Modelization and identification of medical compression stocking: Part 2

Abstract

The goal of this article is to propose a practical and scientific model to complete the state of the art in the techniques of determining the pressure exerted by the medical compression stocking on the limbs. The proposed 3D model is developed to explain the mechanics of materials and in particular the compression phenomenon by the medical compression stocking on complex surfaces. This research includes three main parts: The first part presents the research field and assesses the dependence of the constraint between the crushing stress and local geometry of the medical compression stocking bottom. Meanwhile, a pragmatic strategy is provided to simplify the model substantially, based on the assumption that the medical compression stocking structure is subject to small movements induced by muscular contractions. The second part focuses on solving the exact 3D model as a generalization of Laplace's laws by dividing the complicated model into several simple sub-models, making it easier to reveal the non-trivial relations connecting the crushing stress, the local radius of curvature of the bottom, the membrane forces and the material constituting the bottom. Considering the fact that the bottom has extremely low thickness, the third part further provides the asymptotic model to obtain the limiting behavior of the 3D model, which shows that the bottom resists essentially by its tangential rigidity. Meanwhile, the influences of the radii ratio of curvature on the crushing stress are discussed and highlighted. The simulation results are also verified by the finite element method, aiming to optimize the design of medical compression stockings and increase their therapeutic effect.

Keywords

3D modeling simulation crushing stress finite elements medical compression stocking

The medical compression stocking (MCS) is a kind of medical textile product and has been widely used in compression therapy for relieving symptoms associated with venous disease and lymphatic disorders in the upper and lower limbs.^1,2 The compression principle of MCS is based on the tensioning of a band of elastic material to press on the part of the body to be compressed. The relation P = F/R, with P in hPa, F in N/cm and R in cm, is used and often mentioned by phlebologists and angiologists under the name “Laplace's law”, where R is the radius of a cylindrical body on which a stretched strip is pressed and F is the tension of this strip.^3,4 This model is shown in Figure 1 and refers to a cylindrical body. This approximation (assimilation of the cross-section of a member to a cylinder) considerably simplifies the formalization of the pressure since the radius R is constant. However, there are some questions often asked by the medical profession: what is the validity limit of the model that assimilates the member to a stack of cylinders? Is the theoretical model of a limb really the same as the actual anatomical shape of the leg shown in Figure 2? When another form of cross-section of shape is to be compressed, for example elliptic, one would then have another pressure topography on the periphery of this cross-section as shown in Figure 3. The characterization of the pressure exerted by the MCS, on a given point of the limb, can be summed up to determine the force F on the one hand and the radius of curvature on the other hand. Therefore, it is important to consider the calculation of the interface pressure of MCS according to the continuum mechanics approach, which is much more complex and complete than the surface approach adopted so far.

Figure 1.

Contact pressure by belt tension P = F/R.

Figure 2.

The anatomical shape of the ankle.

Figure 3.

Different situations of contact pressure.

In our previous published articles, we investigated dynamic yarn modeling and proposed explanations about the mechanics of MCS through its components behavior.^5–7 However, we must point out the limits of our procedure: we did not take the friction phenomena into account that are totally non-linear and random. In everyday use of clothing, there is a multitude of causes of the presence of fat (sweat, detergent, softener, pollution, etc.), in nature and in quantity in an absolutely random and unpredictable way, which totally modify inter-fiber friction.^8,9 In the past few decades, many researchers have paid more attention to the friction phenomenon and properties of MCS contact with human skin. For instance, the company Ganzoni France began a thesis to determine the interface pressure exerted on the MCS members using the approaches of the mechanics of continuous materials and in particular the method of finite elements (EF).¹⁰ The basic principle of the EF method is to subdivide the studied domain into simpler-form subdomains. Unlike the textile and surface approach, the mechanical approach of the problem consists in analyzing the stresses and deformations inside the material itself (knitting), considering nevertheless that the textile substrate is continuous and orthotropic. This work can effectively assemble the subdomains and construct the global stiffness matrix and the force vector, as well as achieve the final equation. Wei Ke and his research group not only investigated the relationship between the friction and microscopic contact behavior of MCS at different strains against a mechanical skin model, but also studied the friction behavior of five MCSs against forearm skin in vivo as a function of normal load under dry and wet conditions.^11,12 They found that the friction behavior of the MCS is in accordance with the adhesion friction model and that the contact behavior on the microscopic level can be theoretically described by assuming the fabric surface to be composed of numerous round asperities obeying the Hertz contact model. Stolk et al. developed a new method to investigate the dynamic behavior of MSC during walking, which was simulated using an artificial leg-segment model.¹³ Wang et al. developed a new measuring system that provided useful information on both the dynamic pressure and the stiffness of various compression garments.¹⁴ This new system uses a direct measuring method to monitor the dynamic pressure behavior and introduce a new stiffness index.

However, there are few people to solve a multitude of problems of identification of the constitutive law of the knitted fabric as well as the development of codes of computation. The aim of this study is to propose a practical and scientific 3D model to consider the friction behavior and complete the state of the art in the techniques of determining the pressure exerted by the MCS on the limbs. The 3D model based on the FE method is set up by a mathematical approach. This 3D modeling will deepen our knowledge in the mechanics of materials and in particular on the phenomenon of compression by MCS on complex surfaces (such as those of a cluster of varicose veins whose surface could be associated with that of an ellipsoid) where a portion of knitted MCS fabric will be pressed.

Position of the problem

The purpose of this study is to qualitatively assess the dependence of this constraint on, in particular, the material constituting the bottom and the local geometry of the member. Consider a band or MCS on one leg in Figure 4. The pressure exerted at one point down the leg will later be called “crushing stress”.

Figure 4.

(a) Pressure on the patella region; (b) leg and calf cross-section; (c) field of research.

We refer to Figure 4 for the notation used. Let M be a point belonging to the contact surface S between the leg and the bottom surface of the MCS. Let us denote by Β, a neighborhood of the bottom “centered” in M, which we consider as an opening of R³. We will start by isolating the neighborhood B. For this, we denote by F, the field surface forces on the lateral edge ∂B of B, precisely: $F : M \in \partial B \to F (M) \in R^{3}$ (unit: pressure). The field F represents in fact the efforts exerted by the complement of B on B (the rest of the bottom not isolated). The field of surface contact forces between the bottom and the leg will be designated by: $G : M \in S \to G (M) \in R^{3}$ (unit: pressure). The stress field reigning in B will be characterized by the field of symmetrical endomorphisms of R³: $σ : M \in B \to σ (M) \in R^{3} \times R^{3}$ (unit: pressure).

