Module-3: Kinematics Lecture-20: Material and Spatial Time Derivatives The velocity acceleration and velocity gradient are important quantities of kinematics. Here we discuss the description of these kinematic quantities in both Lagrangian and Eulerian framework. Material and spatial description of velocity field: Let Ω 0 be a reference configuration and Ω be a deformed configuration at an instant of time t. Let x = χx t) be a mapping that represents the motion from Ω 0 to Ω. Since χ is invertible for all X Ω 0 at every instant of time t the inverse map X = χ 1 x t) exists from Ω to Ω 0 which associates every spatial point x to a material point X. The material coordinate X is fixed to a particle which is independent of time see Example-1 in Lecture-17). On the other hand the spatial position of particle x can vary with time. The displacement of particle is defined by the relation u = x X. Therefore the velocity of particle x X) u = = x 1) Substituting the map x = χx t) in u we get velocity ṽ in terms of X as ṽx t) = x χx t) = 2) This represents the velocity field in Lagrangian description material description) as it is a function of material variable X. Using the inverse map X = χ 1 x t) we can obtain the following Eulerian description spatial description) of velocity field vx t) = ṽχ 1 x t) t). 3) In general the function forms of ṽ and v are different while the value of the functions are equal at a material point X and corresponding spatial point x = χx t). We now present an example to understand the material and spatial description of velocity fields. Example 1. Let the mapping function for motion x = χx t) is given by x 1 = X 1 1 + t 2 ) x 2 = X 2 1 + t) x 3 = 0. Then find the velocity in material and spatial descriptions. Solution: Consider given motion x 1 = X 1 1 + t 2 ) x 2 = X 2 1 + t) x 3 = 0. Since material description of velocity ṽ = χ we get ṽ 1 = x 1 = 2X 1t ṽ 2 = x 2 = X 2 ṽ 3 = x 3 = 0. Joint initiative of IITs and IISc Funded by MHRD Government of India 1
Using the motion representation we can obtain X 1 = x 1 /1 + t 2 ) and X 2 = x 2 /1 + t). Substituting in the material description we get the following spatial velocity field v 1 = 2x 1t 1 + t 2 v 2 = x 2 1 + t v 3 = 0. It is clear from the example that ṽ and v are having different forms whereas both the forms give same value of the velocity corresponding material and spatial coordinates i.e. ṽx t) = vx t) For example material description of velocity ṽ = 4 2 0) at a material point X = 1 2 3) at an instant of time t = 2. We now see that x = 5 6 0) is a spatial coordinate corresponding to the material point X = 1 2 3) at an instant of time t = 2. Substituting x in spatial description of velocity we get v = 4 2 0). Therefore the values of velocity are equal both in material or spatial at corresponding points i.e. ṽx t) 123) = vx t) 560) We now introduce the concept of material and spatial time derivatives of more general fields. Later we apply this result to get the acceleration in both descriptions. Material and spatial time derivative of general fields: There is a possibility of confusion in taking material and spatial time derivatives of general fields. Hence we introduce the following notations to avoid the confusion in taking the differentiation of fields. Let φx t) gx t) GX t) be scalar vector and tensor fields in the Lagrangian material) description respectively. Then the material gradient of scalar vector and tensor fields are denoted by X φ X g X G the material divergence of vector and tensor field are denoted by X g X G and the material time derivative of scalar vector and tensor fields are denoted by φ D φ or g or D g D G G or. In component form X φ)i = φ X g = g i φ = D φ = φ X g) ij = g i X G) i = G ij g = D g = g X G)ijk = G ij X k G = D G = G. Let φx t) gx t) Gx t) be scalar vector and tensor fields in the Eulerian or spatial description. Then the spatial gradient of scalar vector and tensor fields are denoted by x φ x g x G the spatial divergence of vector and tensor field are denoted by x g x G and the spatial time derivative of scalar vector and tensor fields are denoted by φ g G. In component form x φ) i = φ x g = g i φ = φ g = g x g) ij = g i x G) i = G ij G = G. x G) ijk = G ij x k Joint initiative of IITs and IISc Funded by MHRD Government of India 2
