.00.51.089 Modes 0.95 1.00−0.20.00.2Figure 6: Top: Strain histories from bonded resistance strain gauge and DIC; load history on second vertical axis. Reference line indicates onset of non-linearity in the load-displacement curve. Three vertical dotted lines indicate times for which the modes and unit circles are plotted below. Middle: Normalized rmodes for the three times. For elastic response, the modes are decaying; when plasticity starts, three of the modes rise dramatically. Bottom: Unit circles and eigenvalues for the three times. When response is elastic all of the eigenvalues are in the unit circle. With onset of plasticity, the eigenvalues fall out of the circle. Second and third row of graphs in the ﬁgure display the modes and eigen- 331 values for the three timestamps, respectively. At 18 s, the Gavish-Donoho rank 332 truncation yields 56 modes and all of the modes are decaying as shown. 333 At 21 s, 64 modes are retained, and there are three eigenvalues out of the unit 334 circle, and this corresponds to the three modes that rise dramatically. Likewise 335 when there is signiﬁcant plastic strain (at 31 s) in the structure, the eigenvalues 336 are still outside the unit circle with corresponding rising modes. 337 The key result of the ﬁgure, is that the onset of global and local plasticity 338 coincide owing to uniform deformation in the gauge section, and the DMD 339 algorithm unambiguously identiﬁes this with both rising modes and location of 340 18 the eigenvalues outside the unit circle. 341 5.3. DMD on plate with hole 342 Next we consider application of DMD on a structure with a ﬂaw. Figure 7 343 presents strain histories at the top; and the strains are from the virtual gauges, 344 and the bonded strain gauge. Owing to the localization of plasticity within the 345 structure, strain values plateaus and does not grow further. In the top graph, 346 there are three vertical lines at 14.8, 22, and 36 s, respectively. Comparison 347 of the strain histories obtained using virtual gauges and the resistance gauge 348 validates our DIC workﬂow. 349 The graph at the top also shows the load history, and a dotted reference line 350 identiﬁes the point of non-linearity in structural response , which occurs 351 around 31 s. The ﬁrst and second vertical lines are at 14.8 and 22 s, respectively 352 and located in the elastic portion of the global structural response; in contrast, 353 the strain histories indicate that the local response is elastic at 14.8 s and 354 plasticity has started at 22 s (owing to the curvature in the strain-history curve). 355 The second and third rows of graphs are the DMD modes and eigenvalues 356 with unit circle. All of the modes in the elastic region (at 14.8 s) decay and the 357 eigenvalues are in the unit circle. In contrast, there is a growing mode at 22 s, 358 and one of the eigenvalues is out of the unit circle. Subsequently at 36 s, three 359 modes are increasing rapidly, with three eigenvalues out of the unit-circle. 360 The key result here is that the DMD algorithm identiﬁes the onset of local 361 plasticity, while the global structural response is elastic. 362 6. Discussion and Summary 363 At the outset, we classiﬁed uses of DMD into the three categories of state 364 estimation and future-state prediction, identiﬁcation of features in experimen- 365 tal measurements, and control. It is well known that standard DMD cannot 366 characterize non-linear phenomena; see sec. 7.2 of Brunton and Kutz (2019). 367 However, as shown in this work, for linear elastic systems, DMD can be used to 368 reconstruct the deformation ﬁelds to very high accuracies of the order of 10−7369 mm, and can serve as a reduced order model. Furthermore, for control of linear 370 structures, the DMD algorithm provides means of constructing accurate models. 371 We have also made the case for the capacity of DMD algorithm for future- 372 state prediction in linear elastic solids. We have shown that the breakdown of 373 DMD in linear systems can be used to identify onset of global and local plasticity. 374 19 0 5 10 15 20 25 30 35 40 Time [s]024Strain1e−3 vg1 vg1 sg 01020 Load [kN]ref. line Load 0 100.00.51.045 Modes 0.95 1.00−0.20.00.20 10 200.00.51.061 Modes 0.95 1.00−0.20.00.20 200.00.51.093 Modes 0.95 1.00 1.05−0.20.00.2Figure 7: Top: Strain histories from bonded resistance strain gauge and DIC; load history on second vertical axis. Reference line indicates onset of non-linearity in the load-displacement curve. Three vertical dotted lines indicate times for which the modes and unit circles are plotted below. Middle: Normalized rmodes for the three times. For elastic response, the modes are decaying; when plasticity starts, three of the modes rise dramatically. Bottom: Unit circles and eigenvalues for the three times. When response is elastic all eigenvalues are in the unit circle. With onset of plasticity, the eigenvalues fall out of the circle. 20 For instance, in the plate-with-hole structure, we could select a window for DIC 375 analysis that is around the hole (as shown in Fig. 5), a second window in 376 the elastic region, and the DMD analysis would accordingly indicate onset of 377 plasticity in the former, and elasticity in the latter. 378 Our result that DMD picks out onset of local plasticity in an unsupervised 379 fashion points to to a new technique for the assessment of structural integrity of 380 structures. Accordingly, there is potential for application on three-dimensional 381 displacement data obtained using Digital Volume Correlation (using tomog- 382 raphy). Both surface and volume data used in conjunction with DMD has 383 potential to detect onset of structural failure and damage. 384 Limitations of the DMD technique are in regards to responses of transient 385 dynamics, and rotational and translation invariance; Kutz et al. (2016). Modi- 386 ﬁcations to the basic DMD algorithm to over come these limitations have been 387 developed; Brunton and Kutz (2019). Rowley et al. (2009) have developed DMD 388 algorithm for non-linear ﬂows using Koopman theory, and it presents signiﬁcant 389 potential for describing the dynamics of non-linear systems; see also Brunton 390 and Kutz (2019). 391 Nayak et al. (2023)Alfatlawi et al. (2020) 392 The highlights of the purely data-driven DMD algorithm applied in this work 393 dePeng et al. (2022)serve emphasis: 394 1.No knowledge of the material properties is required. 395 2.No knowledge of the boundary conditions are required; as we applied the 396 DMD algorithm to a window, only the state of displacement in the window 397 was required. 398 3.We did not evaluate gradients to compute strains. 399 4.No analytical or numerical solution of the conservation equations is re- 400 quired. 401 5.The basic DMD, as presented in this work, can be implemented with a 402 few lines of code (see sec. 7.2 of Brunton and Kutz (2019)). 403 Upon publication of the paper, the codes and data will be available at https: 404 //github.com/csimha/DMD_solid_R0 . 405 7. Acknowledgments 406 Funding from the Natural Sciences and Engineering Research Council of 407 Canada through the Discovery Grant is gratefully acknowledged. The assistance 408 21 of graduate student Randall Vandyk in carrying out the experiments is much 409 appreciated. 410