CH.3. COMPATIBILITY EQUATIONS. Multimedia Course on Continuum Mechanics

Size: px
Start display at page:

Download "CH.3. COMPATIBILITY EQUATIONS. Multimedia Course on Continuum Mechanics"

Transcription

1 CH.3. COMPATIBILITY EQUATIONS Multimedia Course on Continuum Mechanics

2 Overview Introduction Lecture 1 Compatibility Conditions Lecture Compatibility Equations of a Potential Vector Field Lecture 3 Compatibility Conditions for Infinitesimal Strains Lecture 4 Integration of the Infinitesimal Strain Tensor Lecture 5 Integration of the Deformation Rate Tensor Lecture 6

3 3.1 Compatibility Conditions Ch.3. Compatibility Equations 3

4 Introduction Given a displacement field, the corresponding strain field is found: U( X,t) u( x,t) E ε ij ij 1 U U i j Uk U k = + + i, j {1,, 3} X j Xi Xi X j 1 u u i j = + i, j {1,, 3} j i Is the inverse possible? ε ( x,t) u( x,t) 4

5 Compatibility Conditions Given an (arbitrary) symmetric second order tensor field, ε x,t, s a displacement field, u( x,t), fulfilling ux (,) t = ε( x, t) cannot always be obtained: 1 u u i j εij = + i, j {1,, 3} 6 PDEs OVERDETERMINED j 3 unknowns SYSTEM i For ε( x,t) to match a symmetric strain tensor: It must be integrable. There must exist a displacement field from which it comes from. COMPATIBILITY CONDITIONS must be satisfied REMARK Given u x,t, there will always exist an associated strain tensor, ε( x,t), obtainable through differentiation, which will automatically satisfy the compatibility conditions. 5

6 Compatibility Conditions The compatibility conditions are the conditions a symmetric nd order tensor must satisfy in order to be a strain tensor and, thus, exist a displacement field which satisfies: 1 u u i j εij = + i, j {1,, 3} j i They guarantee the continuity of the continuous medium during the deformation process. E( X,t) Incompatible strain field 6

7 3. Compatibility Equations of a Potential Vector Field Ch.3. Compatibility Equations 7

8 Preliminary example: Potential Vector Field v( x,t) A vector field will be a potential vector field if there exists a scalar function φ( x,t) (named potential function) such that: (, t) = φ( x, t) φ( x, t) ( x t) v x { } v i, = i 1,,3 Given a continuous scalar function potential vector field v( x,t). Is the inverse true? v( x,t) (,t) i φ( x,t) there will always exist a φ x such that φ ( x, t) = v( x, t) 8

9 Potential Field v( x,t) (,t) φ x such that φ ( x, t) = v( x, t) In component form, ( x t) ( x t) φ, φ, v i( x, t) = v i( x, t) = 0 i 1,,3 i i { } 3 eqns. 1 unknown OVERDETERMINED Differentiating once these expressions with respect to : SYSTEM x j φ ( x, t) vi = i, j 1,,3 j j i { } 9 eqns. 9

10 Schwartz Theorem The Schwartz Theorem about symmetry of second partial derivatives guarantees that, given a continuous function Φ ( x1, x,..., xn ) with continuous derivatives, the following holds true: Φ Φ = i, j i j j i 10

11 Compatibility Equations Considering the Schwartz Theorem, vx φ vx φ vx φ = = = y xy z xz vy φ vy φ vy φ = = = yx y y z yz vz φ vz φ vz φ = = = z x y z y z z In this system of 9 equations, only 6 different nd derivatives of the unknown φ( x,t) appear: φ φ φ φ φ φ,,,, and y z y z y z They can be eliminated and the following identities are obtained: v vy v v vy v = = = y z z y x x z z 11

12 Compatibility Equations A scalar function φ x,t which satisfies φ x, t = v x, t will exist if the vector field v x,t verifies: v y v z v y INTEGRABILITY (COMPATIBILITY) EQUATIONS of a potential vector field x z def vx = 0 = S y def v z = 0 = S v def y = 0 = S z z y x where v = 0 v v i j = 0 i, j 1,,3 xj x i eˆ1 eˆ ˆ e3 S x S S y v y z S z v v v { } x y z REMARK A functional relation can be established between these three equations. ( v) 0 = 1

13 3.3 Compatibility Conditions for Infinitesimal Strains Ch.3. Compatibility Equations 13

14 Infinitesimal strains case The infinitesimal strain field can be written as: u 1 x u u x y 1 ux u z + + y x z εxx εxy ε xz uy 1 uy u z ε = εxy εyy εyz = + y z y εxz εyz ε zz u z symmetrical z 6 PDEs 3 unknowns 14

15 Infinitesimal strains case The infinitesimal strain field can be written as: ε ε ε xx yy zz u 1 x u u x y = 0 εxy + = 0 y x u y 1 ux u z = 0 εxz + = 0 y z uz 1 u y u z = 0 εyz + = 0 z z y 6 PDEs 3 unknowns The system will have a solution only if certain compatibility conditions are satisfied. 15

16 Compatibility Conditions The compatibility conditions for the infinitesimal strain field are obtained through double differentiation (single differentiation is not enough). u x ε xx,,,,, x y z xy xz yz 1 u y u z ε yz + z y =, y, z, y, z, yz = 6 equations 6 equations 6x6=36 equations 16

17 Compatibility Conditions 17 The compatibility conditions for the infinitesimal strain field are 3 obtained through: 3 3 ε xx u ε yz 1 u x y u z = = + 3 zx yx 18 equations for εxy, εxz, εyz 18 equations for εxx, εyy, εzz ε xx u ε 1 yz u x y u z = = + 3 y xy y zy y ε xx u ε x yz 1 uy u z =... = + 3 z xz z z yz 3 3 ε xx u ε x yz 1 3 uy u z = = + xy y xy zxy y x ε xx u ε x yz 1 uy u z = = + xz z xz z x yxz ε xx u ε x yz 1 uy u z = = + yz xyz yz z y y z

18 Compatibility Conditions All the third derivatives of ux, uy and uz appear in the equations: 3 ux 3 3 3, y, z, y, y x, y z, z, z x, z y, yz = 10 derivatives 3 y u, y, z, y, y x, y z, z, z x, z y, yz 3 uz 3 3 3, y, z, y, y x, y z, z, z x, z y, yz = = 10 derivatives 10 derivatives which constitute 30 of the unknowns in the system of 36 equations: f n 3 u ε i ij, = 0 n 1,,...,36 jkl kl 30 { } 18

19 Compatibility Equations 19 Eliminating the 30 unknowns, 3 ui j k l only strain derivatives) are obtained: def εyy ε ε zz yz Sxx = + = 0 z y yz def ε zz εxx εxz S yy = + = 0 z xz def ε ε xx yy εxy Szz = + = 0 y xy def ε ε zz yz ε ε xz xy Sxy = + + = 0 xy z x y z def εyy εyz ε ε xz xy Sxz = + + = 0 xz y y z def ε ε xx yz ε ε xz xy S yz = = 0 yz y z, 6 equations (involving COMPATIBILITY EQUATIONS for the infinitesimal strain tensor S = ( ε ) = 0

20 Compatibility Equations The six equations are not functionally independent. They satisfy the equation, In indicial notation: S = ε = 0 S S xx xy Sxz + + = 0 y z Sxy Syy Syz + + = 0 y z S S xz yz Szz + + = 0 y z 0

21 Compatibility Equations The compatibility equations can be expressed in terms of the permutation operator,. e ijk S = e e ε, = 0 ml, 1,,3 ml mjq lir ij qr Or, alternatively: { } εij, kl + εkl, ij εik, jl ε jl, ik = 0 i, jkl,, 1,, 3 REMARK Any linear strain tensor (1 st order polynomial) with respect to the spatial variables will be compatible and, thus, integrable. 1

22 3.4 Integration of the Infinitesimal Strain Tensor Ch.3. Compatibility Equations

23 Preliminary Equations Rotation tensor Ω x,t : Rotation vector θ x,t : 1 Ω= skew( u ) = ( u u) 1 u u i j Ω ij = i, j {1,, 3} j i θ1 Ω3 Ω yz 0 θ3 θ 1 θ = u = θ = Ω = Ω [ Ω ] = θ 0 θ θ 31 zx 3 1 θ3 Ω 1 Ω xy θ θ1 0 3

24 Preliminary Equations Differentiating Ω x,t with respect to : 1 u u i j Ωij 1 u u i j Ω ij = = xj x i k k j i x k Adding and subtracting the term k : 1 u Ωij 1 u u i j 1 uk 1 uk = + = k k j i ij ij 1 ui u k 1 u j u k ε ε ik = + + = j xk xi xi xk x j j i = ε ik = ε jk i j jk 4

25 Preliminary Equations Using the previous results, the derivative of θ θ Ω 1 yz ε ε θ Ω ε ε xz xy = = = = x x z x x x y z θ Ω εxy ε zx θ Ω ε ε θ = = y y z x y y y z θ Ω ε ε θ Ω 1 yz ε ε zz zy = = = = z z z x z z y z θ Ω 3 xy ε xy εxx = = x x x y θ Ω 3 xy εyy εxy θ3 = = y y x y θ Ω 3 xy ε yz εxz = = z z x y 1 yz yz yy 1 = = θ x,t is obtained: zx xx xz yz zx xz zz 5

