1 00:00:00,040 --> 00:00:02,470 The following content is provided under a Creative 2 00:00:02,470 --> 00:00:03,880 Commons license. 3 00:00:03,880 --> 00:00:06,920 Your support will help MIT OpenCourseWare continue to 4 00:00:06,920 --> 00:00:10,570 offer high quality educational resources for free. 5 00:00:10,570 --> 00:00:13,470 To make a donation or view additional materials from 6 00:00:13,470 --> 00:00:17,875 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:17,875 --> 00:00:19,125 ocw.mit.edu. 8 00:00:20,896 --> 00:00:23,260 PROFESSOR: Ladies and gentlemen, welcome to this 9 00:00:23,260 --> 00:00:26,400 lecture on nonlinear finite element analysis of solids and 10 00:00:26,400 --> 00:00:27,690 structures. 11 00:00:27,690 --> 00:00:31,340 In the previous lectures, we talked about the basic process 12 00:00:31,340 --> 00:00:34,160 of the incremental solution that is used in 13 00:00:34,160 --> 00:00:35,990 finite element analysis. 14 00:00:35,990 --> 00:00:39,890 And we talked about quite some important continuum mechanics 15 00:00:39,890 --> 00:00:41,980 variables that we are employing in nonlinear finite 16 00:00:41,980 --> 00:00:43,760 element analysis. 17 00:00:43,760 --> 00:00:47,750 This was really preparatory work for the discussion of a 18 00:00:47,750 --> 00:00:52,760 general formulation that is very widely used to analyze 19 00:00:52,760 --> 00:00:56,350 nonlinear systems using finite elements. 20 00:00:56,350 --> 00:00:59,250 And this formulation is called the total Lagrangian 21 00:00:59,250 --> 00:01:00,720 formulation. 22 00:01:00,720 --> 00:01:02,990 This is the formulation that I'd like to discuss with you 23 00:01:02,990 --> 00:01:05,820 now in this lecture. 24 00:01:05,820 --> 00:01:10,250 The total Lagrangian formulation is a formulation 25 00:01:10,250 --> 00:01:17,420 that refers the stress and the strain variables at time t 26 00:01:17,420 --> 00:01:22,700 plus delta t to the original configuration at time zero. 27 00:01:22,700 --> 00:01:25,910 Total, in fact, means reference to the original 28 00:01:25,910 --> 00:01:28,000 configuration here. 29 00:01:28,000 --> 00:01:31,270 And notice we, of course, encountered this equation 30 00:01:31,270 --> 00:01:33,860 already earlier in the early lectures. 31 00:01:33,860 --> 00:01:37,080 We have on the left hand side again the total internal 32 00:01:37,080 --> 00:01:38,125 virtual work. 33 00:01:38,125 --> 00:01:40,880 And on the right hand side, the total external virtual 34 00:01:40,880 --> 00:01:44,200 work all at time t plus delta t. 35 00:01:44,200 --> 00:01:47,210 We have here the second Piola-Kirchhoff stress tensor. 36 00:01:47,210 --> 00:01:50,380 Here we have the Green-Lagrange strain tensor. 37 00:01:50,380 --> 00:01:53,870 And on the right hand side, of course, all the body forces, 38 00:01:53,870 --> 00:01:56,040 surface forces et cetera would enter. 39 00:01:56,040 --> 00:01:58,900 We discussed in a previous lecture how the right hand 40 00:01:58,900 --> 00:02:03,140 side is calculated in general and what it all contains. 41 00:02:03,140 --> 00:02:08,090 We also discussed that this equation here is quite 42 00:02:08,090 --> 00:02:10,860 equivalent to this equation. 43 00:02:10,860 --> 00:02:14,580 And in this equation, we have the Cauchy stress tensor 44 00:02:14,580 --> 00:02:18,450 operating on the virtual strain tensor, an 45 00:02:18,450 --> 00:02:21,050 infinitesimally small strain tensor. 46 00:02:21,050 --> 00:02:24,460 And this product is integrated over the volume at time 47 00:02:24,460 --> 00:02:26,550 t plus delta t. 48 00:02:26,550 --> 00:02:28,200 On the right hand side, we have the same 49 00:02:28,200 --> 00:02:31,380 quantity as up there. 50 00:02:31,380 --> 00:02:34,700 Now notice that we really want to solve, of course, this 51 00:02:34,700 --> 00:02:36,880 equation here in finite element analysis. 52 00:02:36,880 --> 00:02:40,630 But we discussed earlier that in order to do so effectively, 53 00:02:40,630 --> 00:02:43,740 we need to introduce a new stress measure, a new strain 54 00:02:43,740 --> 00:02:46,830 measure, and that, of course, brings us then to this 55 00:02:46,830 --> 00:02:49,400 equation up here. 56 00:02:49,400 --> 00:02:53,370 Let us recall briefly some of the important points that we 57 00:02:53,370 --> 00:02:56,010 made earlier in the other lectures. 58 00:02:56,010 --> 00:03:02,450 We said that this equation here, applied at time t plus 59 00:03:02,450 --> 00:03:05,870 delta t, is an expression of the equilibrium, 60 00:03:05,870 --> 00:03:09,070 compatibility, and the stress-strain law at time t 61 00:03:09,070 --> 00:03:10,440 plus delta t. 62 00:03:10,440 --> 00:03:13,180 Let's just look at this equation once more, where 63 00:03:13,180 --> 00:03:14,660 these quantities enter. 64 00:03:14,660 --> 00:03:20,940 Well, if equilibrium is satisfied locally everywhere 65 00:03:20,940 --> 00:03:26,090 in the continuum, then this equation must be holding for 66 00:03:26,090 --> 00:03:29,760 any virtual displacements that satisfy the displacement 67 00:03:29,760 --> 00:03:33,050 boundary conditions, and corresponding virtual strains. 68 00:03:33,050 --> 00:03:36,810 Notice that the virtual displacements enter on the 69 00:03:36,810 --> 00:03:40,600 right hand side, and these here are the corresponding 70 00:03:40,600 --> 00:03:42,400 virtual strains. 71 00:03:42,400 --> 00:03:47,462 The compatibility enters in here because we would, of 72 00:03:47,462 --> 00:03:50,730 course, use compatible virtual displacement, and we would 73 00:03:50,730 --> 00:03:52,930 calculate also the stresses from compatible displacements. 74 00:03:56,580 --> 00:04:01,410 The stress-strain law enters in the calculation of the 75 00:04:01,410 --> 00:04:04,910 stress here from the strains given. 76 00:04:04,910 --> 00:04:10,670 And once again we apply this at time t plus delta t. 77 00:04:10,670 --> 00:04:15,580 We also noted earlier that we will be using 78 00:04:15,580 --> 00:04:17,430 an incremental solution. 79 00:04:17,430 --> 00:04:21,640 Namely, that the solution at time t plus delta t will be 80 00:04:21,640 --> 00:04:26,300 calculated from the solution at time t by incrementing the 81 00:04:26,300 --> 00:04:28,630 displacements by ui. 82 00:04:28,630 --> 00:04:31,570 Notice that we will assume, of course, that 83 00:04:31,570 --> 00:04:33,370 the tui's are known. 84 00:04:33,370 --> 00:04:36,690 That the displacementts at time t are known, and that 85 00:04:36,690 --> 00:04:38,520 these are the unknown quantities that 86 00:04:38,520 --> 00:04:39,490 we're looking for. 87 00:04:39,490 --> 00:04:42,140 Of course, that gives us then the unknown quantities that we 88 00:04:42,140 --> 00:04:46,050 actually want to solve for. 89 00:04:46,050 --> 00:04:49,430 Our goal is, for the finite element solution to linearize 90 00:04:49,430 --> 00:04:53,530 the equation of the principle of virtual work, so as to 91 00:04:53,530 --> 00:04:58,370 obtain a set of equations, finite element equations, that 92 00:04:58,370 --> 00:04:59,390 read as follows. 93 00:04:59,390 --> 00:05:02,420 Here we have a stiffness matrix, a 94 00:05:02,420 --> 00:05:04,310 tangent stiffness matrix. 95 00:05:04,310 --> 00:05:08,630 Here we have an incremental displacement vector. 96 00:05:08,630 --> 00:05:12,790 And on the right hand side, we have a load vector, a vector 97 00:05:12,790 --> 00:05:15,590 of nodal point forces corresponding to the 98 00:05:15,590 --> 00:05:19,930 externally applied loads at time t plus delta t. 99 00:05:19,930 --> 00:05:23,300 And here we have a nodal point force vector that is 100 00:05:23,300 --> 00:05:26,060 equivalent to the current element stresses, in this 101 00:05:26,060 --> 00:05:30,150 particular instance at time t, as indicated by the 102 00:05:30,150 --> 00:05:32,430 superscript t here. 103 00:05:32,430 --> 00:05:34,916 Now this incremental displacement vector, of 104 00:05:34,916 --> 00:05:38,310 course, will be added to the displacement that we know 105 00:05:38,310 --> 00:05:41,170 already, and that correspond to time t. 106 00:05:41,170 --> 00:05:45,480 And that will give us then a first estimate for the 107 00:05:45,480 --> 00:05:49,400 solution, displacement solution, at time 108 00:05:49,400 --> 00:05:51,060 t plus delta t. 109 00:05:51,060 --> 00:05:53,780 Notice this is also summarized down here. 110 00:05:53,780 --> 00:05:57,550 Notice, however, that this vector here is only an 111 00:05:57,550 --> 00:06:01,240 approximation to the actual vector that we're looking for, 112 00:06:01,240 --> 00:06:05,260 because of the linearization process that leads us to this 113 00:06:05,260 --> 00:06:06,950 system of equations. 