Lecture 16: Design of paper and board packaging Advanced concepts: FEM, Fracture Mechanics After lecture 16 you should be able to describe the finite element method and its use for paper- based industry illustrate how a finite element model is created discuss the results from finite element analysis of: Compression test of a package Creasing and folding discuss concepts such as the J-integral and the fracture toughness account for the procedures of failure predictions in paper materials 1
Literature Pulp and Paper Chemistry and Technology - Volume 4, Paper Products Physics and Technology, Chapters 2 & 12 Östlund, S. and Mäkelä, P., Fracture properties, KTH, 211 Three-dimensional modelling and analysis of paper and board, FEM Adapted from Mikael Nygårds 2
Why use numerical methods? Different types of loadings can be investigated. The effect of different material properties can be investigated. More information about damage and deformation mechanisms can be gained. Material behaviour and structural behaviour can be predicted Properties that t are important t for the manufacturer can be linked to properties that are important for the user. Saves costs and enables more efficient product development FEM Process Solutions Material Performance Material performance Establish the important/critical material properties p Converting Packaging Design Delamination 3
The Finite Element-Method (FEM) is a method to solve partial differential equations procedure to solve field problems with engineering accuracy In field problems the parameters of interest are described by continuous or discrete field variables, e.g. Continuum mechanics Solid mechanics stress, strain Moisture transport Fluid mechanics Heat transfer, temperature Electro magnetic field theory... Finite element method Assume a deformation (displacement field) N u Calculate displacements N u = N N u Calculate strain N ε = β N u Calculate stress (in general non-linear relations) σ = F ( ) ε Principle of virtual work V σ : εdv = S t T uds + V f T udv 4
Way of working 1. Understand the problem (boundary conditions, symmetries, material models, approximations) 2. Modelling, discretisation, input data, selection of elements, Verification (PRE-PROCESSING) 3. Solving of systems of equations and calculation of relevant properties e.g. stress and strain 4. Analysis of output, graphical presentation (POST- PROCESSING) 5. Consequences of analysis Modelling, discretisation, input data, selection of elements, Verification (PRE-PROCESSING) Geometry (discretisation) Boundary conditions Material model σ = f(ε) 5
Challenges within paperboard research Constitutive models for paper that involve the through thickness direction, as well as moisture and temperature dependence is still active research Incomplete knowledge of material data for many materials Simulation tools for many applications are still missing Lack of robustness and convergence problems Compression of boxes Experiments 6
Compression of cylinders Experiments What is the difference compared to rectangular boxes? FE-analysis of compression of box USE OF PRO- DUCT MATERIAL PROPERTIES ANALYSIS LOADS BOX- MODEL (FEM) LOADS ON DETAILS DETAIL MODEL (FEM) CRITICAL LOADS 7
Static compressive load on box FEM and experiment Reference: L. Beldie, Lund University, 21 Top and bottom segments Reference: L. Beldie, Lund University, 2 8
Middle segment Reference: L. Beldie, Lund University, 2 Experimental characterization of crease Reference: L. Beldie, Lund University, 21 9
Static compressive load on box FEM with crease elements and experiment Test of whole package Reference: L. Beldie, Lund University, 21 Development of a three dimensional paperboard model Real process/object Experimental verification Laws of mechanics Model formulation Solution of mathematical problems Numerical method Mathematical model 1
Mechanical behaviour of paperboard Top layer Middle layer/s Bottom layer In-plane behaviour CD 4 MD MD Stress [MPa] 3 2 1 CD -1 2 4-1 Anisotropic elasticity The elastic modulus in MD are 2-3 times larger than the elastic modulus in CD Anisotropic initial yield stress The initial yield stress in MD are 2-3 times greater then the initial yield stress in CD Anisotropic plastic strain hardening Strain [%] Paper hardens more in MD than in the CD Reference: Q. Xia, MIT, Boston, 22. 11
