Finite elastic wrinkling deformations of incompressible fiber-reinforced plates

Abstract

A two-dimensional plate theory, valid for finite elastic deformations with small strains, is derived for incompressible, fiber-reinforced materials. Single-layer plates and two-layer laminates are considered. Numerical simulations illustrate the substantial effect that fiber reinforcement has on wrinkling patterns in the sheet.

1. Introduction

The nonlinear elastic response of thin sheets is of great significance in technology and biology. For example, the mitral valve of the heart may be viewed as a thin bio-elastic sheet with fairly complex mechanical properties conferred by a fibrous, laminated substructure in which each lamina is reinforced by one or two families of fibers (Wenk et al., 2012; Zhang et al., 2016). This structure is the subject of considerable current research (Grashow et al., 2006a,b; Ayoub et al., 2016; Sacks and Yoganathan, 2007), which, however, is typically based on modeling that appears to us to be somewhat ad hoc by the standards of the Mechanics community. This circumstance furnishes impetus for the present work, concerned with the careful development of a two-dimensional framework for thin, fiber-reinforced, incompressible elastic sheets.

We begin in Section 2 with a review of the constitutive structure of three-dimensional elasticity for incompressible materials and its specialization to the small-strain, finite-deformation regime where contact is made with Spencer’s formulation (Spencer, 1984) for transversely isotropic and orthotropic materials. These model materials are reinforced, respectively, by either one family of fibers or by two families of initially orthogonal fibers. A two-dimensional strain energy is then derived in Section 3, for both single laminae and for two-ply laminates, by an asymptotic analysis of the three-dimensional energy for a thin sheet. This is used in Section 4 to derive the equilibrium theory for thin plates on the basis of a virtual-power postulate. The formal similarity of the present theory to the theory of second-grade elasticity is emphasized to aid in its interpretation. In Section 5, we outline a computational procedure, used in a related work (Taylor et al., 2014) concerned with the wrinkling of isotropic plates, for solving the relevant equilibrium equations. For the sake of brevity and to avoid duplication, we confine ourselves to a description of this method and refer the interested reader to that work for a full exposition. Finally, in Section 6 we discuss a number of examples intended to highlight the effects of fiber reinforcement. These illustrate the substantial effect of anisotropy on wrinkling patterns in thin sheets. Indeed, plate wrinkling is currently an active field of research, albeit typically limited to isotropic materials, e.g., (Puntel et al., 2011; Nayyar et al., 2011; Healey et al., 2013; Taylor et al., 2014; Qin et al., 2014; Taylor et al., 2015; Fu et al., 2019; Wang et al., 2019), with notable exceptions including studies of orthotropic sheets using finite elements (Woo et al., 2004; Gerngross and Pellegrino, 2009; Deng and Pellegrino, 2012) and an extended finite Föppl-von Kármán model (Sipos and Fehér, 2016). Our main objective in the present work is to expand the scope of this field to encompass certain types of anisotropy in a framework derived from three-dimensional elasticity.

Standard notation is adopted throughout. Thus, we use bold face for vectors and tensors and indices to denote their Cartesian components. Latin indices take values in {1,2,3}; Greek in {1,2}. The latter are associated with surface coordinates and associated vector and tensor components, and the usual summation convention for repeated subscripts is adopted. The notation ⊗ identifies the standard tensor product of vectors and a dot between bold symbols is used to denote the Euclidean inner product. Thus, if A₁ and A₂ are second-order tensors, then ${\mathbf{A}}_1\cdot {\mathbf{A}}_2=\mathit{tr}({\mathbf{A}}_1{\mathbf{A}}_2^t)$, where tr(⋅) is the trace and the superscript ^t is used to denote the transpose. The norm of a tensor A is $\mid \mathbf{A}\mid =\sqrt{\mathbf{A}\cdot \mathbf{A}}$ and its determinant is det A. Its inverse A⁻¹ exists and is unique if and only if det A ≠ 0. The notation SymA, DevA and SphA is used to denote the symmetric, deviatoric and spherical parts of A, respectively, whereas Sym, Dev and Sph are the linear spaces of symmetric, deviatoric and spherical tensors. If $\mathcal{M}$ is a fourth-order tensor, then $\mathcal{M}[\mathbf{A}]$ is the second-order tensor with Cartesian components ${\mathcal{M}}_{iAjB}{A}_{jB}$. We use Div to denote the three-dimensional divergence operator, and div to denote its two-dimensional counterpart. For example, DivA = A_iA,Ae_i and divA = A_iα,αe_i, where {e_i} is an orthonormal basis in which subscripts preceded by commas are used to denote partial derivatives with respect to Cartesian coordinates, and the unit vector k = e₃ identifies the orientation of the plate midplane prior to deformation. We also use ∇ to denote the two-dimensional gradient on this plane. Finally, F_A stands for the tensor-valued derivative of a scalar-valued function F(A) with respect to its tensor argument.

2. Three-dimensional constitutive theory

We consider hyperelastic materials endowed with strain-energy functions of the form W(F; x), where x is the position of a material point in a reference configuration κ of the body and F is the gradient of the deformation χ(x). Here, the explicit dependence of W on x accommodates a possible non-uniformity of the material due, for example, to a spatial distribution of reinforcing fibers.

The 1st Piola-Kirchhoff stress P is given by

$$\mathbf{P}=W_{\mathbf{F}}-p{\mathbf{F}}^{\ast },$$ (1)

where F* = JF^−t is the cofactor of F, J = det F and p is a Lagrange multiplier associated with the constraint of incompressibility (J = 1). Here, for the purpose of calculating the derivative W_F, W is regarded as a smooth extension of the strain-energy function from the constraint manifold defined by J = 1 to the set defined by det F > 0; the derivative is then evaluated on the manifold. It is well known that the strain-energy function is frame-invariant if and only if it depends on F through the Cauchy-Green deformation tensor C = F^tF or, equivalently, through the Lagrange strain $\mathbf{E}=\frac{1}{2}(\mathbf{C}-\mathbf{I})$, where I is the identify for 3-space; we write W(F; x) = U(E; x). This furnishes the 2nd Piola-Kirchhoff stress S, related to P by

$$\mathbf{P}=\mathbf{FS}.$$ (2)

Thus,

$$\mathbf{S}=U_{\mathbf{E}}-p{\mathbf{C}}^{-1},$$ (3)

where, again, U is a smooth extension from the manifold defined by det(I + 2E) = 1 and the derivative is evaluated, post facto, on the manifold.

It is well known (Steigmann, 2017) that the difference of the derivatives of two extensions of the energy is orthogonal to the local tangent space to the constraint manifold and may therefore be absorbed into the Lagrange multiplier. Thus any one extension may be used without loss of generality.

Consider a one-parameter family χ(x; ϵ) of deformations. This induces the one-parameter families F(x;ϵ) and E(x;ϵ). We differentiate the equation W = U with respect to the parameter, on the constraint manifold, obtaining

$$W_{\mathbf{F}}\cdot \dot{\mathbf{F}}=U_{\mathbf{E}}\cdot \dot{\mathbf{E}},\text{with}\dot{\mathbf{E}}=Sym({\mathbf{F}}^t\dot{\mathbf{F}}),$$ (4)

or, equivalently,

$$W_{\mathbf{F}}\cdot \dot{\mathbf{F}}=\mathbf{F}(U_{\mathbf{E}})\cdot \dot{\mathbf{F}},$$ (5)

in which $\stackrel{.}{\mathbf{F}}$ is restricted by the requirement $\stackrel{.}{J}=0$. With J_F = F* the latter implies that

$${\mathbf{F}}^{*}\cdot \dot{\mathbf{F}}=0$$ (6)

and $\stackrel{.}{\mathbf{F}}$ is otherwise arbitrary.

