ADV COMPUTNL FLOWS & TRANSPORT
ADV COMPUTNL FLOWS & TRANSPORT CAM 397
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A SHORTCOURSE ON NONLINEAR CONTINUUM MECHANICS J Tinsley Oden CAM 397 Introduction to Mathematical Modeling Third Edition Fall 2008 Preface These notes are prepared for a class on an Introduction to Mathematical Modeling7 de signed to introduce CAM students to Area C of the CAM CES Program I View nonlinear continuum mechanics as a vital tool for mathematical modeling of many physical events particularly for developing phenomenological models of thermomechanical behavior of solids and uids I attempt here to present an accelerated course on continuum mechanics acces sible to students equipped with some knowledge of linear algebra7 matrix theory7 and vector calculus 1 supply notes and exercises on these subjects as background material J Tinsley Oden September 2008 CONTENTS Chapter 1 Kinematics The study of the motion of bodies without regard to the causes of the motion Chapter 2 Mass and Momentum The principle of conservation of mass and de nitions of linear and angular momentum Chapter 3 Force and Stress in Deformable Bodies The concept of stress Cauchy s Principle Chapter 4 The Principles of Balance of Linear and Angular Momentum The connection between motion and force Chapter 5 The Principle of Conservation of Energy Chapter 6 Thermodynamics of Continua and the Second Law The Clausius Duhem inequality7 entropy7 temperature and Helmholtz free energy Chapter 7 Constitutive Equations General Principles Chapter 8 Examples and Applications iii CHAPTER 1 Kinematics of Deformable Bodies Continuum mechanics models the physical universe as a collection of deformable bodies77 a concept that is easily accepted from our everyday experiences with observable phenomena Deformable bodies occupy regions in threedimensional Euclidean space E and a given body will occupy different regions at different times The subsets of E occupied by a body 3 are called its con gurations It is always convenient to identify one con guration in which the geometry and physical state of the body are known and to use that as the reference con guration then other con gurations of the body can be characterized by comparing them with the reference con guration in ways we will make precise later For a given body we will assume that the reference con guration is an open bounded connected subset 90 of R3 with a smooth boundary 800 The body is made up of physical points called material points To identify these points we assign each a vector X and we identify the components of X as the coordinates of the place occupied by the material point when the body is in its reference con guration relative to a xed Cartesian coordinate system It is thus important to understand that the body 3 is a non denumerable set of material points X This is the fundamental hypothesis of continuum mechanics matter is not discrete it is continuously distributed in oneto one correspondence with points in some subset of R3 Bodies are thus continuous media77 7 the components of X with respect to some basis are real numbers Symbolically we could write BX N XXiei E 20 for some orthonormal basis e1e2e3 and origin 0 chosen in threedimensional Euclidean space and thus identi ed with R3 Hereafter repeated indices are summed throughout their ranges the summation convention Kinematics is the study of the motion of bodies without regard to the causes of the motion It is purely a study of geometry and is an exact science within the hypothesis of a continuum a continuous media 11 Motion We pick a point 0 in R3 as the origin of a xed coordinate system 1 2 3 x de ned by orthonormal vectors ei i 1 2 3 The system 1 zgz3 is called the spatial coordinate system When the physical body 3 occupies its reference con guration 90 at say timet O the material point X occupies a position place corresponding to the vector X Xiei 1 CHAPTER 1 KINEMATICS OF DEFORMABLE BODIES 1 Figure 11 Motion from the reference con guration 90 to the current con guration t The spatial coordinates X1X2X3 of X are labels that identify the material point The coordinate labels Xi are sometimes called material coordinates see Fig 11 Remark Notice that if there were a countable set of discrete material points such as one might use in models of molecular or atomistic dynamics the particles discrete masses could be labeled using natural numbers n E N as indicated in Figure 12 But the particles material points in a continuum are not countable so the use of a label of three real numbers for each particle corresponding to the coordinates of their position at t 0 in the reference con guration seems to be a very natural way to identify such particles The body moves through 5 over a period of time and occupies a con guration 9 C R3 at time t Thus material points X in 90 the closure of 90 are mapped into positions X in Q by a smooth vector valued mapping see Fig 11 00975 Thus LpX t is the spatial position of the material point X at time t The oneparameter family LpX t of positions is called the trajectory of X We demand that 4p be differen tiable injective and orientation preserving Then LP is called the motion of the body 1 9 is called the current con guration of the body 2 LP is injective except possibly at the boundary 890 of 90 3 LP is orientation preserving which means that the physical material cannot pene trate itself or reverse the orientation of material coordinates 7 Which means that det V4pXt gt 0 11 MOTION e1 Figure 12 A discrete set of material particles Hereafter we will not explicitly show the dependence of 4p and other quantities on time t unless needed this time dependency is taken up later The vector eld 11 40X 7 X is the displacement of point X Note that dx V4pXdX ltie dzl EdXJD The tensor is called the deformation gradient Clearly7 FX IVuX where I is the identity tensor and Wu is the displacement gradient Some De nitions 0 A deformation is homogeneous if F C constant 3 CHAPTER 1 KINEMATICS OF DEFORMABLE BODIES o A motion is M39gz39d if it is the sum of a translation a and a rotation Q 94X a QX where a E R3 Q E 03 with 1 the set of orthogonal matrices of order 3 with determinant equal to 1 0 As noted earlier7 the fact that the motion is orientation preserving means that detV4pX gt 0 v X e 20 0 Recall that Cof F cofactor matrix tensor of F det F F T For any matrix A Ag of order n7 and for each row i and column j let Agj be the matrix of order n 7 1 obtained by deleting the ith row and jth column of A Let dij 71 7 det Agj Then the matrix COfA is the cofactor matrix of A and dij is the i7j7cofactor of A ACof AT Cof ATA detAI 12 Strain and Deformation Tensors A differential material line segment in the reference con guration is 153 dXTdX de dX dX while the same material line in the current con guration is 152 dedx dXTFTFdX The tensor C FTF the right Cauchy Green defamation tensor is thus a measure of the change in 153 due to gradients of the motion 152 7 153 dXTCdX 7 dXTdX C is symmetric7 positive de nite Another deformation measure is simply 152 7 153 dXT2EdX 4 12 STRAIN AND DEFORMATION TENSORS where E C 7 I the Green strain tensor Since F I Wu and C FTF 1 E 5Vu VuT VuTVu The tensor B FFT the left Cauchy Green deformation tensor is also symmetric and positive de nite 121 Interpretation of E Take 150 Xm ie dX dX1OOT Then 152 7 153 152 7 de 2E11dX12 so 1 d8 2 a measure of the stretch of a E11 5 7 1gt material line originally oriented 1 in the Xlidirection in 90 We call el the extension in the Xlidirection at X which is a dimensionless measure of change in length per unit length el Ef dS Xm 12E1171 Xm 2E11 1 51 71 Similar de nitions apply to E22 and E33 Now take dX dX1dX20T and cos0 L Exercise lldxlll lldlel 1 2E11 1 2E22 The shear or shear strain in the Xng plane is de ned by the angle change see Fig ure 137 df39n39 Vlz gie CHAPTER 