Advanced Modelling of Sports Footwear - Paul Gibbs
Contents
1. the Standard job failed to converge on a solution and had to be abandoned most likely due to the required increase in contact calculations 66 4 6 1 Material Models The equations derived to form the range of material models that are available in ABAQUS are numerous and extensive so this section will concentrate on the material models selected for use in shoe modelling These comprise the elastic hyperelastic and hyperfoam models The specific forms of these models are known as strain energy potentials Elastic The elastic material model assumes the stress experienced by an element is directly proportional to its strain This model has been used where there is little information on material properties or the material shows an elastic response up to 5 strain and then deforms plastically Hyperelastic The hyperelastic material model was designed for use with rubber like ma terials at finite strains and provides a general strain energy potential to describe the material behaviour for nearly incompressible elastomers It is a non linear elasticity model suitable for large gt 5 strains and is therefore suitable for modelling of shoe components in partic ular outsoles and the structures and mechanisms of the new generation of shoes preliminary modelling showed strains of up to 40 for materials in both EVA and structure components in simple shoes The hyperelastic model assumes isotropy for the following reasons The in2. 25 Compression mm Figure 4 4 Experimental and FE results from testing of midsoles that contain AS or do not con tain NAS air pouches The no air pouch model is shown in Figure 4 3 Reproduced from Thomson 1999 60 4 4 FEA Models of the Foot Although the forces on the upper surface of the midsole can be measured Section 2 9 the forces recorded are dependent on the shoe used for the test as the foot is flexible Where prototype shoes are available the forces at the ground shoe and shoe foot interfaces can be measured but the application of these boundary forces to models of entirely new shoes must be carefully considered To model the forces on the shoe more accurately some research has gone into modelling the foot Most of the work aimed at foot shoe interaction has been concerned with the effect of the heel pad as this directly affects the peak forces in the heel strike as discussed in Section 4 5 Aerts 1993 looked at the effect of various midsoles on the forces transmitted to the leg using human test subjects but had problems restraining the leg and so the results suffered from inaccuracies The approach of Noe 1993 although slightly less practical was more successful By using fresh human cadaveric lower leg specimens the bone could be bolted directly to a plate eliminating any excess vibration Tests were carried out barefoot and with different midsoles and the report concludes that any evaluation of the shock
3. absorbency of a midsole must take into account their mechanical interaction with the body Nakabe 2002 used a moderately complex FEA model of a foot to assess the rotation of the joints during running with a selection of midsoles with stability correction inserts Figure 4 5 The results were validated with barefoot running and good correlation can be seen for the modelled part of the footstrike Figure 4 6 hard EVA SRIS C 65 soft EVA SRIS C 55 a b Figure 4 5 a FEA model of human foot b FEA model of midsoles with stability correction inserts Nakabe 2002 61 4 4 2 i i Time msec 20 gt 0 F 2 Bc lat med 4 i Was Analysis on 6 5 i Experimental QC post ant 8 i Figure 4 6 Results of FEA combined foot midsole model ac rotation of the foot plantar dorsiflextion 6c inversion eversion Reproduced from Nakabe 2002 The foot model was progressed by Chen 2003 who modelled the larger bones in the foot and used cable elements to simulate the ligaments The bones were covered with skin to allow the determination of plantar stress distributions on simple midsole surfaces Figure 4 7 All biological materials were assumed to be linear elastic the properties are listed in the paper but the method of determination is not specified Figure 4 7 a FEA model of the foot bones with cartilage and major plantar ligaments b complete model of the foot wi
4. hydrostatic elements in a simple bending model top Note For simplifica tion of the diagram only 2D elements are shown but the principles also apply to the 3D 70 To overcome this problem the element must be substituted for the quadratic version C3D10 and C3D20 Figure 4 17 which allows a variation in strain or have its integration modes reduced C3D8R giving a single deformation gradient across each face A fine mesh of reduced elements arranged in layers more than one element thick is recommended for meshes that are likely to see high distortion Figure 4 17 10 node tetrahedral C3D10 and 20 node hexahedral C3D20 elements Reduced integration elements can however exhibit hourglassing modes Figure 4 18 where the stiffness of the element is effectively zero as a change in the shape of the element does not always result in a change in the length or relative angle of the integration lines Figure 4 18 An element exhibiting an hourglassing mode Note the change in shape of the element with no change in the deformation gradients To combat this behaviour a number of controls are available to modify the behaviour of the element The default control in Explicit is Viscoelastic where the rate of volume change of the element is artificially damped at the start of a step after which control is based on the stiffness response of the material This initially stops the hourglassing but assumes that the maximum load ch