According to the principle of virtual powers, the equilibrium of neighborhood B is translated by

\int_{B} T_{r} (σ \cdot \frac{\partial \hat{v}}{\partial OM}) d B = \int_{\partial B} \bar{F} \cdot \hat{v} d \partial B + \int_{S} \bar{G} \cdot \hat{v} d S

(1)

where we used the following notations: Tr denotes the trace of a square matrix (thus an endomorphism). The bar represents the transposition, the point a matrix product (a composition of linear applications). Thus, if (a, b) ∈ R³ × R³, the notation:

\bar{a}

, b denotes the scalar product of a by b (

\bar{a}

is a transposed vector, that is, a linear form on R³); represents the field of the virtual speeds of the bottom:

\hat{v} : M \in B \to \hat{v} (M) \in R^{3}

;

\frac{\partial \hat{v}}{\partial OM}

represents the gradient of virtual velocities, where O is an arbitrary origin of the chosen space once and for all

\frac{\partial \hat{v}}{\partial OM} : M \in B \to \frac{\partial \hat{v}}{\partial OM} (M) \in R^{3} \times R^{3}

Under the effect of the external forces F and G, the points belonging to the isolated neighborhood will undergo displacements. We will denote by u, the vector field displacements. Specifically, if B₀ represents the low at rest (i.e. not slipped on the leg), the field u will connect the lower points before deformations (belonging to B₀) to these same points after deformations (belonging to B): $u : M_{0} \in B \to u (M_{0}) \in R^{3} such as OM = O M_{0} + u (M_{0})$ .

The deformation matrix of Green-Lagrange will be written according to

γ (u) = \frac{1}{2} (\begin{matrix} \frac{\partial u}{\partial O M_{0}} (M_{0}) + \bar{\frac{\partial u}{\partial O M_{0}} (M_{0})} \\ + \bar{\frac{\partial u}{\partial O M_{0}} (M_{0})} \cdot \frac{\partial u}{\partial O M_{0}} (M_{0}) \end{matrix})

(2)

As we know, to have a chance to find a solution to the model (1), we need a constitutive law which is the link between the inner forces at the bottom (the stress matrix σ) and the deformations (the matrix of deformations γ). We can formalize this relationship in the form

σ = f (γ) with f : R^{3} \times R^{3} \to R^{3} \times R^{3}

(3)

The problem to be solved then consists in establishing the relation between the crushing stress G and the local geometry of the bottom by solving the models (1) and (3). For that we consider in all the continuation that the contact between the bottom and the leg is persistent: there is no void in S (no detachment) or S is a connected open.

Simplifying assumptions

In practice, after placing the stocking on the leg, the lower structure is subject to small movements induced by muscular contractions. This assumption allows us to simplify models (1) and (3) substantially.

Let u₀ denote the displacement undergone by the lower points from the configuration B₀ (bottom not yet threaded) to the intermediate configuration B₁ corresponding to the lower pose on the leg at rest (muscles not contracted). The field of displacements then admits the decomposition

u = u_{0} + \tilde{u} with u_{0} : B_{0} \to B and \tilde{u} : B_{1} \to B

(4)

where

\tilde{u}

corresponds to the small displacements induced by the muscular contractions.

We then formally carry out a limited development of the constitutive law (3) in the neighborhood of u₀

σ = f (γ (u_{0})) + \frac{\partial f}{\partial γ} (u_{0}) \cdot γ (\tilde{u}) + \dots

(5)

$f (γ (u_{0})) = σ_{0}$ corresponds to the state of stress of the lower part resting on the leg.

$γ (\tilde{u})$ is the field of deformations induced by small displacements.

$\frac{\partial f}{\partial γ} (u_{0})$ is the bottom stiffness endomorphism resting on the resting leg noted R.

Moreover, given the smallness of $\tilde{u}$ , it is also possible to linearize the deformations of Green-Lagrange (the notation Sym (A) designates the symmetrical part of an endomorphism A)

γ (\tilde{u}) = \frac{1}{2} (\frac{\partial \tilde{u}}{\partial O M_{1}} + \bar{\frac{\partial \tilde{u}}{\partial O M_{1}}} + \dots) = Sym (\frac{\partial \tilde{u}}{\partial O M_{1}}) + \dots

(6)

So around the B₁ configuration, we will retain the following linearized behavior law

\begin{matrix} {\begin{matrix} σ = σ_{0} + \tilde{σ} \\ σ_{0} = f (γ (u_{0})) and \tilde{σ} = R \cdot Sym (\frac{\partial \tilde{u}}{\partial O M_{1}}) \end{matrix} \end{matrix}

(7)

Given this linearization, the bottom equilibrium is written

\begin{matrix} \int_{B} T_{r} (σ_{0} \cdot \frac{\partial \hat{v}}{\partial OM}) d B + \int_{B} T_{r} (\tilde{σ} \cdot \frac{\partial \hat{v}}{\partial OM}) d B \\ = \int_{\partial B} \bar{F} \cdot \hat{v} d \partial B + \int_{S} \bar{G} \cdot \hat{v} d S \end{matrix}

(8)

It is then possible to decompose the external forces F and G in a manner similar to the displacements u. Precisely if F₀ (or G₀) designates the surface forces exerted by the complement of B on B (respectively the crushing stress) during the installation of the bottom on the leg at rest, it is then possible to write

F = F_{0} + \tilde{F} (or G = G_{0} + \tilde{G})

(9)

where

\tilde{F}

(or

\tilde{G}

) denotes the lateral forces (respectively the crushing stress) induced by the muscular contractions. But in the absence of muscular contractions, the balance of the low is written

\int_{B} T_{r} (σ_{0} \cdot \frac{\partial \hat{v}}{\partial OM}) d B = \int_{\partial B} \bar{F_{0}} \cdot \hat{v} d \partial B + \int_{S} \bar{G_{0}} \cdot \hat{v} d S

(10)

We deduce that the stresses and displacements induced by muscular contractions ( $\tilde{σ}$ and $\tilde{}$ ) are the solutions of the following model

\begin{matrix} {\begin{matrix} \int_{B_{1}} T_{r} (\tilde{σ} \cdot \frac{\partial \hat{v}}{\partial O M_{1}}) d B_{1} = \int_{\partial B_{1}} \bar{\tilde{F}} \cdot \hat{v} d \partial B_{1} + \int_{S_{1}} \bar{\tilde{G}} \cdot \hat{v} d S_{1} \\ \tilde{σ} = R \cdot Sym (\frac{\partial \tilde{u}}{\partial O M_{1}}) \end{matrix} \end{matrix}

(11)

which is nothing else than the model in small deformations where we have geometrically confused the configurations B and B₁.

Expression of the crushing stress

The particular geometry of the neighborhood B₁ (low slipped on the leg at rest) leads us naturally to a modeling of the latter through a shell structure.

It is not our intention, in the first place, to resort to the simplified theories of Kirchhoff-Love or its derivatives. In this section we will solve the exact 3D model, which will reveal the non-trivial relations connecting the crushing stress, the local radius of curvature of the bottom of the underlying leg (the radius of curvature is the radius of a bottom circle that best fits a normal section or combinations thereof), the membrane forces and the material constituting the bottom. In short, these relations will constitute a non-trivial generalization of Laplace's laws.