1 KINEMATICS OF DEFORMABLE BODIES dXZ dx 1 Figure 13 Change of angle through the motion 40 Therefore 2E12 M m 1 2E11x1 2E22 Thus7 E12 and7 analogously7 E13 and E23 is a measure of the shear in the X17X2 or Xng and XTXg plane 122 Small Strains The tensor 1 e Vu VuT is called the in nitesimal or small or engineering strain tensor Clearly 1 T E e Vu Vu Note that if E is small ie ltlt 1 51 1 2E1112 71 1E11 710E121 3 E11 611 that is d5 7 Xm etc 5 5 11 1 Xm 7 2e12 sin V12 z V12 etc Thus7 small strains can be given the classical textbook interpretation en is the change in length per unit length and e12 is the change in the right angle between material lines in the X17 and Xgidirections 13 RATES OF MOTION AND DEFORMATION 13 Rates of Motion and Deformation lf 40X7 t is the motion of X at time t7 ie X 90X7 t then X lg 89009 t 8t is the velocity and i 13f 82900 t 822 is the acceleration Since LP is in general bijective7 we can also describe the velocity as a function of the place X in R3 and time t vvmwxw4mww This is called the spatial description of the velocity This leads to two different ways to interpret the rates of motion of continua 1 The Material Description functions are de ned on material points X in R3 2 The Spatial Description functions are de ned on spatial places X in R3 When the equations of continuum mechanics are written in terms of the material descrip tion7 the collective equations are commonly referred to as the Lagrangian form formulation of the equations see Fig 14 When the spatial description is used7 the term Eulerian form formulation is used see Fig 15 There are differences in the way rates of change appear in the Lagrangian and Eulerian formulations In the Lagrangian case wmwwmwwmwe dt 7 8t ex 825 8X but 7 0 because X is simply a label of a material point Thus7 at WOW 890X7 75 dt In the Eulerian case Given a eld 1 11w t7 114967 75 814967 wma dt 3t z xed 8w 8t 7 CHAPTER 1 KINEMATICS OF DEFORMABLE BODIES Xat t0 X3 Xatt1gtt0 dxdt8PXtdt 2 K Fixed in Space X 1 Figure 14 Lagrangian material description of velocity The velocity ofa material point is the time rate of change of the position of the point as it moves along its path its trajectory in R3 8w but a vwt is the velocity at position w and time 25 Thus 77vwti 11496775 WWW 81406775 dt 8t 8w Lagrangian notation 877 8 8X iviGrad 4p7V4piDlvzp Eulen39an notation 8 77 8 8w grad 7w r le v In classical literature some authors write as the material time derivative of a scalar eld a giving the rate of change of 11 at a xed place X at time 75 Thus in the Eulerian formulation the acceleration is aigialv radv 7Dt78t g v being the velocity 14 RATES OF DEFORMATION V at x change with time X3 but x is xed X2 K Fixed in Space X1 Figure 15 Eulerian spatial description of velocity The velocity at a xed place X in R3 is the speed and direction at time t of particles owing through the place X 14 Rates of Deformation The spatial Eulerian eld L Lwt d5f 83V 137t grad Vwt w is the velocity gradient The time rate of change of the deformation gradient F is r 7 8 7 840 F EV Xt7VEXt 8 8V 8w KW Ea grad V F FgradVFLmF where Lm L is written in material coordinates7 so Lm FF l It is standard practice to write L in terms of its symmetric and skew symmetric parts L DW D L LT the deformation rate tensor W L 7 LT the spin tensor Here CHAPTER 1 KINEMATICS OF DEFORMABLE BODIES We can easily show that if V is the velocity eld WVwgtltV where w is the vorticity w curl V Recall Cf Exercise 26 that Ddet A V detAVT A 1 for any invertible tensor A and arbitrary V C LV V Also if fgt fo gt denotes the composition of functions f and g the chain rule of differentiation leads to dfgt deft d dfltglttgtgt 7 Dfltglttgtgt go Combining these expressions we have 8detF d tF7 e a DdetF F detF FTF 1 detF tr L detF diVV Since F F 1 tr FF 1 tr Lm where Lm is L written as a function of the material coordinates and tr L tr grad V div V Summing up det F detF div v 15 The Piola Transformation The situation is this a subdomain CO C 90 of the reference con guration of a body with boundary 8G0 and unit exterior vector no normal to the surfacearea element dAo is mapped by the motion 4p into a subdomain G 40G0 C 9 of the current con guration with boundary 8G with unit exterior vector n normal to the deformed surface area dA see Fig 16 Moreover there is a tensor eld To T0X de ned on G0 that associates with no the vector T0Xn0X at a point X on 8G0 the vector Tono being the flux of T0 across or through 8G0 at X with respect to T0 There is a corresponding tensor eld Tw TLPX de ned on G that associates with n the vector Twnw at a point w 40X on 8G We seek a relationship between T0X and Tw that will result in the same total flux through the surfaces 8G0 and 8G so that T0Xn0XdA0 TwnwdA 300 30 with w This relationship between T0 and T is called the Piola transformation 10 15 THE PIOLA TRANSFORMATION Figure 16 Mapping from reference con guration into current con guration Proposition The above correspondence holds if T0X det FX Tm FX T 7 Two Cof FX Proof This development follows that of Ciarlet We will use the Green s formulas divergence theorems T0 TOHO 1140 Go 300 and div sz TndA G ac where i 8 T Div T0 V T0 walla 8 d1V T Ejei dm d1d2d3 detF dX det F Xmngng We will also need to use the fact that 8X Cof m 0 To show this7 we rst verify by direct calculation that 8 8 8 8 CofFi39CofV lquot 7i 7i 7 7i 7i 7 4P 89 1 QXHZW 2 89 1 lt8XH1 2gt 11 CHAPTER 1 KINEMATICS OF DEFORMABLE BODIES where no summation is used Then a direct computation shows that 8 Next7 set T0X TwCof Noting that 7 c fF T F l det F O and 5i 8Xm 8j 8j we see that Cof F det F F lm det F an 8i Thus7 i 8 8 T DIV T0X ekianT0ijei ej 822 Bi 8 COf 8T 3x 8 0 87 cOf FXmjei TimainWei 6mm 8x7 an d tF 7 8m an e amme 8Tquot ei det F 7 div T det F that is Div T0 detF div T Thus Div TOdX detF diV TdX Go Go TOnOdA0 dideethXdidem TndA 300 CO C 30 as asserted 16 COROLLARIES AND OBSERVATIONS 16 Corollaries and Observations 0 Since G0 is arbitrary symbolically TOHO dAO TH 0 Set T I identity Then det F F Tno dAo n M 0 Since n det FF Tn0 and 17 1A det F HF TnOHdAO Nanson s Formula where denotes the Eulerian norm Thus Cof Fno HCOanOH 17 The Polar Decomposition Theorem A real invertible matrix F can be factored in a unique way as FRUVR where R is an orthogonal matrix and U and V are symmetric positive de nite matrices Proof We will use as a fact the following lemma for every symmetric positive de nite matrix A7 there exists a unique symmetric positive de nite matrix B such that B2 A Suppose F RU where R orthogonal7 U symmetric positive de nite Then FTF UTRTRU UTU U2 Thus U can be the unique matrix whose square is the symmetric positive de nite matrix C FTF Then set R FU l since RTR U TFTFU l U lcU l U lU ZU l 1 Similarly7 we prove that F can be written F VS 13 CHAPTER 1 KINEMATICS OF DEFORMABLE BODIES V TR TR Figure 17 The Polar Decomposition Theorem where V is symmetric positive de nite and S is an orthogonal matrix One can then show that S R Exercise D Thus if C FTF and B FFT are the right and left Cauchy Green deformation tensors and FRURUVR then C UTRTRU UTU U2 B VRRTV vvT v2 where U and V are the right and left stretch tensors respectively Clearly the Polar Decomposition Theorem establishes that the deformation gradient F can be obtained or can be viewed as the result of a stretching followed by a rotation or vice versa see Fig 17 18 Principal Directions and Invariants of Deformation and Strain For a given deformation tensor eld CX and strain eld at point X recall that dXTCdX 2dXTEdX 7 dXTdX is the square 152 of a material line segment in the current con guration Then a measure of the stretch or compression of a unit material element originally oriented along a unit vector In is given by Am d5 71 2mTEm mm me 1 14 18 PRINCIPAL DIRECTIONS AND INVARIANTS OF DEFORMATION AND STRAIN One may ask of all possible