CONTINUUM MECHANICS. lecture notes 2003 jp dr.-ing. habil. ellen kuhl technical university of kaiserslautern

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1 CONTINUUM MECHANICS lecture notes 2003 jp dr.-ing. habil. ellen kuhl technical university of kaiserslautern

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3 Contents Tensor calculus. Tensor algebra Vector algebra Notation Euclidian vector space Scalar product Vector product Scalar triple vector product Tensor algebra Notation Scalar products Dyadic product Scalar triple vector product Invariants of second order tensors Trace of second order tensors Determinant of second order tensors Inverse of second order tensors Spectral decomposition Symmetric skew-symmetric decomposition Symmetric tensors Skew-symmetric tensors Volumetric deviatoric decomposition Orthogonal tensors Tensor analysis Derivatives Frechet derivative Gateaux derivative Gradient Gradient of a scalar valued function Gradient of a vector valued function Divergence Divergence of a vector field Divergence of a tensor field Laplace operator Laplace operator acting on scalar valued function Laplace operator acting on vector valued function i

4 Contents.2.5 Integral transformations Integral theorem for scalar valued fields Integral theorem for vector valued fields Integral theorem for tensor valued fields Kinematics Motion Rates of kinematic quantities Velocity Acceleration Gradients of kinematic quantities Displacement gradient Strain Rotation Volumetric deviatoric decomposition of strain tensor Strain vector Normal shear decomposition Principal strains stretches Compatibility Special case of plane strain Voigt representation of strain Balance equations Basic ideas Concept of mass flux Concept of stress Volumetric deviatoric decomposition of stress tensor Normal shear decomposition Principal stresses Special case of plane stress Voigt representation of stress Concept of heat flux Balance of mass Global form of balance of mass Local form of balance of mass Classical continuum mechanics of closed systems Balance of linear momentum Global form of balance of momentum Local form of balance of momentum Reduction with lower order balance equations Classical continuum mechanics of closed systems Balance of angular momentum Global form of balance of angular momentum Local form of balance of angular momentum Reduction with lower order balance equations ii

5 Contents 3.5 Balance of energy Global form of balance of energy Local form of balance of energy Reduction with lower order balance equations First law of thermodynamics Balance of entropy Global form of balance of entropy Local form of balance of entropy Reduction with lower order balance equations Second law of thermodynamics Generic balance equation Global / integral format Local / differential format Thermodynamic potentials Constitutive equations Linear constitutive equations Mass flux Fick s law Momentum flux Hook s law Heat flux Fourier s law Hyperelasticity Specific stored energy Specific complementary energy Isotropic hyperelasticity Specific stored energy Specific complementary energy Elastic constants Transversely isotropic hyperelasticity A Übungsaufgaben 05 A. Tensoralgebra zweistufiger Tensoren A.2 Tensoranalysis A.3 Ableitungen A.4 Verzerrungs und Spannungsvektor iii

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7 Tensor calculus. Tensor algebra.. Vector algebra... Notation Einstein s summation convention u i 3 j A i j x j b i A i j x j b i (..) summation over indices that appear twice in a term or symbol, with silent (dummy) index j and free index i, and thus u A x A 2 x 2 A 3 x 3 b u 2 A 2 x A 22 x 2 A 23 x 3 b 2 (..2) u 3 A 3 x A 32 x 2 A 33 x 3 b 3 Kronecker symbol δ i j δ i j for i j 0 for i j (..3) multiplication with Kronecker symbol corresponds to exchange of silent index with free index of Kronecker symbol u i δ i j u j (..4)

8 Tensor calculus permutation symbol 3 e i jk 3 e i jk for i, j, k... odd permutation for i, j, k... even permutation 0 for... else (..5)...2 Euclidian vector space consider linear vector space 3 characterized through addition of its elements u, v and multiplication with real scalars α, β α, β... real numbers u, v linear vector space definition of linear vector space axioms α u v α u α v α β u α u β u α β u α β u 3 through the following (..6) zero element and identity 0 u 0 u u (..7) linear independence of elements e, e 2, e 3 3 if α α 2 α 3 0 is the only (trivial) solution to α i e i 0 (..8) 2

9 . Tensor algebra consider linear vector space 3 equipped with a norm n u mapping elements of the linear vector space 3 to the space of real numbers n : 3 norm (..9) definition of norm through the following axioms n u 0 n u 0 u 0 n α u α n u n u v n u n v n 2 u v n 2 u v 2 n 2 u n 2 v (..0) 3 equipped with the Eu- consider Euclidian vector space clidian norm n u u u u u 2 u 2 2 u (..) mapping elements of the Euclidian vector space space of real numbers 3 to the n : 3 Euclidian norm (..2) representation of three dimensional vector a 3 a a i e i a e a 2 e 2 a 3 e 3 (..3) with a, a 2, a 3 coordinates (components) of a relative to the basis e, e 2, e 3 a a, a 2, a 3 t (..4) 3

10 Tensor calculus...3 Scalar product Euclidian norm enables the definition of scalar (inner) product between two vectors u, v and introduces a scalar α u v α (..5) geometric interpretation with 0 ϑ π being the angle enclosed by the vectors u and v, then u cos ϑ can be interpreted as the projection of u onto the direction of v and u v u v cos ϑ PSfrag replacements u u ϑ u cos ϑ cos ϑ v corresponds to the grey area in the picture with the above interpretation with 0 ϑ π, obviously u v u v (..6) properties of scalar product u v v u α u β v w α u w β v w (..7) w α u β v α w u β w v positive definiteness of scalar product u u 0, u u 0 u 0 (..8) orthogonal vectors u and v u v 0 u v (..9) 4

11 . Tensor algebra...4 Vector product vector product of two vectors u, v defines a new vector w 3 u v w (..20) PSfrag replacements geometric interpretation w v v sin ϑ with 0 ϑ π being the ϑ angle enclosed by the vectors u and v, then v sin ϑ u can be interpreted as the height of the grey polygon and u v u v sin ϑ n introduces the vector w orthogonal to u and v whereby its length corresponds to the grey area with the above interpretation, obviously u parallel to v if u v 0 u v (..2) index representation of w u v w w 2 w 3 u 2 v 3 u 3 v 2 properties of vector product u 3 v u v 3 (..22) u v 2 u 2 v u v v u α u β v w α u w β v w u u v 0 u v u v u u v v u v 2 (..23) 5

12 Tensor calculus...5 Scalar triple vector product scalar triple vector product of three vectors u, v, w introduces a scalar α u, v, w u v w α (..24) geometric interpretation with vector product v w v w sin ϑ n PSfrag replacements defining aera of ground surface u, v, w u v w defines volume of parallelepiped obviously v ϑ u v w w α u v w v w u w u v α u w v v u w w v u index representation of α u, v, w (..25) α u v 2 w 3 v 3 w 2 u 2 v 3 w v w 3 u 3 v w 2 v 3 w (..26) properties of scalar triple product u, v, w v, w, u w, u, v u, w, v v, u, w w, v, u αu βv, w, d α u, w, d β v, w, d three vectors u, v, w are linearly independent if (..27) u, v, w 0 (..28) 6

