CHAPTER NUMERICAL MODELING OF HUMAN FOOT ~ If you want to accomplish something in the world, idealism is not enough - you need to choose a method that works to achieve the goal ~ Richard Stallman 29 2.1 Finite element model of foot-ankle complex 2.1.1 Finite element method The FEM is a versatile numerical method which allows stress and strain analyses of complex structures with irregular geometry and material nonlinearities. When a foot structure is loaded, stresses are generated in different materials (i.e. tissues). The distribution of these stresses, their magnitudes and orientations throughout the structure, is a result of complex interplay of the foot skeleton, cartilages, muscles, ligaments, fascia, and the external environment that arises from footground interactions. In such a model that mimics the real structures to a certain degree of refinement, the structural aspects (geometry, material properties, and loading/boundary conditions) are required to be expressed mathematically. In using the FEM, the model of the human foot as a geometrical entity, has to be firstly defined. Modern musculoskeletal imaging techniques such as magnetic resonance imaging (MRI) and computer tomography (CT) can be a source for such complete anatomical structures. The geometrically complex foot structures, are then discretized into finite number of relatively simple elements (i.e. FE mesh generation), connected by nodal points or nodes. Each element can have its own material properties. The equations for describing the mechanical behavior of these elements are known. The computer program (e.g. ABAQUS) can calculate the stiffness matrix of each element, and the stiffness matrix of the whole structure is determined. Knowledge of the structural stiffness matrix, of the loads and the boundary conditions allows the response of a model to any form of external loading to be predicted. 30 2.1.2 3-D reconstruction of foot geometry The three-dimensional soft tissue-skeletal geometry was created based on the volume reconstruction of the coronal computer tomography (CT) images of the right foot (non-weight-bearing condition) of a male subject (27 years old, height of 169 cm and body weight of 65.1 kg). The CT scan has a resolution of 0.25ｘ0.25 mm2 (pixel size) with a slice spacing of mm. This allows a detailed morphological reconstruction of the various anatomical structures of the foot, including bones, cartilages, and a bulk soft tissue boundary (Fig. 2.1.2.1.). Model segmentation was performed by MIMICS program (Materialise Inc., Belgium). Fig. 2.1.2.1. Segmentation of a human foot from individual coronal CT slices. The bones were modeled as articulated parts enveloped into a bulk soft tissue Two pixel phases were defined based on gray-scale threshold computing: the foot skeletal phase and a lumped soft-tissue phase. The hard foot skeleton and soft tissue were then reconstructed into a solid model (Fig. 2.1.2.2.). The 31 solid model was finally imported into a pre-processor, PATRAN (MacNeal Schwendler Corporation, USA), for FE mesh generation. Thirty bony parts, including sesamoids, were created individually and enveloped into the homogenous mass of foot soft-tissue. Particular attention was given to the surface geometry of various foot bones (tibia, fibula, talus, calcaneus, navicular, medial, intermediate and lateral cuneiform, and cuboid, the medial, intermediate and lateral cuneiforms, and the cuboid, metatarsal bones and phalanges) that are internally articulated. Fig. 2.1.2.2. 3-D solid model of foot geometry, including a bulk soft tissue (A) and bones (B). Note that the foot skeletal was stabilized by actually anatomical ligaments. Passive soft-tissue stabilizers of the foot skeleton, including ligaments and plantar fascia, which could not be reconstructed from CT were determined based on the anatomical descriptions from Primal Pictures 2006 (ANATOMY.TV, Primal Pictures Ltd., London, UK). A total of 134 ligaments (i.e., multiple element modeling for major ligament bundles) and a fan-shaped plantar fascia structure with a mm out-of-plane thickness were incorporated into the model. 