Continuum Mechanics

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CONTINUUM MECHANICS

lecture notes 2003 jp dr.-ing. habil. ellen kuhl technical university of kaiserslautern “conti” — 2004/9/6 — 9:53 — page 2 — #2 “conti” — 2004/9/6 — 9:53 — page i — #3

Contents

1 Tensor calculus 1 1.1 Tensor algebra ...... 1 1.1.1 Vector algebra ...... 1 1.1.1.1 Notation ...... 1 1.1.1.2 Euclidian vector space ...... 2 1.1.1.3 Scalar product ...... 4 1.1.1.4 Vector product ...... 5 1.1.1.5 Scalar triple vector product ...... 6 1.1.2 Tensor algebra ...... 7 1.1.2.1 Notation ...... 7 1.1.2.2 Scalar products ...... 10 1.1.2.3 Dyadic product ...... 12 1.1.2.4 Scalar triple vector product ...... 13 1.1.2.5 Invariants of second order tensors ...... 14 1.1.2.6 Trace of second order tensors ...... 15 1.1.2.7 Determinant of second order tensors ...... 16 1.1.2.8 Inverse of second order tensors ...... 17 1.1.3 Spectral decomposition ...... 18 1.1.4 Symmetric – skew-symmetric decomposition ...... 19 1.1.4.1 Symmetric tensors ...... 20 1.1.4.2 Skew-symmetric tensors ...... 21 1.1.5 Volumetric – deviatoric decomposition ...... 22 1.1.6 Orthogonal tensors ...... 23 1.2 Tensor analysis ...... 24 1.2.1 Derivatives ...... 24 1.2.1.1 Frechet derivative ...... 24 1.2.1.2 Gateaux derivative ...... 24 1.2.2 Gradient ...... 25 1.2.2.1 Gradient of a scalar–valued function ...... 25 1.2.2.2 Gradient of a vector–valued function ...... 25 1.2.3 Divergence ...... 26 1.2.3.1 Divergence of a vector ﬁeld ...... 26 1.2.3.2 Divergence of a tensor ﬁeld ...... 26 1.2.4 Laplace operator ...... 27 1.2.4.1 Laplace operator acting on scalar–valued function . . . 27 1.2.4.2 Laplace operator acting on vector–valued function . . . 27

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Contents

1.2.5 Integral transformations ...... 29 1.2.5.1 Integral theorem for scalar–valued fields ...... 29 1.2.5.2 Integral theorem for vector–valued fields ...... 29 1.2.5.3 Integral theorem for tensor–valued fields ...... 29

2 Kinematics 30 2.1 Motion ...... 30 2.2 Rates of kinematic quantities ...... 31 2.2.1 Velocity ...... 31 2.2.2 Acceleration ...... 31 2.3 Gradients of kinematic quantities ...... 32 2.3.1 Displacement gradient ...... 32 2.3.2 Strain ...... 34 2.3.3 Rotation ...... 35 2.3.4 Volumetric–deviatoric decomposition of strain tensor ...... 37 2.3.5 Strain vector ...... 41 2.3.6 Normal–shear decomposition ...... 42 2.3.7 Principal strains – stretches ...... 43 2.3.8 Compatibility ...... 44 2.3.9 Special case of plane strain ...... 45 2.3.10 Voigt representation of strain ...... 45

3 Balance equations 46 3.1 Basic ideas ...... 46 3.1.1 Concept of mass ﬂux ...... 48 3.1.2 Concept of stress ...... 50 3.1.2.1 Volumetric–deviatoric decomposition of stress tensor . 54 3.1.2.2 Normal–shear decomposition ...... 57 3.1.2.3 Principal stresses ...... 58 3.1.2.4 Special case of plane stress ...... 59 3.1.2.5 Voigt representation of stress ...... 59 3.1.3 Concept of heat ﬂux ...... 60 3.2 Balance of mass ...... 62 3.2.1 Global form of balance of mass ...... 62 3.2.2 Local form of balance of mass ...... 63 3.2.3 Classical continuum mechanics of closed systems ...... 63 3.3 Balance of linear momentum ...... 64 3.3.1 Global form of balance of momentum ...... 64 3.3.2 Local form of balance of momentum ...... 64 3.3.3 Reduction with lower order balance equations ...... 65 3.3.4 Classical continuum mechanics of closed systems ...... 66 3.4 Balance of angular momentum ...... 66 3.4.1 Global form of balance of angular momentum ...... 66 3.4.2 Local form of balance of angular momentum ...... 67 3.4.3 Reduction with lower order balance equations ...... 68

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Contents

3.5 Balance of energy ...... 69 3.5.1 Global form of balance of energy ...... 69 3.5.2 Local form of balance of energy ...... 70 3.5.3 Reduction with lower order balance equations ...... 70 3.5.4 First law of thermodynamics ...... 71 3.6 Balance of entropy ...... 73 3.6.1 Global form of balance of entropy ...... 73 3.6.2 Local form of balance of entropy ...... 74 3.6.3 Reduction with lower order balance equations ...... 74 3.6.4 Second law of thermodynamics ...... 75 3.7 Generic balance equation ...... 77 3.7.1 Global / integral format ...... 77 3.7.2 Local / differential format ...... 77 3.8 Thermodynamic potentials ...... 79

4 Constitutive equations 80 4.1 Linear constitutive equations ...... 81 4.1.1 Mass flux – Fick’s law ...... 82 4.1.2 Momentum flux – Hook’s law ...... 83 4.1.3 Heat flux – Fourier’s law ...... 85 4.2 Hyperelasticity ...... 86 4.2.1 Specific stored energy ...... 86 4.2.2 Specific complementary energy ...... 88 4.3 Isotropic hyperelasticity ...... 89 4.3.1 Specific stored energy ...... 89 4.3.2 Specific complementary energy ...... 95 4.3.3 Elastic constants ...... 99 4.4 Transversely isotropic hyperelasticity ...... 100

A Ub¨ ungsaufgaben 105 A.1 Tensoralgebra zweistuﬁger Tensoren ...... 105 A.2 Tensoranalysis ...... 112 A.3 Ableitungen ...... 114 A.4 Verzerrungs– und Spannungsvektor ...... 117

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1 Tensor calculus

1.1 Tensor algebra

1.1.1 Vector algebra

1.1.1.1 Notation

Einstein’s summation convention

3 ui ∑j 1 Ai j x j bi Ai j x j bi (1.1.1) summation over indices that appear twice in a term or symbol, with silent (dummy) index j and free index i, and thus

u1 A11 x1 A12 x2 A13 x3 b1

u2 A21 x1 A22 x2 A23 x3 b2 (1.1.2)

u3 A31 x1 A32 x2 A33 x3 b3

Kronecker symbol δi j

1 for i j δi j (1.1.3) 0 for i j multiplication with Kronecker symbol corresponds to exchange of silent index with free index of Kronecker symbol

ui δi j u j (1.1.4)

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1 Tensor calculus

3 permutation symbol ei jk

1 for i, j, k ... even permutation 3 ei jk 1 for i, j, k ... odd permutation (1.1.5) 0 for ... else

1.1.1.2 Euclidian vector space

consider linear vector space 3 characterized through ad- dition of its elements u, v and multiplication with real scalars α, β

α, β ... real numbers u, v 3 3 ... linear vector space deﬁnition of linear vector space 3 through the following axioms

α u v α u α v α β u α u β u (1.1.6) α β u α β u zero element and identity

0 u 0 1 u u (1.1.7)

3 linear independence of elements e1, e2, e3 if α1 α2 α3 0 is the only (trivial) solution to

αi ei 0 (1.1.8)

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1.1 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 (1.1.9) deﬁnition of norm through the following axioms

n u 0 n u 0 u 0 n α u α n u (1.1.10) n u v n u n v n2 u v n2 u v 2 n2 u n2 v

consider Euclidian vector space 3 equipped with the Eu- clidian norm

2 2 2 1 2 n u u u u u1 u2 u3 (1.1.11) mapping elements of the Euclidian vector space 3 to the space of real numbers

n : 3 Euclidian norm (1.1.12) representation of three–dimensional vector a 3

a ai ei a1 e1 a2 e2 a3 e3 (1.1.13) with a1, a2, a3 coordinates (components) of a relative to the basis e1, e2, e3

t a a1, a2, a3 (1.1.14)

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1 Tensor calculus

1.1.1.3 Scalar product

Euclidian norm enables the deﬁnition of scalar (inner) product between two vectors u, v and introduces a scalar α u v α (1.1.15) geometric interpretation with 0 ϑ π being the angle u enclosed by the vectors u and v, ϑ then u cos ϑ can be interpreted as the projection of u u cos ϑ v onto the direction of v andPSfrag replacements u cos ϑ u v u v cos ϑ corresponds to the grey area in the picture with the above interpretation with 0 ϑ π, obviously u v u v (1.1.16) properties of scalar product

u v v u α u β v w α u w β v w (1.1.17) w α u β v α w u β w v positive deﬁniteness of scalar product u u 0, u u 0 u 0 (1.1.18) orthogonal vectors u and v u v 0 u v (1.1.19)

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1.1 Tensor algebra

1.1.1.4 Vector product vector product of two vectors u, v deﬁnes a new vector w 3 u v w (1.1.20) PSfrag replacements geometric interpretation w v v sinϑ with 0 ϑ π being the ϑ angle enclosed by the vec- u tors u and v, then v sin ϑ 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 (1.1.21) index representation of w u v

w1 u2 v3 u3 v2

w2 u3 v1 u1 v3 (1.1.22)

w3 u1 v2 u2 v1 properties of vector product u v v u α u β v w α u w β v w (1.1.23) u u v 0 u v u v u u v v u v 2

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1 Tensor calculus

1.1.1.5 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 (1.1.24) geometric interpretation with vector product u PSfrag replacements v w v w v w sin ϑ n w deﬁning aera of ground surface v ϑ u, v, w u v w deﬁnes volume of parallelepiped obviously

α u v w v w u w u v (1.1.25) α u w v v u w w v u index representation of α u, v, w

α u1 v2w3 v3w2 u2 v3w1 v1w3 u3 v1w2 v3w1 (1.1.26) properties of scalar triple product

u, v, w v, w, u w, u, v u, w, v v, u, w w, v, u (1.1.27) αu βv, w, d α u, w, d β v, w, d three vectors u, v, w are linearly independent if u, v, w 0 (1.1.28)

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1.1 Tensor algebra

1.1.2 Tensor algebra

1.1.2.1 Notation

Second order tensors tensor (dyadic) product u v of two vectors u and v introduces a second order tensor A A u v (1.1.29) introducing u ui ei and v v j ej yields index representation of three-dimensional second order tensor A

A Ai j ei ej (1.1.30) with Ai j ui v j coordinates (components) of A relative to the tensor basis ei ej, matrix representation of coordinates

A11 A12 A13

Ai j A21 A22 A23 (1.1.31)

A31 A32 A33 transpose of second order tensor At At u v t v u (1.1.32) introducing u ui ei and v v j ej yields index representation of transpose of second order tensor At t A A ji ej ei (1.1.33) with A ji v j ui coordinates (components) of A relative to the tensor basis e j ei, matrix representation of coordinates

A11 A21 A31

A ji A12 A22 A32 (1.1.34)

A13 A23 A33

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1 Tensor calculus second order unit tensor I in terms of Kronecker symbol δi j