Some geometrical details

We will model B₁ by a shell of constant thickness 2h (Figure 5(a)). We consider that the volume occupied by B₁ is generated by the displacement of a segment of length 2h whose middle point travels a left surface ω of the space (which will be the average surface of the hull); the remaining segment perpendicular to ω.

Figure 5.

Benchmarks associated with the area of investigation: (a) geometry of the hull; (b) projection of the vectors.

If O designates an arbitrary origin of space, m a point belonging to the mean surface ω, $e_{3} (\in R^{3})$ the unitary normal to ω in m and M a point belonging to the volume B₁, we write

\begin{matrix} B_{1} = ω \times] - h, + h [= {M \in R^{3}} suchas OM \\ = Om + x e_{3} (m), x \in] - h, h [ \end{matrix}

(12)

Variable x describes the thickness of the shell.

Gradient of a vector field defined on B₁

Let us consider a vector field v defined on B₁ as a function of the points m of ω and of x. This field can represent the movements $\tilde{u}$ generated by the muscular contractions or the virtual velocity field $\tilde{v}$

v : (m, x) \in ω \times] - h, + h [\to v (m, x) \in R^{3}

(13)

Let π denote the tangent plane in m to ω; by compound derivation we have

\begin{matrix} {\begin{matrix} \frac{\partial v}{\partial OM} \cdot dOM = \frac{\partial v}{\partial OM} \cdot dOm + \frac{\partial v}{\partial x} dx \\ with (dOM, dOm, dx) \in R^{3} \times π \times R \end{matrix} \end{matrix}

(14)

The link between the three infinitesimal vectors dOM, dOm and dx is obtained by deriving equation (12)

\begin{matrix} dOM = d (Om + x e_{3} (m)) = μ \cdot dOm + dx e_{3} with μ \\ = l_{2} + x \frac{\partial e_{3}}{\partial Om} \in R^{2} \times R^{2} \end{matrix}

(15)

Note: The gradient of the unit $\frac{\partial e_{3}}{\partial Om}$ normal makes it possible to measure the curvatures of ω, we name curvature endomorphism. $\frac{\partial e_{3}}{\partial Om}$ is an endomorphism of $ℝ^{2}$ applying the plane π in itself. To be convinced of this, it is enough to derive the identity $e_{3} \cdot \bar{e_{3}} = 1$ , which gives

\bar{e_{3}} \cdot \frac{\partial e_{3}}{\partial Om} = 0

(16)

$\frac{\partial e_{3}}{\partial Om}$ is a symmetrical endomorphism; its eigenvalues are positive. We deduce that the endomorphism μ, said shift endomorphism, appearing in equation (15) is symmetrical, its eigenvalues are strictly positive. It is therefore invertible. Introduce the projection operator on the unit normal e₃: $e_{3} \cdot \bar{e_{3}} : v \in R^{3} \to e_{3} \cdot \bar{e_{3}} \cdot v = v_{3} \cdot e_{3}$ as well as its complement in R³, that is to say the operator P of projection on the tangent plane π: $P = l_{3} - e_{3} \cdot \bar{e_{3}} : v \in R^{3} \to P \cdot v = v - v_{3} \cdot e_{3} = v_{t} \in π$ , so that any vector field of R³ admits decomposition (Figure 5(b))

v = v_{t} + v_{3} e_{3}

(17)

The projections of the elementary arc dOM give us according to the equation (15)

{\begin{matrix} P \cdot dOM = μ \cdot dOm \Rightarrow dOm = μ^{- 1} \cdot P \cdot dOM \\ \bar{e_{3}} \cdot dOM = dx \end{matrix}

(18)

so that equation (14) gives us the expression of the gradient of the vector field v

\frac{\partial v}{\partial OM} = \frac{\partial v}{\partial Om} \cdot μ^{- 1} \cdot P + \frac{\partial v}{\partial x} \cdot \bar{e_{3}}

(19)

Let us now specialize this expression by using the decomposition of v in its tangential component v_t and the normal one v₃ (equation (17)), we have in particular

\begin{matrix} \frac{\partial v}{\partial OM} = \frac{\partial v_{t}}{\partial Om} + e_{3} \cdot \frac{\partial v_{3}}{\partial Om} + v_{3} \frac{\partial e_{3}}{\partial Om} \end{matrix}

(20)

\begin{matrix} \begin{matrix} \frac{\partial v}{\partial x} = \frac{\partial v_{t}}{\partial x} + \frac{\partial v_{3}}{\partial x} \cdot e_{3} \end{matrix} \end{matrix}

(21)

so that the gradient of the vector field v is now written precisely

\begin{matrix} \frac{\partial v}{\partial OM} = \frac{\partial v_{t}}{\partial Om} \cdot μ^{- 1} \cdot P + e_{3} \cdot \frac{\partial v_{3}}{\partial Om} \cdot μ^{- 1} \cdot P \\ + v_{3} \frac{\partial e_{3}}{\partial Om} \cdot μ^{- 1} \cdot P + \frac{\partial v_{t}}{\partial x} \cdot e_{3} + \frac{\partial e_{3}}{\partial x} \cdot e_{3} \cdot \bar{e_{3}} \end{matrix}

(22)

Decomposition of the endomorphism fields of R³ defined in B₁

We will now break down the constraints of Cauchy σ and the gradient (equation (22)) by revealing their tangent (i.e. belonging to π), transverse and normal components which will play a determining role in the following.

Consider, therefore, an endomorphism A of R³ defined on B₁

A : M \in B_{1} \to A (M) \in R^{3} \times R^{3}

By definition of the operator P on the tangent plane π, we have

l_{3} = P + e_{3} \cdot \bar{e_{3}}

(23)

of this identity, it is possible to break down A as follows

\begin{matrix} A = l_{3} \cdot A \cdot l_{3} = (P + e_{3} \cdot \bar{e_{3}}) \cdot A \cdot (P + e_{3} \cdot \bar{e_{3}}) \\ = P \cdot A \cdot P + P \cdot A \cdot e_{3} \cdot \bar{e_{3}} + e_{3} \cdot \bar{e_{3}} \cdot A \cdot P + A_{33} \cdot e_{3} \cdot \bar{e_{3}} \end{matrix}

(24)

Let:

P·A·P = A_t. This is the plane restriction of A. A_t indeed appears as an endomorphism of R² applying the tangent plane to itself, A_t: $π \to π$ .

$P \cdot A \cdot e_{3} \cdot \bar{e_{3}} = A_{s} and e_{3} \cdot \bar{e_{3}} \cdot A \cdot P = A'_{s}$ . These are the cross-cutting restrictions of A. A_s (or A'_s) appears as an endomorphism applying the unit normal on the tangent plane (respectively the tangent plane on the unit normal), A_s : $e_{3} \to π (or A_{s}' : π \to e_{3}) .$ Note carefully that if A is a symmetrical endomorphism ( $A = \bar{A}$ ), so $A_{s} = \bar{A_{s}}$ so that $A_{s} + A_{s}' = 2 Sym (A_{s})$ .