32 2.1.3 Model discretization The reconstructed solid foot model was then used to generate three-dimensional finite element mesh using various types of structural elements; this process is known as model discretization. The foot skeleton and the soft-tissue component were meshed by 160,000 and 240,000 tetrahedral elements (C3D4: full integration linear element). Meshes for the bones and the soft-tissue component share same nodes at the interface except those joint space regions where a series of contact conditions were defined between adjacent bones (Fig. 2.1.3.1.). The ligament structures were represented by 3-D truss elements (T3D2) with a “no compression” option, in accordance to their physiological function. This would allow these elements to resist tension-producing forces when stabilizing the foot skeleton. Fig. 2.1.3.1. Finite element mesh of a human foot with (A) soft tissue and (B) internal bony structures 33 For the muscular structures, six major extrinsic plantar flexors were included. To model the gastrocnemius-soleus (G-S) complex, a threedimensional geometry of the Achilles tendon was constructed and incorporated into the posterior extreme of the calcaneus. This facilitates application of G-S muscle forces through the Achilles tendon-bone junction, and ensure more realistic muscle load transfer compared to those in previous models, which only employed nodal points to apply such forces (Gefen et al., 2000, Cheung et al., 2006). The long tendons of the other five muscles were also inserted into the model, at their corresponding anatomical attachment sites. This was done using bar elements based on straight-line approximation, i.e. several bar elements stringed together to represent the actual tendon trajectory inside the foot. A mesh sensitivity analysis was performed to ensure that the mesh density used in the FE model was sufficient to reach the converged numerical results. Mesh refinement process was carried out in a 2-D plane-strain finite element model based on a sagittal section through the 2nd ray of the foot. The total strain energy and displacement served as the convergence criteria, with the tolerance level being set as the change of less than 5%. 2.2 Material properties for finite element modeling 2.2.1 Cortical and cancellous bones, ligaments, and cartilages Mechanical properties of cortical bone have been well documented in the literature. Traditional mechanical testing such as uni-axial tensile or compressive testing has revealed that material property of cortical bone may be considered as 34 having ‘nearly’ linearly elastic behavior. Measuring mechanical properties of cancellous bone tissue is far more difficult than measuring those of cortical bone tissue, due to the extremely small dimension of individual trabeculae. Because of this difficulty, the reported cancellous bone tissue modulus ranges from 0.76 to 20 GPa (Cowin et al., 1998). In the current analysis, a quasi-static loading system is applied to the foot model. According to Huiskes (1996), in the case of quasi-static loading, both cortical and trabecular bones may be linearly elastic for simplification. Consequently, in the present model, bone tissues are assumed to be homogeneous, isotropic, and linear elastic. For the foot bones, Young’s modulus is taken as 7300 N/mm2, a value that was weighted by Nakamura et al. (1981) from human cortical and cancellous bone properties and the Poisson ratio is taken as 0.3. The ligaments, following Cheung et al. (2005), were considered as non-compressive materials and assigned with a Young’s modulus of 260 MPa, a Poisson’s ratio of 0.3, and a cross-section of 18.4 mm2. 2.2.2 Achilles and other flexor tendons In the literature, mechanical properties have been reported for many tendons (e.g. Achilles tendon and anterior cruciate ligament). Modulus values are generally in the range of 500~1,850 MPa (Yamamoto et al., 1992, Danto and Woo, 1993). For the Achilles tendon, uni-axial tension tests have been conducted by Wren et al. (2001) at lower strain rates of 1%/s and relatively higher strain rate of 10%/s. The mean moduli found were 816 MPa (± 218) at 1%/s rate and 822 MPa (±211) 35 at 10%/s rate, respectively. Considering the quasi-static loading system used in the current study, as will be discussed in the following sections, the elastic modus obtained at lower strain rate is chosen and a common Poisson’s ratio of 0.3 was used as the material properties for the Achilles tendon in the current foot FE model. For the flexor tendons, the literature contains the least amount of information pertaining to their mechanical properties. It was stated that the stiffness values of flexor hallucis longus (FHL) and peroneus brevis (PB) were 43.3 N/mm (± 14.1) and 43.6 N/mm (± 18.9) (Maffulli et al., 2008), respectively. Using the relation, a flexor tendon of the foot with a typical cross-sectional area of 12.5 mm2 was calculated to have a Young’s modulus of 450 MPa (GarciaGonzalez et al., 2009). Thus, the Achilles and other flexor tendons, were idealized as isotropic linear elastic materials with different Young’s moduli of 816 MPa (Wren et al., 2001) and 450 MPa (Garcia-Gonzalez et al., 2009), and a common Poisson’s ratio of 0.3. A summary of material properties to define different tissues for finite element modeling is given in Table 2.2.2.1. 