26 Preliminary Equations 6 Considering the displacement gradient tensor J x,t, (, t) u x J = = ε+ Ω = ε ij ui 1 u u i j 1 u u i j Jij = = + + = εij +Ωij i, j 1,,3 j j i j i Introducing the definition of are rewritten: i = 1: i = : i = 3: = Ω ij, the components of { } θ( x,t) J( x,t) j = 1 j = j = 3 ux ux ux = εxx = εxy θ3 = εxz + θ y z uy uy uy = εxy + θ3 = εyy = εyz θ1 y z uz uz uz = εxz θ = εyz + θ1 = εzz y z

27 Integration of the Strain Field The integration of the strain field steps: ε( x,t) is performed in two 1. Integration of derivative of θ x,t using the1 st order PDE system derived for θ, θ and θ. The solution will be of the type: 1 3 ( xyzt,,, ) c( t) i { 1,, 3} θ = θ + i i i The integration constants ci t can be obtained knowing the value of the rotation vector in some points of the medium (boundary conditions). ε( x ). Known,t and θ x,t, u is integrated using the 1 st order PDE system derived for u REMARK. The solution will be: If the compatibility equations ui = u i( xyzt,,, ) + c i ( t) i { 1,, 3} The integration constants c i ( t) can be obtained knowing the value of are satisfied, the displacements in some point of space (boundary conditions) these equations will be integrable. 7

28 Integration of the Strain Field The integration constants that appear imply that an integrable strain tensor ε( x,t) will determine the movement in any instant of not not time except for a rotation c() t = θˆ () t and a translation c () t = uˆ () t : ˆ θ x, t = θ x, t + θ t ε x, t u( x, t) = u ( x, t) + uˆ ( t) A displacement field can be constructed from this uniform rotation and translation: u ( x,) t = Ωˆ ( ˆ θ ()) t x+ uˆ () t u =Ωˆ S * 1 T 1 ˆ ˆ T ( u ) = ( u + ( u ) ) = ( Ω+Ω ) = 0 This corresponds to a rigid solid movement. 8

29 3.5 Integration of the Deformation Rate Tensor Ch.3. Compatibility Equations 9

30 Compatibility Equations in a Deformation Rate Field There is a correspondence between The concept of compatibility conditions can be extended to deformation rate tensor j i ij j i j i ij j i u u x x u u x x ε = + Ω = = u u u ε θ v v 1 v v 1 w 1 j i ij j i j i ij j i d x x x x = + = = v dv v ω d v 30

31 Example Obtain the velocity field corresponding to the deformation rate tensor: such that In point ( 1, 1, 1) ty 0 te 0 ty d( x, t) = te 0 0 tz 0 0 te the following conditions is fulfilled: t v( x, t) = e ω( x t) x= ( 1,1,1) e e t t 0 1, = v= 0 x= ( 1,1,1) t te 31

32 Example - Solution ty 0 te 0 ty d( x, t) = te 0 0 tz 0 0 te Consider the correspondence: u ε( u) 1 θ= u v dv 1 ω= v Take the expressions derived for θ1, θ θ3 substitute,t with and x,t with d x,t : ε ω 1 d d xz xy = = 0 0 x y z ω d d 0 0 y y z ω d 1 d zz zy = = 0 0 z y z 1 yz yy ω 1 = = and θ( x ) ω( x,t) ( t) C ( t) ω = 1 1 3

33 Example - Solution ty 0 te 0 ty d( x, t) = te 0 0 tz 0 0 te ω dxx dxz = = 0 0 x z x ω d d 0 0 y z x ω d xz d zz = = 0 0 z z x xy yz ω = = ω d 3 xy d xx = = 0 0 x x y ω d d y x y ω d 3 yz d xz = = 0 0 z x y 3 yy xy ty ω 3 = = 0 te ( t) C ( t) ω = ty ty ( y, t) t e dy te C ( t) ω = =

34 Example - Solution 1 1 ω =C t ω =C t ty 3 3 ω = te + C t So, For point 1, 1, 1 : 0 1 { ω( x, t) } = v= 0 t te 1 1 ω = 0 = C t ω = 0 = C t t ty ω 3 = te = te + C3 t Therefore, for any point, x = { ω( x t) } ( 1,1,1) 0, = 0 ty te C C C 1 3 ( t) ( t) ( t) = 0 = 0 = 0 34

35 Example - Solution { ω( x t) } 0, = 0 ; ty te ty 0 te 0 ty d( x, t) = te 0 0 tz 0 0 te Taking the expressions i = 1: i = : The components of the velocities can be obtained: v v y x x v z x = d = 0 xx 3 = d ω = te te = te xy = d +ω = 0+ 0 xz j = 1 j = j = 3 v v v = d = d ω = d +ω y z x x x xx xy 3 xz v v v = d +ω = d = d ω y z y y y xy 3 yy yz 1 v v v i = 3: = d ω = d +ω = d y z ty ty ty z z z xz yz 1 zz ty ty = = + v, x y t te dy e C1 t 35

36 Example - Solution { ω( x t) } 0, = 0 ; ty te ty 0 te 0 ty d( x, t) = te 0 0 tz 0 0 te The components of the velocities can be obtained: v v v = d +ω = te + te = 0 y ty ty xy 3 y y y z = d = 0 yy = d ω = 0 0 yz 1 = v y t C t v v y z z v z z = d ω = 0 0 xz = d +ω = 0+ 0 yz zz 1 = d = te tz tz tz = = + v, z z t te dz e C3 t 36

37 Example - Solution v tz z = e + C 3 t v ty x = e + C 1 t v y = C t For point 1, 1, 1 : { v( x, t) } t e t = e t e So, v = = + t v y = e = C ( t) v Therefore, for any point, t ty x e e C1 t = = + t tz z e e C3 t x = x = ( 1,1,1) ( 1,1,1) { v( x, t) } ty e t = e tz e 1 3 ( t) ( t) C = 0 C t = e C = 0 t 37

38 Chapter 3 Compatibility Equations 3.1 Introduction Given a sufficiently regular displacement field U(X, t), it is always possible to find the corresponding strain field (for example, the Green-Lagrange strain field) by differentiating this strain field with respect to its coordinates (in this case, the material ones) 1, E ij = 1 ( Ui + U j + U ) k U k not = 1 Ui, j +U j,i +U k,i U k, j X j X i X i X j (3.1) i, j {1,,3}. In the infinitesimal strain case, given a displacement field u(x, t), the strain field ε ij = 1 ( ui + u ) j not = 1 j i (u i, j + u j,i ) i, j {1,,3} (3.) is obtained. The question can be formulated in reverse, that is, given a strain field ε (x,t), is it possible to find a displacement field u(x,t) such that ε (x,t) is its infinitesimal strain tensor? This is not always possible and the answer provides the socalled compatibility equations. Expression (3.) constitutes a system of 6 (due to symmetry) partial differential equations (PDEs) with 3 unknowns: u 1 (x,t), u (x,t), u 3 (x,t). This system is overdetermined because there exist more conditions than unknowns, and it may not have a solution. Therefore, for a second-order symmetric tensor ε (x, t) to correspond to a strain tensor (and, thus, be integrable and there exist a displacement field from which it comes) it is necessary that this tensor verifies certain conditions. These conditions are denominated compatibility conditions or equations and guarantee Continuum Mechanics for Engineers Theory and Problems X. Oliver and C. Agelet de Saracibar 1 Here, the simplified notation U i / X j not = U i, j is used. 109

39 110 CHAPTER 3. COMPATIBILITY EQUATIONS Figure 3.1: Non-compatible strain field. the continuity of the continuous medium during the deformation process (see Figure 3.1). Definition 3.1. The compatibility conditions are conditions that a second-order tensor must satisfy in order to be a strain tensor and, therefore, for there to exist a displacement field from which it comes. Remark 3.1. Note that, to define a strain tensor, the 6 components of a symmetric tensor cannot be written arbitrarily. These must satisfy the compatibility conditions. Remark 3.. Given a displacement field, one can always obtain, through differentiation, an associated strain field that automatically satisfies the compatibility conditions. Therefore, in this case, there is no sense in verifying that the compatibility conditions are satisfied. Continuum Mechanics for Engineers Theory and Problems X. Oliver and C. Agelet de Saracibar 3. Preliminary Example: Compatibility Equations of a Potential Vector Field A given vector field v(x,t) is a potential field if there exists a scalar function φ (x,t) (named potential function) such that its gradient is v(x,t), v(x,t)= φ (x,t), v i (x,t)= φ(x,t) i i {1,,3}. (3.3) X. Oliver and C. Agelet de Saracibar Continuum Mechanics for Engineers.Theory and Problems doi: /rg

40 Preliminary Example: Compatibility Equations of a Potential Vector Field 111 Therefore, given a scalar (continuous) function φ (x, t), it is always possible to define a potential vector field v(x,t) such that the scalar function is its potential, as defined in (3.3). Now, the reverse question is posed: given a vector field v(x,t), does there exist a scalar function φ (x,t) such that φ (x,t) =v(x,t)? This is written in component form as v x = φ v y = φ y v z = φ z = v x φ = 0, = v y φ y = 0, = v z φ z = 0, (3.4) which corresponds to a system of PDEs with 3 equations and 1 unknown (φ (x,t)), thus, the system is overdetermined and may not have a solution. Differentiating once (3.4) with respect to (x,y,z) yields v x = φ, v y = φ y, v z = φ z, v x y = φ y, v y y = φ y, v z y = φ z y, v x z = φ z, v y z = φ y z, v z z = φ z, (3.5) which represents a system of 9 equations. Considering the equality of mixed partial derivatives, it is observed that 6 different functions (second derivatives) of the unknown φ are involved in these 9 equations, φ, φ y, φ z, φ y, ϕ z and φ y z. (3.6) So, they can be removed from the original system (3.5) and 3 relations, named compatibility conditions, can be established between the first partial derivatives of the components of v(x,t). Hence, for there to exist a scalar function φ (x,t) such that φ (x,t)=v(x,t), the given vector field v(x, t) must satisfy the following compatibility conditions. v y v x y = 0 def = S z ê 1 ê ê 3 v x z v z = 0 def = S y v z y v y z = 0 def = S x Continuum Mechanics for Engineers Theory and Problems X. Oliver and C. Agelet de Saracibar where S not S x S y S z y z v x v y v z not rot v not = v (3.7) X. Oliver and C. Agelet de Saracibar Continuum Mechanics for Engineers.Theory and Problems doi: /rg

