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**Mini-Seminar Dr. James Throne, Instructor**

Advanced Topics in Thermoforming Mini-Seminar Dr. James Throne, Instructor 8:00-8:50 - Technology of Sheet Heating 9:00-9:50 - Constitutive Equations Applied to Sheet Stretching 10:00-10:50 - Trimming as Mechanical Fracture

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**Mini-Seminar Advanced Topics in Thermoforming**

Part 2: 9:00-9:50 Constitutive Equations Applied to Sheet Stretching

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Let’s begin!

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**Mini-Seminar Advanced Topics in Thermoforming**

All materials contained herein are the intellectual property of Sherwood Technologies, Inc., copyright No material may be copied or referred to in any manner without express written consent of the copyright holder All materials, written or oral, are the opinions of Sherwood Technologies, Inc., and James L. Throne, PhD Neither Sherwood Technologies, Inc. nor James L. Throne, PhD are compensated in any way by companies cited in materials presented herein Neither Sherwood Technologies, Inc., nor James L. Throne, PhD are to be held responsible for any misuse of these materials that result in injury or damage to persons or property

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**Mini-Seminar Advanced Topics in Thermoforming**

This mini-seminar requires you to have a working engineering knowledge of heat transfer and stress-strain mechanics Don’t attend if you can’t handle theory and equations Each mini-seminar will last 50 minutes, followed by a 10-minute “bio” break Please turn off cell phones PowerPoint presentations are available at the end of the seminar for downloading to your memory stick

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Outline Fundamentals Definitions General Premise General Premise for Thermoforming Elastic Constitutive Equations Viscoelastic Constitutive Equations Forming Window Measurement Finite Element Analysis Sag

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Fundamentals - Stress - Strain - Rate-of-Strain

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Definitions Stress - Applied load per unit area. Usually given the symbol s Strain - Deformation resulting in applied load per unit area. Usually given the symbol e or l where e = l-1 Stress and strain apply primarily to elastic materials

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Definitions Rate of strain (or strain rate) - The rate of deformation owing to applied stress. Usually given the symbol Rate of strain is usually applied to materials that yield or flow under stress

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Definitions Viscoelasticity - The combination of elastic and viscous behavior. The general form for stress-strain-rate-of-strain is s=f(e, ;T) Linear viscoelasticity - The simple sum of elastic and viscous responses to applied shear. Usually shown as s = f1(e) + f2( )

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Definitions Elasto-Plastic Deformation - Material stretches elastically to a given extension, then rapidly deforms with little additional stress

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

General Premise If a material responds elastically to applied load, it recovers fully and instantaneously once the load is removed (think rubber band) If a material responds viscously to applied load, it remains completely deformed once the load is removed (think pudding) If a material recovers a little but remains somewhat deformed once the load is removed, the material is considered viscoelastic (think silly putty)

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

General Premise For Thermoforming An amorphous polymer is stretched primarily in its rubbery solid state Polyethylene is typically stretched in its elastic melt state Polypropylene is stretched either in its rubbery solid state (solid state forming) or, if it has good melt strength, in its elastic melt state

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

General Premise For Thermoforming Ergo, for most polymers and most stretching regions on a given part, the elastic character of the polymer dominates For certain polymers (PP, for example), and for certain regions on a given part for many other polymers, the viscous character of the polymer influences the local part wall thickness

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

General Premise For Thermoforming There are four general stretching modes Uniaxial stretching - Stretching only in one direction Equal biaxial stretching - Stretching to the same elongation in two directions Biaxial stretching - Stretching in two directions but not necessarily to the same elongation Plane strain stretching - defined below

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Simple Hookean elastic behavior - E is elastic modulus s = E e Power-law behavior s = E e n Simple elongational Newtonian viscous behavior - he is elongational Newtonian viscosity s = he Elongational power-law behavior- me is elongational non-Newtonian viscosity s = me ( ) n

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Elastic Constitutive Equations Stresses in terms of the strain energy function Strain energy function in terms of the principal invariants of the Cauchy strain tensor

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Elastic Constitutive Equations The principal invariants of the Cauchy strain tensor

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Elastic Constitutive Equations Stress-strain relationship in terms of Cauchy invariants For an incompressible solid, or III = 1

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Elastic Constitutive Equations For uniaxial stretching, l1=l, l2=1, l3=l-1/2 For equal biaxial stretching, l1=l2=l, l3=l-2

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Elastic Constitutive Equations The power-law form for the strain energy function The neo-Hookean solid form C10 is a constant related to the elastic modulus

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Elastic Constitutive Equations The Rivlin form (developed for rubber elasticity) The Mooney form (also for rubber elasticity) C01 and C10 are shape constants, described later

