The Finite Element Method for the Analysis of Non-Linear and Dynamic Systems Prof. Dr. Eleni Chatzi Lecture ST1-19 November, 2015 Institute of Structural Engineering Method of Finite Elements II 1
Constitutive Relations Overview so far Material Nonlinearity Large Displacements Dynamic Analysis Special Topics Material Laws The Contact problem Fracture & Special Formulations (XFEM, SBFEM) Institute of Structural Engineering Method of Finite Elements II 2
Constitutive Relations Overview so far Material Nonlinearity Large Displacements Dynamic Analysis Special Topics Material Laws The Contact problem Fracture & Special Formulations (XFEM, SBFEM) Institute of Structural Engineering Method of Finite Elements II 2
Constitutive Relations Previously we examined the kinematic equations formulation (displacement, strain displacement relations) The next step is to determine appropriate constitutive relationships of the form: σ = f(ε) ex. linear analysis in 1D σ = Eε When dealing with higher dimensions & incremental analysis, this is written in tensor form for time t: t σ = t C ijrst ɛ rs Institute of Structural Engineering Method of Finite Elements II 3
Constitutive Relations It is necessary that kinematic and constitutive relations are appropriate. Previously we saw that in the large displacement formulation appropriate tensors need to be defined. e.g. TL Formulation Second Piola-Kirchhoff stress tensor, Green Lagrange strain tensor). Therefore a problem involving large strains should also be combined with a material law that admits large strains. However, we might be examining a problem of large displacements with small strains. In that case we can still use the material laws defined for classic engineering stress and strain measures (for small displacements) but this time combined with the SP-K stress and G-L strain tensors. Institute of Structural Engineering Method of Finite Elements II 4
Notation Main Stress - Strain pairs: Material Nonlinearity (small deformations) Engineering (or Nominal) Stress σ Engineering Strain ε TL formulation (large deformations) 2nd Piola-Kirchhoff Stress S Green-Lagrange Strain ɛ UL formulation (large deformations) Cauchy (or True) Stress τ Almansi Strain ɛ A Note that: τ = L L 0 σ, ε = L L 0 L 0 Institute of Structural Engineering Method of Finite Elements II 5
Solution Flowchart General Solution process in incremental nonlinear FE Known Solution at t: Stresses t σ, strains t ε, Internal material parameters t κ Known Quantities at iterations i- 1 : Nodal Displacements at first Iteration: and hence Element strains t+δt ε i 1 t+δt U i 1 Calculate at t+ Δt: t+δt Stresses σ i 1 Repat till Convergence Tangent stress strain matrix C i 1 Internal material parameters t+δt κ i 1 Elastic Analysis: directly obtain t+δt t+δt σ i 1, C i 1 from ε i 1 Inelastic Analysis: Integrate to get t+δt σ i 1 t t+δti 1 = σ + dσ t Calculate: Incremental Displacement Vector ΔU i: t+δt K i 1 ΔU i = Then, t+δt R t+δt F i 1 t+δt t+δt U i = U i 1 + ΔU i Institute of Structural Engineering Method of Finite Elements II 6
Overview of Material Descriptions We can discriminate amongst the following major classes of material behavior Elastic, linear or nonlinear Hyperelastic Hypoelastic Elastoplastic Creep Viscoplastic Institute of Structural Engineering Method of Finite Elements II 7
Elastic Material For an elastic material the stress is a function of strain only. The stress path is the same both in loading and unloading Linear Elastic The elasticity (constitutive) tensor components, C ijrs are constant Nonlinear Elastic The elasticity (constitutive) tensor components, C ijrs are a function of strain Example: Almost all materials under small stress σ ε Institute of Structural Engineering Method of Finite Elements II 8
Elastic Material Cauchy Elastic Models Institute of Structural Engineering Method of Finite Elements II 9 (a) (b) igure 1.1: Uniaxial Response of an Elastic Material: (a) Linear, (b) Nonlinear
Elastic Material For the case of an elastic material we already saw that the TL Formulation (used for large deformation analysis) yields: t 0S ij = t t 0C ijrs 0ɛ rs The elasticity tensor for 3D stress conditions is defined as: t C ijrs = λδ ij δrs + µ(δ ir δjs + δ is δjr) where λ and µ are the Lamé constants and δ ij is the Kronecker delta, Eν λ = (1 + ν)(1 2ν), µ = E (homogeneous isotropic material) 2(1 + ν) { 0 i j δ ij = 1 i = j Institute of Structural Engineering Method of Finite Elements II 10
Elastic Material Important Note The 2nd Piola-Kirchhoff (PK2) stress and Green-Lagrange strain tensor components are invariant to rigid body motions. For problems with small strains we can take advantage of this observation and use any constitutive relationship that has been developed for engineering stress and strain measures by just substituting with the PK2 stress and Green-Lagrange strain This observation can be extended to all problems with large deformations but small strain conditions such as the elastic or elastoplastic buckling problem and the collapse analysis of slender structures. Institute of Structural Engineering Method of Finite Elements II 11