41 11 CHAPTER 3. COMPATIBILITY EQUATIONS In consequence, from (3.7), the compatibility equations can be written as Compatibility equations of a potential vector field v = 0 v i v j = 0 i, j {1,,3} j i (3.8) Remark 3.3. The 3 compatibility equations (3.7) or(3.8) are not independent of one another and a functional relation can be established between them. Indeed, applying the condition that the divergence of the rotational of a vector field is null, ( v)= Compatibility Conditions for Infinitesimal Strains Consider the infinitesimal strain field ε (x,t) with components ε ij = 1 ( ui + u ) j not = 1 j i (u i, j + u j,i ) i, j {1,,3}, (3.9) which may be written in matrix form as ( u x 1 ux ε xx ε xy ε xz y + u ) ( y 1 ux z + u ) z ( [ε]= ε xy ε yy ε yz = u y 1 uy ε xz ε yz ε zz y z + u ) z. y u z (symm) z (3.10) Due to the symmetry in (3.10), only 6 different equations are obtained, Continuum Mechanics for Engineers Theory and Problems X. Oliver and C. Agelet de Saracibar ε xx u x = 0, ε xy 1 ( ux y + u ) y = 0, ε yy u y y = 0, ε xz 1 ( ux z + u ) z = 0, ε zz u z z = 0, ε yz 1 ( uy z + u ) z = 0. y (3.11) A theorem of differential geometry states that the divergence of the rotational of any field is null, [ ( )] = 0. X. Oliver and C. Agelet de Saracibar Continuum Mechanics for Engineers.Theory and Problems doi: /rg

42 Compatibility Conditions for Infinitesimal Strains 113 Equation (3.11) is a system of 6 PDEs with 3 unknowns, which are the components of the displacement vector u(x,t) not [u x, u y, u z ] T. In general, this problem will not have a solution unless certain compatibility conditions are satisfied. To obtain these conditions, the equations in (3.11) are differentiated twice with respect to their spatial coordinates, ( ε xx u ) x, y, z, y, z, yz = 6 equations providing a total of 36 equations, (.. ε yz 1 ( uy z + u )) z y, y, z, y, z, yz = 6 equations, ε xx = 3 u x 3 ε xx y = 3 u x y ε xx z = 3 u x z ε xx y = 3 u x y ε xx z = 3 u x z ε xx y z = 3 u x y z }{{} (18 eqns for ε xx, ε yy, ε zz ) ε yz = 1 ( 3 u y z + 3 ) u z y ε yz y = 1 ( 3 u y z y + 3 ) u z y 3 ε yz z = 1 ( 3 u y z ) u z y z ε yz y = 1 ( 3 u y z y + 3 ) u z y ε yz z = 1 ( 3 u y z + 3 ) u z y z ε yz y z = 1 ( 3 u y z y + 3 ) u z y z }{{} (18 eqns for ε xy, ε xz, ε yz ) (3.1) (3.13) Continuum Mechanics for Engineers Theory and Problems X. Oliver and C. Agelet de Saracibar All the possible third derivatives of each component of the displacements u x, u y and u z are involved in these 36 equations. Thus, there are 30 different derivatives, 3 u x 3, y, z, y 3, y x, y z, z 3, z x, z = 10 derivatives, y, yz 3 u y 3, y, z, y 3, y x, y z, z 3, z x, z = 10 derivatives, y, yz 3 u z 3, y, z, y 3, y x, y z, z 3, z x, z = 10 derivatives, y, yz (3.14) X. Oliver and C. Agelet de Saracibar Continuum Mechanics for Engineers.Theory and Problems doi: /rg

43 114 CHAPTER 3. COMPATIBILITY EQUATIONS which constitute the 30 unknowns in the system of 36 equations ( ) 3 u i ε ij f n, n {1,... 36} (3.15) j k l k l } {{ } 30 defined in (3.13). Therefore, the 30 unknowns, which are the displacement derivatives 3 u i /( j k l ), can be eliminated from this system and 6 equations are obtained. In these equations, the third derivatives mentioned above do not appear, but there will be 1 second derivatives of the strain tensor ε ij /( k l ). After the corresponding algebraic operations, the resulting equations are Compatibility equations S xx def S yy def S zz def S xy def = ε yy z + ε zz y ε yz y z = 0 = ε zz + ε xx z ε xz z = 0 = ε xx y + ε yy ε xy y = 0 = ε zz y + ( εyz z + ε xz y ε ) xy = 0 z = ε yy z + ( εyz y ε xz y + ε ) xy = 0 z = ε xx y z + ( ε yz + ε xz y + ε ) xy = 0 z S xz def S yz def (3.16) Continuum Mechanics for Engineers Theory and Problems X. Oliver and C. Agelet de Saracibar which constitute the compatibility equations for the infinitesimal strain tensor ε. The compact expression corresponding to the 6 equations in (3.16) is Compatibility equations for the infinitesimal strain tensor { S = (ε )=0 (3.17) Another way of expressing the compatibility conditions (3.16) is in terms of the three-index operator named permutation operator ( e ijk ). In this case, the compatibility equations can be written as S mn = e mjq e nir ε ij,qr = 0. (3.18) X. Oliver and C. Agelet de Saracibar Continuum Mechanics for Engineers.Theory and Problems doi: /rg

44 Compatibility Conditions for Infinitesimal Strains 115 Remark 3.4. The 6 equations (3.16) are not functionally independent and, taking again into account the fact that the divergence of the rotational of a field is intrinsically null, the following functional relations can be established between them. S = ( (ε )) = 0 = S xx + S xy y + S xz z = 0 S xy + S yy y + S yz z = 0 S xz + S yz y + S zz z = 0 Remark 3.5. The three-index operator denominated permutation operator is given by 0 if an index is repeated, i = j or i = k or j = k 1 positive (clockwise) direction of the indexes, e ijk = i, j,k {13,31,31} 1 negative (counterclockwise) direction of the indexes, i, j,k {13,31,13} This definition is summarized in graphic form in Figure 3.. Continuum Mechanics for Engineers Theory and Problems X. Oliver and C. Agelet de Saracibar Figure 3.: Definition of the permutation operator, e ijk. X. Oliver and C. Agelet de Saracibar Continuum Mechanics for Engineers.Theory and Problems doi: /rg

45 116 CHAPTER 3. COMPATIBILITY EQUATIONS Finally, another possible expression of the compatibility conditions is ε ij,kl + ε kl,ij ε ik, jl ε jl,ik = 0 i, j,k,l {1,,3}. (3.19) Remark 3.6. Since the compatibility equations (3.16) only involve the second spatial derivatives of the components of the strain tensor ε (x,t), every strain tensor that is linear (first-order polynomial) with respect to the spatial variables will be compatible and, therefore, integrable. As a particular case, every uniform strain tensor ε (t) is integrable. 3.4 Integration of the Infinitesimal Strain Field Preliminary Equations Consider the rotation tensor Ω(x, t) for the infinitesimal strain case (see Chapter, Section.11.6), Ω = 1 (u u), Ω ij = 1 ( ui u ) (3.0) j i, j {1,,3}. j i and the infinitesimal rotation vector θ (x, t), associated with said rotation tensor, defined as 3 θ = 1 rot u = 1 θ 1 Ω 3 Ω yz u not = =. (3.1) Continuum Mechanics for Engineers Theory and Problems X. Oliver and C. Agelet de Saracibar θ θ 3 Ω 31 Ω 1 Ω zx Ω xy Differentiating the infinitesimal rotation tensor in (3.0) with respect to a coordinate x k yields Ω ij = 1 ( ui u ) j = Ω ij = 1 ( ui u ) j. (3.) j i k k j i 3 The tensor Ω is skew-symmetric, i.e., Ω not 0 Ω 1 Ω 31. Ω 1 0 Ω 3 Ω 31 Ω 3 0 X. Oliver and C. Agelet de Saracibar Continuum Mechanics for Engineers.Theory and Problems doi: /rg

46 Integration of the Infinitesimal Strain Field 117 Adding and subtracting in (3.) the term u k /( i j ) and rearranging the expression obtained results in Ω ij = 1 ( ui u ) j + 1 u k 1 u k = k k j i i j i j = ( 1 ui + u ) k ( 1 u j + u ) k = ε ik ε jk (3.3). j k i i k j j i }{{}}{{} ε ik ε jk This expression can now be used to calculate the Cartesian derivatives of the components of the infinitesimal rotation vector, θ (x, t), given in (3.1), as follows. θ 1 = Ω yz = ε xz y ε xy z θ 1 θ 1 y = Ω yz y = ε yz y ε yy z θ 1 z = Ω yz = ε zz z y ε zy z θ = Ω zx = ε xx z ε xz θ θ y = Ω zx y = ε xy z ε yz θ z = Ω zx = ε xz z z ε zz (3.4) (3.5) Continuum Mechanics for Engineers Theory and Problems X. Oliver and C. Agelet de Saracibar θ 3 θ 3 = Ω xy = ε xy ε xx y θ 3 y = Ω xy y = ε yy ε xy y (3.6) θ 3 z = Ω xy = ε yz z ε xz y Assume the value of the infinitesimal rotation vector θ (x,t) is known and, through it by means of (3.1), the value of the infinitesimal rotation tensor X. Oliver and C. Agelet de Saracibar Continuum Mechanics for Engineers.Theory and Problems doi: /rg