13 . Tensor algebra..2 Tensor algebra..2. Notation Second order tensors tensor (dyadic) product u v of two vectors u and v introduces a second order tensor A A u v (..29) introducing u u i e i and v v j e j yields index representation of three-dimensional second order tensor A A A i j e i e j (..30) with A i j u i v j coordinates (components) of A relative to the tensor basis e i e j, matrix representation of coordinates (..3) A i j A 2 A 22 A 23 A A 2 A 3 A 3 A 32 A 33 transpose of second order tensor A t A t u v t v u (..32) introducing u u i e i and v v j e j yields index representation of transpose of second order tensor A t A t A ji e j e i (..33) with A ji v j u i coordinates (components) of A relative to the tensor basis e j e i, matrix representation of coordinates (..34) A ji A 2 A 22 A 32 A A 2 A 3 A 3 A 23 A 33 7

14 Tensor calculus second order unit tensor I in terms of Kronecker symbol δ i j I δ i j e i e j (..35) matrix representation of coordinates δ i j δ ji (..36) Third order tensors tensor (dyadic) product A vectors u introduces a third order tensor 3 a v of second order tensor A and 3 a A v (..37) introducing A A i j e i e j and u u k e k yields index representation of three-dimensional third order tensor 3 a 3 a 3 a i jk e i e j e k (..38) with 3 a i jk A i j u k coordinates (components) of 3 a relative to the tensor basis e i e j e k third order permutation tensor 3 e in terms of permutation symbol 3 e i jk 3 e 3 e i jk e i e j e k (..39) 8

15 . Tensor algebra Fourth order tensors tensor (dyadic) product A B of two second order tensors A and B introduces a fourth order tensor /A /A A B (..40) introducing A A i j e i e j and B B kl e k e l yields index representation of three-dimensional fourth order tensor /A /A A i jkl e i e j e k e l (..4) with A i jkl A i j B kl coordinates (components) of /A relative to the tensor basis e i e j e k e l fourth order unit tensor II II δ ik δ jl e i e j e k e l (..42) transpose fourth order unit tensor II t II t δ il δ jk e i e j e k e l (..43) symmetric fourth order unit tensor II sym II sym 2 δ ik δ jl δ il δ jk e i e j e k e l (..44) skew-symmetric fourth order unit tensor II skw II skw 2 δ ik δ jl δ il δ jk e i e j e k e l (..45) volumetric fourth order unit tensor II vol II vol 3 δ i j δ kl e i e j e k e l (..46) deviatoric fourth order unit tensor II dev II dev 3 δ i j δ kl 2 δ ik δ jl 2 δ il δ jk e i e j e k e l (..47) 9

16 Tensor calculus..2.2 Scalar products scalar product A u between second order tensor A and 3 vector u defines a new vector v A u A i j e i e j u k e k A i j u k δ jk e i A i j u j e i v i e i v (..48) second order zero tensor 0, second order identity tensor I 0 a 0 I a a (..49) positive semi-definiteness of second order tensor A a A a 0 (..50) positive definiteness of second order tensor A a A a 0 (..5) properties of scalar product A αa βb α A a β A b A B a A a B a α A a α A a (..52) scalar product A B between two second order tensors A and B defines a second order tensor C A B A i j e i e j : B kl e k e l A i j B kl δ ik e i e l A i j B jl e i e l C il e i e l C (..53) second order zero tensor 0, second order identity tensor I 0 A 0 I A A (..54) 0

17 . Tensor algebra properties of scalar product α A B α A B A α B A B C A B A C A B C A C B C (..55) properties in terms of transpose A t of a tensor A a A t b b A a α A β B t α A t β B t A B t B t A t (..56) scalar product A : B between two second order tensors A and B defines a scalar α A : B A i j e i e j : B kl e k e l A i j B kl δ ik δ jl A i j B i j α (..57) scalar product /A : B between fourth order tensor /A and second order tensor B defines a new second order tensor C /A : B A i jkl e i e j e k e l : B mn e m e n A i jkl B mn δ km δ ln e i e j A i jkl B kl e i e j A i j e i e j A (..58)

18 Tensor calculus..2.3 Dyadic product tensor (dyadic) product u v of two vectors u and v introduces a second order tensor A A u v u i e i v j e j u i v j e i e j A i j e i e j (..59) properties of dyadic product u v w v w u α u β v w α u w β v w u α v β w α u v β u w u v w x v w u x A u v A u v u v A u A t v (..60) or in index notation u i v j w j v j w j u i α u i β v i w j α u i w j β v i w j u i α v j β w j α u i v j β u i w j u i v j w j x k v j w j u i x k (..6) A i j u j v k A i j u i v k u i v j A jk u i A kj v j 2

19 . Tensor algebra..2.4 Scalar triple vector product consider the set of Cartesian base vectors e i i,2,3 and an arbitrary second set of base vectors u, v, w with scalar triple product u, v, w, with arbitrary second order tensor A, evaluate A u, v, w u, A v, w u, v, A w u, v, w (..62) with index representation of each term according to A u, v, w A u i e i, v j e j, w k e k u i v j w k A e i, e j, e k (..63) expression (..62) can be rewritten as u i v j w k A e, e 2, e 3 e, A e 2, e 3 e, e 2, A e 3 u, v, w (..64) term in brackets remains unchanged upon cyclic permutation of e i i,2,3, its sign reverses upon non cyclic permutations, thus u i v j w k 3 ei jk A e, e 2, e 3 e, A e 2, e 3 e, e 2, A e 3 u, v, w A e, e 2, e 3 e, A e 2, e 3 e, e 2, A e 3 (..65) the above expression according to (..62) is thus invariant under the choice of base system, it yields the same scalar value I A for arbitrary base systems I A A u, v, w u, A v, w u, v, A w u, v, w A e, e 2, e 3 e, A e 2, e 3 e, e 2, A e 3 e, e 2, e 3 (..66) I A is called the first invariant of the second order tensor A 3

20 Tensor calculus..2.5 Invariants of second order tensors the following property of the scalar triple product u, v, w u v w (..67) introduces three scalar valued quantities I A, II A, III A associated with the second order tensor A A u, v, w u, A v, w u, v, A w I A u, v, w u, A v, A w A u, v, A w A u, A v, w II A u, v, w A u, A v, A w III A u, v, w (..68) the proof of II A, III A being invariant for different base systems u, v, w is similar to the one for I A I A, II A, III A are called the three principal invariants of A which can be expressed as I A tr A A I A I II A 2 tr2 A tr A 2 A II A I A I A (..69) III A det A A III A III A A t alternatively, we could work with the three basic invariants Ī A, II A, III A of A which are more common in the context of anisotropy Ī A II A III A A : I A 2 : I A 3 : I (..70) 4