5. mathematical solvers available This section will present a concise description of the physical and mathematical principles required to understand the workings of the ABAQUS code Exhaustive mathematical proofs of all the required matrices and equations are provided in the ABAQUS instruction manuals so only a few important equations are given here Where necessary some wording and quotes have been taken from the various ABAQUS manuals The bulk of the materials in the shoes to be modelled are classed as hyperelastic That is to say that their stress strain response is non linear and modelling requires information about this response ideally for all strains that are to be modelled This allows ABAQUS to create the deformation gradient matrix F as follows _ Ox F ox where X is the initial position of a material particle for this purpose an integration point on an element and x is its new position For hyperelastic materials because the stress strain response is known over a wide range of strains the gradient matrix can be calculated For elastic materials the assumption that the movement of the point is negligible is made to allow the Cauchy or true stress measure to be used To calculate the solution to these matrices ABAQUS comes with two primary solvers Standard and Explicit The main difference between these solvers is the way they solve the matrices over time Standard is implicit it calculates the value of the result
6. midsole with the model results Figure 4 2 FE mesh of a simple EVA shoe heel Shiang 1997 The response of the foam in car seats was studied by Pajon 1996 Mechanical tests were performed and the test data used to verify the material models The report notes the importance of verifying the simple material models with complex real parts as the car seat foam is in a combination of compression and tension and may not be modelled accurately with uniaxial mechanical tests as can be seen in Figure 3 31 Thomson 1999 used uni axial test data to create a hyperfoam material model and used this to model the heel impact To reduce processing time only a cylindrical section of the heel was modelled this allowed a 2D axisymmetric model as shown in Figure 4 3 The model was pinned at the base and the nodes on the vertical walls were constrained to move in the vertical direction only The accuracy of these boundary conditions is questionable and may require practical investigation although results for this simple model are consistent with the mechanical tests Figure 4 4 59 a ee ee we ee N TEREERE EE gooo222222 Bescoasaae 9 A Kel We OO OOO OOO OO Ne R ih Besoogoos Ee mses cee a L Figure 4 3 2D axisymmetric model of a heel impact numbers shown correspond to node number ing Thomson 1999 2500 NAS FE AS FE Force N 0 T T T T 0 5 10 10 20
7. therefore not being released in full at this time This section discusses models that have been found in the public domain at present A reasonably complex 2D model of a therapeutic shoe was performed by Lewis 2003 The model included some mesh refinement but assumed all the materials behaved elastically A static load case was assumed Figure 4 10 Lewis results show a difference in the behaviour of the shoe when certain materials are substituted However none of the calculations are verified with practical testing 71031 z ppr o Ji jp 996 os l 967 one 09 fl F 942 e OS 70 s22 I i ii Te 34 y 90 me HH if if f na w y 928 y Bahl ye Lhe Cree Ay cea 7 Pee SR wy 59 i We ane a R oe pe Hn pe RE Melt autor re a 79 n3 RZ L ees var EU fa STS TAL J 7 AS SSS Figure 4 10 2D model of a therapeutic shoe Numbers indicate nodes of interest in this model Lewis 2003 Verdejo 2002 created a similar model to that of Thomson 1999 but with the inclusion of the heel pad The Ogden hyperfoam material model was used with data taken from mechanical testing Model results were verified against ASTM mechanical testing Figures 4 11 4 12 Verdejo reports that the ASTM testing produces a different pressure distribution on the upper surface of the EVA midsole compared to that measured in the shoe or predicted by FEA Heel Bone ASTM Stamp Tissue Midsole 300 a
8. 5 kg missile impacting onto midsole specimens the exact nature of these specimens is not specified Tests results were then applied to an isotropic linear viscoelastic material model and FEA performed Good correlation was reported for impact heights of 13 amp 25 mm but from 50 mm upwards the model deviates from the practical results As an example of this deviation the 50 mm result is shown in Figure 4 1 58 1000 Impact Force Response PU 50 mm drop height 800 mean force 0 95 Cl 0 95 Cl pog model fit 400 Force N 200 200 T T T T 1 0 00 0 01 0 02 0 03 0 04 0 05 Time sec Figure 4 1 Drop tests results from 8 5 kg mass falling from 50 mm onto midsole foam average test values and model predications are shown Reproduced from Swigart 1995 Shiang 1997 used FE to study the cushioning response of an unspecified midsole ma terial Material properties were obtained from compression of a cylindrical specimen under loading by a vertical actuator Instron machine Loading values were taken from in shoe measurements and an assumed shear force of 1 10th of the peak vertical loading was applied to the model shown in Figure 4 2 Both linear and hyperfoam material models were used along with different boundary conditions and models that included combinations of mid sole insole insole board It is not clear if the results of the study were verified by comparing real compression of the
9. Advanced Modelling of Sports Footwear by Paul J Gibbs A Doctoral Thesis Submitted in Partial Fulfilment of the Requirements for award of Doctor of Philosophy of Loughborough University Chapter 4 Literature Review Finite Element Analysis 4 1 Introduction This chapter presents a very brief overview of the finite element method of structural analysis a subject which can fill many libraries A short introduction is followed by reviews of the most advanced and applicable models in the public domain at the time of writing It is expected that models of shoes are being created and used by major manufacturers but the competitive nature of the footwear industry means these models are unlikely to be released into the public domain in the near future if at all Modelling of the human foot has been included as most of the work on shoes to date has come out of a need for analysis of medical footwear almost all of which developed as an extension of a foot model The final section goes into more detail regarding the use of ABAQUS the finite element software used in this research This chapter does not aim to assess the required accuracy of the various models that will come out of this research this is to be achieved through physical testing of footwear samples and parts and is discussed in Chapters 7 9 4 2 The Finite Element Method Finite element analysis is a theoretical mathematical method of predicting the solutio
