CVEN 7511 Computational Mechanics of Solids and Structures
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1 CVEN 7511 Computational Mechanics of Solids and Structures Instructor: Kaspar J. Willam Original Version of Class Notes Chishen T. Lin Fall 1990
2 Chapter 1 Fundamentals of Continuum Mechanics Abstract In this section, two topics of Continuum Mechanics will be reviewed: Kinematics of Motion : X, x, u (Strain ɛ ɛ). alance Laws : conservation of mass, linear and angular momenta. (Stress σ σ). 1.1 Kinematics of Motion: The description of motion can be divided into : Lagrange Coordinates : (Material escription) X = X A e A where A =1, 2, 3 (1.1) Euler Coordinates : (Spatial escription) x = x i e i where i =1, 2, 3 (1.2) The scalar temperature field may be represented by : Lagrangian Coordinates : T = T (X,t) - material description. Eulerian Coordinates : T = T (x,t) - spatial description. Jacobian as eformation Gradient : F = x X dx = F dx (1.3) 1
3 Then mapping of the material line element dx from the reference into current configuration is defined by : dx i dx i dx i dx i dx A dx dx C dx dx j = dx j dx j dx j A dx A dx dx C dx (1.4) dx k dx C dx k dx A dx k dx dx k dx C Restriction : for unique one-to-one mapping det F 0 where J =detf. 1.2 Polar ecomposition : Polar decomposition theorem states that the deformation gradient tensor F may be decomposed uniquely into a positive definite tensor and a proper orthogonal tensor, i.e. the right U or left V stretch tensor plus the rotation R tensor. 1. Right escription : F = R U where: det R =1, R t R = 1, R R t = 1, thus U = U t such that λ U > 0and 2. Left escription : F = V R with U 2 = U t R t R U = F t F (1.5) V 2 = F F t = V R R t V (1.6) and λ V = λ U Physical Meaning of U and V : Right Stretch : dx = R U dx, wheredx is first stretched and then rotated. Left Stretch : dx = V R dx, wheredx is first rotated and then stretched. If F is nonsingular det F 0, then there is a unique decomposition into a proper orthogonal tensor R and a positive definite tensor U or V. Logarithmic Hencky Strain [1928]: in the uniaxial case the logarithmic strain is defined by integrating the stretch rate: ɛ ln = L L 0 dl l = ln L L 0 = lnλ where λ = L L 0. (1.7) Triaxial extension of logarithmic strain: 2
4 Lagrangian format : ɛ U ln = lnu, with principal coordinates which are defined by e U ) Eulerian format : ɛ V ln = lnv, with principal Coordinates which are defined by e V = R e U. Spectral Representation : U = V = 3 λ i e i U ei U and 3 ɛ U ln = (lnλ i ) e i U ei U (1.8) i=1 i=1 3 3 λ i e i V e i V and ɛ V ln = (lnλ i ) e i V e i V (1.9) i=1 i=1 1.3 Lagrangian Strain Measure : Extensional deformation ds 2 ds 2 = dx t dx dx t dx = dx t (F t F 1) dx (1.10) The Green-Lagrange strain is related to the right stretch tensor by E G = 1 2 (Ft F 1) = 1 2 (U2 1) (1.11) In indicial form, F = x i X A e i e A (1.12) U 2 = x i x i X A X (1.13) EA G = 1 2 ( x i x i δ A ) X A X (1.14) Generalized Lagrangian Strain [oyle-erickson 1956]: for m=0 and for m=1,2,... E 0 = lnu (1.15) E m = 1 m (Um 1) (1.16) 1-dim examples with λ = L L 0 : E 0 = lnλ E 1 = λ 1 E 2 = 1 2 (λ2 1) (1.17) 3