114 00:06:06,950 --> 00:06:10,600 The equation tku equals r minus f is, of course, 115 00:06:10,600 --> 00:06:14,700 applicable for a single element as well as a total 116 00:06:14,700 --> 00:06:16,150 assemblage of elements. 117 00:06:16,150 --> 00:06:20,670 In other words, this equation here, tk delta u1 equals t 118 00:06:20,670 --> 00:06:25,130 plus delta r minus tf, would be applicable to an element in 119 00:06:25,130 --> 00:06:28,710 which n then would mean just the degrees of freedom, the 120 00:06:28,710 --> 00:06:31,350 number of degrees of freedom of that element. 121 00:06:31,350 --> 00:06:34,510 And also for a total structure in which n then, of course 122 00:06:34,510 --> 00:06:37,910 means the total number of structural degrees of freedom. 123 00:06:37,910 --> 00:06:42,010 The tk matrix, the r vector, and the f vector would be 124 00:06:42,010 --> 00:06:44,780 constructed using the direct stiffness method the way we 125 00:06:44,780 --> 00:06:49,370 are used to it in the linear elastic analysis. 126 00:06:49,370 --> 00:06:51,060 In other words, there is nothing new here to be 127 00:06:51,060 --> 00:06:54,820 discussed really, as far as a construction of these matrices 128 00:06:54,820 --> 00:07:01,910 go, when we talk about a total element assemblage from the 129 00:07:01,910 --> 00:07:04,660 individual element matrices. 130 00:07:04,660 --> 00:07:06,850 However, an important point is that we cannot simply 131 00:07:06,850 --> 00:07:09,140 linearize the principle of virtual work when it is 132 00:07:09,140 --> 00:07:11,210 written in this form here. 133 00:07:11,210 --> 00:07:15,650 And the reason was, of course, that we cannot integrate over 134 00:07:15,650 --> 00:07:16,560 an unknown volume. 135 00:07:16,560 --> 00:07:21,680 The volume t plus delta t is unknown, or the volume at time 136 00:07:21,680 --> 00:07:24,890 t plus delta t is unknown, and we cannot directly increment 137 00:07:24,890 --> 00:07:29,150 the Cauchy stresses the way we discussed it already earlier. 138 00:07:29,150 --> 00:07:32,410 To linearize then, we need to choose a known reference 139 00:07:32,410 --> 00:07:33,700 configuration. 140 00:07:33,700 --> 00:07:37,630 And one known reference configuration that is a very 141 00:07:37,630 --> 00:07:39,850 natural one to use really, is the one 142 00:07:39,850 --> 00:07:41,710 corresponding to time zero. 143 00:07:41,710 --> 00:07:44,880 In this case, if we use that reference configuration, we 144 00:07:44,880 --> 00:07:47,830 talk about the total Lagrangian formulation. 145 00:07:47,830 --> 00:07:51,440 If we use as a reference configuration the 146 00:07:51,440 --> 00:07:54,560 configuration corresponding to time t, then we talk about the 147 00:07:54,560 --> 00:07:56,380 updated Lagrangian formulation. 148 00:07:56,380 --> 00:07:58,120 I'd like to make one point here. 149 00:07:58,120 --> 00:08:02,180 Namely, that of course we have calculated already from all 150 00:08:02,180 --> 00:08:05,430 the configurations from time zero to time t. 151 00:08:05,430 --> 00:08:08,590 So you may ask yourself, why is he not choosing as a 152 00:08:08,590 --> 00:08:12,350 reference configuration, for example, the configuration t 153 00:08:12,350 --> 00:08:13,740 minus delta t? 154 00:08:13,740 --> 00:08:17,040 In other words, one configuration prior to time t 155 00:08:17,040 --> 00:08:18,150 as a possibility. 156 00:08:18,150 --> 00:08:21,580 Well, it would be a possibility, but if you choose 157 00:08:21,580 --> 00:08:25,290 to do so, you would lose the advantages that you have in 158 00:08:25,290 --> 00:08:27,880 the updated Lagrangian formulation, and the 159 00:08:27,880 --> 00:08:30,370 advantages that you would have in the total Lagrangian 160 00:08:30,370 --> 00:08:31,580 formulation. 161 00:08:31,580 --> 00:08:33,140 Of course, there are differences here and we will 162 00:08:33,140 --> 00:08:36,140 talk about the advantages and disadvantages of either of 163 00:08:36,140 --> 00:08:37,870 these formulations. 164 00:08:37,870 --> 00:08:41,539 And you would be left with that choice of t minus delta t 165 00:08:41,539 --> 00:08:45,700 reference configuration, with basically all the 166 00:08:45,700 --> 00:08:47,170 disadvantages only. 167 00:08:47,170 --> 00:08:50,740 And therefore, we generally want to choose either this one 168 00:08:50,740 --> 00:08:56,130 or that one as the reference configuration. 169 00:08:56,130 --> 00:09:00,330 Let us now discusses the total Lagrangian formulation. 170 00:09:00,330 --> 00:09:03,190 In the total Lagrangian formulation, we use the 171 00:09:03,190 --> 00:09:07,470 reference configuration zero, and therefore we talk about 172 00:09:07,470 --> 00:09:08,230 this stress. 173 00:09:08,230 --> 00:09:11,130 We use this stress and that strain, this was a second 174 00:09:11,130 --> 00:09:13,610 Piola-Kirchhoff stress referred to the configuration 175 00:09:13,610 --> 00:09:14,630 at time zero. 176 00:09:14,630 --> 00:09:17,060 This is the Green-Lagrange strain referred to the 177 00:09:17,060 --> 00:09:19,140 configuration at time zero. 178 00:09:19,140 --> 00:09:23,240 The principle of virtual work once again, originally in this 179 00:09:23,240 --> 00:09:29,215 form, is now written in this form. 180 00:09:29,215 --> 00:09:32,700 We have had that already before on viewgraphs, and I 181 00:09:32,700 --> 00:09:34,990 don't think I need to go through those details again. 182 00:09:34,990 --> 00:09:37,280 But the important point is that this is the starting 183 00:09:37,280 --> 00:09:41,730 point of the total Lagrangian formulation. 184 00:09:41,730 --> 00:09:44,880 The formulation proceeds as follows. 185 00:09:44,880 --> 00:09:51,130 We know the solution at time t, therefore this stress, this 186 00:09:51,130 --> 00:09:54,090 displacement derivative, all the static and kinematic 187 00:09:54,090 --> 00:09:58,180 variables in fact, are known corresponding to the 188 00:09:58,180 --> 00:10:00,180 configuration at time t. 189 00:10:00,180 --> 00:10:04,210 And we can now decompose the stress corresponding to time t 190 00:10:04,210 --> 00:10:10,420 plus delta t into one known value and one unknown value. 191 00:10:10,420 --> 00:10:13,190 Similarly we proceed with the strain measures. 192 00:10:13,190 --> 00:10:16,110 The strain measures, the Green-Lagrange strain at time 193 00:10:16,110 --> 00:10:19,460 t plus delta t is decomposed into one value that we know 194 00:10:19,460 --> 00:10:21,620 already, and an increment. 195 00:10:21,620 --> 00:10:25,805 And, of course, remember these two increments are unknown. 196 00:10:25,805 --> 00:10:29,740 They are unknown, whereas these are known because we 197 00:10:29,740 --> 00:10:31,880 have calculated already the configuration 198 00:10:31,880 --> 00:10:35,150 corresponding to time t. 199 00:10:35,150 --> 00:10:39,720 In terms of displacements, we can directly develop that the 200 00:10:39,720 --> 00:10:45,350 Green-Lagrange strain at time t is written as shown here. 201 00:10:45,350 --> 00:10:49,120 We went over these different components already earlier. 202 00:10:49,120 --> 00:10:51,990 Notice once again that we have a product of displacement 203 00:10:51,990 --> 00:10:55,132 derivatives here, that this is, of 204 00:10:55,132 --> 00:10:57,310 course, a nonlinear term. 205 00:10:57,310 --> 00:11:00,160 And once again, as I pointed out in an earlier lecture, we 206 00:11:00,160 --> 00:11:02,130 have nothing neglected here. 207 00:11:02,130 --> 00:11:04,070 You see only quadratic terms. 208 00:11:04,070 --> 00:11:06,590 There are no cubic terms or higher-order terms. 209 00:11:06,590 --> 00:11:07,590 We have nothing neglected. 210 00:11:07,590 --> 00:11:11,030 This is a strain measure that holds for any amount of 211 00:11:11,030 --> 00:11:14,250 deformation, any amount of strain. 212 00:11:14,250 --> 00:11:17,870 Then we can also write this same strain, Green-Lagrange 213 00:11:17,870 --> 00:11:21,560 strain, corresponding to time t plus delta t, and this would 214 00:11:21,560 --> 00:11:23,290 be the result. 215 00:11:23,290 --> 00:11:27,880 Notice whatever we had here as t superscript has now become a 216 00:11:27,880 --> 00:11:30,470 t plus delta t superscript. 