A further differentiation yields

$$W_{\mathbf{FF}}[\dot{\mathbf{F}}]\cdot \dot{\mathbf{F}}+W_{\mathbf{F}}\cdot \ddot{\mathbf{F}}=U_{\mathbf{EE}}[\dot{\mathbf{E}}]\cdot \dot{\mathbf{E}}+U_{\mathbf{E}}\cdot \ddot{\mathbf{E}},\text{with}\ddot{\mathbf{E}}={\dot{\mathbf{F}}}^t\dot{\mathbf{F}}+\mathit{Sym}({\mathbf{F}}^t\ddot{\mathbf{F}}),$$ (7)

or, equivalently,

$$W_{\mathbf{FF}}[\dot{\mathbf{F}}]\cdot \dot{\mathbf{F}}+W_{\mathbf{F}}\cdot \ddot{\mathbf{F}}={\mathbf{F}}^{\mathit{t}}\dot{\mathbf{F}}\cdot U_{\mathbf{EE}}[{\mathbf{F}}^{\mathit{t}}\dot{\mathbf{F}}]+\mathbf{F}(U_{\mathbf{E}})\cdot \ddot{\mathbf{F}}+U_{\mathbf{E}}\cdot {\dot{\mathbf{F}}}^{\mathit{t}}\dot{\mathbf{F}},$$ (8)

in which the minor symmetries of U_EE have been used, all derivatives are evaluated on the constraint manifold, and $\ddot{\mathbf{F}}$ is restricted by the requirement $\ddot{J}=0$, i.e.

$${\mathbf{F}}^{\text{*}}\cdot \ddot{\mathbf{F}}+{\mathbf{F}}_{\mathbf{F}}^{*}[\dot{\mathbf{F}}]\cdot \dot{\mathbf{F}}=0.$$ (9)

The 4th-order tensor ${\mathbf{F}}_{\mathbf{F}}^{\ast }$ is given, in terms of the Cartesian components F_iA of F, by (Steigmann, 2017)

$$J\partial F_{iA}^{*}/\partial F_{jB}=F_{iA}^{*}{F}_{jB}^{*}-F_{jA}^{*}{F}_{iB}^{*}.$$ (10)

With J = 1, this furnishes

$${\mathbf{F}}_{\mathbf{F}}^{*}[\dot{\mathbf{F}}]=({\mathbf{F}}^{*}\cdot \dot{\mathbf{F}}){\mathbf{F}}^{*}-F_{iB}^{*}{F}_{jA}^{*}{\dot{F}}_{jB},$$ (11)

which combines with (6) and (9) to give

$${\mathbf{F}}^{\text{*}}\cdot \ddot{\mathbf{F}}=F_{iB}^{*}{F}_{jA}^{*}{\dot{F}}_{jB}{\dot{F}}_{iA},$$ (12)

whereas the part of $\ddot{\mathbf{F}}$ orthogonal to F* is arbitrary.

Consider the rank-one form $\stackrel{.}{\mathbf{F}}=\mathbf{p}\otimes \mathbf{q}$. This yields ${\mathbf{F}}^{\ast }\cdot \ddot{\mathbf{F}}={(\mathbf{p}\cdot {\mathbf{F}}^{\ast }\mathbf{q})}^2$, which vanishes because

$$\mathbf{p}\cdot {\mathbf{F}}^{*}\mathbf{q}=0$$ (13)

in accordance with (6). Because of (5) and (6), we then have

$$W_{\mathbf{F}}\cdot \ddot{\mathbf{F}}=\mathbf{F}(U_{\mathbf{E}})\cdot \ddot{\mathbf{F}}$$ (14)

and (8) reduces to

$$W_{\mathbf{FF}}[\mathbf{p}\otimes \mathbf{q}]\cdot \mathbf{p}\otimes \mathbf{q}={\mathbf{F}}^t\mathbf{p}\otimes \mathbf{q}\cdot U_{\mathbf{EE}}[{\mathbf{F}}^t\mathbf{p}\otimes \mathbf{q}]+{\left|\mathbf{p}\right|}^2\mathbf{q}\cdot (U_{\mathbf{E}})\mathbf{q}.$$ (15)

The strong-ellipticity condition is

$$W_{\mathbf{FF}}[\mathbf{p}\otimes \mathbf{q}]\cdot \mathbf{p}\otimes \mathbf{q}>0$$ (16)

for all p ⊗ q ≠ 0 subject to (13) (Fosdick and MacSithigh, 1986). Accordingly, if the deformation is such that the strong-ellipticity condition is satisfied, then the right-hand side of (15) is positive with the same proviso. Supposing strong ellipticity to be satisfied at the undeformed state F = I, and that U_E vanishes at E = 0, we then have

$$\mathbf{p}\otimes \mathbf{q}\cdot \mathcal{C}[\mathbf{p}\otimes \mathbf{q}]>0$$ (17)

for all p ⊗ q ≠ 0 subject to p ⋅ q = 0, where $\mathcal{C}=U_{\mathbf{EE}}(0)$ is the classical elasticity tensor. This makes sense because p ⊗ q vanishes if and only if Sym(p ⊗ q) vanishes. Conventionally, (17) is ensured by requiring that $\mathcal{C}$ be positive definite in the sense that

$$\mathbf{A}\cdot \mathcal{C}[\mathbf{A}]>0$$ (18)

for all non-zero symmetric A with trA = 0. This in turn follows from the inequality $\ddot{U}>0$, evaluated at F = I, with U_E = 0 at E = 0 and with $\mathit{Sym}\stackrel{.}{\mathbf{F}}=\mathbf{A}$ restricted in accordance with (6), i.e. I ⋅ A = 0.

For small strains of incompressible materials, it is well known that E is deviatoric at leading order. This follows by expanding J² = det(I + 2E) with J = 1, yielding

$$tr\mathbf{E}={\left|\mathbf{E}\right|}^2-{(tr\mathbf{E})}^2-4\text{det}\mathbf{E},$$ (19)

which implies that E = DevE + SphE with |SphE| = O(|E|²). The leading-order small-strain approximation to the strain energy is then given by

$$U(\mathbf{E})=Q(\mathit{Dev}\mathbf{E})+o({\left|\mathbf{E}\right|}^2),$$ (20)

with

$$Q(\mathbf{E})=\frac{1}{2}\mathbf{E}\cdot \mathcal{C}[\mathbf{E}].$$ (21)

The latter expression furnishes the obvious extension of the leading-order energy Q(DevE) for use in computing the derivative U_E.