5 Strain-isplacement Relationship The displacement description of motion defines the new location of the material particle X in terms of : x = x(x,t)=u(x,t)+x (1.18) The deformation gradient is given as x X = u x + 1 = H + 1 where u X = H (1.19) With F = H + 1, we may develop the Lagrangian strain-displacement relationship U 2 = F t F = 1 + H + H t + H t H (1.20) 1.4 Eulerian Strain Measure : Spatial description of extensional deformation ds 2 ds 2 = dx t (1 F t F 1 ) dx (1.21) The Almansi strain tensor is in terms of the left stretch tensor e A = 1 (1 2 F t F 1 )= 1 (1 2 (V2 ) 1 ) (1.22) Generalized Eulerian Strain [oyle-erickson 1956]: for m=0 and for m=-1,-2,... e 0 = lnv (1.23) e m = 1 m (Vm 1) (1.24) 1-dim examples when 1 λ = L 0 L : e 0 = ln 1 λ e 1 = 1 1 λ e 2 = 1 2 (1 1 λ 2 ) (1.25) Strain-isplacement Relationship In the spatial description the motion is described by: The spatial displacement gradient is defined as: u(x,t)=x X(x,t) (1.26) h = u x = 1 X x = 1 F 1 (1.27) 4
6 or F 1 = 1 h (1.28) V 2 = F t F 1 = 1 h h t + h t h (1.29) Substituting into the definition of the Almansi strain e A leads to the Almansi strain-displacement relationship e A = 1 2 (h + ht h t h) (1.30) 1.5 Infinitesimal eformations: If det H << 1 det(h t H) 0, then ɛ = 1 2 (H + Ht ) 1 2 (h + ht ) (1.31) For Infinitesimal eformations the spatial and the material displacement gradients coincide, u X u (1.32) x Additive decomposition into symmetric and skew-symmetric components leads to H = u X = 1 2 (H + Ht )+ 1 2 (H Ht ) (1.33) where the symmetric component defines the traditional linearized strain tensor ɛ ij = 1 2 ( u i X j + u j X i ) (1.34) and where the skew-symmetric component defines the infinitesimal rotation tensor such that ɛ ij = ɛ ji and ω ij = ω ji. ω ij = 1 2 ( u i X j u j X i ) (1.35) 1.6 The Velocity Gradient: Considering the Eulerian format of the velocity field: v = v(x,t) (1.36) the Velocity Gradient is defined as : l = v x Additive decomposition into symmetric and skew-symmetric components leads to (1.37) l = d + w = 1 2 (l + lt )+ 1 2 (l lt ) (1.38) 5
7 The rate of deformation tensor is the symmetric part The spin tensor is the skew-symmetric part 1.7 The Rate of eformation Tensor: Given d = 1 2 ( v i x j + v j x i ) (1.39) w = 1 2 ( v i x j v j x i ) (1.40) ds 2 = dx t dx = dx t (F t F) dx (1.41) then the material time derivative is t (ds2 )=dx t (F t Ḟ + Ḟt F)dX (1.42) Given F = x v the material time derivative yields Ḟ = = v x. X X x X Relationship between rate of deformation gradient and velocity gradient: Ḟ = l F (1.43) The difference between the spatial description of the temperature field T = T (x,t)andthe material description T = T (X,t) leads to the important difference of derivatives as follows: Spatial time derivative : T = T (x,t) Material time derivative : (T )= T + T ẋ t x Considering the material time derivative of the line element ds we have t (ds2 )=dx t (F t l F + F t l t F) dx (1.44) and in spatial form t (ds2 )=dx t (l + l t ) dx =2dx t d dx (1.45) Hence the rate of deformation tensor measures the rate of extensional deformation of ds 1 2 t (ds2 )=dx t d dx (1.46) 6
8 1.8 Reynold s Transport Theorem : The material time derivative of a physical quantity in the spatial description involves two terms = t t X = t x +v grad (1.47) For the scalar temperature field T = T (x,t)wehave T t = T(X,t) t = T(x,t) t + T x x t = T x +grad T v (1.48) Recalling that Ḟ = l F and Ḟt = F t l t, we consider the material time derivative of the right polar decomposition U 2 = F t F : t (U2 )= t (Ft F) =Ḟt F+F t F = F t l t F+F t l F = F t (l t +l) F =2F t d F (1.49) From the Green-Lagrange strain E G = 1 2 (U2 1), we find the relationship with the rate of deformation tensor Ė G = 1 2 (U U + U U) =F t d F (1.50) 1.9 Lagrangian Strain Rate: From the right polar decomposition of F = R U, the time rate of the deformation gradient is Ḟ = Ṙ U + R U (1.51) With Ḟ = l F, the left hand side expands into l R U = Ṙ U + R U (1.52) After multiplication with U 1 and R t we get l = Ṙ Rt + R U U 1 R t (1.53) where Ṙ R t = Ω = Ω t denotes the Rate of the Material Rotation Tensor. The velocity gradient decomposes into symmetric and skew-symmetric components d = 1 2 R ( U U 1 + U 1 U) R t (1.54) and w = Ṙ Rt R ( U U 1 U 1 U) R t (1.55) Remark : The tensor Ω = Ṙ Rt defines the rate of material rotation without stretching. Only when there is no change of stretch, the spin tensor w coincides with the rate of material 7
9 rotation tensor Ω. In general, the rate of deformation tensor d does not coincide with the normalized rate of the right stretch tensor U.U 1 because of the superposed rotation R 0. This leads to the concept of the rotation-neutralized intermediate configuration which should be used to define rotation-free constitutive rate equations. d RN = R t d R = 1 2 ( U U 1 + U 1 U) (1.56) 1.10 Eulerian Strain Rate: From the left polar decomposition of the Jacobian : F = V R, the time rate of the Jacobian is Ḟ = V R + V Ṙ (1.57) Using Ḟ = l F, then the left hand side expands into or after multiplication with R t and V 1 we get l V R = V R + V Ṙ (1.58) l = V V 1 + V Ṙ Rt V 1 (1.59) Then, the velocity gradient decomposes into symmetric and skew-symmetric components d = 1 2 ( V V 1 + V 1 V) (1.60) w = 1 2 ( V V 1 V 1 V)+V Ω V 1 (1.61) Remark : The rate of deformation tensor d coincides with the normalized rate of the left stretch tensor, while the spin tensor w does not agree with the rate of material rotation Ω when stretching takes place with V 1 and V 0 8
10 Chapter 2 alance Laws: The balance laws comprise statements as follows: 1. alance of Linear Momentum. 2. alance of Angular Momentum. 3. alance of Mass. 4. alance of Energy (First Law of Thermodynamics). Forces are measured indirectly through their action on deformable solids. istributed forces include: b : body force/unit volume(i. e. density). t n : surface traction. Integrating the distributed forces over part I of the body defines the resultant force vector f I = b dv + I t n da I (2.1) 2.1 alance of Linear Momentum: Linear momentum is defined as: i = ρ ẋ dv (2.2) I Application of Newton s second law f = m a to the control volume of the body t i = f (2.3) ynamic equilibrium or the balance of linear momentum may be expressed as t (ρẋ)dv = b dv + t n da (2.4) I I I 9
11 or in terms of I (b t (ρẋ))dv + t n da = 0 (2.5) Cauchy Lemma : states pointwise balance of surface tractions across any surface in the interior of the body. t n (x, n) +t n (x, n) = 0 (2.6) or t n (x, n) = t n (x, n) (2.7) 2.2 Cauchy s First Theorem: The stress tensor is a linear mapping of the stress vector t n onto the normal vector n. t n = σ t (x) n (2.8) In indicial notation, Considering elementary tetrahedron: t i = σ ji n j (2.9) t n = n 1 t 1 + n 2 t 2 + n 3 t 3 (2.10) The stress tensor σ is simply composed of the coordinates of stress vectors on three mutually orthogonal planes σ 11 σ 12 σ 13 σ = σ 21 σ 22 σ 23 (2.11) σ 31 σ 32 σ 33 In our notation of σ ij, the first subscript i refers to normal direction of the area element and the subscript j refers to the direction of the traction. Considering the equilibrium in x 1 direction : f x1 =0. t 1 da = σ 11 n 1 da + σ 21 n 2 da + σ 31 n 3 da (2.12) then t 1 = σ 11 n 1 + σ 21 n 2 + σ 31 n 3 (2.13) With the help of the ivergence Theorem we find σ ji n j da = σ ji,j dv (2.14) The equation of motion by Cauchy states the balance between the body forces and surface tractions when inertia forces remain negligible: 10
