MECHANICS OF POLYMERS

Transcription

1 MECHANICS OF POLYMERS by Stefan Kolling Technische Universität Darmstadt, WS 3/4 contact:

2 Course Contents and Goals Topics Basics on continuum mechanics (no chemistry!) Classification of polymers Material modeling Damage and failure Composites (glass fiber reinforces plastics) Goals General aspects of structural analysis with focus on numerical simulation Background and fundamentals Experimental requirements and input data for numerical simulation Limits of the chosen formulations

3 . Introduction

4 stress Polymer Structures Plastics is a group name comprising many different materials Mechanical response at room temperature may be glassy or rubbery ε thermoset plastic (duroplast) thermoplastic elastomer strain glasslike behaviour plastic or viscous flow 3 low ductility 4 high ductility 5 rubbery crystalline thermoplastic amorphous thermoplastic 4

5 Glass Transition and Temperature Strain Rate Relation Example: polyethylene G glass rubber Tc Tg T Arrhenius law A e KT Ref log C T T Roughly: for certain thermoplastics, C decrease of temperature corresponds to an increase of one order of magnitude in strain rate, so the rate effects have a higher relative importance than in the case of metals Ref 5

6 Temperature Glass transition of PVB (Saflex): Room temperature Source: Hooper, Blackman, Dear The mechanical behaviour of poly(vinyl butyral) at different strain magnitudes and strain rates, J Mater Sci () 6

7 Temperature 7

8 Consistency Glass transistion and melting temperature thermoplastics elastomers amorphous = glassy T g [ C] PS 5 PMMA PVC 8 PC 5 PET 85 semicrystalline T m [ C] PET 65 PBT 5 PA6 PA66 65 PE -35 PP 65 T g [ C] BR -9 SBR -5 PEA -5 PBA -6 8

9 Usage of Polymers Overview PE PS PS PVC PP 9

10 Usage of Polymers In Automotive Industry,5% 7,% 4,7% 4,4%,%,3% Stahl steel und Eisen Polymere polymers Leichtmetalle Al / Mg Betriebsstoffe Percentage of polymeric materials in a middle class car (3) Sonstige Werkstoffe 7,% 63,8% Buntmetalle other materials Pozeßpolymere Elektrik/Elektronik

11 Usage of Polymers Construction PMMA PE-PTFE

12 Summary Polymers make life easier, cheaper and more comfortable They have a wide range of application in engineering Basic knowhow of polymers is just a must for an analysis engineer (at least for a good one) For structural parts made from polymers, computational methods are still topic of ongoing research and development projects In what follows we care for analysis methods in the context of modern mechanics, i,e. with focus on numerical simulations

13 . Basics on Continuum Mechanics

14 Basics on Continuum Mechanics - Contents Introduction Measure of Stress Beam Definition Plane Stress Principal Stress True and Engineering Stress Measure of Strain Definition Plane Strain Principal Strain True and Engineering Strain Strain Rate 4

15 Introduction Newton s nd Law The alteration of motion is ever proportional to the motive force impressed, and is made in the direction of the right line in which that force is impressed. Sir Isaac Newton (643 77) Consequences hereof: the force depends on direction and is thus a vector quantity there is no motive force in the case of statics, i.e. we have neither translation nor rotation of the system! This is the fundamental law of mechanics and contains the conservation of momentum ( st law) and action-reaction (3 rd law) as special cases 5

16 Introduction Newton s nd Law (modern interpretation) The change of momentum of a body is proportional to the impulse impressed on the body, and happens along the straight line on which that impulse is impressed dp force = rate of momentum: F dt d dt ( mv) Sir Isaac Newton (643 77) Case of statics: v which leads to F In the case of statics vanishes the resulting force acting on a body (equilibrium of forces). 6

17 Introduction And for rotation of a body: moment = rate of angular momentum dl d M r F ( r mv) dt dt Sir Isaac Newton (643 77) Case of statics: v which leads to M In the case of statics vanishes the resulting moment acting on a body (equilibrium of moments). 7

18 Introduction Cartesian coordinates x y z x y z x right-hand system, i.e. here: ex ey ez 8

19 Introduction Definition of stress beam under uniaxial tension DA DN s lim DA DN DA dn da A N Stress is defined as the local force DN over local area DA Usually, the unit of stress is N/mm² = MPa Under uniaxial stress s we may thus compute the normal force N by dn sda N s da A 9

