 In this lecture and for the remaining part of this course we will discuss some issues related to development of non-linear financial models, so we will begin in today's lecture a brief introduction of the what are the main issues and we will start reviewing principles of continuum mechanics as much as is needed for purpose of illustrating the model development. So first we can ask ourselves the question what are non-linear systems, so a simple answer would be to begin by defining what is linear system and a negation of that definition would tell us what is non-linear system. So if you have a system with input X1 producing output Y1 and X2 of T producing output Y2 of T, if you were to apply X1 and X2 together the response would be some of Y1 and Y2 and this is known as additivity property. Similarly if we apply an input which is a scalar multiple A of X1 of T the response will be A of Y1 of T, so if these two properties are satisfied we say that the system is linear, a non-linear system is a system that is not linear, so one of these conditions would not be satisfied, so this is a non-specific description it does not tell you what exactly is the nature of non-linearity it simply tells that linear property is not obeyed. Now some simple examples, suppose if you consider a system whose input is X and output is Y and they are related through this equation Y equal to MX plus C it's a scalar equation, so if X1 is the input let Y1 be the output, now if I multiply the input by a factor A we say that we see that this is not equal to the principle of superposition is not valid here, suppose Y1 is response to AX1 and if we consider the response AY which is not equal to Y1, so the system is not linear since it does not obey the scaling property, so a lesson from this is that 0 input produces 0 output in a linear system, so this is not satisfied in this case. Now to fix the idea we can consider a simple single degree linear vibrating system with input X of T and output Y of T, so let us assume that system starts from rest, suppose Y1 is a response to X1 and Y2 is a response to X2, if we now add these two equations we will see that the output gets added up X1 plus X2, so by inspection we see that Y1 plus Y2 is a solution of this equation therefore principle of first condition is satisfied. If I multiply this equation input by A we can again by inspection we see that AY1 is a solution, so this system is linear, now if you now include a cubic nonlinear term alpha Y cube let X for under X1 let Y1 be the response and under X2 let Y2 be the response, let us assume for the purpose of illustration that we have 0 initial conditions. Now again as before if I add the inputs I mean add these two equations I see that I get this equation whereas if I now apply an excitation X2 plus X3 that is X1 plus X2 if our response is denoted by Y3 we see that Y3 is not equal to Y1 plus Y2 therefore system is nonlinear. Now we can as an exercise you can examine the two examples that is a single degree linear system and single degree nonlinear system by including the effect of nonzero initial condition so you can see what happens, another system X of T is the input and Y of T is the output so if I now supply an input which is a scalar multiple of X of T say A of X of T we see that Y1 of T will be A of Y of T that means scaling property is satisfied but additivity property is not satisfied therefore the system is still nonlinear, it is not linear although one of the property is satisfied. Now we can start discussing about what is the implication of system being nonlinear on response analysis, so response to all the relevant loads need to be analyzed simultaneously so you cannot analyze response to individual loads and superpose the response because principle of superposition is not valid. Now I will illustrate with some simple cases but I will just enunciate some of the properties undamped in undamped free vibration frequency of oscillations depend on initial conditions which is again an unusual feature if you are thinking this property for linear systems. Harmonic inputs at frequency omega can produce harmonic responses at frequency is not equal to omega, it can produce nonharmonic responses and it can also produce a periodic responses so we can think of primary subharmonic and superharmonic resonances whereas in a linear single degree freedom system there is one natural frequency and one resonant frequency whereas for a single degree freedom nonlinear system under a single frequency excitation there can be several resonances. Then reciprocity relations which are one of the major features of linear system are not valid for nonlinear system so this property helps us to verify in a laboratory for example if a system is behaving linearly or not then so large responses can occur at frequencies other than the driving frequency this again refers to the subharmonic and superharmonic resonances that I mentioned. Then steady state responses depend on initial conditions whereas in a linear systems under harmonic excitations steady state when they exist they are independent of initial conditions whereas this is not true here then there can be multiplicity of steady state solutions for nonlinear system if you simply consider an algebraic quadratic equation AX square plus BX plus C equal to 0 even for such simple algebraic equation we already know there are two solutions. So if it is a differential equation with nonlinear terms one can still expect that there will be multiplicity of solutions that indeed happens then system can process multiple equilibrium states and display a wide range of bifurcations this we have discussed when we talked