10. ange will occur at the beginning of the step which is not always the case This problem can be overcome by using the Enhanced hourglassing control which com bines a stiffness based and viscoelastic based control of the deformation throughout the step This results in a small decrease in element efficiency but is vital for some jobs to run with realistic results the use of a purely viscoelastic control while more efficient can result in severe hourglassing in quasi static conditions during a step Another method of avoiding the shear locking problem is the use of incompatible mode elements C3D8I Figure 4 19 Additional integration modes are added on the diagonals of the element to control the trapezoidal distortion associated with locking In general all elements produce more accurate results the closer to their ideal shape they are e g square faces at 90 for hexahedral elements Generally tetrahedral elements have lower sensitivity to distortion so are recommended for meshes undergoing high strains though 71 Deformation gradient Figure 4 19 Face of a C3D8 element with normal left and incompatible right density gradient modes the increased numbers required to fill the same volume will create a slower running job The modelling of incompressible substances where the Poisson s ratio v of the material is close to 0 5 can give rise to problems due to round off errors as the stress increase for a given defor
11. b Figure 4 11 Contours kPa of stress distribution for a Heel bone and pad on EVA midsole b ASTM heel on EVA midsole Reproduced from Verdejo 2002 64 0 7 0 6 5 ri 05 ASTM Pa Force kN 12 14 16 Deflection mm Figure 4 12 Comparison of model results and ASTM mechanical testing for model in Figure 4 11 b Reproduced from Verdejo 2002 The only FEA model found to use realistic 3D midsole geometry was found in Nishiwaki 2002 It is shown as a practical implementation of a system for predicting human response to midsoles using human drop tests see Section 2 9 Very little information is given about the FEA model but the report concludes the simulation method is a very powerful tool in the design of shoe soles The result of the FEA model is shown in Figure 4 13 Results from an investigation into the effect on cushioning of varying sandwiched EVA layers are also presented in Nishiwaki s paper 0 010 f gt Cushioning Parameter 0 00 groove Model 1 Model 2 Model 3 Model 4 Figure 4 13 FEA model of a midsole using realistic component geometry Black areas indicated on the chart represent areas of reduced stiffness in the model Nishiwaki 2002 65 4 6 Modelling in ABAQUS There are a number of FEA packages available on the market at present ABAQUS was selected as the solver primarily as it is used by adidas but it also has a reputation as one of the most accurate
12. ber TPU and hyperfoam high strain highly compressible e g EVA foam Details on which material models have been used in the shoe models in this thesis are given in Chapter 7 Material modelling with FEA can be divided into constitutive and macro scale models where the former considers the atomic molecular response of idealised materials while the latter works on a larger scale overall response The equations used in constitutive modelling are normally based on the mechanics of the molecular structure while macro scale models use curve fitting techniques Knowledge of how the models work in principle is essential to using them correctly in a model however this research will only use current material models mathematics and not attempt to develop new equations or fits D Agati 1993 performed linear and non linear elastic FEA on a simple representation of a midsole using force input data collected from in shoe measurements vertical force only A single 20 node cube quadratic element was used to assess the stress strain curve and then this data was used in the midsole model using the same element type No numerical results were given but the paper quotes Differences between the linear model s peak stresses and strains varied from 98 to 680 with respect to the non linear model s results This is attributed to the non linear response of the EVA material used Swigart 1995 used a vertical impact drop test with an 8
13. contact between objects are also assigned known as boundary conditions to either both nodes or elements The elements are assigned mechanical properties e g stiffness and the problem is then solved in matrix form with appropriate manipulations to give the solution The size of the matrices directly affects the time taken to process the solution and this in turn is affected by the number of degrees of freedom and complexity of the boundary conditions as well as of course the processing capabilities of the computer used Incorrect specification of any of the material parameters geometry or boundary condi tions coupled with some unavoidable assumptions that must be made when using the FE technique mean that inaccuracies will be present in the model It is the job of the FE analyst to be aware of quantify minimise and where possible correct for these inaccuracies 4 3 Material FEA The first step in creating an accurate model of a midsole is to model the basic materials that it is made of in this case mostly EVA foam and TPU If mechanical tests with simple geometry cannot be modelled correctly then a complex full shoe model will be inaccurate and therefore of limited use unless the inconsistencies can be assessed and quantified A material model is usually created by fitting an equation to test data the main equation types being linear e g metals in the elastic region hyperelastic high strain incompressible materials e g rub
