Advance Publication by J-STAGE Mechanical Engineering Journal

© The Japan Society of Mechanical Engineers

Advance Publication by J-STAGE

Mechanical Engineering Journal

DOI:10.1299/mej.21-00200

Received date : 21 May, 2021

Accepted date : 20 August, 2021

J-STAGE Advance Publication date : 5 September, 2021


Developing a three-dimensional numerical foot model andidentifying the loading condition for designing a stable sole for

running shoesMai NONOGAWA∗, Seigo NAKAYA∗, Masashi ISOBE∗, Kenzen TAKEUCHI∗∗ and Hideyuki AZEGAMI∗∗∗

∗ ASICS Corporation6-2-1, Takatsukadai, Nishi-ku, Kobe 651-2271, Japan

E-mail: [email protected]∗∗ Faculty of Engineering and Design, Kagawa University

1-1, Saiwai-cho, Takamatsu, 760-8521, Japan∗∗∗Graduate School of Informatics, Nagoya University

Furo-cho, Chikusa-ku, Nagoya 464-8601, Japan

AbstractThis study presents methods to develop a three-dimensional numerical foot model and to identify the loadingcondition that is used to design a stable sole for running shoes. In a previous study, the authors proposed a methodto optimize the shape of the sole to increase stability while maintaining the cushioning property. In the problemformulation, the loading condition was given as a boundary force distributed on the top surface of the sole. Theaim of this study is to replace the loading condition with the force and moment at the origin of the ankle jointcoordinate (AJC) system by modeling a foot with a finite element model. A finite element model of a foot isconstructed using X-ray CT image data, and consists of bony structures, soft tissue, and plantar fascia. The plantarfascia is set at the bottom of the bony structures. The force and moment used in the finite element analysis areidentified by inverse dynamic analysis using an experimental measurement in the practical operation of the groundreaction force (GRF) when the GRF in the direction of the foot length becomes minimum. In the finite elementanalysis, the finite deformation containing the contact condition between the bottom surface of the foot and theground representing a sole made of resin is considered. For an index of the shoe stability, we define a heel eversionangle (HEA) by the rotational angle of the heel with respect to an axis in the foot length direction and evaluate itby finite element analysis. The validity of the finite element foot model as well as the force and moment obtainedin this study are confirmed based on the agreement in the HEA results between the experiment and finite elementanalysis.

Keywords : Foot model, Running, hoe, Finite element analysis, Inverse dynamic analysis

1. Introduction

As running is a popular sport in the world, running shoes are made to protect runners from injuries and improveperformance (Cavanagh, 1980; Nishiwaki, 2008). Considering injury prevention, the concept of shoe stability has beenproposed to maintain the correct posture of the foot while running (e. g., Nigg (1980); Woensel and Cavanagh (1992);Areblad et al. (1990); Stacoff et al. (1992)). In the previous studies, pronation has been reported to occur in the foot whilerunning. Pronation is a series of movements consisting of the inward (eversion) rotation of the calcaneus, falling of thearch, and inward rotation of the lower extremity. Excessive pronation can lead to running injuries. Shoe stability wasevaluated experimentally using the rotational angle of the heel with respect to an axis in the foot length direction (gradientof the heel in the foot width direction) while running when the ground reaction force (GRF) in the foot length directionbecomes minimum (Woensel and Cavanagh, 1992; Areblad et al., 1990). In this study, the rotational angle is referred tothe heel eversion angle (HEA).

S


Fig. 1 Typical example of X-ray CT images of foot

To design a sole that maximizes shoe stability while maintaining a constant cushioning property, the authors formu-lated a shape optimization problem for shoe soles and presented its solution using a finite element model (Nonogawa et al.,2020). In that study, the loading condition was assumed to be given as a boundary force distributed on the top surface of asole measured by an experiment of practical running, and it was assumed that the boundary force is invariable in the shapeoptimization process. However, in reality the boundary force would change as the shape of the sole changes. Moreover,shoe stability is defined by the HEA, while the angle of an approximation plane determined from the deformation of thetop surface of the sole was used as the index of stability. To use the original definition of shoe stability, it is necessary tointroduce a finite element model of a foot containing a heel.