$A_{33} \cdot e_{3} \cdot \bar{e_{3}} = A_{n}$ is the normal restriction of A. We check immediately that A_n applies the unit normal to itself, $A_{n} : e_{3} \to e_{3}$ .

So every endomorphism of A of R³ admits decomposition

A = A_{t} + A_{s} + A_{s}' + A_{n}

(25)

To fix the ideas, let us complete the unit normal e₃ by two vectors e₁ and e₂ so that (e₁, e₂, e₃) forms an orthonormal basis attached to $m \in ω$ (e₁ and e₂ then form a base in the vector space associated with π). In this base the endomorphisms A, P and $e_{3} \cdot \bar{e_{3}}$ are explained by the components of their representative matrix

\begin{matrix} A = [\begin{matrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{matrix}], P = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}], \\ e_{3} \cdot \bar{e_{3}} = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}] \end{matrix}

It is then easy to see

\begin{matrix} A_{t} = [\begin{matrix} A_{11} & A_{12} & 0 \\ A_{21} & A_{22} & 0 \\ 0 & 0 & 0 \end{matrix}], A_{s} = [\begin{matrix} 0 & 0 & A_{13} \\ 0 & 0 & A_{23} \\ 0 & 0 & 0 \end{matrix}], \\ A'_{s} = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ A_{31} & A_{32} & 0 \end{matrix}], A_{n} = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & A_{33} \end{matrix}] \end{matrix}

In the case where A is the matrix of the constraints the previous denomination is justified immediately. We have indeed (Figure 6(a))

$σ_{i} = [\begin{matrix} σ_{11} & σ_{12} & 0 \\ σ_{21} & σ_{22} & 0 \\ 0 & 0 & 0 \end{matrix}]$ : these are the tangential or membrane stresses

$σ_{s} = [\begin{matrix} 0 & 0 & σ_{13} \\ 0 & 0 & σ_{23} \\ 0 & 0 & 0 \end{matrix}]$ : these are the transverse shear stresses (in the thickness of the shell)

$σ_{n} = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & σ_{33} \end{matrix}]$ : this is the normal stress. It will be noted that the latter coincides with the crushing stress $\tilde{G_{3}}$ for x = −h. (Figure 6(b))

Figure 6.

Decomposition of constraints: (a) constraints; (b) contact bottom/leg.

It will be noted that by defining the transverse shear vector $σ_{tr} = σ_{13} e_{1} + σ_{23} e_{2}$ , one can write

σ_{s} = \bar{σ_{s}} = σ_{tr} \cdot \bar{e_{3}}

(26)

So the decomposition of the endomorphism of the constraints can be written

σ = σ_{t} + 2 Sym (σ_{tr} \cdot \bar{e_{3}}) + σ_{33} e_{3} \cdot \bar{e_{3}}

(27)

In the case where A is the gradient of the vector field v, we have (equation (22))

\begin{matrix} {\begin{matrix} (\frac{\partial v}{\partial Om})_{t} = P \cdot \frac{\partial v_{t}}{\partial Om} \cdot μ^{- 1} \cdot P + v_{3} \frac{\partial e_{3}}{\partial Om} \cdot μ^{- 1} \cdot P \\ (\frac{\partial v}{\partial Om})_{s} = \frac{\partial v_{t}}{\partial x} \cdot e_{3} \\ (\frac{\partial v}{\partial Om})_{s} = e_{3} \cdot (\frac{\partial v_{3}}{\partial Om} - \bar{v_{t}} \frac{\partial e_{3}}{\partial Om}) \cdot μ^{- 1} \cdot P \\ (\frac{\partial v}{\partial Om})_{n} = \frac{\partial v_{3}}{\partial x} e_{3} \cdot \bar{e_{3}} \end{matrix} \end{matrix}

(28)

Equilibrium equation of the bottom

Let us consider the model (11) expressing the equilibrium of a neighborhood of the bottom under the effect of small displacements induced by muscular contractions. Given the decompositions made for σ and the gradient of a vector field, the equilibrium can now be explained precisely by

\begin{matrix} {\begin{matrix} \int_{B_{1}} T_{r} (\tilde{σ_{t}} \cdot [\frac{\partial {\hat{v}}_{t}}{\partial OM} \cdot μ^{- 1} + {\hat{v}}_{3} \cdot \frac{\partial e_{3}}{\partial OM} \cdot μ^{- 1}]) d B_{1} \\ + \int_{B_{1}} \bar{{\tilde{σ}}_{tr}} \cdot [\frac{\partial {\hat{v}}_{t}}{\partial x} + μ^{- 1} (\frac{\bar{\partial {\hat{v}}_{3}}}{\partial OM} - \frac{\partial e_{3}}{\partial OM} \cdot \tilde{v} t)] d B_{1} \\ + \int_{B_{1}} {\tilde{σ}}_{33} \frac{\partial {\hat{v}}_{3}}{\partial x} d B_{1} \\ = \int_{\partial B_{1}} (\bar{{\tilde{F}}_{t}} \cdot {\hat{v}}_{t} + \bar{{\tilde{F}}_{3}} \cdot {\hat{v}}_{3}) d \partial B_{1} + \int_{S_{1}} (\bar{{\tilde{G}}_{t}} \cdot {\hat{v}}_{t} + \bar{{\tilde{G}}_{3}} \cdot {\hat{v}}_{3}) d S_{1} \\ \tilde{σ} = R \cdot (\begin{matrix} Sym (\frac{\partial \tilde{u}}{\partial OM})_{t} + Sym (\frac{\partial \tilde{u}}{\partial OM})_{S} \\ + Sym (\frac{\partial \tilde{u}}{\partial OM})_{S} + \frac{\partial {\tilde{u}}_{3}}{\partial x} e_{3} \cdot \bar{e_{3}} \end{matrix}) \end{matrix} \end{matrix}

(29)

To access the generalization of Laplace's law, we will translate the equilibrium locally. For this, we will need the following calculation rules:

Calculation of the elementary volume dB₁

Let three infinitesimal vectors dOM₁, dOM₂, dOM₃ be linearly independent and belonging to the volume B₁. The elementary volume generated by these three vectors is by definition obtained by mixed product

d B_{1} = \bar{dO M_{1}^dO M_{2}} \cdot dO M_{3}

(30)

Then choose dOM₃ collinear to the unit normal: dOM₃ = dxe₃ and dOM₁ = dOM₂ so that it corresponds to them according to equation (18), two vectors of dOm₁ and dOm₂ of the tangent plane π

dO M_{1} = μ \cdot dO m_{1} and dO M_{2} = μ \cdot dO m_{2}

(31)

In these conditions

\begin{matrix} d B_{1} = \bar{μ \cdot dO m_{1}^μ \cdot dO m_{2}} dx e_{3} \\ = det (μ) dx \bar{dO m_{1}^dO m_{2}} \cdot e_{3} = \det (μ) d ω d x \end{matrix}

(32)

GREEN formula

Let v_t be a vector field of the tangent plane and A_t a field of endomorphisms of the tangent plane. According to the formula of GREEN (integration by parts), we have

\begin{matrix} \int_{B_{1}} T_{r} (A_{t} \cdot P \cdot \frac{\partial v_{t}}{\partial Om}) d B_{1} = - \int_{B_{1}} div A_{t} \cdot v_{t} d B_{1} \\ + \int_{\partial B_{1}} \bar{N} \cdot A_{t} \cdot v_{t} d \partial B_{1} \end{matrix}

(33)

where div(A_t) is the divergence of A_t. It is a field of linear forms (of vectors transposed) defined on the tangent plane: div(A_t):

π \to R

. N is the unitary normal with the lateral contour

\partial B_{1}

coming out of B₁.