217 00:11:30,470 --> 00:11:33,510 That's in fact the only difference. 218 00:11:33,510 --> 00:11:37,410 If you subtract from this strain, that strain, we get 219 00:11:37,410 --> 00:11:39,900 the increment, and that increment is 220 00:11:39,900 --> 00:11:41,880 written out right here. 221 00:11:41,880 --> 00:11:44,590 Now, there are some interesting points regarding 222 00:11:44,590 --> 00:11:46,710 this incremental strain. 223 00:11:46,710 --> 00:11:52,420 This here is an increment in the strain that is linear in 224 00:11:52,420 --> 00:11:54,700 the displacement components. 225 00:11:54,700 --> 00:11:56,140 Similarly this one. 226 00:11:56,140 --> 00:12:01,010 Notice this one here, is, of course a product, product and 227 00:12:01,010 --> 00:12:04,010 may look, at first sight, as a nonlinear term. 228 00:12:04,010 --> 00:12:07,100 When you see products of displacements, you think of a 229 00:12:07,100 --> 00:12:07,840 nonlinearity. 230 00:12:07,840 --> 00:12:09,710 However, if you look closer, you find that 231 00:12:09,710 --> 00:12:11,250 there is a t here. 232 00:12:11,250 --> 00:12:13,790 Therefore, this part is known. 233 00:12:13,790 --> 00:12:19,150 Since this part is known, and this part is unknown, this 234 00:12:19,150 --> 00:12:23,100 total term is really linear in the incremental displacement. 235 00:12:23,100 --> 00:12:25,390 In fact, the incremental displacements don't even go in 236 00:12:25,390 --> 00:12:27,790 here, because we're differentiating here the 237 00:12:27,790 --> 00:12:31,220 displacements at time t with respect to the original 238 00:12:31,220 --> 00:12:32,930 coordinates. 239 00:12:32,930 --> 00:12:36,320 Therefore, this total term here is really 240 00:12:36,320 --> 00:12:38,300 still linear in ui. 241 00:12:38,300 --> 00:12:41,210 Linear in the incremental displacements. 242 00:12:41,210 --> 00:12:44,680 We call this part here, as given here, the initial 243 00:12:44,680 --> 00:12:45,900 displacement effect. 244 00:12:45,900 --> 00:12:48,360 The initial displacement effect because the initial 245 00:12:48,360 --> 00:12:52,490 displacements go into these derivatives here. 246 00:12:52,490 --> 00:12:55,390 We are differentiating the displacement at time t with 247 00:12:55,390 --> 00:12:59,060 respect to the original coordinates. 248 00:12:59,060 --> 00:13:02,440 This term, on the other hand here, is nonlinear in ui. 249 00:13:02,440 --> 00:13:06,130 It is nonlinear because here we're taking one term that 250 00:13:06,130 --> 00:13:09,880 depends on ui, and we multiply it by another term that 251 00:13:09,880 --> 00:13:11,140 depends on ui. 252 00:13:11,140 --> 00:13:16,850 Notice this a differentiation of uk, the incremental 253 00:13:16,850 --> 00:13:20,230 displacement, into the k direction with respect to the 254 00:13:20,230 --> 00:13:23,320 original coordinates. 255 00:13:23,320 --> 00:13:26,120 Coordinate axis xj here particular. 256 00:13:26,120 --> 00:13:29,590 So this is here, clearly a nonlinear term in ui. 257 00:13:29,590 --> 00:13:31,190 A quadratic term, as a matter of fact. 258 00:13:33,810 --> 00:13:38,490 We will look at this strain, of course, a little 259 00:13:38,490 --> 00:13:40,080 bit more just now. 260 00:13:40,080 --> 00:13:44,360 But keep in mind that there is one linear term that contains 261 00:13:44,360 --> 00:13:48,110 an initial displacement effect and a nonlinear term. 262 00:13:48,110 --> 00:13:49,960 I might add here that, of course, this initial 263 00:13:49,960 --> 00:13:53,560 displacement effect is zero if the initial 264 00:13:53,560 --> 00:13:54,580 displacements are zero. 265 00:13:54,580 --> 00:13:57,010 In other words, in our incremental solution, for 266 00:13:57,010 --> 00:14:00,870 example, if we are just starting the solution process, 267 00:14:00,870 --> 00:14:03,280 then the initial displacements are still zero. 268 00:14:03,280 --> 00:14:04,530 This term drops out. 269 00:14:07,350 --> 00:14:10,620 We note now regarding the formulation that the variation 270 00:14:10,620 --> 00:14:14,100 on the total Green-Lagrange strain tensor corresponding to 271 00:14:14,100 --> 00:14:18,080 time t plus delta t, is really equal to the variation in the 272 00:14:18,080 --> 00:14:20,020 increment of the Green-Lagrange strain tensor, 273 00:14:20,020 --> 00:14:24,170 from time t to time t plus delta t. 274 00:14:24,170 --> 00:14:28,200 In this picture we show what we mean by that. 275 00:14:28,200 --> 00:14:31,520 Here on the left hand side in black, you have the body in 276 00:14:31,520 --> 00:14:33,300 its original configuration. 277 00:14:33,300 --> 00:14:36,820 It moves to a configuration at time t. 278 00:14:36,820 --> 00:14:41,040 And then to a configuration, here shown in green, that 279 00:14:41,040 --> 00:14:43,490 corresponds to time t plus delta t. 280 00:14:43,490 --> 00:14:47,920 Notice here we have the displacement corresponding to 281 00:14:47,920 --> 00:14:51,590 time t, and then the displacement corresponding 282 00:14:51,590 --> 00:14:54,230 from time t to time t plus delta t. 283 00:14:54,230 --> 00:14:56,630 The total displacement, of course, here is the 284 00:14:56,630 --> 00:15:00,680 displacement from time zero to time t plus delta t. 285 00:15:00,680 --> 00:15:03,230 What we are doing, of course in the principle of virtual 286 00:15:03,230 --> 00:15:09,350 work, is to impose a variation in displacements about the 287 00:15:09,350 --> 00:15:11,700 configuration at time t plus delta t. 288 00:15:11,700 --> 00:15:17,490 And that variation is here indicated by the blue line. 289 00:15:17,490 --> 00:15:22,230 And if we impose this variation here, since the 290 00:15:22,230 --> 00:15:27,370 displacements at time t are constant, surely whatever 291 00:15:27,370 --> 00:15:30,610 these displacements were should not matter. 292 00:15:30,610 --> 00:15:33,810 And that is, in fact, expressed by this 293 00:15:33,810 --> 00:15:35,450 relationship here. 294 00:15:35,450 --> 00:15:39,270 If we vary here, since the displacements to ui are 295 00:15:39,270 --> 00:15:43,110 constant, they do not affect the variation on the strains. 296 00:15:45,790 --> 00:15:50,660 And therefore, we can simply vary the incremental strain, 297 00:15:50,660 --> 00:15:54,910 and we find that variation is indeed equal to the variation 298 00:15:54,910 --> 00:15:56,530 of the total strain. 299 00:15:56,530 --> 00:15:59,260 Of course, you can also prove that to yourself 300 00:15:59,260 --> 00:16:02,080 mathematically by just going through the arithmetic. 301 00:16:02,080 --> 00:16:06,200 In other words, applying the variation to this quantity, 302 00:16:06,200 --> 00:16:09,230 and then applying the variation to that quantity, 303 00:16:09,230 --> 00:16:11,560 which I've given you on the previous viewgraph, and you 304 00:16:11,560 --> 00:16:13,980 would find that indeed you get identically the same 305 00:16:13,980 --> 00:16:15,230 expressions. 306 00:16:16,960 --> 00:16:24,850 We can define a linear strain term, linear strain increment, 307 00:16:24,850 --> 00:16:32,290 as 0 eij, and the nonlinear strain increment as 0 eta ij. 308 00:16:32,290 --> 00:16:35,990 Notice that once again, here we have the quadratic terms. 309 00:16:35,990 --> 00:16:38,770 And, of course, when we add these two terms together, we 310 00:16:38,770 --> 00:16:42,820 get back our total strain increment from time t to time 311 00:16:42,820 --> 00:16:44,660 t plus delta t. 312 00:16:44,660 --> 00:16:47,820 If we take a variation on that increment, that is equivalent 313 00:16:47,820 --> 00:16:51,130 to taking a variation on the linear term and 314 00:16:51,130 --> 00:16:53,870 the nonlinear term. 315 00:16:53,870 --> 00:16:56,240 Notice that so far we have only talked 316 00:16:56,240 --> 00:16:58,550 about continuum mechanics. 317 00:16:58,550 --> 00:17:02,220 Of course we want to use the continuum mechanics principles 318 00:17:02,220 --> 00:17:04,720 that we discussed so far in the finite element 319 00:17:04,720 --> 00:17:05,810 discretization. 320 00:17:05,810 --> 00:17:10,200 But notice that these strain terms here that we talked 321 00:17:10,200 --> 00:17:13,109 about, are really continuum mechanics variables. 322 00:17:13,109 --> 00:17:15,589 There is an interesting observation. 