HOMOGENIZATION OF A VISCOELASTIC MODEL FOR PLANT CELL WALL BIOMECHANICS

DOI:10.1051/cocv/2016060 www.esaim-cocv.org

HOMOGENIZATION OF A VISCOELASTIC MODEL FOR PLANT CELL WALL BIOMECHANICS

Mariya Ptashnyk

and Brian Seguin

Abstract.The microscopic structure of a plant cell wall is given by cellulose microfibrils embedded in a cell wall matrix. In this paper we consider a microscopic model for interactions between viscoelastic deformations of a plant cell wall and chemical processes in the cell wall matrix. We consider elastic deformations of the cell wall microfibrils and viscoelastic Kelvin–Voigt type deformations of the cell wall matrix. Using homogenization techniques (two-scale convergence and periodic unfolding methods) we derive macroscopic equations from the microscopic model for cell wall biomechanics consisting of strongly coupled equations of linear viscoelasticity and a system of reaction-diffusion and ordinary differential equations. As is typical for microscopic viscoelastic problems, the macroscopic equations governing the viscoelastic deformations of plant cell walls contain memory terms. The derivation of the macroscopic problem for the degenerate viscoelastic equations is conducted using a perturbation argument.

Mathematics Subject Classification. 35B27, 35Q92, 35Kxx, 74Qxx, 74A40, 74D05.

Received November 10, 2015. Accepted August 31st, 2016.

1. Introduction

To obtain a better understanding of the mechanical properties and development of plant tissues it is important to model and analyse the interactions between the chemical processes and mechanical deformations of plant cells. The main feature of plant cells are their walls, which must be strong to resist high internal hydrostatic pressure (turgor pressure) and flexible to permit growth. The biomechanics of plant cell walls is determined by the cell wall microstructure, given by microfibrils, and the physical properties of the cell wall matrix. The orientation of microfibrils, their length, high tensile strength, and interactions with wall matrix macromolecules strongly influence the wall’s stiffness. It is also supposed that calcium-pectin cross-linking chemistry is one of the main regulators of cell wall elasticity and extension [30]. Pectin can be modified by the enzyme pectin methylesterase (PME), which removes methyl groups by breaking ester bonds. The de-esterified pectin is able to form calcium-pectin cross-links, and so stiffen the cell wall and reduce its expansion, see e.g. [29]. It has been shown that the modification of pectin by PME and the control of the amount of calcium-pectin cross-links

Keywords and phrases.Homogenization, two-scale convergence, periodic unfolding method, viscoelasticity, plant modelling.

∗M. Ptashnyk and B. Seguin gratefully acknowledge the support of the EPSRC First Grant EP/K036521/1 “Multiscale mod- elling and analysis of mechanical properties of plant cells and tissues”.

1 Department of Mathematics, University of Dundee. Dundee, DD1 4HN, UK.[email protected]

2 Department of Mathematics and Statistics, Loyola University Chicago, 60660 Chicago, USA.[email protected]

c M. Ptashnyk and B. Seguin. Published by EDP Sciences, SMAI 2017

This is an Open Access article distributed under the terms of theCreative Commons Attribution License(http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

M. PTASHNYK AND B. SEGUIN

greatly influence the mechanical deformations of plant cell walls [23,24], and the interference with PME activity causes dramatic changes in growth behavior of plant cells and tissues [31].

To address the interactions between microstructure, chemistry and mechanics, in the microscopic model for plant cell wall biomechanics we consider the influence of the microscopic structure, associated with the cellulose microfibrils, and the calcium-pectin cross-links on the mechanical properties of plant cell walls. We model the cell wall as a three-dimensional continuum consisting of a polysaccharide matrix and cellulose microfibrils. It was observed experimentally that plant cell wall microfibrils are anisotropic, seee.g. [10], and the cell wall matrix, in addition to elastic deformations, exhibits viscous behaviour, seee.g. [14]. Hence we model the cell wall matrix as a linearly viscoelastic Kelvin–Voigt material, whereas microfibrils are modelled as an anisotropic linearly elastic material. Within the matrix, we consider the dynamics of the enzyme PME, methylesterfied pectin, demethylesterfied pectin, calcium ions, and calcium-pectin cross-links. A model for plant cell wall biomechanics in which the cell wall matrix was assumed to be linearly elastic was derived and analysed in [26]. The interplay between the mechanics and the cross-link dynamics comes in by assuming that the elastic and viscous properties of the cell wall matrix depend on the density of the cross-links and that stress within the cell wall can break calcium-pectin cross-links. The stress-dependent opening of calcium channels in the cell plasma membrane is addressed in the flux boundary conditions for calcium ions. The resulting microscopic model is a system of strongly coupled four diffusion-reaction equations, one ordinary differential equation, and the equations of linear viscoelasticity. Since only the cell wall matrix is viscoelastic we obtain degenerate elastic-viscoelastic equations.