about stability problems. Then concept of normal modes natural frequencies and natural coordinates are no longer applicable for nonlinear systems. So you cannot uncouple the equations of motion through a transformation then band limited excitation can produce responses with frequency content outside the bandwidth of excitation. So this again is a very important aspect of nonlinear system behavior both in response analysis as well as in experimental work. So in fact a narrow band for example a single frequency harmonic excitation can produce a very broad band response and that type of behavior is known as chaos we will not be touching upon that I am just mentioning for sake of completeness. Now some simple analytical solutions to illustrate some of the features this is not an exhaustive coverage but just to motivate you to special features that you should expect when you are dealing with nonlinear problems. First we consider an undamped single griffitum system with cubic nonlinear terms and it starts with an initial condition A X of 0 is A and the initial velocity is 0, mu is a parameter if mu is 0 system is linear so we can see that solution is A cos omega N T but omega N is this frequency. Now if mu is not 0 one can expect that the frequency of oscillation and the nature of oscillations would change it need not be at omega N it will be influenced by mu. So what I will do is I will represent the solution in a series X0 of T plus mu X1 plus mu square X2 plus so on and so forth. Then omega square which is the frequency of oscillation I will again expand in a similar series. Now the unknowns here are X0 and omega N are known X1 X2 alpha 1 alpha 2 they are not known. What we do is we substitute this assumed solution into this form into the given equation and then collect terms you know on both sides which are powers of mu, so if you collect terms on mu to the power of 0 I get this equation, mu to the power of 1 I get this equation. Now if we can examine this solution to these equations so if we will consider the first equation it is X0 double dot plus omega square X0 equal to 0 from which we get X0 of T is A cos omega T. Now X1 the equation for X1 has X0 of T on the right hand side or here it is present here. So if we now take that into account and solve the next level of equation which is again linear these all these equation for X0 X1 etcetera will be linear but solution at the previous type drives the solution at the next step. So if we solve this problem we get X1 of T to be given by something like this. Now if let me have expanded cos cube omega T I am writing the right hand side. Now if we examine this equation if this omega and this omega are the same so as time becomes large the solutions will become unbounded because we are in a resonant kind of situation but system is undamped it is conservative therefore this type of solutions are not possible so we demand that this multiplicating term alpha 1 minus 3 4th A square must be 0 for simulating the expected qualitative behavior of the system. So we get alpha 1 to be 3 4th of A square so from based on this I can construct this solution and using the initial conditions I will be able to write this solution for X1 of T. So if we restrict our solution analysis to only the first two terms I get X of T as A cos omega T plus mu A cube by 32 omega square this cos 3 omega T minus cos omega. Omega curiously is given by this this omega depends on A what is A? A is the initial condition so this is what I was telling the frequency of free vibration can depend on initial condition and the response here is periodic but it is not harmonic okay. So period depends on initial condition if mu is greater than 0 omega will be greater than omega naught we call that hardening behavior and mu is less than 0 we call softening here. Now we can of course include higher order terms and do this the objective of this discussion is not to illustrate the solution method but to highlight the qualitative feature of the response. Now a similar system under harmonic excitation it is damped nonlinear system under harmonic excitation if mu is 0 we know that response is harmonic for large times that is a harmonic steady state now we want to examine what should be the solution. So if mu is 0 I have X naught of T is A cos omega T. Now I will assume that for some A and omega this for some A this could be the solution so I will substitute this assumed solution into the equation of motion this is only solution in the steady state. So I will collect terms containing sine cosine terms do what is the harmonic balance and I get a pair of equations which are given by this. Now this resulting equation is the frequency response function of the system. So this is a forcing function F amplitude omega n is a natural frequency omega the driving frequency A is the amplitude of response in linear systems this on the right hand side we will not get this A cubed term okay. So now if I plot this frequency response function we start getting curves like in a linear system we know that the typically this curve will look like this. Now in a nonlinear system things change in certain bandwidth of excitation frequencies there are three possible routes and which one is actually realized in a solution depends on initial conditions and all three solutions need not be stable okay. So now the question is we have to examine stability of these multiple steady state solutions and only those solutions which are stable will be realized. Now if you were to do a thought experiment where you are increasing the driving frequency and you are starting with initial conditions which are on this branch. So as you progress