323 00:17:15,589 --> 00:17:21,800 Namely, if we talk about this strain measure and that strain 324 00:17:21,800 --> 00:17:28,950 measure, rather the linear incremental strains and the 325 00:17:28,950 --> 00:17:34,360 nonlinear incremental strains, linear and nonlinear in ui, 326 00:17:34,360 --> 00:17:39,470 then if we apply these in finite element analysis, we 327 00:17:39,470 --> 00:17:43,470 have to remember that the displacements are interpolated 328 00:17:43,470 --> 00:17:45,910 in terms of nodal point variables. 329 00:17:45,910 --> 00:17:48,250 And what we're really looking for in finite element analysis 330 00:17:48,250 --> 00:17:51,630 are the nodal points variables. 331 00:17:51,630 --> 00:17:58,170 In isoparametric finite element analysis, which is the 332 00:17:58,170 --> 00:18:02,100 final element procedure that we will be using extensively, 333 00:18:02,100 --> 00:18:07,300 we interpolate the internal element displacements, tui, in 334 00:18:07,300 --> 00:18:10,460 terms of the nodal point displacements. 335 00:18:10,460 --> 00:18:13,120 Let me define a little bit more what this is here. 336 00:18:13,120 --> 00:18:16,940 This is, of course, the displacement from time zero to 337 00:18:16,940 --> 00:18:20,520 time t into the i's direction. 338 00:18:20,520 --> 00:18:24,370 We are summing here over all the finite element nodal 339 00:18:24,370 --> 00:18:27,310 points, the N nodal points. 340 00:18:27,310 --> 00:18:30,670 We have interpolation functions, hk, that we will 341 00:18:30,670 --> 00:18:32,700 talk more about later on. 342 00:18:32,700 --> 00:18:38,430 And we have here the nodal point displacement at time t 343 00:18:38,430 --> 00:18:41,780 into the i direction of the nodal point k. 344 00:18:41,780 --> 00:18:44,560 This k goes with that k. 345 00:18:44,560 --> 00:18:47,430 And we're summing over all the nodal points. 346 00:18:47,430 --> 00:18:51,150 Now notice that this is the linear relationship in 347 00:18:51,150 --> 00:18:53,380 isoparametric finite element analysis of 348 00:18:53,380 --> 00:18:55,930 solids, I should say. 349 00:18:55,930 --> 00:18:58,760 This is a linear relationship. 350 00:18:58,760 --> 00:19:03,610 And therefore, our linear strain terms that we talked 351 00:19:03,610 --> 00:19:06,670 earlier about, being linear in the incremental continuum 352 00:19:06,670 --> 00:19:12,210 mechanics displacements, are also linear in the incremental 353 00:19:12,210 --> 00:19:13,440 nodal point displacements. 354 00:19:13,440 --> 00:19:19,520 Similarly, this term here, which was nonlinear in the 355 00:19:19,520 --> 00:19:23,810 increments of displacements within the domain, from a 356 00:19:23,810 --> 00:19:26,800 continuum mechanics point of view, will also be nonlinear 357 00:19:26,800 --> 00:19:29,540 in the nodal point displacement increments. 358 00:19:29,540 --> 00:19:33,690 However, in the formulation of structural elements, with the 359 00:19:33,690 --> 00:19:36,230 incremental displacements, in fact also the total 360 00:19:36,230 --> 00:19:39,230 displacements, I interpolated using nodal point 361 00:19:39,230 --> 00:19:43,100 displacements and nodal point rotations. 362 00:19:43,100 --> 00:19:45,860 In other words, we also deal with nodal point rotations in 363 00:19:45,860 --> 00:19:50,640 order to calculate the total displacements of the element. 364 00:19:50,640 --> 00:19:55,710 And in that case, we have to recognize that the exact 365 00:19:55,710 --> 00:20:00,580 linear strain increment, the way we have defined it earlier 366 00:20:00,580 --> 00:20:06,200 linear in the incremental displacements, is still 367 00:20:06,200 --> 00:20:09,980 properly given as a linear strain increment in the nodal 368 00:20:09,980 --> 00:20:12,270 point variables. 369 00:20:12,270 --> 00:20:17,270 However, the term that we defined earlier as a nonlinear 370 00:20:17,270 --> 00:20:21,980 strain increment, this one, is not the full story of all the 371 00:20:21,980 --> 00:20:23,620 nonlinear strain increments. 372 00:20:23,620 --> 00:20:28,750 Because the rotations will put additional nonlinear strain 373 00:20:28,750 --> 00:20:31,420 increments into this term here. 374 00:20:31,420 --> 00:20:35,950 The reason being, that for large rotations, of course, we 375 00:20:35,950 --> 00:20:42,580 have cosine sine terms in these rotations and when we 376 00:20:42,580 --> 00:20:45,790 evaluate the total nonlinear strain term, we find that 377 00:20:45,790 --> 00:20:47,750 there are additional terms coming up here. 378 00:20:47,750 --> 00:20:50,840 Well, what is the effect of these two statements? 379 00:20:50,840 --> 00:20:56,900 It means that the right hand side force vector, or rather 380 00:20:56,900 --> 00:21:01,450 the vector t 0 f, the vector of nodal point forces 381 00:21:01,450 --> 00:21:06,920 corresponding to the internal elements stresses, is always 382 00:21:06,920 --> 00:21:09,420 correctly calculated. 383 00:21:09,420 --> 00:21:16,410 However, the stiffness matrix, t 0 k, for the formulation is 384 00:21:16,410 --> 00:21:20,000 only approximated because some of the nonlinear terms here 385 00:21:20,000 --> 00:21:24,400 have been dropped in the case of structural elements. 386 00:21:24,400 --> 00:21:27,390 Of course, you can also include the nonlinear terms. 387 00:21:27,390 --> 00:21:31,220 Then you would have to add these terms to this expression 388 00:21:31,220 --> 00:21:36,980 here, and then you would get a true tangent stiffness matrix. 389 00:21:36,980 --> 00:21:41,560 This means, of course, for an actual analysis, that when you 390 00:21:41,560 --> 00:21:47,500 form the equations, we have on the left hand side-- let me 391 00:21:47,500 --> 00:21:50,100 just put all of it in here-- 392 00:21:52,700 --> 00:21:57,720 for the first iteration, we talked about the fact that we 393 00:21:57,720 --> 00:22:00,810 have a stiffness matrix times an increment in displacements. 394 00:22:00,810 --> 00:22:03,480 I already put a 1 there because it will be only an 395 00:22:03,480 --> 00:22:06,540 approximation to the total incremental 396 00:22:06,540 --> 00:22:08,590 displacement vector. 397 00:22:08,590 --> 00:22:13,460 Well, we notice then that this matrix here is approximated if 398 00:22:13,460 --> 00:22:16,020 you were to drop these additional terms. 399 00:22:16,020 --> 00:22:20,950 But the right hand side vector f here is properly calculated. 400 00:22:20,950 --> 00:22:25,220 Therefore if you iterate, you always get the exact solution, 401 00:22:25,220 --> 00:22:26,350 the correct solution. 402 00:22:26,350 --> 00:22:30,530 But the convergence in the iteration might be a bit 403 00:22:30,530 --> 00:22:34,080 slower than what you could have obtained if you had 404 00:22:34,080 --> 00:22:37,510 included these additional terms here that I referred to. 405 00:22:37,510 --> 00:22:40,450 Let us now continue with the continuum mechanics 406 00:22:40,450 --> 00:22:41,900 formulation. 407 00:22:41,900 --> 00:22:45,630 The equation of the principle of virtual work becomes, if we 408 00:22:45,630 --> 00:22:49,810 substitute the quantities that I defined earlier, directly 409 00:22:49,810 --> 00:22:51,120 this equation. 410 00:22:51,120 --> 00:22:55,270 Notice here we have now the increment in stress times the 411 00:22:55,270 --> 00:22:59,280 variation of the incremental Green-Lagrange strain, from 412 00:22:59,280 --> 00:23:01,710 time t to t plus delta t. 413 00:23:01,710 --> 00:23:05,510 Of course, it's integrated over the original volume plus 414 00:23:05,510 --> 00:23:06,900 this integral here. 415 00:23:06,900 --> 00:23:12,190 Here we have the total stress, second Piola-Kirchhoff stress, 416 00:23:12,190 --> 00:23:16,690 corresponding to time t operating on the nonlinear 417 00:23:16,690 --> 00:23:20,430 strain increment of the total 418 00:23:20,430 --> 00:23:21,710 Green-Lagrange strain increment. 419 00:23:21,710 --> 00:23:24,600 And once again, we are integrating over the total 420 00:23:24,600 --> 00:23:27,220 volume corresponding to time zero. 421 00:23:27,220 --> 00:23:30,750 And on the right hand side, we have the external virtual work 422 00:23:30,750 --> 00:23:36,880 minus the stress at time t, operating on the linear strain 423 00:23:36,880 --> 00:23:39,880 increment corresponding to the total 424 00:23:39,880 --> 00:23:43,010 Green-Lagrange strain increment. 