In our model we focus on the interactions between the chemical reactions within the cell wall and its deformation and, hence, do not consider the growth of the cell wall.

To analyse the macroscopic mechanical properties of the plant cell wall we rigorously derive macroscopic equa- tions from the microscopic description of plant cell wall biomechanics. The two-scale convergence,e.g. [4,21], and the periodic unfolding method,e.g. [7,8], are applied to obtain the macroscopic equations. For the viscoelas- tic equations the macroscopic momentum balance equation contains a term that depends on the history of the strain represented by an integral term (fading memory effect). Due to the coupling between the viscoelastic properties and the biochemistry of a plant cell wall, the elastic and viscous tensors depend on space and time variables. This fact introduces additional complexity in the derivation and in the structure of the macroscopic equations, compered to classical viscoelastic equations.

The main novelty of this paper is the multiscale analysis and derivation of the macroscopic problem from a microscopic description of the mechanical and chemical processes. This approach allows us to take into account the complex microscopic structure of a plant cell wall and to analyse the impact of the heterogeneous distribution of cell wall structural elements on the mechanical properties of plants. The main mathematical difficulty arises from the strong coupling between the equations of linear viscoelasticity for cell wall mechanics and the system of reaction-diffusion and ordinary differential equations for the chemical processes in the wall matrix. Also the degeneracy of the viscoelastic equations, due to the fact that only the cell wall matrix is assumed to be viscoelastic and microfibrils are assumed to be elastic, induces additional technical difficulties in the multiscale analysis of the microscopic model. To derive the macroscopic equations for the viscoelastic model for cell wall biomechanics we consider perturbed equations by introducing an inertial term. Once the macroscopic problem of the perturbed equations is derived, the perturbation parameter is sent to zero. By showing that the limit problem (as the perturbation parameter tends to zero) of the two-scale macroscopic problem for the perturbed microscopic equations is the same as the two-scale macroscopic problem for the original microscopic equations, we obtain the effective homogenized equations for the original viscoelastic problem coupled with reaction-diffusion and ordinary differential equations. A perturbation approach, by considering a viscosity term multiplied by a small perturbation parameter in the elastic inclusions, was also used in [11] to derive a macroscopic model for an elastic-viscoelastic problem.

A multiscale analysis of the viscoelastic equations with time-independent coefficients was considered previ- ously in [12,13,18,27]. Macroscopic equations for scalar elastic-viscoelastic equations with time-independent coef- ficients were derived in [11] by applying the H-convergence method [19]. A microscopic viscoelastic Kelvin–Voigt

model with time-dependent coefficients in the context of thermo-viscoelasticity was analysed in [1] and macro- scopic equations were derived by applying the method of asymptotic expansion.

The paper is organised as follows. In Section 2 we formulate a mathematical model for plant cell wall biomechanics in which the cell wall matrix is assumed to be viscoelastic. In Section3 we summarise the main results of the paper. The well-possedness of the microscopic model is shown in Section4. The multiscale analysis of the microscopic model is conducted in Section5.

2. Microscopic model for viscoelastic deformations of plant cell walls

The main feature of plant cells are their walls, which must be strong to resist high internal hydrostatic pressure and flexible to permit the growth. To better understand the interplay between these in some sense conflicting functions, we consider a mathematical model describing the interactions between the mechanical properties and the chemical processes in cell walls, surrounding plant cells. Plant cell walls are separated from the inside of the cell by the plasma membrane, modelled as an internal boundary of the cell wall (see Fig.1a).

Individual cells in plant tissues are joined together by a pectin network of middle lamella. The primary wall of a plant cell consists mainly of oriented cellulose microfibrils imbedded in the cell wall matrix, which is composed of pectin, hemicellulose, structural proteins, and water. It was observed experimentally that in addition to elastic deformations the plant cell wall matrix exhibits viscoelastic behaviour [14]. Hence, in contrast to the model considered in [26], here we assume that the deformations of the plant cell wall matrix are determined by the equations of linear viscoelasticity.