21 . Tensor algebra..2.6 Trace of second order tensors trace tr A of a second order tensor A u v introduces a scalar tr A tr u v u v (..7) such that tr A is the sum of the diagonal entries A ii of A with tr A tr A i j e i e j A i j tr e i e j A i j e i e j (..72) A i j δ i j A ii A A 22 A 33 I A Ī A tr A (..73) properties of the trace of second order tensors tr I 3 tr A t tr A tr A B tr B A tr α A β B α tr A β tr B tr A B t A : B tr A tr A I A : I (..74) 5

22 Tensor calculus..2.7 Determinant of second order tensors determinant det A scalar det A of second order tensor A introduces a det A det A i j 6 e i jk e abc A ia A jb A kc with A A 22 A 33 A 2 A 32 A 3 A 3 A 2 A 23 (..75) A A 23 A 32 A 22 A 3 A 3 A 33 A 2 A 2 III A det A (..76) determinant defining vector product u v u v det u v e u 2 v 2 e 2 u 3 v 3 e 3 u 2 v 3 u 3 v 2 u 3 v u v 3 (..77) u v 2 u 2 v determinant defining scalar triple vector product u, v, w u, v, w u v w det u v w u 2 v 2 w 2 (..78) u 3 v 3 w 3 properties of determinant of a second order tensors det I det A t det A det α A α 3 det A det A B det A det B det u v 0 (..79) 6

23 . Tensor algebra..2.8 Inverse of second order tensors if det A 0 existence of inverse A of second order tensor A A A A A I (..80) in particular v A u A v u (..8) properties of inverse of two second order tensors A A α A α A A B B A (..82) determinant det A of inverse of A det A det A (..83) adjoint A adj of a second order tensor A A adj det A A (..84) cofactor A cof of a second order tensor A A cof det A A t A adj t (..85) with A det A det A A t III A A t A cof (..86) 7

24 Tensor calculus..3 Spectral decomposition eigenvalue problem of arbitrary second order tensor A A n A λ A n A A λ A I n A 0 (..87) solution introduces eigenvector(s) n Ai and eigenvalue(s) λ Ai det A λ A I 0 (..88) alternative representation in terms of scalar triple product A u λ A u, A v λ A v, A w λ A w 0 (..89) removal of arbitrary factor u, v, w yields characteristic equation λ 3 A I A λ 2 A II A λ A III A 0 (..90) roots of characteristic equations are principal invariants of A I A tr A II A 2 tr2 A tr A 2 III A det A (..9) spectral decomposition of A A 3 λ Ai n Ai n Ai (..92) i Cayleigh Hamilton theorem: a tensor A satisfies its own characteristic equation A 3 I A A 2 II A A III A I 0 (..93) 8

25 . Tensor algebra..4 Symmetric skew-symmetric decomposition symmetric skew-symmetric decomposition of second order tensor A A 2 A At 2 A At A sym A skw (..94) with symmetric and skew-symmetric second order tensor A sym and A skw A sym A sym t A skw A skw t (..95) symmetric second order tensor A sym A sym 2 A At II sym : A (..96) upon double contraction symmetric fourth order unit tensor II sym extracts symmetric part of second order tensor II sym 2 II IIt II sym 2 δ (..97) ikδ jl δ il δ jk e i e j e k e l skew symmetric second order tensor A skw A skw 2 A At II skw : A (..98) upon double contraction skew-symmetric fourth order unit tensor II skw extracts skew-symmetric part of second order tensor II skw 2 II IIt II skw 2 δ (..99) ikδ jl δ il δ jk e i e j e k e l 9

26 Tensor calculus..4. Symmetric tensors symmetric part A sym of a second order tensor A A sym 2 A At A sym A sym t (..00) alternative representation A sym II sym : A (..0) whereby symmetric fourth order tensor II sym extracts symmetric part A sym of second order tensor A a symmetric second order tensor S A sym processes three real eigenvalues λ Si i,2,3 and three corresponding orthogonal eigenvectors n Si i,2,3, such that the spectral representation of S takes the following form 3 S λ Si n Si n Si (..02) i three invariants I S, II S, III S of symmetric tensor S A sym I S λ S λ S2 λ S3 II S λ S2 λ S3 λ S3 λ S λ S λ S2 (..03) III S λ S λ S2 λ S3 square root S, inverse S, exponent exp S and logarithm ln S, of positive semi-definite symmetric tensor S for which λ Si 0 S 3 i λ Si n Si n Si S 3 i λ Si n Si n Si exp S 3 i exp λ Si n Si n Si (..04) ln S 3 i ln λ Si n Si n Si 20

27 . Tensor algebra..4.2 Skew-symmetric tensors skew-symmetric part A skw of a second order tensor A A skw 2 A At A skw A skw t (..05) alternative representation A skw II skw : A (..06) whereby skew-symmetric fourth order tensor II skw extracts skew-symmetric part A skw of second order tensor A a skew-symmetric second order tensor W A skw posses three independent entries, three entries vanish identically, three are equal to the negative of the independent entries, these define the axial vector w w 2 3 e: W w 3 e w (..07) associated with each skew-symmetric tensor W A skw W p w p (..08) three invariants I W, II W, III W of skew-symmetric tensor W I W tr W 0 II W w w III W det W 0 (..09) 2

28 Tensor calculus..5 Volumetric deviatoric decomposition volumetric deviatoric decomposition of second order tensor A A A vol A dev (..0) with volumetric and deviatoric second order tensor A vol and A dev tr A vol tr A tr A dev 0 (..) volumetric second order tensor A vol A vol 3 A : I I IIvol : A (..2) upon double contraction volumetric fourth order unit tensor II vol extracts volumetric part of second order tensor II vol 3 I I II vol 3 δ i jδ kl e i e j e k e l (..3) deviatoric second order tensor A dev A dev A 3 A : I I IIdev : A 0 (..4) upon double contraction deviatoric fourth order unit tensor II dev extracts deviatoric part of second order tensor II dev II sym II vol II sym 3 I I II dev 2 δ ikδ jl 2 δ ilδ jk 3 δ i jδ kl e i e j e k e l (..5) 22

29 . Tensor algebra..6 Orthogonal tensors a second order tensor Q is called orthogonal if its inverse Q is identical to its transpose Q t Q Q t Q t Q Q Q t I (..6) a second order tensor A can be decomposed multiplicatively into a positive definite symmetric tensor U t U or V t V with a U a 0 and a V a 0 and an orthogonal tensor Q t Q as A Q U V Q (..7) with S0(3) being the special orthogonal group, Q det Q, then Q is called proper orthogonal S0(3) if a proper orthogonal tensor Q S0(3) has an eigenvalue equal to one λ Q introducing an eigenvector n Q such that Q n Q n Q (..8) let n Qi i,2,3 be a Cartesian basis containing the vector n Q, then matrix representation of coordinates Q i j Q i j 0 cosϕ sinϕ sinϕ cosϕ (..9) geometric interpretation: Q characterizes a finite rotation around the axis n Q with Q n Q n Q, i.e. associated with λ Q 23