I δi j ei ej (1.1.35) matrix representation of coordinates δi j

1 0 0

δ ji 0 1 0 (1.1.36) 0 0 1

Third order tensors tensor (dyadic) product A v of second order tensor A and 3 vectors u introduces a third order tensor a 3 a A v (1.1.37) introducing A Ai j ei ej and u uk ek yields index repre- 3 sentation of three-dimensional third order tensor a 3 3 a ai jk ei e j ek (1.1.38)

3 3 with ai jk Ai j uk coordinates (components) of a relative to the tensor basis ei ej ek

3 third order permutation tensor e in terms of permutation 3 symbol ei jk

3 3 e ei jk ei ej ek (1.1.39)

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1.1 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 (1.1.40) introducing A Ai j ei ej and B Bkl ek el yields index representation of three-dimensional fourth order tensor /A

/A Ai jkl ei ej ek el (1.1.41) with Ai jkl Ai j Bkl coordinates (components) of /A relative to the tensor basis ei ej ek el fourth order unit tensor II

II δik δ jl ei e j ek el (1.1.42) transpose fourth order unit tensor IIt

t II δil δ jk ei ej ek el (1.1.43) symmetric fourth order unit tensor IIsym

sym 1 (1.1.44) II 2 δik δ jl δil δ jk ei ej ek el skew-symmetric fourth order unit tensor IIskw

skw 1 (1.1.45) II 2 δik δ jl δil δ jk ei ej ek el volumetric fourth order unit tensor IIvol vol 1 (1.1.46) II 3 δi j δkl ei ej ek el deviatoric fourth order unit tensor IIdev dev 1 1 1 (1.1.47) II 3 δi j δkl 2 δik δ jl 2 δil δ jk ei ej ek el

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1 Tensor calculus

1.1.2.2 Scalar products

scalar product A u between second order tensor A and vector u deﬁnes a new vector v 3

A u Ai jei ej uk ek (1.1.48) Ai j uk δ jk ei Ai j u j ei viei v second order zero tensor 0, second order identity tensor I 0 a 0 I a a (1.1.49) positive semi-deﬁniteness of second order tensor A a A a 0 (1.1.50) positive deﬁniteness of second order tensor A a A a 0 (1.1.51) properties of scalar product

A αa βb α A a β A b A B a A a B a (1.1.52) α A a α A a scalar product A B between two second order tensors A and B deﬁnes a second order tensor C

A B Ai j ei ej : Bkl ek el

Ai j Bkl δikei el (1.1.53)

Ai j Bjl ei el Cilei el C second order zero tensor 0, second order identity tensor I 0 A 0 I A A (1.1.54)

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1.1 Tensor algebra properties of scalar product

α A B α A B A α B A B C A B A C (1.1.55) A B C A C B C properties in terms of transpose At of a tensor A

a At b b A a α A β B t α At β Bt (1.1.56) A B t Bt At

scalar product A : B between two second order tensors A and B deﬁnes a scalar α

A : B Ai j ei ej : Bkl ek el (1.1.57) Ai j Bkl δik δ jl Ai j Bi j α scalar product /A : B between fourth order tensor /A and second order tensor B deﬁnes a new second order tensor C

/A : B Ai jkl ei ej ek el : Bmn em en

Ai jkl Bmn δkm δln ei ej (1.1.58)

Ai jkl Bklei ej Ai jei ej A

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1 Tensor calculus

1.1.2.3 Dyadic product

tensor (dyadic) product u v of two vectors u and v introduces a second order tensor A

A u v ui ei v jej ui v j ei ej Ai j ei ej (1.1.59) 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 (1.1.60) u v w x v w u x A u v A u v u v A u At v or in index notation

ui v j w j v j w j ui

α ui β vi w j α ui w j β vi w j

ui α v j β w j α ui v j β ui w j (1.1.61) ui v j w j xk v j w j uixk

Ai j u j vk Ai j ui vk

ui v j A jk ui Akj v j

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1.1 Tensor algebra

1.1.2.4 Scalar triple vector product consider the set of Cartesian base vectors ei i 1,2,3 and an arbitrary second set of base vectors u, v, w with scalar triple product u, v, w , with arbitrary second order tensor A, eval- uate A u, v, w u, A v, w u, v, A w u, v, w (1.1.62) with index representation of each term according to

A u, v, w A ui ei , v j ej , wk ek ui v j wk A ei, ej, ek expression (1.1.62) can be rewritten as (1.1.63)

uiv jwk A e1, e2, e3 e1, A e2, e3 e1, e2, A e3 u, v, w (1.1.64) term in brackets remains unchanged upon cyclic permutation of ei i 1,2,3, its sign reverses upon non–cyclic permu- tations, thus 3 uiv jwkei jk A e1, e2, e3 e1, A e2, e3 e1, e2, A e3 u, v, w

A e1, e2, e3 e1, A e2, e3 e1, e2, A e3 (1.1.65) the above expression according to (1.1.62) is thus invariant under the choice of base system, it yields the same scalar value IA for arbitrary base systems

IA A u, v, w u, A v, w u, v, A w u, v, w

A e1, e2, e3 e1, A e2, e3 e1, e2, A e3 e1, e2, e3 (1.1.66)

IA is called the ﬁrst invariant of the second order tensor A

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1 Tensor calculus

1.1.2.5 Invariants of second order tensors the following property of the scalar triple product

u, v, w u v w (1.1.67) introduces three scalar–valued quantities IA, IIA, IIIA associated with the second order tensor A

A u, v, w u, A v, w u, v, A w IA u, v, w

u, A v, A w A u, v, A w A u, A v, w IIA u, v, w

A u, A v, A w IIIA u, v, w (1.1.68) the proof of IIA, IIIA being invariant for different base systems u, v, w is similar to the one for IA

IA, IIA, IIIA are called the three principal invariants of A which can be expressed as

IA tr A ∂A IA I 1 2 2 (1.1.69) IIA 2 tr A tr A ∂A IIA IA I A t IIIA det A ∂A IIIA IIIA A alternatively, we could work with the three basic invariants

I¯A, I¯I A, II¯I A of A which are more common in the context of anisotropy

1 I¯A A : I 2 I¯I A A : I (1.1.70) 3 II¯I A A : I

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1.1 Tensor algebra

1.1.2.6 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 (1.1.71) such that tr A is the sum of the diagonal entries Aii of A

tr A tr Ai j ei ej

Ai j tr ei ej Ai j ei ej (1.1.72)

Ai j δi j Aii A11 A22 A33 with

IA I¯A tr A (1.1.73) properties of the trace of second order tensors

tr I 3 tr At tr A tr A B tr B A (1.1.74) tr α A β B α tr A β tr B tr A Bt A : B tr A tr A I A : I

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1 Tensor calculus

1.1.2.7 Determinant of second order tensors determinant det A of second order tensor A introduces a scalar det A

1 det A det Ai j 6 ei jk eabc Aia A jb Akc

A11 A22 A33 A21 A32 A13 A31 A12 A23 (1.1.75)

A11 A23 A32 A22 A31 A13 A33 A12 A21 with

IIIA det A (1.1.76) determinant deﬁning vector product u v

u1 v1 e1 u2v3 u3v2

u v det u2 v2 e2 u3v1 u1v3 (1.1.77)

u3 v3 e3 u1v2 u2v1 determinant deﬁning scalar triple vector product u, v, w

u1 v1 w1

u, v, w u v w det u2 v2 w2 (1.1.78)

u3 v3 w3 properties of determinant of a second order tensors

det I 1 det At det A det α A α3 det A (1.1.79) det A B det A det B det u v 0

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1.1 Tensor algebra

1.1.2.8 Inverse of second order tensors if det A 0 existence of inverse A 1 of second order tensor A

A A 1 A 1 A I (1.1.80) in particular

v A u A 1 v u (1.1.81) properties of inverse of two second order tensors

A 1 1 A α A 1 1 α 1 A (1.1.82) A B 1 B 1 A 1 determinant det A 1 of inverse of A

det A 1 1 det A (1.1.83) adjoint Aadj of a second order tensor A

Aadj det A A 1 (1.1.84) cofactor Acof of a second order tensor A

Acof det A A t Aadj t (1.1.85) with

t t cof ∂Adet A det A A IIIA A A (1.1.86)

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1 Tensor calculus

1.1.3 Spectral decomposition eigenvalue problem of arbitrary second order tensor A

A nA λA nA A λA I nA 0 (1.1.87) solution introduces eigenvector(s) nAi and eigenvalue(s) λAi

det A λA I 0 (1.1.88) alternative representation in terms of scalar triple product

A u λAu, A v λAv, A w λAw 0 (1.1.89) removal of arbitrary factor u, v, w yields characteristic equation

3 2 λA IA λA IIA λA IIIA 0 (1.1.90) roots of characteristic equations are principal invariants of A

IA tr A 1 2 2 (1.1.91) IIA 2 tr A tr A

IIIA det A spectral decomposition of A

3 A ∑ λAi nAi nAi (1.1.92) i 1 Cayleigh–Hamilton theorem: a tensor A satisﬁes its own characteristic equation

3 2 A IA A IIA A IIIA I 0 (1.1.93)

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1.1 Tensor algebra

1.1.4 Symmetric – skew-symmetric decomposition symmetric – skew-symmetric decomposition of second order tensor A 1 1 A A At A At Asym Askw (1.1.94) 2 2 with symmetric and skew-symmetric second order tensor Asym and Askw

Asym Asym t Askw Askw t (1.1.95)

symmetric second order tensor Asym 1 Asym A At IIsym : A (1.1.96) 2 upon double contraction symmetric fourth order unit tensor IIsym extracts symmetric part of second order tensor

IIsym 1 II IIt 2 (1.1.97) sym 1 II 2 δikδ jl δilδ jk ei ej ek el skew–symmetric second order tensor Askw 1 Askw A At IIskw : A (1.1.98) 2 upon double contraction skew-symmetric fourth order unit tensor IIskw extracts skew-symmetric part of second order tensor IIskw 1 II IIt 2 (1.1.99) skw 1 II 2 δikδ jl δilδ jk ei e j ek el

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1 Tensor calculus

1.1.4.1 Symmetric tensors symmetric part Asym of a second order tensor A 1 Asym A At Asym Asym t (1.1.100) 2 alternative representation Asym IIsym : A (1.1.101) whereby symmetric fourth order tensor IIsym extracts symmetric part Asym of second order tensor A a symmetric second order tensor S Asym processes three real eigenvalues λSi i 1,2,3 and three corresponding orthogonal eigenvectors nSi i 1,2,3, such that the spectral representation of S takes the following form 3 S ∑ λSi nSi nSi (1.1.102) i 1 sym three invariants IS, IIS, IIIS of symmetric tensor S A

IS λS1 λS2 λS3

IIS λS2 λS3 λS3 λS1 λS1 λS2 (1.1.103)

IIIS λS1 λS2 λS3 square root S, inverse S 1, exponent exp S and logarithm ln S , of positive semi-deﬁnite symmetric tensor S for which

λSi 0 3 S ∑i 1 λSi nSi nSi 1 3 1 S ∑ λ nSi nSi i 1 Si (1.1.104) 3 exp S ∑i 1 exp λSi nSi nSi 3 ln S ∑i 1 ln λSi nSi nSi

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1.1 Tensor algebra

1.1.4.2 Skew-symmetric tensors skew-symmetric part Askw of a second order tensor A 1 Askw A At Askw Askw t (1.1.105) 2 alternative representation

Askw IIskw : A (1.1.106) whereby skew-symmetric fourth order tensor IIskw extracts skew-symmetric part Askw of second order tensor A a skew-symmetric second order tensor W Askw posses three independent entries, three entries vanish identically, three are equal to the negative of the independent entries, these deﬁne the axial vector w 1 3 3 w e: W w e w (1.1.107) 2 associated with each skew-symmetric tensor W Askw