47 118 CHAPTER 3. COMPATIBILITY EQUATIONS Ω(x,t) is also known. Then, the displacement gradient tensor J(x,t) (see Chapter, Section.11.6) becomes J = u(x,t) = ε + Ω J ij = u i = 1 ( ui + u ) j + 1 ( ui u ) j = ε ij + Ω ij j j i j i }{{}}{{} ε ij Ω ij i, j {1,,3}. (3.7) Finally, writing in explicit form the different components in (3.7) and taking into account (3.1), the following is obtained 4. i = 1: i = : i = 3: j = 1 j = j = 3 u x = ε u x xx y = ε u x xy θ 3 z = ε xz + θ u y = ε u y xy + θ 3 y = ε u y yy z = ε yz θ 1 u z = ε u z xz θ y = ε u z yz + θ 1 z = ε zz (3.8) 3.4. Integration of the Strain Field Consider ε (x,t) is the infinitesimal strain field one wants to integrate. This operation is performed in two steps: 1) Using (3.4) through (3.6), the infinitesimal rotation vector θ (x, t) is integrated. The integration, with respect to space, of the infinitesimal rotation vector in (3.4) through (3.6) leads to a solution of the type θ i = θ i (x,y,z,t)+c i (t) i {1,,3}, (3.9) Continuum Mechanics for Engineers Theory and Problems X. Oliver and C. Agelet de Saracibar where the integration constants c i (t), which, in general, may be a function of time, can be determined if the value (or the evolution along time) of the infinitesimal rotation vector at some point of the medium is known. ) Once the infinitesimal strain tensor ε (x, t) and the infinitesimal rotation vector θ (x,t) are known, the displacement field u(x,t) is integrated. The system of first-order PDEs defined in (3.8) is used, resulting in 4 According to (3.1), Ω not u i = ũ i (x,y,z,t)+c i (t) i {1,,3}. (3.30) 0 Ω 1 Ω 31 Ω 1 0 Ω 3 = Ω 31 Ω θ 3 θ θ 3 0 θ 1 θ θ 1 0 X. Oliver and C. Agelet de Saracibar Continuum Mechanics for Engineers.Theory and Problems doi: /rg

48 Integration of the Infinitesimal Strain Field 119 Again, the integration constants c i (t) that appear, which, in general, will be a function of time, are determined when the value (or the evolution along time) of the displacements at some point of space is known. Remark 3.7. The integration processes in steps 1) and ) involve integrating systems of first-order PDEs. If the compatibility equations in (3.16) are satisfied, these systems will be integrable (without leading to contradictions in their integration process) and will finally allow obtaining the displacement field. Remark 3.8. The presence of the integration constants in (3.9) and (3.30) shows that an integrable strain tensor, ε (x, t), determines the motion of each instant of time except for a rotation c(t) not = ˆθ (t) and a translation c (t) not = û(t). { θ (x,t)= ε (x,t) θ (x,t)+ ˆθ (t) u(x,t)=ũ(x,t)+û(t) From these uniform rotation ˆθ (t) and translation û(t) the displacement field u (x,t)= ˆΩ(t)x + û(t) = u = ˆΩ can be defined, which corresponds to a rigid body motion 5. Indeed, the strain associated with this displacement is null, ε (x,t)= s u = 1 (u + u )= 1 ( ˆΩ + }{{} ˆΩ T ) = 0, ˆΩ as corresponds to the concept of rigid body (without deformation). Consequently, it is concluded that every compatible strain field determines the displacements of the continuous medium except for a rigid body motion, which must be determined by means of the appropriate boundary conditions. Continuum Mechanics for Engineers Theory and Problems X. Oliver and C. Agelet de Saracibar 5 The rigid body rotation tensor ˆΩ(t) (antisymmetric) is defined based on the rotation vector 0 ˆΩ 1 ˆΩ 31 0 ˆθ 3 θ not ˆθ (t) as ˆΩ ˆΩ 1 0 ˆΩ 3 = θ 3 0 θ 1. ˆΩ 31 ˆΩ 3 0 θ θ 1 0 X. Oliver and C. Agelet de Saracibar Continuum Mechanics for Engineers.Theory and Problems doi: /rg

49 10 CHAPTER 3. COMPATIBILITY EQUATIONS Example 3.1 A certain motion is defined by the infinitesimal strain tensor 8x y 3 x z ε (x,t) not y x 0 3. x z 0 x 3 Obtain the corresponding displacement vector u(x, t) and the infinitesimal not rotation tensor Ω(x,t) taking into account that u(x,t) x=[0,0,0] T [3t, 0, 0] T and Ω(x,t) x=[0,0,0] T = 0. Solution Infinitesimal rotation vector Posing the systems of equations defined in (3.4) through (3.6) results in θ 1 = 0 ; θ 1 y = 0; θ 1 z = 0 θ 1 = C 1 (t), θ = 3xz ; θ y = 0; θ z = 3 x θ = 3 x z +C (t), θ 3 = 0 ; θ 3 y = 3 ; θ 3 z = 0 θ 3 = 3 y +C 3 (t). The integration constants C i (t) are determined by imposing that Ω(x,t) x=(0,0,0) T = 0 (and, therefore, the infinitesimal rotation vector θ (x,t) x=(0,0,0) T = 0), that is, [ C 1 (t)=c (t)=c 3 (t)=0 = θ (x) not 0, 3 x z, 3 y Continuum Mechanics for Engineers Theory and Problems X. Oliver and C. Agelet de Saracibar and the infinitesimal rotation tensor is 0 θ 3 θ Ω(x) not θ 3 0 θ 1 = θ θ y 3 x z 3 y x z 0 0 ] T X. Oliver and C. Agelet de Saracibar Continuum Mechanics for Engineers.Theory and Problems doi: /rg

50 Compatibility Equations and Integration of the Strain Rate Field 11 Displacement vector Posing, and integrating, the systems of equations in (3.8) produces u 1 = 8x ; u 1 y = y ; u 1 z = 0 u 1 = 4x y +C 1 (t), u = y ; u y = x ; u z = 0 u = xy +C (t), u 3 = u 3 3x z ; y = 0 ; u 3 z = x3 u 3 = x 3 z +C 3 (t). and imposing that u(x,t) x=(0,0,0) T [3t, 0, 0] T yields not C 1 (t)=3t ; C (t)=c 3 (t)=0 = u(x) not [ 4x y + 3t, xy, x 3 z ] T. 3.5 Compatibility Equations and Integration of the Strain Rate Field Given the definitions of the infinitesimal strain tensor ε, the infinitesimal rotation tensor Ω and the infinitesimal rotation vector θ, there exists a clear correspondence between these magnitudes and a) the strain rate tensor d, b) the rotation rate (or spin) tensor w and c) the spin vector ω given in Chapter. These correspondences can be established in the following manner: u ε (u) ε ij = 1 ( ui + u ) j j i Ω ij = 1 ( ui u ) j j i Continuum Mechanics for Engineers Theory and Problems X. Oliver and C. Agelet de Saracibar d ij = 1 w ij = 1 v d(v) ( vi j + v j i ( vi j v j i ) ) (3.31) θ = 1 u ω = 1 v Then, it is obvious that the concept of compatibility of a strain field ε introduced in Section 3.1 can be extended, by virtue of the correspondence with (3.31), to the compatibility of a strain rate field d(x,t). X. Oliver and C. Agelet de Saracibar Continuum Mechanics for Engineers.Theory and Problems doi: /rg

51 1 CHAPTER 3. COMPATIBILITY EQUATIONS To integrate this field, the same procedure as that seen in Section 3.4. can be used, replacing ε by d, u by v, Ω by w and θ by ω. Certainly, this integration can only be performed if the compatibility equations in (3.16) are satisfied for the components of d(x,t). Remark 3.9. The resulting compatibility equations and the integration process of the strain rate vector d(x,t) are not, in this case, restricted to the infinitesimal strain case. Continuum Mechanics for Engineers Theory and Problems X. Oliver and C. Agelet de Saracibar X. Oliver and C. Agelet de Saracibar Continuum Mechanics for Engineers.Theory and Problems doi: /rg

52 Problems and Exercises 13 PROBLEMS Problem 3.1 Determine the spatial description of the velocity field that corresponds to the strain rate tensor te tx 0 0 d(x,t) not 0 0 te y te y For x = 0, ω 0 not [t 1, 0, 0] T and v 0 not [t, 0, t] T for t is satisfied. Solution The problem is solved by integrating the corresponding differential equations, taking into account the existent parallelism between the variables: u v ε d θ ω Angular velocity of the rotation vector ω 1 = 0; ω 1 y = ω 1 tey ; z = 0 ω 1 = C 1 (t)+te y, ω = 0; ω y = 0; ω z = 0 ω = C (t), ω 3 = 0; ω 3 y = 0; ω 3 z = 0 ω 3 = C 3 (t). The boundary conditions are imposed for x = 0, t 1 t +C 1 C 1 = 1 not ω 0 0 = C = C = 0 0 C 3 = 0, Continuum Mechanics for Engineers Theory and Problems X. Oliver and C. Agelet de Saracibar C 3 X. Oliver and C. Agelet de Saracibar Continuum Mechanics for Engineers.Theory and Problems doi: /rg