30 Tensor calculus.2 Tensor analysis.2. Derivatives consider smooth, differentiable scalar field Φ with scalar argument Φ : ; Φ x α 3 vectorial argument Φ : ; Φ x α 3 3 tensorial argument Φ : ; Φ X α.2.. Frechet derivative scalar vectorial tensorial DΦ x D Φ x D Φ X Φ x x Φ x x Φ X X x Φ x x Φ x X Φ X (.2.).2..2 Gateaux derivative Gateaux derivative as particular Frechet derivative with respect to directions u, u and U d scalar DΦ x u dɛ Φ x ɛ u ɛ 0 u d vectorial D Φ x u dɛ Φ x ɛ u 3 ɛ 0 u tensorial DΦ X :U d dɛ Φ X ɛ U 3 3 ɛ 0 U (.2.2) in what follows in particular vectorial arguments, i.e. point position x or displacement u 24

31 .2 Tensor analysis.2.2 Gradient consider scalar and vector valued field f x and f x on domain 3 f : f : x f x f : 3 f : x f x.2.2. Gradient of a scalar valued function gradient f x of scalar valued field f x f and thus f x x f x f,i x e i (.2.3) x i f, f,2 (.2.4) f,3 gradient of scalar valued field renders a vector field Gradient of a vector valued function gradient f x of vector valued field f x f x and thus f x f i x f i,j x e i e j (.2.5) x j f, f,2 f,3 f 2, f 2,2 f 2,3 (.2.6) f 3, f 3,2 f 3,3 gradient of vector valued field renders a (second order) tensor field 25

32 Tensor calculus.2.3 Divergence consider vector and second order tensor field f x and F x on domain 3 f : 3 f : x f x F : 3 3 F : x F x.2.3. Divergence of a vector field divergence f x of vector field f x div f x tr f x f x : (.2.7) with f x f i,j x e i e j div f x f i,i x f, f 2,2 f 3,3 (.2.8) divergence of a vector field renders a scalar field Divergence of a tensor field divergence F x of tensor field F x div F x tr F x F x : (.2.9) with F x F i,jk x e i e j e k div F x F i j,j x F, F 2,2 F 3,3 F 2, F 22,2 F 23,3 (.2.0) F 3, F 32,2 F 33,3 divergence of a second order tensor field renders a vector field 26

33 .2 Tensor analysis.2.4 Laplace operator consider scalar and vector valued field f x and f x on domain 3 f : f : x f x f : 3 f : x f x.2.4. Laplace operator acting on scalar valued function Laplace operator f x acting on scalar valued field f x and thus f x div f x (.2.) f x f,ii f, f,22 f,33 (.2.2) Laplace operator acting on on scalar valued field renders a scalar field Laplace operator acting on vector valued function Laplace operator f x acting on vector valued field f x and thus f x div f x (.2.3) (.2.4) f x f i,jj f 2, f 2,22 f 2,33 f, f,22 f,33 f 3, f 3,22 f 3,33 Laplace operator acting on on vector valued field renders a vector field 27

34 Tensor calculus Useful transformation formulae consider scalar, vector and second order tensor field α x, u x, v x and A x on domain 3 α : α : x α x u : 3 u : x u x v : 3 v : x v x A : 3 3 A : x A x important transformation formulae α u u α α u u v u v v u div α u α div u u α div α A α div A A α div u A u div A A : u div u v u div v v u t (.2.5) or in index notation α u i,j u i α,j α u i,j u i v i,j u i v i,j v i u i,j α u i,i α u i,i u i α,i α A i j,j α A i j,j A i j α,j (.2.6) u i A i j,j u i A i j,j A i j u i,j u i v j,j u i v j,j v j u i,j 28

35 .2 Tensor analysis.2.5 Integral transformations integral theorems define relations between surface integral and volume integral...dv...da X n consider scalar, vector and second order tensor field α x, u x and A x on domain 3 α : α : x α x u : 3 u : x u x A : 3 3 A : x A x.2.5. Integral theorem for scalar valued fields (Green theorem) α n da α dv α n i da α,i dv (.2.7) Integral theorem for vector valued fields (Gauss theorem) u n da div u dv u i n i da u i,i dv (.2.8) Integral theorem for tensor valued fields (Gauss theorem) A n da div A dv A i j n j da A i j,j dv (.2.9) 29

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37 2 Kinematics restriction to geometrically linear theory, valid if local strains remain small 2. Motion consider a material body as a simply connected subset of the Euclidian space 3 as 3, with the boundary being denoted as, a material point is defined as a point of the body x PSfrag replacements x u x, t e i i,2,3 motion of a body 3 characterized through time dependent vector field of displacements u 3 parameterized in terms of position x and time t u : 3 u x, t u i x, t e i (2..) 3

38 2 Kinematics 2.2 Rates of kinematic quantities 2.2. Velocity vector field of velocities v position x and time t 3 parameterized in terms of v : 3 v x, t v i x, t e i (2.2.) velocity field v defined through rate of change of displacement field u v x, t velocity vector D t u x, t u x, t t xfixed (2.2.2) v v, v 2, v 3 t D t u, u 2, u 3 t (2.2.3) common notation in the literature v u Acceleration vector field of accelerations a of position x and time t 3 parameterized in terms a : 3 a x, t a i x, t e i (2.2.4) acceleration field a defined through rate of change of velocity field v a x, t D t v x, t acceleration vector v x, t t xfixed 2 u x, t t 2 xfixed (2.2.5) a a, a 2, a 3 t D t v, v 2, v 3 t D 2 t u, u 2, u 3 t (2.2.6) common notation in the literature a v ü 32

39 2.3 Gradients of kinematic quantities 2.3 Gradients of kinematic quantities 2.3. Displacement gradient second order tensor field of displacement gradient H 3 3 parameterized in terms of position x and time t H : 3 3 H x, t H i j x, t e i e j (2.3.) displacement gradient H defined through gradient of displacement field u H x, t u x, t u x, t x tfixed (2.3.2) index representation H i j e i e j u i X j e i e j u i,j e i e j (2.3.3) matrix representation of coordinates H i j H i j H 2 H 22 H 23 H H 2 H 3 H 3 H 32 H 33 u, u,2 u,3 u 2, u 2,2 u 2,3 (2.3.4) u 3, u 3,2 u 3,3 displacement gradient H u is non symmetric, H H t u does not vanish for point- displacement gradient H wise rigid body motion 33