along this line when you come here the response suddenly drops this is steady state response this you won't see in the time history of a response it is two different time histories but with initial conditions in the neighborhood of this response. So it drops and it will follow this path so this branch will not be realized in such a calculation. But on the other hand if you start with this side of the solution and you progress this way keeping initial conditions in the neighborhood of the steady state solutions here then you will see that you will move along this path and there will be an upward jump and it will be like. So if you ignore presence of nonlinearity you may think that this is the resonant amplitude frequency whereas it will be this and that can lead to unconservative estimate of the response. Now how do we see in regions where there are multiple solutions or even elsewhere how do you know that the solutions are realizable? So what we can do we did this stability analysis earlier what we will do is we will perturb the assumed solution by a small perturbation substitute back into this and we get now a time varying system where the assumed periodic solution appears as a coefficient. So for a given value of A from this graph for a given pair of values of omega and A you come here and do a flow case analysis and find out whether the perturbations grows in time or not. So that establishes whether the assumed solution is stable or not. So if there are multiple steady state solutions possible then which one will be realized is a question that you have to answer and in fact it turns out that in regions where there are multiple steady state solutions the responses will be dependent on initial conditions. So this again is a newer feature. Now there is another way of looking at this problem I will quickly run through this we again consider a similar system, system with cubic nonlinearity and I will seek the solution in the neighborhood of capital omega being in the neighborhood of omega naught. So what we do is I perturb omega naught by a small parameter known as detuning, okay this is called as detuning, epsilon is a small parameter. Now I will arrange the amplitude of excitations and damping in this manner and this method is known as method of multiple scales. The discussion on this is not focused on how to implement the method but what the method tells us if you do it. So I will skip this the details of this implementation of this solution but what I will show is in this method I get solution in the form A of T cos omega naught T plus beta of T where A and B are slowly varying functions of time which are governed by these two differential equation A and beta, okay. Now I can rewrite this equation in this form and if a steady state exists you can see that A and beta should become constants. So in which case A prime and B prime must be equal to 0, so the condition for steady state is that these are 0, these are nothing but fixed points of these two differential equations. So corresponding to each fixed point of the equation for amplitude and phase there is one steady state possible and this again turns out to be multiple valued and how do we study stability here? What we do is we study the stability of the fixed points that also we have discussed earlier. So in these two approaches in the first approach we are using flocase theory and the system that we are studying is a system with time varying coefficients whereas here in the next approach we get the steady state as fixed points of certain simplified set of differential equations and we investigate the stability of the solution by studying the stability of the fixed points. Now see this gives you a kind of a brief overview of what are the qualitative features that you could expect in a typical nonlinear system. This is just a tip of an iceberg there are many more complicating issues but given that you have an understanding of linear system behavior this is a list of items that you should start thinking about. Now why nonlinear analysis is important? There are many reasons although many engineering systems are designed to behave linearly one might think that nonlinearity is not that important in engineering design but that is not factually correct. For example in earthquake engineering we design the structures to display controlled inelastic response. There are certain preferred modes of failures and certain failure modes that are not preferred that means we design the structure to fail in a particular mode. For example we want to have strong columns and weak beams that means if a multi-story building is going to fail under an earthquake the failure should begin with failure of beams that is slabs and things like that and not for example the ground floor column. If a ground floor column fails no matter how strong the structure is the whole structure will collapse. Then similarly for industrial structures like piping systems and things like that we use certain supporting devices known as snubbers and nonlinear energy dissipation devices and things like that these again impart nonlinearity to the system. For example heat exchanger pipe in a nuclear reactor conveys fluid at hot temperature so in the normal course of its operation it should be able to withstand the thermal loads because of the conveyance of hot fluids but in the event of an earthquake there will be additional support motion. To cope up with thermal loads the structure needs to be flexible but that very flexibility brings a natural frequency of the system into the range of earthquake excitation frequencies. So supports like snubbers in the event of an earthquake they lock the