425 00:23:43,010 --> 00:23:47,290 We have so far made no approximations, and this is a 426 00:23:47,290 --> 00:23:52,140 relationship that has to hold for any arbitrary variation in 427 00:23:52,140 --> 00:23:56,950 displacements which go in here, or any arbitrary virtual 428 00:23:56,950 --> 00:23:59,140 displacement that go in here. 429 00:23:59,140 --> 00:24:03,040 And of course, corresponding strains that appear here. 430 00:24:05,840 --> 00:24:11,040 In essence, what we have been doing is that we have 431 00:24:11,040 --> 00:24:17,170 established a variation on the configuration at time 432 00:24:17,170 --> 00:24:19,790 t plus delta t. 433 00:24:19,790 --> 00:24:22,310 We have looked at the principle of virtual work 434 00:24:22,310 --> 00:24:25,400 corresponding to time t plus delta t, but we have 435 00:24:25,400 --> 00:24:29,210 introduced this increment in displacement. 436 00:24:29,210 --> 00:24:32,210 Notice that this increment in displacement is measured from 437 00:24:32,210 --> 00:24:33,760 the displacement at time t. 438 00:24:36,650 --> 00:24:40,200 All we have done therefore, is to rewrite the principle of 439 00:24:40,200 --> 00:24:45,010 virtual work in terms of tu and u instead of dealing with 440 00:24:45,010 --> 00:24:46,870 t plus delta tu. 441 00:24:46,870 --> 00:24:49,700 No approximation being done yet so far. 442 00:24:49,700 --> 00:24:51,760 Of course, we really haven't been talking about finite 443 00:24:51,760 --> 00:24:55,510 elements either yet, when we just look at this basic 444 00:24:55,510 --> 00:24:58,540 equation of principle of virtual work. 445 00:24:58,540 --> 00:25:02,210 It is this principle or the equation that we are dealing 446 00:25:02,210 --> 00:25:05,700 with, is in general, a very complicated function of the 447 00:25:05,700 --> 00:25:08,610 unknown displacement increment. 448 00:25:08,610 --> 00:25:12,490 We obtain an approximate relationship in finite element 449 00:25:12,490 --> 00:25:16,816 analysis, by linearizing the governing equation, and, of 450 00:25:16,816 --> 00:25:20,800 course, then in the finite element sense cast it into 451 00:25:20,800 --> 00:25:24,280 this set of linear equations. 452 00:25:24,280 --> 00:25:29,725 The process of linearizing is a very important process, 453 00:25:29,725 --> 00:25:31,880 interesting process for us to study, and that's what 454 00:25:31,880 --> 00:25:34,680 I want to do now. 455 00:25:34,680 --> 00:25:38,900 We begin to look, in this linearization at all the 456 00:25:38,900 --> 00:25:42,150 individual terms that appear in the integral. 457 00:25:42,150 --> 00:25:45,720 And let us look first at this term here. 458 00:25:45,720 --> 00:25:50,160 This term is actually already linear in ui. 459 00:25:50,160 --> 00:25:52,910 So there is nothing much to linearize. 460 00:25:52,910 --> 00:25:56,090 t 0 sij does not contain ui. 461 00:25:56,090 --> 00:25:57,460 It's a known quantity. 462 00:25:57,460 --> 00:25:59,290 It's a known stress value. 463 00:25:59,290 --> 00:26:02,560 And notice that when we take the variation on this 464 00:26:02,560 --> 00:26:06,670 nonlinear strain increment, we directly get this expression, 465 00:26:06,670 --> 00:26:12,640 where this part here is a variation on the 466 00:26:12,640 --> 00:26:14,350 displacement increment. 467 00:26:14,350 --> 00:26:15,990 Similarly here. 468 00:26:15,990 --> 00:26:19,090 And of course, these are going to be constants. 469 00:26:19,090 --> 00:26:22,310 Therefore, this term here is linear in the incremental 470 00:26:22,310 --> 00:26:23,640 displacements. 471 00:26:23,640 --> 00:26:24,740 Linear in ui. 472 00:26:24,740 --> 00:26:26,920 So nothing to linearize there. 473 00:26:26,920 --> 00:26:32,950 The term on this viewgraph here, contains an incremental 474 00:26:32,950 --> 00:26:36,430 stress and an incremental strain. 475 00:26:36,430 --> 00:26:39,170 Notice this incremental strain contains the linear and the 476 00:26:39,170 --> 00:26:40,940 nonlinear term. 477 00:26:40,940 --> 00:26:46,285 And this, of course, this product is a linear functional 478 00:26:46,285 --> 00:26:49,280 off ui, sorry it contains linear terms, 479 00:26:49,280 --> 00:26:52,330 nonlinear terms as well. 480 00:26:52,330 --> 00:26:56,530 This stress here is, in general, a nonlinear function 481 00:26:56,530 --> 00:26:58,950 of 0 epsilon ij. 482 00:26:58,950 --> 00:27:02,295 The variation that we are taking here means we are 483 00:27:02,295 --> 00:27:04,420 taking a variation on the linear part and 484 00:27:04,420 --> 00:27:06,080 the nonlinear part. 485 00:27:06,080 --> 00:27:08,660 We will see actually that this one here is constant. 486 00:27:08,660 --> 00:27:13,550 But this one carries also terms of ui, so this total 487 00:27:13,550 --> 00:27:16,090 expression is linear in ui. 488 00:27:16,090 --> 00:27:20,750 And if we multiply these terms here, we want to end up in 489 00:27:20,750 --> 00:27:24,360 just one expression that has neglected all higher-order 490 00:27:24,360 --> 00:27:28,040 terms in ui, but just contains ui. 491 00:27:28,040 --> 00:27:31,350 Well, let us do so. 492 00:27:31,350 --> 00:27:36,820 The objective is expressed once more here, and we 493 00:27:36,820 --> 00:27:41,440 recognize that, of course, the variation on 0 epsilon ij 494 00:27:41,440 --> 00:27:44,680 contains only constant and linear terms in ui, the way I 495 00:27:44,680 --> 00:27:45,930 just expressed it. 496 00:27:49,220 --> 00:27:54,530 0 sij can be written as a Taylor series expansion in 0 497 00:27:54,530 --> 00:27:56,510 epsilon ij. 498 00:27:56,510 --> 00:27:58,490 This is the general expression. 499 00:27:58,490 --> 00:28:02,400 Here we take the partial of the stress at time t, with 500 00:28:02,400 --> 00:28:06,220 respect to the Green-Lagrange strain at time t, here we are 501 00:28:06,220 --> 00:28:08,790 having this bar with the t that denotes that we are 502 00:28:08,790 --> 00:28:12,370 actually doing this evaluation at time t. 503 00:28:12,370 --> 00:28:16,020 And here we have an increment in this strain, and of course, 504 00:28:16,020 --> 00:28:17,610 there are higher-order terms. 505 00:28:17,610 --> 00:28:20,930 These higher-order terms we will neglect, and therefore we 506 00:28:20,930 --> 00:28:25,260 get directly for this stress here, this expression. 507 00:28:25,260 --> 00:28:29,190 Notice we already have substituted for 0 epsilon rs, 508 00:28:29,190 --> 00:28:31,880 these two quantities. 509 00:28:31,880 --> 00:28:36,810 This one here is quadratic in ui, this one is linear in ui. 510 00:28:36,810 --> 00:28:44,490 And if we linearize this term, we get directly this one here. 511 00:28:44,490 --> 00:28:48,360 Notice that this is the appropriate stress-strain law 512 00:28:48,360 --> 00:28:50,230 that has to be used corresponding 513 00:28:50,230 --> 00:28:51,790 to this stress measure. 514 00:28:51,790 --> 00:28:54,900 And we will later on talk more about how we evaluate this 515 00:28:54,900 --> 00:28:57,340 stress-strain law. 516 00:28:57,340 --> 00:29:01,600 As an example, we may look here at this schematic 517 00:29:01,600 --> 00:29:05,630 solution, computed solution, over a number of time steps. 518 00:29:05,630 --> 00:29:12,210 We're going time 1, 2, 3, 4, 5, up to t minus delta 519 00:29:12,210 --> 00:29:15,040 t and then to t. 520 00:29:15,040 --> 00:29:21,550 The stress-strain law relates basically the stress increment 521 00:29:21,550 --> 00:29:23,970 here to the strain increment. 522 00:29:23,970 --> 00:29:28,230 And notice it represents a tangent to this this curve. 523 00:29:28,230 --> 00:29:32,860 This green line here is a tangent to the red or the blue 524 00:29:32,860 --> 00:29:36,730 curve, the blue one now overlapping the red curve 525 00:29:36,730 --> 00:29:39,060 right there. 526 00:29:39,060 --> 00:29:43,310 So we have here the tangent, or the slope of this tangent 527 00:29:43,310 --> 00:29:47,460 is giving us the material tensor 0c. 528 00:29:47,460 --> 00:29:49,690 Of course, here we're talking just about one element. 529 00:29:49,690 --> 00:29:53,370 In general we have many elements corresponding to the 530 00:29:53,370 --> 00:29:57,560 strain and stress measures that we are dealing with. 531 00:29:57,560 --> 00:30:02,080 If we now substitute the result that we just looked at 532 00:30:02,080 --> 00:30:03,790 into the general equation-- 533 00:30:03,790 --> 00:30:06,130 sorry to take this one down-- 534 00:30:06,130 --> 00:30:10,460 into the general equation here, we see directly that 535 00:30:10,460 --> 00:30:15,370 this part here, of course, is the stress. 