40 2 Kinematics Symmetric-skew-symmetric decomposition of displacement gradient symmetric skew-symmetric decomposition of displacement gradient H u H 2 H Ht 2 H Ht ɛ ω (2.3.5) with symmetric and skew-symmetric second order tensor ɛ H sym and ω H skw ɛ ɛ ɛ t ω ω t (2.3.6) geometrically linear strain tensor ɛ 2 H Ht 2 u t u sym u II sym : u (2.3.7) upon double contraction symmetric fourth order unit tensor II sym extracts symmetric part of second order tensor II sym 2 II IIt II sym 2 δ (2.3.8) ikδ jl δ il δ jk e i e j e k e l geometrically linear rotation tensor ω ω 2 H Ht 2 u t u skw u II skw : u (2.3.9) upon double contraction skew-symmetric fourth order unit tensor II skw extracts skew-symmetric part of second order tensor II skw 2 II IIt II skw 2 δ (2.3.0) ikδ jl δ il δ jk e i e j e k e l 34

41 2.3 Gradients of kinematic quantities Strain second order tensor field of (geometrically linear) strains ɛ 3 3 parameterized in terms of position x and time t ɛ : 3 3 ɛ x, t ɛ i j x, t e i e j (2.3.) strain tensor ɛ defined through symmetric part of gradient of displacement field u ɛ x, t sym u x, t index representation u x, t x sym (2.3.2) ɛ i j e i e j 2 u i,j u j,i e i e j (2.3.3) matrix representation of coordinates ɛ i j ɛ i j ɛ ɛ 2 ɛ 3 ɛ 2 ɛ 22 ɛ 23 ɛ 3 ɛ 32 ɛ u, u,2 u 2, u,3 u 3, u 2, u,2 2u 2,2 u 2,3 u 3,2 u 3, u,3 u 3,2 u 2,3 2u 3,3 (2.3.4) strain tensor ɛ sym u is symmetric, ɛ ɛ t ɛ i j ɛ ɛ 2 ɛ 3 ɛ 2 ɛ 22 ɛ 23 ɛ 3 ɛ 32 ɛ 33 ɛ ɛ 2 ɛ 3 ɛ 2 ɛ 22 ɛ 32 ɛ ji (2.3.5) ɛ 3 ɛ 23 ɛ 33 ɛ i j for i ɛ i j for i j... diagonal entries: normal strain j... off diagonal entries: shear strain 35

42 2 Kinematics Rotation second order tensor field of (geometrically linear) rotation ω 3 3 parameterized in terms of position x and time t ω : 3 3 ω x, t ω i j x, t e i e j (2.3.6) rotation tensor ω defined through skew symmetric part of gradient of displacement field u ω x, t skw u x, t index representation u x, t x skw (2.3.7) ω i j e i e j 2 u i,j u j,i e i e j (2.3.8) matrix representation of coordinates ω i j ω i j 0 ω 2 ω 3 ω 2 0 ω 23 ω 3 ω u,2 u 2, u,3 u 3, u 2, u,2 0 u 2,3 u 3,2 u 3, u,3 u 3,2 u 2,3 0 (2.3.9) rotation tensor ω skw u is skew-symmetric, ω ω t ω i j 0 ω 2 ω 3 ω 2 0 ω 23 ω 3 ω 32 0 corresponding axial vector w 0 ω 2 ω 3 ω 2 0 ω 32 ω 3 ω e: 3 ω ω ji (2.3.20) w w, w 2, w 3 t ω 23, ω 3, ω 2 t (2.3.2) 36

43 2.3 Gradients of kinematic quantities Symmetric-skew-symmetric decomposition of displacement gradient symmetric skew-symmetric decomposition of displacement gradient H u H 2 u t u 2 u t u ɛ ω (2.3.22) with symmetric and skew-symmetric second order tensor ɛ sym u and ω skw u ɛ ɛ t ω ω t (2.3.23) geometric interpretation: representation of symmetric and skew symmetric part of displacement gradient for two-dimensional case strain rotation e 2 e 2 ag replacements u,2 ɛ PSfrag replacements u,2 ω e e u 2, u 2, ɛ 2 2 u,2 u 2, ω 2 2 u,2 u 2, symmetric part ɛ sym u represents strain while skewsymmetric ω skw u part represents rotation 37

44 2 Kinematics Volumetric deviatoric decomposition of strain tensor a material volume element can deform volumetrically and deviatorically, volumetric deformation conserves the shape (i.e. no changes in angles, no sliding) while deviatoric (isochoric) deformation conserves the volume volumetric deviatoric decomposition of strain tensor ɛ ɛ ɛ vol ɛ dev (2.3.24) with volumetric and deviatoric strain tensor ɛ vol and ɛ dev tr ɛ vol tr ɛ tr ɛ dev 0 (2.3.25) volumetric second order tensor ɛ vol ɛ vol 3 ɛ : I I IIvol : ɛ (2.3.26) upon double contraction volumetric fourth order unit tensor II vol extracts volumetric part ɛ vol of strain tensor II vol 3 I I II vol 3 δ i jδ kl e i e j e k e l (2.3.27) deviatoric second order tensor ɛ dev ɛ dev ɛ 3 ɛ : I I IIdev : ɛ (2.3.28) upon double contraction deviatoric fourth order unit tensor II dev extracts deviatoric part of strain tensor II dev II sym II vol II sym 3 I I II dev 2 δ ikδ jl 2 δ ilδ jk 3 δ (2.3.29) i jδ kl e i e j e k e l 38

45 2.3 Gradients of kinematic quantities Volumetric deformation volumetric deformation is characterized through the volume dilatation e, i.e. difference of deformed volume and original volume dv dv scaled by original volume dv e dv dv dv ɛ ɛ 22 ɛ 33 ɛ ɛ 22 ɛ 33 ɛ 2 i j (2.3.30) neglection of higher order terms: trace of strain tensor tr ɛ ɛ : I as characteristic measure for volume changes e div u u : I ɛ : I tr ɛ (2.3.3) volumetric part ɛ vol of strain tensor ɛ ɛ vol 3 e I 3 ɛ : I I 3 I I : ɛ II vol : ɛ (2.3.32) index representation ɛ vol ɛ vol i j e i e j (2.3.33) matrix representation of coordinates ɛ vol i j 0 0 ɛ vol i j e e tr ɛ (2.3.34) incompressibility is characterized through div u 0 volumetric strain tensor ɛ vol is a spherical second order tensor as ɛ vol 3 ei volumetric strain tensor ɛ vol contains the volume changing, shape preserving part of the total strain tensor ɛ 39