We now derive the relation between the material and spatial gradients. Relation between material and spatial gradients: Let Ω 0 be a reference configuration and Ω be a deformed configuration. Let x = χx t) be a map from Ω 0 to Ω and F = X χ be the deformation gradient. Let φx t) and gx t) be scalar and vector fields defined over Ω. Then we have or equivalently X φ) i = φ = φ χ j = F ji φ X g) ij = g i = g i x k χ k = g i x k F kj X φ = F T x φ 4) X g = x gf. 5) We now use the relation between material and spatial gradients to represent Green and Almansi strain tensors in terms of displacement gradients. We know that the displacement u = x X. Furthermore the deformation gradient X u = F I. Therefore the Green strain tensor E = 1 2 F T F I ) = 1 2 X u + X u T + X u T X u ). 6) The Green strain tensor in component form E ij = 1 ui + u j + u ) k u k. 7) 2 Using Eq. 5) we get spatial gradient of displacement x u = X u) F 1 = F I)F 1 = I F 1. 8) Substituting F 1 = I x u in Almansi strain tensor see Eq. 6) in Lecture-18) we get Ẽ = 1 2 I F T F 1) = 1 2 x u + x u T x u T x u ). 9) The Almansi strain tensor in component form Ẽ ij = 1 ui + u j u ) k u k. 10) 2 The relations between spatial and material gradients are shown in Eqs. 4) and 5). We now want to derive relation between material and spatial time derivatives. In order to get the relation between time derivatives we need to have the following chain rule. Chain rule: Let ξt) be a scalar valued function of time t. Let ψξt) t) be a scalar valued function which depends explicitly and implicitly through ξ on t. Then the total derivative with respect to time t can be written as dψ dt = ψ + ψ dξ ξ dt. 11) Joint initiative of IITs and IISc Funded by MHRD Government of India 3
Proof: Consider Taylor s expansion of ψξt + t) t + t). ) ψξt + t) t + t) = ψξt + t) t) + + o t) ξt+ t) = ψ ξt) + dξ ) ) t + o t) t + + o t) ξt+ t) dt ψ = ψξt) t) + ξ = ψξt) t) + ψ ξ Using definition of differentiation we get ) dξ dt t + ) dξ dt + ) + o t) ξt+ t) ) t + o t) ξt+ t) dψ dt ψξt + t) t + t) ψξt) t) = lim t 0 t = ψ + ψ ξ dξ dt. we should treat ξ as fixed while computing ψ ψ. Similarly t is fixed for evaluating ξ. This formula works for scalar function we now extend this to vector fields. Let gt) be a vector valued function of time t. Let Φgt) t) be a vector valued function which depends explicitly and implicitly through g on t. Then the following chain rule can be obtained for the total derivative of Φ. dφ i dt = Φ i + Φ i g j dg j dt. 12) This result can be proved similar to the previous case of scalar valued function. Relation between material and spatial time derivatives: Let φx t) gx t) and GX t) be scalar vector and tensor valued function over Ω 0 respectively. Let φx t) gx t) and Gx t) be scalar vector and tensor valued function over Ω respectively. Let these functions are related by the map x = χx t) i.e. φx t) = φx t) gx t) = gx t) and GX t) = Gx t) for x = χx t). Then the material time derivative of scalar field D φ scalar field φ are related by and the spatial time derivative of D φx t) = Dφx t) = φ + φ χ i = φ + xφ) v. the material time derivative of vector field D g vector field g are related by D gx t) = Dgx t) and the spatial time derivative of = g + g χ i = g + xg)v. Joint initiative of IITs and IISc Funded by MHRD Government of India 4