directions In at X which choice results in the largest or smallest value of Am This is a constrained maximizationminimization problem nd m mmam or mmm that makes Am as large or small as possible subject to the constraint me 1 To resolve this problem we use the method of Lagrange multipliers Denote by Lm A Am 7 Ame 7 1 A being the Lagrange multiplier The maxima on minimize and maximize points of L satisfy 8Lm A 0 4 Em 7 Am 8m Thus unit vectors m that maximize or minimize Am are associated with multipliers A and satisfy Em Am me 1 That is m A are eigenvectoreigenvalue pairs of the strain tensor E and m is normalized so that me 1 or 1 The following fundamental properties of the above eigenvalue problem can be listed 1 There are three real eigenvalues and three eigenvectors of E at X we adopt the ordering A1 2 A2 2 A3 to For A 7 A the corresponding eigenvectors are orthogonal for pairs mlAi and mi Aj miij 6 as can be seen as follows mTA7mj miTij mngi mAm M jmiij 0 ifx xj mej6 1943 i if A Aj we can always construct mj so that it is orthogonal to m 3 Let N be the matrix with the mutually orthogonal eigenvectors as rows Then A1 0 0 NTC N 0 A2 0 diagAi123 0 0 A3 The coordinate system de ned by the mutually orthogonal triad of eigenvectors de ne the principal directions and values of C at X For this choice of a basis 3 E ZAimi mi i1 15 CHAPTER 1 KINEMATICS OF DEFORMABLE BODIES F If 1 2 2 2 A3 1 corresponds to the maximum7 3 t0 the minimum7 and 2 to a mini max principal value of E or of An U The characteristic polynomial Of E is detE 7 I 7A3 E2 7 11E llIE where I7 II7 IZIare the principal invariants of E E trace E E tr E En E11 E22 E33 1 2 3 1 2 1 2 IIE 5tr E 7 EtrE tr Cof E 12 23 13 IZIE detE 1 8tr E373tr Etr E22 tr E3 123 An invariant of a real matrix C is any real valued function aC with the property aC MA lCA for all invertible matrices A CHAPTER 2 Mass and Momentum Mass the property of a body that is a measure of the amount of material it contains and causes it to have weight in a gravitational eld In continuum mechanics the mass of a body is continuously distributed over its volume and is an integral of a density eld 9 90 a RJV called the mass density The total mass MOS of a body is independent of the motion 40 but the mass density 9 can of course change as the volume of the body changes while in motion Symbolically MBthdz where dz volume element in the current con guration 9 of the body Given two motions 4p and 1 see Figure 21 let 9 and gw denote the mass densities in the con gurations 4090 and 190 respectively Since the total mass is independent of the motion MB 99 dm yd dm 9090 M90 This fact represents the principle of consemation of mass The mass of a body 3 is thus an invariant property measuring the amount of material in B the weight of B is de ned as gMB where g is a constant gravity eld Thus a body may weigh differently in different gravity elds eg the earth s gravity as opposed to that on the moon but its mass is the same 21 Local Forms of the Principle of Conservation of Mass Let go 90X be the mass density of a body in its reference con guration and g 9w t the mass density in the current con guration 9 Then 90 90XdX 9mm 17 CHAPTER 2 MASS AND MOMENTUM P i Figure 21 Two motions 4p and 1 where the dependence of g on t has been suppressed But dz det olX7 so 9 9000 e QMX detFXl dX 0 o and7 therefore 90X gardet This is the material description or the Lagrangian formulation of the principle of con servation of mass To obtain the spatial description or Eulerian formulation7 we observe that the invariance of total mass can be expressed as d i t d 0 dt Qt 9967 90 Changing to the material coordinates gives 0 1 9w7tdetFXtdX g detF gdetFdX dt 90 90 where ddt Recalling that det F det F div V7 we have i 89 0 detF gdivvivgradg 1X 90 8t from which we conclude 89 8t 22 MOMENTUM 22 Momentum The momentum of a material body is a property the body has by virtue of its mass and its velocity Given a motion LP of a body 3 of mass density 9 the linear momentum 23 t of B at time t and the angular momentum HB7 t of B at time 25 about the origin 0 of the spatial coordinate system are de ned by 5775 4 9V dm t HBt wxgvdz 9 Again7 dz dmldmgdzg is the volume element in Q The rates of change of momenta both I and H are of fundamental importance To calculate rates7 rst notice that for any smooth eld w wwt d d dw dw a Qt wgdz E 90 w4pXttgwtdetFXt dX 00 E90 dX at E9 dz Thus7 d103 t 7 dv dt 7 Qt 9 d3 dHltBltgt 7 m X yd dm dt Qt dt CHAPTER 2 MASS AND MOMENTUM CHAPTER 3 Force and Stress in Deformable Bodies The concept of force is used to characterize the interaction of the motion of a material body with its environment More generally as will be seen later force is a characterization of interactions of the body with agents that cause a change in its momentum In continuum mechanics there are basically two types of forces 1 contact forces representing the contact ofthe boundary surfaces of the body with the exterior universe ie its exterior environment or the contact of internal parts of the body on surfaces that separate them and 2 body forces acting on material points of the body by its environment Body Forces Examples of body forces are the weight per unit volume exerted by the body by gravity or forces per unit volume exerted by an external magnetic eld Body forces are a type of external force naturally characterized by a given vector valued eld f called the body force density per unit volume The total body force is then Qtfwt dz Alternatively we can measure the body force with a density b per unit mass f gb Then fdm gbdm Q 9 Contact Forces Contact forces are also called surface forces because the contact of one body with another or with its surroundings must take place on a material surface Contact forces fall into two categories 1 External contact forces representing the contact of the exterior boundary surface of the body with the environment outside the body and to Internal contact forces representing the contact of arbitrary parts of the body that touch one another on parts of internal surfaces they share on their common boundary The Concept of Stress There is essentially no difference between the structure of external or internal contact forces they differ only in what is interpreted as the boundary 21 CHAPTER 3 FORCE AND STRESS IN DEFORMABLE BODIES exterior external contact forces internal contact forces Figure 31 External and internal contact forces that separates a material body from its surroundings Portion I of the partitioned body in Fig 31 could just as well been de ned as body 3 and portion ll would then be part of its exterior environment Fig 32 is an illustration of the discrete version of the various forces a collection of rigid spherical balls of weight W each resting in a rigid bowl and pushed downward by balancing a book on the top ball of weight P Explode the collection of balls into free bodies as shown The ve balls are the body 3 The exterior contact with the outside environment is represented by the force P and the contact forces N representing the fact that the balls press against the bowl and the bowl against the balls in an equal and opposite way Then P N1 N2 N3 are external contact forces The weights W are the body forces Internally the balls touch one another on exterior surfaces of each ball The action of a given ball on another is equal and opposite to the action of the other balls on the given balls These contact forces are internal They cancel out balance when the balls are reassembled into the whole body 3 In the case of a continuous body the same idea applies except that the contact of any part of the body part I say with the complement part H is continuous as there are now a continuum of material particles in contact along the contact surface and the nature of these contact forces depends upon how we visualize the body is partitioned