structure and reduce the spans of the piping and increase the natural frequency so that it attract lesser seismic loads. Similarly in wind engineering the interaction between structures and the flow past vibrating structures induces highly nonlinear forcing functions that I will just briefly mention in the next one of the next slides. Materials like concrete and soil which are important in civil engineering application display nonlinear behavior even at low strength so the response need not be very large before nonlinearity is switched on. Also they display differing behavior in tension and compression then response depends upon entire time history duration over which the load is applied and ambient effects such as temperature. Next if a structure is cracked and it starts vibrating the closing and opening of cracks induces certain type of nonlinearity that we have to think of and if you as engineers we are always interested in study of failures because we want to design structures to prevent failures so to understand failures we have to enter nonlinear regimes of responses, a linear system in principle can never fail you know it can withstand infinite stresses. There could be problems of loss of stability this we have discussed like buckling and snap through and so on and so forth. In many application model application prototype testing using nonlinear FE models have become popular for example crash analysis in automotive design, simulation of draft test in electronic industry etc. instead of performing costly experimental studies cheaper finite element simulations are being used. The sources of nonlinearity in structural mechanics problems can originate from nonlinear strain displacement relations this type of nonlinearity is called geometric nonlinearity or the relative the constitutive relations typically relating stress and strain it could be temperature as well could be nonlinear then nonlinearity associated with boundary conditions as in contact related problems and so on and so forth and similarly nonlinear energy dissipation mechanisms these are some sources of nonlinearity. So a structure can behave in different ways for example the material of the structure could be nonlinear but there could be small displacements so the typical force response diagram may look like this. So the deformations are not large but the material of the structure has already entered a nonlinear regime so it is nonlinear retain constitutive relations. In the other type of behavior material may be linear or nonlinear but large rotations will occur and there are small deformations so there are small strains but large rotations. Now of course one can have large rotations as well as large strains so these are the most difficult problems to deal with. There could be special conditions like for example the same structure is supported through a gap and a loaded spring like this so this till the time this gap is negotiated this spring won't come into action and moment that happens the there will be a bilinear nonlinearity and this type of boundary conditions create what are known as nonlinear boundary conditions. If you write the equation of motion for a simple case again say M X double dot plus C X plus a nonlinear term the function G is a function of instantaneous value of displacement and velocity this is inelastic this is nonlinearly elastic system that means upon unloading this force would go to zero and there won't be any residual displacement the loading and unloading path will be tracing each other. On the other hand there could be forces of nonlinear forces which are dependent on entire history of the response up to the current time so this is typically originates from material nonlinearity and this originates from geometric nonlinear. So a typical force displacement you know graph for a system exhibiting so called hereditary nonlinear behavior or hysteretic nonlinear behavior is shown here so this is something like a force and a displacement and for a given value of displacement you can see that there are multiple forces possible which one will be realized depends on how we have reached that point that means it depends on the history of the response up to that time. So this type of systems are more difficult to analyze as you could expect than this model if this is a difficult to model as well as to more difficult to analyze. Now there are nonlinear effects in I mentioned about wind engineering problems so if you imagine that there is a chimney which is subjected to say a flow this is planar view the chimney is something like this and the flow is taking place like this. The flow past this chimney will create a pressure field on the object and if you integrate the pressure field over the surface area you get a force which can be resolved in line and across the flow directions and we can show that for certain flow velocities and certain geometries there will be vertices that will be shed and because of that there will be dynamic excitations predominantly harmonic on this chimney and these are known as across wind oscillation. So and if the chimney is flexible the nature of these excitations become fairly complicated so flow past a flexible object can create severe interaction fluid structure interaction and for that type of systems there will be a special type of nonlinearity as shown here I discuss this when we discuss limit cycle oscillations in the one of the earlier lectures you can see here that for a small x dot the term inside the parenthesis will be less than 1 and the net effect of this term will be negative it is like a negatively damped system and small oscillations tend to grow. For large