53 14 CHAPTER 3. COMPATIBILITY EQUATIONS and the final result is Velocity vector ω (x,t) not te y v 1 = v 1 tetx ; y = 0; v 1 z = 0 v 1 = C 1 (t)+etx, v = 0; v y = 0; v z = v = C (t)+z, v 3 = 0; v 3 y = v 3 tey ; z = 0 v 3 = C 3 (t)+tey. The boundary conditions are imposed for x = 0, t 1 +C 1 C 1 not v 0 0 = C = C t t +C 3 C 3 = t, and the spatial description of the velocity field is e tx +t 1 v(x) not z. te y t Continuum Mechanics for Engineers Theory and Problems X. Oliver and C. Agelet de Saracibar X. Oliver and C. Agelet de Saracibar Continuum Mechanics for Engineers.Theory and Problems doi: /rg

54 Problems and Exercises 15 EXERCISES 3.1 Deduce the displacement field that corresponds to the infinitesimal strain tensor 0 te ty 0 ε (x,t) not te ty te tz At point (1,1,1), u not [e t, e t, e t ] T and θ not [0, 0, te t ] T is verified. 3. Determine the spatial description of the velocity field that corresponds to the strain rate tensor 0 0 te tz d(x,t) not 0 te ty 0. te tz 0 0 The following is known: { for z = 0: v x = v z = 0, t, x,y for y = 1: v y = 0, t, x,z Continuum Mechanics for Engineers Theory and Problems X. Oliver and C. Agelet de Saracibar X. Oliver and C. Agelet de Saracibar Continuum Mechanics for Engineers.Theory and Problems doi: /rg

PEAT SEISMOLOGY Lecture 2: Continuum mechanics

PEAT SEISMOLOGY Lecture 2: Continuum mechanics PEAT8002 - SEISMOLOGY Lecture 2: Continuum mechanics Nick Rawlinson Research School of Earth Sciences Australian National University Strain Strain is the formal description of the change in shape of a

More information

CH.6. LINEAR ELASTICITY. Multimedia Course on Continuum Mechanics

CH.6. LINEAR ELASTICITY. Multimedia Course on Continuum Mechanics CH.6. LINEAR ELASTICITY Multimedia Course on Continuum Mechanics Overview Hypothesis of the Linear Elasticity Theory Linear Elastic Constitutive Equation Generalized Hooke s Law Elastic Potential Isotropic

More information

CH.11. VARIATIONAL PRINCIPLES. Multimedia Course on Continuum Mechanics

CH.11. VARIATIONAL PRINCIPLES. Multimedia Course on Continuum Mechanics CH.11. ARIATIONAL PRINCIPLES Multimedia Course on Continuum Mechanics Overview Introduction Functionals Gâteaux Derivative Extreme of a Functional ariational Principle ariational Form of a Continuum Mechanics

More information

CH.9. CONSTITUTIVE EQUATIONS IN FLUIDS. Multimedia Course on Continuum Mechanics

CH.9. CONSTITUTIVE EQUATIONS IN FLUIDS. Multimedia Course on Continuum Mechanics CH.9. CONSTITUTIVE EQUATIONS IN FLUIDS Multimedia Course on Continuum Mechanics Overview Introduction Fluid Mechanics What is a Fluid? Pressure and Pascal s Law Constitutive Equations in Fluids Fluid Models

More information

CH.1. DESCRIPTION OF MOTION. Multimedia Course on Continuum Mechanics

CH.1. DESCRIPTION OF MOTION. Multimedia Course on Continuum Mechanics CH.1. DESCRIPTION OF MOTION Multimedia Course on Continuum Mechanics Overview 1.1. Definition of the Continuous Medium 1.1.1. Concept of Continuum 1.1.2. Continuous Medium or Continuum 1.2. Equations of

More information

3D Elasticity Theory

3D Elasticity Theory 3D lasticity Theory Many structural analysis problems are analysed using the theory of elasticity in which Hooke s law is used to enforce proportionality between stress and strain at any deformation level.

More information

KINEMATICS OF CONTINUA

KINEMATICS OF CONTINUA KINEMATICS OF CONTINUA Introduction Deformation of a continuum Configurations of a continuum Deformation mapping Descriptions of motion Material time derivative Velocity and acceleration Transformation

More information

By drawing Mohr s circle, the stress transformation in 2-D can be done graphically. + σ x σ y. cos 2θ + τ xy sin 2θ, (1) sin 2θ + τ xy cos 2θ.

By drawing Mohr s circle, the stress transformation in 2-D can be done graphically. + σ x σ y. cos 2θ + τ xy sin 2θ, (1) sin 2θ + τ xy cos 2θ. Mohr s Circle By drawing Mohr s circle, the stress transformation in -D can be done graphically. σ = σ x + σ y τ = σ x σ y + σ x σ y cos θ + τ xy sin θ, 1 sin θ + τ xy cos θ. Note that the angle of rotation,

More information

GG303 Lecture 6 8/27/09 1 SCALARS, VECTORS, AND TENSORS

GG303 Lecture 6 8/27/09 1 SCALARS, VECTORS, AND TENSORS GG303 Lecture 6 8/27/09 1 SCALARS, VECTORS, AND TENSORS I Main Topics A Why deal with tensors? B Order of scalars, vectors, and tensors C Linear transformation of scalars and vectors (and tensors) II Why

More information

NONLINEAR CONTINUUM FORMULATIONS CONTENTS

NONLINEAR CONTINUUM FORMULATIONS CONTENTS NONLINEAR CONTINUUM FORMULATIONS CONTENTS Introduction to nonlinear continuum mechanics Descriptions of motion Measures of stresses and strains Updated and Total Lagrangian formulations Continuum shell

More information

CH.5. BALANCE PRINCIPLES. Multimedia Course on Continuum Mechanics

CH.5. BALANCE PRINCIPLES. Multimedia Course on Continuum Mechanics CH.5. BALANCE PRINCIPLES Multimedia Course on Continuum Mechanics Overview Balance Principles Convective Flux or Flux by Mass Transport Local and Material Derivative of a olume Integral Conservation of

More information

CH.3. COMPATIBILITY EQUATIONS. Continuum Mechanics Course (MMC) - ETSECCPB - UPC

CH.3. COMPATIBILITY EQUATIONS. Continuum Mechanics Course (MMC) - ETSECCPB - UPC CH.3. COMPATIBILITY EQUATIONS Connuum Mechancs Course (MMC) - ETSECCPB - UPC Overvew Compably Condons Compably Equaons of a Poenal Vecor Feld Compably Condons for Infnesmal Srans Inegraon of he Infnesmal

More information

Getting started: CFD notation

Getting started: CFD notation PDE of p-th order Getting started: CFD notation f ( u,x, t, u x 1,..., u x n, u, 2 u x 1 x 2,..., p u p ) = 0 scalar unknowns u = u(x, t), x R n, t R, n = 1,2,3 vector unknowns v = v(x, t), v R m, m =

More information

Mechanics PhD Preliminary Spring 2017

Mechanics PhD Preliminary Spring 2017 Mechanics PhD Preliminary Spring 2017 1. (10 points) Consider a body Ω that is assembled by gluing together two separate bodies along a flat interface. The normal vector to the interface is given by n

More information

12. Stresses and Strains

12. Stresses and Strains 12. Stresses and Strains Finite Element Method Differential Equation Weak Formulation Approximating Functions Weighted Residuals FEM - Formulation Classification of Problems Scalar Vector 1-D T(x) u(x)

More information

CH.1. DESCRIPTION OF MOTION. Continuum Mechanics Course (MMC)

CH.1. DESCRIPTION OF MOTION. Continuum Mechanics Course (MMC) CH.1. DESCRIPTION OF MOTION Continuum Mechanics Course (MMC) Overview 1.1. Definition of the Continuous Medium 1.1.1. Concept of Continuum 1.1.. Continuous Medium or Continuum 1.. Equations of Motion 1..1

More information

A short review of continuum mechanics

A short review of continuum mechanics A short review of continuum mechanics Professor Anette M. Karlsson, Department of Mechanical ngineering, UD September, 006 This is a short and arbitrary review of continuum mechanics. Most of this material

More information

CH.2. DEFORMATION AND STRAIN. Continuum Mechanics Course (MMC) - ETSECCPB - UPC

CH.2. DEFORMATION AND STRAIN. Continuum Mechanics Course (MMC) - ETSECCPB - UPC CH.. DEFORMATION AND STRAIN Continuum Mechanics Course (MMC) - ETSECCPB - UPC Overview Introduction Deformation Gradient Tensor Material Deformation Gradient Tensor Inverse (Spatial) Deformation Gradient

More information

Mechanics of materials Lecture 4 Strain and deformation

Mechanics of materials Lecture 4 Strain and deformation Mechanics of materials Lecture 4 Strain and deformation Reijo Kouhia Tampere University of Technology Department of Mechanical Engineering and Industrial Design Wednesday 3 rd February, 206 of a continuum

More information

Course 2BA1: Hilary Term 2007 Section 8: Quaternions and Rotations

Course 2BA1: Hilary Term 2007 Section 8: Quaternions and Rotations Course BA1: Hilary Term 007 Section 8: Quaternions and Rotations David R. Wilkins Copyright c David R. Wilkins 005 Contents 8 Quaternions and Rotations 1 8.1 Quaternions............................ 1 8.