Thus at a point m if we separate 3 conceptually into bodies I and H with a surface AA de ned with an orientation given by a unit vector n the distribution of contact forces at a point w on the surface will be quite different than that produced by a different partitioning of the body 22 P F 39 internal forces W weight Figure 32 Illustrative example of the stress concept Figure 33 The Cauchy hypothesis de ned by a different surface BB though the same point m but with orientation de ned by a different unit vector In see Fig 33 These various possibilities are captured by the so called Cauchy hypothesis there exists a vector valued surface contact force density 0n7 96775 giving the force per unit area on an oriented surface F through m with unit normal n at time t The convention is that Un w 25 de nes the force per unit area on the negative side of the material n is a unit exterior or outward normal exerted by the material on the opposite side thus the direction of 0 on body H is opposite to that on I because the exterior normals are in opposite directions 7 see Figure 34 Thus if the vector eld Un wt were known one could pick an arbitrary point w in the body or equivalently in the current con guration t at time t and pass a surface through m with orientation given by the unit normal n The vector Un w 25 would then represent the contact force per unit area on this surface at point m at time t The surface F through w partitions the body into two parts the orientation of the vector Un wt on 23 CHAPTER 3 FORCE AND STRESS IN DEFORMABLE BODIES Figure 34 The stress vector one part at w is opposite to that on the other part The vector eld 039 is called the stress vector eld and 039n7 at is the stress vector at w and t for orientation n The total force on surface T is 039n7 at 01A7 r dA being the surface area element The total force acting on body 3 and total moment about the origin 07 at time t When the body occupies the current con guration Qt are7 respectively7 I7l3tQtfdxant039n dA Mzst wxfdx mudi Q 39 where we have suppressed the dependence of 7quot and 039 on w and t CHAPTER 4 The Principles of Balance of Linear and Angular Momentum The momentum balance laws are the fundamental axioms of mechanics that connect motion and force The Principle of Balance of Linear Momentum The time rate of change of linear momentum 23 t of a body 3 at time t equals or is balanced by the total force 7B7 t acting on the body 1123 t dt TB7t The Principle of Balance of Angular Momentum The time rate of change of angular momentum HQ7 t of a body 3 at time t equals or is balanced by the total force MB7 t acting on the body was t dt MB t Thus7 for a continuous media gdldz fdz UndA 9 dt 9 39 dv wxgidz wxfdz wxondA at dt 9 ant 25 CHAPTER 4 THE PRINCIPLES OF BALANCE OF LINEAR AND AN G ULAR M OMENTUM Figure 41 Tetrahedron r for the proof of Cauchy s Theorem 41 Cauchy s Theorem The Cauchy Stress Tensor Theorem At each time t let the body force density f Qt gt R3 be a continuous function of w and the stress vector eld 039 039n w t be continuously di erentiable with respect to n for each m 6 t and continuously di erentiable with respect to w for each n Then the principles of balance of linear and angular momentum imply that there exists a continuously di erential tensor eld T Qt gt M3 the set of square matrices of order three such that Unwt Twtn Vw 6 Qt Vn TwtTwtT Vw 6 Qt Proof of Cauchy s Theorem The proof is classical pick w 6 9 Since 9 is open we can construct a tetrahedron r with vertex at m with three faces parallel to the m coordinate planes and with an exterior face F with unit normal n n ei n gt 0 see Fig 41 The faces of r opposite to the vertices i i 123 are denoted F and areaF niareaF According to the principle of balance of linear momentum d fdm UndA7givdz0 739 3r 739 dt Let f fiei Un aine V viei Then for each component of the above equation 26 42 THE EQUATIONS OF MOTION LINEAR MOMENTUM and using the mean value theorem Tln w dA 37 Tl737 717 areaF 0n Q areaF ai7ejw dA ainw dA FFn F for EFJ and Q E F so that ATUnwdATltgd w 710490 dz 123 Thus since volume739 C areaF32 C a constant and i 1 23 and 7 mac Minn737 0n areaF x areaF32 Keeping n xed we shrink the tetrahedron 739 to the vertex w by collapsing the vertices to w areaF a 0 and obtain 039139Il 7njai7ejw or since Un aine Now for each vector 039ej w de ne functions such that 08j 13 Tijwe Then Tlmm Tijwnj or Un w Twn as asserted by continuity these hold for all m 6 t The tensor T is of course the Cauchy stress tensor We will take up the proof that T is symmetric later as an exercise which follows from the principle of balance of angular momentum 42 The Equations of Motion Linear Momentum According to the divergence theorem TndA diVTdm 35L 9 Thus UndA TndA diVsz ant ant at 27 CHAPTER 4 THE PRINCIPLES OF BALANCE OF LINEAR AND AN G ULAR M OMENTUM Figure 42 Mapping of volume and surface elements It follows that the principle of balance of linear momentum can be written gdldm fdiV Tdm 9 dt 9 where T is the Cauchy stress tensor Thus dv d T f7 10 flV gdtz But this must also hold for any arbitrary subdomain G C 9 Thus div Twt fwt gwt dv i t d w gt Returning to the proof of Cauchy s Theorem we apply the principle of balance of angular momentum to the tetrahedron and use the fact that diV T f7 gdvdt O This leads to the conclusion that TTT Details are left as an exercise 43 The Equations of Motion Referred to the Reference Con guration The Piola Kirchhoff Stress Tensors In the current con guration div Tm m 9w 83 Va grad vwgt 28 43 THE EQUATIONS OF MOTION REFERRED TO THE REFERENCE CONFIGURATION THE PIOLA KIRCHHOFF STRESS TENSORS where we have again not expressed the dependence on time t for simplicity Here div and grad are de ned with respect to w ie the dependent variables are regarded as functions of spatial position w and t We now refer the elds to the reference con guration foX fwdetFX 96 000 90X 9w det FX PX detFX Tm FX T Tm Cof FX by the Piola transformation The tensor PX is called the First Piola Kirchho Stress Tensor Note that P is not symmetric however PFT FPT since T is symmetric Recalling earlier proof7 we have 8 8 8 8mg 7T 8 0 7P7TCfF thiT7F Tic 8X t 8X 0 lt gtgt e 8X it 8X7 t WWW detF i kdethivT ka l ie Div P detF div T so i 1 dv Wu 1 d1vTw fw mD1vP f0X QE 90WwdetF Thus7 the equations of motion linear and angular momentum referred to the reference con guration are Div PX f0X 90X X PX FTX FX PTX Here the dependence of P7 f07 ii7 and F on time t is not indicated for simplicity but note that go is independent of t Alternatively these equations can be written in terms of a symmetric tensor SX sXt FX 1PX det FF lTF T F lT CofF The tensor S is the so called Second Piola Kirchho Stress Tensor Then Div FXSX foX 9009mm CHAPTER 4 THE PRINCIPLES OF BALANCE OF LINEAR AND AN G ULAR M OMENTUM In summary Cauchy Stress T det F 1 P FT First Piola Kirchhoff Stress P det F T F T Second Piola Kirchhoff Stress S 44 Power A fundamental property of a body in motion subjected to forces is power7 the work per unit time developed by the forces acting on the body Work is force times distance77 and power is force times velocity ln continuum mechanics7 the work per unit time 7 the power 7 is the function 73 73t de ned by 73 fVdm 039nVdA Qt 39 where V is the velocity Since 039nVdA TnVdAVdiVTTgradVdm ant ant Qt wehave i dV 73 Vd1VTfdmTDdzVg7dzTde Qt Qt 9 dt 9 d 1 gt dr 7 7 gVde Tde7 TDdz dtlt2 n 9 dt 9 where a is the kinetic energy 1 a 7 gV V dz 2 9 D grad V5ym grad V grad VT Note that T grad V T D W T D The quantity T D is called the stress power 44 POWER In summary the total power of a body 3 in motion is the sum of the time rate of change of the kinetic energy and the total stress power Equivalently fvdz TnVdAdjTde nt ant dt 9 In terms of quantities referred to the reference con guration foudX PnoudAo PFdXil oomudX no ano no 1752 no In