12 b i dv + t n i da = 0 (2.15) can be rewritten as σ ji,j + b i =0 (2.16) Cauchy s First Theorem states pointwise equilibrium in the interior of the body. In the dynamic case, the balance equations generalize to 2.3 alance of Mass: σ ji,j + b i = t (ρx i) (2.17) The material time derivative of volume integral is comprised of two terms, one volume and one surface integral t ( Φρdv) = t (Φρ) dv + Φ ρ v n da (2.18) Using the divergence theorem the above equation reduces to: t ( Φρ)dv = [ Φ[ ρ t + div (ρv)] + ρ[dφ dt + v grad Φ] ] dv (2.19) If we assume that the function Φ = 1, then this reduces to an integral statement of mass conservation [ ] ρ ρdv = + div (ρ v) dv = 0 (2.20) t t or pointwise it must hold ρ + div(ρv) =0 (2.21) t Expanding the divergence term ρ +(gradρ) v + ρdivv = 0 (2.22) t Finally, the Continuity Condition at each point is simply ρ t + ρdivv =0 (2.23) In indicial notation ρ t + v i = 0 (2.24) x i where ρ t = ρ t + v ρ i (2.25) x i The continuity condition reduces the material time derivative of linear momentum to ρẋ = t t (ρẋ)dv + ẋ ρẋ nda = 11 ρ ẋ dv (2.26) t
13 Material escription of Mass alance: Considering dx = F dx, with det F = J = dv dv,then dv = JdV (2.27) 0 There is no mass flow, but only mapping of geometry. For mass conservation, With ρ 0 dv = ρjdv we find ρdv = 0 ρ 0 dv (2.28) J = ρ 0 (2.29) ρ When mass flow is involved; the material time derivative of the mass balance must vanish, t (Jρ ρ 0) = 0 (2.30) With ρ J + J ρ = 0 we find that t t J = Jdivv (2.31) t For incompressible behavior, ρ = 0. and thus div v = 0, or v t i,i = 0. Therefore, the incompressibility condition reduces to, tr d = 0 (2.32) 2.4 alance of Angular Momentum The angular momentum involves h 0 = I (x ρẋ) dv (2.33) where the pole is assumed to coincide with the origin x 0 = 0. The moment of the distributed forces is m 0 = (x b)dv + (x t n ) da (2.34) The balance of angular momentum states (x ρẋ)dv = t h 0 t = m 0 (2.35) (x b)dv + 12 (x t n ) da (2.36)
14 The divergence theorem yields for the last term above [x (b + div σ t ρẍ)]dv +2 (1 σ t )dv = 0 (2.37) Application of the first theorem of Cauchy : b + div σ t ρ tẋ = 0, we get 1 σ t = 0 σ = σ t (2.38) which states that the stress tensors symmetric, i.e. σ t = σ, i.e. the oltzmann Postulate of a symmetric stress tensor. In index notation : e ijk σ jk =0 σ jk = σ kj (2.39) this results in σ 23 σ 32 =0;σ 31 σ 13 =0;σ 12 σ 21 = 0 (2.40) 2.5 Alternative Stress Measures: Consider that the force vector is the same on the deformed and undeformed surface areas f = t n da = σ t n da =Σ t N da (2.41) then Σ denotes the First Piola-Kirchhoff stress tensor with respect to the undeformed surface area. From Nanson s formula n da = J F t N da we get Jσ t F t N da =Σ t N da (2.42) or Σ = J F 1 σ (2.43) which shows the loss of symmetry of the first Piola-Kirchhoff stress, Σ Σ t. The Second Piola-Kirchoff stress is defined as S =Σ F t = J F 1 σ F t (2.44) or S = F 1 τ F t (2.45) in which τ = Jσ denotes the Kirchoff stress. From σ = 1 J FSFt, we can relate the Kirchhoff stress to the second Piola-Kirchhoff stress τ = FSF t (2.46) alance of Linear Momentum: 13