More information

Variational principles in mechanics

Variational principles in mechanics CHAPTER Variational principles in mechanics.1 Linear Elasticity n D Figure.1: A domain and its boundary = D [. Consider a domain Ω R 3 with its boundary = D [ of normal n (see Figure.1). The problem of

More information

3.2 Hooke s law anisotropic elasticity Robert Hooke ( ) Most general relationship

3.2 Hooke s law anisotropic elasticity Robert Hooke ( ) Most general relationship 3.2 Hooke s law anisotropic elasticity Robert Hooke (1635-1703) Most general relationship σ = C ε + C ε + C ε + C γ + C γ + C γ 11 12 yy 13 zz 14 xy 15 xz 16 yz σ = C ε + C ε + C ε + C γ + C γ + C γ yy

More information

3. The linear 3-D elasticity mathematical model

3. The linear 3-D elasticity mathematical model 3. The linear 3-D elasticity mathematical model In Chapter we examined some fundamental conditions that should be satisfied in the modeling of all deformable solids and structures. The study of truss structures

More information

M E 320 Professor John M. Cimbala Lecture 10

M E 320 Professor John M. Cimbala Lecture 10 M E 320 Professor John M. Cimbala Lecture 10 Today, we will: Finish our example problem rates of motion and deformation of fluid particles Discuss the Reynolds Transport Theorem (RTT) Show how the RTT

More information

Basic Equations of Elasticity

Basic Equations of Elasticity A Basic Equations of Elasticity A.1 STRESS The state of stress at any point in a loaded bo is defined completely in terms of the nine components of stress: σ xx,σ yy,σ zz,σ xy,σ yx,σ yz,σ zy,σ zx,andσ

More information

Stress, Strain, Mohr s Circle

Stress, Strain, Mohr s Circle Stress, Strain, Mohr s Circle The fundamental quantities in solid mechanics are stresses and strains. In accordance with the continuum mechanics assumption, the molecular structure of materials is neglected

More information

V (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t)

V (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t) IV. DIFFERENTIAL RELATIONS FOR A FLUID PARTICLE This chapter presents the development and application of the basic differential equations of fluid motion. Simplifications in the general equations and common

More information

2.20 Fall 2018 Math Review

2.20 Fall 2018 Math Review 2.20 Fall 2018 Math Review September 10, 2018 These notes are to help you through the math used in this class. This is just a refresher, so if you never learned one of these topics you should look more

More information

Lecture 8. Stress Strain in Multi-dimension

Lecture 8. Stress Strain in Multi-dimension Lecture 8. Stress Strain in Multi-dimension Module. General Field Equations General Field Equations [] Equilibrium Equations in Elastic bodies xx x y z yx zx f x 0, etc [2] Kinematics xx u x x,etc. [3]

More information

3 2 6 Solve the initial value problem u ( t) 3. a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1

3 2 6 Solve the initial value problem u ( t) 3. a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1 Math Problem a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1 3 6 Solve the initial value problem u ( t) = Au( t) with u (0) =. 3 1 u 1 =, u 1 3 = b- True or false and why 1. if A is

More information

LOWELL JOURNAL. MUST APOLOGIZE. such communication with the shore as Is m i Boimhle, noewwary and proper for the comfort

LOWELL JOURNAL. MUST APOLOGIZE. such communication with the shore as Is m i Boimhle, noewwary and proper for the comfort - 7 7 Z 8 q ) V x - X > q - < Y Y X V - z - - - - V - V - q \ - q q < -- V - - - x - - V q > x - x q - x q - x - - - 7 -» - - - - 6 q x - > - - x - - - x- - - q q - V - x - - ( Y q Y7 - >»> - x Y - ] [

More information

Group, Rings, and Fields Rahul Pandharipande. I. Sets Let S be a set. The Cartesian product S S is the set of ordered pairs of elements of S,

Group, Rings, and Fields Rahul Pandharipande. I. Sets Let S be a set. The Cartesian product S S is the set of ordered pairs of elements of S, Group, Rings, and Fields Rahul Pandharipande I. Sets Let S be a set. The Cartesian product S S is the set of ordered pairs of elements of S, A binary operation φ is a function, S S = {(x, y) x, y S}. φ

More information

Mathematical Preliminaries

Mathematical Preliminaries Mathematical Preliminaries Introductory Course on Multiphysics Modelling TOMAZ G. ZIELIŃKI bluebox.ippt.pan.pl/ tzielins/ Table of Contents Vectors, tensors, and index notation. Generalization of the concept

More information

DEFORMATION PATTERN IN ELASTIC CRUST

DEFORMATION PATTERN IN ELASTIC CRUST DEFORMATION PATTERN IN ELASTIC CRUST Stress and force in 2D Strain : normal and shear Elastic medium equations Vertical fault in elastic medium => arctangent General elastic dislocation (Okada s formulas)

More information

2 Introduction to mechanics

2 Introduction to mechanics 21 Motivation Thermodynamic bodies are being characterized by two competing opposite phenomena, energy and entropy which some researchers in thermodynamics would classify as cause and chance or determinancy

More information

CH.11. VARIATIONAL PRINCIPLES. Continuum Mechanics Course (MMC)

CH.11. VARIATIONAL PRINCIPLES. Continuum Mechanics Course (MMC) CH.11. ARIATIONAL PRINCIPLES Continuum Mechanics Course (MMC) Overview Introduction Functionals Gâteaux Derivative Extreme of a Functional ariational Principle ariational Form of a Continuum Mechanics

More information

MA 201: Partial Differential Equations Lecture - 2

MA 201: Partial Differential Equations Lecture - 2 MA 201: Partial Differential Equations Lecture - 2 Linear First-Order PDEs For a PDE f(x,y,z,p,q) = 0, a solution of the type F(x,y,z,a,b) = 0 (1) which contains two arbitrary constants a and b is said

More information

Tensors, and differential forms - Lecture 2

Tensors, and differential forms - Lecture 2 Tensors, and differential forms - Lecture 2 1 Introduction The concept of a tensor is derived from considering the properties of a function under a transformation of the coordinate system. A description

More information

Problem Set 2 Due Tuesday, September 27, ; p : 0. (b) Construct a representation using five d orbitals that sit on the origin as a basis: 1

Problem Set 2 Due Tuesday, September 27, ; p : 0. (b) Construct a representation using five d orbitals that sit on the origin as a basis: 1 Problem Set 2 Due Tuesday, September 27, 211 Problems from Carter: Chapter 2: 2a-d,g,h,j 2.6, 2.9; Chapter 3: 1a-d,f,g 3.3, 3.6, 3.7 Additional problems: (1) Consider the D 4 point group and use a coordinate

More information

Lecture 8 Analyzing the diffusion weighted signal. Room CSB 272 this week! Please install AFNI

Lecture 8 Analyzing the diffusion weighted signal. Room CSB 272 this week! Please install AFNI Lecture 8 Analyzing the diffusion weighted signal Room CSB 272 this week! Please install AFNI http://afni.nimh.nih.gov/afni/ Next lecture, DTI For this lecture, think in terms of a single voxel We re still

More information

Mechanics of Earthquakes and Faulting

Mechanics of Earthquakes and Faulting Mechanics of Earthquakes and Faulting www.geosc.psu.edu/courses/geosc508 Overview Milestones in continuum mechanics Concepts of modulus and stiffness. Stress-strain relations Elasticity Surface and body

More information

Chapter 4: Fluid Kinematics

Chapter 4: Fluid Kinematics Overview Fluid kinematics deals with the motion of fluids without considering the forces and moments which create the motion. Items discussed in this Chapter. Material derivative and its relationship to

More information

Fundamentals of Linear Elasticity

Fundamentals of Linear Elasticity Fundamentals of Linear Elasticity Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research of the Polish Academy

More information

4.3 Momentum Balance Principles

4.3 Momentum Balance Principles 4.3 Momentum Balance Principles 4.3.1 Balance of linear angular momentum in spatial material description Consider a continuum body B with a set of particles occupying an arbitrary region Ω with boundary

More information

Rigid body dynamics. Basilio Bona. DAUIN - Politecnico di Torino. October 2013

Rigid body dynamics. Basilio Bona. DAUIN - Politecnico di Torino. October 2013 Rigid body dynamics Basilio Bona DAUIN - Politecnico di Torino October 2013 Basilio Bona (DAUIN - Politecnico di Torino) Rigid body dynamics October 2013 1 / 16 Multiple point-mass bodies Each mass is

More information

A.1 Appendix on Cartesian tensors

A.1 Appendix on Cartesian tensors 1 Lecture Notes on Fluid Dynamics (1.63J/2.21J) by Chiang C. Mei, February 6, 2007 A.1 Appendix on Cartesian tensors [Ref 1] : H Jeffreys, Cartesian Tensors; [Ref 2] : Y. C. Fung, Foundations of Solid

More information

Analytical Mechanics: Elastic Deformation

Analytical Mechanics: Elastic Deformation Analytical Mechanics: Elastic Deformation Shinichi Hirai Dept. Robotics, Ritsumeikan Univ. Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 1 / 60 Agenda Agenda

More information

Classical Mechanics. Luis Anchordoqui

Classical Mechanics. Luis Anchordoqui 1 Rigid Body Motion Inertia Tensor Rotational Kinetic Energy Principal Axes of Rotation Steiner s Theorem Euler s Equations for a Rigid Body Eulerian Angles Review of Fundamental Equations 2 Rigid body

More information

Wave and Elasticity Equations

Wave and Elasticity Equations 1 Wave and lasticity quations Now let us consider the vibrating string problem which is modeled by the one-dimensional wave equation. Suppose that a taut string is suspended by its extremes at the points

More information

Cartesian Tensors. e 2. e 1. General vector (formal definition to follow) denoted by components