Numerical foot models have been used in the investigation of running shoes. Cheung et al. (2004) reconstructed afinite element foot model using magnetic resonance image (MRI) and evaluated the effects of varying the elastic modulusof the plantar fascia on the stress distribution of the bony, ligamentous and soft tissue structures. Hsu et al. (2008) createda finite element foot model using a modeler in a finite element analysis software and used the model to analyze the stresson the junction between the plantar fascia and calcaneus when wearing different shapes of insoles. In Cheung and Nigg(2008), a finite element foot model was reconstructed from MRI, and sensitivity analyses were conducted to study theload distribution and spatial motion of the ankle-foot structures with posterior tibial tendon dysfunction (PTTD). Gu et al.(2011) reconstructed a finite element foot model from X-ray computed tomography (CT) images and studied the use ofthe numerical model in subject-specific midsole design and optimization for heel pressure reduction using a midsole plugunder the calcaneus area. In these studies, the prescribed Achilless tendon force and GRF were used as loading conditions.Meanwhile, Chen et al. (2019) obtained the loading condition using motion capture analysis. They constructed a finiteelement foot model from MRI data and analyzed the foot arch deformation and plantar fascia tension in different footstrike techniques while running. In their analysis, the boundary conditions such as segmental kinematics, foot muscleforce, and ankle joint reaction force were analyzed using the captured motion data.

In this study, we developed a finite element foot model to evaluate shoe stability, which provides a bony structure tocalculate the HEA, as well as a method to identify the external load at the origin of the ankle joint coordinate system (AJC)based on the experimental data of the GRF during practical running on a sheet of resin foam laid on the running course.We used X-ray CT images to reconstruct a finite element foot model consisting of bony structures and encapsulated softtissue and added truss elements representing the stiffness of the plantar fascia. To identify the force and the momentapplied at the origin of the AJC, we used the inverse dynamic analysis from the experimentally obtained GRF. Finally, thevalidity of the proposed methods is confirmed by comparing the results of the experimentally measured HEA with thoseobtained by the finite element analysis considering that the finite deformation of the contact between the bottom surfaceof the foot and the ground consists of resin foams.

2. Construction of a numerical foot model

The objective of this study is to present a scheme to evaluate the HEA using a numerical foot model coupled withthe external load identified from experimental data. In this section, we present a scheme to construct a finite element footmodel that contains a bony structure and plantar fascia, which comes into contact with the ground made of resin foams,and is representative of the soles used in running shoes. To maintain the geometrical accuracy of the bony structure andboundary shape of the foot, we used X-ray CT images of the foot of a healthy woman who is in the supine position andwhose foot is unloaded. In the supine position, the foot tends to bend plantarly at rest. Considering the angle of the footand that the lower extremity is approximately vertical when evaluating the stability, the CT images were measured bybending the foot dorsiflexively using a jig within a reasonable range. A typical example of an X-ray CT images is shown


(a) Perspective view of the total elements (b) Bony structure and plantar fascia

Fig. 2 Finite element model of a foot and the ground representing a sole.

in Fig. 1. The subject is 25 years old with a weight of 45 kg and height of 1.58 m, and has no history of a foot injury.According to the National Health and Nutrition Survey (Ministry of Health, Labor and Welfare, 2017), the average weightand height of Japanese women in their 20s are 51 kg and 1.58 m. With the image data, we created standard triangulatedlanguage (STL) data of the outer boundary and bony structure of the foot using a medical image processor (SimplewareSoftware, Synopsys, Inc.). From the STL data, a finite element foot model using four-node tetrahedral elements wasconstructed by a preprocessor of finite element analysis (HyperMesh, Altair Engineering, Inc.). The role of the plantarfascia is to support the arch in the foot length direction, connecting the calcaneus with the ends of the metatarsal bone,which are the tips of the arch. In addition, from micro observation, the force in the foot length direction is dominant (Guoet al., 2018; Kitaoka et al., 1994). Then, we replicated the stress induced by the plantar fascia using the truss elementsconnecting the calcaneus and the ends of the metatarsal bony structures. The model does not consider contact with softtissue, referring to the study by Wright and Rennels (1964); Cheung et al. (2004). We assumed that these truss elementsresist a tensile force and become invalid to the compression force. The resin foam contacting the bottom of the footconsists of eight-node hexahedral elements. Figure 2 presents the foot and ground models constructed by the indicatedscheme. For the foot model consisting of bony structure, soft tissue, and plantar fascia, 35,705 elements and 7,840 nodeswere used.