terms of the second Piola Kirchhoff stress tensor the stress power is TDdz TgradVdethX Do 90 But 7 a 8 m T grad VdetF 7 TWTXk lt at gt WjdetF rm det F FkTdet Fngl FTF s and i 1 i i 1 i i i i FTFS 5FTFFTF FTF7FTF SESOSE Thus Tde SEdX Do 90 and nally A d 1 f0ildX FSn0ildASEdXilti goll11dXgt no ano no dt 2 no CHAPTER 4 THE PRINCIPLES OF BALANCE OF LINEAR AND AN G ULAR M OMENTUM CHAPTER 5 The Principle of Conservation of Energy Energy is a quality of a physical system eg a deformable body measuring its capacity to do work a change in energy causes work to be done by the forces acting on the system A change in energy in time produces a rate of work 7 ie power So the rate of change of the total energy of a body due to mechanical processes77 those without a change in temperature is equal to the mechanical power developed by the forces on the body due to its motion The total energy is the sum of the kinetic energy the energy due to motion n and the internal energy 5th due to deformation Total energy n th The internal energy depends on the deformation temperature gradient and other physical entities The precise form of this dependency varies from material to material and depends upon the physical constitution of the body In continuum mechanics it is assumed that a speci c intemal energy density energy density per unit mass exists so that Elm 9050Xt dX gewt dm Do at Recall that the kinetic energy is given by nl gouudXl gvvdz 2 Do 2 at and that the power 73 is w dn dn 7 TDd 7 FSF dX 7D dtn 3 dtno The change in unit time of the total energy the time rate of change of n5mt produces power and heating of the body The heating per unit time Q is of the form Q 7qndArdm 7q0n0dA0 rodm 35 t 390 0O 33 CHAPTER 5 THE PRINCIPLE OF CONSERVATION OF ENERGY where q is the heat ux entering the body across the surface 89 minus q indicated heat entering and not leaving the body and r is the heat per unit volume generated by internal sources eg chemical reactions and 10 and r0 are their counterparts referred to the reference con gurations q detF F T q CofF 10 r det F To The principle of conservation of energy asserts that the timerateof change of the total energy is balanced by equals the power plus the heating of the body 51 Local Forms of the Principle of Conservation of Energy Recalling the de nitions of a 5W 73 and O we have gvvdz gedmgt TDdzdji qndA rdm dt 2 nt nt nt dt ant nt Thus 15 i 977TDdivqir dz0 9 dt TD7divqr For equivalent results referred to the reference con guration 90 we have ilto goeodXgt sEdXdi7 q0n0dA0 rodX dt no no dt ano no ltgo oisEDivq07r0gtdX0 no go 0SE7Div q0ro Thus7 locally7 CHAPTER 6 Thermodynamics of Continua and the Second Law ln contemplating the thermal and mechanical behavior of the physical universe it is con venient to think of thermomechanical systems as some open region S of threedimensional Euclidean space containing perhaps one or more deformable bodies Such a system is closed if it does not exchange matter with the complement of 5 called the exterior of 5 There can be an exchange of energy between S and its exterior due to work of external forces and heating cooling of S by the transfer of heat from S to its exterior A system is a thermodynamic system if the only exchange of energy with its exterior is a possible exchange of heat and of work done by body and contact forces acting on S The thermody namic state of a system S is characterized by the values of so called thermodynamic state uariables such as temperature mass density etc which re ect the mechanical and thermal condition of the system we may write TS t for the thermodynamic state of system S at time t If the thermodynamic system does not evolve in time it is in thermodynamic equilibrium The transition from one state to another is a thermodynamic process Thus we may think of the thermodynamic state of a system as the values of certain elds that provide all of the information needed to characterize the system stress strain velocity etc and quantities that measure the hotness or coldness of the system and possible rates of change of these quantities The absolute temperature 9 E R 9 gt 0 provides a measure of the hotness of a system and a characterization of the thermal state of a system Two closed systems 51 and 52 are in thermal equilibrium with each other and with a third system 53 if they share the same value of 0 Thus 9 is a state variable The second quantity needed to de ne the thermodynamic state of a system is the entropy which represents a bound on the amount of heating a system can receive at a given temperature 0 ln continuum mechanics the total entropy H of a body is the integral of a speci c entropy density n entropy density per unit mass H grwt dm 9 In classical thermodynamics the change in entropy between two states TS t1 and TS t2 of a system measures the quantity of heat received per unit temperature When 9 constant the classical condition is H57t1 H57t2 0 35 CHAPTER 6 THERMODYNAMICS OF CONTINUA AND THE SECOND LAW for reversible processes and Q H57t1 H57t2 5 Z 0 for all possible processes HS if being the total entropy of system S at time t and Q being the heating ln continuum mechanics we analogously require E lqndA7idx20 dt 390 90 This is called the Second Law of Thermodynamics the total entropy production per unit time is always 2 O Locally we have dn q r i d 7quot d gt0 Altgdt 1V0 0 a In the material description i i qo r0 D i7igt0 QOUO l 1V 0 0 7 This relation is called the Clausius Duhem inequality Material Spatial LAGRANGIAN EULERIAN Conservation of Mass a 909detF 87idivgv0 Conservation of Linear Momentum a Div FSf0 goazuQtz divTfglt87vgrad v Conservation of Angular Momentum s sT T TT Conservation of Energy l A l 35 l 906082E7D1Vq07 0 gavgradeTDidlvqr Second Law of Thermodynamics The ClausiusDuhem Inequality 90770Div7020 ggvgradndivgig20 Figure 61 Summary CHAPTER 6 THERMODYNAMICS OF CONTINUA AND THE SECOND LAW CHAPTER 7 Constitutive Equations Given initial data ie given all of the information needed to describe the body 3 and its thermodynamic state 5 while it occupies its reference con guration at time t 0 20 890 f0X t t E 0T V0 uX0 00X 0X0 we wish to use the equations of continuum mechanics conservation of mass energy balance of linear and angular momentum the second law of thermodynamics to determine the behavior of the body the motion deformation temperature stress heat ux entropy at each X E 90 for any time t in some interval 0 T Unfortunately we do not have enough information to solve this problem The balance laws apply to all materials and we know that different materials respond to the same stimuli in different ways depending on their constitution To complete the problem to close the system of equations we must supplement the basic equations with constitutiue equations that characterize the materials of which the body is composed ln recognizing that additional relations are needed to close the system we ask what variable can we identify as natural dependent variables and which are independent Our choice is to choose as primitive state variables those features of the response naturally experienced by observation 7 by our physical senses the motion or displacement the rate of motion the deformation or rate of deformation the hotness or coldness the temperature or its gradients etc or the histories ofthese quantities If we then knew how for example the stress heat ux internal energy and entropy were dependent on the state variables for a giuen material we would hope to have suf cient information to completely characterize the behavior of the body under the action of given stimuli loads heating etc Thus we seek constitutive equations of the type T Twt q Qw7t7 e SWIM 7 Hw7t7 where for the moment denotes everything we might expect to in uence the stress heat ux internal energy and entropy at a point w 6 9 at time t for example F C 0V0 L D The functions