Cartesian Tensors. e 2. e 1. General vector (formal definition to follow) denoted by components Cartesian Tensors Reference: Jeffreys Cartesian Tensors 1 Coordinates and Vectors z x 3 e 3 y x 2 e 2 e 1 x x 1 Coordinates x i, i 123,, Unit vectors: e i, i 123,, General vector (formal definition to

More information

Chapter 7. Kinematics. 7.1 Tensor fields

Chapter 7. Kinematics. 7.1 Tensor fields Chapter 7 Kinematics 7.1 Tensor fields In fluid mechanics, the fluid flow is described in terms of vector fields or tensor fields such as velocity, stress, pressure, etc. It is important, at the outset,

More information

Continuum mechanics V. Constitutive equations. 1. Constitutive equation: definition and basic axioms

Continuum mechanics V. Constitutive equations. 1. Constitutive equation: definition and basic axioms Continuum mechanics office Math 0.107 ales.janka@unifr.ch http://perso.unifr.ch/ales.janka/mechanics Mars 16, 2011, Université de Fribourg 1. Constitutive equation: definition and basic axioms Constitutive

More information

Rigid body simulation. Once we consider an object with spatial extent, particle system simulation is no longer sufficient

Rigid body simulation. Once we consider an object with spatial extent, particle system simulation is no longer sufficient Rigid body dynamics Rigid body simulation Once we consider an object with spatial extent, particle system simulation is no longer sufficient Rigid body simulation Unconstrained system no contact Constrained

More information

Elements of Rock Mechanics

Elements of Rock Mechanics Elements of Rock Mechanics Stress and strain Creep Constitutive equation Hooke's law Empirical relations Effects of porosity and fluids Anelasticity and viscoelasticity Reading: Shearer, 3 Stress Consider

More information

INTEGRALSATSER VEKTORANALYS. (indexräkning) Kursvecka 4. and CARTESIAN TENSORS. Kapitel 8 9. Sidor NABLA OPERATOR,

INTEGRALSATSER VEKTORANALYS. (indexräkning) Kursvecka 4. and CARTESIAN TENSORS. Kapitel 8 9. Sidor NABLA OPERATOR, VEKTORANALYS Kursvecka 4 NABLA OPERATOR, INTEGRALSATSER and CARTESIAN TENSORS (indexräkning) Kapitel 8 9 Sidor 83 98 TARGET PROBLEM In the plasma there are many particles (10 19, 10 20 per m 3 ), strong

More information

Waves in Linear Optical Media

Waves in Linear Optical Media 1/53 Waves in Linear Optical Media Sergey A. Ponomarenko Dalhousie University c 2009 S. A. Ponomarenko Outline Plane waves in free space. Polarization. Plane waves in linear lossy media. Dispersion relations

More information

Chapter 6: Rational Expr., Eq., and Functions Lecture notes Math 1010

Chapter 6: Rational Expr., Eq., and Functions Lecture notes Math 1010 Section 6.1: Rational Expressions and Functions Definition of a rational expression Let u and v be polynomials. The algebraic expression u v is a rational expression. The domain of this rational expression

More information

CH.7. PLANE LINEAR ELASTICITY. Multimedia Course on Continuum Mechanics

CH.7. PLANE LINEAR ELASTICITY. Multimedia Course on Continuum Mechanics CH.7. PLANE LINEAR ELASTICITY Multimedia Course on Continuum Mechanics Overview Plane Linear Elasticit Theor Plane Stress Simplifing Hpothesis Strain Field Constitutive Equation Displacement Field The

More information

A DARK GREY P O N T, with a Switch Tail, and a small Star on the Forehead. Any

A DARK GREY P O N T, with a Switch Tail, and a small Star on the Forehead. Any Y Y Y X X «/ YY Y Y ««Y x ) & \ & & } # Y \#$& / Y Y X» \\ / X X X x & Y Y X «q «z \x» = q Y # % \ & [ & Z \ & { + % ) / / «q zy» / & / / / & x x X / % % ) Y x X Y $ Z % Y Y x x } / % «] «] # z» & Y X»

More information

Lecture notes on introduction to tensors. K. M. Udayanandan Associate Professor Department of Physics Nehru Arts and Science College, Kanhangad

Lecture notes on introduction to tensors. K. M. Udayanandan Associate Professor Department of Physics Nehru Arts and Science College, Kanhangad Lecture notes on introduction to tensors K. M. Udayanandan Associate Professor Department of Physics Nehru Arts and Science College, Kanhangad 1 . Syllabus Tensor analysis-introduction-definition-definition

More information

Introduction to Seismology Spring 2008

Introduction to Seismology Spring 2008 MIT OpenCourseWare http://ocw.mit.edu 12.510 Introduction to Seismology Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Stress and Strain

More information

Chapter 9: Differential Analysis

Chapter 9: Differential Analysis 9-1 Introduction 9-2 Conservation of Mass 9-3 The Stream Function 9-4 Conservation of Linear Momentum 9-5 Navier Stokes Equation 9-6 Differential Analysis Problems Recall 9-1 Introduction (1) Chap 5: Control

More information

CP1 REVISION LECTURE 3 INTRODUCTION TO CLASSICAL MECHANICS. Prof. N. Harnew University of Oxford TT 2017

CP1 REVISION LECTURE 3 INTRODUCTION TO CLASSICAL MECHANICS. Prof. N. Harnew University of Oxford TT 2017 CP1 REVISION LECTURE 3 INTRODUCTION TO CLASSICAL MECHANICS Prof. N. Harnew University of Oxford TT 2017 1 OUTLINE : CP1 REVISION LECTURE 3 : INTRODUCTION TO CLASSICAL MECHANICS 1. Angular velocity and

More information

MATH45061: SOLUTION SHEET 1 II

MATH45061: SOLUTION SHEET 1 II MATH456: SOLUTION SHEET II. The deformation gradient tensor has Cartesian components given by F IJ R I / r J ; and so F R e x, F R, F 3 R, r r r 3 F R r, F R r, F 3 R r 3, F 3 R 3 r, F 3 R 3 r, F 33 R

More information

Lesson Rigid Body Dynamics

Lesson Rigid Body Dynamics Lesson 8 Rigid Body Dynamics Lesson 8 Outline Problem definition and motivations Dynamics of rigid bodies The equation of unconstrained motion (ODE) User and time control Demos / tools / libs Rigid Body

More information

The Kinematic Equations

The Kinematic Equations The Kinematic Equations David Roylance Department of Materials Science and Engineering Massachusetts Institute of Technology Cambridge, MA 0139 September 19, 000 Introduction The kinematic or strain-displacement

More information

4 NON-LINEAR ANALYSIS

4 NON-LINEAR ANALYSIS 4 NON-INEAR ANAYSIS arge displacement elasticity theory, principle of virtual work arge displacement FEA with solid, thin slab, and bar models Virtual work density of internal forces revisited 4-1 SOURCES

More information

Useful Formulae ( )

Useful Formulae ( ) Appendix A Useful Formulae (985-989-993-) 34 Jeremić et al. A.. CHAPTER SUMMARY AND HIGHLIGHTS page: 35 of 536 A. Chapter Summary and Highlights A. Stress and Strain This section reviews small deformation

More information

The Matrix Representation of a Three-Dimensional Rotation Revisited

The Matrix Representation of a Three-Dimensional Rotation Revisited Physics 116A Winter 2010 The Matrix Representation of a Three-Dimensional Rotation Revisited In a handout entitled The Matrix Representation of a Three-Dimensional Rotation, I provided a derivation of

More information

Chapter 9: Differential Analysis of Fluid Flow

Chapter 9: Differential Analysis of Fluid Flow of Fluid Flow Objectives 1. Understand how the differential equations of mass and momentum conservation are derived. 2. Calculate the stream function and pressure field, and plot streamlines for a known

More information

DEFORMATION PATTERN IN ELASTIC CRUST

DEFORMATION PATTERN IN ELASTIC CRUST DEFORMATION PATTERN IN ELASTIC CRUST Stress and force in 2D Strain : normal and shear Elastic medium equations Vertical fault in elastic medium => arctangent General elastic dislocation (Okada s formulas)

More information

FMIA. Fluid Mechanics and Its Applications 113 Series Editor: A. Thess. Moukalled Mangani Darwish. F. Moukalled L. Mangani M.

FMIA. Fluid Mechanics and Its Applications 113 Series Editor: A. Thess. Moukalled Mangani Darwish. F. Moukalled L. Mangani M. FMIA F. Moukalled L. Mangani M. Darwish An Advanced Introduction with OpenFOAM and Matlab This textbook explores both the theoretical foundation of the Finite Volume Method (FVM) and its applications in

More information

NIELINIOWA OPTYKA MOLEKULARNA

NIELINIOWA OPTYKA MOLEKULARNA NIELINIOWA OPTYKA MOLEKULARNA chapter 1 by Stanisław Kielich translated by:tadeusz Bancewicz http://zon8.physd.amu.edu.pl/~tbancewi Poznan,luty 2008 ELEMENTS OF THE VECTOR AND TENSOR ANALYSIS Reference

More information

Deformable Materials 2 Adrien Treuille. source: Müller, Stam, James, Thürey. Real-Time Physics Class Notes.