In the present work we consider materials reinforced by a single family of fibers and by two families of initially orthogonal fibers with distinct mechanical properties. The first of these types is transversely isotropic, with (Spencer, 1984)

$$\mathcal{C}[\mathbf{E}]=2{\mu }_T\mathbf{E}+2({\mu }_L-{\mu }_T)(\mathbf{E}\mathbf{a}\otimes \mathbf{a}+\mathbf{a}\otimes \mathbf{E}\mathbf{a})+\beta (\mathbf{a}\cdot \mathbf{E}\mathbf{a})\mathbf{a}\otimes \mathbf{a},$$ (22)

where a is the unit-tangent field to the fiber trajectories in κ. The moduli μ_T and μ_L respectively are the shear moduli for shearing in planes transverse and parallel to the fibers, and β contributes to the extensional stiffness along the fibers. The second type of material is orthotropic, with (Spencer, 1984)

$$\mathcal{C}[\mathbf{E}]=2\mu \mathbf{E}+2{\mu }_a(\mathbf{E}\mathbf{a}\otimes \mathbf{a}+\mathbf{a}\otimes \mathbf{E}\mathbf{a})+2{\mu }_b(\mathbf{E}\mathbf{b}\otimes \mathbf{b}+\mathbf{b}\otimes \mathbf{E}\mathbf{b})+({\beta }_a\mathbf{a}\cdot \mathbf{E}\mathbf{a}+{\beta }_{\mathit{ab}}\mathbf{b}\cdot \mathbf{E}\mathbf{b})\mathbf{a}\otimes \mathbf{a}+({\beta }_{\mathit{ab}}\mathbf{a}\cdot \mathbf{E}\mathbf{a}+{\beta }_{\mathit{b}}\mathbf{b}\cdot \mathbf{E}\mathbf{b})\mathbf{b}\otimes \mathbf{b},$$ (23)

where a and b, with a ⋅ b = 0, are the unit-tangent fields to the two fiber families. Here μ, μ_a and μ_b are shear moduli, and we refer to Spencer (1984) for interpretations of these and of β_a, β_b and β_ab in terms of the conventional engineering moduli.

We have

$$U_{\mathbf{E}}=\mathcal{C}[\mathbf{E}]+o(\left|\mathbf{E}\right|),$$ (24)

and with C⁻¹ = I + O(|E|), the leading-order approximation to the stress S is

$$\mathbf{S}\simeq \mathcal{C}[\mathit{Dev}\mathbf{E}]-p\mathbf{I},$$ (25)

in which we have evaluated the derivative U_E on the constraint manifold, as required by (3). The latter is approximated at leading order by its tangent space at E = 0, i.e. by the linear space Dev. Of course the spherical part of $\mathcal{C}[\mathit{Dev}\mathbf{E}]$ may be absorbed into the multiplier p, but it is quite unnecessary to do so. The expression (25) for the stress coincides precisely with that given in Spencer (1984).

In this work we take the fibers to lie parallel to a fixed plane with unit normal k, so that a ⋅ k = 0 and b ⋅ k = 0. With A = Sym(p ⊗ q) and p ⋅ q = 0, the strong-ellipticity condition (17) for transversely isotropic materials becomes

$$0<{\mu }_T{\left|\mathbf{p}\right|}^2{\left|\mathbf{q}\right|}^2+({\mu }_L-{\mu }_T)[{\left|\mathbf{p}\right|}^2{(\mathbf{q}\cdot \mathbf{a})}^2+{\left|\mathbf{q}\right|}^2{(\mathbf{p}\cdot \mathbf{a})}^2]+\frac{1}{2}\beta {(\mathbf{p}\cdot \mathbf{a})}^2{(\mathbf{q}\cdot \mathbf{a})}^2\cdot$$ (26)

To derive necessary conditions, we choose q = k with p = p_aa + p_bb, obtaining

$${\mu }_L{p}_a^2+{\mu }_T{p}_b^2>0$$ (27)

for all p_a, p_b with $p_a^2+p_b^2>0$; thus,

$${\mu }_L>0\text{and}{\mu }_T>0.$$ (28)

In the case of orthotropy, after some calculation we find that (17) becomes

$$0<\mu {\left|\mathbf{p}\right|}^2{\left|\mathbf{q}\right|}^2+{\mu }_a[{\left|\mathbf{p}\right|}^2{(\mathbf{q}\cdot \mathbf{a})}^2+{\left|\mathbf{q}\right|}^2{(\mathbf{p}\cdot \mathbf{a})}^2]+{\mu }_b[{\left|\mathbf{p}\right|}^2{(\mathbf{q}\cdot \mathbf{b})}^2+{\left|\mathbf{q}\right|}^2{(\mathbf{p}\cdot \mathbf{b})}^2]+\frac{1}{2}{\beta }_a{(\mathbf{p}\cdot \mathbf{a})}^2{(\mathbf{q}\cdot \mathbf{a})}^2+\frac{1}{2}{\beta }_b{(\mathbf{p}\cdot \mathbf{b})}^2{(\mathbf{q}\cdot \mathbf{b})}^2+{\beta }_{\mathit{ab}}(\mathbf{p}\cdot \mathbf{a})(\mathbf{q}\cdot \mathbf{a})(\mathbf{p}\cdot \mathbf{b})(\mathbf{q}\cdot \mathbf{b}).$$ (29)

Choosing q and p as before, we derive

$$(\mu +{\mu }_a)p_a^2+(\mu +{\mu }_b)p_b^2>0$$ (30)

and conclude that

$$\mu +{\mu }_a>0\text{and}\mu +{\mu }_b>0.$$ (31)

Of course the necessary conditions (28) and (31) are not sufficient for strong ellipticity, but they do suffice for our purposes. Detailed discussions of strong ellipticity for transversely isotropic and orthotropic materials are given in Payton (1983); Merodio and Ogden (2003) and Aguiar (2019), respectively. We discuss inequality (18) for the two material types in Section 3.2.

3. Asymptotic derivation of the leading-order plate energy for combined bending and stretching

3.1. Descent from three dimensions to two

Position in the reference placement of the plate may be written

$$\mathbf{x}=\mathbf{u}+\varsigma \mathbf{k},$$ (32)

where u ∈ Ω, Ω is the midplane of the plate, ς ∈ [−h/2, h/2] is a through-thickness coordinate and h is the plate thickness, assumed to be much smaller than the next smallest length scale, l say, in a given problem. We simplify the notation by adopting l as the unit of length (l = 1); then, the dimensionless thickness h << 1.

The strain energy associated with a given deformation is

$$\mathcal{E}={\int }_{\kappa }U(\tilde{\mathbf{E}}(\mathbf{x}))dv={\int }_{\Omega }{\int }_{-h/2}^{h/2}U(\tilde{\mathbf{E}}(\mathbf{u}+\varsigma \mathbf{k}))d\varsigma da,$$ (33)

where $\tilde{\mathbf{E}}(\mathbf{x})$ is the three-dimensional strain. If this is sufficiently smooth, then by Leibniz’ Rule and Taylor’s Theorem, applied to h (Shirani and Steigmann, 2019), we find that the areal energy density on Ω is given, for small h, by

$${\int }_{-h/2}^{h/2}U(\tilde{\mathbf{E}}(\mathbf{u}+\varsigma \mathbf{k}))d\varsigma =hU+\frac{1}{24}{h}^{3}U″+o(h^3),$$ (34)

where primes are used to denote derivatives with respect to ς, $\mathbf{E}(\mathbf{u})={\tilde{\mathbf{E}}}_{\mid \Omega }$ is the midplane strain, U is evaluated at E(u) and, by the chain rule,

$$U\prime =U_{\mathbf{E}}\cdot \mathbf{E}\prime \text{and}U″=U_{\mathbf{EE}}[\mathbf{E}\prime ]\cdot \mathbf{E}\prime +U_{\mathbf{E}}\cdot \mathbf{E}″$$ (35)

in which the derivatives U_E and U_EE are evaluated at E(u), and

$${\mathbf{E}}^{(n)}={\tilde{\mathbf{E}}}_{|\varsigma =0}^{(n)}.$$ (36)

Here, we assume that there is no variation of material properties in the direction orthogonal to Ω, so that U depends on ς only implicitly, via $\tilde{\mathbf{E}}(\mathbf{x})$. The material moduli, the thickness h, and the fiber directions a and b may, however, depend on u ∈ Ω.