36 Table 2.2.2.1. Summary of FE model listing element type and material properties for different model entities. E (MPa), v Crosssection (mm2) 7300, 0.3  1.01, 0.4  Ligament 2-node tension-only truss 260, 0.4 18.4 Achilles tendon 4-node tetrahedral continuum 816, 0.3  Other flexor tendons 2-node tension-only cable 450, 0.3 12.5 350, 0.4 290.7 Entity Bone Cartilage Plantar fascia Plantar soft-tissue Element type (ABAQUS) 4-node tetrahedral continuum 4-node tetrahedral continuum 2-node tension-only truss 4-node tetrahedral continuum Hyperelastic  Reference Cortical and cancellous bone properties weighted by Nakamura et al. (1981) Indentation test conducted on 1st MTP joint by Athanasiou et al. (1998) Properties obtained from collateral ligaments of the human ankle joint by Siegler et al. (1988) Tensile modulus obtained at lower strain rate (Wren et al., 2001) Calculation based on the stiffness values of flexor hallucis longus and peroneus brevis (GarciaGonzalez et al., 2009) Wright (1964)) and Rennels, Stress-strain curve determined experimentally by Chen et al. (2011) 37 2.2.3 Constitutive model for plantar soft tissue 2.2.3.1 Hyperelastic material model The plantar soft-tissue is a typical load-bearing soft tissue that undergoes very large strains/deformation (large-strain elasticity) with strongly non-linear stressstrain behavior when subject to high loads during gait. Furthermore, the plantar soft-tissue in normal foot is often rich in fluid content, and its material behavior under compression often resembles those of most elastomers (i.e. solid rubberlike materials), which exhibit very little compressibility compared to their shear flexibility. For these materials, their stress-strain relationship is suitable to be derived from a ‘strain-energy density function’. This function defines the strain energy stored in the material per unit of reference volume (volume in the initial configuration) as a function of the strain at that point in the material. For this reason, these rubberlike materials are usually referred to as hyperelastic materials, as is the case for the plantar soft tissue. And various hyperelastic material models have been successfully used to model plantar soft tissue behavior in finite element simulation (Lemmon et al., 1997, Cheung et al., 2005, Erdemir et al., 2006). There are several forms of strain energy potentials available in ABAQUS to model approximately incompressible isotropic elastomers: the Arruda-Boyce form, the Marlow form, the Mooney-Rivlin form, the neo-Hookean form, the Ogden form, the polynomial form, the reduced polynomial form, the Yeoh form, and the Van der Waals form (Hibbert and Karssonn, 2006). This study focuses on the Ogden-form hyperelastic constitutive model that are widely used in the 38   U  2.6 The models are described below. Once a particular strain-energy function is chosen, it is straightforward to obtain explicit relations between stress and stretch from Eqs. (2.1) and (2.6). 2.2.3.2 Ogden model A very common form of strain energy function is that proposed by Ogden (1972): U 1 ,  , 3     i 1 i N i 1   2 i  3 i /  i 2.7 where N is a positive integer (normally it can be taken smaller than 3, depending on its capacity for nonlinear stress-strain curve-fitting), and αi are real parameters (i.e. material constants) that describe the behavior of this rubber-like material model. These material constants, and αi, can be positive or negative, satisfying the condition that: N   i i   k i 1 2.8 where is the ground state (i.e. initial) shear modulus. Introducing Eq. (2.7) into Eq. (2.6) and imposing N = (1st-order model), we obtain the form of hyperelastic Ogden formulation with stress () and stretch () relation given as:    1   ( / 2) 1   2 2.9 For 1st-order Ogden model, we only have two unknown material constants, and α, that have to be determined by curve fitting procedure. 40 In the literature, the stress-strain curves of the plantar soft tissue are often obtained based on tests of the fat pad under the heel bone (i.e. heel pad) (Erdemir et al., 2005, Cheung et al., 2005, Lemmon et al., 1997). Since the primary focus of the current model is the sub-MTH region, the material behavior has to be determined for soft tissues specific to that location. For this purpose, an instrument-driven indentation device was developed for realistic in vivo mechanical characterization (i.e. tissue stiffness and force relaxation behavior) of the forefoot plantar soft tissue. The indentation force-displacement curve was directly measured from soft tissue under the metatarsal heads, and was used for extraction of the material constants to model such tissue. To determine these material constants, we performed additional simulations of indentation experiments on human plantar soft tissue under the 2nd MTH, based on this customized tissue tester. The parameters were chosen such that the reaction forces upon indentation optimally fit the experimental observations; this yielded = 3.75 ｘ10-2 MPa and α = 5.5. (See chapter 3, section 3.1.4.3, for detailed descriptions of plantar soft tissue material property characterization.) 