To model mechanical deformations of plant cell walls, we consider a domainΩ = (0, a₁)×(0, a₂)×(0, a₃) representing a flat section of a cell wall, where a_i, withi= 1,2,3, are positive numbers. We assume that the microfibrils are oriented in thex₃-direction (see Fig.1b). We shall distinguish between six disjoint parts of the boundary ∂Ω of the domainΩ. The interior boundaryΓ_I ={0} ×(0, a₂)×(0, a₃) represents the cell plasma membrane, the exterior boundary Γ_E = {a1} ×(0, a₂)×(0, a₃) denotes the side of the cell wall which is in contact with the middle lamella, on the top and bottom boundaries Γ_U = (0, a₁)× {0} ×(0, a₃)∪(0, a₁)× {a₂} ×(0, a₃) we will prescribe traction boundary conditions, reflecting the turgor pressure. On the boundaries Γ_P = (0, a₁)×(0, a₂)× {0} ∪(0, a₁)×(0, a₂)× {a₃}we consider periodic boundary conditions.

To determine the microscopic structure of the cell wall given by cell wall microfibrils, we consider Y = (0,1)²×(0, a₃) and define ˆY = (0,1)², together with the subdomain ˆY_F, with ˆY_F ⊂Yˆ, and ˆY_M = ˆY \Yˆ_F. Then Y_F = ˆY_F×(0, a₃) and Y_M = ˆY_M ×(0, a₃) represent the cell wall microfibrils and cell wall matrix, rescaled to the ‘unit cell’Y (see Fig. 1c). We also define ˆΓ =∂Yˆ_F ∩∂Yˆ_M and Γ =∂Y_F ∩∂Y_M.

We assume that the microfibrils in the cell wall are distributed periodically and have a diameter on the order ofε, where the small parameterεcharacterise the size of the microstructure,i.e. the ratio between the diameter of the microfibrils and the thickness of the cell wall. The domains

Ω_F^ε =

ξ∈Z²

ε( ˆY_F +ξ)×(0, a₃) | ε( ˆY +ξ)⊂(0, a₁)×(0, a₂)

and Ω_M^ε =Ω \Ω^ε_F

denote the parts of Ω occupied by the microfibrils and by the cell wall matrix, respectively. The boundary between the cell wall matrix and the microfibrils is denoted by

Γ^ε=∂Ω_M^ε ∩∂Ω^ε_F.

We adopt the following notation:Ω_T = (0, T)×Ω,Ω_M,T^ε = (0, T)×Ω^ε_M,Γ_I,T = (0, T)×Γ_I,Γ_T^ε= (0, T)×Γ^ε, Γ_U,T = (0, T)×Γ_U,Γ_E,T = (0, T)×Γ_E, andΓ_EU,T = (0, T)×(Γ_E∪Γ_U), and define

W(Ω) ={u∈H¹(Ω;R³)

Ωudx=0,

Ω

[(∇u)₁₂−(∇u)₂₁] dx= 0 and uisa₃-periodic inx₃}, V(Ω_M^ε ) ={n∈H¹(Ω_M^ε ) nisa₃-periodic inx₃}.

M. PTASHNYK AND B. SEGUIN

) c ( )

b ( )

a ( cell nucleus

-

mitochandrion -

vacule -

cell wall -

middle lamella - plasma membrane

- Γ_I

-

Γ_E Ω^ε_F -

Ω^ε_M -

-

a₁

6

? a₂

- -

a₃ -

1 Y_M -

- Y_F

-

- a₃

6

? 1 Γ_U

Γ_U -

Figure 1.(a) A schematic of a plant cell with an indication of the domainΩas a part of the cell wall. (b) A depiction of the domainΩ with the subsets representing the cell wall matrix Ω_M^ε and the microfibrils Ω_F^ε. The (hidden) surface Γ_I corresponds to the plasma membrane and is in contact with the interior of the cell, the surface Γ_E is facing the outside of the cell and is in contact with the middle lamella, andΓ_U is the union of the surfaces on the top and bottom ofΩ. (c) A depiction of the ‘unit cell’Y.

By Korn’s second inequality, theL²-norm of the strain defines a norm onW(Ω)

u_W(Ω)=e(u)_L2(Ω) for allu∈ W(Ω),

seee.g. [6,17,22]. For more details see also [26].