If $\tilde{\mathbf{F}}(\mathbf{x})$ is the gradient of the three-dimensional deformation $\tilde{\chi }(\mathbf{x})$, then

$$\tilde{\mathbf{F}}=\nabla \tilde{\chi }+\tilde{\chi }\prime \otimes \mathbf{k},$$ (37)

where ∇ is the (two-dimensional) gradient with respect to u on Ω. Its first and second derivatives with respect to ς are

$$\tilde{\mathbf{F}}\prime =\nabla \tilde{\chi }\prime +\tilde{\chi }″\otimes \mathbf{k}\text{and}\tilde{\mathbf{F}}″=\nabla \tilde{\chi }″+\tilde{\chi }‴\otimes \mathbf{k},$$ (38)

and these reduce, on the midplane, to

$$\mathbf{F}=\nabla \mathbf{r}+\mathbf{d}\otimes \mathbf{k},\mathbf{F}\prime =\nabla \mathbf{d}+\mathbf{g}\otimes \mathbf{k},\text{and}\mathbf{F}″=\nabla \mathbf{g}+\mathbf{h}\otimes \mathbf{k},$$ (39)

where

$$\mathbf{r}(\mathbf{u})={\tilde{\chi }}_{|\Omega }$$ (40)

is the midplane deformation, and

$$\mathbf{d}(\mathbf{u})={\tilde{\chi }}_{\mid \Omega }^{\prime },\mathbf{g}(\mathbf{u})={\tilde{\chi }}_{\mid \Omega }^{″}\text{and}\mathbf{h}(\mathbf{u})={\tilde{\chi }}_{\mid \Omega }^{‴}.$$ (41)

are independent functions of u. These are the coefficient vectors in the order - ς³ expansion

$$\tilde{\chi }(\mathbf{u}+\varsigma \mathbf{k})=\mathbf{r}(\mathbf{u})+\varsigma \mathbf{d}(\mathbf{u})+\frac{1}{2}{\varsigma }^2\mathbf{g}(\mathbf{u})+\frac{1}{6}{\varsigma }^3\mathbf{h}(\mathbf{u})+\dots .$$ (42)

Here, r(u) is the position of a material point on the deformed image ω of the midplane Ω; its gradient ∇r maps the translation space Ω′ of Ω to the tangent plane T_ω to the surface ω at the material point u. The functions d(u), g(u) and h(u) provide information about the three-dimensional deformation in the vicinity of the midplane.

Further, in (34) and (35) we have (compare (4)₂, (7)₂)

$$\mathbf{E}=\frac{1}{2}({\mathbf{F}}^{\mathit{t}}\mathbf{F}-\mathbf{I}),\mathbf{E}\prime =\mathit{Sym}({\mathbf{F}}^{\mathit{t}}\mathbf{F}\prime )\text{and}\mathbf{E}″={(\mathbf{F}\prime )}^{\mathit{t}}\mathbf{F}\prime +\mathit{Sym}({\mathbf{F}}^{\mathit{t}}\mathbf{F}″),$$ (43)

in which F, F′ and F″ are restricted by bulk incompressibility, i.e. by the constraints J = 1, J′ = J″ = 0. In particular, we may use J = Fk ⋅ F*k with the Piola-Nanson formula F*k = αn, where n is the unit normal to T_ω and α = |F*k| is the areal stretch of Ω, together with J = 1 and (39)₁, to conclude that (Steigmann, 2017)

$$\mathbf{d}={\alpha }^{-1}\mathbf{n}+(\nabla \mathbf{r})\mathbf{e},$$ (44)

for some 2-vector e ∈ Ω′. Here we note that F*k = Fe₁ × Fe₂ = (∇r)e₁ × (∇r)e₂, where {e_α} is any orthonormal basis for Ω′ such that e₁ × e₂ ⋅ k = 1. Thus, d is determined by ∇r and e.

In the same way, we use 0 = J′ = F* ⋅ F′ with (39)₂, concluding that g ⋅ n = −α⁻¹F* ⋅ ∇d and hence that

$$\mathbf{g}=(\nabla \mathbf{r})\mathbf{f}-{\alpha }^{-1}({\mathbf{F}}^{*}\cdot \nabla \mathbf{d})\mathbf{n}$$ (45)

for some 2-vector f ∈ Ω′. A similar procedure may be used to derive a representation for h, etc., if desired.

Corresponding to (44), the Cauchy-Green deformation tensor C = F^tF is

$${\mathbf{C}=\mathbf{c}+\mathbf{ce}\otimes \mathbf{k}+\mathbf{k}\otimes \mathbf{ce}+({\alpha }^{-2}+\mathbf{e}\cdot \mathbf{ce})\mathbf{k}\otimes \mathbf{k},\text{where}\mathbf{c}=(\nabla \mathbf{r})}^{\mathit{t}}(\nabla \mathbf{r})$$ (46)

is the surfacial Cauchy-Green deformation tensor; this has the property

$$\det \mathbf{c}={\alpha }^2.$$ (47)

The associated strain is

$$\mathbf{E}=\epsilon +\mathit{Sym}(\gamma \otimes \mathbf{k})+\overline{E}\mathbf{k}\otimes \mathbf{k},$$ (48)

where

$$\gamma =\mathbf{ce},\overline{E}=\frac{1}{2}({\alpha }^{-2}-1+\mathbf{e}\cdot \mathbf{ce})\text{and}\epsilon =\frac{1}{2}(\mathbf{c}-1)$$ (49)

is the surface strain, in which

$$1=\mathbf{I}-\mathbf{k}\otimes \mathbf{k}$$ (50)

is the projection onto Ω′ and γ is evidently the transverse shear strain. Accordingly, for small |E|, incompressibility yields Ē = −trε + O(|E|²), so that DevE = Ē + O(|E|²), with

$$\overline{\mathbf{E}}=\epsilon -(tr\epsilon )\mathbf{k}\otimes \mathbf{k}+\mathit{Sym}(\gamma \otimes \mathbf{k}).$$ (51)

We are interested in applications involving finite deformations with small-to-moderate midplane strains. This suffices for most engineering applications (Koiter, 1960, 1966; Ciarlet, 2005). Further applications of contemporary interest in biomechanics include mitral heart valves, which typically experience principal stretches that are close to unity (Grashow et al., 2006a,b). In such circumstances, it is appropriate to adopt an estimate of the energy (34) in which the separate order - h and order - h³ terms are replaced by their leading-order approximations for small |E| . For example, in (34) we have

$$U=Q(\overline{\mathbf{E}})+o({\left|\mathbf{E}\right|}^2),$$ (52)

where Ē is given by (51). Similarly, from (24) we have $U_{\mathbf{E}}=\mathcal{C}[\mathbf{E}]+o(\mid \mathbf{E}\mid )=O(\mid \mathbf{E}\mid )$, whereas $U_{\mathbf{EE}}=\mathcal{C}+O(\mid \mathbf{E}\mid )$. From (35)₂ we then have $U^{″}={\mathbf{E}}^{\prime }\cdot \mathcal{C}[{\mathbf{E}}^{\prime }]+O(\mid \mathbf{E}\mid )$, in which E′ - given by (43)₂ - is estimated by observing that the right-stretch factor U in the polar decomposition F = RU of the deformation gradient, in which R is a rotation, satisfies U = I + O(|E|). Then,

$$\mathbf{E}\prime =\mathit{Sym}({\mathbf{R}}^{\mathit{t}}\mathbf{F}\prime )+O(\left|\mathbf{E}\right|),$$ (53)

yielding

$$U″={\mathbf{R}}^{\mathit{t}}\mathbf{F}\prime \cdot \mathcal{C}[{\mathbf{R}}^{\mathit{t}}\mathbf{F}\prime ]+O(\left|\mathbf{E}\right|).$$ (54)

Combining this with (33) and (34), we obtain

$$\mathcal{E}=E+o(h^3),$$ (55)

where

$$E={\int }_{\Omega }W\mathit{da},$$ (56)

with

$$W\simeq hQ(\overline{\mathbf{E}})+\frac{1}{12}{h}^{3}Q({\mathbf{R}}^{\mathit{t}}\mathbf{F}\prime ),$$ (57)

in which both terms are valid to leading order in |E|.