W p w p (1.1.108) three invariants IW, IIW, IIIW of skew-symmetric tensor W

IW tr W 0

IIW w w (1.1.109)

IIIW det W 0

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1 Tensor calculus

1.1.5 Volumetric – deviatoric decomposition volumetric – deviatoric decomposition of second order tensor A

A Avol Adev (1.1.110) with volumetric and deviatoric second order tensor Avol and Adev

tr Avol tr A tr Adev 0 (1.1.111)

volumetric second order tensor Avol 1 Avol A : I I IIvol : A (1.1.112) 3 upon double contraction volumetric fourth order unit tensor IIvol extracts volumetric part of second order tensor

IIvol 1 I I 3 (1.1.113) vol 1 II 3 δi jδkl ei ej ek el deviatoric second order tensor Adev 1 Adev A A : I I IIdev : A 0 (1.1.114) 3 upon double contraction deviatoric fourth order unit tensor IIdev extracts deviatoric part of second order tensor

dev sym vol sym 1 II II II II 3 I I dev 1 1 1 II 2 δikδ jl 2 δilδ jk 3 δi jδkl ei ej ek el (1.1.115)

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1.1 Tensor algebra

1.1.6 Orthogonal tensors a second order tensor Q is called orthogonal if its inverse Q 1 is identical to its transpose Qt

Q 1 Qt Qt Q Q Qt I (1.1.116) 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 Qt Q 1 as

A Q U V Q (1.1.117) with S0(3) being the special orthogonal group, Q S0(3) if det Q 1, then Q is called proper orthogonal a proper orthogonal tensor Q S0(3) has an eigenvalue equal to one λQ 1 introducing an eigenvector nQ such that

Q nQ nQ (1.1.118) let nQi i 1,2,3 be a Cartesian basis containing the vector nQ, then matrix representation of coordinates Qi j

1 0 0

Qi j 0 cosϕ sinϕ (1.1.119) 0 sinϕ cosϕ geometric interpretation: Q characterizes a ﬁnite rotation around the axis nQ with Q nQ nQ, i.e. associated with λQ 1

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1 Tensor calculus

1.2 Tensor analysis

1.2.1 Derivatives consider smooth, differentiable scalar ﬁeld Φ with

scalar argument Φ : ; Φ x α vectorial argument Φ : 3 ; Φ x α tensorial argument Φ : 3 3 ; Φ X α

1.2.1.1 Frechet derivative ∂Φ x scalar DΦ x ∂ Φ x ∂x x ∂Φ x vectorial DΦ x ∂ Φ x (1.2.1) ∂x x ∂Φ X tensorial DΦ X ∂ Φ X ∂X X

1.2.1.2 Gateaux derivative

Gateaux derivative as particular Frechet derivative with respect to directions u, u and U d scalar DΦ x u Φ x ) u ! u d) 0 d 3 vectorial DΦ x u Φ x ) u ! u d) 0 d 3 3 tensorial DΦ X :U Φ X ) U ! U d) 0 (1.2.2) in what follows in particular vectorial arguments, i.e. point position x or displacement u

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1.2 Tensor analysis

1.2.2 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

1.2.2.1 Gradient of a scalar–valued function gradient f x of scalar–valued ﬁeld f x ∂ f x f x f,i x ei (1.2.3) ∂xi and thus

f,1

f x f,2 (1.2.4)

f,3 gradient of scalar–valued ﬁeld renders a vector ﬁeld

1.2.2.2 Gradient of a vector–valued function gradient f x of vector–valued ﬁeld f x

∂ fi x f x fi,j x ei ej (1.2.5) ∂x j and thus

f1,1 f1,2 f1,3

f x f2,1 f2,2 f2,3 (1.2.6)

f3,1 f3,2 f3,3 gradient of vector–valued ﬁeld renders a (second order) tensor ﬁeld

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1 Tensor calculus

1.2.3 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

1.2.3.1 Divergence of a vector ﬁeld divergence f x of vector ﬁeld f x

div f x tr f x f x : 1 (1.2.7) with f x fi,j x ei ej

div f x fi,i x f1,1 f2,2 f3,3 (1.2.8) divergence of a vector ﬁeld renders a scalar ﬁeld

1.2.3.2 Divergence of a tensor ﬁeld divergence F x of tensor ﬁeld F x

div F x tr F x F x : 1 (1.2.9) with F x Fi,jk x ei ej ek

F11,1 F12,2 F13,3

div F x Fi j,j x F21,1 F22,2 F23,3 (1.2.10)

F31,1 F32,2 F33,3 divergence of a second order tensor ﬁeld renders a vector ﬁeld

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1.2 Tensor analysis

1.2.4 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

1.2.4.1 Laplace operator acting on scalar–valued function

Laplace operator f x acting on scalar–valued ﬁeld f x

f x div f x (1.2.11) and thus

f x f,ii f,11 f,22 f,33 (1.2.12) Laplace operator acting on on scalar–valued ﬁeld renders a scalar ﬁeld

1.2.4.2 Laplace operator acting on vector–valued function

Laplace operator f x acting on vector–valued ﬁeld f x

f x div f x (1.2.13) and thus

f1,11 f1,22 f1,33

f x fi,jj f2,11 f2,22 f2,33 (1.2.14)

f3,11 f3,22 f3,33

Laplace operator acting on on vector–valued ﬁeld renders a vector ﬁeld

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1 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 (1.2.15) div α A α div A A α div u A u div A A : u div u v u div v v ut or in index notation

α ui ,j ui α,j α ui,j

ui vi ,j ui vi,j vi ui,j

α ui ,i α ui,i ui α,i (1.2.16) α Ai j ,j α Ai j,j Ai j α,j

ui Ai j ,j ui Ai j,j Ai j ui,j

ui v j ,j ui v j,j v j ui,j

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1.2 Tensor analysis

1.2.5 Integral transformations integral theorems deﬁne relations n between surface integral ...dA ∂ ∂ X and volume integral ...dV 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

1.2.5.1 Integral theorem for scalar–valued ﬁelds (Green theorem)

α n dA α dV ∂ (1.2.17) α ni dA α ,i dV ∂

1.2.5.2 Integral theorem for vector–valued ﬁelds (Gauss theorem)

u n dA div u dV ∂ (1.2.18) ui ni dA ui,i dV ∂

1.2.5.3 Integral theorem for tensor–valued ﬁelds (Gauss theorem)

A n dA div A dV ∂ (1.2.19) Ai j n j dA Ai j,j dV ∂

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2 Kinematics

restriction to geometrically linear theory, valid if local strains remain small

2.1 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

∂ x PSfrag replacements u x, t

ei i 1,2,3 motion of a body 3 characterized through time de- pendent vector ﬁeld of displacements u 3 parameterized in terms of position x and time t

3 u : u x, t ui x, t ei (2.1.1)

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2 Kinematics

2.2 Rates of kinematic quantities

2.2.1 Velocity vector ﬁeld of velocities v 3 parameterized in terms of position x and time t

3 v : v x, t vi x, t ei (2.2.1) velocity field v defined through rate of change of displacement field u ∂u x, t v x, t Dtu x, t (2.2.2) ∂t xfixed velocity vector

t t v v1, v2, v3 Dt u1, u2, u3 (2.2.3) common notation in the literature v u˙

2.2.2 Acceleration vector ﬁeld of accelerations a 3 parameterized in terms of position x and time t

3 a : a x, t ai x, t ei (2.2.4) acceleration field a defined through rate of change of velocity field v ∂v x, t ∂2u x, t a x, t Dtv x, t 2 (2.2.5) ∂t xfixed ∂t xfixed acceleration vector

t t 2 t a a1, a2, a3 Dt v1, v2, v3 Dt u1, u2, u3 (2.2.6) common notation in the literature a v˙ u¨

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2.3 Gradients of kinematic quantities

2.3.1 Displacement gradient second order tensor ﬁeld of displacement gradient H 3 3 parameterized in terms of position x and time t

3 3 H : H x, t Hi j x, t ei ej (2.3.1) displacement gradient H defined through gradient of displacement field u ∂u x, t H x, t u x, t (2.3.2) ∂x tfixed index representation

∂ui Hi jei ej ei ej ui,j ei ej (2.3.3) ∂Xj matrix representation of coordinates Hi j

H11 H12 H13 u1,1 u1,2 u1,3

Hi j H21 H22 H23 u2,1 u2,2 u2,3 (2.3.4)

H31 H32 H33 u3,1 u3,2 u3,3

displacement gradient H u is non–symmetric, H H t displacement gradient H u does not vanish for pointwise rigid body motion

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2 Kinematics

Symmetric-skew-symmetric decomposition of displacement gradient symmetric–skew-symmetric decomposition of displacement gradient H u 1 1 H H Ht H Ht ! ω (2.3.5) 2 2 with symmetric and skew-symmetric second order tensor ! Hsym and ω Hskw

! !t ω ωt (2.3.6)

geometrically linear strain tensor ! 1 1 ! H Ht u tu symu IIsym : u (2.3.7) 2 2 upon double contraction symmetric fourth order unit tensor IIsym extracts symmetric part of second order tensor

IIsym 1 II IIt 2 (2.3.8) sym 1 II 2 δikδ jl δilδ jk ei ej ek el geometrically linear rotation tensor ω 1 1 ω H Ht u tu skwu IIskw: u (2.3.9) 2 2 upon double contraction skew-symmetric fourth order unit tensor IIskw extracts skew-symmetric part of second order tensor IIskw 1 II IIt 2 (2.3.10) skw 1 II 2 δikδ jl δilδ jk ei e j ek el

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2.3 Gradients of kinematic quantities

2.3.2 Strain second order tensor ﬁeld of (geometrically linear) strains ! 3 3 parameterized in terms of position x and time t

3 3 ! : ! x, t )i j x, t ei ej (2.3.11) strain tensor ! deﬁned through symmetric part of gradient of displacement ﬁeld u ∂u x, t sym ! x, t symu x, t (2.3.12) ∂x index representation 1 ) e e u u e e (2.3.13) i j i j 2 i,j j,i i j matrix representation of coordinates )i j

)11 )12 )13 2u1,1 u1,2 u2,1 u1,3 u3,1 1 )i j ) ) ) u u 2u u u 12 22 23 2 2,1 1,2 2,2 2,3 3,2 )13 )32 )33 u3,1 u1,3 u3,2 u2,3 2u3,3 (2.3.14) strain tensor ! symu is symmetric, ! !t

)11 )12 )13 )11 )21 )31

)i j )21 )22 )23 )12 )22 )32 ) ji (2.3.15)

)31 )32 )33 )13 )23 )33

)i j for i j ... diagonal entries: normal strain )i j for i j ... off–diagonal entries: shear strain

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2 Kinematics

2.3.3 Rotation second order tensor field of (geometrically linear) rotation ω 3 3 parameterized in terms of position x and time t 3 3 ω : ω x, t ωi j x, t ei ej (2.3.16) rotation tensor ω defined through skew–symmetric part of gradient of displacement field u ∂u x, t skw ω x, t skwu x, t (2.3.17) ∂x index representation 1 ω e e u u e e (2.3.18) i j i j 2 i,j j,i i j matrix representation of coordinates ωi j

0 ω12 ω13 0 u1,2 u2,1 u1,3 u3,1 1 ωi j ω 0 ω u u 0 u u 21 23 2 2,1 1,2 2,3 3,2 ω31 ω32 0 u3,1 u1,3 u3,2 u2,3 0 (2.3.19) rotation tensor ω skwu is skew-symmetric, ω ωt

0 ω12 ω13 0 ω21 ω31

ωi j ω21 0 ω23 ω12 0 ω32 ωji

ω31 ω32 0 ω13 ω23 0 (2.3.20) 3 corresponding axial vector w 1 2 e: ω t t w w1, w2, w3 ω23, ω31, ω12 (2.3.21)

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2.3 Gradients of kinematic quantities

Symmetric-skew-symmetric decomposition of displacement gradient

symmetric–skew-symmetric decomposition of displacement gradient H u 1 1 H u tu u tu ! ω (2.3.22) 2 2 with symmetric and skew-symmetric second order tensor ! symu and ω skwu

! !t ω ωt (2.3.23)

geometric interpretation: representation of symmetric and skew–symmetric part of displacement gradient for two-dimensional case

strain rotation

e2 e2

u1,2 u1,2 ! PSfrag replacements PSfrag replacements ω

e1 e1

u2,1 u2,1

) 1 ω 1 12 2 u1,2 u2,1 12 2 u1,2 u2,1

symmetric part ! symu represents strain while skew- symmetric ω skwu part represents rotation

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2 Kinematics

2.3.4 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 (iso- choric) deformation conserves the volume volumetric – deviatoric decomposition of strain tensor !