14. ct and slowing the overall job time down If parts of the model have a negligible effect on the current test then they can be removed or substituted For example if the interest is in the performance of a heel part under heel cushioning then the forefoot of the model could be meshed coarsely or removed entirely This can give significant reductions in job times but the effects on the results should be checked 4 6 4 2 Mass Scaling As discussed at the start of this section for simple problems ABAQUS Standard is a very fast solver but it drastically loses efficiency as the models increase in complexity and in some con tact models cannot provide a solution To provide faster solutions using ABAQUS Explict 72 a number of solution controls are used the most powerful and complex of these being mass scaling The total time a job will take to process is roughly dependant on the number of degrees of freedom to be solved in each increment and the number of increments to be solved The solver automatically limits the size of the time increment based on the global stable time At of the model Fundamentally this is a measure of the shortest time it takes a stress wave to propagate across any of the elements this is based on factors including stiffness size and damping following the equation At min z ne 4 1 A 2p where L is the characteristic length of an element in an undamped material this is the length of the shortes
15. descriptions between modelling steps e General contact has a built in surface smoothing algorithm for element based surfaces Each of the functions above are not available when using contact pairs Conversely the following functions are only available with contact pairs Contact Pair e 2D surfaces can be defined e Analytical rigid surfaces can be defined e Thermal contact can be defined The limitations on each method are reduced with each new version of ABAQUS for example both methods have to be used in shoe models as the structures have too many surfaces to define as pairs but the floor is usually modelled as an analytical rigid body requiring a contact pair the latter limitation is to be removed in ABAQUS version 6 6 The default setting for the contact algorithms are also different general contact uses the penalty enforcement algorithm while contact pair uses kinematic 4 6 5 2 Contact Algorithms ABAQUS includes two main methods for processing surface contact small sliding and finite sliding Small sliding assumes a negligibly small displacement of one surface along another and does not allow separations after contact but is much more computationally efficient Finite sliding allows any finite movement of the two surfaces including separations with a reduction in efficiency To correctly model any interactions between surfaces in shoes the finite sliding method must be used as separation can occur after contact The E
16. g practical mechanical testing e Determination of the overall variability of the shoes modelled With enough time this kind of analysis should be possible but for it to be applicable to industry it must be able to be carried out within a production time scale e g a few weeks with little manpower resources The following issues are believed to be critical in achieving the correct speed accuracy balance for the FEA models e The level of model complexity specifically mesh densities use of component bonds contact and complex material behaviour such as Mullins effect and viscoelasticity e The level of accuracy of geometry required to produce adequate meshes as the geometry may need repair as discussed in Chapter 8 e The sensitivity of the model to slight changes in any of the aforementioned model parameters and geometry e The level of repeatability of performance of manufactured shoes Investigations into all of the above are carried out in Chapters 7 11 to address these questions The use of FE models within a production time scale along with suggested workflows is discussed in Section 12 5 1 TT
17. he uni axial data only specifies the material property in one compression tension mode so assumptions are made about the other modes unless extra test data from bi axial and tri axial experiments are included The use of tri axial data is recommended when the material being modelled is compressible Once data has been inputted ABAQUS calculates the required energy potential form coefficients and allows the user to compare results for a given range of strain useful for checking if any of the forms becomes unstable in the expected range of strain Hyperfoam The ABAQUS hyperfoam model is similar to that of the hyperelastic model but it accounts for the difference in deformation modes in compression and tension exhibited by most elastomeric foams see Section 3 3 1 The model is designed for materials that can deform elastically to large strains including up to 90 in compression Isotropy is assumed 67 4 6 2 Material Model Modifications In addition to a non linear elastic stress strain response shoe materials especially foams can contain a number of energy loss mechanisms as discussed in Section 3 3 The resulting overall response of the material is vital for providing the correct performance during a running strike and so any computer model that attempts to simulate the entire footstrike accurately must contain representations of these mechanisms 4 6 2 1 Hysteresis and Viscoelasticity A repeatable change in the stress strain re