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Elastic Constitutive Equations In a recent paper by Hosseini and Berdyshev, “A Solution for Rupture of Polymeric Sheet in Plug-Assist Thermoforming,” presented at the 2006 SPE ANTEC, they propose the following constitutive equation: Where G(T) is the temperature-dependent tensile modulus W = (G(T)/4)[(I-3)+(II-3)]

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Elastic Constitutive Equations The Mooney stress-strain equation - uniaxial The Mooney stress-strain equation - equal biaxial

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Elastic Constitutive Equations The coefficients C01 and C10 are curve-fit to stress-strain curves They are also highly temperature-dependent In the limit as the constants are determined from the elastic modulus

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Elastic Constitutive Equations For the Mooney model Typically for many polymers

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Elastic Constitutive Equations If C01=0, the value for C10 is just the elastic modulus This is usually valid for low levels of deformation When C10=0, the model seems to correlate with PP creep data

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Elastic Constitutive Equations The Ogden Model an and mn are curve-fitting constants Usually m<3 yielding 2, 4, or 6 constants when m=2, a1=2 and a2=-2, the Mooney equation results

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Elastic Constitutive Equations - Plug Stretching Plane strain - No relative effect of stretching is seen from the vertical

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Elastic Constitutive Equations - Plug Stretching Plane strain - No relative effect of stretching is seen from the vertical

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Elastic Constitutive Equations - Plug Stretching Mooney-Rivlin constitutive equation for plane strain where F is the applied force, r is the instant location between the edge of the plug and the rim, and ho is the initial sheet thickness

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Elastic Constitutive Equations - Plug Stretching Comparison of plane strain model with FEA models that include viscoelasticity

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Elastic Constitutive Equations The Ogden model is the favorite for model builders today The Mooney-Rivlin models are considered classical and are not usually used for model building

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Viscoelastic Constitutive Equations A simple way of including time-dependency in stress-stain equations The current way of including fading memory

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Viscoelastic Constitutive Equations is the memory function where Gi and li are material parameters Typically only the first term of the series is used B(q,q’) is the Finger strain tensor

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Viscoelastic Constitutive Equations h(I,II) is the damping function of the two strain invariants, in the Wagner form for simple equal biaxial stretching where e(q)=ln L(q), e0 and m are called Wagner constants, L(q) is the stretch ratio at q related to time q‘.

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**Homework assignment for TF Conference 2007**

Analyze the four papers presented by Hosseini and Berdyshev at the 2006 SPE ANTEC, to wit: “A Solution for Warpage in Polymeric Products by Plug-Assist Thermoforming” “A Solution for Rupture of Polymeric Sheet in Plug-Assist Thermoforming” “Modeling of Deformation Processes in Vacuum Thermoforming of Prestretched Sheet” “Rheological Modeling of Warpage in Polymeric Products Under High Temperature”

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**Homework assignment for TF Conference 2007**

Their first paper, “A Solution for Warpage in Polymeric Products by Plug-Assist Thermoforming,” was reprinted in TF Quarterly, 3rd Quarter 2006. [Note: As Tech Editor of the Quarterly, I made the comment that the authors had tacitly assumed that warpage could be described as uniaxial deformation and recovery. In other words, the authors used the scalar forms for the Cauchy, Hencky, and the flow strain rate terms. Is this correct? Should they have used the tensor forms as they have in their other papers?]

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Typical temperature-dependent stress-strain curves for an amorphous polymer

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

ABS temperature-dependent stress-strain curves

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

The forming window overlay on the stress-strain field

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Forming Window Measurement Hot tensile testing - Very difficult to get repeatable data at elevated temperatures Dynamic mechanical testing - Yields temperature-dependent modulus through frequency of oscillation or more commonly, by direct measurement Hot creep measurement - Instrumented device similar to HDT device, measuring viscosity Instrumented plug - By changing the rate of plugging, the role of viscosity can be ascertained

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Instrumented plug device for measuring thermoformability - Transmit Technology Group

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Temperature-dependent elastic modulus as a determinant for the forming window

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Temperature-dependent elastic modulus as a determinant for the forming window

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

The maximum applied stress restricts the forming window to the cross-hatched region

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Finite Element Analysis

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Finite Element Analysis Two-dimensional Replaces sheet surface with triangular grid connected through nodes As grid stretches, triangles remain planar but increase in surface area Material assumed to have constant volume; thus increase in local surface area means decrease in local thickness Model works best for thin sheet

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Finite Element Analysis Three-dimensional Replaces sheet surface with triangular grid connected through nodes Sheet thickness accounted for by several initially parallel grids, connected through initially parallel node junctions Often called brick model; volume in each thin brick constant Model allows for localized compression, shear

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Finite Element Analysis Two-dimensional

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Finite Element Analysis Two-dimensional Location of each node of triangle at time q [Xq, Yq, Zq] After differential deformation, location of each node of triangle at time q+Dq [Xq+Dq, Yq+Dq, Zq+Dq]