! !vol !dev (2.3.24) with volumetric and deviatoric strain tensor !vol and !dev

tr !vol tr ! tr !dev 0 (2.3.25)

volumetric second order tensor !vol 1 !vol ! : I I IIvol : ! (2.3.26) 3 upon double contraction volumetric fourth order unit tensor IIvol extracts volumetric part !vol of strain tensor

IIvol 1 I I 3 (2.3.27) vol 1 II 3 δi jδkl ei ej ek el deviatoric second order tensor !dev 1 !dev ! ! : I I IIdev : ! (2.3.28) 3 upon double contraction deviatoric fourth order unit tensor IIdev extracts deviatoric part of strain tensor

IIdev IIsym IIvol IIsym 1 I I 3 (2.3.29) dev 1 1 1 II 2 δikδ jl 2 δilδ jk 3 δi jδkl ei ej ek el

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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 dv dV e 1 )11 1 )22 1 )33 1 dV (2.3.30) 2 )11 )22 )33 )i j neglection of higher order terms: trace of strain tensor tr ! ! : I as characteristic measure for volume changes e div u u : I ! : I tr ! (2.3.31) volumetric part !vol of strain tensor !

vol 1 1 1 vol (2.3.32) ! 3 e I 3 ! : I I 3 I I : ! II : ! index representation

vol vol ! )i j ei ej (2.3.33)

vol matrix representation of coordinates )i j

1 0 0 )vol 1 tr ! (2.3.34) i j 3 e 0 1 0 e 0 0 1

incompressibility is characterized through div u 0 volumetric strain tensor !vol is a spherical second order !vol 1 tensor as 3 eI volumetric strain tensor !vol contains the volume changing, shape preserving part of the total strain tensor !

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2 Kinematics

Deviatoric deformation deviatoric strain tensor !dev preserves the volume and contains the remaining part of the total strain tensor ! deviatoric part !dev of the strain tensor !

!dev ! !vol ! 1 ! : IIdev : ! (2.3.35) 3 I I index representation

dev dev ! )i j ei ej (2.3.36)

dev matrix representation of coordinates )i j

2)11 )22 )33 )12 )13 )dev 1 i j 3 )21 2)22 )11 )33 )13

)31 )32 2)33 )11 )22 (2.3.37) trace of deviatoric strains tr !dev

!dev 1 tr 3 2)11 )22 )33 1 (2.3.38) 3 2)22 )11 )33 1 3 2)33 )11 )22 0 deviatoric strain tensor !dev is a traceless second order tensor as tr !dev 0 deviatoric strain tensor !dev contains the shape changing, volume preserving part of the total strain tensor !

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2.3 Gradients of kinematic quantities

Volumetric–deviatoric decomposition of strain tensor

examples of purely volumetric deformation 1 !vol ! : I I IIvol : ! tr !vol tr ! (2.3.39) 3 expansion compression

PSfrag replacements PSfrag replacements e1 e e2 1 ω e2 u ω 1,2 u2,1 u1,2 u2,1 e 0 and !dev 0 e 0 and !dev 0

examples of purely deviatoric deformation 1 !dev ! ! : I I IIdev : ! tr !dev 0 (2.3.40) 3 pure shear simple shear

PSfrag replacements PSfrag replacements β γ α β α β γ α β α e 0 and !dev 0 e 0 and !dev 0

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2 Kinematics

2.3.5 Strain vector assume we are interested in strain on a plane characterizedPSfragthroughreplacements t! !n its normal n, strain vector t! acting n on plane given through normal pro- !t jection of strain tensor !

t! ! n (2.3.41) index representation

t! )i j ei ej nk ek (2.3.42) )i j nkδ jk ei )i j n j ei t!i ei representation of coordinates t!i

t!1 )11 n1 )12 n2 )13 n3

t!2 )21 n1 )22 n2 )23 n3 (2.3.43)

t!3 )31 n1 )32 n2 )33 n3 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 1 stretch as change of displacement vector u in the direction of n given through the Gateaux derivative 1.2.1.2 d D u x n u x ) n ! d) 0 (2.3.44) u x ) n n ! 0 u x n outer derviative inner derivative recall that u symu skwu ! ω whereby rotation ω skwu does not induce strain, thus sym t! u n ) n (2.3.45)

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2.3 Gradients of kinematic quantities

2.3.6 Normal–shear decomposition assume we are interested in strain along a particular ﬁber PSfragcharacterizedreplacements t! !n through its normal n, stretch of n ﬁber )n given through normal pro- !t jection of strain vector t!

)n t! 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

)n n u n n ! n ! : n n (2.3.47) normal-shear (tangential) decomposition of strain vector t!

t! !n !t (2.3.48) normal strain vector – stretch of ﬁbers in direction of n

!n ! : n n n (2.3.49) shear (tangential) strain vector – sliding of ﬁbers parallel to n sym !t t! !n ! : II n n n n (2.3.50) amount of sliding γn

2 γn 2 !t 2 !t !t 2 t! t! )n (2.3.51) in general, i.e. for an arbitrary direction n, we have normal and shear contributions to the strain vector, however, three particular directions n! i i 1,2,3 can be identiﬁed, for which t! !n and thus !t 0, the corresponding n! i i 1,2,3 are called principal strain directions and )n i i 1,2,3 λ!i i 1,2,3 are the principal strains or stretches

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2 Kinematics

2.3.7 Principal strains – stretches assume strain tensor ! to be known at x , principal strains λ! i i 1,2,3 and principal strain directions n! i i 1,2,3 can be derived from solution of special eigenvalue problem according to 1.1.3

! n! i λ! i n! i ! λ! i n! i 0 (2.3.52) solution

det ! λ! I 0 (2.3.53) or in terms of roots of characteristic equation

3 2 λ! I! λ! II! λ! III! 0 (2.3.54) roots of characteristic equations in terms of principal invariants of !

I! tr ! λ!1 λ!2 λ!3 1 2 ! !2 (2.3.55) II! 2 tr tr λ!2λ!3 λ!3λ!1 λ!1λ!2

III! det ! λ!1 λ!2 λ!3 spectral representation of !

3 ! ∑ λ!i n!i n!i (2.3.56) i 1 principal strains (stretches) λ!i are purely normal, no shear deformation (sliding) γn in principal directions, i.e. t!i !n λ!i n!i and !t 0 thus γn 0 due to symmetry of strains ! !t, strain tensor possesses three real eigenvalues λ! i i 1,2,3, corresponding eigendirec- tions n! i i 1,2,3 are thus orthogonal n!i n!j δi j

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2.3 Gradients of kinematic quantities

2.3.8 Compatibility until now, we have assumed the displacement field u x, t to be given, such that the strain field ! symu 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 ! x, t , we want to know whether these strains ! are compatible with a contin- uous single–valued displacement field u symmetric second order incompatibility tensor η crl crl ! (2.3.57) index representation of incompatibility tensor 3 3 η ηi jei ej eikm )kn,ml ejln ei ej 0 (2.3.58) coordinate representation of compatibility condition

)kl,mn )mn,kl )ml,kn )kn,ml 0 (2.3.59) valid k, l, m, n, thus 81 equations which are partly redun- dant, six independent conditions

St. Venant compatibility conditions

η11 )22,33 )33,22 2)23,32 0

η22 )33,11 )11,33 2)31,13 0

η33 )11,22 )22,11 2)12,21 0 (2.3.60) η12 )13,32 )23,31 )33,12 )12,33 0

η23 )21,13 )31,12 )11,23 )23,11 0

η31 )32,21 )12,23 )22,31 )31,22 0 incompatible displacement ﬁeld, e.g. in dislocation theory

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2 Kinematics

2.3.9 Special case of plane strain dimensional reduction in case of plane strain with vanishing strains )13 )23 )31 )32 )33 0 in out of plane direction, e.g. in geomechanics

! )i j ei ej (2.3.61) matrix representation of coordinates )i j

)11 )12 0

)i j )21 )22 0 (2.3.62) 0 0 0

2.3.10 Voigt representation of strain three dimensional second order strain tensor !

! )i j ei ej (2.3.63) matrix representation of coordinates )i j

)11 )12 )13

)i j )21 )22 )23 (2.3.64)

)31 )23 )33 due to symmetry )i j ) ji and thus )12 )21, )23 )32, )31 )13, strain tensor ! contains only six independent components )11,)22,)33,)12,)23,)31,it proves convenient to represent second order tensor ! through a vector ) t ) )11,)22,)33, 2)12, 2)23, 2)31 (2.3.65) vector representation ) of strain ! in case of plane strain t ) )11,)22,)33, 2)12 (2.3.66)

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3 Balance equations

3.1 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

PSfrag replacements

isolation rn tσ n qn ¯ x ¯

∂ ∂ ¯ ¯ x

ei i 1,2,3

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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 tσ, 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 ¯

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3.1 Basic ideas

3.1.1 Concept of mass ﬂux

the contact mass ﬂux rn at a point x is a scalar of the unit [mass/time/surface area]

the contact mass ﬂux rn characterizes the transport of matter PSfragnormalreplacementsto the tangent plane to an imaginary surface passing through this point with normal vector n ¯ r ∂

x ¯ ∂ ¯ ¯ x n rn

isolation ei i 1,2,3

definition of contact heat flux qn in analogy to Cauchy’s postulate, lemma and theorem originally introduced for the momentum flux in 3.1.2

Cauchy’s postulate

rn rn x, n (3.1.1)

Cauchy’s lemma

rn x, n rn x, n (3.1.2)

Cauchy’s theorem

the contact mass ﬂux rn can be expressed as linear function of the surface normal n and the mass ﬂux vector r

rn r n (3.1.3)

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3 Balance equations

Mass flux vector the vector field r is called mass flux vector

r ri ei (3.1.4) Cauchy’s theorem

rn r n (3.1.5) index representation

rn ri ei n j ej ri n j δi j ri ni (3.1.6) geometric interpretation

r2 e3 PSfrag replacements e1 e2

the coordinates ri 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

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3.1 Basic ideas

3.1.2 Concept of stress traction vector f d f tσ lim (3.1.7) a 0 a da interpretation as surface force per unit surface area

Cauchy’s postulate the traction vector tσ at a point x can be expressed exclu-

tσ

PSfrag replacements x

ei i 1,2,3 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

tσ tσ x, n (3.1.8)