The microscopic model for elastic-viscoelastic deformations u^ε of plant cell walls and for the densities of esterified pectinp^ε₁, PME enzyme p^ε₂, de-esterified pectinn^ε₁, calcium ionsn^ε₂, and calcium-pectin cross-linksb^ε reads

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩

div (E^ε(b^ε, x)e(u^ε) +V^ε(b^ε, x)∂_te(u^ε)) =0 in Ω_T, (E^ε(b^ε, x)e(u^ε) +V^ε(b^ε, x)∂_te(u^ε))ν=−pIν onΓ_I,T, (E^ε(b^ε, x)e(u^ε) +V^ε(b^ε, x)∂_te(u^ε))ν=f onΓ_EU,T,

u^ε a₃-periodic inx₃, u^ε(0, x) =u0(x) in Ω,

(2.1)

and

∂_tp^ε= div(D_p∇p^ε)−Fp(p^ε) in Ω_M,T^ε ,

∂_tn^ε= div(D_n∇n^ε) +Fn(p^ε,n^ε) +Rn(n^ε, b^ε,Nδ(e(u^ε))) in Ω_M,T^ε ,

∂_tb^ε=R_b(n^ε, b^ε,Nδ(e(u^ε))) in Ω_M,T^ε ,

(2.2)

where p^ε = (p^ε₁, p^ε₂)^T, n^ε = (n^ε₁, n^ε₂)^T, and div(D_p∇p^ε) = (div(D¹_p∇p^ε₁),div(D²_p∇p^ε₂))^T, and div(D_n∇n^ε) = (div(D¹_n∇n^ε₁),div(D_n²∇n^ε₂))^T, together with the initial and boundary conditions

D_p∇p^εν=Jp(p^ε) on Γ_I,T,

D_p∇p^εν=−γpp^ε on Γ_E,T,

D_n∇n^εν=Nδ(e(u^ε))G(n^ε) on Γ_I,T,

D_n∇n^εν=Jn(n^ε) on Γ_E,T,

D_p∇p^εν=0, D_n∇n^εν=0 on Γ_T^ε andΓ_U,T,

p^ε, n^ε a₃-periodic inx₃,

p^ε(0, x) =p0(x), n^ε(0, x) =n0(x), b^ε(0, x) =b₀(x) in Ω^ε_M.

(2.3)

Here Nδ(e(u^ε)), defined as Nδ(e(u^ε)) =

−

Bδ(x)∩Ωtr (E^ε(b^ε,x)˜ e(u^ε)) d˜x ₊

in (0, T)×Ω, forδ >0, (2.4) represents the nonlocal impact of mechanical stresses on the calcium-pectin cross-links chemistry, whereB_δ(x) is a ball of a fixed radius δ > 0 at x ∈ Ω. From a biological point of view the nonlocal dependence of the chemical reactions on the displacement gradient is motivated by the fact that pectins are long molecules and hence cell wall mechanics has a nonlocal impact on the chemical processes. The positive part in the definition of Nδ(e(u^ε)) reflects the fact that extension rather than compression causes the breakage of cross-links. The boundary condition (2.3)₃reflects the fact that the flow of calcium ions between the interior of the cell and the cell wall depends on the displacement gradient, which corresponds to the stress-dependent opening of calcium channels in the plasma membrane [28].

The elasticity and viscosity tensors are defined as E^ε(ξ, x) =E(ξ,x/ε) andˆ V^ε(ξ, x) =V(ξ,x/ε), where theˆ Yˆ-periodic inyfunctionsEandVare given byE(ξ, y) =EM(ξ)χ_Yˆ

M(y)+EFχ_Yˆ

F(y) andV(ξ, y) =VM(ξ)χ_Yˆ

M(y).

For a given measurable set A we use the notation φ1, φ_{2 A} =

Aφ₁φ₂dx, where the product of φ₁ and φ₂ is the scalar-product if they are vector valued. By ψ₁, ψ_{2 V,V} we denote the dual product betweenψ₁ ∈ L²(0, T;V(Ω_M^ε )) andψ₂∈L²(0, T;V(Ω_M^ε )). We also denoteI_μ^k = (−μ,+∞)^k, for an arbitrary fixedμ >0 and k∈N.

Throughout the text we shall use boldface letters, either upper or lower case, to denote vectors. However, matrices are not denoted with bold letters. Blackboard bold characters, with the exception of the standard symbols for the real numbers and the integers, denote fourth-order tensors.

Assumption 2.1.

1. D^j_α∈R^3×3 is symmetric, with (D^j_αξ,ξ)≥d_α|ξ|² for allξ∈R³ and somed_α>0, where α=p, n,j = 1,2, andγ_p≥0.

2. Fp : R² → R² is continuously differentiable in I_μ², with F_p,1(0, η) = 0, F_p,2(ξ,0) = 0, F_p,1(ξ, η) ≥0, and

|F_p,2(ξ, η)| ≤g₁(ξ)(1 +η) for allξ, η∈R₊ and someg₁∈C¹(R₊;R₊).