46 2 Kinematics Deviatoric deformation deviatoric strain tensor ɛ dev preserves the volume and contains the remaining part of the total strain tensor ɛ deviatoric part ɛ dev of the strain tensor ɛ ɛ dev ɛ ɛ vol ɛ 3 ɛ : I I IIdev : ɛ (2.3.35) index representation ɛ dev ɛ dev i j e i e j (2.3.36) matrix representation of coordinates ɛ dev i j 2ɛ ɛ 22 ɛ 33 ɛ 2 ɛ 3 ɛ dev i j 3 ɛ 2 2ɛ 22 ɛ ɛ 33 ɛ 3 ɛ 3 ɛ 32 2ɛ 33 ɛ ɛ 22 (2.3.37) trace of deviatoric strains tr ɛ dev tr ɛ dev 3 2ɛ ɛ 22 ɛ ɛ 22 ɛ ɛ ɛ 33 ɛ ɛ 22 0 (2.3.38) deviatoric strain tensor ɛ dev is a traceless second order tensor as tr ɛ dev 0 deviatoric strain tensor ɛ dev contains the shape changing, volume preserving part of the total strain tensor ɛ 40

47 2.3 Gradients of kinematic quantities Volumetric deviatoric decomposition of strain tensor examples of purely volumetric deformation ɛ vol 3 ɛ : I I IIvol : ɛ tr ɛ vol tr ɛ (2.3.39) expansion compression PSfrag replacements e e 2 ω u,2 u 2, PSfrag replacements e e 2 ω u,2 u 2, e 0 and ɛ dev 0 e 0 and ɛ dev 0 examples of purely deviatoric deformation ɛ dev ɛ pure shear 3 ɛ : I I IIdev : ɛ tr ɛ dev 0 (2.3.40) simple shear PSfrag replacements PSfrag replacements β γ α β α β γ α β α e 0 and ɛ dev 0 e 0 and ɛ dev 0 4

48 2 Kinematics Strain vector assume we are interested in strain on a plane characterized through its normal n, strain vector t ɛ acting on plane given through normal projection of strain tensor ɛ PSfrag replacements t ɛ ɛ n (2.3.4) index representation t ɛ ɛ i j e i e j n k e k ɛ i j n k δ jk e i ɛ i j n j e i t ɛi e i (2.3.42) representation of coordinates t ɛi t ɛ t ɛ2 t ɛ3 ɛ n ɛ n ɛ 2 n 2 ɛ 3 n 3 ɛ 2 n ɛ 22 n 2 ɛ 23 n 3 (2.3.43) ɛ 3 n ɛ 32 n 2 ɛ 33 n 3 alternative interpretation: assume we are interested in strains along a particular material direction, i.e. the stretch of a fiber at x characterized through its normal n with n stretch as change of displacement vector u in the direction of n given through the Gateaux derivative.2..2 D u x n d dɛ u x ɛ n ɛ 0 u x ɛ n outer derviative n inner derivative n ɛ t t ɛ ɛ 0 u x n (2.3.44) sym skw recall that u u u ɛ ω whereby rotation ω skw u does not induce strain, thus t ɛ sym u n ɛ n (2.3.45) 42

49 2.3 Gradients of kinematic quantities Normal shear decomposition assume we are interested in strain along a particular fiber characterized through its normal n, stretch of fiber ɛ n given through normal projection of strain vector t ɛ PSfrag replacements ɛ n n ɛ t t ɛ ɛ n t ɛ n (2.3.46) alternative interpretation: stretch of a line element can be understood as the projection of change of displacement in the direction of n as Du n u n onto the direction n ɛ n n u n n ɛ n ɛ : n n (2.3.47) normal-shear (tangential) decomposition of strain vector t ɛ t ɛ ɛ n ɛ t (2.3.48) normal strain vector stretch of fibers in direction of n ɛ n ɛ : n n n (2.3.49) shear (tangential) strain vector sliding of fibers parallel to n ɛ t t ɛ ɛ n ɛ : II sym n n n n (2.3.50) amount of sliding γ n γ n 2 ɛ t 2 ɛ t ɛ t 2 t ɛ t ɛ ɛ 2 n (2.3.5) in general, i.e. for an arbitrary direction n, we have normal and shear contributions to the strain vector, however, three particular directions n ɛ i i,2,3 can be identified, for which t ɛ ɛ n and thus ɛ t 0, the corresponding n ɛ i i,2,3 are called principal strain directions and ɛ n i i,2,3 λ ɛi i,2,3 are the principal strains or stretches 43

50 2 Kinematics Principal strains stretches assume strain tensor ɛ to be known at x, principal strains λ ɛ i i,2,3 and principal strain directions n ɛ i i,2,3 can be derived from solution of special eigenvalue problem according to..3 ɛ n ɛ i λ ɛ i n ɛ i ɛ λ ɛ i n ɛ i 0 (2.3.52) solution det ɛ λ ɛ I 0 (2.3.53) or in terms of roots of characteristic equation λ 3 ɛ I ɛ λ 2 ɛ II ɛ λ ɛ III ɛ 0 (2.3.54) roots of characteristic equations in terms of principal invariants of ɛ I ɛ tr ɛ λ ɛ λ ɛ2 λ ɛ3 II ɛ 2 tr2 ɛ tr ɛ 2 λ ɛ2 λ ɛ3 λ ɛ3 λ ɛ λ ɛ λ ɛ2 (2.3.55) III ɛ det ɛ λ ɛ λ ɛ2 λ ɛ3 spectral representation of ɛ ɛ 3 λ ɛi n ɛi n ɛi (2.3.56) i principal strains (stretches) λ ɛi are purely normal, no shear deformation (sliding) γ n in principal directions, i.e. t ɛi ɛ n λ ɛi n ɛi and ɛ t 0 thus γ n 0 due to symmetry of strains ɛ ɛ t, strain tensor possesses three real eigenvalues λ ɛ i i,2,3, corresponding eigendirections n ɛ i i,2,3 are thus orthogonal n ɛi n ɛj δ i j 44

51 2.3 Gradients of kinematic quantities Compatibility until now, we have assumed the displacement field u x, t sym to be given, such that the strain field ɛ u could have been derived uniquely as partial derivative of u with respect to the position x at fixed time t assume now, that for a given strain field ɛ x, t, we want to know whether these strains ɛ are compatible with a continuous single valued displacement field u symmetric second order incompatibility tensor η crl crl ɛ (2.3.57) index representation of incompatibility tensor η η i j e i e j 3 eikm ɛ kn,ml 3 ejln e i e j 0 (2.3.58) coordinate representation of compatibility condition ɛ kl,mn ɛ mn,kl ɛ ml,kn ɛ kn,ml 0 (2.3.59) valid k, l, m, n, thus 8 equations which are partly redundant, six independent conditions St. Venant compatibility conditions η ɛ 22,33 ɛ 33,22 2ɛ 23,32 0 η 22 ɛ 33, ɛ,33 2ɛ 3,3 0 η 33 ɛ,22 ɛ 22, 2ɛ 2,2 0 η 2 ɛ 3,32 ɛ 23,3 ɛ 33,2 ɛ 2,33 0 η 23 ɛ 2,3 ɛ 3,2 ɛ,23 ɛ 23, 0 η 3 ɛ 32,2 ɛ 2,23 ɛ 22,3 ɛ 3,22 0 (2.3.60) incompatible displacement field, e.g. in dislocation theory 45