2.2.4 Material model for foot-supporting surface 2.2.4.1 Insoles as supporting interface for foot Current practice in the prevention of neuropathic foot ulcers often involves prescription of accommodative in-shoe orthoses or insoles which reduce plantar pressure levels at locations of bony prominences, particularly under the metatarsal heads (MTHs) (Cavanagh et al., 2000, Lott et al., 2007, Bus et al., 41 2008). Those insoles are often fabricated using solid foams that are made up of polyhedral cells that pack in three dimensions. The foam cells can be either open (e.g. plastazote) or closed (e.g. Ethylene-vinyl acetate (EVA) foam). Soft insoles made from such foam materials have excellent energy absorption properties: the energy absorbed by foams is substantially greater than that absorbed by ordinary stiff elastic materials for a certain stress level (Nigg et al., 1988). Typical stressstrain (σε) curve of four types of foam padding material that are commonly used for fabrication of insoles in therapeutic footwear are shown in Fig. 2.2.4.1. Values were calculated from the force (F) and displacement (D) data obtained from uni-axial compression test on foam materials using INSTRON machine (HighWycombe, UK) at biomechanics lab NUS.   4F s , D L 2.10 Fig. 2.2.4.1. Stress-strain curve obtained from uni-axial compression test of deformable foam padding materials 42 The samples prepared were 30 mm in diameter (s) and 13.6 mm thick (L). Each sample was compressed up to a nominal strain of 50% at a displacement rate of mm/s. From the foam stress-strain curve, three stages can be distinguished during compression: 1. At small strains (5%) the foam deforms in a linear elastic manner due to cell wall bending. 2. The next stage is a plateau of deformation at almost constant stress, caused by the elastic buckling of the columns or plates that make up the cell edges or walls. In closed cells the enclosed gas pressure and membrane stretching increase the level and slope of the plateau. 3. Finally, a region of densification occurs, where the cell walls crush together, resulting in a rapid increase of compressive stress. 2.2.4.2 ABAQUS® hyperfoam material model In the present model, the foot-supporting surface that the foot makes contact with was considered to consists of two layers of material – a concrete base modeled as rigid body, and a 6.8 mm thick, flat, foam pad. Due to the highly-deformable nature of the foam pad, it was modeled as a compressible, hyperfoam material (2nd-order) with a strain energy function U of:      2 i   i J el  i  i   1  2 i  3 i    i  i 1  i  U Where,  i  i  2 i 2.11 2.12 43 Values of these elastomeric foam parameters of the hyperfoam model (i;αi; βi) were obtained through nonlinear least squares (NLS) minimization provided in ABAQUS (Hibbert and Karssonn, 2006). Table 2.2.4.2. shows the calculated hyperelastic constants that fitted to the set of stressstrain (σε) curve data of Plastazote, which will be used in the modeling study. Table 2.2.4.2. The material constants of the foam pad used as the foot-supporting interface from the hyperfoam strain energy function α1 (MPa) β1 1 α2 (MPa) β2 2 0.467 15.5 0.021 -0.11 3.62 -1.05 2.3 Loading and boundary conditions To load the foot FE model physiologically, realistic musculoskeletal loads needs to be defined. Although previous FE modeling of the foot has demonstrated the potential of such numerical modeling to investigate a number of topics in foot biomechanics, such as plantar pressure prediction (Lemmon et al., 1997, Cheung et al., 2005), ligament failure (Wu, 2007), plantar fascia release (Gefen, 2002), and internal tissue stresses (Gefen, 2003, Chen et al., 2010b), most existing models primarily focus on static standing posture, and modeling of the muscles has been largely ignored. Thus, compared to the majority of existing foot models which focus on standing (Gefen, 2002, Cheung et al., 2005, Chen et al., 2010a), the current model attempted to simulate a muscle-demanding posture during heel rise, and forefoot sub-MTH stress/strain responses is the focus. Moreover, to study the human