Out-of-plane behaviour Nor rmal Stress [MPa].4.3.2.1 1 She ear Stress [MPa] 1.2.9.6. 3.5 1. 1.5.5 1. 1.5 2. Normal displacement [mm] Shear displacement [mm] Reference: N. Stenberg, STFI/KTH, Stockholm, 2. Out-of-plane behaviour Summary Small amount of non-linearity before pre-peakpeak load Dominating softening behavior after the peak load Shear strength is pressure dependent Shearing causes normal dilatation Residual shear-load remains under normal compressive loading. Reference: N. Stenberg, STFI, Stockholm, 2. Q. Xia et. al., MIT, Boston, 22. 12
Model z Constitutive model: damaged region x T x T z T z δ z δ x z T x Continuum model Stresses: σ x, σy, σz, τxy, τzx, τyz Strains: ε x, εy, εz, γxy, γzx, γ yz interface y x Two dimensional view of tractions and displacements at interface Interface model Tractions: Tx, Ty, Tz Displacements: δ, δ, δ x y z In-plane: elastic-plastic continuum Anisotropic elasticity Anisotropic initial yield Anisotropic plastic strain hardening Out-of-plane: interfacial model Post-peak softening tensile and shear behavior. Pressure dependent shear resistance Normal dilation under shearing Existence of shear friction History dependent Reference: Q. Xia et. al., MIT, Boston, 22. Verification Uniaxial tension F F References: N. Stenberg, STFI, Stockholm, 22 Q. Xia, MIT, Boston, 22. 13
Verification Biaxial, compression Out-of-plane compression In-plane compression Reference: N. Stenberg, STFI, Stockholm, 22 Q. Xia, MIT, Boston, 22. Delamination model Failure surface Reference: N. Stenberg, STFI, Stockholm, 22 14
Delamination model ZD tension Reference: N. Stenberg, STFI, Stockholm, 22 Q. Xia, MIT, Boston, 22. Delamination model Shear Thickness increase under shear Reference: N. Stenberg, STFI, Stockholm, 22 Q. Xia, MIT, Boston, 22. 15
Creasing and folding Model setup Top ply Middle ply Bottom ply Creasing of Paperboard Reference: H. Dunn, MIT, 2. Reference: STFI-Packforsk/Tetra Pak 24 16
Creasing of Paperboard Reference: H. Dunn, MIT, 2. Reference: STFI-Packforsk/Tetra Pak 24 Creasing of Paperboard Reference: H. Dunn, MIT, 2. Reference: STFI-Packforsk/Tetra Pak 24 17
Creasing of Paperboard Reference: H. Dunn, MIT, 2. Reference: STFI-Packforsk/Tetra Pak 24 Verification Creasing Measurements Reaction force, F Displacement, u 18
Verification Folding Measurements Reaction force, F Rotation ti angle, θ Centre of rotation Specimen Load cell Clamps Creasing and folding Mikael Nygårds, STFI-Packforsk 19
Creasing and folding Comparison with experiments MD folding at two different creasing depths Fracture Mechanics Adapted from Petri Mäkelä Innventia AB 2
The A4-example Load: 145 N Elongation: 5.2 mm Load: 75 N Elongation: 1.4 mm 1 mm edge crack 48 % reduction of load carrying capacity and 73 % reduction of strain at break Reduced effective width of a paper web 21
Corresponding stress distribution Reduced strength Fracture mechanics Geometry Fracture toughness Loading 22
Paperboard applications Cut-outs and cracks in corrugated board Failure of sacks Crack Perforation Nicks Crack initiation spots Web breaks K-cracks Web breaks σˆ nom σ nom σ interior crack σ σ edge crack σ R p R p t 2a σ a) b) 23
Delamination in printing nips a) b) c) Crack tip modelling FPZ appearance and modes of crack opening 3 4 mm Crack Tip Fracture Process Zone of considerable size Damage Modes I and II are predominant under in-plane loading and mode I is considered most severe. 24
Linear elastic fracture mechanics LEFM σ E ε Stress state in the crack tip region σ ij KI = KI k ij r ( ϕ ) = f(material, geometry, loading) Failureprocess zone Singularity dominated zone controlled by K in Equations (6.1) Log (σ yy ) 1 2 Log (r/r p ) Fracture criterion, LEFM Stress Tensile strength K I Critical K I Strain Strain 25
Linear elastic fracture mechanics MD 12 1.2 12 1.2 CD σ nom /σ b, ε nom /ε b 1..8.6.4.2. Stress at break, exp. Strain at break, exp. Stress at break, num. Strain at break, num. 5 1 15 2 Crack size [mm] σ nom /σ b, ε nom /ε b 1..8.6.4.2. Stress at break, exp. Strain at break, exp. Stress at break, num. Strain at break, num. 5 1 15 2 Crack size [mm] Conclusions, LEFM Does not apply to paper materials in general. Large cracks in very large structures are required in order for LEFM to be applicable to paperboard. 26
Non-linear fracture mechanics, NLFM The HRR crack tip fields σ E E, N, ε Stress state in the crack tip region 1 n+ 1 J σ ij = a fij ( n, ϕ ) r J = f(material,geometry,loading) Log (σ yy ) Failure process zone J dominated zone K dominated zone 1 n + 1 1 2 Log (r/r p ) Hutchinson 1968, Rice, Rosengren 1968 J is the energy release-rate in a non-linear elastic material In the special case of a linear elastic material, J is proportional to K 2 I This means that LEFM is a special case of NLFM. J is defined as a path-independent line- integral around the crack-tip involving expressions containing stress, strain and displacement. 27