amplitude oscillations the term here becomes negative and induces a positive sign on this and large amplitudes tend to decay so in free vibration the system displays what is known as the limit cycle behavior. So and that's a periodic solution it's an isolated periodic solution and when such systems are driven by external excitations there can be complex interactions between the periodic solutions which are highly nonlinear periodic solutions in free vibration and the components due to external excitation and there are very many complex behavior one something known as entrainment and things like that these are again characterizes nonlinear resonances and if you want to understand the peak response in such systems you have to understand the basic entrainment phenomena. Now so with this brief background we will come to the objectives of the discussion on nonlinear systems so the idea here is the subject is very vast it cannot be covered in few lectures that I am planning to you know dedicate to this topic the idea here is to provide a brief review of background concepts and present a flavor of treatment of nonlinear structural mechanics problems using finite element method. The focus is on geometrically nonlinear problems we will not be talking about material nonlinearity by and large we will be focusing on geometrically nonlinear problems. We can begin with whatever background we have without asking too many newer questions for example if you are talking about a planar Euler-Bernoulli beam if you assume that there are large transverse displacement but small strains and there are moderate rotations then the changes in geometry due to deformation need not be accounted for while defining stress. So the point here is if a structure undergoes large amplitude oscillations the geometry of the structure also would change so that needs to be taken into account while defining stress so that modifies many of the basic formulations that we will use for analyzing this system but suppose we do not get into that under these conditions probably one can overlook those complications then we will be able to proceed with whatever background we have. For example if we assume the invoke the Euler-Bernoulli hypothesis that upon deformation the line segment Mn remains straight and normal to the neutral axis and its length does not change we get the displacement field this we have discussed a few times where U1, U2, U3 are displacement along XYZ respectively and U0 of X and W0 of X are the displacement of the point on neutral axis. The strain displacement relations that we will be using are nonlinear we are not using the infinitesimal definition of the strain we are including the nonlinear terms as well this I have discussed the definitions I have discussed in one of the previous lectures. So the nonlinear strain displacement relations are again displayed here the quantities that appear in the red are the nonlinear terms so for this assume displacement field I get epsilon 11 I will retain the first term dou U1 by dou X1 but this quadratic term we are not including because we assume strains are small but dou U3 by dou X1 is a rotation that is included so I get this term for epsilon 11 that mean dou U1 by dou X1 whole square is ignored but dou U3 by dou X1 is retained. Now only this strain will be nonzero all other strains will be zero and stress again I assume isotropic elastic material so stress is related to strain through this relation I will be able to write the expression for strain energy and the kinetic energy using the assumed you know form of displacement and the consequence strains and I get strain energy in this form and if I now use the assumed displacement form I get this you see now there are quadratic terms in displacement in the expression for strain energy. So the terms appearing in red are the newer terms now if we assume the beam to possess symmetric cross section the terms appearing in red are not the I mean it does not connote nonlinear terms but the terms which will go to zero if you assume beam section to be symmetric so if that happens these two terms will go to zero and I am left with U which is this and the second set of terms are the new terms due to presence of nonlinear. Kinetic energy is given by this I can write the Lagrangian and again I will take a two nodded element with 3 degrees of freedom per node and we will again interpolate the axial displacements using linear interpolation functions and transverse displacement using Hermite polynomials I get this set of equations and if I run through the Lagrangian's equation I will get the required equations of motion. So if you examine the Lagrangian the first few terms were already encountered when you do the linear analysis so they lead to typical mass and stiffness matrices that we have already derived and the newer terms will originate from these two nonlinear terms so suppose if you will focus on one of this suppose the first term 1 by 8 AE dou W by dou X for DX and substitute for the assumed displacement form I get this and when we run the Lagrangian on this we will get cubic terms here and this IJMK is a new integral that has to be determined so here we would not get matrices we will get vectors. So similarly the other term involving the other nonlinear term which is this if we do that again we get newer nonlinear term which could be quadratic or cubic. So I again name some of the integrals that appear here through notation KIR, SK, etc. So the final form of equation of motion at the element level will be MEUE double dot plus KEUE plus vector of quadratic and cubic terms in U of t okay and the multipliers that appear here are properties