20 True and Engineering Stress Consider a bar with constant cross section along the entire length l l initial cross section A F F actual cross section A Engineering stress True stress (Cauchy stress) Relationship s s s for incompressible materials, i.e. A l =Al, else F A s F A, () necking, i.e. s s ( ),

21 True and Engineering Strain Engineering strain l l Dl l l l Dl F x F l Another definition of strain: true strain (Hencky s strain) Heinrich Hencky (885-95) An infinitesimal change of elongation dl related to the actual length l defines the true strain increment d d dl l l dl l l ln l ln l ln l ln l l l l ln l l * Hencky studied at the TH Darmstadt where he also received his PhD

22 True and Engineering Values True Strain vs. Engineering Strain Relationship between true and engineering strain Dl l ln l l l l Dl Dl l l l, l small strain region l ln

23 relative volume [-] Increase of Volume during Tensile Tests Relative Volume for =const. : V V exp Example: Terblend N NM9 (ABS/PA blend by BASF) volume [mm³] relative volume V(t)/V total volume V(t) st principle strain Courtesy of FAT AK7, experiments performed at EMI, Freiburg 3

24 True and Engineering Values Engineering values are always related to the initial geometry True values are always related to the actual geometry The difference vanishes for small strain problems (linear calculations) Nonlinear FE analysis (e.g. crash) is based on true values and gives thus true values in the output data! All material data have to be converted to true data for input in such FE-computations (see example in section mechanics of materials ) 4

25 Stress as a Tensor Quantity Consider now a beam under angular cross section and compute Cauchy s stress F F F F Fcos n F Fsin t F F s n Stress components depend on (the normal of the) cross section! Later on we will therefore define stress by a second rank tensor 5

26 3D Stress State: Transformation Transformation in y-z-plane y P z,,,, P x y z P x ycos zsin ysin zcos Transformation matrix Φ is valid for all vectors in R 3 For transformation of matrices, e.g. A R 3x3 : Φ T AΦ x x cos sin y sin cos z rotary matrix in y-z-plane Φ It seems that stresses are transformed rather than matrices than vectors! 6

27 Stress Tensor Definition of stress Stress vector DF t n x z y DA t lim DA DF DA df da The stress vector is defined as the local force over local area Stress components depend on the normal of the cross section! This mapping is given by the nd rank stress tensor s: t σn that is sometimes called Cauchy s relation 7

28 Stress Tensor Three-dimensional stress state (Cauchy stress) In general, we have the following stress components z x y dz dx zx xz s z zy yz yx xy s y σ s x xy xz s yx y yz zx zy s z Augustin Louis Cauchy s x dy Balance of angular momentum results in the symmetry of the stress tensor, i.e. 6 components are independently: σ s x xy xz T σ xy s y yz xz yz s z Alternatively: σ s 3 T σ s 3 s

29 Stress Tensor Balance of angular momentum dy dz M x ' yz dx dz zy dx dy yz dx dy dz zy dx dy dz yz zy z s y yz zy s z zy x s z yz s y y dx dy M z ' xy dy dz yx dx dz xy dx dy dz yx dx dy dz xy yx x s x xz zx zx s z y s z z xz s x dz dx M y ' zx dx dy xz dz dy zx dx dy dz xz dx dy dz zx xz s y yx x s x xy z s x xy yx s y y 9

30 Stress Tensor Computation of the stress vector t Stress vector DF t n x z y DA t lim DA DF DA df da Example: stress in x-z-plane s x xy xz xy n t σn xy s y yz s y xz yz s z yz 3

31 Stress Tensor Transformation (z-rotation) T Φ σφ y s y s yx xy s s x x s sx sy sx sy cos( ) xy sin( ) s sx sy sx sy cos( ) xy sin( ) sx sy sin( ) xy cos( ) 3

32 Objectivity of the Stress Tensor The components of the stress tensor depend on the section under consideration (tensor property) Transformation of the coordinate system, however, may not affect the state of stress (principle of objectivity), i.e. uniaxial stress remains uniaxial stress : * y x* x y* 3

33 Stress Tensor Principal stress A unique transformation state for that no shear stress exist of the stress tensor s results in a stress We call this state principal stress and the corresponding axis principal axis s x xy xz s xy y yz xz yz s z Φ T Φ σφ s s s 3 s z s x z y zx xz xy zy yx yz s y s s x 3 s 3 33