15 In the current reference configuration t ( ρ ẋ dv) = b dv + t n da (2.47) in which t n = σ t n. In the initial undeformed reference configuration this reads ẋ ρ 0 t dv = b 0 dv + Σ t N da 0 0 (2.48) Mechanical Stress Power: Conjugate forms of kinematic and static measures lead to an inner product form of stress and deformation rate. Using the divergence theorem the spatial description of momentum balance leads to the local statement of differential equilibrium : div σ t + b = ρ v (2.49) t If this equation of motion is scalar multiplied with v and integrated over the entire body, we get (div σ t ) v dv + b v dv = ρ v v dv (2.50) With (div σ t ) v = div(σ t v) σ : d, weget b v dv + t v da = σ : d dv + t ρ v.v dv (2.51) Using the divergence theorem the material description of momentum balance leads to the local statement of differential equilibrium : iv Σ t V + b 0 = ρ 0 t (2.52) If this equation is multiplied with the weighing function v and integrated over the entire body in the reference configuration, we find (iv Σ t ) v dv + b 0 v dv = ρ t V v dv (2.53) After analogous calculation to the spatial description we get b 0 v dv + (Σ t N) v da = Σ t : Ḟ dv + 0 ρ 0 V v dv (2.54) Stress Power: Considering τ = ρσ, the inner product of stress and the rate of deformation leads to alternative representations in terms of conjugate values Ẇ σ = 1 ρ 0 τ : d = 1 ρ σ : d = 1 ρ 0 Σ t : Ḟ = 1 ρ 0 S : ĖG (2.55) 14
16 Chapter 3 Lagrangian and Eulerian escriptions Abstract This section develops Lagrangean and Eulerian settings for the finite element description of motion in terms of linearized incremental statements of virtual work. 3.1 Strong Equilibrium Statements Considering the balance of momentum neglecting inertia effects: 1. Reference configuration: iv Σ t + b 0 = 0 (3.1) and the traction boundary condition T = Σ t N in the reference configuration; The balance of angular momentum results in symmetry of F Σ =Σ t F t 2. Current configuration: div σ t + b = 0 (3.2) and the traction boundary condition t = σ t n in the current configuration. The balance of angular momentum results in symmetry of the Cauchy stress tensor, i.e. σ = σ t. 3.2 Weak Equilibrium Statements: δw i = δw e The weak form of equilibrium is expressed in terms of the virtual work principle, with the internal work being defined in the reference configuration as δw i = Grad(δu) :Σ dv 0 (3.3) and in the current configuration as δw i = grad(δu) :σ dv (3.4) Two descriptions will be disscussed for linearizing these weak equilibrium statements: 15
17 1. The Lagrangian Approach: Total Lagrange Method. 2. The Eulerian Approach: Updated Lagrange Method. 3.3 Lagrangian Approach: The incremental form of weak equilibrium derives directly from the weighted form of pointwise balance of incremental linear momentum. In the reference configuration this results in the increment of virtual work δ U = δẇ with δẇi = Grad(δu) : Σ dv (3.5) Recall that Ḟ = Grad( u) = l F, and that the tangential material law in the reference configuration is defined as Ṡ = C 0 : ĖG, With (iv t Σ) = iv Σ, andwithσ = S F t we may substitute these expressions into the incremental form of the internal virtual work, which leads to δẇi = Grad(δu) :(Ṡ Ft + S Ḟt )dv = Grad(δu) :(F C 0 : ĖG + S F t l t )dv (3.6) β 1 where ĖG = 1 2 ( F t F + F t Ḟ). This expression may be decomposed into two separate terms, δẇi = Grad(δu) :S Grad( u) t dv + 1 Grad(δu) :F : C 0 :(F t