18. iatoric and volumetric strain The variables for the damage model can be inputted directly or derived from multi ple strain level cyclic test data by way of a least squares fit as is done for specifying the primary behaviour via HYPERELASTIC or HYPERFOAM However the ABAQUS doc umentation does strongly recommend that the resulting fitted coefficients are verified by single element models against the test data 4 6 2 3 Combining Material Model Modifications The materials used in shoes are subject to hysteresis viscoelasticity and the Mullins effect Unfortunately these energy loss mechanisms are currently incompatible within ABAQUS MULLINS EFFECT cannot be used in conjunction with VISCOELASTIC or HYSTERE SIS Even if the material data are collected from samples pre strained to the average maximum strain seen in use this data will be too stiff or too soft for the majority of the material This is not so much of a problem in simple geometry shoes where the strain is fairly constant but it is a major issue in structure shoes As the Mullins effect can greatly reduce the stiffness of a section of material there is a real risk of under prediction of stress in a component that could result in material failure While the material model could be created from pre stressed material data this will only predict correct stresses in simple homogenously strained parts 68 Components with highly varying strain fields such as the Ultraride str
19. itial orientation of the long chain molecules in elastomeric materials is assumed to be random therefore the material is initially isotropic As the material is stretched the molecules orient themselves giving rise to anisotropy However the development of this anisotropy follows the direction of straining hence the material can be considered to be isotropic throughout the deformation history As a consequence the strain energy potential can be formulated as a function of the strain invariants The strain energy potential defines the strain energy stored in the material per unit of reference volume volume in the original configuration as a function of the strain at that point in the material ABAQUS provides a number of different forms of strain energy potential to model the approximately incompressible elastomers including Arrunda Boyce Mooney Rivlin neo Hookean Ogden and Yeoh forms Some of these are derived from purely polynomial curve fitting while others are phenomenological they are based on extrapolations of the underlying cellular molecular structure of the material The exception is the Marlow form which is designed to fit any test data to a fairly smooth non polynomial curve The choice of a particular form should be based primarily on verified model results All the energy potential forms require some kind of sample data to formulate their co efficients the most basic requirement being uni axial tests see Section 3 4 T
20. made rigid either as a new rigid part or using RIGID and mass scaling can be applied to all of the model excluding that part Using the fixed time increment drives the job along at a constant rate ignoring the limitations of the stiff elements in the rigid part This method can give very fast processing jobs but will fail if there is moderate to severe mesh distortion anywhere within the model 4 6 5 Contact Contact in FE models should occur when two or more objects collide However objects will occupy the same physical space without having an effect on each other unless a specific contact interaction has been defined between two or more surfaces external parts of objects which must also be pre defined ABAQUS allows surfaces to be specified as node based element based or analytical rigid Node based surfaces are non continuous as nodes are intersections of element edges and are mathematically considered discrete Element surfaces are continuous and contain more information about a given surface as the curvature and area of each element is known this allows more accurate calculation of the transfer of forces between two element based surfaces and leads to much better results for contact problems Analytical rigid surfaces are a special case as they are described using straight and curved line sections either rotated or swept to form simple shapes They allow much better definition of curved surfaces as they are not discritised b
21. mation approaches infinity as v approaches 0 5 To avoid this hybrid elements are offered that calculate the pressure stress and displacement stiffness as separate but coupled variables 4 6 4 Job Controls It is often the case that a more accurate finite element model requires the addition of more physical processes into the calculations with a resulting increase in the time taken to process Although performance of easily available desktop computers is increasing at a steady rate there still may be a need to reduce the time taken to achieve a useful result from FEA These controls are discussed here in theory and in Section 8 4 2 with relation to practical application 4 6 4 1 DOF Reduction The most obvious way of reducing the processing time of a job is to reduce the amount of calculations required This can be achieved by reducing the number of degrees of freedom DOF in the model Replacing quadratic elements with linear ones is a quick method of achieving this provided any interactions or boundary conditions are specified on an element basis though there is a notable performance difference between some types of elements as discussed in Section 4 6 3 If a reduction in accuracy is acceptable then physical processes can also be removed such as viscoelasticity However if removing these process destabilises the model causing vibration and extra movement the number of iterations a job needs can increase having the reverse effe