foot mechanism, the complex interplay or interrelationship among muscular control, internal joint movement, and plantar 44 loading transfer needs to be considered. In this study, relative articulating movements of the bony joints were simulated in the entire foot (see below). This is considered as a significant improvement as compared to previous FE models which have joints that were over-constrained by “fusing” them (Jacob, 2001b, Gefen, 2002, Wu, 2007, Gefen et al., 2000, Gefen, 2003) or using kinematiccontrolled ‘connectors’ (Budhabhatti et al., 2007, Lemmon and Cavanagh, 1997). Neither of these two reflect the anatomical joint constraints that are actually governed by the bony congruence facets and passive stabilizers (i.e. ligaments and fascia) (Leardini et al., 2000). The modeling strategy utilized here enabled the foot bones to be self-adapted to the static-equilibrium positions in response to the external loading boundary conditions. Moreover, a model that includes joint articulation could be potentially validated via joint kinematics and contact forces obtained experimentally (Gefen et al., 2000). 2.3.1 Articular joint movement Relative articular joint movements were included in the foot model. These are located at the ankle (i.e., joint between articular surface of the tibia and fibula and talus), tarsus (i.e., adjacent articulations among talus, calcaneus, navicular, medial, intermediate and lateral cuneiform, and cuboid), tarsometatarsal (i.e., the medial, intermediate and lateral cuneiforms, and the cuboid, articulating with the bases of the metatarsal bones) and metatarsophalangeal (MTP) joints, with the aim to represent the physiological condition experienced by a weight-bearing foot. To simulate these joint interactions, ABAQUS surface-to-surface “contact” 45 elements (Master/Slave approach, see Fig. 2.3.1.1.) were created for the corresponding potential contact regions (i.e., articular surfaces) of these bony joints. Nodes on one surface (the slave surface) contact the discretized segments on the other surface (the master surface). This allows the bones to slide over one another without friction. Fig. 2.3.1.1. Master/Slave approach to model contact interactions. The contact behavior strictly follows the kinematic implications that slave nodes cannot penetrate master surface segments (Hibbert and Karssonn, 2006). Twenty pairs of articulations were modeled as a whole; for ankle joint interaction, for tarsus joint interactions (including for subtalar joint), for tarsometatarsal joint interactions and for metatarsal-phalangeal joint interactions. As the friction was neglected (because physiologically, synovial fluid lubricates the articulating joints which significantly reduces friction), only compression forces were transmitted through adjacent bone contacts. The geometry of individual articular cartilage of these bony joints was not accounted for; instead, their mechanical properties that contribute to normal joint loading transfer were modeled. Thus, the governing stiffness during contact was chosen 46 to be 1.01 MPa, based on the compressive material properties of the foot joint cartilage obtained from the 1st MTP joints (Athanasiou et al., 1998). 2.3.2 Plantar flexor muscle forces 2.3.2.1 Roles of the extrinsic plantar flexor muscles The functional roles of the extrinsic plantar flexor muscles are to provide stability of the foot during the stance phase (Sutherland et al., 1980, O'Connor and Hamill, 2004). Anatomically, these musculotendinous units often have specialized longtendon structures, which allow them to affect multiple joints inside the foot. This multi-articular characteristic in extrinsic flexors facilitate control and stabilizing of major bony joints, including the ankle, the subtalar, and indirectly the metatarsophalangeal (MTP) joints, and thus provide primary coordination of the stance-phase placement of the foot, which is essential for normal weight-bearing of the foot during gait (Perry, 1992). The gastrocnemius-soleus (G-S) complex associated with the Achilles tendon is the most dominant extrinsic plantar flexor. Various studies using cadaveric foot models have established the important biomechanical linkage among the Achilles tendon, plantar fascia and MTP joints, both statically (Carlson et al., 2000) and dynamically (Erdemir et al., 2004), early described by (Hicks, 1955) as the Windlass mechanism. With the foot stabilized by flexor muscles upon heel-rise, substantial dynamic ground reaction forces (GRF) are imposed solely onto the forefoot, generating highly localized stresses, particularly underneath the metatarsal heads (MTHs) (Chen et al., 2010b). Such potentially 47 ‘detrimental’ stresses may only last for a short-dwelling time in a normal foot, provided that the coordinative muscles involved activate the right amount of force at the exact right time during gait (Perry, 1992, Hayafune et al., 1999). 