We use the constitutive laws of a linear elastic material presented by Gefen (2003) for the bony structure and ahyperelastic material proposed by Lemmon et al. (1997) for the soft tissue. The surface of the bony structure is connectedto the soft tissue by sharing nodes. The adjacent bones are connected by the friction-free contact condition. The massdensities of each part were determined referring to the study by Clauser et al. (1969). For the constitutive law of theground, we prepared three types of resin foam with hardness properties of 45◦, 55◦, and 65◦ with a Type E durometeraccording to ISO 48-4. In this study, the three types are shown as EH45, EH55, and EH65. The Young’s moduli of theresin foams were measured by the uniaxial pressure test. The strain rates in the running motion and material test were 100

s−1 and 101 s−1, respectively. The effect of the difference on material rigidity is considered to be miniscule (Jeong et al.,2012). Table 1 lists the material properties used in this study. We assume that the coefficient of friction between the footand resin foam is 0.3.

Table 1 Material properties used in this study.

Part Young’s modulus Poisson’s ratio sectional area mass density(MPa) (mm2) (mg/mm3)

Bony structures 7300 0.3 - 1.66Soft tissue Hyper elastic∗ Incompressible - 0.942Plantar fascia 700 - 290.7 1.30Resin foam EH45 3.96 0.2 - 0.17Resin foam EH55 5.78 0.2 - 0.2Resin foam EH65 7.90 0.2 - 0.23∗ Parameters of Ogden model: α1 = −0.011, µ1 = 3.0, α2 = 0.00027, µ2 = 23, α3 = 0.092,

µ3 = −1.2, D1 = 0.0049, D2 = 0.0, D3 = 0.0


(a) Perspective view (b) Top view

Fig. 3 Ankle joint coordinate (AJC) system.

(a) Forces and moments acting on foot (b) Points on a rigid body representing the foot.

Fig. 4 Forces and moments at specified points used in the inverse dynamic analysis.

3. Identification of the loading condition

To analyze the HEA based on the numerical results obtained using the finite element method, we need to developa method to identify the external load on the numerical model from the experimental data measured during practicalrunning. In this section, we use the idea of inverse dynamic analysis.

Preparing for the following discussions, we defined a coordinate system for a foot model, as shown in Fig. 3. Theorigin is defined at the midpoint of the malleolus in the lateral view, and the center point in the foot width direction in thehorizontal cross section of the foot. We assume that the direction from the origin to the third distal phalanx is defined asthe y axis, and the horizontal and vertical directions are defined as the x and z axes, respectively. In this study, we refer tothis coordinate system as the ankle joint coordinate system (AJC).

The external force and moment acting on a foot can be evaluated using various methods, for which one of the mostreliable data is the GRF measured while running. In this study, we developed a method to identify the external force andmoment acting at the origin of the AJC from the GRF using inverse dynamic analysis (Kuitunen et al., 2002; Hansenet al., 2004). We assume that the foot shown in Fig. 4 (a) is a rigid body as defined in Fig. 4 (b), and the GRF at the centerof pressure (CoP) and the trajectories of the two markers attached to the lateral and medial points of the malleolus aremeasurable. We denote the force and moment of the GRF by (FG,MG) : (0, tT)→ R6, and those at the origin of the AJCby (FA,MA) : (0, tT) → R6, where tT is a positive constant representing the time limit of the measurement. Moreover,we denote the velocity and the angular velocity at the center of gravity (CoG) of the foot obtained from the trajectories by(u,w) : (0, tT) → R6 and the accelerations by (u, w) : (0, tT) → R6. Using the definitions and the equation of motion, the(FA,MA), which is needed in the finite element analysis, is given by the following:

FA = mu + FG − mg, (1)


Fig. 5 Result of (FA,MA) identified from the experimental measurement data.

MA = Iw + w × (Iw) − rA × FA + rG × FG

+ MG. (2)

Here, we use (rA, rG) : (0, tT)→ R3×3 as the vectors from the CoG to the origin of the AJC and CoP, m ∈ R, and I ∈ R forthe mass and moment of inertia, respectively, determined by referring to Ae et al. (1992) and g ∈ R3 as the accelerationof gravity.