536 00:30:15,370 --> 00:30:20,290 And this part here gives us that part. 537 00:30:20,290 --> 00:30:24,710 We simply multiply out and look at the individual terms. 538 00:30:24,710 --> 00:30:28,530 We notice that this one does not contain ui. 539 00:30:28,530 --> 00:30:31,530 We notice that this one is linear in ui. 540 00:30:31,530 --> 00:30:35,130 So since this one is already linear in ui, this term will 541 00:30:35,130 --> 00:30:36,700 be quadratic in ui. 542 00:30:36,700 --> 00:30:39,220 We will have to drop it. 543 00:30:39,220 --> 00:30:44,260 This term here is linear in ui, but this one is constant, 544 00:30:44,260 --> 00:30:47,940 so this total term is linear in ui. 545 00:30:47,940 --> 00:30:51,820 And this is the one we keep, and which is then the 546 00:30:51,820 --> 00:30:54,260 linearized result. 547 00:30:54,260 --> 00:30:58,240 In other words, in summary once more, this total term 548 00:30:58,240 --> 00:31:04,490 here linearized is obtained via this expression here. 549 00:31:07,220 --> 00:31:11,200 The final linearized equation that we are then dealing with, 550 00:31:11,200 --> 00:31:14,330 now that we're substituting all these results into the 551 00:31:14,330 --> 00:31:19,880 original equation that we have developed, is as follows. 552 00:31:19,880 --> 00:31:23,800 Here we have one term that comes from the incremental 553 00:31:23,800 --> 00:31:29,550 stress times the variation on the total incremental 554 00:31:29,550 --> 00:31:30,650 Green-Lagrange strain. 555 00:31:30,650 --> 00:31:34,140 That all has reduced to this term. 556 00:31:34,140 --> 00:31:36,670 Here we did not have to linearize at all. 557 00:31:36,670 --> 00:31:40,620 We kept what we already had. 558 00:31:40,620 --> 00:31:46,520 And on the right hand side, did not linearize either, 559 00:31:46,520 --> 00:31:49,280 because this was a term that we had already there, and of 560 00:31:49,280 --> 00:31:52,430 course this is the external virtual work. 561 00:31:52,430 --> 00:31:57,660 Now it's interesting to note that these two terms here 562 00:31:57,660 --> 00:32:00,350 result into this expression, where this is the tangent 563 00:32:00,350 --> 00:32:01,660 stiffness matrix. 564 00:32:01,660 --> 00:32:05,420 Notice that tangent stiffness matrix contains the material 565 00:32:05,420 --> 00:32:07,945 tensor, as well as the current stresses. 566 00:32:11,590 --> 00:32:16,510 This incremental displacement here, vector, comes from this 567 00:32:16,510 --> 00:32:22,090 strain part and that strain part. 568 00:32:22,090 --> 00:32:26,400 Notice that this virtual displacement vector here comes 569 00:32:26,400 --> 00:32:31,246 from that strain part and also from that one here. 570 00:32:34,900 --> 00:32:38,270 Below here we have, of course, that the external virtual work 571 00:32:38,270 --> 00:32:44,410 results into a vector of nodal point forces times these 572 00:32:44,410 --> 00:32:46,190 virtual displacements. 573 00:32:46,190 --> 00:32:52,150 And this one here results into the force vector corresponding 574 00:32:52,150 --> 00:32:54,850 to the internal element stresses. 575 00:32:54,850 --> 00:32:59,130 And this is, of course, a very important quantity that we 576 00:32:59,130 --> 00:33:02,070 have to calculate accurately, as I pointed out earlier. 577 00:33:02,070 --> 00:33:07,040 Because we want ultimately in the iteration that this vector 578 00:33:07,040 --> 00:33:10,250 is equilibriating that vector, and if we do make a mistake in 579 00:33:10,250 --> 00:33:13,710 calculating this vector, then we might have converged, but 580 00:33:13,710 --> 00:33:15,280 we have converged to the wrong solution. 581 00:33:15,280 --> 00:33:18,150 So it's very important to recognize that this vector 582 00:33:18,150 --> 00:33:21,590 must be accurately calculated, by all means. 583 00:33:21,590 --> 00:33:26,740 However, the k matrix here, if we go back once more to the 584 00:33:26,740 --> 00:33:30,400 discussion of this matrix, the k matrix here is a tangent 585 00:33:30,400 --> 00:33:31,980 stiffness matrix. 586 00:33:31,980 --> 00:33:37,370 And this tangent stiffness matrix is selected such as to 587 00:33:37,370 --> 00:33:40,680 obtain, of course, in the incremental solution, an 588 00:33:40,680 --> 00:33:43,670 appropriate incremental displacement. 589 00:33:43,670 --> 00:33:47,640 We will see later on that this matrix is 590 00:33:47,640 --> 00:33:50,170 updated in the iteration. 591 00:33:50,170 --> 00:33:53,530 It depends on what kind of intuitive scheme we are using, 592 00:33:53,530 --> 00:33:55,960 depending on that scheme we are updating this matrix 593 00:33:55,960 --> 00:33:57,750 differently. 594 00:33:57,750 --> 00:34:01,680 In any case, it's a matrix that we will in quotes, "play 595 00:34:01,680 --> 00:34:05,550 around with", in order to accelerate the convergence. 596 00:34:05,550 --> 00:34:09,570 Therefore there is no unique matrix that can be used here, 597 00:34:09,570 --> 00:34:11,940 that should be used here. 598 00:34:11,940 --> 00:34:14,540 There are different possibilities that we will 599 00:34:14,540 --> 00:34:16,170 discuss in later lectures. 600 00:34:16,170 --> 00:34:20,159 But on the right hand side once again, this F vector is 601 00:34:20,159 --> 00:34:24,800 the F vector obtained from the current stresses, and that one 602 00:34:24,800 --> 00:34:28,880 has to be calculated uniquely in the correct way. 603 00:34:28,880 --> 00:34:34,100 Well, an important point is that this relationship here on 604 00:34:34,100 --> 00:34:37,969 the left hand side, is equal to the relationship given here 605 00:34:37,969 --> 00:34:39,429 on the right hand side. 606 00:34:39,429 --> 00:34:42,620 And this holds because this term here is 607 00:34:42,620 --> 00:34:45,030 equal to that term. 608 00:34:45,030 --> 00:34:48,830 We interpret, in fact, this right hand side as an out of 609 00:34:48,830 --> 00:34:51,440 balance virtual work term. 610 00:34:51,440 --> 00:34:54,619 Let's look once why this holds, because this might be a 611 00:34:54,619 --> 00:34:55,770 bit of a surprise. 612 00:34:55,770 --> 00:34:59,740 The mathematical explanation is given on this viewgraph. 613 00:34:59,740 --> 00:35:02,510 We had earlier already that the variation on the total 614 00:35:02,510 --> 00:35:04,840 Green-Lagrange strain is nothing else than the 615 00:35:04,840 --> 00:35:09,930 variation on the incremental total Green-Lagrange strain. 616 00:35:09,930 --> 00:35:15,610 Now if we look down here, we recognize that this variation 617 00:35:15,610 --> 00:35:20,940 here at ui equal to 0, is nothing else than the 618 00:35:20,940 --> 00:35:23,430 variation on the Green-Lagrange 619 00:35:23,430 --> 00:35:26,630 strain at time t. 620 00:35:26,630 --> 00:35:30,930 If ui is zero, then this term here is nothing else than the 621 00:35:30,930 --> 00:35:32,640 Green-Lagrange strain at time t. 622 00:35:32,640 --> 00:35:35,410 So we're looking here at the variation of the 623 00:35:35,410 --> 00:35:40,420 Green-Lagrange at time, t if I put ui equal to 0. 624 00:35:40,420 --> 00:35:43,190 Now let's look at these expressions here. 625 00:35:43,190 --> 00:35:50,840 If I evaluate this term here, at ui equal to 0, we find that 626 00:35:50,840 --> 00:35:55,010 this term here, being a constant, is recovered. 627 00:35:55,010 --> 00:35:58,540 But notice that this term here, with ui equal to 0, 628 00:35:58,540 --> 00:36:00,560 simply turns out to be 0. 629 00:36:00,560 --> 00:36:06,440 Therefore, this term is equal to that term. 630 00:36:06,440 --> 00:36:10,660 And this means, therefore, that the variation on t zero 631 00:36:10,660 --> 00:36:15,940 epsilon ij is equal to the variation on 0 eij, which we 632 00:36:15,940 --> 00:36:17,950 wanted to prove, of course. 633 00:36:17,950 --> 00:36:24,190 So this is the reason why this term here is equal to that 634 00:36:24,190 --> 00:36:28,700 term, as we have used it on the previous viewgraph. 635 00:36:28,700 --> 00:36:33,190 This result also makes physical sense, because if we 636 00:36:33,190 --> 00:36:38,350 look at the governing equation, we find that this 637 00:36:38,350 --> 00:36:42,140 term here, replaced, is nothing else than tr. 638 00:36:42,140 --> 00:36:44,090 Should be tr. 639 00:36:44,090 --> 00:36:46,820 And of course, the script r the external virtual 640 00:36:46,820 --> 00:36:49,030 work at time t. 641 00:36:49,030 --> 00:36:52,280 And this, of course, is here the out of balance load term. 642 00:36:52,280 --> 00:36:57,930 Now if, for example, the material is elastic, and the 643 00:36:57,930 --> 00:37:02,610 external virtual work has not changed, then clearly the 644 00:37:02,610 --> 00:37:04,830 displacements would not change. 