of the structural system okay so this is the equation so in free vibration this will be 0. Now energies in different elements can be added so Lagrangian can be constructed for the built up structure using the approach that we have used there is no change in that aspect of our work so assembly of element level matrices and vector can be done as before to obtain global equations of motion then derivation of external forces and imposition of boundary conditions again follows the earlier developed procedure there is nothing new there. The resulting equation of motion for the structure after imposing boundary conditions and after computing the external forces will be of this form in this case. So this G of U is the nonlinear term that is arising in this model and this will as we have seen which will have cubic and quadratic terms in U. Now we already discussed how to solve this equation see for example we during earlier lecture, lecture number 16 we have developed a you know operator splitting methods and other methods to tackle these equations so that can be used I am not going to discuss the solution procedures at this juncture again. Similar analysis can be done for Timoshenko beam I this I leave as an exercise so this is the assumed displacement field and this will be the strain and there will be one more strain which is epsilon 13 and you will have to use this expression and construct the Lagrangian now you have to include kinetic energy due to translation and the rotation the rotor inertial effect also has to be included here. So once you do that following a similar procedure you will be able to derive the equation for Timoshenko beam. Now this is alright but how about a more general theory? Now in a more general theory we need to allow for measures of strain and stresses to be defined consistent with the deformations and also we need to allow for material nonlinear this will not be doing but this we will discuss now as a structure undergoes large deformation the cross sectional properties might change so when we define stress we will have questions on some kind of a force divided by an area which area are you talking about is it the structure in its undeformed configuration or in the deformed configuration. If you say the area is to be computed based on deformed configuration when you are defining stress you would not know what the deformed configuration is right so and then similarly when you define strains you have to think about large rotations I will show either during this lecture or the next lecture that if you use the infinitesimal definition for strains a structure undergoing regent rotation the strains would not be 0 so that is not acceptable so say you cannot stick with infinitesimal strain definition that also needs to be modified. So some of these issues need to be addressed and to do all these systematically we need to get into the subject of continuum mechanics and develop all the language and the notations in a systematic way before we even we can address as simple class of problems. This subject is very vast as I already said we are not going to discuss many aspects of this will not be discussed I have given a list of references which cover this subject in good detail and I will be using some of these references during the lecture. Now the subject of non-linear analysis of structures is mathematically lot more refined than a linear analysis there are many issues associated with notations and as I already said definition of stress strain and the balance laws all of them we need to revisit so there are issues about notations there are four sets of notations that one has to use one has to understand to understand literature on this subject. The indicial notations I will quickly review this a set of variables X1 to X2 XN is simply denoted as XI that means the indicial here in the name refers to the index to the variables that we assign the range of values taken by the index I needs to be specified typically I runs from 1, 2, 3 if they are in a Cartesian space but it need not be so. Now repeated indices imply summation for example if I have a term like alpha equal to I equal to 1 to N AI XI this is simply written as alpha AI XI and I I have to specify what range it has to be use I is 1 to N since I is a dummy index instead of writing AI XI as well I can write ASXS so that I or S is not very important it is a dummy index similarly a term like this is written as half K IJ UI OJ you can see that I and J are repeated therefore a summation on I and J are implied from I equal to 1 to N. This Kroniker delta is a symbol that is used delta IJ is 1 if I equal to J otherwise it is 0, so using that for example the length of an infinitesimal element DS square which is given by DX1 square plus DX2 square plus DX2 square is written as delta IJ DXI DXJ, the symbol known as permutation symbol epsilon IJK it is defined as shown here so you can keep this figure in mind if you run from 1, 2, 3 or 2, 3, 1, 2 epsilon IJK is 1 on the other hand you run in the other way 1, 3, 2, 3, 2, 1 or 2, 1, 3 it is minus 1 for all other combinations it is 0. Now if AI is a 3 by 3 matrix the determinant of AI can be written using the permutation symbol in this way there is a small exercise there is an identity known as epsilon delta identity you can show that epsilon IJK and Kroniker delta related through this identity. Now there is a symbol for differentiation suppose if you consider a function which is F of X1, X2 up to XN and if DF is what I am looking at it is given by this this is written compactly as DF equal to dou F by dou XI into DXI the index I repeats and it has to be summed over 1 to N and this comma