Cauchy’s lemma the traction vectors acting on opposite sides of a surface are equal in magnitude and opposite in sign

tσ1 x, n1 tσ2 x, n2 (3.1.9)

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3 Balance equations

tσ1

n PSfrag replacements 1 n2

x tσ2

ei i 1,2,3

generalization with n n1 n2 and tσ tσ1

tσ x, n tσ x, n (3.1.10)

Cauchy’s theorem

the traction vector tσ can be expressed as a linear map of the surface normal n mapped via the transposed stress tensor σ t

t tσ σ n (3.1.11)

accordingly with n n1 n2 and tσ tσ1

t t tσ1 σ n1 σ n tσ (3.1.12) t t tσ2 σ n2 σ n tσ Cauchy tetraeder

balance of momentum (pointwise)

tσ n da tσ ni dai tσ ei dai tσi dai (3.1.13) surface theorem, area fractions from Gauss theorem da nda n da e da i e n cos e , n (3.1.14) i i i i da i i

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3.1 Basic ideas

tσ1 PSfrag replacements n2 e2 n1 e1 x e3

tσ2 tσ n

e1 e2

n3 e3 tσ3

traction vector as linear map of surface normal

dai tσ n tσ tσ cos e , n tσ e n tσ e n i da i i i i i i (3.1.15)

t compare tσ n σ n t interpretation of second order stress tensor as σ tσi ei

Stress tensor Cauchy stress (true stress)

t σ tσi ei σ jiej ei σ ei tσi σi jei ej (3.1.16)

Cauchy theorem

t tσ σ n (3.1.17)

index representation

tσ σ jie j ei nkek σ jinkδikej σ jiniej t jej (3.1.18)

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3 Balance equations

matrix representation of tensor coordinates of σi j

t σ11 σ12 σ13 tσ1 σ t i j σ21 σ22 σ23 tσ2 (3.1.19) t σ31 σ32 σ33 tσ3 geometric interpretation

σ 33 PSfrag replacements σ 32

σ 31 σ 23 σ 22 e3

σ 13 e1 e2 σ 21

σ 12 σ 11

with traction vectors on surfaces

t tσ1 σ11 σ12 σ13 t tσ2 σ21 σ22 σ23 (3.1.20) t tσ3 σ31 σ32 σ33 ﬁrst index ... surface normal second index ... direction (coordinate of traction vector) diagonal entries ... normal stresses non–diagonal entries .. shear stresses

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3.1 Basic ideas

3.1.2.1 Volumetric–deviatoric decomposition of stress tensor in analogy to the strain tensor !, 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 (3.1.21) with volumetric and deviatoric stress tensor σ vol and σ dev

tr σ vol tr σ tr σ dev 0 (3.1.22)

volumetric second order tensor σ vol 1 σ vol σ : I I IIvol : σ (3.1.23) 3 upon double contraction volumetric fourth order unit tensor IIvol extracts volumetric part σ vol of stress tensor

IIvol 1 I I 3 (3.1.24) vol 1 II 3 δi jδkl ei ej ek el deviatoric second order tensor σ dev 1 σ dev σ σ : I I IIdev : σ (3.1.25) 3 upon double contraction deviatoric fourth order unit tensor IIdev extracts deviatoric part of stress tensor

IIdev IIsym IIvol IIsym 1 I I 3 (3.1.26) dev 1 1 1 II 2 δikδ jl 2 δilδ jk 3 δi jδkl ei ej ek el

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3 Balance equations

Volumetric stress volumetric part σ vol of stress tensor σ

vol 1 1 vol (3.1.27) σ 3 σ : I I 3 I I : σ II : σ interpretation of trace as hydrostatic pressure

1 1 1 (3.1.28) p 3tr σ 3 σ : I 3 σ11 σ22 σ33 index representation

vol vol σ σi j ei ej (3.1.29)

vol matrix representation of coordinates σi j

1 0 0 1 σ vol p 0 1 0 p tr σ (3.1.30) i j 3 0 0 1 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

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3.1 Basic ideas

Deviatoric stress deviatoric stress tensor σ dev preserves the volume and con- taines the remaining part of the total stress tensor σ deviatoric part σ dev of the stress tensor σ

σ dev σ σ vol σ 1 σ : IIdev : σ (3.1.31) 3 I I index representation

dev dev σ σi j ei ej (3.1.32)

dev matrix representation of coordinates σi j

2σ11 σ22 σ33 σ12 σ13 σ dev 1 i j 3 σ21 2σ22 σ11 σ33 σ13

σ31 σ32 2σ33 σ11 σ22 (3.1.33) trace of deviatoric stresss tr σ dev

σ dev 1 tr 3 2σ11 σ22 σ33 1 (3.1.34) 3 2σ22 σ11 σ33 1 3 2σ33 σ11 σ22 0 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

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3 Balance equations

3.1.2.2 Normal–shear decomposition assume we are interested in the stress σn normal to a particularPSfrag replacements tσ σ n plane characterized through its n normal n, i.e. the normal pro- σ t jection of the stress vector tσ

t t t σn tσ n σ n n σ : n n σ : N (3.1.35) normal–shear (tangential) decomposition of stress vector tσ

tσ σ n σ t (3.1.36) normal stress vector – stress in direction of n t t σ n σ : n n n σ : n n n (3.1.37) shear (tangential) stress vector – stress in the plane

t t σ t tσ σ n σ n σ : n n n (3.1.38) σ t : IIsym n n n n σ t : T amount of shear stress τn 2 2 τn tσ σ n tσ σ n tσ tσ 2tσ σ n σnn n (3.1.39) and thus 2 τn σt σ t σ t tσ tσ σn (3.1.40) in general, i.e. for an arbitrary direction n, we have normal and shear contributions to the stress vector, however, three particular directions nσ i i 1,2,3 can be identiﬁed, for which tσ σ n and thus σ t 0, the corresponding nσ i i 1,2,3 are called prinicpal stress directions and σn i i 1,2,3 λσi i 1,2,3 are the principal stresses

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3.1 Basic ideas

3.1.2.3 Principal stresses assume stress tensor σ t to be known at x , principal stresses λσ i i 1,2,3 and principal stress directions nσ i i 1,2,3 can be derived from solution of special eigenvalue problem according to 1.1.3 t t σ nσ i λσ i nσ i σ λσ i nσ i 0 (3.1.41) solution t det σ λσ I 0 (3.1.42) or in terms of roots of characteristic equation 3 2 λσ Iσ λσ IIσ λσ IIIσ 0 (3.1.43) roots of characteristic equation in terms of principal invariants of σ t

t Iσ tr σ λσ1 λσ2 λσ3 1 2 t t 2 IIσ 2 tr σ tr σ λσ2λσ3 λσ3λσ1 λσ1λσ2 t IIIσ det σ λσ1 λσ2 λσ3 (3.1.44) spectral representation of σ 3 t σ ∑ λσi nσi nσi (3.1.45) i 1 principal stresses λσi are purely normal, no shear stress τn in principal directions, i.e. tσi σ n λσi nσi and σ t 0 thus τn 0 due to symmetry of stresses σ σ t, stress tensor posseses three real eigenvalues λσ i i 1,2,3, corresponding eigendi- rections nσ i i 1,2,3 are thus orthogonal nσi nσ j δi j

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3 Balance equations

3.1.2.4 Special case of plane stress dimensional reduction in case of plane stress with vanishing stresses σ13 σ23 σ31 σ32 σ33 0 in out of plane direction, e.g. for ﬂat sheets

σ σi j ei ej (3.1.46) matrix representation of coordinates σi j

σ11 σ12 0

σi j σ21 σ22 0 (3.1.47) 0 0 0

3.1.2.5 Voigt representation of stress three dimensional second order stress tensor σ

σ σi j ei ej (3.1.48) matrix representation of coordinates σi j

σ11 σ12 σ13

σi j σ21 σ22 σ23 (3.1.49)

σ31 σ23 σ33 due to symmetry σi j σ ji and thus σ12 σ21, σ23 σ32, σ31 σ13, stress tensor σ contains only six independent components σ11,σ22,σ33,σ12,σ23,σ31,it proves convenient to represent second order tensor σ through a vector σ t σ σ11,σ22,σ33,σ12,σ23,σ31 (3.1.50) vector representation σ of stress σ in case of plane stress t σ σ11,σ22,σ33,σ12 (3.1.51)

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3.1 Basic ideas

3.1.3 Concept of heat ﬂux

the contact heat ﬂux qn at a point x is a scalar of the unit [energy/time/surface area]

the contact heat ﬂux qn characterizes the energy transport PSfragnormalreplacementsto the tangent plane to an imaginary surface passing through this point with normal vector n ¯ q ∂

x ¯ ∂ ¯ ¯ x n qn

isolation ei i 1,2,3

definition of contact heat flux qn in analogy to Cauchy’s postulate, lemma and theorem originally introduced for the momentum flux in 3.1.2

Cauchy’s postulate

qn qn x, n (3.1.52)

Cauchy’s lemma

qn x, n qn x, n (3.1.53)

Cauchy’s theorem

the contact heat ﬂux qn can be expressed as linear function of the surface normal n and the heat ﬂux vector q

qn q n (3.1.54)

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3 Balance equations

Heat flux vector the vector field q is called heat flux vector

q qi ei (3.1.55) Cauchy’s theorem

qn q n (3.1.56) index representation

qn qi ei n j ej qi n j δi j qi ni (3.1.57) geometric interpretation

q2 e3 PSfrag replacements e1 e2

the coordinates qi 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

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3.2 Balance of mass

3.2 Balance of mass total mass m of a body ¯

m : d m (3.2.1) ¯ mass density m dm ρ lim d m ρ dV (3.2.2) V 0 V dv

m ρ dV (3.2.3) ¯ mass exchange of body ¯ with environment msur and mvol

sur vol m : rn d A m : d V (3.2.4) ∂ ¯ ¯ with contact mass ﬂux rn r n and mass source

3.2.1 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 msur and the at-a-distance mass exchange mvol.”

sur vol Dtm m m (3.2.5) and thus

Dt ρ dV rn d A d V (3.2.6) ¯ ∂ ¯ ¯

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3 Balance equations

3.2.2 Local form of balance of mass modiﬁcation of rate term Dt m

¯ﬁxed Dt m Dt ρ dV Dtρ dV (3.2.7) ¯ ¯ modiﬁcation of surface term msur

sur Cauchy Gauss m rn d A r n d A div r d V ∂ ¯ ∂ ¯ ¯ (3.2.8) and thus

Dtρ dV div r d V d V (3.2.9) ¯ ¯ ¯ for arbitrary bodies ¯ local form of mass balance

Dtρ div r (3.2.10)

3.2.3 Classical continuum mechanics of closed systems in classical closed system continuum mechanics (here), r 0 and 0, such that the mass density ρ is constant in time

Dt ρ x, t 0 ρ ρ x const (3.2.11) typically r 0 and 0 only in bio– or chemomechanics

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3.3 Balance of linear momentum

3.3 Balance of linear momentum total momentum p of a body ¯

p : Dtu d m v d m ρ v dV (3.3.1) ¯ ¯ ¯ momentum exchange of body ¯ with environment through contact forces f sur and at-a-distance forces f vol

sur vol f : tσ d A f : b d V (3.3.2) ∂ ¯ ¯ t with contact/surface force tσ σ n and volume force b

3.3.1 Global form of balance of momentum

”The time rate of change of the total momentum p of a body ¯ is balanced with the momentum exchange due to contact momentum ﬂux / surface force f sur and the at-a-distance momentum exchange / volume force f vol.” sur vol Dt p f f (3.3.3) and thus