2.3.2.2 Calculation of musculoskeletal loads As the areas under the MTHs are of particular clinical interest, the instant the forefoot force reaches a peak (i.e. second peak GRF probably corresponding to the sub-MTH peak pressure) was chosen for simulation. At the commencement of the simulation, forefoot kinematic data was used to assist in replicating the foot pose relative to the ground. Specifically, the second metatarsal shaft was angled at approximately 25 degrees to the horizontal to reflect mean forefoot orientation in the sagittal plane (i.e. the main loading plane) at push-off (Fauth, 2002). The tibia, fibula and the superior surfaces of the soft tissue were fully constrained. Application of extrinsic muscle forces were simulated via force vectors in axial alignment with the tendons attached (Fig. 2.3.2.2.). During push-off, the ground reaction force (GRF) vector, acting at its center of pressure at the forefoot segment, causes a bending moment along the ankle joint. This depends on the joint positions and is balanced by the plantar flexor muscles comprising the gastrocnemius-soleus (G-S) complex, Tibialis posterior (TIBP), Flexor hallucis longus (FHL), Flexor digitorum longus (FDL), Peroneus brevis (PB) and Peroneus longus (PL). Note that the G-S complex is dominant, since the other five extrinsic flexors have only small lever arms. 48 Analysis commenced with this baseline model, driven by musculoskeletal loading. A maximum vertical GRF of 623.1 N, approximating the second GRF peak during walking (measured for this particular subject who provides the foot geometry using Kistler force plates during gait analysis), was generated by contracting the plantar flexors at the prescribed kinematic configuration. This loading protocol mimicked the manipulations of the cadaveric study conducted by Sharkey et al. (1995), which simulated the activity of the flexor tendons by the use of linear actuators. Fig. 2.3.2.2. Cut through the element mesh of the finite element model of muscular foot and ankle complex, incorporating internal soft tissue, skeletal structures, ligaments, plantar fascia, and musculotendinous units for push-off simulation. Application of muscle forces were simulated by force vectors align with the tendons attached. FAT = Gastrocnemius-soleus complex, FTIBP = Tibialis posterior, FFHL= Flexor hallucis longus, FFDL = Flexor digitorum longus, FPB = Peroneus brevis, and FPL = Peroneus longus. 49 The initial configurations of the metatarsals were affected slightly because of considerable deformation of the soft tissues and articulation of various bony joints. Solution of the model converged with the 2nd metatarsal shaft oriented at 26.8 degrees to the horizontal. Table 2.3.2.2. The input forces in the muscles applied through the nodes connected to tendon elements to drive the foot finite element model th Muscle forces i muscle No. of nodes #1 Gastrocnemius Soleus FAT 1620N 30 #2 Tibialis posterior FTIBP 267N #3 Flexor hallucis longus FFHL 130N #4 Flexor digitorum longus FFDL 81N #5 Peroneus brevis FPB 91N #6 Peroneus longus FPL 193N The corresponding total muscle force (TMF) was computed and the forces in individual muscles are proportional to their physiological cross-sectional areas (PCSA); i.e. the ith muscle force (MF) is given by: MFi  PCSAi  PCSA j  TMF i  1,2,3,4,5,6  2.13 j 1 This method was similar to the one used by Salathe and Arangio, (2002) in an analytical foot structure model for estimation of foot flexor muscle forces. The forces calculated in the present study are listed in Table 2.3.2.2. This corresponds to approximately 50% of the maximum contractile stresses (~0.40 MPa) in plantar flexor muscles conservatively estimated by Wickiewicz et al., 50 (1983). Additional analysis was also undertaken, whereby activation of toe flexors was not considered (i.e. forces applied to FHL & FDL approach zero), in order to determine the significance of toe flexor muscle forces in realistic toe load-bearing. The models were solved using the general purpose FE analysis package ABAQUS (SIMULIA). 2.3.3 Foot-ground interaction with frictional contact 2.3.3.1 Coulomb friction To model the foot-ground interaction, ABAQUS contact capacities that handle general frictional contact conditions were utilized. Sliding contact algorithm was implemented in ABAQUS to model such interaction at the foot-supporting interface (Fig. 2.3.3.1.). Coulomb's law of friction is enforced globally over a surface segment within a contact pair: known as a master/slave approach. Fig. 2.3.3.1. Relative sliding of points with contact constraint. Note the possible evolution of contact between node 101 and its master surface, BSURF, involving sliding contact conditions with friction. 