symbol that's a differentiation symbol if I again have this function F1, F2, F3 to be functions of X1, X2, X3 if I write F of I, J it is dou FI by dou XJ. Similarly sigma IJ, K is delta of sigma IJ by dou XK this comma K means you have to differentiate with respect to the Kth variable. So this is like a mathematical shorthand for writing the long expressions physics will get buried inside these notations so it is not very convenient if you are understanding the subject for the first time but it is very useful in compactly expressing the results. The algebraic notations suppose I have vector with components X1, X2, X3 and Y1, Y2, Y3 X dot Y is X1, Y1 plus X2, Y2 plus X3, Y3 and X cross Y is given by this and this is written as Zi is epsilon IJK XJ YK. Then grad that is this in the delta, inverted delta is this operation E1 dou by dou X1 plus E2 dou by dou X2 plus this which is written EI dou XI. Now this is scalar function if you take a grad of a scalar function it is given by dou F by dou XI that my ith component is this. Now if you apply the grad operation on a vector valued function F we can define what is our divergence which is del dot F which is given by this, this is a dot symbol, the curl is delta cross F and that is given by this. Now if you apply a grad function on a vector valued function if this is written on delta, F this is given by this and F itself is a vector therefore I have to write this. So you can see that grad of a vector valued function will have these gradients present in their representation. In matrix notations we arrange the components of vector in a column like this and even stress is arranged as a column like this, right. So stress is a tensor represented at 3 by 3 matrix but in this so called white notations we write it as 6 cross 1 vector similarly strain. So the strain energy is written as half sigma transpose epsilon so we represent the terms like this for example R square is written as X transpose X and so on and so on. So when doing this we don't write explicitly the connective symbols for example when I say X transpose X I am not putting in between any dot or a multiplication or any other simple symbol. In tensor notations indices are not shown this is applicable to Cartesian and other coordinate systems XI, YI that means it is a summation of X1, Y1 plus X2, Y2 plus X2 is simply written as X dot Y, AIJ, BIJ is written as A double dot B this is a new symbol that we will use column denotes contraction of pair of repeated indices whereas a dot denotes a single dot denotes contraction of inner indices whereas this double dot denotes contraction of pair of repeated indices. So this relation sigma IJ, CIJ, KL, epsilon KL is in tensor notations is written as C double dot epsilon. So I have given some examples of writing different expressions in alternative notations this you can examine. So if we have something like Phi transpose K Phi in tensor notation it is written as Phi dot K dot Phi, in indicial it is Phi I KIJ Phi J, similarly you have half epsilon transpose C epsilon this is written as half epsilon double dot C double dot epsilon whereas this epsilon I CIJ epsilon J so on and so forth. This and equilibrium equations for elasticity is in the long hand in full notation form it is given by this, in matrix form it is given by this, in indicial it is given by this. Actually the full notation is a notation of last result where everything is spelt out without cutting on any you know you write all the terms and this clearly becomes cumbersome if you have to deal with this type of equations to of. Now I will start now a quick review of continuum mechanics there is what is the continuum hypothesis. So according to this hypothesis material is infinitely divisible and each infinitesimal element retains all the properties of the material so that Newtonian mechanics is directly applicable that means calculus works that means concept of elementary strip and things like that work and you can derive the governing physical laws can be expressed as partial differential equations or as ordinary differential equations or through variational arguments. Now obviously we know matter is not infinitely divisible it breaks down to elementary particles if we do that but that we are ignoring in this hypothesis. So consequently we need to focus our attention to characteristic dimensions which is about greater than about 10 to the power of minus 6 centimeter. So if you are dealing with dimensions less than this then continuum hypothesis needs to be I mean you have to look at other possible effects that are present in the physics of the problem. Just to give in this context diameter of a water molecule is about 10 to the power of minus 8 centimeter so if you are dealing with fluid mechanics problem in which the medium is water the fluid is water then you cannot think of sizes less than 10 to the power of minus 6 characteristic lengths less than this. The continuum mechanics theory is valid for both solids and fluids it doesn't distinguish between the two and due to the assumption of existence of continuum notions of density temperature pressure at a point make sense. The primary aim is to model macroscopic behavior of solids and fluids so just to emphasize again it ignores the atomic structure of the matter and also matter consists of discrete particles which are perpetually in motion even this motion is not included in our analysis. Then questions on treatment of molecular grain or crystal structure are not addressed in continuum mechanics. There are different themes in continuum mechanics we talk about kinematics where we talk about motion and deformation, kinetics where we talk about