52 2 Kinematics Special case of plane strain dimensional reduction in case of plane strain with vanishing strains ɛ 3 ɛ 23 ɛ 3 ɛ 32 ɛ 33 0 in out of plane direction, e.g. in geomechanics ɛ ɛ i j e i e j (2.3.6) matrix representation of coordinates ɛ i j ɛ i j ɛ 2 ɛ 22 0 ɛ ɛ (2.3.62) Voigt representation of strain three dimensional second order strain tensor ɛ ɛ ɛ i j e i e j (2.3.63) matrix representation of coordinates ɛ i j (2.3.64) ɛ i j ɛ 2 ɛ 22 ɛ 23 ɛ ɛ 2 ɛ 3 ɛ 3 ɛ 23 ɛ 33 due to symmetry ɛ i j ɛ ji and thus ɛ 2 ɛ 2, ɛ 23 ɛ 32, ɛ 3 ɛ 3, strain tensor ɛ contains only six independent components ɛ,ɛ 22,ɛ 33,ɛ 2,ɛ 23,ɛ 3,it proves convenient to represent second order tensor ɛ through a vector ɛ ɛ ɛ,ɛ 22,ɛ 33, 2ɛ 2, 2ɛ 23, 2ɛ 3 t vector representation ɛ of strain ɛ in case of plane strain ɛ ɛ,ɛ 22,ɛ 33, 2ɛ 2 t (2.3.65) (2.3.66) 46

53 3 Balance equations 3. Basic ideas until now: kinematics, i.e. characterization of deformation of a material body without studying its physical cause now: balance equations, i.e. general statements that characterize the cause of cause of the motion of any body rag replacements isolation r n t σ n q n x x e i i,2,3 47

54 3 Balance equations Basic strategy isolation of an arbitrary subset of the body characterization of the influence of the remaining body on through phenomenological quantities, i.e. the contact mass flux r, the contact stress t σ, the contact heat flux q definition of basic physical quantities, i.e. the mass m, the linear momentum I, the moment of momentum D and the energy E of subset postulate of balance of these quantities renders global balance equations for subset localization of global balance equations renders local balance equations at point x 48

55 3. Basic ideas 3.. Concept of mass flux the contact mass flux r n at a point x is a scalar of the unit [mass/time/surface area] the contact mass flux r n characterizes the transport of matter PSfrag normal replacements to the tangent plane to an imaginary surface passing through this point with normal vector n r x x n r n isolation e i i,2,3 definition of contact heat flux q n in analogy to Cauchy s postulate, lemma and theorem originally introduced for the momentum flux in 3..2 Cauchy s postulate r n r n x, n (3..) Cauchy s lemma r n x, n r n x, n (3..2) Cauchy s theorem the contact mass flux r n can be expressed as linear function of the surface normal n and the mass flux vector r r n r n (3..3) 49

56 3 Balance equations Mass flux vector the vector field r is called mass flux vector r r i e i (3..4) Cauchy s theorem r n r n (3..5) index representation r n r i e i n j e j r i n j δ i j r i n i (3..6) geometric interpretation r 3 e 3 r 2 PSfrag replacements e e 2 r the coordinates r i characterize the transport of matter through the planes parallel to the coordinate planes in classical closed system continuum mechanics (here) the mass flux vector vanishes identically examples of mass flux: transport of chemical reactants in chemomechanics or cell migration in biomechanics 50

57 3. Basic ideas 3..2 Concept of stress traction vector f d f t σ lim a 0 a da interpretation as surface force per unit surface area (3..7) Cauchy s postulate the traction vector t σ at a point x can be expressed exclu- t σ n PSfrag replacements x e i i,2,3 sively in terms of the point x and the normal n to the tangent plane to an imaginary surface passing through this point traction vector t σ t σ x, n (3..8) Cauchy s lemma the traction vectors acting on opposite sides of a surface are equal in magnitude and opposite in sign t σ x, n t σ2 x, n 2 (3..9) 5

58 3 Balance equations t σ PSfrag replacements n n 2 x t σ2 e i i,2,3 generalization with n n n 2 and t σ t σ t σ x, n t σ x, n (3..0) Cauchy s theorem the traction vector t σ can be expressed as a linear map of the surface normal n mapped via the transposed stress tensor σ t t σ σ t n (3..) accordingly with n n n 2 and t σ t σ t σ σ t n σ t n t σ t σ2 σ t n 2 σ t n t σ (3..2) Cauchy tetraeder balance of momentum (pointwise) t σ n da t σ n i da i t σ e i da i t σi da i (3..3) surface theorem, area fractions from Gauss theorem nda n i da i e i da i da i da e i n cos e i, n (3..4) 52

59 3. Basic ideas PSfrag replacements t σ x n 2 e 2 e 3 n e t σ2 n t σ e e 2 n 3 e 3 t σ3 traction vector as linear map of surface normal t σ n t σi da i da t σi cos e i, n t σi e i n t σi e i n (3..5) compare t σ n σ t n interpretation of second order stress tensor as σ t t σi e i Stress tensor Cauchy stress (true stress) σ t t σi e i σ ji e j e i σ e i t σi σ i j e i e j (3..6) Cauchy theorem t σ σ t n (3..7) index representation t σ σ ji e j e i n k e k σ ji n k δ ik e j σ ji n i e j t j e j (3..8) 53

60 3 Balance equations matrix representation of tensor coordinates of σ i j σ i j σ 2 σ 22 σ 23 σ σ 2 σ 3 σ 3 σ 32 σ 33 geometric interpretation t t σ t t σ2 t t σ3 (3..9) PSfrag replacements σ 33 σ 3 σ 32 σ 23 e 3 σ 22 σ 3 e e 2 σ 2 σ σ 2 with traction vectors on surfaces t σ σ σ 2 σ 3 t t σ2 σ 2 σ 22 σ 23 t (3..20) t σ3 σ 3 σ 32 σ 33 t first index... surface normal second index... direction (coordinate of traction vector) diagonal entries... normal stresses non diagonal entries.. shear stresses 54

61 3. Basic ideas Volumetric deviatoric decomposition of stress tensor in analogy to the strain tensor ɛ, the stress tensor σ can be additively decomposed into a volumetric part σ vol and a traceless deviatoric part σ dev volumetric deviatoric decomposition of stress tensor σ σ σ vol σ dev (3..2) with volumetric and deviatoric stress tensor σ vol and σ dev tr σ vol tr σ tr σ dev 0 (3..22) volumetric second order tensor σ vol σ vol 3 σ : I I IIvol : σ (3..23) upon double contraction volumetric fourth order unit tensor II vol extracts volumetric part σ vol of stress tensor II vol 3 I I II vol 3 δ i jδ kl e i e j e k e l (3..24) deviatoric second order tensor σ dev σ dev σ 3 σ : I I IIdev : σ (3..25) upon double contraction deviatoric fourth order unit tensor II dev extracts deviatoric part of stress tensor II dev II sym II vol II sym 3 I I II dev 2 δ ikδ jl 2 δ ilδ jk 3 δ (3..26) i jδ kl e i e j e k e l 55