Dt ρ v dV tσ d A b d V (3.3.4) ¯ ∂ ¯ ¯

3.3.2 Local form of balance of momentum modiﬁcation of rate term Dt p

¯ﬁxed Dt p Dt ρ vdV Dt ρ v dV (3.3.5) ¯ ¯

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3 Balance equations modiﬁcation of surface term f sur

sur Cauchy Gauss t f tσ d A σ t n d A div σ d V ∂ ¯ ∂ ¯ ¯ (3.3.6) and thus

t Dt ρ v dV div σ d V b d V (3.3.7) ¯ ¯ ¯ for arbitrary bodies ¯ local form of momentum balance

t Dt ρ v div σ b (3.3.8)

3.3.3 Reduction with lower order balance equations balance equations are typically modiﬁed with the help of lower order balance equations

t Dt ρ v v Dt ρ ρ Dt v div σ b (3.3.9) with balance of mass multiplied by velocity v

v Dt ρ v div r v 0 div v r v r v 0 (3.3.10) local momentum balance in reduced format

t ρ Dtv div σ v r b v r v 0 (3.3.11)

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3.4 Balance of angular momentum

3.3.4 Classical continuum mechanics of closed systems in classical closed system continuum mechanics (here), r 0 and 0, such that the mass density ρ is constant in time, i.e. Dt ρ v ρ Dtv and thus

t ρ Dtv div σ b (3.3.12) the above equation is referred to as ’Cauchy’s ﬁrst equation of motion’, Cauchy [1827]

3.4 Balance of angular momentum total angular momentum l of a body ¯

l : x Dtu d m x v d m ρ x v dV (3.4.1) ¯ ¯ ¯ angular momentum exchange of body ¯ with environment through momentum from contact forces lsur and momentum from at-a-distance forces lvol

sur vol l : x tσ d A l : x b d V (3.4.2) ∂ ¯ ¯ with momentum due to contact/surface forces x tσ x σ t n and volume forces x b, assumption: no additional external torques

3.4.1 Global form of balance of angular momentum

”The time rate of change of the total angular momentum l of a body ¯ is balanced with the angular momentum exchange

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3 Balance equations due to contact momentum ﬂux / surface force lsur and due to the at-a-distance momentum exchange / volume force lvol.”

sur vol Dtl l l (3.4.3) and thus

Dt ρ x v dV x tσ d A x b d V (3.4.4) ¯ ∂ ¯ ¯

3.4.2 Local form of balance of angular momentum modiﬁcation of rate term Dt p

¯ﬁxed Dt l Dt ρ x vdV Dt ρ x v dV (3.4.5) ¯ ¯ modiﬁcation of surface term lsur

sur Cauchy t l x tσ d A x σ n d A ∂ ¯ ∂ ¯ Gauss div x σ t d V (3.4.6) ¯ x div σ t d V x σ t d V ¯ ¯ and thus

t Dt ρ x v dV x div σ d V ¯ ¯ (3.4.7) I σ t d V x b d V ¯ ¯ for arbitrary bodies ¯ local form of ang. mom. balance

t t Dt ρ x v x div σ I σ x b (3.4.8)

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3.4 Balance of angular momentum

3.4.3 Reduction with lower order balance equations modiﬁcation with the help of lower order balance equations

Dt ρ x v Dt x ρ v x Dt ρ v (3.4.9) x div σ t I σ t x b

position vector x ﬁxed, thus Dt x 0, with balance of linear momentum multiplied by position x

t x Dt ρ v x div σ x b (3.4.10) local angular momentum balance in reduced format

3 I σ t e: σ 2 axl σ 0 σ σ t (3.4.11)

the above equation is referred to as ’Cauchy’s second equation of motion’, Cauchy [1827] in the absense of couple stresses, surface and body couples, the stress tensor is symmetric, σ σ t, else: micropolar / Cosserat continua 3 vector product of second order tensors A B e: A Bt

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3 Balance equations

3.5 Balance of energy total energy E of a body ¯ as the sum of the kinetic energy K and the internal energy I

E : e dV k i dV K I ¯ ¯ 1 K : k dV ρ v v dV (3.5.1) ¯ ¯ 2 I : i dV ¯ energy exchange of body ¯ with environment esur and evol

sur E : v tσ d A qn d A ∂ ¯ ∂ ¯ (3.5.2) Evol : v b d V d V ¯ ∂ ¯ with contact heat ﬂux qn r n and heat source

3.5.1 Global form of balance of energy

”The time rate of change of the total energy E of a body ¯ is balanced with the energy exchange due to contact energy ﬂux Esur and the at-a-distance energy exchange Evol.”

sur vol DtE E E (3.5.3) and thus

Dt e dV v tσ qn dA v b dV (3.5.4) ¯ ∂ ¯ ¯

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3.5 Balance of energy

3.5.2 Local form of balance of energy modification of rate term Dt E ¯fixed Dt E Dt e dV Dte dV (3.5.5) ¯ ¯ modification of surface term Esur

sur E v tσ qn d A ∂ ¯ Cauchy v σ t n q n d A (3.5.6) ∂ ¯ Gauss div v σ t div q d V ¯ and thus

t Dte dV div v σ q d V v b d V (3.5.7) ¯ ¯ ¯ for arbitrary bodies ¯ local form of energy balance

t Dte div v σ q v b (3.5.8)

3.5.3 Reduction with lower order balance equations modiﬁcation with the help of lower order balance equations with

Dt e Dt k i v Dt ρ v Dt i (3.5.9) div v σ t q v b with

1 1 1 Dt k Dt 2ρv v 2Dt ρv v 2 v Dt ρv v Dt ρv

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3 Balance equations

(3.5.10) with balance of momentum multiplied by velocity v

t t t v Dt ρ v v div σ v b div v σ v : σ v b (3.5.11) with stress power

t t t v : σ σ : v σ : Dtϕ (3.5.12) t t sym t σ : Dt ϕ σ : Dt ϕ σ : Dt! local energy balance in reduced format/balance of int. energy

t Dt i σ : Dt! div q (3.5.13)

3.5.4 First law of thermodynamics alternative deﬁnitions pext : div v σ t v b ... external mechanical power int t t p : σ : v σ : Dt! ... internal mechanical power qext : div q ... external thermal power (3.5.14)

Balance of total energy / ﬁrst law of thermodynamics the rate of change of the total energy e, i.e. the sum of the kinetic energy k and the potential energy i is in equilibrium with the external mechanical power pext and the external thermal power qext

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3.5 Balance of energy

ext ext Dt e Dt k Dt i p q (3.5.15) the above equation is typically referred to as ”principle of interconvertibility of heat and mechanical work” , Carnot [1832], Joule [1843], Duhem [1892]

ﬁrst law of thermodynamics does not provide any informa- tion about the direction of a thermodynamic process balance of kinetic energy

ext int Dt k p p (3.5.16) the balance of kinetic energy is an alternative statement of the balance of linear momentum balance of internal energy

ext int Dt i q p (3.5.17) kinetic energy k and internal energy i are no conservation properties, they exchange the internal mechanical energy pint

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3 Balance equations

3.6 Balance of entropy total entropy H of a body ¯

H : h dV (3.6.1) ¯ entropy exchange of body ¯ with environment through Hsur and Hvol

sur vol H : hn d A H : d V (3.6.2) ∂ ¯ ¯ with contact entropy ﬂux hn h n and entropy source internal entropy production of body ¯ as Hpro

Hpro : hpro d V 0 (3.6.3) ¯ whereby Hpro is strictly non-negative

3.6.1 Global form of balance of entropy

”The time rate of change of the total entropy H of a body ¯ is balanced with the energy exchange due to contact energy ﬂux Hsur, the at-a-distance entropy exchange Hvol and the non–negative entropy production Hpro with Hpro 0”.

sur vol pro DtH H H H (3.6.4) and thus

pro Dt hdV hn dA dV h dV (3.6.5) ¯ ∂ ¯ ¯ ¯

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3.6 Balance of entropy

3.6.2 Local form of balance of entropy modiﬁcation of rate term Dt H

¯ﬁxed Dt H Dt h dV Dth dV (3.6.6) ¯ ¯ modiﬁcation of surface term Hsur

sur Cauchy Gauss H hnd A h nd A div h d V ∂ ¯ ∂ ¯ ¯ (3.6.7) and thus

pro Dth dV div h d V d V h d V ¯ ¯ ¯ ¯ (3.6.8) for arbitrary bodies ¯ local form of entropy balance

pro Dth div h h (3.6.9)

3.6.3 Reduction with lower order balance equations modiﬁcation with the help of lower order balance equations assumption of existience of absolute temperature theta 0 which renders the following constitutive assumption

1 1 h θ q θ (3.6.10)

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3 Balance equations such that

1 1 pro Dth div θ q θ h 1 1 pro θ Dth θ θdiv q q θ θ θ h 1 pro Dt θ h θ θdiv q q ln θ θ h h Dtθ (3.6.11) with balance of internal energy

ext int t Dt i q p div q σ : Dt! (3.6.12) combination of balance of entropy and internal energy

t pro Dt i θ h σ : Dt! h Dtθ q ln θ θ h (3.6.13) Legendre transform: free Helmholz energy ψ i θ h (3.6.14) and thus

t pro Dt ψ σ : Dt! h Dtθ q ln θ θ h (3.6.15)

3.6.4 Second law of thermodynamics

”The production of entropy hpro is non-negative.” hpro 0 (3.6.16) entropy is a measure of microscopic disorder of a system, positive entropy production gives a preferred direction to thermodynamic processes Clausius: ’heat never ﬂows from a colder to a warmer system’ introduction of dissipation (-rate) : θ hpro 0 (3.6.17)

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3.6 Balance of entropy dissipation inequality

t σ : Dt! h Dtθ Dt ψ q ln θ 0 (3.6.18) the above equation is referred to as ’Clausius–Duhem inequality’, Clausius [1822-1888]

0 ... reversible process (3.6.19) 0 ... irreversible process conjugate pairs

stress σ vs. strain ! (3.6.20) entropy h vs. temperature θ decomposition into local and conductive part

loc con 0 (3.6.21) local part / Clausius–Planck inequality

loc t σ : Dt! h Dtθ Dt ψ 0 (3.6.22) conductive part / Fourier inequality

con q ln θ 0 (3.6.23) example: Fourier law of heat conduction

q κ θ isotropic κ κ I (3.6.24) Fourier inequality a priori statisﬁed by construction

θ κ θ 0 (3.6.25)

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3 Balance equations

3.7 Generic balance equation any balance law can be expressed a general format

3.7.1 Global / integral format

Dt (3.7.1) or alternatively

Dt dV n dA dV γ dV (3.7.2) ¯ ∂ ¯ ¯ ¯ whereby

... balance quantity ... surface transport through ∂ ¯ (3.7.3) ... volume source in ¯ ... production in ¯

3.7.2 Local / differential format local format of balance law follows from application of Gauss’ theorem localization theorem, i.e. ¯ arbitrarily small

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3.7 Generic balance equation

PSfrag replacements

balance quantity ¯ x γ x , n ¯ ∂ ¯

ei i 1,2,3

Dt div γ (3.7.4)

quantity ﬂux source production γ mass ρ r – lin. mom. ρ v σ t b – ang. mom. x ρ v x σ t x b – energy e v σ t q v b – t t kin. energy k v σ v b σ : Dt" t int. energy i q σ : Dt" entropy h h pro

Table 3.1: generic balance law

mass, linear momentum, angular momentum and total energy are conservation properties, while kinetic energy, internal energy and entropy are not, they possess a production term

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3 Balance equations

3.8 Thermodynamic potentials internal energy

i i !, h, ... σ D!i and θ Dhi (3.8.1) Legendre-Fenchel transform h θ ψ !,θ inf i !, h θ h i !, h θ θ h θ (3.8.2) h Helmholtz free energy

ψ ψ !,θ, ... σ D!ψ and h Dθψ (3.8.3) Legendre-Fenchel transform ! σ

g σ,θ sup σ : ! ψ !,θ σ : ! σ ψ ! σ ,θ ! (3.8.4)

Gibbs free energy

g g σ,θ, ... ! Dσ g and h Dθg (3.8.5) Legendre-Fenchel transform θ h η σ, h inf g σ,θ hθ g σ,θ h hθ h (3.8.6) θ enthalpy

η η !, h, ... ! Dσ η and θ Dhη (3.8.7) Legendre-Fenchel transform σ !

i !, h sup ! : σ η σ, h ! : σ ! η σ ! , h σ (3.8.8)

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4 Constitutive equations

motivation: unknowns density ρ 1 displacement u 3 temperature θ 1 mass flux r 3 stress σ 9 heat flux q 3 in total 20 equations balance of mass 1 balance of momentum 3 balance of angular momentum 3 balance of energy 1 in total 8 balance of unknowns vs. number of equations: 20-8=12 equations are missing, introduction of 12 material specific constitutive equations, i.e. equations for the mass flux r (3 eqns.), the symmetric stress σ σ t (6 eqns.) and the heat flux q (3 eqns.)