Given these calculation rules, an integration by parts of the left-hand member of the principle of virtual powers (powers of internal forces) makes it possible to rewrite the latter in the form

\begin{matrix} \int_{B_{1}} [- div (adj (μ) \cdot {\tilde{σ}}_{t}) - \frac{\partial}{\partial x} (\bar{{\tilde{σ}}_{tr}} det (μ)) - \bar{{\tilde{σ}}_{tr}} \cdot \bar{adj (μ)} \cdot \frac{\partial e_{3}}{\partial Om}]_{1} \cdot {\hat{v}}_{t} d ω d x \\ + \int_{B_{1}} [- div (adj (μ) \cdot {\tilde{σ}}_{tr}) - \frac{\partial}{\partial x} ({\tilde{σ}}_{33} det (μ)) - T_{r} \cdot (\bar{adj (μ)} \cdot {\tilde{σ}}_{t} \cdot \frac{\partial e_{3}}{\partial Om})]_{1} \cdot {\hat{v}}_{3} d ω d x \\ + \int_{\partial ω \times] - h, h [} \bar{N} \cdot (\bar{adj (u) \cdot {\tilde{σ}}_{tr}}) {\hat{v}}_{t} d \partial ω d x + \int_{ω \times] - h, h [} (\bar{{\tilde{σ}}_{tr}} \det (μ)) {\hat{v}}_{t} d \partial ω \times {\pm h} + \int_{\partial ω \times] - h, h [} (\bar{N} \cdot (\bar{adj (u) \cdot {\tilde{σ}}_{tr}}) {\hat{v}}_{3} d \partial d x \\ + \int_{ω \times] - h, h [} (\bar{{\tilde{σ}}_{33}} \det (μ)) {\hat{v}}_{3} d \partial ω \times {\pm h} = \int_{\partial B_{1}} ({\bar{\tilde{F}}}_{t} \cdot {\hat{v}}_{t} + {\tilde{F}}_{3} {\hat{v}}_{3}) d \partial B_{1} + \int_{S_{1}} ({\bar{\tilde{G}}}_{t} \cdot {\hat{v}}_{t} + {\tilde{G}}_{3} {\hat{v}}_{3}) d S_{1} \end{matrix}

(34)

The notation adj(μ) designates the adjoint endomorphism of μ. We have effect according to the formulas of Cramer, $adj (μ) = μ^{- 1} \det (μ)$ . Given the arbitrariness of the virtual velocity field, we deduce equations of local equilibrium

\begin{matrix} \forall M_{1} \in B_{1} {\begin{matrix} \begin{matrix} div (\bar{adj (μ)} \cdot {\tilde{σ}}_{t}) + \frac{\partial}{\partial x} (\bar{{\tilde{σ}}_{tr}} d \cdot et (μ)) \\ + \bar{{\tilde{σ}}_{tr}} \cdot \bar{adj (μ)} \cdot \frac{\partial e_{3}}{\partial Om} = 0 \end{matrix} \\ \begin{matrix} div (\bar{adj (μ)} \cdot {\tilde{σ}}_{tr}) + \frac{\partial}{\partial x} (\bar{{\tilde{σ}}_{33}} \cdot \det (μ)) \\ + T_{r} (\bar{adj (μ)} \cdot {\tilde{σ}}_{t} \cdot \frac{\partial e_{3}}{\partial Om}) = 0 \end{matrix} \end{matrix} \\ \forall M_{1} \in \partial ω \times] - h, h [{\begin{matrix} \bar{N} \cdot (\bar{adj (μ)} \cdot {\tilde{σ}}_{t}) = \bar{{\tilde{F}}_{t}} \\ \bar{N} \cdot (\bar{adj (μ)} \cdot σ_{tr}) = \bar{{\tilde{F}}_{3}} \end{matrix} \\ \forall M_{1} \in \partial ω \times {- h} {\begin{matrix} \bar{{\tilde{σ}}_{tr}} \det (μ) = \bar{{\tilde{G}}_{t}} \\ \bar{{\tilde{σ}}_{33}} \det (μ) = \bar{{\tilde{G}}_{3}} \end{matrix} \\ \forall M_{1} \in \partial ω \times {h} {\begin{matrix} \bar{{\tilde{σ}}_{tr}} = 0 \\ \bar{{\tilde{σ}}_{33}} = 0 \end{matrix} \end{matrix}

(35)

A simple way to explicitly connect the crush stress G to the curvature of the leg and Cauchy's constraints is to integrate these local equations on the thickness:

For this, define the following quantities:

$M = \int_{- h}^{h} \bar{adj (u)} \cdot σ_{t} d x$ which will be interpreted as the resultant in the thickness of the tangential stresses.

$T = \int_{- h}^{h} \bar{adj (μ)} \cdot σ_{tr} d x$ which will be interpreted as the resultant in the thickness of the transverse shear.

Then by quadrature of equation (35) in the thickness, we get

\forall m \in ω \begin{matrix} {\begin{matrix} divM + \bar{T} \cdot \frac{\partial e_{3}}{\partial Om} + {\bar{\tilde{G}}}_{t} = 0 \\ divT + T_{r} (\bar{M} \cdot \frac{\partial e_{3}}{\partial Om}) + {\bar{\tilde{G}}}_{3} = 0 \end{matrix} \end{matrix}

(36)

\forall m \in ω \begin{matrix} {\begin{matrix} \bar{N} \cdot M = \int_{- h}^{h} {\bar{\tilde{F}}}_{t} d x \\ \bar{N} \cdot T = \int_{- h}^{h} {\bar{\tilde{F}}}_{3} d x \end{matrix} \end{matrix}

(37)

Let us place ourselves at a point where the curvature is non-zero, the first equation of equilibrium gives

\bar{T} = (- divM - \bar{{\tilde{G}}_{t}}) \cdot \frac{\partial {e_{3}}^{- 1}}{\partial Om}