645 00:37:04,830 --> 00:37:09,830 In other words, t plus delta tui would be equal to tui, and 646 00:37:09,830 --> 00:37:12,190 the incremental displacements would be 0. 647 00:37:12,190 --> 00:37:14,660 Therefore, equilibrium would be satisfied. 648 00:37:14,660 --> 00:37:18,820 Hence, this term, which we have here, must actually be 649 00:37:18,820 --> 00:37:25,160 equal to t 0 sij times variation t 0 epsilon ij, the 650 00:37:25,160 --> 00:37:28,570 way we discussed it just now. 651 00:37:28,570 --> 00:37:32,240 We may rewrite the linearized governing equation as given on 652 00:37:32,240 --> 00:37:33,520 this viewgraph. 653 00:37:33,520 --> 00:37:36,650 And I am rewriting it in this form because we anticipate 654 00:37:36,650 --> 00:37:38,990 that we need an iteration. 655 00:37:38,990 --> 00:37:43,140 We simply introduce here, an iteration counter 1, with the 656 00:37:43,140 --> 00:37:48,680 delta in front of the incremental strain value. 657 00:37:48,680 --> 00:37:52,520 And similarly we introduce a delta here, and an iteration 658 00:37:52,520 --> 00:37:54,520 counter 1 there. 659 00:37:54,520 --> 00:37:57,880 And on the right hand side, we introduce also an iteration 660 00:37:57,880 --> 00:38:00,090 counter, but one lower. 661 00:38:00,090 --> 00:38:03,510 In other words, 0 here, whereas we have a 1 there. 662 00:38:03,510 --> 00:38:05,610 Notice 0 here, a 1 here. 663 00:38:05,610 --> 00:38:09,900 And notice furthermore, that this term is nothing else than 664 00:38:09,900 --> 00:38:13,550 that term, and this term here is nothing else than the 665 00:38:13,550 --> 00:38:17,450 variation on t 0 epsilon ij. 666 00:38:17,450 --> 00:38:20,840 This would be the equation that we have developed 667 00:38:20,840 --> 00:38:23,820 already, just having introduced now a different 668 00:38:23,820 --> 00:38:27,920 notation which leads us towards what we want to deal 669 00:38:27,920 --> 00:38:31,450 with in the iteration. 670 00:38:31,450 --> 00:38:35,860 The governing equations from a finite element point of view, 671 00:38:35,860 --> 00:38:38,300 then would be as shown here. 672 00:38:38,300 --> 00:38:42,880 Notice tangent stiffness matrix, del u1, iteration 673 00:38:42,880 --> 00:38:46,810 counter, the external loads, which we still assume to be 674 00:38:46,810 --> 00:38:48,440 constant, by the way. 675 00:38:48,440 --> 00:38:51,680 We still assume deformation independent loading if we have 676 00:38:51,680 --> 00:38:54,170 deformation dependent loading, these would also 677 00:38:54,170 --> 00:38:56,440 change in the iteration. 678 00:38:56,440 --> 00:39:01,140 And here we have the vector corresponding to the internal 679 00:39:01,140 --> 00:39:03,540 element stresses at time t. 680 00:39:03,540 --> 00:39:08,260 We write that vector as t plus delta t 0, 0 there. 681 00:39:08,260 --> 00:39:11,280 And that means that we are really talking, at this point, 682 00:39:11,280 --> 00:39:12,590 about this vector. 683 00:39:12,590 --> 00:39:16,510 We calculate this incremental displacement vector and add it 684 00:39:16,510 --> 00:39:19,430 to the previous displacements, which, of course, in this 685 00:39:19,430 --> 00:39:22,760 particular instance, in the first iteration, are nothing 686 00:39:22,760 --> 00:39:23,690 else than tu. 687 00:39:23,690 --> 00:39:29,750 We add those, and obtain our first estimates of 688 00:39:29,750 --> 00:39:32,650 displacements, namely the estimate at the end of 689 00:39:32,650 --> 00:39:35,050 iteration 1. 690 00:39:35,050 --> 00:39:39,170 Having obtained now this estimate on displacements, we 691 00:39:39,170 --> 00:39:41,890 can repeat the process. 692 00:39:41,890 --> 00:39:45,860 And that is shown here on the next viewgraph. 693 00:39:45,860 --> 00:39:50,470 Notice that now I am talking about the right hand side, 694 00:39:50,470 --> 00:39:53,790 about the Green-Lagrange strains corresponding to time 695 00:39:53,790 --> 00:39:57,920 t plus delta t, and the first iteration. 696 00:39:57,920 --> 00:40:00,150 In other words, these are the Green-Lagrange strains 697 00:40:00,150 --> 00:40:02,990 corresponding to the end of the first iteration. 698 00:40:02,990 --> 00:40:05,740 They include now the incremental displacements that 699 00:40:05,740 --> 00:40:07,780 we just calculated. 700 00:40:07,780 --> 00:40:10,500 Similarly, these are the stresses corresponding to the 701 00:40:10,500 --> 00:40:13,680 end of the first iteration. 702 00:40:13,680 --> 00:40:18,610 And this is an out of balance virtual work term. 703 00:40:18,610 --> 00:40:23,130 If that is non-zero, we will have further increment in 704 00:40:23,130 --> 00:40:23,880 displacements. 705 00:40:23,880 --> 00:40:27,080 And that increment results, of course, into an 706 00:40:27,080 --> 00:40:29,590 increment of strains. 707 00:40:29,590 --> 00:40:31,230 Namely, the second increment. 708 00:40:31,230 --> 00:40:34,380 Here is the 2 that denotes the increment. 709 00:40:34,380 --> 00:40:38,080 Here is the 2 that denotes the increment. 710 00:40:38,080 --> 00:40:41,950 The finite element equations then would look as shown here. 711 00:40:41,950 --> 00:40:44,510 Same equation as before, but now with a 1 712 00:40:44,510 --> 00:40:47,100 here and a 2 there. 713 00:40:47,100 --> 00:40:50,510 We calculate the incremental displacements, we add them to 714 00:40:50,510 --> 00:40:53,630 what we had already and get a better approximation to the 715 00:40:53,630 --> 00:40:58,150 displacements corresponding to time t plus delta t. 716 00:40:58,150 --> 00:41:01,980 This process, of course, can now be repeated, and on this 717 00:41:01,980 --> 00:41:05,120 viewgraph we show how it's being done for 718 00:41:05,120 --> 00:41:06,840 every iteration k. 719 00:41:06,840 --> 00:41:10,120 In a particular iteration, we have now on the right hand 720 00:41:10,120 --> 00:41:13,440 side the external virtual work corresponding to time t plus 721 00:41:13,440 --> 00:41:17,740 delta t, the stresses corresponding to time t plus 722 00:41:17,740 --> 00:41:20,580 delta t and iteration k minus 1. 723 00:41:20,580 --> 00:41:23,770 At the end of iteration k minus 1, the strains 724 00:41:23,770 --> 00:41:27,200 corresponding to time t plus delta t, and at the end of 725 00:41:27,200 --> 00:41:29,590 iteration k minus 1. 726 00:41:29,590 --> 00:41:32,680 This is the out of balance virtual work term, and on the 727 00:41:32,680 --> 00:41:35,900 left hand side we have the same quantities as before, but 728 00:41:35,900 --> 00:41:39,310 now with the iteration counter k. 729 00:41:39,310 --> 00:41:43,160 When discretized, we obtain these equations here. 730 00:41:43,160 --> 00:41:49,690 Delta uk, the increment in the iteration k of displacements, 731 00:41:49,690 --> 00:41:53,420 r minus f, computed, of course, from the current 732 00:41:53,420 --> 00:41:54,630 displacements. 733 00:41:54,630 --> 00:41:58,450 From the current displacements we calculate that vector here. 734 00:41:58,450 --> 00:42:04,170 Notice this vector is obtained from this integral here. 735 00:42:04,170 --> 00:42:09,880 We repeat this process for iteration k 1, 2, 3 et cetera 736 00:42:09,880 --> 00:42:10,870 until convergence. 737 00:42:10,870 --> 00:42:12,740 We will have to talk about how we measure 738 00:42:12,740 --> 00:42:14,360 convergence and so on. 739 00:42:14,360 --> 00:42:17,210 But in this iteration we're adding up the total 740 00:42:17,210 --> 00:42:21,330 incremental displacements to obtain always a better 741 00:42:21,330 --> 00:42:25,170 estimate for the displacements corresponding to time t plus 742 00:42:25,170 --> 00:42:28,790 delta t, and here, of course, iteration k. 743 00:42:28,790 --> 00:42:32,780 Notice that the counter here goes from j equals 1 to k, 744 00:42:32,780 --> 00:42:36,820 that is this j and this k, of course, is nothing else than 745 00:42:36,820 --> 00:42:40,260 the k that we have on the left hand side here. 746 00:42:40,260 --> 00:42:44,890 Notice once again that the first iteration, when k is 747 00:42:44,890 --> 00:42:48,140 equal to 1, it amounts to nothing else than what we 748 00:42:48,140 --> 00:42:51,740 discussed already at the very beginning a little earlier in 749 00:42:51,740 --> 00:42:53,290 the lecture. 