T Q E H de ning the constitutive equations are 39 CHAPTER 7 CONSTITUTIVE EQUATIONS uts l l l l l 1 s F l l t the current state Figure 71 Time history called response functions or response functionals 71 Rules and Principles for Constitutive Equations What rules of thumb or even fundamental principles govern the forms of the response func tions for real materials We list some of the most important rules and principles 1 The Principle of Determinism The behavior of a material at a point X occupying m at time t is determined by the history of the primitive variables A In other words the conditions that prevail at wt depend upon the past behavior of the array not the future In general a history of a function u uw 25 up to current time t is the set uts uwti s 3 Z 0 The history from the reference con guration is then uw tis t Z s 2 0 see Fig 71 Then we should in general replace by ts in the constitutive equations unless conditions prevail in the constitution of the material that suggest the response depends only on values at the current time 2 The Principle of Material Frame Indifference 7 0r Principle of Objectivity The form of the constitutive equations must be invariant under changes of the frame of reference they must be independent of the observer For example if we rotate the coordinates w into a system mi and translate the origin a Change of the Observer77 40 71 RULES AND PRINCIPLES FOR CONSTITUTIVE EQUATIONS and change the clock 25 t 7 a the constitutive functions should remain invariant For example if T Twt then T Twt where T is the same function in both equations and T QlttgtTQlttT 3 The Principle of Physical Consistency The constitutive equations cannot violate or contradict the physical principles of mechanics the conservation of mass energy balance of momentum or the Clausius Duhem inequal ity 4 The Principle of Material Symmetry For every material there is a group g of unimodular transformations of the material coor dinates called the isotropy group of the material under which the forms of the constitutive functions remain invariant o The unimodular tensors H are those for which det H i1 0 The group g is a group of unimodular linear transformations with group operation composition matrix multiplication Thus 1 AB QAB Q 2 ABC ABC 3 31 EquchthatA1 A 4 VA 6 g EN1 6 g such that AA l 1 0 Let 9 be the group of rotations the proper orthogonal group If at a material point X the symmetry group g 9 then the material is isotropic at X otherwise it is anisotropic 5 The Principle of Local Action The dependent primitive constitutive variables at X are not affected by the actions of independent variables T q 5 7 for example at points distant from X There are other rules For example Dimensional consistency Terms in the constitutive equations must of course be di mensionally consistent this can be interpreted as a corollary to Principle 3 Beyond this dimensional analysis can be used to extract information on the constitutive variables Existence wellposedness The constitutive equations must be such that there exist 41 CHAPTER 7 CONSTITUTIVE EQUATIONS solutions to properly posed boundary and initial value problems resulting from use of the equations of continuum mechanics Equipresence When beginning to characterize the response functions of a material7 be sure that the dependent variables in are equally presen 7 7 in other words7 until evi dence from some other source suggests otherwise7 assume that all the constitutive functions for T7 17 e 7 depend on the same full list of variables A CHAPTER 8 Examples and Applications 81 Principle of Material Frame Indifference 811 Solids Consider a change in the observer Qt 1 QWT where wt LpXt Clearly F Vw and F Vw so that F QF and det F det F If n Qn and Un QUn then T TF Suppose Local Action T depends onVLp7 not 4p Then T TF Thus7 the material response function is independent of the observer7 if7 VQt7 QtT Qt 17 we have 7QF QTFQT Application If TF T then TF RTURT since TRTF RTRTRURT RTURT This shows that we can also express T as a function ofp or7 since U2 C7 of C or E ie T TF RTURT FU 1TCl2U 1FT FTCFT D 43 CHAPTER 8 EXAMPLES AND APPLICATIONS 812 Fluids Suppose T JquotL a uid and 7quot0 7p I7 p pressure when there is no motion7 we want the stress eld to be a hydrostatic pressure7 p 19w7 Suppose also that tr L div V 0 Then T 7191 70D 700 0 If To is linear in D7 then a necessary and suf cient condition that 70D is invariant under a change of the observer is that there exists a constant or scalar u ut gt 07 such that 70D 2pD Then T7pl2luD This is the classical constitutive equation for Cauchy stress in a Newtonian uid a viscous incompressible uid where u is the viscosity of the fluid 82 The Navier Stokes Equations for Incompressible Flow Conservation of Mass at d1v gv 0 For an incompressible uid7 we have 01dmdethXdethivvdXdivvdz 1750t no no Qt which implies that div V 0 Thus 8 8 d 087Vgradggdivv vgradg gisconstant Conservation of Momentum 8V i ga gvgradvid1vTf7 Vwt QtXOT TTT 44 82 THE NAVIER STOKES EQUATIONS FOR IN COMPRESSIBLE FLOW Constitutive Equation T i PIJrQID 2970 D 7 grad V grad vT 2 The Navier Stokes Equations 8 gaigvgradviluAvgradp f divv 0 where A vector Laplacian div grad Application An InitialBoundaryValue Problem B C gt H a a a A F E D Figure 81 Geometry of the backstep channel ow Initial conditions vw0 v0w where the initial eld must satisfy div v0 07 which implies that v0notA07 iev0n00n890 390 Boundary conditions 1 On segment BC and DE U EF U FA7 we have the no slip boundary condition v O 45 CHAPTER 8 EXAMPLES AND APPLICATIONS 2 On the in ow boundary77 AB7 we prescribe the Poiseuille ow velocity pro le see Fig 82 2 11107 270 7 gt U0 a U207 27 If 0 3 On the out ow boundary77 CD7 there are several possibilities A commonly used one is 8111 7 7 Te 0 29 M 8351 mid limpr 112L7 27 If 0 where L is the length of the channel7 ie L lBCl U0 Figure 82 Poiseuille ow velocity pro le a a y H V Z J A x recirculation Figure 83 Recirculation in backstep channel ow 83 Application of the Principle of Physical Consistency Instead of writing the equations governing the conservation of energy or the Clausius Duhem inequality in terms of the internal energy 5 per unit volume7 it is often convenient to replace 5 by the Helmholtz free energy w6n0 46 84 HEAT CONDUCTION Then the Clausius Duhem inequality becomes eg i i i 1 90110 7 907100 S E 7 a 10 V0 2 0 See Exercise Suppose we have a constitutive equation for 110 which we initially take to be of the form we IIEt97 V0 Then A 8 1 A 811 8 11 1 wo E0TV V0 and 8 1 1 8 1 8 11 1 1 i 7 7 7 i i if gt goltn080gt0lts 908EgtE908WW eqo V070 This implies that 7 8 1 sf 8 1 8 1 70 7 0 7 go ave Thus7 the Clausius Duhem inequality allows us to 1 conclude that 1 does not depend on V0 and 2 conclude that no and S can be charac terized through a single energy functional 11 Remark The quantity 60 8137 901ZJO Clo 7 TO 1 900770 is called the internal dissipation According to the second law Clausius Duhem 1 60 7 ago Z 0 This is the Clausius Planck inequality If 60 07 it asserts that heat must ow from hot to cold But 60 may not be zero 84 Heat Conduction lgnoring motion and deformation for the moment7 consider a rigid body being heated by some outside source The constitutive equations are 1 0 10 5002 c constant 10 kV0 Fourier s Law 47 CHAPTER 8 EXAMPLES AND APPLICATIONS Then EL 770 i 80 In a reversible process 60 0 So 700 and S go 0 irrelevant 83 QE DiV Clo To 090770 0 Setting 0770 7009 z 70000 where 00 reference temperature gt O we get 80 i 7 k 0 7 9060 at V V r0 where 00 000 This is the classical heat conduction diffusion equation 85 Theory of Elasticity We consider a deformable body 3 under the action of forces body forces f and prescribed contact forces Un w gw on 890 The body is constructed on a material which is homogeneous and isotropic and is subjected to only isothermal 0 const and