In this study, we set a course on which a sheet made of resin foam EH65 is laid, and the subject is allowed to runat a speed of 2.8 m/s. The GRF is measured at the CoP by a pressure measurement sensor (Multi component force plate,Kistler Holding AG), and the trajectories of the two markers attached to the lateral and medial malleolus are obtained bya motion capture system (VICON-MX, Vicon motion systems, Ltd.).

Figure 5 presents the result of (FA,MA) calculated by (1) and (2) using the measurement data.

4. Numerical analysis of stability

In this study, we propose a method to evaluate stability, which is defined by the HEA, by finite element analysis usingthe numerical model developed in Section 2 and the external load obtained in Section 3. The static hyperelastic problemwas solved using a nonlinear finite element analysis solver (Abaqus 2018, Dassault Systemes).

The loading condition was provided as follows: we set an independent node at the origin of the AJC, which isconnected to all nodes on the boundary of the talus by the rigid elements, and applied the external force and moment tothe independent node. The external force and moment were evaluated at the time when the y-axis component of the GRFbecomes minimum as follows:

FA = (8.65, 153.08,−756.09) N, (3)

MA = (−33.51, 3.22, 0.44) Nm. (4)

The body force due to the whole body weight was applied to all elements for standardization. The centrifugal and inertialforces were neglected. For the finite element analysis, the nonlinearities of finite deformation, hyperelasticity, and thecontact condition between the bottom surface of the foot and the ground representing the sole were considered.

The HEA was evaluated by the rotational angle of the straight line through the two points shown in Fig. 6 around they-axis of the AJC. The bar for EH65 in Fig. 7 indicates the result of HEA when the FA and MA in (3) and (4), respectively,were used. Moreover, the bars for EH45 and EH55 in Fig. 7 indicate the results of HEA when the same FA and MA,which were obtained using the resin foam EH65 and not obtained using the resin foam EH45 or EH55.


(a) Perspective view (b) Top view

Fig. 6 Two points used to evaluate HEA.Fig. 7 Comparison of HEAs obtained by numerical analysis and experiment.

(a) EH45 (b) EH55 (c) EH65

Fig. 8 Foot pressure distributions measured (upper) on the top surface of resin foams and evaluated (lower).

For comparison, the experimental results of the HEA on each resin foam are shown in Fig. 7. The experimentswere performed in the same manner as indicated in Section 3. The HEA was measured at the same points from theresults obtained by the motion capture. The plotted results are the average values and variances of the two trials for eachhardness. Considering the experimental results of the HEA in Fig. 7, we can confirm that when the hardness of the resinfoam decreased, the HEA increased. A similar variation in the HEA with respect to the hardness of the resin foam wasobtained from the numerical analysis. The foot pressure distributions were also measured using a pressure sensor sheet(F-scan, Nitta Corporation) with a thickness of 0.15 mm attached the bottom of foot. The experiment was carried out inthe same protocol shown in Section 3. Figure 8 shows the pressure distributions of experiments and numerical analysis.In Fig. 9, the contact areas and maximum values of pressure distributions in the experiments and numerical analyses arecompared. From these results, the contact area tended to be larger and the maximum value was smaller as the hardnessof resin foam decreased. The location of the maximum value was different. In the experiment, it is the outside of themidfoot, but in the numerical analysis, it is the forefoot or heel.

5. Discussion

A decrease in the hardness of the resin foam worsens shoe stability (Clarke et al., 1983; Hamill et al., 1992; Milani


(a) Contact area (b) Maximum value of pressure

Fig. 9 Comparisons of contact areas and maximum values of pressures obtained by numerical analysis andexperiment.

et al., 1997), which is shown by the results in Fig. 7. In addition, the present approach has indicated that the numericalmodel and the external force and moment, which were identified only by the resin foam (EH65), are applicable to a widerange of resin foam hardness. In reality, when the load conditions obtained from an experiment using EH45 and EH55were used in the numerical analysis, the same tendency for stability was observed. The foot pressure distributions shownin Fig. 8 were also able to reproduce the tendency of the experimental results that verified the difference in the hardness ofresin foam, that is, when the hardness is low, the contact area is large and the maximum value is small. This applicability isuseful in the design of the sole. As indicated in the Introduction, the authors are developing an optimal design system forthe sole (Nonogawa et al., 2020). In the present version of the system, the loading condition is given as a fixed boundaryforce on the top surface of the sole. However, by using the finite element foot model and the external force and momentinstead of the fixed boundary force, we can quantify the effect of the change in the shape of a sole on the boundary force.