Note that tr(R^tF′) = R ⋅ F′ = R* ⋅ F′, where R* is the cofactor of R; this coincides with R because R is a rotation. Then, because F* = R*U*, and with U* = I + O(|E|), we have R* = F* + O(|E|), yielding tr(R^tF′) = O(|E|) on account of F* ⋅ F′(= J′) = 0. Thus R^tF′ is deviatoric at leading order, i.e. at order unity. Because Ē is also deviatoric, this means that the plate energy W is fully specified by (21), in which $\mathcal{C}[\mathit{Dev}\mathbf{E}]$ is given by (22) or (23) as appropriate.

The energy W involves the vector fields d and g via (39)_1,2. These in turn involve the 2-vectors e and f, which at this stage are kinematically independent of the midplane deformation r. This suggests a strategy whereby we attempt to render the energy stationary with respect to these fields a priori. For example, E is stationary with respect to e at fixed ∇r and f if and only if

$$h{Q}_{\mathbf{E}}(\overline{\mathbf{E}})\cdot {(\overline{\mathbf{E}})}^{\cdot }+O(h^3)=0,$$ (58)

where the superposed dot refers to a variational derivative, and (cf. (51))

$${(\overline{\mathbf{E}})}^{\cdot }=\mathit{Sym}(\dot{\gamma }\otimes \mathbf{k}),\text{with}\dot{\gamma }=\mathbf{c}\dot{\mathbf{e}}.$$ (59)

Dividing (58) by h and passing to the limit, we obtain

$$\dot{\gamma }\cdot (\mathcal{C}[\overline{\mathbf{E}}])\mathbf{k}=0$$ (60)

at leading order, in which $\stackrel{.}{\gamma }\in {\Omega }^{\prime }$ is arbitrary. Thus,

$$1(\mathcal{C}[\overline{\mathbf{E}}])\mathbf{k}=0,$$ (61)

where 1 is the projection onto Ω′.

For transversely isotropic materials, (22) is used to reduce this to

$${\mu }_T\gamma +({\mu }_L-{\mu }_T)(\mathbf{a}\cdot \gamma )\mathbf{a}=0.$$ (62)

Decomposing γ in the basis {a, b}, with b = k × a, and invoking the strong-ellipticity inequalities (28), we conclude that γ = 0. Moreover, a further consequence of strong-ellipticity is that this solution is energetically optimal (Steigmann, 2017; Shirani and Steigmann, 2019).

For orthotropic materials, (23) is used to reduce (61) to

$$(\mu +{\mu }_a)(\mathbf{a}\cdot \gamma )\mathbf{a}+(\mu +{\mu }_b)(\mathbf{b}\cdot \gamma )\mathbf{b}=0,$$ (63)

and the strong-ellipticity conditions (31) again yield γ = 0, which again is energetically optimal. Then, because c is positive definite, it follows from (49)₁ that e = 0, yielding

$$\overline{\mathbf{E}}=\epsilon -(tr\epsilon )\mathbf{k}\otimes \mathbf{k}$$ (64)

and

$$\mathbf{d}=\lambda \mathbf{n},\text{with}\lambda ={\alpha }^{-1}.$$ (65)

Thus, we have derived the well known Kirchhoff-Love hypothesis with thickness distension as the asymptotic leading-order approximation. Further, Q(Ē) is then determined entirely by ∇r. We combine (44) and (49)₂, reaching

$$\frac{1}{2}({\lambda }^2-1)=\overline{E}=-tr\epsilon +O({\left|\mathbf{E}\right|}^2).$$ (66)

Accordingly λ should be replaced by unity in the coefficient of h³ in the energy (57).

To elaborate, from (39)₂ and (65) we have that

$$\mathbf{F}\prime =\lambda \nabla \mathbf{n}+\mathbf{n}\otimes \nabla \lambda +\mathbf{g}\otimes \mathbf{k},$$ (67)

where ∇n is the referential gradient, defined by dn = (∇n)du. Let b be the curvature tensor of the deformed surface ω; this is the symmetric 2-tensor, defined by dn = −bdr, that maps the tangent plane T_ω at a particular material point to itself. Thus,

$$\nabla \mathbf{n}=-\mathbf{b}(\nabla \mathbf{r}).$$ (68)

In the course of arriving at (57), to be consistent with (53), we have approximated F by R. To this order of approximation, E vanishes and λ = 1. Moreover, with

$$\mathbf{R}=\mathbf{R}1+\mathbf{Rk}\otimes \mathbf{k},$$ (69)

the same approximation, in combination with (39)₁, gives

$$\mathbf{R}1\simeq \nabla \mathbf{r}\text{and}\mathbf{Rk}=\mathbf{n}.$$ (70)

Then the consistent-order approximation to R^tF′ is

$${\mathbf{R}}^{\mathit{t}}\mathbf{F}\prime \simeq \kappa +\mathbf{k}\otimes \nabla {\lambda +\mathbf{R}}^{\mathit{t}}\mathbf{g}\otimes \mathbf{k},$$ (71)

where

$$\kappa =-{(\nabla \mathbf{r})}^{\mathit{t}}\mathbf{b}(\nabla \mathbf{r})$$ (72)

is a symmetric 2-tensor on Ω′.

Recalling that consistency of (59) with bulk incompressibility requires that R^tF′ be deviatoric at leading order, we impose tr(R^tF′) = 0 and conclude that

$$\mathbf{k}\cdot {\mathbf{R}}^{\mathit{t}}\mathbf{g}\simeq -tr\kappa .$$ (73)

This agrees with the consistent-order approximation to (45) on noting that F* ⋅ ∇d = F*1 ⋅ ∇d with F* ≃ R; namely,

$$\mathbf{n}\cdot \mathbf{g}\simeq -\mathbf{R}1\cdot (\nabla \mathbf{n}+\mathbf{n}\otimes \nabla \lambda ),$$ (74)

in which R1 ⋅ n ⊗ ∇λ = n ∇ (∇r)∇λ vanishes because (∇r)∇λ ∈ T_ω. Writing

$${\mathbf{R}}^{\mathit{t}}\mathbf{g}=1({\mathbf{R}}^{\mathit{t}}\mathbf{g})+(\mathbf{k}\cdot {\mathbf{R}}^{\mathit{t}}\mathbf{g})\mathbf{k},$$ (75)

we then have

$${\mathbf{R}}^{\mathit{t}}\mathbf{F}\prime \simeq \kappa -(tr\kappa )\mathbf{k}\otimes \mathbf{k}+\mathbf{k}\otimes \nabla \lambda +1({\mathbf{R}}^{\mathit{t}}\mathbf{g})\otimes \mathbf{k}.$$ (76)

The energy (57) involves only the symmetric part:

$$\mathit{Sym}({\mathbf{R}}^{\mathit{t}}\mathbf{F}\prime )\simeq \kappa -(tr\kappa )\mathbf{k}\otimes \mathbf{k}+\mathit{Sym}(\mu \otimes \mathbf{k}),$$ (77)

where

$$\mu =1({\mathbf{R}}^{\mathit{t}}\mathbf{g})+\nabla \lambda .$$ (78)

This has a structure similar to (51).