adiabatic q 0 processes The sole constitutive equation is 11 free energy X 250 E The constitutive equation for stress is thus 8w s 7 8E sym In this case the free energy is called the stored energy function or the strain energy function Since and 8118E must be formiinvariant under changes of the observer and since 3 is isotropic must depend on invariants of E E WIE7 HE7 ME Or since E C 7 I2 we could also write 1 as a function of invariants of C 1 171710 1101210 The constitutive equation for stress is then 8W 81E 8W 811E 8W 8111 QIE QE QUE QE QME QE 8W 810 8 8110 8 81210 Q10 QC QHC QC Q1110 QC 48 85 THEORY OF ELASTICITY and we note that 81E 7 1 BE 88 tr E 117E TCofE 8111 W 7 CofE Materials for which the stress is derivable from a stored energy potential are called hypere lastic materials The governing equations are 8W 82u D I V f 7 1V u8E gym 0 Qoatz with F 1 Vu 8W S 7 7113113111313 8E gym E Vu VuT VuTVu Linear Elasticity 1 E z e 5Vu VuT 1 i39jkl W 5E EM 517 8W 81 SH 1 Ewklekl EWMTXZ Hooke s Law Eijkl Eji39kl Eijlk Ekli j Then7 for small strainsdisplacements7 3 E 81 1017 321 an ljklaXl Oi Q atz For isotropic n1aterials7 Eijkl 35171379 Mlt6ik6jl 6il6jk 49 CHAPTER 8 EXAMPLES AND APPLICATIONS where 7 u are the Lame constants A7 11E 7 E HzM14147 quot 21V E is the Young s Modulus and 1 the Poisson s Ratio Then S tr EI 2pE div 11 1 2MVusym i H 814k 31 814739 S A 6 ltan M ltan 8X1 and the Lame equations of elastostatics Wu8252 0 are 821 lt 32m 32m 7 7 39 lt 4 4 lt Aanan M mm anaXl f0 1 W 7 3 8 6 EXERCISES 86 Exercises 1 to It is often considered useful to write the rst and second laws of thermodynamics in terms of the so called free energy rather than the internal energy The scalar eld A e i 0n is called the Helmholtz free energy per unit volume Show that dd dn EiTDid1vqri0E7 d0 ndt and that dd 10 1 777 7 TD77 0gt0 dt ndt l 001 V i or equivalently i i i 1 110 7100 l SCE gCIo V020 Consider the small deformations and heating of a thermo elastic solid constructed of a material characterized by the following constitutive equations Free energy 110 tr e2 ue e Ctr e0 07002 Heat Flux qO kV0 where 1 e 5Vu VuT the in nitesimal strain tensor z E u the displacement eld 9 the temperature eld u c 00 k material constants A body 3 is constructed of such a material and is subjected to body forces f0 and to surface contact forces g on a portion Pg of its boundary Pg C 800 On the remainder of its boundary Pu 1390Pg the displacements u are prescribed as zero u 0 on Pu The mass density of the body is go and when in its reference con guration at time t O uw0 u0w 8uw 08t V0w w E 90 where uo and V0 are given functions A portion Fq of the boundary is heated resulting in a prescribed heat ux h q n and the complementary boundary F9 1300Pq is subjected to a prescribed temperature 0wt Twt w 6 F9 eg is immersed in ice water Develop a mathematical model of this physical phenomena a set of partial differential equations boundary and initial conditions the dynamic thermomechanical behavior of a thermoelastic solid CHAPTER 8 EXAMPLES AND APPLICATIONS CHAPTER 9 Assignments Things you should know 0 Linear algebra and matrix theory 0 Vector calculus 0 Index notation 0 Introductory real analysis Index Notation and Symbolic Notation 0 Let ei i 1 23 be an orthonormal basis ie 1 ifz j ei39ej6 0 ifz y j 0 Let a be a vector a ai ei repeated indices are summed Two vectors a and b are equal ie a b if a bi i123 0 Cross product a x b gigk aibj ek 1 if ijk even permutation gigk 71 if ijk odd permutation 0 if ijk is not a permutation o Nabla V 13187 azk o Divergence of a vector denoted div V or V V 3 311739 311739 811k 7 7 7 7 8 V V ek k U787 8k 8k k 8k Uhk or kvk 8 ek 8739 o Curl of a vector denoted curl V or V x V 8 V gtlt V Eijkavjek Eijkvmek Eijk8wjek l 53 o The following relations hold 6 3 6ij6jk 5 gijkgijm 26km 7 gijkgijk 6 0 Typical identities 1 VVV VVV7Vgtlt Vx V 2 Vx VgtltWWVVVVW7WVV7VVW o Tensors ei X ej tensor product of ei and e A second order tensor Z Aijei X ej A ei X ej Al7 ei Aej M B third order tensor Bigk 5139 X ej X ek Assignment 1 Vectors and Index Notation 1 Prove that VVV VltVV7Vgtlt Vx V 2 Prove that Vx wa WVVVVw7wVV7VVw Vectors and Inner Product Spaces 3 Give a complete de nition and a non trivial example of a a real vector space 10 an inner product space c a linear transformation from a vector space U into a vector space V Tensors 4 Let V be an inner product space A tensor is a linear transformation from V into V If T is a tensor7 TV denotes the image of the vector V in V TVTV E V Show that the class LVV of all linear transformations of V into itself is also a vector space with vector addition and scalar multiplication de ned as follows ST E LVV STRltgtRVSVTV as RltgtRVaSV VV 6 V Va 6 R 0 E LVV 0V0 E V Tensor Product 5 The tensor product of two vectors a and b is the tensor7 denoted a X b7 that assigns to each vector c the vector b ca that is a bc b ca 51 Show that a X b is a tensor and that a X bT b X a a bc d b ca d 55 52 If 81132 133 is an orthonormal basis 13 ej 6ij1 ij 3 MelH2 eiei 1 then show that 0 ifi 79 e ee o e p I 4 Zjel 8 3 ei if z j 53 For an arbitrary tensor A and for the orthonormal basis 81132 33 AZAi7 ei ej M where A 8139 A8739 The array Alj is the matrix characterizing A for this particular choice of a basis for V R3 If A and B are two second order tensors and Alj Blj are their matrices corresponding to a basis e1 e2 eg of V de ne construct the rules of matrix algebra a A B C 5 Mel Bijl W b ABC ABAoB A0 C C 7 0 the zero element of LV CL c AC I I is the identity tensor AI IA A and C A71 e AT C C 7 AT is the transpose of A it is the unique tensor such that AV u V ATu 77 being the vector inner product in R3 6 The inner product dot product of two vectors u V E V R3 is denoted uV It is a symmetric positive de nite bilinear form on V lfu 21 uiei and V 21 Viei for an orthonormal basis ei then uV 21 mm Moreover the Euclidean norm ofu v is Hull m M W The space LV V of second order tensors can be naturally equipped with an inner product as well and hence a norm The construction is as follows i The trace of the tensor product of two vectors u and V is the linear operation tru V if u V Likewise the trace of a tensor A E LV V is de ned by trAtr ei ej ZAijtrei 8jZAij6ijZAii ij ij ij i 56 ii The trace of the composition of two tensors A7 B E LV7 V is then trAB Z Aiiji ii We denote A B trATB AljBlj ii a Show that A B the operation de nes an inner product on LV7 V b De ne the associated norn1 of A E IV7 V c Show that tr A tr AT d Show that trivially I A tr A I identity tensor u V t q p uqVp Assignment 2 3 A 1252 I 1 i X a I r I 90 IX l quot39quotquotquot i 83 82 a m 81 K 1 O lt gt l a l Figure 91 The cube 0 07 13 K39 quot of Continual Media 1 The reference con guration of a deformable body 3 is the cube 0 07 13 with the origin of the spatial and material coordinates at the corner7 as shown in Fig 91 Consider a motion LP of the body de ned by 3 WK ZWO W i1 where Where A is a real number a parameter possibly depending on time t For this motion7 a Sketch the deformed shape ie sketch the current con guration in the Xng plane for A a Then compute the following b the displacement eld u c the deformation gradient F d the deformation tensor C e the Green strain tensor E f the extensions ei i 1 23 g sin V12 where V12 is the shear in the X17X2 plane Determinants 2 Let A E LV V be a second order tensor with a matrix Alj relative to a basis egg1 ie the A are components of A with respect to 13 X ej For n 3 the determinant of AM is de ned by 1 detlAil 8 Z gigksmtAiAjsAkt ijk 39rst where 1 if ij k is an even permutation of ij k gigk 71 if ij k is an odd permutation of ij k 0 if ij k is not a permutation of ij k ie if at least two indices are equal The determinant of the tensor A is de ned as the determinant of its matrix compo nents AU det A det This de nition is independent of