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Finite Element Analysis Two-dimensional Force balance is made at each node between time q and time q+Dq. If Fi,ext is the external force applied to node i, being the pressure, p, times the normal to the element, n, then If Fi,int is the internal force applied to node i, W is the internal energy function and u is the displacement coordinate at node i, then

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Finite Element Analysis Two-dimensional An equilibrium force balance is applied over the entire surface of the sheet (no acceleration, please!) This equation set is then combined with the appropriate material constitutive equation of state, localized for each triangular element

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Finite Element Analysis Two-dimensional In addition to FEA, very accurate identification of mold surface is needed [X’j, Y’j, Z’j] In certain FEA models, a coefficient of friction is needed between the sheet and the mold surface (this subject is under review and will be the topic at future Conferences, including this one!) Triangles can rotate, translate, and grow in surface areas Triangles cannot flex, bend, or fold

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Finite Element Analysis Two-dimensional Consider a node fixed when its X,Y,Z coordinate approximates a mold X’,Y’,Z’ coordinate location Keep in mind that although one or two nodes of a triangular element are affixed to the mold surface, the triangle can continue to stretch Keep in mind that the local triangle nodes do not need to be affixed to the mold surface if all the nodes of adjacent triangles are affixed

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Finite Element Analysis Two-dimensional When dealing with a plug, a tag is assigned to each node that is affixed to the plug This allows the analysis to include these nodes when the mesh is mathematically stripped from the plug The FEA is complete when no nodes are moveable or viable Newer algorithms address only those nodes that are not immoveable or are tagged, in this way rapidly accelerating the analysis

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Finite Element Analysis Two-dimensional Keys to successful FEA include Selection of correct (small) Dq step Predetermination of local mesh size (too small will generate excessive computer time, too large will generate strange surface bumps, instabilities) Very accurate mold surface replication and mapping Selection of a time-conservative iterative method such as Newton-Raphson iteration Galerkin weighted residuals

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Finite Element Analysis Two-dimensional Recommended practice If the part is symmetric (viz, five sided box), select only one portion rather than solving the entire structure If the part is axisymmetric (viz, drink cup), select only a wedge portion rather than the entire structure Select coarse mesh initially Refine mesh in local areas Repeat computation Continue to refine mesh until (nearly) all nodes are at rest on mold surface

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Finite Element Analysis Two-dimensional HOWEVER... If your sheet is not uniformly heated OR… If your sheet is sagging OR… If your mold is not uniformly cooled OR… Use the entire sheet and mold surface!

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Finite Element Analysis Two-dimensional Keep in mind that the computer display of the stretching mechanism is NOT real time, ONLY COMPUTER TIME If the model being used is elastic-only, keep in mind that the final computer prediction of wall thickness represents what happens in an instant! (There is no time parameter in Mooney-Rivlin or Ogden models) If the model is viscoelastic and/or if the model includes a moving plug, there will be a time factor included… Make certain this matches real time!

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Sag

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Sag As sheet heats, it tends to drape or sag under its own weight If the sheet is of uniform temperature and is clamped on only two edges (think roll-fed), sag shape can be predicted by considering it to be a catenary or chain

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Sag The sheet weight, m, is the sheet density times its local thickness, m = rh The tension, T, in the sheet is factored into vertical and horizontal components, where s is the sheet length: T sin q = ms T cos q = To

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Sag The extent of deflection, y, below the horizontal is Now (ds)2 = (dx)2 + (dy)2

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Sag The extent is then given as And integrated to yield

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Sag The position along the sheet is And integrated to yield the total sheet length

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Sag Review: To is temperature-dependent tensile strength m is sheet unit weight L is total initial span of sheet As sheet heats, To decreases (linearly to perhaps exponentially, depending on the polymer) Sheet sag increases as To decreases, as observed

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Sag Mathematics Advantage: Very compact, easy to obtain local slope as a function of horizontal position (important when calculating the view factor for radiant heat to sagging sheet) Disadvantage: As sheet sags, S increases, but total sheet weight remains constant. Sheet must thin, meaning that m, local unit sheet weight, must decrease. Can be rectified by trial-and-error or by making m = m(s) in equation and solving arithmetically.

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Sag Mathematics One approach to the sag problem is given in Thermoforming Quarterly, 4Q06 where the view factor is determined as a function of two-dimensional (catenary) sheet sag

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Sag Mathematics Caveat - Problem can be solved using FEA BUT to effectively use the FEA model, we need to know the temperature of every element. We can only get that by calculating the view factor - as we did in the first lecture – but now for a sagging sheet. This is NOT done in the FEA model.

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

End of Part 2 Constitutive Equations Applied to Sheet Stretching

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**Part 2: Constitutive Equations Applied to Sheet Stretching**

Trimming as Mechanical Fracture Begins promptly at 10:00!

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ME 520 Fundamentals of Finite Element Analysis

ME 520 Fundamentals of Finite Element Analysis

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