(38)

as a result, we deduce from the second equation of local equilibrium

\begin{matrix} - div (\frac{\partial {e_{3}}^{- 1}}{\partial Om} \cdot \bar{divM}) - div (\frac{\partial {e_{3}}^{- 1}}{\partial Om} \cdot {\tilde{G}}_{t}) \\ + T_{r} (\bar{M} \cdot \frac{\partial e_{3}}{\partial Om}) + {\tilde{G}}_{3} = 0 \end{matrix}

(39)

It is this last expression that will provide us with the relationship sought. If we neglect the friction constraints ${\tilde{G}}_{t}$ before the crushing stress ${\tilde{G}}_{3}$ , we obtain

{\tilde{G}}_{3} = div ({\frac{\partial e_{3}}{\partial Om}}^{- 1} \cdot \bar{divM}) - T_{r} (\bar{M} \cdot \frac{\partial e_{3}}{\partial Om})

(40)

In a way, it is a generalization of the “boilermakers” formula. It should be noted carefully that the crushing stress depends on: (a) the intensity and the variation of the membrane forces; and (b) all the curvatures of the studied neighborhood.

The influence of the material is exhibited by the constitutive law (29). Precisely, we have (taking into account equation (28)):

\begin{matrix} {\tilde{σ}}_{t} = P \cdot \tilde{σ} \cdot P = P \cdot R \cdot (P \cdot \frac{\partial {\tilde{u}}_{t}}{\partial Om} + {\tilde{u}}_{3} \frac{\partial e_{3}}{\partial Om}) \cdot μ^{- 1} \cdot P \\ + e_{3} (\frac{\partial {\tilde{u}}_{3}}{\partial Om} - {\bar{\tilde{u}}}_{t} \cdot \frac{\partial e_{3}}{\partial Om}) \cdot μ^{- 1} \cdot P \end{matrix}

(41)

Subsequently one obtains the constitutive law for membrane forces

\begin{matrix} M = \int_{- h}^{h} \bar{adj (μ)} P \cdot R \cdot P (\frac{\partial {\tilde{u}}_{t}}{\partial Om} + {\tilde{u}}_{3} \frac{\partial e_{3}}{\partial Om}) \cdot μ^{- 1} P d x \\ + \int_{- h}^{h} \bar{adj (μ)} P \cdot R \cdot e_{3} (\frac{\partial {\tilde{u}}_{3}}{\partial Om} - {\bar{\tilde{u}}}_{t} \frac{\partial e_{3}}{\partial Om}) μ^{- 1} P d x \end{matrix}

(42)

The asymptotic model

Although complete and without any prior assumptions, models (40) and (42) are not easy to interpret, particularly because of the complex dependence of μ⁻¹ and det (μ) on the variable of thickness. On the other hand, these models (40) and (42) remain valid for thick hulls, such as the stacks of bands and compression foams.

Here we can take advantage of the fact that the bottom has an extremely low thickness 2h compared with the other surface dimensions (Figure 6(b)). A convenient way to obtain the limiting behavior of the 3D model when h → 0, is to make the following change on the thickness variable

x = zh; z \in [- 1, 1]

(43)

and to formally go to the limit (h → 0). Note, however, that it is necessary to scale the crushing stress

\tilde{G}

so that the problem is well posed, that is to say that its power is of the same order of magnitude as the power of the effort's interiors. Precisely, we will ask

{\tilde{G}}_{t} = h {\tilde{G}}_{t} and {\tilde{G}}_{3} = h {\tilde{G}}_{3}

(44)

The 3D model (equation (29)) is then written in the dominant order

\begin{matrix} {\begin{matrix} \frac{\partial}{\partial z} {\tilde{σ}}_{tr} = 0 \\ \frac{\partial}{\partial z} {\tilde{σ}}_{33} = 0 \end{matrix}, \forall z \in] - 1, 1 [and {\begin{matrix} {\tilde{σ}}_{tr} (\pm h) = 0 \\ {\tilde{σ}}_{33} (\pm h) = 0 \end{matrix} \end{matrix}

(45)

We then deduce that ${\tilde{σ}}_{tr} = 0$ , ${\tilde{σ}}_{33} = 0$ and consequently

T = 0 and M = \int_{- h}^{h} σ_{t} d x

(46)

Moreover, the constitutive law shows that

(\frac{\partial \tilde{u}}{\partial OM})_{S} = 0, (\frac{\partial \tilde{u}}{\partial OM})_{S'} = 0, (\frac{\partial \tilde{u}}{\partial OM})_{N} = 0

(47)

which implies in particular that the field of displacements is in the dominant order surface area (insofar as it depends only on the points belonging to the average surface ω)

{\tilde{u}}_{t} = {\tilde{u}}_{t} (m), {\tilde{u}}_{3} = {\tilde{u}}_{3} (m)

Since the dominant order shear force is zero, the equilibrium equations integrated three-dimensional thickness provide us with the simplified model

\forall m \in ω, \begin{matrix} {\begin{matrix} divM + {\bar{\tilde{G}}}_{t} = 0 \\ T_{r} (\bar{M} \cdot \frac{\partial e_{3}}{\partial Om}) + {\tilde{G}}_{3} = 0 \\ \forall m \in \partial ω, \bar{N} \cdot M = \int_{- h}^{h} {\bar{F}}_{t} d x \end{matrix} \end{matrix}

(48)

With the constitutive law, we get the following substantial simplifications

M = \int_{- h}^{h} P \cdot R \cdot P d x (\frac{\partial {\tilde{u}}_{t}}{\partial Om} + {\tilde{u}}_{3} \frac{\partial e_{3}}{\partial Om})

(49)

It will be noted that only the plane restriction of the stiffness endomorphism (P·R·P) occurs. The bottom therefore resists essentially by its “tangential” rigidity.

Influencing parameters and discussion

The objective of this section is to highlight the magnitudes influencing significantly the crushing stress and consequently to deduce an optimal use of compression stockings in order to increase their therapeutic effect.