the choice of basis ei ie detA is a property of A invariant under changes of basis Show that a detAB det Adet B for n3 is suf cient b detAT detA det AB det A det B Take n 3 det A gklmgabcAkaAlemc det B gklmgdekadBleBmf But skim det A gabcAkaAlemc gum det B EdekadBleAmc for example det A EabcAmAszsg 59 Note that Eklmgkgm 6 Thus gum det A sum det B 6 det A det B 6 gdefgabcAkaBkd Aleel Achfm 6 det AB Cofactor and Inverse 3 For A E LV V and A the components of A relative to a basis ei i 1 2 n let Ag be the elements of a matrix of order n 7 1 obtained by deleting the ith row and jth column of AM The scalar 1 71 739 dew11 is called the ij7cofactor of Alj and the matrix of cofactors Cof A digl is called the cofactor matrix of A a Show that ACof AT detAI it is suf cient to show this for n 3 by construction b Show trivially that for A invertible A l detA 1CofAT c For n 3 show that 1 Cof A 5 Z sijksmAlTAjS 1 g k t 3 ijk 39rst Orthogonal Transformation 4 A tensor Q E LV V is orthogonal if it preserves inner products in the sense that Qu Qv u v Vuv E V R3 Show that a necessary and suf cient condition that Q be orthog onal is that or equivalently 5 Con rm that detA is an invariant of A in the following sense if A 14 8i 3 2AM i i M M where e Qei i 1 2 3 Q an orthogonal tensor then detlAijl detlAijl Review of Vector and Tensor Calculus 6 Let U and V be nite dimensional normed spaces and let f be a function from U into V We say that f is differentiable at u E U if there exists a linear functional D f on V such that 1 Dfu V 6limj 6fu 0V 7 fu Dfu is the derivative of f at u Let 4p be a scalar valued function de ned on the set S invertible tensors A E LU U ie 4p S a R de ned by MA det A Show that it is suf cient to consider the 3D case D A V detAVT A l Hint Note that detA 0V detI 0VA 1A detAdetI 0VA 1 and that detI 013 1 0 tr B 002 7 Let Q be an open set in R3 and 4p be a smooth function mapping 9 into R The vector g with the property D4pw V gw V VV 6 R3 is the gradient of V at point w E Q We use the classical notation we WM 61 00 to Show that VltLpVgt ApVV V X Vap div4pV 4p div V V th divV X w V div w VVW Let 4p and u be CZ scalar and vector elds Show that a curl th 0 b div curl V 0 Let Q be an open connected smooth domain in R3 with boundary 89 Let n be a unit exterior normal to 89 Recall the Green divergence theorem diVVdm VndA Q 39 for a vector eld V Show that divAdz AndA 9 an Note that for arbitrary vector u u AndA ATundA 39 39 If myz is a Cartesian coordinate system with origin at the corner of a cube 0 0 13 and axes along edges of the cube compute V n dAo 390 where 890 is the exterior surface of 90 n is the unit exterior normal vector and V is the eld V zel yeg l 3233 Hint 1diV v diVergence of v tr Vv avg8m if v viei diV A is the unique Vector eld such that diV A a diVATa for all Vector a 62 Assignment 3 1 The cube 07 13 is the reference con guration 90 of a body subjected to simple to EJgtOJ U a shear77 A ul 3X27 U2 U3 07 A constant The unit exterior normal to 890 is no The vector no at boundary point 017 0127 012 is mapped into the unit exterior normal 11 on 89 Compute n and sketch the deformed body Suppose A in Problem 1 above is a function of time t Compute a the Lagrangian description of the velocity and acceleration elds7 b the Eulerian descriptions of these elds7 c L grad V Show that WV w x V Suppose that at a material point X E 90 the Green Strain tensor is given by E Z Eijlei 8739 193733 where ei is an orthonormal basis in R3 and Eijl 0amp0 ow HOD Determine a the principal directions of E7 b the principal values of E7 c the transformation Q that maps ei into the vectors de ning the principal directions of E7 d the principal invariants of C 2E I Recall that an invariant of C is any real valued function MC such that MC MA lCA for all invertible matrices A Show that tr C7 tr Cof C7 and det C are invariants of C Construct a detailed proof of the relations 1 a WV 5w x V7 where w curl V b det F det F div v Assignment 4 1 OJ F Reproduce the proof of Cauchy s Theorem for the existence of the stress tensor for the two dimensional case plane stress to simplify geometric issues Thus for Un 01ne1 02ne2 show that 3T such that Un Tn The Cauchy stress tensor in a body 3 is Tt Tijt 8139 8739 where 1000095 7 7000951 x2 7000951 x3 2000953 700013 300035 100ml 2000353 100351 1000353 11 t 610710t a At point w 111 at time t 1 compute the stress vector Un in the direction n niei n 1x3 i 1 2 3 b At t 1 what is the total contact force on the plane surface 1 1 0 mg g 1 0 mg g 1 Let T Tw t be the Cauchy stress at w 6 Q at time t If n is a direction a unit vector such that Tn an T 1 then in analogy with principal values and directions of C a is a principal stress and n is a principal direction of T eigenvalues and eigenvectors of T Continuing let P be a plane through a point m with unit normal n The normal stress an at w is an n Tnn and the shear stress is at Tnio39nTn7 nTnn Show that if n were a principal direction of T then at 0 Newton s Law of action and reaction asserts that for each m 6 t and each t Un wt 707n wt Prove this law under the same assumptions as Cauchy s theorem 64 distance 1 es Figure 92 Tetrahedral element Hint a Construct the tetrahedral element as shown in Fig 92 b Compute total forces on tetrahedral element 1 1 3 7 UmdA 7 07eidA body force 0d A6 Ad Adi Adz c Take limit as d a 0 giving 039n7 w t 7 21m eiaieiwt d Take In 31 then In 32 then In eg to establish that 171317 w infiei7 w for any 15 Then the assertion follows e Complete the proof of Cauchy s Theorem Show Tw TwT This proof makes use of 1 the principle of balance of angular momentum7 and 2 the equa tions of motion diV T f ngdt Assignment 5 1 Let P be a material surface associated with the reference con guration P C 89 Let g be an applied force per unit area acting on Hg gw7 t7 w E P The traction boundary condition on P at each m E P is Tn g Show that FSno go on 90711 where no is the unit exterior normal to F0 P LpP0 and 00975 detFXHF TXHoH 9196 to Consider an Eulerian description of the ow of a uid in a region of R3 The ow is characterized by the triple V7 9 T velocity eld7 density eld7 Cauchy stress eld The ow is said to be potential if the velocity is derivable as the gradient of a scalar eld 4p V grad 4p The body force eld acting on the uid is said to be conservative if there is also a potential U such that f g grad U The special case in which the stress T is of the form T ipl where p is a scalar eld and I is the unit tensor7 is called a pressure eld p the uid pressure or hydrostatic pressure a Show that for potential ow7 a pressure eld T 1917 and conservative body forces7 the momentum equations imply that grad 8ilVVU lgradp0 8t 2 g This is Bernoulli s Equation for potential ow Hint show that 1 div T igrad 197 2 dVdt 8V8t grad V V 66 03 F b If the motion is steady ie if 8V8t O7 Vw7 t is invariant with respect to 257 and 8f8t 07 so 8U8t 07 then the equations of part a reduce to 1 1 Vgradlt VVUgt7VgradV0 9 Notice that V dVdt V grad V Let u uX7 t 40X t 7 X be the displacement eld De ne the quantity 1 11 7 you udX 2 90 Show that if f0 07 1 gouthi PVudX uPn0dA0 no 90 390 A cylindrical rubber plug 1 cm in diameter and 1 cm long see Fig 93 is glued to a rigid foundation Then it is pulled by external forces so that the at cylindrical upper face F0 X17X27X3 X3 17 X12 1 X 12 12 is squeezed to a at circular face P of diameter 14 cm with normal n 32 as shown at position w w Suppose that the stress vector at w w is uniform and normal to P Unw Uegw 1000e2 kgcmz w E P Suppose that the corresponding Piola Kirchhoff stress p0 Pno is uniform on F0 and normal to To a Determine the Piola Kirchhoff stress vector p0 Pno on P0 b Determine one possible tensor Cof FX17 X27 1 for this situation X2 Figure 93 Illustrative sketch for question 4 of assignment 5
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