3. Jp:R²→R² is continuously differentiable inI_μ², with J_p,1(0, η)≥0,J_p,2(ξ,0)≥0,|J_p,1(ξ, η)| ≤γ_J(1 +ξ), and|Jp,2(ξ, η)| ≤g(ξ)(1 +η) for allξ, η∈R+ and someγ_J >0 andg∈C¹(R+;R+).

4. Fn:R⁴→R² is continuously differentiable inI_μ⁴, withF_n,1(ξ,0, η₂)≥0,F_n,2(ξ, η₁,0)≥0, and

|F_n,1(ξ,η)| ≤γ_F¹(1 +g₂(ξ) +|η|), |F_n,2(ξ,η)| ≤γ_F²(1 +g₂(ξ) +|η|), for allξ= (ξ₁, ξ₂)^T,η= (η₁, η₂)^T ∈R²₊and someγ_F¹, γ_F² >0, andg₂∈C¹(R²₊;R₊).

M. PTASHNYK AND B. SEGUIN

5. Rn:R³×R₊→R² andR_b:R³×R₊→Rare continuously differentiable inI_μ³×R₊ and satisfy R_n,1(0, ξ₂, η, ζ)≥0, |R_n,1(ξ, η, ζ)| ≤β₁(1 +|ξ|+η)(1 +ζ),

R_n,2(ξ₁,0, η, ζ)≥0, |R_n,2(ξ, η, ζ)| ≤β₂(1 +|ξ|+η)(1 +ζ),

R_b(ξ,0, ζ)≥0, |Rb(ξ, η, ζ)| ≤β₃(1 +|ξ|+η)(1 +ζ), (R_b(ξ, η, ζ))⁺≤β₄ for some β_j >0, j= 1, . . . ,4, and allξ= (ξ₁, ξ₂)^T ∈R²₊, η, ζ∈R₊.

6. Jn:R²→R² is continuously differentiable inI_μ², withJ_n,1(0, η)≥0,J_n,2(ξ,0)≥0,|J_n,1(ξ, η)| ≤γ_n¹(1 +ξ), and|J_n,2(ξ, η)| ≤γ_n²(1 +ξ+η) for allξ, η∈R₊ and someγ_n¹, γ_n²>0.

7. G(ξ, η) :R²→R², withG(ξ, η) = (0, γ₁−γ₂η)^T forη∈Rand someγ₁, γ₂≥0.

8. VM ∈C¹(R) possesses major and minor symmetries,i.e.VM,ijkl =VM,klij =VM,jikl =VM,ijlk, and there exists ω_V >0 such thatVM(ξ)A·A≥ω_V|A|²for all symmetricA∈R^3×3andξ∈R₊.

9. EM ∈ C¹(R), EF, EM possess major and minor symmetries, i.e. EL,ijkl =EL,klij =EL,jikl =EL,ijlk, for L=F, M, and there existsω_E >0 such thatEFA·A≥ω_E|A|²,EM(ξ)A·A≥ω_E|A|², andE_M(ξ)A·A≥0 for all symmetric A∈R^3×3 andξ∈R+. There existsγ_M >0 such that|EM(ξ)| ≤γ_M for allξ∈R+. 10. The initial conditions p0 = (p_0,1, p_0,2)^T,n0 = (n_0,1, n_0,2)^T ∈ L^∞(Ω)², b₀ ∈ H¹(Ω)∩L^∞(Ω) are non-

negative, andu0∈ W(Ω).

11. f ∈H¹(0, T;L²(Γ_E ∪Γ_U))³ andp_I ∈H¹(0, T;L²(Γ_I)).

Remark 2.2. Notice that Assumption2.1.9 is not restrictive from a physical point of view, since every biolog- ical material will have a maximal possible stiffness. Also, in contrast to [26], we assume that (R_b(ξ, η, ζ))⁺ is bounded, see Assumption 2.1.5. This assumption is used to derivea prioriestimates for solutions of the equa- tions of linear viscoelasticity, independent ofb^ε, and to prove the global in time existence of a weak solution of (2.1) and (2.3) for arbitrary initial data and boundary conditions satisfying Assumptions2.1.10 and2.1.11. The local in time existence of a weak solution or the existence of a weak solution for small data can be shown by considering the same assumptions as in [26],i.e. without the assumption of the boundedness of (R_b(ξ, η, ζ))⁺. Notice that possible biologically relevant forms for reaction terms in (2.2) are given by Fp(p) = (R_eE(p),0)^T, Fn(p,n) = (R_eE(p)−2R_dc(n)−R_dn₁,−Rdc(n))^T, Rn(n, b,Nδ(e(u))) = (2R_bb(b)Nδ(e(u)), R_bb(b)Nδ(e(u)))^T, and R_b(n, b,Nδ(e(u))) = R_dc(n)−R_bb(b)Nδ(e(u)). Then the boundedness of (R_b(ξ, η, ζ))⁺, assumed in As- sumption2.1.5, is ensured if (R_dc(ξ))⁺ is bounded for nonnegativeξ₁andξ₂,e.g.R_dcis a Hill function.