34 Mohr s Circles for a three-dimensional state of stresses Stress tensor in principal axes σ diag s, s, s 3 Principal shear stresses s s 3 Centers of the three Mohr s circles s s s m m m3 s s3 ss3 ss s3s 3 ss s s s 3 3 s m s m s m3 s Christian Otto Mohr

35 Mohr s Circles for a three-dimensional state of stresses Uniaxial stress σ diag s,, s, 3 s s m, s m s m s m3 s m s s Biaxial stress s σ diag ss,,, 3 s s m s m s m, s m3 s s m s s Christian Otto Mohr Hydrostatic stress σ diag sss,, 3 s m s m s m3 s 35

36 Equilibrium Conditions Method of section: Consider a closed Volume V (boundary V) of a deformed body B loaded by external forces and body forces F i f i B da n V Equilibrium is fulfilled if Applying Gauß divergence theorem leads finally to B t q V t da f dv, i,,3 i V i with Cauchy s relation t i σn t da f dv σn da f dv divσdv f dv i i i i i V V V V V V divσ f dv V i i Johann Carl Friedrich Gauß

37 Equilibrium Conditions The global equation divσ fi dv V is fulfilled for arbitrary volumes if divσ f s, f ji i i and as fully written out symbol equation: s x s y s y dy y y xy xy dy y xy xy dx f x x s x dy s x dx f x y dx x xy xy s y Note the time consuming derivation in engineering mechanics! 37

38 Stress Tensor Principal stress Eigenvalue problem t σn s n σ sn Characteristic equation / Cayley-Hamilton-Theorem det σs s s x xy xz s s s I s I s I xy y yz s s xz yz z where I, I, I 3 are the so-called invariants of the stress tensor s Roots of the characteristic equation leads to 3 (real) eigenvalues, the principal stresses s > s > s 3 da n t 3 3 Arthur Cayley William Rowan Hamilton

39 Stress Tensor Invariants of the stress tensor s An invariant is a property of a system which remains unchanged under some transformation I tr( σ) s x s y s z s s s3 I tr( σ ) tr( σ ) s xs y s xs z s ys z xy yz xz ss ss 3 s s 3 I 3 s x xy xz det( σ) s s s s xy y yz s xz yz z 3 39

40 Stress Tensor Volumetric-deviatoric split The average normal stress gives the pressure as Thus, the hydrostatic stress state is given by The difference between stress tensor s and hydrostatic stress state defines the deviatoric stress tensor s 3 s x s y s z s m 3 s x s y s z 3 s x s y s z p s m s x s y s z 3 3 s x s y s z xy xz s σs m xy 3 s x s y s z yz xy yz 3 s x s y s z 4

41 Stress Tensor Invariants of the deviator s An invariant is a property of a system which remains unchanged under some transformation x m y m z m J tr( s) s s s s s s 6 6 J s: s s x s y s y s z s z s x xy yz xz J 3 det( s) s s s s 3 s 3 s I, J (and J 3 ) play an important role in the theory of plasticity 4

42 Stress Tensor Equivalent stress s e The equivalent stress is a single stress value that can be compared to the admissible stress of the material The equivalent stress represents yield condition for ductile materials and failure criterion for brittle materials In common use is the equivalent stress that is related to maximum principal stress s = max s, s, s e 3 maximum shear stress octahedral shear stress oct s = = max s s, s s, s s e max 3 3 oct VonMises stress s vm 4

43 Stress Tensor VonMises stress s vm Most popular criterion for equivalent stress The vonmises stress is written in stress components as s vm s s s s s s 3 3 s s s s s s x y y z x z 6 xy yz zx s s s s s s s s s 3 x y z x y y z z x xy yz zx The vonmises stress is directly related to the second invariant of the deviatoric stress tensor and to the octahedral shear stress Richard von Mises ( ) 3 s vm 3 J s: s s vm 3 oct 43

44 Stress Tensor Illustration of vonmises stress for two components s and (e.g. beam) the vonmises stress over shear stress represents a circle vonmises stress experiments (steel) s s 3 vm s 3 For arbitrary 3D-stress state vonmises stress represents a cylinder in the principal stress space. The direction of the cylinder is given by the hydrostatic axis, i.e. s s s 3 s s 44

45 Stress Tensor Illustration of vonmises stress A cut through the s -s -plane of the cylinder gives an ellipse that represents the state of plane stress s 3 s plane stress s s s s s s s s vm 45