Grad( u) +Grad ( u t )F)dV (3.7) 2 in which the term Grad(δu) :S Grad( u) t gives rise to the geometric stiffness, andthe second term Grad(δu) :F C 0 :(F t Grad( u) + Grad( u) F) gives rise to the material stiffness δw G = Grad(δu) : S Grad( u) t dv (3.8) δẇm = Eulerian Approach: β 1 Grad(δu) :F : C 0 :(F t Grad( u) +Grad( u) t F) dv (3.9) We start from the definition of Nominal Stress in terms of the first Piola-Kirchhoff stress Σ = J F 1 σ. The material time derivative involves three terms because of the chain rule of differentiation t (Σ) = Σ = J F 1 σ + J Ḟ 1 σ + J F 1 σ (3.10) 16
18 with the definition J = J (trl) andwithf F 1 = 1, weobtainḟ 1 = F 1 l.the material derivative of the nominal stress is henceforth Σ = J F 1 ((trl) σ l σ + σ) (3.11) The nominal stress rate in the Current Configuration is obtained by push-forward operation and updating the reference configuration to the current configuration. since F = 1 and J = 1. Σ c = (trl) σ l σ + σ (3.12) Adopting an objective description of the hypo-elastic material law: σ = c : d (3.13) in terms of the objective rate of deformation and an objective stress rate such as the corotational Jaumann rate of Cauchy-stress σ = σ w σ + σ w t (3.14) with the spin defined as w t = w = 1 2 (l lt ), we obtain for the nominal stress rate an expansion as follows Σ c = σ +(trl)σ l σ + w σ σ w t (3.15) or Σ c = c : d +(trl) σ l σ (l σ l t σ + σ l t σ l) (3.16) With l = Ḟ F 1 and F 1 = 1, we may develop the desired relationship between the nominal stress state Σ c and the rate of the deformation gradient Ḟ = l F. Σ c = c T : l (3.17) The effective tangential material tensor c T is comprised of two terms which involve the tangential moduli tensor c and the current stress state σ. Explicit format : or expanded c ijkl = c T ijkl + σ ij δ kl δ ik σ lj (δ ikσ lj δ il σ kj + δ jk σ il δ jl σ ik ) (3.18) Σ c ij = c T ijkl d kl + σ ij l kk l il σ lj (l ilσ lj l ik σ kj + l jl σ il l jk σ ik ) (3.19) 17
19 Substituting the rate of the nominal stress tensor into the weak equilibrium statement, the virtual internal work statement in the current configuration reduces to δẇi = Grad(δu) : Σ dv grad(δu) : Σ c dv (3.20) β 1 with l = grad( u), the incremental virtual work expands into δẇ = grad(δu) : c T : l dv = grad(δu) : c T : grad( u) dv (3.21) β β β This incremental equilibrium statement gives rise to the Tangential Material Stiffness and the Geometric Stiffness operators defined by: δẇi = δẇm + δẇg (3.22) With the tangential material law σ = c : d, we recover the material stiffness operator of the small displacement theory δẇm = grad(δu) :c : d dv = δɛ : c : d dv (3.23) β and the geometric stiffness of the initial stress operator δẇg = grad(δu) :ˆσ : grad( u) dv (3.24) β In the finite element sense, these virtual work terms translate into matrix notation: and β δẇ h i = δu t I (k E + k G ) u I (3.25) δẇh e = δu t I ḟi (3.26) The Tangential Stiffness relationship describes a quasi-linear statement of incremental equilibrium which reads at the element level as; (k E + k G ) u I = ḟi (3.27) With G = N, and the material tangent E x T = c, we retrieve the format of the Eulerian tangent stiffness operators, k E = t L E T L dv (3.28) Ω e k G = G t ˆσ G dv (3.29) Ω e in which L denotes the traditional linear rate of deformation-nodal velocity operator and G the spatial gradient of the finite element velocity expansion. 18
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