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4 Constitutive equations

4.1 Linear constitutive equations

for the mass flux r with rn r n t t for the momentum flux / stress σ σ with tn σ n for the heat flux q with qn q n

PSfrag replacements

isolation rn tσ n qn ¯ x ¯

∂ ∂ ¯ ¯ x

ei i 1,2,3

in the simplest case, we could introduce ad hoc definitions of the mass flux, the momentum flux and the heat flux in terms of the spatial gradients of the density, the deformation and the temperature

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4.1 Linear constitutive equations

4.1.1 Mass flux – Fick’s law linear relation between mass flux r (vector) and density gradient ρ (vector) in terms of mass conduction coefficient R (second order tensor)

r R ρ (4.1.1) index representation

ri ei Ri j ei e j ρ,kek Ri jeiδ jkρ,k Ri jρ,jei (4.1.2) matrix representation of coordinates

r1 R11 ρ,1 R12 ρ,2 R13 ρ,3

ri r2 R21 ρ,1 R22 ρ,2 R23 ρ,3 (4.1.3)

r3 R31 ρ,1 R32 ρ,2 R33 ρ,3 special case of isotropie

r1 ρ,1

R R I r R ρ ri r2 R ρ,2 (4.1.4)

r3 ρ,3 a linear relation between the ﬂux of matter r and the gradient of concentrations ρ is referred to as Fick’s law

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4 Constitutive equations

4.1.2 Momentum ﬂux – Hook’s law linear relation between momentum ﬂuxσ (second order tensor) and displacement gradient u or rather! symu (second order tensor) in terms of elasticity tensor IE (fourth order tensor)

σ IE : symu IE : ! (4.1.5) index representation

σi j ei ej Ei jkl ei ej ek el )mnem en (4.1.6) Ei jklei ejδkmδln)mn Ei jkl)kl ei e j special case of isotropie i.e. identical Eigenbasis of stress & strain

3 3 σ ∑ λσinσi nσi ! ∑ λ!in!i n!i (4.1.7) i1 i1 representation theorem for isotropic tensor–valued tensor- functions

σ ! f1 I f2! f3! ! (4.1.8) with fi fi I!, II!, III! function of strain invariants

I! tr ! λ!1 λ!2 λ!3 1 2 2 (4.1.9) II! 2 tr ! tr ! λ!2λ!3 λ!3λ!1 λ!1λ!2

III! det ! λ!1 λ!2 λ!3 a linear relation between the momentum ﬂuxσ and the strains ! represents the generalized form of Hook’s law

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4.1 Linear constitutive equations

Hypoelasticity / Cauchy Elasticity a hypoelastic / Cauchy elastic constitutive law can be repre- sented in the following form

σ σ ! (4.1.10)

invertible relation between stress σ and strain ! (rates) possible dissipation of energy in closed strain circles

loc dt σ : ! dt Dtψ dt σ : ! dt (4.1.11)

homogeneous strain path from !t1 to !t2

! α 1 α !t1 α !t2 d! !t2 !t1 dα (4.1.12) stress work for linear elastic material

t2 1 σ : d! ! α : IE : !t2 !t1 dα t1 0 (4.1.13) 1 ! ! : IE : ! ! 2 t2 t1 t2 t1 dissipation in isothermal closed strain cycle t1 t2 t3 t1

loc skw skw skw dt !t1 : IE : !t2 !t2 : IE : !t3 !t3 : IE : !t1 0 (4.1.14)

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4 Constitutive equations

4.1.3 Heat flux – Fourier’s law linear relation between heat flux q (vector) and temperature gradient θ (vector) in terms of heat conduction coefficient κ (second order tensor)

q κ θ (4.1.15) index representation

qi ei κi j ei ej θ,kek κi jeiδ jkθ,k κi jθ,jei (4.1.16) matrix representation of coordinates

q1 κ11 θ,1 κ12 θ,2 κ13 θ,3

qi q2 κ21 θ,1 κ22 θ,2 κ23 θ,3 (4.1.17)

q3 κ31 θ,1 κ32 θ,2 κ33 θ,3 special case of isotropie

q1 θ,1

κ κ I q κ θ qi q2 κ θ,2 (4.1.18)

q3 θ,3 a linear relation between the heat ﬂux vector q and the temperature gradient θ is referred to as Fourier’s law which goes back to Fourier [1822]

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4.2 Hyperelasticity

4.2.1 Speciﬁc stored energy a hyperelastic / Green elastic constitutive law can be repre- sented in the following form

loc σ σ ! and Dtψ 0 (4.2.1)

invertible relation between stress σ and strain ! (and stress and strain rates Dtσ and Dt!) based on a potential potential corresponds to elastically stored speciﬁc energy by construction no dissipation of energy in closed strain circles stress power . σ : Dt! Dtψ (4.2.2)

loc ensuring Dtψ 0 by construction, thus

ψ ψ ! and Dtψ D!ψ : Dt! (4.2.3) speciﬁc stored energy W as path independent integral of stress power

W ! ψ ! (4.2.4) with

t2 t2 t2 W !t2 W !t1 DtWdt dt σ : d! t1 t1 t1 (4.2.5)

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4 Constitutive equations generic hyperelastic / Green elastic constitutive law

σ D!W with W W ! (4.2.6)

t2 path independent W !t2 W !t1 d W t1 no dissipation d W 0 D2W symmetric D! D! relation between stress rates and strain rates deﬁnes continuum tangent stiffness (fourth order tensor) IEtan

tan Dtσ IE : Dt! (4.2.7) fourth order tangent stiffness / elastic material tangent D2W IEtan E e e e e (4.2.8) D! D! i jkl i j k l minor and major symmetries: reduction from 34 81 to 62 36 to 21 coefﬁcients

Ei jkl E jikl E jilk Ei jlk and Ei jkl Ekli j (4.2.9)

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4.2 Hyperelasticity

4.2.2 Speciﬁc complementary energy speciﬁc stored energy

σ D!W with W W ! (4.2.10)

Legendre-Fenchel tranform σ ! W σ sup σ : ! W ! (4.2.11) ! speciﬁc complementary stored energy W W σ σ : ! σ W ! σ (4.2.12) general hyperelastic constitutive law

! DσW with W W σ (4.2.13) relation between strain rates and stress rates deﬁnes continuum tangent compliance (fourth order tensor)CI tan

tan tan tan 1 Dt! CI : Dtσ with CI IE (4.2.14) fourth order tangent compliance D2W CI tan C e e e e (4.2.15) Dσ Dσ i jkl i j k l minor and major symmetries: reduction from 34 81 to 62 36 to 21 coefﬁcients

Ci jkl Cjikl Cjilk Ci jlk and Ci jkl Ckli j (4.2.16)

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4 Constitutive equations

4.3 Isotropic hyperelasticity

4.3.1 Speciﬁc stored energy isotropy: identical eigenbasis of stress and strain 3 3 σ ∑ λσinσi nσi ! ∑ λ!in!i n!i (4.3.1) i1 i1 representation theorem for isotropic tensor–valued tensor functions

σ ! f1 I f2! f3! ! (4.3.2)

with fi fi I!, II!, III! function of strain invariants

I! tr ! 1 2 ! !2 (4.3.3) II! 2 tr tr

III! det ! representation theorem for isotropic scalar–valued tensor functions

W ! W I!, II!, III! (4.3.4) with W ! W Q ! Qt Q SO 3 stress for hyperelastic material

DW DI! DW DII! DW DIII! σ D!W (4.3.5) DI! D! DII! D! DIII! D!

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4.3 Isotropic hyperelasticity with derivatives of invariants I!, II!, III! with respect to second order tensor !

D! I! I

D! II! ! I! I (4.3.6) t 2 D! III! III! ! ! I!! II! I general representation of stress

2 σ DI!W I DII!W ! I! I DIII!W ! I!! II! I comparison of coefﬁcients (4.3.7)

f1 DI!W I! DII!W II! DIII!W

f2 DII!W I! DIII!W (4.3.8)

f3 DIII!W assumption of linearity (quadratic term vanishes), two Lame´ constants λ and µ

f1 I!λ ! : I λ f2 2 µ f3 0 (4.3.9) speciﬁc stored energy (quadratic in strains)

1 1 W ! : IE : ! λ ! : I 2 µ !2 : I (4.3.10) 2 2 stress tensor (linear in strains)

σ D!W IE : ! f1 I f2! λ ! : I I 2 µ ! (4.3.11)

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4 Constitutive equations matrix representation of coordinates

λI! 2 µ)11 2 µ)12 2 µ)13

σi j 2 µ)21 λI! 2 µ)22 2 µ)23 (4.3.12)

2 µ)31 2 µ)32 λI! 2 µ)33 linear elastic continuum tangent stiffness (constant in strains)

tan sym tan IE λI I 2 µII Dtσ IE : Dt! (4.3.13) linear elastic continuum secant stiffness

IE λI I 2 µIIsym σ IE : ! (4.3.14)

Voigt representation of stiffness tensor

λ 2µ λ λ 0 0 0 λ λ 2µ λ 0 0 0 λ λ λ 2µ 0 0 0 IE (4.3.15) 0 0 0 µ 0 0 0 0 0 0 µ 0 0 0 0 0 0 µ

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4.3 Isotropic hyperelasticity volumetric - deviatoric decomposition speciﬁc stored energy (quadratic in volumetric and deviatoric strains)

W Wvol !vol Wdev !dev (4.3.16) 1 vol 2 dev2 2 κ ! : I µ ! : I with !vol 1 ! : I I IIvol : ! 3 (4.3.17) !dev ! 1 ! dev ! 3 : I I II : stress tensor

σ D!W IE : ! 3κ !vol 2 µ !dev (4.3.18) matrix representation of coordinates

dev dev dev κI! 2 µ)11 2 µ)12 2 µ)13 σ dev dev dev i j 2 µ)21 κI! 2 µ)22 2 µ)23 dev dev dev 2 µ)31 2 µ)32 κI! 2 µ)33 (4.3.19) linear elastic continuum tangent stiffness

tan vol dev tan IE 3κII 2 µII Dtσ IE : Dt! (4.3.20)