22. matrix based on the values of the input matrices and the result matrix itself that is the process is itera tive Explicit as its name suggests solves the matrix explicitly assuming a negligibly small increment between one state and the next using the result matrix with no further checks Each of the solvers allows a computational job to be split into steps separate calculations where the final result of one step forms the initial result of the next In each step various conditions of the model can be altered Standard lends itself to efficient calculation of static and quasi static problems but for problems that require the inclusion of inertia Explicit is better suited The Explicit solver also has a more robust set of methods for the solution of complex geometry and contact problems and is used almost exclusively in this research for the modelling of shoes Standard could be used for the modelling of small simple parts but the need to predict the behaviour of a part or material sample in the Explicit whole shoe model outweighs the performance ad vantage of using Standard for such jobs As an example the initial modelling of a simple shoe was performed in both Standard and Explicit a forefoot cushioning test on the Supernova model taking 11 1 minutes in Standard and 274 5 minutes in Explicit These jobs modelled the boundary conditions as prescribed constraints but when physical experimental parts such as the stamp and floor were added
23. n to a physical problem by dividing discritising the geometry of the problem into small sections elements It is normally used when the precise analytical solution is too complex to be solved in the time available and practical testing is either inefficient or not possible Each element is analysed using relatively simple mathematics and will normally have a linear or quadratic variation of the desired property with length The large number of simple calculations required for a FEA solution are suited to processing by computer and the vast majority of FE analyses are now done in this way except for teaching purposes where the problems used are very simple A major advantage of FE over practical testing is that once a model has been verified proven to be satisfactorily accurate there is a high likelihood that the internal stresses in the model are similar to that in the real test situation allowing the user to probe the inner workings of whatever is being modelled This can be extremely difficult to do and sometimes impossible with real testing A FE model is normally constructed from geometry which is discritised into a mesh a representation of all the elements in the model connected by nodes Each node has a given 57 number of degrees of freedom variables in the model Generally these are the 6 spatial degrees of freedom but others are available for example temperature conductivity and inertia External forces fixings and
24. of knowledge in the mathematical modelling of hyperelastic materials however the application of these models into FEA has been limited to more traditional industrial applications Specific modelling of footwear has been concentrated on the biomechanical or medical orthopedic use of models with complex foot assemblies but only very simple midsoles Looking at the techniques offered in one of the most popular FE programs ABAQUS shows that the technology is available to model the materials geometry and interactions involved in footwear Simple models have given reasonable results in the past and the appli cation of FE mechanism modelling used in other engineering areas should allow the modelling of the increasingly complex modern shoes It must be noted that while there is a range of data available for the biomechanical inputs into the system and some general data on material properties there is no apparent quantifiable information detailing the variation between one shoe and another of the same size make and model This information is lacking both in the public domain and within adidas The literature review in Chapters 2 4 proves that the methods proposed in this research will provide new insights and fill in vital gaps in knowledge in this field e FEA modelling of simple and complex running shoes e Laboratory determination of the properties of the materials in said shoes e Material model validation with FEA e Shoe model validation usin
25. plicit The application of bulk viscosity is discussed in Section 4 6 4 2 4 6 3 Element Performance and Selection Various element types are available in ABAQUS but this section will pay attention only to those to be used in the modelling of shoes three dimensional tetrahedral and hexahedral The basic elements C3D4 and C3D8 are shown in Figure 4 15 Across each face and between the faces lines of integration exist where the properties and response of the element are calculated during the analysis The more integration points and lines an element contains the more detailed the changes in property across the element can be modelled 69 Figure 4 15 4 node tetrahedral C3D4 and 8 node hexahedral C3D8 elements These are the most mathematically simplistic of the 3D solid elements and as such are the most computationally efficient These elements cannot be used in all circumstances however For example the linear C3D8 element will become shear locked when used in a direct bending model Figure 4 16 The linear hexahedral element begins the step as shape a In the physical solution the element should become b but as the element formulation has a constant strain the integration lines dashed must remain of equal length and the sides of the linear element cannot bend generating large shear stresses c resulting in a stiffness modulus that approaches infinity f a b Figure 4 16 Linear
26. s To this end penalty contact treats the penetration of surfaces more accurately but the pressure distribution becomes less ac curate This increase in stiffness also has a detrimental effect on the stable time increment 75 slowing down the job although the stiffness values are chosen automatically to balance the accuracy of contact with the increment time The use of hard contact also reduces pen etration but while there is likely to be less penetration than using kinematic the penalty algorithm does not guarantee there will be no penetration The penalty algorithm accounts for all the energy during contact so the inertia of the contacting nodes is retained useful for jobs where rebound is likely Both algorithms can have problems resolving contact between surfaces of significantly different masses due to high velocities being generated in the lighter material in an attempt to conserve overall energy across the contact In Figure 4 20 a a less dense master surface has a high velocity of contact and penetrates the slave surface in the next increment b The algorithm then resolves the contact based on the slave surface deforming and providing a pressure that moves the master surface back slightly c Master J o_ _Y__ f i a b c Figure 4 20 Surface contact errors due to mismatched material densities To overcome this problem the master surface should be specified as the coarser denser mesh If this i