46 Stress Tensor Illustration of vonmises stress Invariant plane (BURZYŃSKI-plane) uniaxial tension s vm shear uniaxial compression I p, J s 3 vm 3 3 State of pure shear at p= Uniaxial tension p σ s x p tr σ s x 3 3 s vm 3p svm s x 46

47 Stress Tensor Triaxiality The relation of pressure over vonmises stress can be used as a measure of triaxiality:.577 plane strain p s vm.6.4. p s vm 3 3 biaxial tension uniaxial tension shear 47

48 Stress Tensor State of plane stress Most of structural parts made from plastic are modeled by shell elements that assume a so-called plane stress state, i.e. in principal axis s (, ) s s(, ) s s 3 y s y xy xy s x xy s x s y xy x What does this mean for the von Mises stress in the invariant plane? 48

49 Plane Stress s σ ks s ( k ) k s triaxiality vm p s ( k ) ( k) 3 ( k) k s vm 3 ( k) ks s s k s bounds: uniaxial tension p s vm ( k ) lim lim lim k k k 3 ( k) k 3 biaxial tension uniaxial tension k ( k ) lim lim k k 3 ( k) k 3 biaxial tension 49

50 Plane Stress lower bound: s vm 3 p (biaxial tension) s s k s s (, ) s (, ) 5

51 Plane Stress - Summary For shell-like structures, i.e. plane stress state, the vonmises stress is bounded by s vm 3 p uniaxial tension s vm shear uniaxial compression Triaxiality thus becomes biaxial tension biaxial compression p s vm 3 p This defines the requirements/restrictions for experimental work 5

52 Strain Tensor Reminder: D-definition of strain Lets consider a bar with initial length l under uniaxial loading F l F xu, F dx F F dx+du the local uniaxial strain is then defined as ε du dx If (and only if) the cross section of the bar and the material properties (Young s modulus) are constant along the length l, we may write xl xl du ε dx Dl du ε dx ε dx εl ε x x Dl l 5

53 Strain Tensor In a fully 3-dimesional structural part, the local strain is defined as yv, zw, xu, T ε u u u, v x y, 33 z w x y z u v xy, u w 3 xz, v w 3 yz y x z x z y And the strain tensor is given analogue to the stress tensor as ε x xy xz xy y yz xz yz z in total 6 independent strain components symmetric by definition rotation of a body does not cause any strain 53

54 Strain Tensor Principal strains ε 3 For plane stress, we may expect the following deformation modes y s x s y s x dy s x.5 yx dy.5 xx dx s x.5 yy dy s y.5 xy dx.5 s y dx s y x This gives the state of plane strain, i.e. all z-components vanish: ε x yx y yx and ε respectively 54

55 Comparison: Plane Stress / Plane Strain Plane strain ε vs. plane stress s σ s Dog bone model (fracture mechanics) y plain stress plane strain plane strain crack plain stress x In practice: plane stress = shell structures; plane strain = thick shells 3D stress state can only be computed by solids! 55

56 Generalizations Initial (reference) and current (actual) configuration We consider an undeformed body B. A material point X within this body is defined by its spatial coordinates. This state is called reference configuration. The deformed body B is characterized by its coordinates x. This state is called current or actual configuration. z x y 56

57 Generalizations Displacement Deformation gradient Transformation of a volume element dv by the Jacobian J Relation between the deformation gradient and the displacement 57

58 Generalizations Measure of strain the deformation gradient can be uniquely decomposed into a proper orthogonal rotation tensor R = R T with det R = and a symmetric and positive definite right stretch tensor U = U T and left stretch tensor V = V T respectively The squares of the right and left stretch tensor define the second rank left and right Cauchy-Green tensor With these geometrical measures we may define measures of strain 58

59 Generalizations Green-Lagrange strain Euler-Almansi strain Logarithmic strain If the displacement gradient is small enough, the difference between current configuration and reference configuration vanishes. Then, the symmetric part of the displacement gradient describes the infinitesimal strain Leonhard Euler (77-783) 59

60 Generalizations Measure of stress Stress vector in current and reference configuration The symmetric Cauchy stress tensor σ is also referred to as true stress tensor. The Piola-Kirchhoff stress tensor P is the nominal or engineering stress: Gustav Robert Kirchhoff (84-887) Since P is unsymmetric, it is instructive to introduce the so-called second Piola-Kirchhoff stress S as This quantity has no physical meaning but is just defined as a symmetric measure in the reference configuration 6