Proceeding as in the foregoing, we seek 1(R^tg) (equivalently, f) that renders the energy stationary at fixed ∇r and e. With reference to (60), this implies that at leading order the former should satisfy

$$\dot{\mu }\cdot (\mathcal{C}[{\mathbf{R}}^{\mathit{t}}\mathbf{F}\prime ])\mathbf{k}=0$$ (79)

for all $\stackrel{.}{\mu }\in {\Omega }^{\prime }$; i.e.,

$$1(\mathcal{C}[{\mathbf{R}}^{\mathit{t}}\mathbf{F}\prime ])\mathbf{k}=0,$$ (80)

which is easily seen to reduce to (62) or (63), with γ replaced by μ, for transversely isotropic or orthotropic materials respectively. We conclude that μ = 0, and again strong ellipticity implies that this solution is energetically optimal (Steigmann, 2010). Thus,

$$\mathit{Sym}({\mathbf{R}}^{\mathit{t}}\mathbf{F}\prime )\simeq \kappa -(tr\kappa )\mathbf{k}\otimes \mathbf{k},$$ (81)

and the strain energy (57) reduces to

$$W=h{Q}^{*}(\epsilon )+\frac{1}{12}{h}^3{Q}^{*}(\kappa ),$$ (82)

where, for any symmetric 2-tensor α on Ω′,

$$Q^{*}(\alpha )=Q(\alpha -(tr\alpha )\mathbf{k}\otimes \mathbf{k}).$$ (83)

This is an additive decomposition of the strain energy into pure membrane and bending energies, as in Koiter’s well established model (Koiter, 1960, 1966) for isotropic materials undergoing finite deformations with small midplane strains. Indeed, (82) provides the extension of Koiter’s energy to incompressible transversely isotropic or orthotropic materials. It is determined entirely by the 1st and 2nd gradients of the midsurface deformation r(u).

Remark:

Our procedure for eliminating e and f relied on asymptotic estimates of the associated stationarity conditions at leading order in h and at fixed midplane deformation. On the other hand, the leading-order stationarity condition for the energy (82) furnishes membrane theory, which is known to be ill-posed as a minimization problem due to the failure of Q*(ε) to satisfy the operative Legendre-Hadamard necessary condition. This in turn is due to the presence of compressive in-plane 2nd Piola-Kirchhoff stresses in the stress-deformation relation derived from Q*(ε) (Steigmann, 1986; Shirani and Steigmann, 2019). In this case a well-posed formulation is achieved on replacing Q*(ε) by its quasiconvexification (Pipkin, 1994; Dacarogna, 1989; Le Dret and Raoult, 1996), which automatically furnishes equilibria having no compressive stress. These equilibria also furnish the least possible membrane energy that can be attributed to a given midplane deformation. This model was adapted to anisotropic membranes in Pipkin (1994) and further specialized to orthotropic bio-elastic membranes in Atai and Steigmann (2014). However, the relaxed membrane problem lacks the intrinsic length scale h required to resolve the spatial distribution of wrinkle patterns. For this reason we retain the full energy (82) in the ensuing variational treatment. In Section 5, we show that this procedure also yields an effective regularization of membrane theory.

For transversely isotropic laminae, (22) and (83) may be used to obtain

$$Q^{*}(\alpha )={\mu }_T[\alpha \cdot \alpha +{(tr\alpha )}^2]+2({\mu }_L-{\mu }_T)\alpha \mathbf{a}\cdot \alpha \mathbf{a}+\frac{1}{2}\beta {(\mathbf{a}\cdot \alpha \mathbf{a})}^2.$$ (84)

The derivative $Q_{\alpha }^{\ast }$, needed in Section 4, is

$$Q_{\alpha }^{*}=2{\mu }_T[\alpha +(tr\alpha )1]+2({\mu }_L-{\mu }_T)(\alpha \mathbf{a}\otimes \mathbf{a}+\mathbf{a}\otimes \alpha \mathbf{a})+\beta (\mathbf{a}\cdot \alpha \mathbf{a})\mathbf{a}\otimes \mathbf{a}.$$ (85)

For orthotropic laminae, we find, using (23), that

$$Q^{*}(\alpha )=\mu [\alpha \cdot \alpha +{(tr\alpha )}^2]+2{\mu }_a\alpha \mathbf{a}\cdot \alpha \mathbf{a}+2{\mu }_b\alpha \mathbf{b}\cdot \alpha \mathbf{b}+\frac{1}{2}{\beta }_a{(\mathbf{a}\cdot \alpha \mathbf{a})}^2+\frac{1}{2}{\beta }_b{(\mathbf{b}\cdot \alpha \mathbf{b})}^2+{\beta }_{\mathit{ab}}(\mathbf{a}\cdot \alpha \mathbf{a})(\mathbf{b}\cdot \alpha \mathbf{b}),$$ (86)

with derivative

$$Q_{\alpha }^{*}=2\mu [\alpha +(tr\alpha )1]+2{\mu }_a(\alpha \mathbf{a}\otimes \mathbf{a}+\mathbf{a}\otimes \alpha \mathbf{a})+2{\mu }_b(\alpha \mathbf{b}\otimes \mathbf{b}+\mathbf{b}\otimes \alpha \mathbf{b})+[{\beta }_a(\mathbf{a}\cdot \alpha \mathbf{a})+{\beta }_{\mathit{ab}}(\mathbf{b}\cdot \alpha \mathbf{b})]\mathbf{a}\otimes \mathbf{a}+[{\beta }_{\mathit{ab}}(\mathbf{a}\cdot \alpha \mathbf{a})+{\beta }_b(\mathbf{b}\cdot \alpha \mathbf{b})]\mathbf{b}\otimes \mathbf{b}.$$ (87)

3.2. Positive-definiteness conditions

Inequality (18), combined with (83), implies that Q*(α) is positive definite. This in turn imposes restrictions on the moduli, which we pause to derive here. To this end we decompose α in the form

$$\alpha ={\alpha }_{\mathit{aa}}\mathbf{a}\otimes \mathbf{a}+{\alpha }_{\mathit{bb}}\mathbf{b}\otimes \mathbf{b}+{\alpha }_{\mathit{ab}}(\mathbf{a}\otimes \mathbf{b}+\mathbf{b}\otimes \mathbf{a}).$$ (88)

We combine this with (84) and (85) to deduce, after some some algebra, that for both the transversely isotropic and orthotropic laminae,

$$Q^{*}(\alpha )=A{\alpha }_{aa}^2+B{\alpha }_{bb}^2+2C{\alpha }_{aa}{\alpha }_{bb}+D{\alpha }_{ab}^2,$$ (89)

where A − D depend only on the moduli. This is sum of two independent quadratic forms. Accordingly, Q*(α) is positive definite if and only if both quadratic forms are positive definite. Necessary and sufficient conditions are

$$\mathit{AB}-C^2>0,B>0\text{and}D>0.$$ (90)