27. s not possible then the priority balance between master and slave surfaces can be manually over ridden If the default value of 1 0 gives surface A as the master then changing the priority to 0 0 gives surface B as the master Any value between 1 0 and 0 0 requires a weighted average of the results with 0 5 giving a result generated from one calculation with A as master and one with B as master This method gives the most accurate results but is the most computationally intensive 4 6 6 Connectors For the efficient modelling of mechanisms ABAQUS provides a set of specialist connector elements These act as packaged boundary constraints specifying a particular linked rela tionship between two nodes in the model While the aim of the connector elements is to provide a quick method of simulating joints etc they have proven to be unreliable in models with complex geometry They are very sensitive to overconstraints where a single degree of freedom is constrained more than once As overconstraints can be formed from a change in contact conditions throughout the job the use of connectors requires very careful job set up and if a job fails lengthy debugging and adjustments to the model are needed to allow successful execution see Chapter 9 76 4 7 Chapter Summary This chapter has reviewed the state of knowledge of finite element analysis applied to a footwear modelling application It is apparent that there is a good depth
28. se to around 2 useful qualitative results have been achieved with scalings of up to 1 000 000 corresponding to a reduction in job time of around 1000 times see Section 10 4 While this contradicts the advice against using super stiff materials with shoe modelling the scaling is applied mostly to the steel impactors which undergo very small deformations and if their movement is specified by position then the extra mass has a negligible effect on the result Mass scaling is automatically applied to all Explicit jobs in the form of bulk viscosity By adjusting the density of the material slightly a similar effect to that of the element floating in a viscous fluid is achieved this is very similar to directly applied mass scaling but on a much smaller scale Bulk viscosity p is calculated thus Pov bLc vot 4 2 where b is the inputted co efficient and vo the volumetric strain rate The automatically 73 calculated bulk viscosity has been found to be ineffective in most of the shoe models as the time increments are quite large to begin with Fixed Time Increment Generally it is best to allow ABAQUS to calculate the time increment in an Explicit job automatically this allows the time increment to reduce in response to very small elements but also to increase when the computation becomes less demanding However during some models where there are stiff parts that undergo little strain such as drop tests the part can be
29. sponse between the loading and unloading cycle and the resulting energy loss can be modelled either with the HYSTERESIS or VIS COELASTIC functions within ABAQUS Both models take a defined stress strain response and modify it with relation to the strain rate the material experiences The main difference between the two models is that HYSTERESIS can only be used with large strain hypere lastic models not hyperfoams due to the model assuming no strain rate dependance with a change in volume but only in shear The viscoelasticity model allows a range of inputs including specification of the strain rate response using a Prony series normally obtained from multiple creep tests or DMA Dynamic Measurement Analysis which provides a much faster physical test and is described in detail in Section 5 2 4 2 4 6 2 2 Mullins Effect As discussed in depth in Section 3 3 3 the Mullins effect provides a way of dissipating energy from the model via permanent deformation of the material It is strain and strain history dependant To incorporate this damage into the material model ABAQUS uses a damage calculation that removes a given amount of strain energy upon loading of the material This can be expressed UTotal Ostorea oh Urost Deviatoric T Ustored Utost Volumetric When used with HYPERELASTIC the MULLINS EFFECT calculation assumes that only the deviatoric strain is subject to loss When using HYPERFOAM damage occurs on both dev
30. t side p is the density and and are the effective Lam s constants which are derived A vE oe 1 v 1 2 eS E H M Fv where v is Poisson s ratio and E the Young s modulus of the material ABAQUS Analysis User Manual 6 3 3 As a result of this any model with unrealistic super light or stiff materials will produce a very small time increment Elements with super light or soft properties are likely to be deformed and crushed also producing a short wave speed and thus a small time increment However using mass scaling the time increment can be controlled by varying the density of the material with values selected to either fit to a user specified time increment or level of scaling The level of scaling is specified as a multiple and reported as a percentage increase in overall mass Mass scaling is excluded from gravitational calculations so the extra mass cannot be weighed in the model it only causes an effect through a change in inertia ABAQUS Analysis User Manual 7 6 1 The automatic controls can be overridden in a number of ways ranging from specifying a set increment for example in jobs involving only rigid bodies to giving merely a start value and allowing the solver to adjust values throughout the job This can decrease job time by orders of magnitude but can affect the output significantly and may cause job failure or non physical results While the ABAQUS manual recommends limiting the mass increa