61 Material Modeling Rheological Models Classification of Materials Linear Elasticity Hyperelasticity Visco-Elasticity Plasticity and Visco-Plasticity 6

62 Rheological Basic Models Spring s s s linear (elasticity) non-linear (hyperelasticity) E E Damper (dashpot) s s s linear non-linear (not available in most FE packages) Slider s y non-linear hardening linear hardening s s y s perfect plasticity 6

63 Rheological Models These basic models in combination allow for the formulation of all kind of stress-strain relations Examples: s s s s visco-elasticity elasto-visco-plasticity s s 63

64 Classification of Materials From an engineering point of view (focus on simulation), it is instructive to subdivide materials according to their mechanical behavior into elastic materials viscous materials plastic materials Examples usually we have a combination of them Nearly all materials are elastic for small deformations. At large deformations, rubber-like materials and recoverable foams are (nonlinear) elastic Polymers are strain rate dependent to some degree. Under cyclic loading they convert mechanical energy to heat which causes material damping Plastics show permanent deformation above yield stress 64

65 Classification of Materials To characterize a material phenomenologically, we consider a uniaxial tensile/compression test with unloading F l F F l A l cross section A F t Hereby we use engineering stresses and engineering strains for a rough subdivision where A is the initial cross section and l the initial length. For the dynamic response, strain rate dependent tests are performed subsequently (pendulum test, drop tower test,...) 65

66 Linear Elastic Materials The loading path follows a straight stress-strain path The deformations remain small and the unloading path corresponds to the loading path Slope corresponds to Young s modulus E F s F This behaviour is called linear elastic E the idea can be traced back to a paper by Leonhard Euler published in 77, some 8 years before Young's publication Thomas Young (773-89) 66

67 Linear Elasticity: Elasticity Tensor The linear relation between stress s ij and strain kl is mapped by the 4 th rank elasticity tensor C ijkl : σ Cε sij Cijklkl Symmetry: Karl Hermann Amandus Schwarz (843 9) C ijkl = C jikl because of the stress tensor symmetry s ij = s ji C ijkl = C ijlk because of the strain tensor symmetry kl = lk C ijkl = C klij holds if a strain energy density W exists, such that s ij s W ij W W, Cijkl ij kl ij kl kl ij (Schwarz theorem) 67

68 Linear Elasticity: Voigt s Notation Because of the elasticity tensor symmetry we may rearrange all stress tensor components in a 6-dimensional vector (and also for the strain tensor components) The relation between these two vectors is mapped by a 6x6 matrix which contains all elastic constants: Woldemar Voigt (85-99) 68

69 Linear Elasticity In anisotropic materials, the number of constants depends on the internal structure triclinic rhombic tetragonal 69

70 Linear Elasticity In anisotropic materials, the number of constants depends on the internal structure hexagonal (transverse isotropic) cubic isotropic 7

71 Linear Elasticity Isotropic linear elastic materials can be expressed by parameters where l and m are Lamé constants; alternatively E = Young s modulus and = Poisson s ratio m m m m l l l l m l l l l m l C Gabriel Lamé (795-87) 3 E m l m l m l l m Siméon Denis Poisson (78-84) 7

72 Relation between Elastic Constants whereby 7

73 Plane Stress vs. Plane Strain: Note the Difference! Plane stress (ESZ) Plane strain (EVZ) 73

74 Non-Linear Elastic Materials The loading path follows a non-linear stress-strain curve The unloading path ideally corresponds to the loading path. F s F This behaviour is called hypoelastic or hyperelastic depending on the theoretical formulation 74

75 Non-Linear Elastic Materials Examples: Rubber-like materials where internal damping can be neglected s engine mount head impactor Hardy-disc tire 75

76 Hyperelasticity Definition In a hyperelastic material, both stress and strain energy are pathindependent functions of the current deformation t W : E dt W ( E) W ( E) t Consequently, the strain energy function W per unit undeformed volume can be used as a potential function to compute the stress by derivation: W W E C In hyperelasticity it exists a strain energy function from which the stresses can be computed by derivation with respect to the strain 76

77 Hyperelasticity The given definition of hyperelasticity is equivalently fulfilled by the first law of thermodynamics (energy balance, no dissipation): rate of internal work rate of free energy Inserting yields: W W C E 77