Comparing the foot pressure distribution obtained from the experiment and the numerical analysis shown in Fig. 8and 9, these results have same tendency that the contact area is large and the maximum value is small when the hardnessof the foam material is low. On the other hand, the location of the maximum pressure value occurred was different. Inthe experiment, the maximum value occurred on the lateral side of the midfoot, but in the numerical analysis, it occurredon the forefoot or heel. It is presumed that the maximum value is occurred in the midfoot due to the large verticaldisplacement on the midfoot. The proposed foot model is composed of bony structures, plantar fascia and soft tissuefor simplification, but the actual foot has lots of ligaments and muscles. Since lots of ligaments and muscles were notconsidered in detail, the amount of vertical displacement of the midfoot could not be reproduced appropriately. Even ifthe soft tissue material was softened within the range of common sense, the vertical displacement of the midfoot could notbe reproduced in detail. However, it is also known that the majority of runners have the maximum value in the forefoot orheel as shown in the lower figures in Fig. 8, not in the upper figures shown in Fig. 8 (De Cock et al., 2006; Razeghi andBatt, 2002; Hughes et al., 1991). From the numerical simulation and literature, the foot model we developed in this studywas able to qualitatively evaluate the overall rigidity and the tendency of the distribution of contact pressure. However,in order to obtain a quantitatively accurate foot model, it turned out that the composition and placement of ligaments andmuscles must be considered more faithfully.

Regarding the time to evaluate the external force and moment, we considered the following. In the previous studiesby Woensel and Cavanagh (1992) and Areblad et al. (1990), shoe stability was evaluated when the GRF in the foot lengthdirection was minimized. In this study, we also confirmed that the same time is suitable for evaluating the external forceand moment because the contact status between the bottom of the foot model and the resin foam hardly change and theinfluence of the inertia force is small, while only the posture of the lower extremity changes.

In the modeling of the finite element foot model, the effect of the joints is considered in the following manner. Theactual joints consist of the bony structure, the cartilage surrounding the heads of bones, and the ligaments connectingthe heads of the adjacent bones. Considering the difficulty and inefficiency of modeling them with high precision, wemade a simplified model in which the heads of the adjacent bones contact friction free. The rigidities of the cartilage andligaments are reflected in the material properties of the soft tissue. Therefore, to control the rigidity of the joints, we canchange the rigidity of the soft tissue around the joints. For example, it may be effective to partially reduce the stiffness ofthe soft tissue in order to increase the vertical displacement of the midfoot.


In this study, we focused on the stability of running shoes. However, other performances of shoes other than therunning shoes (for example, walking shoes or track spikes) can be evaluated simply by changing the force and momentapplied to the origin of the AJC. For the other category, the loading condition considered differs based on the performancerequired for the shoes. In this case, we can identify the force and moment applied to the origin of the AJC at the timeconsidered by the same method presented in Section 3, and analyze an index of the performance by finite element analysisusing the foot model developed in Section 2.

6. Conclusion

In this study, we presented methods to develop a three-dimensional finite element foot model to identify the loadingcondition that is used to design a sole of a running shoe with stability. The finite element foot model consisting of bonystructure, soft tissue, and plantar fascia was constructed from X-ray CT image data. The force and moment used in thefinite element analysis were identified by inverse dynamic analysis using an experimental measurement of the GRF at thetime when the GRF in the foot length direction becomes minimum during practical running. The validity of the numericalmodel and the external load was confirmed by analyzing the finite deformations containing the contact condition betweenthe bottom surface of the foot and the ground representing a sole made of resins by a nonlinear finite element analysissolver and comparing the results of the HEA, which was defined as an index of the shoe stability, with the experimentalmeasurements. From the results, the applicability of the present approach to a wide range of resin foam hardness wasconfirmed. This applicability is useful in the design of soles. By using the numerical model and the external load, we canshorten the development lead time by reducing the experiments by numerical analyses with respect to design changes ofsoles.

Acknowledgement

We thank Nobuhara Hospital and Institute of Biomechanics, Tatsuno, Japan, for facilitating the X-ray CT scanning.

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