Deformable Materials 2 Adrien Treuille. source: Müller, Stam, James, Thürey. Real-Time Physics Class Notes. Deformable Materials 2 Adrien Treuille source: Müller, Stam, James, Thürey. Real-Time Physics Class Notes. Goal Overview Strain (Recap) Stress From Strain to Stress Discretization Simulation Overview Strain

More information

MECH 5312 Solid Mechanics II. Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso

MECH 5312 Solid Mechanics II. Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso MECH 5312 Solid Mechanics II Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso Table of Contents Preliminary Math Concept of Stress Stress Components Equilibrium

More information

Macroscopic theory Rock as 'elastic continuum'

Macroscopic theory Rock as 'elastic continuum' Elasticity and Seismic Waves Macroscopic theory Rock as 'elastic continuum' Elastic body is deformed in response to stress Two types of deformation: Change in volume and shape Equations of motion Wave

More information

Multiple Integrals and Vector Calculus (Oxford Physics) Synopsis and Problem Sets; Hilary 2015

Multiple Integrals and Vector Calculus (Oxford Physics) Synopsis and Problem Sets; Hilary 2015 Multiple Integrals and Vector Calculus (Oxford Physics) Ramin Golestanian Synopsis and Problem Sets; Hilary 215 The outline of the material, which will be covered in 14 lectures, is as follows: 1. Introduction

More information

Multiple Integrals and Vector Calculus: Synopsis

Multiple Integrals and Vector Calculus: Synopsis Multiple Integrals and Vector Calculus: Synopsis Hilary Term 28: 14 lectures. Steve Rawlings. 1. Vectors - recap of basic principles. Things which are (and are not) vectors. Differentiation and integration

More information

CONSERVATION OF MASS AND BALANCE OF LINEAR MOMENTUM

CONSERVATION OF MASS AND BALANCE OF LINEAR MOMENTUM CONSERVATION OF MASS AND BALANCE OF LINEAR MOMENTUM Summary of integral theorems Material time derivative Reynolds transport theorem Principle of conservation of mass Principle of balance of linear momentum

More information

Artificial Intelligence & Neuro Cognitive Systems Fakultät für Informatik. Robot Dynamics. Dr.-Ing. John Nassour J.

Artificial Intelligence & Neuro Cognitive Systems Fakultät für Informatik. Robot Dynamics. Dr.-Ing. John Nassour J. Artificial Intelligence & Neuro Cognitive Systems Fakultät für Informatik Robot Dynamics Dr.-Ing. John Nassour 25.1.218 J.Nassour 1 Introduction Dynamics concerns the motion of bodies Includes Kinematics

More information

GG303 Lecture 15 10/6/09 1 FINITE STRAIN AND INFINITESIMAL STRAIN

GG303 Lecture 15 10/6/09 1 FINITE STRAIN AND INFINITESIMAL STRAIN GG303 Lecture 5 0609 FINITE STRAIN AND INFINITESIMAL STRAIN I Main Topics on infinitesimal strain A The finite strain tensor [E] B Deformation paths for finite strain C Infinitesimal strain and the infinitesimal

More information

Finite Element Method in Geotechnical Engineering

Finite Element Method in Geotechnical Engineering Finite Element Method in Geotechnical Engineering Short Course on + Dynamics Boulder, Colorado January 5-8, 2004 Stein Sture Professor of Civil Engineering University of Colorado at Boulder Contents Steps

More information

NDT&E Methods: UT. VJ Technologies CAVITY INSPECTION. Nondestructive Testing & Evaluation TPU Lecture Course 2015/16.

NDT&E Methods: UT. VJ Technologies CAVITY INSPECTION. Nondestructive Testing & Evaluation TPU Lecture Course 2015/16. CAVITY INSPECTION NDT&E Methods: UT VJ Technologies NDT&E Methods: UT 6. NDT&E: Introduction to Methods 6.1. Ultrasonic Testing: Basics of Elasto-Dynamics 6.2. Principles of Measurement 6.3. The Pulse-Echo

More information

DIFFERENTIAL MANIFOLDS HW Exercise Employing the summation convention, we have: [u, v] i = ui x j vj vi. x j u j

DIFFERENTIAL MANIFOLDS HW Exercise Employing the summation convention, we have: [u, v] i = ui x j vj vi. x j u j DIFFERENTIAL MANIFOLDS HW 3 KELLER VANDEBOGERT. Exercise.4 Employing the summation convention, we have: So that: [u, v] i = ui x j vj vi x j uj [w, [u, v]] i = wi x [u, k v]k x j x k wk v j ui v j x j

More information

Continuum Mechanics. Continuum Mechanics and Constitutive Equations

Continuum Mechanics. Continuum Mechanics and Constitutive Equations Continuum Mechanics Continuum Mechanics and Constitutive Equations Continuum mechanics pertains to the description of mechanical behavior of materials under the assumption that the material is a uniform

More information

Dynamics of Glaciers

Dynamics of Glaciers Dynamics of Glaciers McCarthy Summer School 01 Andy Aschwanden Arctic Region Supercomputing Center University of Alaska Fairbanks, USA June 01 Note: This script is largely based on the Physics of Glaciers

More information

In this section, mathematical description of the motion of fluid elements moving in a flow field is

In this section, mathematical description of the motion of fluid elements moving in a flow field is Jun. 05, 015 Chapter 6. Differential Analysis of Fluid Flow 6.1 Fluid Element Kinematics In this section, mathematical description of the motion of fluid elements moving in a flow field is given. A small

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING CAMBRIDGE, MASSACHUSETTS 02139

MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING CAMBRIDGE, MASSACHUSETTS 02139 MASSACHUSETTS NSTTUTE OF TECHNOLOGY DEPARTMENT OF MATERALS SCENCE AND ENGNEERNG CAMBRDGE, MASSACHUSETTS 39 3. MECHANCAL PROPERTES OF MATERALS PROBLEM SET SOLUTONS Reading Ashby, M.F., 98, Tensors: Notes

More information

MATH 425, FINAL EXAM SOLUTIONS

MATH 425, FINAL EXAM SOLUTIONS MATH 425, FINAL EXAM SOLUTIONS Each exercise is worth 50 points. Exercise. a The operator L is defined on smooth functions of (x, y by: Is the operator L linear? Prove your answer. L (u := arctan(xy u

More information

Rotational & Rigid-Body Mechanics. Lectures 3+4

Rotational & Rigid-Body Mechanics. Lectures 3+4 Rotational & Rigid-Body Mechanics Lectures 3+4 Rotational Motion So far: point objects moving through a trajectory. Next: moving actual dimensional objects and rotating them. 2 Circular Motion - Definitions

More information

CH.4. STRESS. Continuum Mechanics Course (MMC)

CH.4. STRESS. Continuum Mechanics Course (MMC) CH.4. STRESS Continuum Mechanics Course (MMC) Overview Forces Acting on a Continuum Body Cauchy s Postulates Stress Tensor Stress Tensor Components Scientific Notation Engineering Notation Sign Criterion

More information

Course MA2C02, Hilary Term 2010 Section 4: Vectors and Quaternions

Course MA2C02, Hilary Term 2010 Section 4: Vectors and Quaternions Course MA2C02, Hilary Term 2010 Section 4: Vectors and Quaternions David R. Wilkins Copyright c David R. Wilkins 2000 2010 Contents 4 Vectors and Quaternions 47 4.1 Vectors...............................

More information

Lecture notes: Introduction to Partial Differential Equations

Lecture notes: Introduction to Partial Differential Equations Lecture notes: Introduction to Partial Differential Equations Sergei V. Shabanov Department of Mathematics, University of Florida, Gainesville, FL 32611 USA CHAPTER 1 Classification of Partial Differential

More information

Continuum mechanism: Stress and strain

Continuum mechanism: Stress and strain Continuum mechanics deals with the relation between forces (stress, σ) and deformation (strain, ε), or deformation rate (strain rate, ε). Solid materials, rigid, usually deform elastically, that is the

More information

On Vector Product Algebras

On Vector Product Algebras On Vector Product Algebras By Markus Rost. This text contains some remarks on vector product algebras and the graphical techniques. It is partially contained in the diploma thesis of D. Boos and S. Maurer.

More information

Elastic Fields of Dislocations in Anisotropic Media

Elastic Fields of Dislocations in Anisotropic Media Elastic Fields of Dislocations in Anisotropic Media a talk given at the group meeting Jie Yin, David M. Barnett and Wei Cai November 13, 2008 1 Why I want to give this talk Show interesting features on

More information

Engineering Sciences 241 Advanced Elasticity, Spring Distributed Thursday 8 February.

Engineering Sciences 241 Advanced Elasticity, Spring Distributed Thursday 8 February. Engineering Sciences 241 Advanced Elasticity, Spring 2001 J. R. Rice Homework Problems / Class Notes Mechanics of finite deformation (list of references at end) Distributed Thursday 8 February. Problems

More information

CSE 167: Introduction to Computer Graphics Lecture #2: Linear Algebra Primer

CSE 167: Introduction to Computer Graphics Lecture #2: Linear Algebra Primer CSE 167: Introduction to Computer Graphics Lecture #2: Linear Algebra Primer Jürgen P. Schulze, Ph.D. University of California, San Diego Spring Quarter 2016 Announcements Project 1 due next Friday at

More information

CE-570 Advanced Structural Mechanics - Arun Prakash

CE-570 Advanced Structural Mechanics - Arun Prakash Ch1-Intro Page 1 CE-570 Advanced Structural Mechanics - Arun Prakash The BIG Picture What is Mechanics? Mechanics is study of how things work: how anything works, how the world works! People ask: "Do you

More information

16.21 Techniques of Structural Analysis and Design Spring 2003 Unit #5 - Constitutive Equations

16.21 Techniques of Structural Analysis and Design Spring 2003 Unit #5 - Constitutive Equations 6.2 Techniques of Structural Analysis and Design Spring 2003 Unit #5 - Constitutive quations Constitutive quations For elastic materials: If the relation is linear: Û σ ij = σ ij (ɛ) = ρ () ɛ ij σ ij =

More information