51 Using this modeling approach, the contact tractions, including shear interaction, over the whole plantar surface segments can be computed. Moreover, this algorithm also accounts for the stick, slip or separation of contacting surfaces that may be present during foot-ground interaction. In terms of frictional/shear interactions, the Coulomb's law basically describes the general mechanical principles of friction, and it belongs to the classical Laws of Friction. Thus, Coulomb's law follows the three rules of classical Laws of Friction:  Friction is independent of sliding velocity.  Friction is independent of the contact area  Friction is proportional to the normal force, with constant of proportionality, µ. According to Coulomb friction, the relationship between friction force (Fy) and normal force (Fz) can be simply expressed as: Fy   Fz ,   Fy Fz  Tan( ) 2.14 Coulomb friction is used to model the frictional contact between the foot and its supporting surface. As a result of friction, a shear stress proportional to the local contact pressure would be applied to the surface of the deforming plantar soft tissue. However, in reality, the frictional contact at foot-supporting interface can be more complex, as such contact can be an extremely non-linear dynamic problem (see below). 2.3.3.2 Contact problems at foot supporting interface 52 The classical laws of friction offer a paradigm in which the resistance to relative motion is determined by the coefficient of friction, µ. As discussed above, in this traditional paradigm, coefficients of friction are material-dependant constants that are independent of time, velocity and contact pressure. In practice, the classical friction paradigm adequately describes only the time averaged behavior of some simple interactions between uniform, rigid surfaces. As Marpett (2002) argues, Coulomb’s “Laws” are not fundamental physical laws but rather “abstractions” that have been widely taught and adopted. Classical friction does not adequately describe the complex interactions between compliant, resilient, non-uniform surfaces such as those found in typical foot and its support’s interfacial contact problems. In these problems, friction is actually affected by the normal force, by contact area, by sliding velocity, temperature, humidity and other factors. Considering the complex nature of such a contact problem, to produce accurate 3-D contact stresses in the numerical modeling, the coefficient of friction or shear traction ratio, µ, has to be measured experimentally and input into the foot model. This is, in fact, one of the primary sources of complexity that is typically encountered in modeling foot-ground interaction. Previous modeling studies have largely relied on those empirical values of µ. This is probably one of the reasons that majority of existing finite element models of foot-supporting interface are exclusively focused on plantar pressure prediction, while only few researchers explored potential for modeling plantar shear interaction. In this study, however, experimental methods will be developed to measure and 53 calculate the actual shear traction ratio, µ, during foot-ground interaction. Details about this experimental study will be given in Chapter 2. 2.3.4 Finite element model outputs With full muscular loads applied, this baseline foot model was successfully solved in a typical geometry at heel rise, with the ankle and metatarsophalangeal joints maintained at plantar-flexed and extended configurations, respectively (Fig. 2.3.4.1). The ABAQUS CAE/post-processor was used to report results from finite element calculations. There were several model outcome measures of primary interest, the contact stresses at foot-supporting interface, bony joint movement, and von Mises stresses (VMS) in bony structures and internal tissue strains in ligaments, plantar fascia and plantar soft tissue, and the model’s experimental validation will be given in Chapter 4. 54 Fig. 2.3.4.1. The finite element predicted stress response of a whole human foot subjected to comprehensive musculoskeletal loading corresponding to a heel-rise posture. Outcome measures of primary interest in foot biomechanics, including (A) stress distributions of bulk soft tissue, (B) metatarsal bones, (C) ligaments and plantar fascia, can be obtained. 55 [...]