A weak solution of (2.1)–(2.3) is defined in the following way.

Definition 2.3. A weak solution of the microscopic model (2.1)–(2.3) is a tuple (p^ε,n^ε, b^ε,u^ε), such that b^ε∈H¹(0, T;L²(Ω_M^ε )),p^ε,n^ε∈L²(0, T;V(Ω_M^ε ))², ∂_tp^ε, ∂_tn^ε∈L²(0, T;V(Ω_M^ε ))² and satisfy the equations

∂tp^ε,φ_p V,V+Dp∇p^ε,∇φ_p Ω_M,T^ε =−Fp(p^ε),φ_p Ω_M,T^ε +Jp(p^ε),φ_p ΓI,T − γpp^ε,φ_p ΓE,T, ∂_tn^ε,φ_n V,V+D_n∇n^ε,∇φ_n Ω_M,T^ε =

Fn(p^ε,n^ε) +Rn(n^ε, b^ε,Nδ(e(u^ε))),φ_n

Ω_M,T^ε

+

Nδ(e(u^ε))G(n^ε),φ_n

ΓI,T +Jn(n^ε),φ_n ΓE,T

(2.5)

for allφ_p,φ_n∈L²(0, T;V(Ω_M^ε ))²,

∂_tb^ε=R_b(n^ε, b^ε,Nδ(e(u^ε))) a.e. inΩ_M,T^ε , (2.6) andu^ε∈L²(0, T;W(Ω)), with∂_te(u^ε)∈L²((0, T)×Ω_M^ε )³, satisfies

E^ε(b^ε, x)e(u^ε) +V^ε(b^ε, x)∂_te(u^ε),e(ψ)

ΩT =f,ψ ΓEU,T − p_Iν,ψ ΓI,T (2.7) for allψ∈L²(0, T;W(Ω)). Furthermore,p^ε,n^ε,b^εsatisfy the initial conditions inL²(Ω_M^ε ) andu^εsatisfies the initial condition inW(Ω),i.e.u^ε(t,·)→u0inW(Ω),p^ε(t,·)→p0,n^ε(t,·)→n0 inL²(Ω_M^ε )², andb^ε(t,·)→b₀ in L²(Ω^ε_M) ast→0.

3. Main results

The main result of this paper is the derivation of the macroscopic equations for the microscopic viscoelastic model for plant cell wall biomechanics. The main difference between the homogenization results presented here and those in [26] is due to the presence of a degenerate viscous term in the equations for the mechanical deformations of a cell wall. The fact that only the cell wall matrix is viscoelastic and the dependence of the viscosity tensor on the time variable, via the dependence on the cross-links density b^ε, make the multiscale analysis nonclassical and complex.

First we formulate the well-posedness result for the model (2.1)–(2.3).

Theorem 3.1. Under Assumption 2.1there exists a unique weak solution of (2.1)–(2.3)satisfying thea priori estimates

b^εL^∞(0,T;L^∞(Ω_M^ε))+(∂_tb^ε)⁺L^∞(0,T;L^∞(Ω_M^ε ))≤C₁, (3.1) where the constant C₁ is independent of εandδ,

u^ε_L^∞_(0,T;W(Ω))+∂te(u^ε)_L²_((0,T)×Ω^ε_M)≤C₂, (3.2) where the constant C₂ is independent of ε, and

p^ε_L^∞_(0,T;L^∞_(Ω^ε_M))+∇p^ε_L²_(Ω^ε_M,T)+n^ε_L^∞_(0,T;L^∞_(ΩM^ε₎₎+∇n^ε_L²_(Ω^ε_M,T)≤C₃,

∂tb^εL^∞(0,T;L^∞(Ω^ε_M))≤C₃, θhp^ε−p^ε_L²_(ΩM,T−h^ε ₎+θhn^ε−n^ε_L²_(Ω^ε_M,T−h)≤C₃h^1/4

(3.3)

for any h > 0, where θ_hv(t, x) = v(t+h, x) for (t, x) ∈ Ω_M,T−h^ε , with h ∈ (0, T), and the constant C₃ is independent ofε andh.