Let us consider the asymptotic model (equation (49)). Under the effect of the small displacements generated by muscular contractions, the crushing stress has for expression

\begin{matrix} {\begin{matrix} - {\tilde{G}}_{3} = T_{r} (\bar{M} \cdot \frac{\partial e_{3}}{\partial Om}) \\ with M = \int_{- h}^{h} P \cdot R \cdot Pdx (\frac{\partial {\tilde{u}}_{t}}{\partial Om} + {\tilde{u}}_{3} \frac{\partial e_{3}}{\partial Om}) \end{matrix} \end{matrix}

(50)

Recall also that in our modeling, the bottom is considered a homogenized orthotropic continuous medium. If e_T and e_C are the unit vectors of the tangent plane defining the principal orthotropy coordinate system (e_T parallel to the frame direction and e_C parallel to the column direction), we have

\begin{matrix} \int_{- h}^{h} P \cdot R \cdot P d x = 2 h (\begin{matrix} \begin{matrix} \frac{E_{T}}{1 - V_{TC} V_{CT}} e_{T} \otimes e_{T} \otimes e_{T} \otimes e_{T} \\ + \frac{V_{CT} E_{T}}{1 - V_{TC} V_{CT}} e_{T} \otimes e_{T} \otimes e_{T} \otimes e_{T} \end{matrix} \\ \begin{matrix} + \frac{V_{TC} E_{C}}{1 - V_{TC} V_{CT}} e_{T} \otimes e_{C} \otimes e_{T} \otimes e_{T} \\ + \frac{E_{C}}{1 - V_{TC} V_{CT}} e_{C} \otimes e_{C} \otimes e_{C} \otimes e_{C} + \end{matrix} \\ G_{TC} e_{T} \otimes e_{C} \otimes e_{T} \otimes e_{C} \end{matrix}) \end{matrix}

(51)

Locally, a muscle contraction mainly results in a normal displacement ${\tilde{u}}_{3}$ imposed at the bottom (swelling). The tangential displacement ${\tilde{u}}_{t}$ can reasonably be considered negligible.

Let us complete the unit normal e₃ by two vectors e₁ and e₂ tangent to the main coordinate lines of ω. In this base, attached to a point $m \in ω$ , we have

\frac{\partial e_{3}}{\partial Om} = \begin{matrix} [\begin{matrix} \frac{1}{R_{1}} & 0 \\ 0 & \frac{1}{R_{2}} \end{matrix}] \end{matrix}

(52)

where R₁ and R₂ are the main radii of curvature in m. These radii of curvature appear as extremal values. Without restricting the generality, we can suppose that R₁ > R₂.

The crushing constraint is written

- {\tilde{G}}_{3} = T_{r} (\bar{M} \cdot \frac{\partial e_{3}}{\partial Om}) = \frac{M_{11}}{R_{1}} + \frac{M_{22}}{R_{2}}

(53)

As for the constitutive law, it is translated by

M = \int_{- h}^{h} P \cdot R \cdot P d x ({\tilde{u}}_{3} \frac{\partial e_{3}}{\partial Om})

(54)

In the directions of the principal curvatures, the tangential rigidity has, with exception, all its non-zero components, precisely

\int_{- h}^{h} P \cdot R \cdot P d x = 2 h (\begin{matrix} R_{1111} (e_{1} \otimes e_{1} \otimes e_{1} \otimes e_{1}) \\ + R_{2222} (e_{2} \otimes e_{2} \otimes e_{2} \otimes e_{2}) \\ + R_{1122} (\begin{matrix} e_{1} \otimes e_{1} \otimes e_{2} \otimes e_{2} \\ + e_{1} \otimes e_{1} \otimes e_{1} \otimes e_{1} \end{matrix}) \\ + R_{1111} (\begin{matrix} e_{1} \otimes e_{1} \otimes e_{1} \otimes e_{2} \\ + e_{1} \otimes e_{2} \otimes e_{1} \otimes e_{1} \end{matrix}) \\ + R_{2212} (\begin{matrix} e_{2} \otimes e_{2} \otimes e_{1} \otimes e_{2} \\ + e_{1} \otimes e_{2} \otimes e_{2} \otimes e_{2} \end{matrix}) \\ + R_{1212} (e_{1} \otimes e_{2} \otimes e_{1} \otimes e_{2}) \end{matrix})

(55)

So that the constitutive law is rewritten

\begin{matrix} M = 2 h {\tilde{u}}_{3} (\begin{matrix} (\frac{R_{1111}}{R_{1}} + \frac{R_{1112}}{R_{2}}) e_{1} \otimes e_{1} \\ + (\frac{R_{1122}}{R_{1}} + \frac{R_{2222}}{R_{2}}) e_{2} \otimes e_{2} \\ + (\frac{R_{1211}}{R_{1}} + \frac{R_{1222}}{R_{2}}) e_{1} \otimes \pm e_{2} \end{matrix}) \end{matrix}

(56)

We then deduce the following expression of the crushing constraint

- {\tilde{G}}_{3} = 2 h {\tilde{u}}_{3} (\frac{R_{1111}}{{R_{1}}^{2}} + \frac{R_{2222}}{{R_{2}}^{2}} + \frac{R_{1122}}{R_{1} R_{2}})

(57)

In order to show the ratio of the radii of curvature, let us posit

η = \frac{R_{1}}{R_{2}} > 1

So we will retain the following expression for the crush constraint

- {\tilde{G}}_{3} = 2 h \frac{{\tilde{u}}_{3}}{{R_{2}}^{2}} (\frac{R_{1111}}{η^{2}} + R_{2222} + 2 \cdot \frac{R_{1122}}{η})

(58)

It is now easy to qualitatively translate the behavior of ${\tilde{G}}_{3}$ . Several situations are to consider:

1. The radii of curvature have orders of comparable magnitude: $η = 0 (1)$

In that case

- {\tilde{G}}_{3} = 2 h \frac{{\tilde{u}}_{3}}{{R_{2}}^{2}} (R_{1111} + R_{2222} + 2 R_{1122})

(59)

Only frames of higher rigidity generate the expected therapeutic pressure.

2. The radii of curvature are different: $η > 1$

Suppose that the bottom has a much higher rigidity in the direction of the frames than in the direction of the columns of meshes (which is generally the case). In this case, it is possible to modulate the pressure generated by playing on the phase shift ϕ between the direction of the columns of the meshes and the parametric line associated with the smallest radius of curvature (at the highest curvature).

Let R_T, R_C and R_TC respectively denote the tangential rigidities in the weft, column and cross direction. These are the components of the tangential stiffness endomorphism (equation (51)) in the main axes of orthotropy, precisely

\begin{matrix} {\begin{matrix} R_{T} = \frac{E_{T}}{1 - v_{TC} v_{CT}} \\ R_{C} = \frac{E_{C}}{1 - v_{TC} v_{CT}} \end{matrix}, R_{TC} = \frac{v_{TC} E_{C}}{1 - v_{TC} v_{CT}} = \frac{v_{CT} E_{T}}{1 - v_{TC} v_{CT}} \end{matrix}

(60)

with: R_T > R_C since E_T > E_C. So that we can rewrite the stiffness endomorphism in a more concise way

\begin{matrix} \int_{- h}^{h} P \cdot R \cdot P d x = 2 h {\begin{matrix} R_{T} (e_{T} \otimes e_{T} \otimes e_{T} \otimes e_{T}) \\ + R_{TC} e_{T} \otimes e_{T} \otimes e_{T} \otimes e_{C} \\ + R_{TC} (e_{T} \otimes e_{C} \otimes e_{C} \otimes e_{C}) \\ + R_{C} e_{C} \otimes e_{C} \otimes e_{C} \otimes e_{C} \\ + G_{TC} e_{T} \otimes e_{C} \otimes e_{T} \otimes e_{C} \end{matrix}} \end{matrix}

(61)

To obtain the components of the tangential rigidity in the base (e₁, e₂), it suffices to apply to the tangential rigidity expressed in the principal directions of orthotropy (e_T, e_C), a base change of angle $ϕ - π / 2$ .