750 00:42:53,290 --> 00:42:58,520 We simply generalized that discussion now to an equation 751 00:42:58,520 --> 00:43:02,460 in which we iterate and always try to calculate another 752 00:43:02,460 --> 00:43:04,850 increment in displacements until the right 753 00:43:04,850 --> 00:43:05,950 hand side is 0. 754 00:43:05,950 --> 00:43:09,490 Notice when the right hand side is 0, the external 755 00:43:09,490 --> 00:43:15,440 virtual work is equilibrated by the internal virtual work, 756 00:43:15,440 --> 00:43:17,510 and of course, that is what we want to reach. 757 00:43:17,510 --> 00:43:21,730 That is what our criterion is for the analysis. 758 00:43:21,730 --> 00:43:24,660 The external virtual work must be equilibrated by the 759 00:43:24,660 --> 00:43:25,910 internal virtual work. 760 00:43:28,120 --> 00:43:30,780 In the finite element discretization, the whole 761 00:43:30,780 --> 00:43:34,900 process is as summarized on this viewgraph. 762 00:43:34,900 --> 00:43:40,470 Initially, we are given tu, the displacements 763 00:43:40,470 --> 00:43:44,270 corresponding to time t, and the externally applied loads 764 00:43:44,270 --> 00:43:46,380 corresponding to time t plus delta t. 765 00:43:46,380 --> 00:43:50,440 We compute the stiffness matrix, the tangent stiffness 766 00:43:50,440 --> 00:43:52,440 matrix corresponding to time t. 767 00:43:52,440 --> 00:43:56,450 We calculate this vector, the nodal point forces 768 00:43:56,450 --> 00:43:59,990 corresponding to the stresses at time t. 769 00:43:59,990 --> 00:44:03,010 Notice this 0 here refers always to the total Lagrangian 770 00:44:03,010 --> 00:44:05,310 formulation. 771 00:44:05,310 --> 00:44:09,400 Notice that this vector here is really the initial 772 00:44:09,400 --> 00:44:12,140 condition for the iteration. 773 00:44:12,140 --> 00:44:16,170 Similar, the displacements at time t are the initial 774 00:44:16,170 --> 00:44:18,900 conditions for the iteration for the displacements. 775 00:44:18,900 --> 00:44:19,960 Zero here. 776 00:44:19,960 --> 00:44:21,440 Zero there, meaning the initial 777 00:44:21,440 --> 00:44:22,930 conditions for our iteration. 778 00:44:22,930 --> 00:44:26,840 We set the iteration counter, k equal to 1, and we now go 779 00:44:26,840 --> 00:44:28,800 into the following loop. 780 00:44:28,800 --> 00:44:32,640 Notice on the right hand side, an out of balance load vector 781 00:44:32,640 --> 00:44:34,190 is calculated. 782 00:44:34,190 --> 00:44:37,360 And that out of balance load vector gives us an incremental 783 00:44:37,360 --> 00:44:39,630 displacement vector. 784 00:44:39,630 --> 00:44:41,680 Of course, the tangent stiffness matrix is involved 785 00:44:41,680 --> 00:44:43,600 in that calculation. 786 00:44:43,600 --> 00:44:48,100 Notice that this incremental displacement vector is added 787 00:44:48,100 --> 00:44:51,720 to the displacements that we had previously calculated, to 788 00:44:51,720 --> 00:44:54,570 obtain a new estimate on the displacements. 789 00:44:54,570 --> 00:45:01,100 And like that, we have now obtained a displacement vector 790 00:45:01,100 --> 00:45:05,350 that, of course, hopefully is closer to satisfying certain 791 00:45:05,350 --> 00:45:06,770 convergence criteria. 792 00:45:06,770 --> 00:45:08,860 We will have to talk about these convergence 793 00:45:08,860 --> 00:45:10,150 criteria later on. 794 00:45:10,150 --> 00:45:13,370 Basically, we check whether equilibrium is satisfied 795 00:45:13,370 --> 00:45:15,480 within a certain tolerance. 796 00:45:15,480 --> 00:45:19,190 If the equilibrium is not satisfied, we go here. 797 00:45:19,190 --> 00:45:22,810 We use these displacements that we just have calculated 798 00:45:22,810 --> 00:45:27,960 to compute a new vector of nodal point forces 799 00:45:27,960 --> 00:45:30,450 corresponding to the element stresses. 800 00:45:30,450 --> 00:45:36,070 Notice this vector now contains the information that 801 00:45:36,070 --> 00:45:40,930 we have just calculated in solving this equation here. 802 00:45:40,930 --> 00:45:43,370 This was information that we obtained. 803 00:45:43,370 --> 00:45:47,980 And that information is contained in this vector, and 804 00:45:47,980 --> 00:45:51,645 that vector is used right here to calculate this nodal point 805 00:45:51,645 --> 00:45:54,190 force vector. 806 00:45:54,190 --> 00:45:57,005 Having calculated that, we increase our iteration 807 00:45:57,005 --> 00:46:00,910 counter, come back here and like that we continue the 808 00:46:00,910 --> 00:46:04,930 iteration, looping around through here 809 00:46:04,930 --> 00:46:07,000 as I have just described. 810 00:46:07,000 --> 00:46:09,830 Notice at convergence once again, we want, of course, 811 00:46:09,830 --> 00:46:16,150 that r is equal to f, meaning that the external loads are 812 00:46:16,150 --> 00:46:19,230 equilibrated by the nodal point forces corresponding to 813 00:46:19,230 --> 00:46:21,530 the internal element stresses. 814 00:46:21,530 --> 00:46:26,200 Well, this brings us to the end of this lecture. 815 00:46:26,200 --> 00:46:30,140 You might recognize that this last viewgraph really contains 816 00:46:30,140 --> 00:46:34,440 a lot of information that, of course, we discussed in this 817 00:46:34,440 --> 00:46:38,960 lecture, but that also I referred to already in the 818 00:46:38,960 --> 00:46:40,200 first lecture. 819 00:46:40,200 --> 00:46:44,880 At that time, I tried to introduce you to this 820 00:46:44,880 --> 00:46:49,680 iteration process using rather physical concepts, not going 821 00:46:49,680 --> 00:46:52,480 through a lengthy mathematical derivation. 822 00:46:52,480 --> 00:46:55,470 We did not talk about continuum mechanics variables. 823 00:46:55,470 --> 00:46:58,790 We did not talk about total Lagrangian formulation, second 824 00:46:58,790 --> 00:47:00,430 Piola-Kirchhoff stresses, Green-Lagrange 825 00:47:00,430 --> 00:47:01,990 strains and so on. 826 00:47:01,990 --> 00:47:05,020 I hoped at that time to give you just a physical feel of 827 00:47:05,020 --> 00:47:06,680 how we are iterating. 828 00:47:06,680 --> 00:47:10,610 The same kind of equations, the same equations we looked 829 00:47:10,610 --> 00:47:14,290 at at that time, now I hope with this lecture, I have 830 00:47:14,290 --> 00:47:19,140 given you the mathematical basis of these equations. 831 00:47:19,140 --> 00:47:23,140 I hope you have learned how to derive these equations, how to 832 00:47:23,140 --> 00:47:25,370 work with these equations a little bit. 833 00:47:25,370 --> 00:47:29,130 And in some of the next lectures, we will look at how 834 00:47:29,130 --> 00:47:32,110 actually we do construct this k matrix 835 00:47:32,110 --> 00:47:33,930 for different elements. 836 00:47:33,930 --> 00:47:37,430 That is, of course, a very important consideration. 837 00:47:37,430 --> 00:47:43,130 How do we calculate this k matrix for 1D, 2D, 3D 838 00:47:43,130 --> 00:47:45,890 elements, shell elements, beam elements? 839 00:47:45,890 --> 00:47:48,120 How do we calculate this force vector for 840 00:47:48,120 --> 00:47:49,400 these different elements? 841 00:47:49,400 --> 00:47:51,510 We will have to discuss that as well. 842 00:47:51,510 --> 00:47:55,310 Notice that in this k matrix goes the material law. 843 00:47:55,310 --> 00:47:58,560 Whether we are dealing with an elasto-plastic material, 844 00:47:58,560 --> 00:48:02,220 rubber type material, all that will affect the actual 845 00:48:02,220 --> 00:48:03,950 ingredients, the actual elements 846 00:48:03,950 --> 00:48:05,690 here in that k matrix. 847 00:48:05,690 --> 00:48:11,820 We we also have to discuss how we can possibly make this 848 00:48:11,820 --> 00:48:14,340 convergence that I'm talking about fast. 849 00:48:14,340 --> 00:48:16,570 In other words, how can we accelerate the convergence of 850 00:48:16,570 --> 00:48:17,910 these iterations? 851 00:48:17,910 --> 00:48:22,690 So in essence, I like to just convey to you that what you 852 00:48:22,690 --> 00:48:26,160 see here you will see a number of times again. 853 00:48:26,160 --> 00:48:29,630 We will talk about the different parts that you have 854 00:48:29,630 --> 00:48:33,130 been looking at here already briefly once again in the 855 00:48:33,130 --> 00:48:34,260 upcoming lectures. 856 00:48:34,260 --> 00:48:35,820 Thank you very much for your attention.