62 3 Balance equations Volumetric stress volumetric part σ vol of stress tensor σ σ vol 3 σ : I I 3 I I : σ II vol : σ (3..27) interpretation of trace as hydrostatic pressure p 3 tr σ 3 σ : I 3 σ σ 22 σ 33 (3..28) index representation σ vol σ vol i j e i e j (3..29) matrix representation of coordinates σ vol i j σ vol i j p p tr σ (3..30) 3 volumetric stress tensor σ vol is a spherical second order tensor as σ vol p I volumetric stress tensor σ vol contains the hydrostatic pressure part of the total stress tensor σ 56

63 3. Basic ideas Deviatoric stress deviatoric stress tensor σ dev preserves the volume and containes the remaining part of the total stress tensor σ deviatoric part σ dev of the stress tensor σ σ dev σ σ vol σ 3 σ : I I IIdev : σ (3..3) index representation σ dev σ dev i j e i e j (3..32) matrix representation of coordinates σ dev i j 2σ σ 22 σ 33 σ 2 σ 3 σ dev i j 3 σ 2 2σ 22 σ σ 33 σ 3 σ 3 σ 32 2σ 33 σ σ 22 (3..33) trace of deviatoric stresss tr σ dev tr σ dev 3 2σ σ 22 σ σ 22 σ σ σ 33 σ σ 22 0 (3..34) deviatoric stress tensor σ dev is a traceless second order tensor as tr σ dev 0 deviatoric stress tensor σ dev contains the hydrostatic pressure free part of the total stress tensor σ 57

64 3 Balance equations Normal shear decomposition assume we are interested in the stress σ n normal to a particular plane characterized through its normal n, i.e. the normal projection of the stress vector t σ PSfrag replacements σ n n σ t t σ σ n t σ n σ t n n σ t : n n σ t : N (3..35) normal shear (tangential) decomposition of stress vector t σ t σ σ n σ t (3..36) normal stress vector stress in direction of n σ n σ t : n n n σ t : n n n (3..37) shear (tangential) stress vector stress in the plane σ t t σ σ n σ t n σ t : n n n σ t : II sym n n n n σ t : T (3..38) amount of shear stress τ n τ n 2 and thus t σ σ n t σ σ n t σ t σ 2t σ σ n σ 2 n n n (3..39) τ n σ t σ t σ t t σ t σ σ 2 n (3..40) in general, i.e. for an arbitrary direction n, we have normal and shear contributions to the stress vector, however, three particular directions n σ i i,2,3 can be identified, for which t σ σ n and thus σ t 0, the corresponding n σ i i,2,3 are called prinicpal stress directions and σ n i i,2,3 λ σi i,2,3 are the principal stresses 58

65 3. Basic ideas Principal stresses assume stress tensor σ t to be known at x, principal stresses λ σ i i,2,3 and principal stress directions n σ i i,2,3 can be derived from solution of special eigenvalue problem according to..3 σ t n σ i λ σ i n σ i σ t λ σ i n σ i 0 (3..4) solution det σ t λ σ I 0 (3..42) or in terms of roots of characteristic equation λ 3 σ I σ λ 2 σ II σ λ σ III σ 0 (3..43) roots of characteristic equation in terms of principal invariants of σ t I σ tr σ t λ σ λ σ2 λ σ3 II σ 2 tr2 σ t tr σ t 2 λ σ2 λ σ3 λ σ3 λ σ λ σ λ σ2 III σ det σ t λ σ λ σ2 λ σ3 (3..44) spectral representation of σ σ t 3 i λ σi n σi n σi (3..45) principal stresses λ σi are purely normal, no shear stress τ n in principal directions, i.e. t σi σ n λ σi n σi and σ t 0 thus τ n 0 due to symmetry of stresses σ σ t, stress tensor posseses three real eigenvalues λ σ i i,2,3, corresponding eigendirections n σ i i,2,3 are thus orthogonal n σi n σ j δ i j 59

66 3 Balance equations Special case of plane stress dimensional reduction in case of plane stress with vanishing stresses σ 3 σ 23 σ 3 σ 32 σ 33 0 in out of plane direction, e.g. for flat sheets σ σ i j e i e j (3..46) matrix representation of coordinates σ i j σ i j σ 2 σ 22 0 σ σ (3..47) Voigt representation of stress three dimensional second order stress tensor σ σ σ i j e i e j (3..48) matrix representation of coordinates σ i j (3..49) σ i j σ 2 σ 22 σ 23 σ σ 2 σ 3 σ 3 σ 23 σ 33 due to symmetry σ i j σ ji and thus σ 2 σ 2, σ 23 σ 32, σ 3 σ 3, stress tensor σ contains only six independent components σ,σ 22,σ 33,σ 2,σ 23,σ 3,it proves convenient to represent second order tensor σ through a vector σ σ σ,σ 22,σ 33,σ 2,σ 23,σ 3 t (3..50) vector representation σ of stress σ in case of plane stress σ σ,σ 22,σ 33,σ 2 t (3..5) 60

67 3. Basic ideas 3..3 Concept of heat flux the contact heat flux q n at a point x is a scalar of the unit [energy/time/surface area] the contact heat flux q n characterizes the energy transport PSfrag normal replacements to the tangent plane to an imaginary surface passing through this point with normal vector n q x x n q n isolation e i i,2,3 definition of contact heat flux q n in analogy to Cauchy s postulate, lemma and theorem originally introduced for the momentum flux in 3..2 Cauchy s postulate q n q n x, n (3..52) Cauchy s lemma q n x, n q n x, n (3..53) Cauchy s theorem the contact heat flux q n can be expressed as linear function of the surface normal n and the heat flux vector q q n q n (3..54) 6

68 3 Balance equations Heat flux vector the vector field q is called heat flux vector q q i e i (3..55) Cauchy s theorem q n q n (3..56) index representation q n q i e i n j e j q i n j δ i j q i n i (3..57) geometric interpretation q 3 e 3 q 2 PSfrag replacements e e 2 q the coordinates q i characterize the heat energy transport through the planes parallel to the coordinate planes in continuum mechanics of adiabatic systems the heat flux vector vanishes identically 62

69 3.2 Balance of mass 3.2 Balance of mass total mass m of a body m : mass density ρ m d m (3.2.) lim V 0 m V dm dv d m ρ dv (3.2.2) ρ dv (3.2.3) mass exchange of body with environment m sur and m vol m sur : rn d A m vol : with contact mass flux r n r n and mass source d V (3.2.4) 3.2. Global form of balance of mass The time rate of change of the total mass m of a body is balanced with the mass exchange due to contact mass flux m sur and the at-a-distance mass exchange m vol. D t m m sur m vol (3.2.5) and thus D t ρ dv rn d A d V (3.2.6) 63