For transversely isotropic laminae, we derive

$$A=2{\mu }_L+\frac{1}{2}\beta ,B=2{\mu }_T,C={\mu }_T\text{and}D=2{\mu }_L,$$ (91)

and Q*(⋅) is then positive definite if and only if

$${\mu }_T>0,\beta >{\mu }_T-4{\mu }_L\text{and}{\mu }_L>0.$$ (92)

For orthotropic laminae, we find that

$$A=2(\mu +{\mu }_a)+\frac{1}{2}{\beta }_a,B=2(\mu +{\mu }_b)+\frac{1}{2}{\beta }_b,C=\mu +\frac{1}{2}{\beta }_{\mathit{ab}}\text{and}D=2(\mu +{\mu }_a+{\mu }_b),$$ (93)

and Q*(⋅) is positive definite if and only if

$$\mu +{\mu }_a+{\mu }_b>0,4(\mu +{\mu }_b)+{\beta }_b>0\text{and}[4(\mu +{\mu }_a)+{\beta }_a][4(\mu +{\mu }_b)+{\beta }_b]>4{(\mu +\frac{1}{2}{\beta }_{ab})}^2.$$ (94)

3.3. Laminated plates

Our framework may be used to construct a two-dimensional model of laminates. The utility of such models in structural mechanics is of course well known, but the model is also applicable to the mitral valve, for example, which is thought to have a layered structure (Wenk et al., 2012; Zhang et al., 2016). We confine attention to laminates consisting of two laminae of the kind considered in the foregoing, and identify the plane Ω, on which ς = 0, with the interface between them. The upper lamina occupies the interval ς ∈ (0, ηh] and the lower lamina the interval ς ∈ [−(1 − η)h, 0), where η ∈ [0,1] is a fixed constant and h is again the thickness of the laminate. The areal energy density on Ω is then given by (compare (34))

$$\begin{gathered} {\int }_0^{\eta h}{U}_{+}({\tilde{\mathbf{E}}}_{+}(\mathbf{u}+\varsigma \mathbf{k}))d\varsigma +{\int }_{-(1-\eta )h}^0{U}_{-}({\tilde{\mathbf{E}}}_{-}(\mathbf{u}+\varsigma \mathbf{k}))d\varsigma \\ =h[\eta U_{+}+(1-\eta )U_{-}]+\frac{1}{2}{h}^2[{\eta }^2{U}_{+}^{\prime }-{(1-\eta )}^2{U}_{-}^{\prime }] \\ +\frac{1}{6}{h}^3[{\eta }^3{U}_{+}^{″}+{(1-\eta )}^3{U}_{-}^{″}]+o(h^3), \end{gathered}$$ (95)

where again Leibniz’ Rule has been combined with a Taylor expansion in each integral (Steigmann, 2012) and the subscripts ± pertain to the upper and lower laminae respectively.

Estimating the coefficients of h, h² and h³ at leading order in |E₊| in the upper lamina and |E₋| in the lower lamina, we arrive at (56), with

$$W=h[\eta Q_{+}({\overline{\mathbf{E}}}_{+})+(1-\eta )Q_{-}({\overline{\mathbf{E}}}_{-})]+\frac{1}{2}{h}^2\{{\eta }^2{\mathcal{C}}_{+}[{\overline{\mathbf{E}}}_{+}]\cdot {\mathbf{R}}_{+}^{\mathit{t}}{\mathbf{F}}_{+}^{\prime }-{(1-\eta )}^2{\mathcal{C}}_{-}[{\overline{\mathbf{E}}}_{-}]\cdot {\mathbf{R}}_{-}^{\mathit{t}}{\mathbf{F}}_{-}^{\prime }\}+\frac{1}{3}{h}^3[{\eta }^3{Q}_{+}({\mathbf{R}}_{+}^{\mathit{t}}{\mathbf{F}}_{+}^{\prime })+{(1-\eta )}^{3}Q\_({\mathbf{R}}_{-}^{\mathit{t}}{\mathbf{F}}_{-}^{\prime })],$$ (96)

where

$${\overline{\mathbf{E}}}_{\pm }=\epsilon -(tr\epsilon )\mathbf{k}\otimes \mathbf{k}+\mathit{Sym}({\gamma }_{\pm }\otimes \mathbf{k})$$ (97)

and

$$Sym({\mathbf{R}}_{\pm }^{\mathit{t}}{\mathbf{F}}_{\pm }^{\prime })=\kappa -(tr\kappa )\mathbf{k}\otimes \mathbf{k}+\mathit{Sym}({\mu }_{\pm }\otimes \mathbf{k}).$$ (98)

Here, we assume the laminae to be perfectly bonded at the plane Ω, each then having a common interfacial deformation $\mathbf{r}(\mathbf{u})={\tilde{\chi }}_{\pm \mid \Omega }$. This implies that ε and κ, which are determined by the derivatives of r(u), are also common to the two laminae.

The leading-order asymptotic analysis culminating in (60), (79) may be applied to the fields γ_±(= ce_±) separately to conclude, as there, that γ_± (and hence e_±) vanish, yielding Ē_± = Ē, given by (64). Thus the restriction of the strain to Ω is the same for both laminae. In view of the structure of (84) and (86), and because the fiber axes a_± and b_± are assumed to be orthogonal to the unit normal, k, to Ω′, it follows that ${\mathcal{C}}_{\pm }[{\stackrel{‒}{\mathbf{E}}}_{\pm }]\cdot {\mathbf{R}}_{\pm }^t{\mathbf{F}}_{\pm }^{\prime }$ do not involve μ_±. Accordingly, the asymptotic procedure culminating in (80) applies here as well, and with the same conclusion; namely, that μ_± vanish. Finally, the areal energy density reduces to

$$W=h[\eta Q_{+}(\overline{\mathbf{E}})+(1-\eta )Q_{-}(\overline{\mathbf{E}})]+\frac{1}{2}{h}^2\{{\eta }^2{\mathcal{C}}_{+}[\overline{\mathbf{E}}]-{(1-\eta )}^2\mathcal{C}\_[\overline{\mathbf{E}}]\}\cdot \mathit{Sym}({\mathbf{R}}^{\mathit{t}}\mathbf{F}\prime )+\frac{1}{3}{h}^3[{\eta }^3{Q}_{+}({\mathbf{R}}^{\mathit{t}}\mathbf{F}\prime )+{(1-n)}^{3}Q\_({\mathbf{R}}^{\mathit{t}}\mathbf{F}\prime )],$$ (99)

where Ē and Sym(R^tF′) are given respectively by (64) and (81). This energy is again fully specified by the derivatives of the deformation r(u) of the interfacial plane. It may be expressed in terms of the reduced energy Q* defined in (83); thus,

$$W=h[\eta Q_{+}^{*}(\epsilon )+(1-\eta )Q_{\_}^{*}(\epsilon )]+\frac{1}{2}{h}^2\{{\eta }^2{\mathcal{C}}_{+}[\overline{\mathbf{E}}]-{(1-\eta )}^2\mathcal{C}\_[\overline{\mathbf{E}}]\}\cdot \mathit{Sym}({\mathit{R}}^{\mathit{t}}\mathbf{F}\prime )+\frac{1}{3}{h}^3[{\eta }^3{Q}_{+}^{*}(\kappa ){(1-\eta )}^3{Q}_{\_}^{*}(\kappa )].$$ (100)

The coefficient of h² in this expression couples the extensional and bending strains ε and κ. It vanishes if the laminate is apportioned to both laminae equally (η = 1/2), and if they have identical properties; the energy reduces, in this case, to (82), as expected.