the material time derivative of tensor field D G vector field G are related by and the spatial time derivative of D GX t) = DGx t) = G + G χ i = G + xg)v. We note that the material time derivative in reference domain can be obtained by D φx t) = φ D gx t) = g and D GX t) = G. We now present an example to understand the relation between material and spatial time derivatives. Recall from Lecture-17 about the example of temperature distribution of bar under uniform motion presented in Lagrangian and Eulerian description. We now use the same example to explain the relation between material time derivative and spatial time derivative. Example 2. Consider a bar of initial length of four units i.e. the reference domain Ω 0 = [0 4]. Let the bar is extending with time t and the extension is given by the mapping function x = 1+t)X. The bar is experiencing the temperature distribution in Lagrangian description θx t) = X4 X)t 2 or in the Eulerian description θx t) = x4 + 4t x)t 2 /1 + t) 2 shown in Fig. 1). t = 3 t = 2 t = 1 t = 0 X = 1 X = 2 X = 3 θ = 27 θ = 36 θ = 27 X = 1 X = 2 X = 3 θ =12 θ =16 θ =12 X = 1 X = 2 X = 3 θ =3 θ =4 θ =3 x = 1 + t)x X = 1 X = 2 X = 3 θ = 0 θ = 0 θ = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 x Figure 1: Temperature distribution in moving bar It is easy to see that θx t) = θx t) for every x = χx t) = 1 + t)x. Consider the temperature distribution in Lagrangian framework θx t). definition of material derivative we get Using the D θx t) = θ = 2X4 X)t. We now consider the spatial description of temperature θ = x4+4t x)t/1+t) 2. Using the definition we get the spatial time derivative as θ = 2xt 1 + t) 3 4 + 6t x + 2t2 ). Joint initiative of IITs and IISc Funded by MHRD Government of India 5
We need the following terms to get total derivative in spatial description. θ x = 2t2 1 + t) 2 + 2t x) and x 2 = Substituting above terms we get total derivative Dθ = θ + θ x x = It can be observed that for x = 1 + t)x D θ = Dθ. x 1 + t). 2xt 4 + 4t x). 1 + t) 2 Thus the material time derivative is equal to the total derivative in spatial description. In general material time derivative and spatial time derivative are not equal. Material and spatial description of acceleration field: Let Ω 0 be a reference configuration and let Ω be a deformed configuration. Let x = χx t) be a mapping from Ω 0 to Ω. Let ṽx t) and vx t) be velocity fields described in the Lagrangian and Eulerian framework i.e. vx t) = ṽx t) for every x = χx t). Then the acceleration field is defined by ax t) = Dv = v + xv)v. 13) The second term in the equation x v)v is known as convective acceleration. Therefore the acceleration is sum of the spatial time derivative and the convective acceleration. The material description of acceleration is given by ãx t) = Dṽ = ṽ. 14) The material description of acceleration field is equivalent to spatial description i.e. ãx t) = ax t) for x = χx t). Velocity gradient: The velocity gradient L is defined by L = x v 15) where v is the velocity field. Using L the acceleration can be written as ax t) = v + Lv. 16) The velocity gradient L is a second-order tensor whose action on velocity gives the convective acceleration. Relation between F and L: The velocity gradient L is defined as the spatial gradient of velocity. On the other hand the material time derivative of deformation gradient gives the material gradient of velocity i.e. = χx t) XχX t)) = X = X v. Joint initiative of IITs and IISc Funded by MHRD Government of India 6
Using Eq. 5) we can obtain the following relation between spatial and material gradient of velocity. = Xv = x v) F = LF. Since the tensor F is invertible we have F F 1 = I. Taking the material time derivative on both sides we get D F F 1 ) = 1 = F 1 1 + F = O 1 = F F 1 = F 1 L. The material derivative of J i.e. detf )) in can be written as DJ = cof F ) : see an epression in Example-1 of Lecture-16) = cof F ) : LF ) = F cof F ) T : L T = JI : L T = J trl) = J x v. Since J represents locally the ratio of deformed volume to reference volume the isochoric motion J = 1) can be characterized by In summary we have the relations 1 DJ J = x v = 0. 17) 1 1 J DJ = LF 18) = F 1 L 19) = trl) = x v 20) References 1. C. S. Jog Continuum Mechanics: Foundations and Applications of Mechanics Volume I Third edition 2015 Cambridge University Press. 2. M. E. Gurtin E. Fried and L. Anand The Mechanics and Thermodynamics of Continua 2010 Cambridge University Press New York. 3. J. Bonet and R. D. Wood Nonlinear Continuum Mechanics for Finite Element Analysis 1997 Cambridge University Press Cambridge. Joint initiative of IITs and IISc Funded by MHRD Government of India 7