A classical calculation gives us the relations

\begin{matrix} {\begin{matrix} R_{1111} = R_{T} sin^{4} ϕ + R_{C} cos^{4} ϕ \\ + 2 (R_{TC} + 2 G_{TC}) sin^{2} ϕ cos^{2} ϕ \\ R_{2222} = R_{T} sin^{4} ϕ + R_{C} sin^{4} ϕ \\ + 2 (R_{TC} + 2 G_{CT}) sin^{2} ϕ cos^{2} ϕ \\ R_{1122} = R_{T} (sin^{4} ϕ + cos^{4} ϕ) \\ + (R_{T} + R_{C} - 4 G_{CT}) sin^{2} ϕ cos^{2} ϕ \end{matrix} \end{matrix}

(62)

On the other hand, the stiffness endomorphism being defined as positive, we show the following inequality concerning the Poisson ratio (this is a measure of the Poisson effect, the phenomenon in which a material tends to expand in directions perpendicular to the direction of compression)

| v_{CT} | < \sqrt{\frac{E_{C}}{E_{T}}} \Rightarrow R_{TC} < R_{T}

(63)

We then deduce the shape of the graphs of these tangential rigidities, which are shown in Figure 7. It can be seen from Figure 7 that the values of R₁₁₁₁, R₂₂₂₂ and R₁₁₂₂ show different trends, as the phase angle increases. Besides, it is clear that $- {\tilde{G}}_{3}$ reaches its maximum when the bottom frame coincides with the parametric line of stronger curvature, according to the relation of equation (59).

Figure 7.

Variation of tangential rigidities versus phase shift: (a) R₁₁₁₁; (b) R₂₂₂₂; (c) R₁₁₂₂; (d) R₁₁₁₁+R₂₂₂₂+2R₁₁₂₂.

Simulation results

The purpose of this last section is to illustrate, numerically, by finite elements, the influence of the ratio of the radii of curvature on the stress of crushing. The vicinity of the leg centered at the point M is here modeled using a rigid ellipsoid whose main radii of curvature R₁ and R₂ are such that their ratio satisfies $η = R_{1} / R_{2} > 1$ . The neighborhood B is modeled using a membrane sample made of a homogenized orthotropic material, embedded on its entire border (Figure 8(a)). The muscular swelling is here simulated by a vertical displacement of rigid solid of the ellipsoid.

Figure 8.

(a) Mesh of the membrane; (b) ellipsoid mesh; (c) initial position; (d) imposed displacements.

In order to make the variational formulation underlying the finite element method be well posed, it is essential to suppress rigid solid displacements. These are core elements of the variational formulation and will hinder singularities in the global stiffness matrix. To do this, the ellipsoid will be embedded in a very flexible beam subjected to bending (Figure 8(b)). The ellipsoid is meshed using 2376 tetrahedral elements with three nodes and nine degrees of freedom (three nodal components of displacement). For these elements, an isotropic material with higher rigidity is used at the bottom in order to simulate a rigid solid. The bottom is meshed using 1125 membrane elements quadrilateral with four nodes and 12 degrees of freedom. It is thus possible to generate a vertical movement without inducing stiffening movements by applying surface forces on the lower surface of the beam. At rest, the bottom is tangent to the top of the ellipsoid (Figure 8(c)). Figure 8(d) shows the map of imposed vertical displacement.

It will be noted that the influence of the orientation of the bottom threads is not taken into account: the frames remain here parallel to the direction of stronger curvature. To stay within the framework of the small deformations, we calculate the surface force to be applied to the beam in order to have an arrow of the order of the thickness of the membrane (of the order of millimeter). The membrane is then applied to the ellipsoid in order to detect the crushing stress (Figure 9). For each studied case, we find the screenshots in Figure 10, which shows the mesh and the geometry of the ellipsoid and the map of the crushing stress at different ratio η. The unit of the crushing stress is hPa. It can be clearly seen that the value of the crushing stress gradually reduces as the radii ratio of curvature R₁ and R₂ decreases.

Figure 9.

Deformed ellipsoid/bottom set.

Figure 10.

The map of the crushing stress of the modeled ellipsoid with different radii ratio: (a) η = 20; (b) η = 10; (c) η = 6.

Conclusion

Our focus in this article is to provide a new practical and scientific model to study the pressure changes imposed by MCS on the human leg with a 3D interface. The proposed models (1) and (3) are successfully simplified based on the assumption that the MCS structure is subject to small movements induced by muscular contractions. The stresses and displacements induced by muscular contractions are the solutions of model (11), which is the model in small deformations where we have geometrically confused the configurations B and B₁. Next, we model the particular geometry of the B₁ through a shell structure by dividing the complicated model into several simple sub-models. The simplified process is derived strictly according to the mathematical formulas. The obtained exact 3D model also reveals the relations connecting the crushing stress, the local radius of curvature of the bottom, the membrane forces and the material constituting the bottom. In short, these relations will constitute a non-trivial generalization of Laplace's laws, which shows that the crushing stress depends on: (a) the intensity and the variation of the membrane forces; and (b) all the curvatures of the studied neighborhood. Finally, considering the fact that the bottom has extremely low thickness, the asymptotic model is provided to obtain the simplified law of behavior for membrane forces, which shows that the bottom resists essentially by its tangential rigidity. Meanwhile, the influences of the radii ratio of curvature (η = R₁/R₂) on the crushing stress are discussed and highlighted in order to increase the MCS therapeutic effect. The results of the asymptotic model show that (a) when η = 1, only frames of higher rigidity generate the expected therapeutic pressure; and (b) when η > 1, the crushing stress reaches its maximum when the bottom frame coincides with the parametric line of stronger curvature. In addition, the simulation results are verified by finite element method and show that the value of the crushing stress gradually reduces as η decreases.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was funded by the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning (Grant No. TP2017074), the National Natural Science Foundation of China (Grant No. 61906129), China Postdoctoral Science Foundation (Grant No. 2019M661929), and Jiangsu Postdoctoral Science Foundation (Grant No. 2019Z285).

ORCID iDs

Yufa Sun

Yan Hong

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Modelization and identification of medical compression stocking: Part 2 – 3D interface pressure modelization

Abstract

Keywords

Position of the problem

Simplifying assumptions

Expression of the crushing stress

Some geometrical details

Gradient of a vector field defined on B1

Decomposition of the endomorphism fields of R3 defined in B1

Equilibrium equation of the bottom

Calculation of the elementary volume dB1

GREEN formula

The asymptotic model

Influencing parameters and discussion

Simulation results

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

References

Gradient of a vector field defined on B₁

Decomposition of the endomorphism fields of R³ defined in B₁

Calculation of the elementary volume dB₁