Hyperelastic Material Hyperelastic (rubberlike) materials exhibit an incompressible response (volume preserving), path independence and no energy dissipation. The stress is now calculated through the strain energy functional W t 0S ij = W t 0ɛ ij Figure: Stress-strain curves for various hyperelastic material models. Institute of Structural Engineering Method of Finite Elements II 12
Hyperelastic Material Hyperelastic Material Models Saint Venant-Kirchhoff model W (ɛ) = λ 2 [tr(ɛ)]2 + µtr(ɛ 2 ) and the second Piola-Kirchhoff stress can be derived as S = λ[tr(ɛ)]i + 2µɛ λ, µ are the Lamé constants Mooney-Rivlin model W (ɛ) = C 1 (I 1 3) + C 2 (I 2 3) where C1 and C2 are empirically determined material constants and I 1 = tr(c) = C 11 + C 22 + C 33 where C is the Cauchy-Green deformation tensor (see Lecture 4) and I 2 = 1 2 [(I 1) 2 tr(c) 2 ] Institute of Structural Engineering Method of Finite Elements II 13
Inelasticity Elastoplasticity, Creep and Viscoplasticity are types of Inelastic behavior Elastic behavior stresses can be directly calculated from the strain Inelastic behavior the stress at time t depends on the stress strain history In the incremental analysis of inelastic response we had three main scenarios Small displacements-rotations / small strains use linear elastic solution, engineering stress and strain measures Large displacements-rotations / small strains use TL formulation by substituting the appropriate stress - strain measures (PK2, Green-Lagrange) in the place of the engineering stress and strain measures Large displacements-rotations / large strains use either TL or UL formulation, more complex constitutive laws Institute of Structural Engineering Method of Finite Elements II 14
Elastoplasticity In this formulation we encounter a linearly elastic behavior until yield and usually a hardening post yield behavior Examples: Metals, soild and Rocks when subjected to high stresses. Institute of Structural Engineering Method of Finite Elements II 15
Elastoplasticity The strain and stress increments are given by: dɛ rs = dɛ E rs + dɛ P rs dσ ij = C E ijrs (dɛ rs dɛ P rs) where C E ijrs are the components of the elastic constitutive tensor and dɛ rs, dɛ E rs, dɛ P rs are the components of the total strain increment. To calculate the plastic strains we use the following three properties: Yield Function f y(σ, ɛ P ) f y < 0 Elastic behavior f y >= 0 Plastic or elastic behavior depending on the loading condition Flow rule The yield function is used in the flow rule in order to obtain the plastic strain increments λ is a scalar to be determined dɛ P ij = λ fy σ ij Hardening rule This specifies how the yield function is modified during the progression of loading. Institute of Structural Engineering Method of Finite Elements II 16
Elastoplasticity Example: Von Mises yield criterion (in 3D): f y = 0 (σ 11 σ 22) 2 + (σ 22 σ 33) 2 + (σ 11 σ 33) 2 + 6(σ 2 12 + σ 2 23 + σ 2 31) 2σ 2 y = 0 Institute of Structural Engineering Method of Finite Elements II 17
Elastoplasticity Isotropic & Kinematic hardening Rules In the case of isotropic hardening, the yield surface expands uniformly. In the case of kinematic hardening, the size of the yield surface remains unchanged and the center location of the yield surface is shifted. (Bauschinger effect) Institute of Structural Engineering Method of Finite Elements II 18
Elastoplasticity s TICITY. Flow rule for kinematic hardening C Fig. 1 Cyclic loading ly, for a reversed loading process like the one in the cyclic loading diagram of Fig. 1, the hardening will lead to a cyclic test behaviour according to the solid line OABCDE of Fig. 2 h the length of line segment BC is the same as that of line segment AB). It is, however, a well- Cyclic Loading A E Hardening A,E s O B D t D O B,D s ig. 1 Cyclic loading C C Fig. 2 diagram of uniaxial cyclic test. Isotropic hardening Kinematic hardening established fact that in most materials there is a Bauschinger effect, by which C Institute of Structural Engineering Method of Finite Elements II 19
Thermoelastoplasticity and Creep This behavior exhibits time effect of increasing strains under constant loads or decreasing stress under constant deformations (relaxation) Typical examples of such behavior are metals at high temperatures The thermal strain(ɛ = α T ) and the creep strain now enter the formulation of the stress strain relationships. Institute of Structural Engineering Method of Finite Elements II 20
Viscoplasticity Viscoplasticity describes the rate-dependent inelastic behavior of solids. Rate-dependence in this context means that the deformation of the material depends on the rate at which loads are applied. The inelastic behavior that is the subject of viscoplasticity is plastic deformation which means that the material undergoes unrecoverable deformations when a load level is reached. Rate-dependent plasticity is important for transient plasticity calculations. The main difference between rate-independent plastic and viscoplastic material models is that the latter exhibit not only permanent deformations after the application of loads but continue to undergo a creep flow as a function of time under the influence of the applied load. Typical examples of such behavior are Polymers and Metals Institute of Structural Engineering Method of Finite Elements II 21
Viscoplasticity Institute of Structural Engineering Method of Finite Elements II 22