concept of stress and there are different balance laws which basically enunciate certain physical laws I will come to some of them which are common to both fluids and solids. In the context of nonlinear structural mechanics problem what is crucial to gain a reasonable understanding of the subject is to understand how rotations are dealt with. Rotations are very crucial in problems of nonlinear analysis and what is the need for defining alternatively what is the need for alternative definitions for stresses and strains and then how to treat material nonlinear behavior. So we will start some simple questions about kinematics. Kinematics is study of motion and deformation without concerning with causes of motion and deformation. We do not talk about forces which create the motion and deformation we simply focus on geometry. So here we talk about a reference configuration say let's assume body B at time 0, this omega naught is a domain gamma naught is a boundary and we consider Cartesian coordinate system so capital X1, X2, X3 is for body at time t equal to 0. Now during the process of deformation every point here with position vector OP which is X gets mapped to another point P whose position vector is X. This X is related to capital X through this relation so this is a mapping of the deformation. The reference frame the origin is at 0 and there is a orthonormal basis E1, E2, E3 this is a coordinate system. The body B occupies different regions omega naught, omega 1 etc. omega at times t equal to 0, T1, T2, T3 and T. The regions omega naught, omega 1 etc. omega occupied the body at different time instants are known as configurations of the body at the respective time instants. At time t equal to 0 we say that omega naught is the initial state of the body or the initial configuration. It could also be taken as reference configuration with respect to which motion is described. There are other names like it is taken to be undeformed configuration. It is an idealization nobody is truly undeformed because the gravity and things always act on them so what you see as a reference configuration is already deformed due to some one or the other effects. Now gamma naught is a boundary of the initial configuration. At time t this is a current state of the body the current deformed configuration. Gamma is the boundary of the current configuration so this is gamma 1. There are two coordinate systems that we can think of using to describe the problem one is known as Eulerian, the other one is Lagrangian. In Lagrangian description we take capital X1, X2, X3, T as independent variables that means the point P is described by its position in the initial configuration that is capital X1, X2, X3. Though those X1, X2, X3 are taken as independent variables. In Eulerian description we take the lower case X1, X2, X3 as independent variables. Now X is written position vector X is written as XIEI which is nothing but X1E1 plus X2E2 plus X3E3. So this is a position vector of a material point in the initial configuration. This does not change with time because initial configuration is some reference position that does not change with time. It labels all material points whereas X the lower case X which is again XIEI this provides the position of a point in the current configuration. Changes as configurations evolve in time. In problems of solid mechanics we adopt Lagrangian description. The Lagrangian description is also known as material description and the Eulerian description is also known as spatial description. The motion itself that is this function Phi of XT is defined the motion that is a coordinate in the current configuration, a point in the current configuration that is a position vector of a point in the current configuration is related to where the point was in the reference configuration through this function. So this is in longhand it is there are three functions X1, X2, X3 Phi1, Phi2, Phi3 that relate the capital X1, X2, X3 to the lower case X1 and X2 and X3. When reference and initial configurations coincide at T equal to 0 X of X, 0 is capital X so that would mean which is Phi of X, 0. So then XI of XI X, 0 this is the definition which is PhiI of X, 0 and this is an identity transformation. So in material coordinates displacement is given by X minus capital X which is nothing but Phi of X, T minus Phi of X, 0 or Phi of X, T minus capital X. Velocity is its time gradient, capital X does not change with time therefore the gradient is simply DU by DT as shown here. There is no dou U by dou X term which gets multiplied by dou X by dou T that is not there because capital, dou of capital X by dou T is 0. So similarly acceleration also can be defined. On the other hand in the spatial coordinates if you want to define the gradient of a function say Phi of X, T this is dou Phi by dou T plus dou Phi by dou XJ and DXJ by DT so this is there will be a new term. So acceleration gets defined like this and this is a definition of acceleration if you are looking at spatial coordinates. Now a primary quantity of interest in discussing deformation is what is known as deformation gradient. So the problem is the question is the issue is this. This is a configuration of the body at time T equal to 0 and PQ is a line segment. Upon deformation capital P goes to small p and capital Q goes to capital Q and this line segment DX gets mapped to this lower case DX. The question is how this DX is related to this capital DX and that will be through a matrix known as deformation gradient. So we will take up this discussion on deformation gradient and follow up this topic in the next lecture. So we will close this lecture at this point.