The proof of Theorem3.1is similar to the proof of the corresponding existence and uniqueness results in [26].

Thus here we will only sketch the main ideas of the proof and emphasise the steps that are different from those of the proof in [26].

To formulate the macroscopic equations for the microscopic model (2.1)–(2.3), first we define the macroscopic coefficients which will be obtained in the derivation of the limit equations. The macroscopic coefficients coming from the elasticity tensor are given by

E_hom,ijkl(b) =−

Yˆ

Eijkl(b, y) +

E(b, y) ˆey(w^ij)

kl

dy, Kijkl(t, s, b) =−

Yˆ

E(b(t+s), y) ˆey(v^ij(t, s))

kl dy,

(3.4)

and the macroscopic elasticity and viscosity tensors and the memory kernel read:

E_hom,ijkl(b) =E_hom,ijkl(b) + 1

|Yˆ|

YˆM

VM(b)∂_tˆey(w^ij)

kl dy, Vhom,ijkl(b) = 1

|Yˆ|

YˆM

VM,ijkl(b) +

VM(b) ˆey(χ^ij_V)

kl

dy,

Kijkl(t, s, b) =Kijkl(t, s, b) + 1

|Yˆ|

YˆM

VM(b(t+s))∂_teˆy(v^ij(t, s))

kl dy,

(3.5)

M. PTASHNYK AND B. SEGUIN

wherew^ij, χ^ij_V, andv^ij, withi, j= 1,2,3, are solutions of the ‘unit cell’ problems divˆ _y

E(b, y)(ˆey(w^ij) +bij) +V(b, y)∂tˆey(w^ij)

=0 in ˆY_T, w^ij(0, x, y) =0 in ˆY , divˆ _y

VM(b)(ˆey(χ^ij_V) +bij)

=0 in ˆY_M, VM(b)(ˆey(χ^ij_V) +bij)ν=0 on ˆΓ ,

Yˆw^ijdy =0,

YˆM

χ^ij_Vdy=0, w^ij, χ^ij_V Yˆ-periodic,

(3.6)

wherebjk= ¹₂(bj⊗bk+bk⊗bj), with{bj}1≤j≤3 being the canonical basis ofR³, and divˆ _y

E(b(t+s, x), y)ˆey(v^ij) +V(b(t+s, x), y)∂_teˆy(v^ij)

=0 in ˆY_T−s, v^ij(0, s, x, y) =χ^ij_V(s, x, y)−w^ij(s, x, y) in ˆY ,

Yˆ

v^ijdy=

Yˆ

χ^ij_Vdy, v^ij Yˆ-periodic,

(3.7)

for x ∈ Ω and s ∈ [0, T], where χ^ij_V is an extension of χ^ij_V from ˆY_M into ˆY. Here for a vector function v = (v₁, v₂, v₃)^T we denote ˆdiv_yv = ∂_y1v₁+∂_y2v₂ and ˆey(v) is defined in the following way: ˆey(v)₃₃ = 0, ˆey(v)_3j = ˆey(v)_j3 =¹₂∂_yjv₃forj= 1,2, and ˆey(v)_ij= ¹₂(∂_yiv_j+∂_yjv_i) fori, j= 1,2.

The macroscopic diffusion coefficients are defined by D_α,ij^l =−

YˆM

D^l_α,ij+ (D_α^l∇ˆyv_α,l^j )_i

dy for i, j= 1,2,3, α=p, n, l= 1,2, (3.8) where ˆ∇_yv_α,l^j = (∂_y1v^j_α,l, ∂_y2v^j_α,l,0)^T and the functions v^j_α,l are solutions of the ‘unit cell’ problems

div_ˆy( ˆD^l_α∇y_ˆv_α,l^j ) = 0 in ˆY_M, j= 1,2,3,

( ˆD_α^l∇y_ˆv^j_α,l+ ˜D_α^lbj)·ν= 0 on ˆΓ , v^j_α,l Yˆ − periodic,

YˆM

v_α,l^j dy= 0, (3.9) where∇yˆ= (∂_y1, ∂_y2)^T, ˆD_α^l = (D^l_α,ik)_i,k=1,2 and ˜D^l_α= (D_α,ik^l )i=1,2,k=1,2,3, withl= 1,2 and α=p, n.

Applying techniques of periodic homogenization we obtain the macroscopic equations for plant cell wall biomechanics.