78 Hyperelasticity W can be expressed in terms of total strain E or principal stretch ratios λ i = l i l i respectively True stress in global reference frame: T E C F F W E σ J x F X Grad x T σ F F T True stress in principal (true stress) reference frame: J W V W E V F F σ i λ λ j k W λ i W W W i.e. σ, σ and σ 3 λ λ3 λ λ λ3 λ λ λ λ3 78

79 Hyperelasticity For an isotropic material the energy function should depend only on the strain invariants W W I, I, I, e.g. in terms of C = F T F: 3 W W W W C C CC I II III I III C C C σ J or on the principal stretches: W W l, l, l 3 σ i λ λ j k W λ i Invariants of the right Cauchy-Green tensor: I I I : C tr C l l l 3 I CC : l l l l l l V det( C) J lll 3 V 79

80 Hyperelastic Material Laws Conditions for incompressible behavior.5 G E E 3 E lim K lim Full incompressibility cannot be enforced in explicit codes To treat the rubber as an unconstrained material, a hydrostatic work term is included in the strain energy function, e.g. W K J ln J vol J ll l3 This work term is also known as Equation of State (EOS) A confined compression test allows validation of the hydrostatic penalty K lateral confinement 8

81 One-Parameter Laws: Blatz-Ko General form for polyurethane foam rubbers (96): Simplification (e.g. implemented in LS-DYNA): I I I G I I G W I I G W T G J J σ FF

82 Two-Parameter Laws: Mooney-Rivlin General form of the energy function: W A( I 3) B( I 3) C I D I 3 3 T B σ A BI DI I CI 3 J FF J C J Clearly a stress-free initial state requires: C A B D is a penalty coefficient related to hydrostatic response: F l l l D A 5 B 5 A Poisson ratio >.497 will usually work fine 8

83 Two-Parameter Laws: Mooney-Rivlin engineering stress s in uniaxial tension or compression: l s B A l l l / F l / l This allows to determine A and B by fitting a test result Linearization: A Bl s 6 gives the small strain modulus E=6(A+B) 83

84 Two-Parameter Laws: Mooney-Rivlin Uniaxial tensile test eng. stress vs. eng. strain s l s B A l 84

85 Multiple Parameter Models: Ogden s Law In Ogden s energy function deviatoric and volumetric stresses are uncoupled: m j 3 n j * * /3 i W li K J ln J, where J ll l3 and li li J /3 i j j J Principal true stresses follow from derivation n m j s i j J l K 3 * j * lk J i k 3 J j The summed part is clearly deviatoric If we assume full incompressibility, the deviatoric stresses are a function of the principal stretch ratios * * * J i id i, j, k, i id i, j, k s s l l l J K s s l l l p J l 85

86 Multiple Parameter Models: Ogden s Law for n=, = and =-, we get Mooney-Rivlin s material for the incompressible case: J l l l l l * 3, i i W j j j li li li 3 m 3 3 j m m i j j i i m m l l l3 3 l l l3 3 m m m m 3 3 I 3 l l l l l l 3 I 3 I 3 3 B I 3 A I A B 86

87 Example of a windshield interlayer (PVB) 87

88 Further Strain Energy Functions in Terms of Invariants Blatz-Ko Material W μ v v v I 3 ( J ) b v v v τ Jσ m J b Mooney-Rivlin Material W.5 ( I,II, J) A(I 3) B(II 3) C( J ) D( J b b b b 4 (A BI ) B 4 (D ( ) C ) b τ Jσ b b J J J ) Yeoh Material / Neo-Hooke Material W τ~ C 3 (I 3) C(I 3) C3(I 3) b b b C 4C ( I 3) 6C3( I b b 3) b 88

89 Further Strain Energy Functions in Terms of Invariants 89 Van der Waals Material (User Defined) 3 3 ~ 3 ln 3 I a W m l m b b II I ~ I 3) 3) /( ~ ( I l m b b b τ ' ) ( I ' ~ W W 3 ~ ~ ' I a I W W m

90 Further Strain Energy Functions in Terms of Invariants Arruda Boyce Material (MT 7) W I 3 I 9 I 3 m 7 b b b N 5N 4 59 I 8 I b 4 b 7N 67375N ~ τ m Ib N I b 35N 3 9 Ib 75N Ib 3475N 4... b 9

91 Further Strain Energy Functions in Terms of Principal Stretches j j j j nb b b b m j j j J n b C W l l l Simplified Rubber Model (material no. 8) i,,3 n nb b m j j j j C J l l τ Uniaxial Tension: n nearly incompressibility:

92 Further Strain Energy Functions in Terms of Principal Stretches 9 3 β i λ 3)) (I ln( 3) (I 3) (I ) ( i G c G e W b b b Extended Tube Material (User Defined) l l l l l l - 4-3)) ( ( ) ( G c G e J τ Uniaxial Tension:

93 Applications: Non-Homogenous Shear Simulation of complex deformation Parameter identification of material models based on additional uniaxial tensile tests satisfactory approximation up to 5 % strain Mat_8 result identical to experiment 93

94 Applications: Non-Homogenous Shear good agreement for displacements in x-direction satisfactory agreement in z-direction 94

95 Applications: Non-Homogenous Shear well reproduced displacements in x-direction misleading results in z-direction quite reasonable agreement with uniaxial experiments, but not capable to simulate combined non-homogeneous deformation 95

96 Applications - Hardy Disc System: compression test bending test Iterative validation of compression test with Ogden material Based on Ogden material parameters creation of uniaxial tests Based on this uniaxial tests, parameter identification 96

97 Applications - Hardy Disc under Compression Good approximation in x-direction Neo-Hooke material shows slight stiffening 97

98 Applications - Hardy Disc under Bending Load Well approximations in both loading cases with Extended Tube-, Yeoh- and Van der Waals-material Mooney-Rivlin / Mat8 based on uniaxial tests may lead to differences 98

99 Example of a transmission-disk Hardy-disc: Du Bois, Faßnacht & Kolling, LS-DYNA Forum, Bad Mergentheim, Timmel, Kaliske & Kolling, LS-DYNA Forum, Bamberg 4 Validation compression test deformation EuroNCAP Courtesey of Daimler AG, Sindelfingen 99

100 Non-Linear Viscous Materials The loading path follows again a non-linear stress-strain curve Now, the unloading path does not correspond to the loading path and a hysteresis loop is formed F s F We will call this behaviour visco-hyperelastic or strain rate dependent hyperelastic respectively

101 Example: PU foam, strain rate dependency (RG5) Dynamic test, loading only (no unloading) nonlinear viscosity! [EMI 999, Mills 3]

102 Example: PU-Foam Extremely high compression up to 98% Stability problems Time step size! Contact problems Sharp impactors cause high deformation gradients in foam parts Lagrangean finite elements cannot follow the corresponding deformed shapes unlimitedly

103 Linear Viscous Materials The loading path follows a slightly non-linear stress-strain curve Now, the unloading path does not correspond to the loading path and a hysteresis loop is formed F s F We will call this behaviour visco-elastic Example: thermoplastics at small strains 3

104 Linear Viscoelasticity Maxwell element (867) s s E Equilibrium Total strain Material law s s S s D S D s s s S E S s S E S E s D D James Clerk Maxwell (83 879), Scottish theoretical physicist and mathematician 4

105 Linear Viscoelasticity Maxwell element sudden loading: ε t E s s E = ε = const E.9 Exponential decay equation E s( t) s exp t s E dimensionless stress s / E In this dimensionless diagram, the response is independent on the chosen material parameters dimensionless time t E/ [-] 5

106 Linear Viscoelasticity Maxwell element sudden loading: ε t Notation: E t s ( t) s exp t s exp s expt = ε = const E wherw τ = η E =: β is the exponential time constant and is the decay rate. Thus the relaxation function yields s () t s () t E( t) exp t E exp t 6

107 Linear Viscoelasticity Maxwell Element + Spring in parallel E E / E( t) E E exp t logarithmic plotting of the relaxation function gives a better overview E +E log Et ( ) E logt Alternatively: G( t) G G exp t E 7

108 Linear Viscoelasticity Generalized Maxwell Element G N G( t) G G exp t i i i G G In 3D expressed by convolution integrals: G 3 G / G / G 3 / 3 σ ( t) s ( t) p ( t) visc visc visc s visc visc N t β i t Gie εd d i t t N i i t βki p t K e ε d The relaxation functions are represented in terms of Prony series consisting of a set of v material parameters that have to be identified (measured) Gaspard de Prony ( ) 8