31. th soft tissues Chen 2003 At the time of writing Cheung 2006 has produced the most realistic foot model Using images from magnetic resonance scans of a foot all bones ligaments and covering flesh are modelled Figure 4 8 All materials are linear elastic except the soft tissue and midsole which are modelled as a hyperelastic and hyperfoam material respectively actual values and sources are given The paper does not specifically state the verification process used but in a presentation to the 2006 ABAQUS User s Conference the author described experimental loadings of cadaveric foot samples with tendons loaded to simulate various stances The shoe used in the report is a simple extrusion of hexahedral elements from the flat floor to a surface of a medical insert for a shoe Figure 4 9 62 Figure 4 8 Left to right Surface modelled of a MRI scanned foot FE mesh of encapsulated soft tissue FE mesh of bony structures Cheung 2006 foot surface model orthosis surface model solid model of foot orthosis rz MATLAB The MathWorks Inc SolidWorks 2001 SolidWorks Corporation Figure 4 9 Procedures for creating the midsole foot orthosis model Cheung 2006 63 4 5 FEA Shoe Models There is little published research on the use of FEA to model complex 3D shoe models due to the fact that this approaches the limits of complexity of the available technology and so is likely to be commercially sensitive information and
32. ucture plate under a heel stamp test Figure 4 14 will not have their stress predicted correctly using a single material curve lt 5 Strain gt 50 Strain Figure 4 14 Strain result from a heel stamp test performed on the structure plate of the Ultraride showing a large range of strains occurring in the same part Recent work by Bergstr m 2006a has produced a phenomenological material model that extends the functionality of the ABAQUS HYSTERESIS command to incorporate the Mullins effect and include rate dependency This model was not available during the research for this thesis but it has the potential to be very useful for future work A less practical method of including rate dependency and the Mullins effect is described in Section 8 5 where a model is run using VISCOELASTIC and the results analysed for strain levels and the material model adjusted accordingly for the next iteration This method has two main drawbacks the material models are discrete and each iteration requires a complete re run of the analysis extending development time 4 6 2 4 Bulk Viscosity If neither viscoelasticity hysteresis nor Mullins effect is specified a material in a dynamic temperature independent elastic model has no method of dissipating energy which can result in unstable models To this end ABAQUS applies an energy loss mechanism known as bulk viscosity to reduce the likelihood of elements under high strain rates collapsing in Ex
33. ut their use has restrictions Contact between one node based surface and another or contact with itself is not per mitted in ABAQUS the same is true for analytical rigid surfaces Element based surfaces are allowed to interact relatively freely Each interaction requires a definition of the interaction and an algorithm for calculating the result 4 6 5 1 Interaction Definition Contact is a large set of complex mathematical descriptions and is treated differently within Explicit and Standard As Explicit is used in the vast majority of the models created in this research only this is discussed in brief here Full details of all the theory and formulations is found in the ABAQUS Analysis User Manual 21 1 1 There are two methods available for defining contact between surfaces in Explicit general contact and contact pairs General contact sets up every element in the model as a single surface and allows this master surface to contact itself The contact pair method requires the User to specify a pair of surfaces The advantages of each method are summarised General Contact e A single surface can span multiple objects e A surface can have both rigid and deformable regions e A surface can have solid and shell elements e Nodes can have contact with more than one surface at a time very useful for modelling objects with corners 74 e T junctions of shell elements can be modelled with one surface e Changes can be made to surface
34. xplicit solver provides many options for the algorithms that deal with the finite sliding motion but the main choice in job set up is the use of kinematic or penalty contact enforcement Kinematic For each time increment the kinematic algorithm will first displace nodes then calculate the pressure increase generated by the penetration of one surface into the other a non physical condition Corrections are then made to satisfy the pressure balance but not necessarily the penetration The sub options of hard and soft contact can also be specified ABAQUS Analysis User s Manual 22 1 2 where hard contact transmits forces between surfaces with zero distance between them Soft contact gives a linear relationship between the percentage of the force transmitted and the distance between the surfaces this can be a positive or negative distance Hard contact is designed to reduce the penetration of the surfaces but the level of final penetration depends heavily on the quality of refinement of the mesh Hard contact can leave small gaps between contacting surfaces as the kinematic algorithm disregards the inertia of the nodes on the contacting surfaces hence they can bounce back off the surface Penalty Penalty contact will first look for penetrations in the current configuration then apply deformations to remove that penetration These deformations are generated by intro ducing artificial stiffness increases into the slave node
Download Pdf Manuals
Related Search