Theorem 3.2. A sequence of solutions of the microscopic model (2.1)–(2.3) converges to a solution of the macroscopic equations

∂_tp= div(Dp∇p)−Fp(p) in Ω_T,

∂_tn= div(Dn∇n) +Fn(p,n) +Rn(n, b,N_δ^eff(e(u))) in Ω_T,

∂_tb=R_b(n, b,N_δ^eff(e(u))) in Ω_T,

(3.10)

together with the initial and boundary conditions

Dp∇pν=θ_M⁻¹Jp(p), Dn∇nν=θ_M⁻¹G(n)N_δ^eff(e(u)) onΓ_I,T, D_p∇pν=−θ⁻¹_Mγ_pp, D_n∇nν=θ_M⁻¹Jn(n) onΓ_E,T, Dp∇pν=0, Dn∇nν=0 onΓ_U,T,

p, n a₃-periodic in x₃,

p(0) =p0, n(0) =n0, b(0) =b₀ inΩ,

(3.11)

whereθ_M =|Yˆ_M|/|Yˆ|, and the macroscopic equations of linear viscoelasticity div

Ehom(b)e(u) +Vhom(b)∂_te(u) + _t

0 K(t−s, s, b)∂_se(u) ds

=0 in Ω_T,

E_hom(b)e(u) +V_hom(b)∂_te(u) + _t

0 K(t−s, s, b)∂_se(u) ds

ν=f on Γ_EU,T,

E_hom(b)e(u) +V_hom(b)∂_te(u) + _t

0 K(t−s, s, b)∂_se(u) ds

ν=−p_Iν on Γ_I,T,

u a₃-periodic inx₃, u(0) =u0 in Ω.

(3.12)

Here

N_δ^eff(e(u)) =

−

Bδ(x)∩Ω

tr

Ehom(b)e(u) + _t

0

K(t−s, s, b)∂_se(u)ds

d˜x ₊

for (t, x)∈(0, T)×Ω. (3.13)

4. Existence of a unique weak solution of the microscopic problem (2.1) – (2.3) . a priori estimates

In the derivation of a priori estimates for solutions of the microscopic problem (2.1)–(2.3) we shall use an extension of a function defined on a connected perforated domainΩ_M^ε to Ω. Applying classical extension results [2,9,15,22], we obtain the following lemma.

Lemma 4.1. There exists an extension v^ε of v^ε from W^1,p(Ω_M^ε )intoW^1,p(Ω), with 1≤p <∞, such that v^ε_L^p_(Ω)≤μ₁v^ε_L^p_(ΩM^ε₎ and ∇v^ε_L^p_(Ω)≤μ₁∇v^ε_L^p_(Ω^ε_M),

where the constant μ₁ depends only onY andY_M, andY_M ⊂Y is connected.

There exists an extension w^ε ofw^ε fromH¹(Ω_M^ε )³ intoH¹(Ω)³ such that

w^ε_L^p_(Ω)≤μ₂w^ε_L^p_(ΩM^ε₎, ∇w^ε_L^p_(Ω)≤μ₂∇w^ε_L^p_(ΩM^ε₎, e(w^ε)_L^p_(Ω)≤μ₂e(w^ε)_L^p_(ΩM^ε₎, where the constant μ₂ does not depend onw^εandε.

Remark 4.2. Notice that the microfibrils do not intersect the boundariesΓ_I,Γ_U, andΓ_E, and near the bound- ariesΓ_P =∂Ω\(Γ_I∪Γ_E∪Γ_U) it is sufficient to extendv^εandw^εby reflection in the directions normal to the microfibrils and parallel to the boundary. Thus, classical extension results [2,9,15,22,25] apply to Ω_M^ε .

In the sequel, we identify p^ε andn^εwith their extensions.

First we show the well-posedness and a priori estimates for equations (2.2) and (2.3) for a given u^ε ∈ L^∞(0, T;W(Ω)). Next for a givenb^εwe show the existence of a unique solution of the viscoelastic problem (2.1).

Then using the fact that the estimates for b^ε can be obtained independently ofu^ε and applying a fixed point argument we show the well-posedness of the coupled system.

Lemma 4.3. Under Assumption2.1 and foru^ε∈L^∞(0, T;W(Ω))such that

u^ε_L^∞_(0,T;W(Ω))≤C, (4.1)

where the constant C is independent of ε, there exists a unique weak solution (p^ε,n^ε, b^ε) of the microscopic problem (2.2)and (2.3), with p^ε= (p^ε₁, p^ε₂)^T andn^ε= (n^ε₁, n^ε₂)^T, satisfying

p^ε_j(t, x)≥0, n^ε_j(t, x)≥0, b^ε(t, x)≥0 for (t, x)∈(0, T)×Ω_M^ε , j= 1,2, and thea priori estimates (3.1)and (3.3).