109 How to Measure the Relaxation Curves? Time-temperature shift function (WLF shift) Analytical approximation by Williams, Landel and Ferry Relaxation Tests G G T 4 T 3 T T WLF t t G() t t f at with a T A( T T ) g B T T g A = 7.44K and B = 5.6K for amorphous polymers M.L. Williams, R.F. Landel, J.D. Ferry: The Temperature Dependence of Relaxation Mechanisms in Amorphous Polymers and Other Glass-forming Liquids. Journal of the American Chemical Society 77, 955, pp. 37 9

110 How to Measure the Relaxation Curves? Dynamic mechanical analysis (DMA) x( t) xˆ sin t G( T ), G( ), G glass e.g. DMA/SDTA86e by Mettler Toledo f max = Hz T= -5 C - 5 C Tg rubber T

111 Plasticity: Thermoplastics Visco-plastic Materials (Thermoplastics, Thermosets, ) Permanent (plastic) deformation in contrast to elastic materials The unloading path ideally follows a straight line (Hooke) s F All polymers are strain rate dependent to some degree, i.e. they are visco-plastic F Von Mises yield criterion is usually used in practise (OK for most of the metals not for plastics!)

112 Localization and Increase of Volume during plastic flow Strain at break depends on the chosen mean area The decrease of the stress after yield strength is a artefact of the mean area, not a material property.

113 Crazing Crazing Crazing is the notion of formation of surface cracks As a consequence thereof: change of colour to white detectable crazing leads to plastic (permanent) deformation with increase of volume crazing leads to low yield stress values in uniaxial/biaxial tension seems to occur under high values of hydrostatic tension 3

114 wahre Spannung True Stress wahre Spannung True Stress ,,,3,4,5,6, 5,, 5,, 5,3,3 5,4,4 5,5 Mechanical Behavior of Thermoplastics (Uniaxial Loading).mm/s 7mm/s 5mm/s Yield curve AND Young s Modulus are strain rate dependent Non-linear elasticity Uniform necking due to stabilisation Visco-elasto-visco-plasticity True strain wahre Dehnung True strain wahre Dehnung Tensile Zug Test Compression Test Druck Different yielding under tension/compression (and shear) Plastic incompressibility for compression only ( p.5) Under tension foam-like ( p ) No Von Mises type of plasticity! 4

115 Material Models Yield Criteria Define a yield function f that is zero on the yield surface and less than zero for elastic states: In terms of the stress tensor f fˆ( s ij ) In terms of invariants f f ˆ( I, I, I ) 3 Pressure independent yielding: von Mises s 3 f f ˆ( J ) s J where s y 3 s 5

116 Material Models Pressure dependent yielding (Thermoplastics, ) For materials that show dependency on triaxiality (most of the thermoplastics) special yield criteria have to be used s t SAMP s s tension vonmises s vm shear shear s t t biaxial tension tension compression s c s biaxial tension t 3 p compression c f s A A p A p vm 6

117 Experimental Data vs. SAMP - Polyvinyl Chloride (PVC) exp. results taken from Bardenheier 98 7

118 Experimental Data vs. SAMP - Polystyrene (PS) exp. results taken from Bardenheier 98 8

119 Experimental Data vs. SAMP - Polycarbonate (PC) exp. results taken from Bardenheier 98 9

120 Material Models Isotropic Hardening Curve f ( s ) ( p ) f(s) = corresponds the initial yield surface ( p ) is a monotonically increasing function of the hardening parameter p (usually the effective plastic strain) s s s y s s 3 p

121 force Example: Validation of a Component Test (PP-T) displacement

122 force Example: Validation of a Component Test (PP-T) Typical behaviour for thermoplastics: material cards that are fitted for uniaxial tension yield a too soft responds under bending and compression Different yield curves under compression and tension are necessary! displacement

123 Associated and Non-Associated Flow Rules Practical stability condition (Drucker s first postulate) Convex yield surface ds d Associated flow (Drucker s second postulate) ij p ij Plastic strain increments and plastic stress increments are in same direction: d f p dl s i.e. plastic flow occurs perpendicular to the yield surface Non-associated flow Plastic potential is defined for the direction of yielding d g p dl s Thus, plastic Poisson s ratio can be influenced directly! 3

124 Material Models for Polymers Summary Many robust and reliable material models for polymer materials are available (most of them in commercial FE-packages, too) The most popular models are implemented in a tabular way due to their user-friendliness (no parameter identification) Using the presented models, the deformation behavior of the structure can be reproduced pretty good Consideration of the manufacturing process (molding for plastic components) gives a further improvement of the simulation Anisotropy is a topic of recent development (integrative simulation) 4