... were included in the foot model These are located at the ankle (i.e., joint between articular surface of the tibia and fibula and talus), tarsus (i.e., adjacent articulations among talus, calcaneus, navicular, medial, intermediate and lateral cuneiform, and cuboid), tarsometatarsal (i.e., the medial, intermediate and lateral cuneiforms, and the cuboid, articulating with the bases of the metatarsal bones)... Karssonn, 20 06) Table 2. 2.4 .2 shows the calculated hyperelastic constants that fitted to the set of stressstrain (σε) curve data of Plastazote, which will be used in the modeling study Table 2. 2.4 .2 The material constants of the foam pad used as the foot- supporting interface from the hyperfoam strain energy function α1 (MPa) β1 1 2 (MPa) 2 2 0.467 15.5 0. 021 -0.11 3. 62 -1.05 2. 3 Loading and boundary... during gait (Perry, 19 92, Hayafune et al., 1999) 2. 3 .2. 2 Calculation of musculoskeletal loads As the areas under the MTHs are of particular clinical interest, the instant the forefoot force reaches a peak (i.e second peak GRF probably corresponding to the sub-MTH peak pressure) was chosen for simulation At the commencement of the simulation, forefoot kinematic data was used to assist in replicating the. .. joints inside the foot This multi-articular characteristic in extrinsic flexors facilitate control and stabilizing of major bony joints, including the ankle, the subtalar, and indirectly the metatarsophalangeal (MTP) joints, and thus provide primary coordination of the stance-phase placement of the foot, which is essential for normal weight-bearing of the foot during gait (Perry, 19 92) The gastrocnemius-soleus... 1 At small strains (5%) the foam deforms in a linear elastic manner due to cell wall bending 2 The next stage is a plateau of deformation at almost constant stress, caused by the elastic buckling of the columns or plates that make up the cell edges or walls In closed cells the enclosed gas pressure and membrane stretching increase the level and slope of the plateau 3 Finally, a region of densification... replicating the foot pose relative to the ground Specifically, the second metatarsal shaft was angled at approximately 25 degrees to the horizontal to reflect mean forefoot orientation in the sagittal plane (i.e the main loading plane) at push-off (Fauth, 20 02) The tibia, fibula and the superior surfaces of the soft tissue were fully constrained Application of extrinsic muscle forces were simulated via force... complex associated with the Achilles tendon is the most dominant extrinsic plantar flexor Various studies using cadaveric foot models have established the important biomechanical linkage among the Achilles tendon, plantar fascia and MTP joints, both statically (Carlson et al., 20 00) and dynamically (Erdemir et al., 20 04), early described by (Hicks, 1955) as the Windlass mechanism With the foot stabilized... characterization.) 2. 2.4 Material model for foot- supporting surface 2. 2.4.1 Insoles as supporting interface for foot Current practice in the prevention of neuropathic foot ulcers often involves prescription of accommodative in- shoe orthoses or insoles which reduce plantar pressure levels at locations of bony prominences, particularly under the metatarsal heads (MTHs) (Cavanagh et al., 20 00, Lott et al.,... ordinary stiff elastic materials for a certain stress level (Nigg et al., 1988) Typical stressstrain (σε) curve of four types of foam padding material that are commonly used for fabrication of insoles in therapeutic footwear are shown in Fig 2. 2.4.1 Values were calculated from the force (F) and displacement (D) data obtained from uni-axial compression test on foam materials using INSTRON machine... proportional to the local contact pressure would be applied to the surface of the deforming plantar soft tissue However, in reality, the frictional contact at foot- supporting interface can be more complex, as such contact can be an extremely non-linear dynamic problem (see below) 2. 3.3 .2 Contact problems at foot supporting interface 52 The classical laws of friction offer a paradigm in which the resistance . tarsus joint interactions (including 1 for subtalar joint), 5 for tarsometatarsal joint interactions and 5 for metatarsal- phalangeal joint interactions. As the friction was neglected (because. metatarsal shaft was angled at approximately 25 degrees to the horizontal to reflect mean forefoot orientation in the sagittal plane (i.e. the main loading plane) at push-off (Fauth, 20 02) . The. (tibia, fibula, talus, calcaneus, navicular, medial, intermediate and lateral cuneiform, and cuboid, the medial, intermediate and lateral cuneiforms, and the cuboid, metatarsal bones and phalanges)