Fracture criterion, NLFM Stress Tensile strength J Citi Critical ljj Strain Strain NLFM The J-integral method MD 12 1.2 12 1.2 CD σ nom /σ b, ε nom /ε b 1..8.6.4.2. Stress at break, exp. Strain at break, exp. Stress at break, num. Strain at break, num. 5 1 15 2 Crack size [mm] σ nom /σ b, ε nom /ε b 1..8.6.4.2. Stress at break, exp. Strain at break, exp. Stress at break, num. Strain at break, num. 5 1 15 2 Crack size [mm] 28
Conclusions, NLFM (J-integral method) Does predict failure quantitatively for large cracks and qualitatively ti l for short cracks Numerically cheap Easy calibration What information does J carry? J is a loading parameter J expresses the severity of the stresses at the crack-tip When J reaches a critical value, the crack starts to grow 29
Critical value of J (or K) is called fracture toughness 1. In order to formulate a fracture criterion, we need to know how severe stress states the material is able to withstand. 2. We need to know the critical value of J for the material, i.e. the fracture toughness (J c ) of the material. 3. The information from material testing on a test piece with a man-made crack is required to evaluate the fracture toughness. Predictions of failure Generally requires numerical methods 1. Material behaviour 3. 2. Fracture toughness Full-scale predictions of failure FE-analysis FE-analysis 3
Verification of transferability J-integral method for 1 m wide paper webs 12 6 Critical elongati ion / mm 1 8 6 4 2 Fluting Predictions Experiments 5 4 3 2 1 Critical force / kn Critical elongatio on / mm 2 15 1 5 Sackpaper Predictions Experiments 5 4 3 2 1 Critical force e / kn Critical elongation / mm 5 1 15 2 25 3 Crack length / mm 12 2.5 1 Newsprint Predictions Experiments 2. 8 1.5 6 1. 4.5 2 Critical force / kn Critical elongation / mm 14 12 1 8 6 5 1 15 2 25 3 Crack length / mm Testliner Predictions Experiments 4 1 2 4 3 2 Critical force / kn Critical elongation / mm 16 14. 1 2 3 4 Crack length / mm MWC 4 Predictions Experiments 12 3 1 8 2 6 4 1 2 5 1 15 2 25 3 Crack length / mm Crotical force / kn Critical elongation / mm 5 1 15 2 25 3 Crack length / mm 12 2. 1 SC Predictions Experiments 1.5 8 6 1. 4.5 2. 1 2 3 4 Crack length / mm Critical force / kn Stress state in vicinity of crack tip Model Real material 31
Damage behaviour Elastic unloading supports energy Damage evolution consumes energy Elastic unloading supports energy Tensile Stress / M Pa Tensile testing 6 5 4 3 2 1..5 1. 1.5 Strain / % Instability when rate of supported energy from elastic unloading exceeds consumed energy during damage evolution Tensile test results Long and short test pieces Ordinary tensile test piece Te ensile Stress [MPa] 7 6 5 4 3 2 1 Short strip Long strip Short tensile test piece 2 4 6 8 1 Apparant strain [%] 32
Modelling of damage cohesive zone w σ σ (w) σ (w) Stress s (σ ) w L+δ δ Elongation (δ ) σ σ δ δ + L E = r Damage zone Stress (σ ) σ( w) = σ e α β w y a v w (r) σ y (r) Widening (w) x Elastic-plastic material + cohesive zone MD 12 1.2 12 1.2 CD σ nom /σ b, ε nom /ε b 1..8.6.4.2. Stress at break, exp. Strain at break, exp. Stress at break, num. Strain at break, num. 5 1 15 2 Crack size [mm] σ nom /σ b, ε nom /ε b 1..8.6.4.2. Stress at break, exp. Strain at break, exp. Stress at break, num. Strain at break, num. 5 1 15 2 Crack size [mm] 66 33
Conclusions, Cohesive crack model Accurate predictions of failure for all crack sizes Numerically expensive Expensive, cumbersome and Time-consuming calibration No explicit fracture criterion needed Final remarks FPZ size important for the choice of fracture mechanics model. If FPZ is small crack tip fields are singular J-integral method predicts mode I in-plane failure of notched paper structures well. Cohesive stress models excellently predicts mode I inplane failure. Such models are also applicable for out-of-plane failure where the crack tip singularity concept is not unambiguously applicable. Fracture mechanics can be used for damage tolerance analysis of structures containing assumed defects. 34
After lecture 16 you should be able to Illustrate the finite element method and its use for paperbased industry Illustrate how a finite element model is created Discuss the results from finite element analysis of: Compression test of a package Creasing and folding Discuss concepts such as the J-integral and the fracture toughness Account for the procedures of failure predictions in paper materials 35