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4 Constitutive equations linear elastic continuum secant stiffness

IE 3κIIvol 2 µIIdev σ IE : ! (4.3.21)

Voigt representation of stiffness tensor

4 2 2 κ 3µ κ 3µ κ 3µ 0 0 0 2 4 2 κ 3µ κ 3µ κ 3µ 0 0 0 κ 2µ κ 2µ κ 4µ 0 0 0 IE 3 3 3 (4.3.22) 0 0 0 µ 0 0 0 0 0 0 µ 0 0 0 0 0 0 µ transformation with fourth order unit tensors

IIvol 1 I I 3 (4.3.23) dev sym vol sym 1 II II II II 3 I I thus

IE IEtan 3κ IIvol 2 µ IIdev 3 κ 2µ IIvol 2 µ IIsym 3 (4.3.24) 3 λ IIvol 2 µ IIsym λI I 2 µ IIsym bulk modulus and shear modulus 2 κ λ µ and µ (4.3.25) 3

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4.3 Isotropic hyperelasticity comparison of coordinates E1111 and E1122 4 2 4 κ µ λ µ µ λ 2 µ 3 3 3 2 2 2 (4.3.26) κ µ λ µ µ λ 3 3 3 restrictions to elastic constants from positive stored energy, i.e. positive deﬁnite elastic stiffness 2 ! : IE : ! 0 ! 0 κ, µ 0, λ µ (4.3.27) 3

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4 Constitutive equations

4.3.2 Speciﬁc complementary energy fourth order elasticity tensor

IE 2 µIIsym λI I σ IE : ! (4.3.28) inversion by making use of Sherman-Morrison-Woodbury theorem

/A IB αC D (4.3.29) inverse of rank 1 modiﬁed fourth order tensor IB 1 : C D : IB 1 /A 1 IB 1 α (4.3.30) 1 α D : IB 1 : C with /A IE, /A 1 CI , IB 2 µIIsym, α λ, C I and D I we obtain the fourth order compliance tensor

1 λ CI IIsym I I ! CI : σ (4.3.31) 2 µ 2 µ 2 µ 3 λ or rather 1 λ CI γI I IIsym with γ (4.3.32) 2 µ 2µ 2µ 3λ check with IIsym : IIsym IIsym, IIsym : I I I I and I I : I I 3 I I 1 λ IE : IE 1 2 µIIsym λI I : IIsym I I 2 µ 2 µ 2 µ 3 λ λ 2µλ 3 λ2 IIsym I I IIsym 2µ 2µ 2µ 3λ 2µ 2µ 3λ (4.3.33)

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4.3 Isotropic hyperelasticity

Voigt representation of compliance tensor

1 γ 2µ γ γ 0 0 0 1 γ γ 2µ γ 0 0 0 γ γ γ 1 0 0 0 CI 2µ (4.3.34) 1 0 0 0 4µ 0 0 1 0 0 0 0 4µ 0 1 0 0 0 0 0 4µ strain tensor (linear in stresses)

1 ! DσW CI : σ γ σ : I I σ (4.3.35) 2µ matrix representation of coordinates

1 1 1 γIσ 2µσ11 2µσ12 2µσ13 ) 1 1 1 (4.3.36) i j 2µσ21 γIσ 2µσ22 2µσ23 1 1 1 2µσ31 2µσ32 γIσ 2µσ33 speciﬁc complementary energy (quadratic in stresses)

1 1 1 W σ :CI : σ γ σ : I 2 σ 2 : I (4.3.37) 2 2 4µ

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4 Constitutive equations

volumetric - deviatoric decomposition fourth order elasticity tensor

IE 2 µIIdev 3κIIvol σ IE : ! (4.3.38) inversion by making use of orthogonality of IIvol and IIdev yields fourth order compliance tensor

1 1 CI IIdev IIvol ! CI : σ (4.3.39) 2 µ 3κ check with IIdev : IIdev IIdev, IIvol : IIvol IIvol and IIdev : IIvol I0

1 1 IE : IE 1 2 µIIdev 3κ IIvol : IIdev IIvol 2 µ 3κ (4.3.40) IIdev IIvol II

Voigt representation of compliance

1 2 1 1 1 1 9κ 6µ 9κ 6µ 9κ 6µ 0 0 0 1 1 1 2 1 1 9κ 6µ 9κ 6µ 9κ 6µ 0 0 0 1 1 1 1 1 2 0 0 0 CI 9κ 6µ 9κ 6µ 9κ 6µ (4.3.41) 1 0 0 0 4µ 0 0 1 0 0 0 0 4µ 0 1 0 0 0 0 0 4µ

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4.3 Isotropic hyperelasticity comparison of coordinates C1111 and C1122 1 2 1 2 1 1 3 9κ 6µ 9λ 6µ 6µ 3 3λ 2µ 3 2µ 3 2µ 2µ 3λ 2µ 1 3 2µ 3λ 2µ 2µ λ 1 1 γ 2µ 2µ 3λ 2µ 2µ 1 1 1 1 1 1 9κ 6µ 9λ 6µ 6µ 3 3λ 2µ 3 2µ 2µ 3λ 2µ λ γ 3 2µ 3λ 2µ 2µ 2µ 3λ strain tensor (linear in stresses) (4.3.42)

1 ! DσW CI : σ κ σ : I I σ dev (4.3.43) 4µ matrix representation of coordinates dev dev dev κI! 2 µ)11 2 µ)12 2 µ)13 ) dev dev dev i j 2 µ)21 κI! 2 µ)22 2 µ)23 dev dev dev 2 µ)31 2 µ)32 κI! 2 µ)33 (4.3.44) speciﬁc stored energy (quadratic in volumetric and deviatoric stresses)

W W vol σ vol W dev σ dev (4.3.45) 1 vol2 dev2 2 κ ) µ ! : I

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4 Constitutive equations

4.3.3 Elastic constants isotropic linear elasticity can be characterized by only two elastic constants

E, ν E, µ λ, µ κ, µ µ 3λ 2µ 9κµ E E E λ µ 3κ µ E 2µ λ 3κ 2µ ν ν 2µ 2 λ µ 6κ 2µ E µ µ µ µ 2 1 ν Eν µ E 2µ 2 λ λ κ µ 1 ν 1 2ν 3µ E 3 E Eµ 2 κ λ µ κ 3 1 2ν 3 3µ E 3

Table 4.1: relations betweeb elastic constants

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4.4 Transversely isotropic hyperelasticity

ﬁber direction n and structural tensor N

N n n with n 1 (4.4.1) speciﬁc stored energy: isotropic tensor function with two arguments

W W !, n W !, n W !, N (4.4.2) W Q ! Qt, Q N Qt Q SO 3 representation theorem for isotropic tensor functions with two arguments

W W !, N W i!, iN, i!N, (4.4.3) irreducible set of ten invariants, integrity basis

2 3 i¯! ! : I,! : I,! : I 2 3 iN N : I, N : I, N : I (4.4.4) 2 2 2 2 i!,N ! : N,! : N ,! : N,! : N recall different representation of set of three invariants

2 3 i¯! ! : I,! : I,! : I basic invariants 1 2 2 i! tr ! , 2 tr ! tr ! , det ! principal invariants

i! λ!1 λ!2 λ!3 , λ!2λ!3 λ!3λ!1 λ!1λ!3 , λ!1λ!2λ!3 eigenvalue representation of principal invariants (4.4.5) until now: principal invariants i!, now: basic invariants i¯!

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4 Constitutive equations with properties of structural tensor, idempotence & normal- ization

N2 N N n n n n n n N (4.4.6) N : I n n n 2 1 reduced representation with ﬁve invariants

W W !, N W I¯!, I¯I!, I¯II!, I¯V!, V¯! (4.4.7) representation theorem for isotropic tensor functions with two arguments

σ !, N f1 I f2! f3! ! f4N f5 ! N N ! (4.4.8) stress for transversaly isotropic hyperelastic material

DW DI¯! DW DI¯I! DW DII¯I! σ D!W DI¯! D! DI¯I! D! DII¯I! D! DW DIV¯ ! DW DV¯! DI¯V! D! DV¯! D! (4.4.9) with derivatives of invariants I¯!, I¯I!, I¯II!, IV¯ !,V¯! with respect to second order tensor !

I¯! ! : I linear D!I¯! I 2 I¯I! ! : I quadr. D!I¯I! 2! 3 2 II¯I! ! : I cubic D!I¯II! 3! (4.4.10)

IV¯ ! ! : N linear D!I¯V! N 2 V¯! ! : N quadr. D!V¯! ! N N !

2 with V¯! ! : N ! ! : N ! : ! N ! : N !

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4.4 Transversely isotropic hyperelasticity alternatively 2 2 I¯V! ! : N n ! n and V¯! ! : N n ! n general represenation of stress

2 σ D!W D ¯ W I 2 D ¯ W ! 3 D ¯ W ! I! II! III! (4.4.11) DI¯V!W N DV¯!W ! N N ! comparison of coefﬁcients assumption of linearity (quadratic term vanishes), six material parameters f11, f14, f2, f41, f44, f5 ¯ ¯ f1 DI¯!W f11 I! f14 IV!

f2 2 DI¯I!W const.

f3 3 DI¯II!W 0 (4.4.12) ¯ ¯ f4 DI¯V!W f41 I! f44 IV!

f5 DV¯!W const. speciﬁc stored energy (quadratic in strains)

1 1 2 1 1 W σ : ! f11 I¯! f14 I¯V! I¯! f2 I¯I! 2 2 2 2 (4.4.13) 1 1 2 2 f I¯! IV¯ ! f I¯V f V¯ 2 41 2 44 ! 5 ! stress tensor (linear in strains)

σ D!W f11I¯! f14 I¯V! I f2 ! (4.4.14) f41I¯! f44 I¯V! N f5 ! N N !

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4 Constitutive equations linear elastic continuum tangent stiffness (constant in strains) IEtan D!σ

tan sym IE f11 I I f14 I N f41N I f44N N f2II f5/A (4.4.15) whereby /A 2 D! ! N N ! due to symmetry reduction to ﬁve material parameters f14 f41

tan sym IE f11 I I f14 I N N I f44N N f2II f5/A (4.4.16) linear elastic continuum secant stiffness

sym IE f11 I I f14 I N N I f44N N f2II f5/A (4.4.17)

Physical interpretation of parameters interpretation of f11, f14 f41, f44, f2 and f5 Spencer [1984]

1 2 1 2 2 W λ I¯ α I¯! IV¯ ! β I¯V µ I¯I! 2 µ µ V¯ 2 ! 2 ! ! (4.4.18)

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4.4 Transversely isotropic hyperelasticity stress tensor (linear in strains)

σ D!W λ I¯! α I¯V! I 2 µ !

α I¯! β I¯V! N 2 µ µ ! N N ! (4.4.19)

Voigt representation of stiffness tensor for n 1, 0, 0 t, i.e. transverse isotropy with respect to the e1 axis

λ 2α 4µ λ α λ α 0 0 0 2µ β λ α λ 2µ λ 0 0 0 IE λ α λ λ 2µ 0 0 0 0 0 0 µ 0 0 0 0 0 0 µ 0 0 0 0 0 0 µ (4.4.20) comparison of coefﬁcients

f11 λ f2 2 µ f14 α f41 f44 β f5 2 µ µ (4.4.21) with shear moduli µ for shear in ﬁber direction and µ for shear parallel to the ﬁber direction

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