4. Equilibrium

Consider a one-parameter family r(u; ϵ) of deformations and let $\mathbf{v}(\mathbf{u})=\stackrel{.}{\mathbf{r}}$, where $\stackrel{.}{\mathbf{r}}=\partial \mathbf{r}(\mathbf{u};ϵ)∕\partial {ϵ}_{\mid ϵ=0}$; this is the virtual velocity field. We identify equilibria with those deformations r(u) = r(u;0) that satisfy the virtual-power statement

$$\dot{E}=P$$ (101)

for all kinematically admissible v, where E is the strain energy (56) and

$$\dot{E}={\int }_{\Omega }\dot{W}da$$ (102)

in which the superposed dot is the derivative with respect to ϵ at the equilibrium state ϵ = 0; and P, the form of which is made explicit below, is the virtual power supplied by an external agency. The meaning of kinematic admissibility in this context is discussed below. Here we observe, from (49)₃, (81) and (82) or (100), that the strain energy W depends on the deformation through its first and second spatial derivatives on Ω. Therefore, plate theory is subsumed under second-grade elasticity theory (Toupin, 1962, 1964). Indeed, this circumstance affords a much clearer conceptual framework for plate theory than is typically found in the text and monograph literatures.

To elaborate, we adopt a Cartesian-coordinate parametrization of Ω. Thus, with u = u_αe_α, where u_α are the coordinates, we have r(u) = r_i(u_α)e_i, where r_i(u_α) = e_i ⋅ r(u_βe_β), and

$$\nabla \mathbf{r}={\mathbf{a}}_{\alpha }\otimes {\mathbf{e}}_{\alpha },\text{with}{\mathbf{a}}_{\alpha }=r_{i,\alpha }{\mathbf{e}}_i;$$ (103)

and, from (49)₃,

$$\epsilon ={\epsilon }_{\alpha \beta }{\mathbf{e}}_{\alpha }\otimes {\mathbf{e}}_{\beta },\text{with}{\epsilon }_{\alpha \beta }=\frac{1}{2}(a_{\alpha \beta }-{\delta }_{\alpha \beta }),$$ (104)

where a_αβ = r_i,αr_i,β is the metric induced by the (convected) coordinates on ω; here δ_αβ is the Kronecker delta. The metric furnishes the representation

$$\mathbf{c}=a_{\alpha \beta }{\mathbf{e}}_{\alpha }\otimes {\mathbf{e}}_{\beta }$$ (105)

of the surface Cauchy-Green tensor via (46)₂, and this is positive definite on Ω′ by virtue of the positive definiteness of the three-dimensional Cauchy-Green tensor (46)₁. The matrix (a_αβ) is accordingly positive definite.

From the Gauss and Weingarten equations of differential geometry (Flügge, 1972), we also have

$$r_{i;\alpha \beta }=n_i{b}_{\alpha \beta ,}$$ (106)

where n_i are the components of the unit normal n to T_ω, b_αβ are the components of the curvature tensor thereon, and

$$r_{i;\alpha \beta }=r_{i,\alpha \beta }-{\Gamma }_{\alpha \beta }^{\mu }{r}_{i,\mu }$$ (107)

is the covariant derivative on ω, in which ${\Gamma }_{\alpha \beta }^{\mu }$ are the Christoffel symbols, given by

$${\Gamma }_{\alpha \beta }^{\lambda }=\frac{1}{2}{a}^{\lambda \mu }(a_{\alpha \mu ,\beta }+a_{\beta \mu ,\alpha }-a_{\alpha \beta ,\mu }),$$ (108)

with (a^λμ) = (a_λμ)⁻¹. The curvature tensor is b = b_αβa^α ⊗ a^β, where a^α = a^αβa_β, and the surface normal may be computed from the deformation using the formula ϵ_αβn = a_α, × a_β, where ${ϵ}_{\alpha \beta }=\sqrt{a}{e}_{\alpha \beta }$ in which a = det(a_αβ) and e_αβ is the permutation symbol (e₁₂ = −e₂₁ = 1, e₁₁ = e₂₂ = 0). From (106), (107) and n_ir_i,α = 0 we have

$$b_{\alpha \beta }=n_i{r}_{i,\alpha \beta },$$ (109)

and (72), (103) combine to give

$$\kappa =-b_{\alpha \beta }{\mathbf{e}}_{\alpha }\otimes {\mathbf{e}}_{\beta }.$$ (110)

Recall that the strain energy is a function, F say, of the components r_i,α and r_i,αβ. Then,

$$\dot{W}=N_{i\alpha }{v}_{i,\alpha }+M_{i\alpha \beta }{v}_{i,\alpha \beta },$$ (111)

with

$$N_{i\alpha }=\partial F/\partial r_{i,\alpha }\text{and}{M}_{i\alpha \beta }=M_{i\beta \alpha }=\partial F/\partial r_{i,\alpha \beta }.$$ (112)

We define a 2-vector φ with components

$${\phi }_{\alpha }=T_{i\alpha }{v}_i+M_{i\alpha \beta }{v}_{i,\beta },\text{where}{T}_{i\alpha }=N_{i\alpha }-M_{i\alpha \beta ,\beta }.$$ (113)

Then,

$$\dot{W}={\phi }_{\alpha ,\alpha }-v_i{T}_{i\alpha ,\alpha },$$ (114)

and it follows from the Green-Stokes theorem that

$$\dot{E}={\int }_{\partial \Omega }{\phi }_{\alpha }{\nu }_{\alpha }ds-{\int }_{\Omega }{v}_i{T}_{i\alpha ,\alpha }da,$$ (115)

where ν_α is the rightward unit normal to the edge ∂Ω, traversed counter-clockwise.

Let u_α(s) be the arclength parametrization of ∂Ω. The unit tangent to ∂Ω is the vector with components τ_α = du_α/ds, and ν_α = e_αβτ_β. We use these with δ_αβ = τ_ατ_β + ν_αν_β to decompose the derivatives v_i,β(= v_i,αδ_αβ) on ∂Ω in the form

$$v_{i,}{}_{\beta }=T_{\beta }{v}_i^{\prime }+{\nu }_{\beta }{v}_{i,\nu },$$ (116)

where $v_i^{\prime }(s)={\tau }_{\alpha }{v}_{i,a}={\mathit{dv}}_i(\mathbf{u}(s))∕\mathit{ds}$ and v_i,ν(s) = ν_αv_i,α respectively are the independent tangential and normal derivatives of v_i on ∂Ω. Substituting into (113)₁ and (115), we find that

$${\int }_{\partial \Omega }{\phi }_{\alpha }{\nu }_{\alpha }ds={\int }_{\partial \Omega }\{(T_{i\alpha }{\nu }_{\alpha }-(M_{i\alpha \beta }{\nu }_{\alpha }{\tau }_{\beta })\text{'})v_i+M_{i\alpha \beta }{\nu }_{\alpha }{\nu }_{\beta }{v}_{i,\nu }\}ds-{\sum _n[M_{i\alpha \beta }{\nu }_{\alpha }{\tau }_{\beta }]}_{(n)}{v}_{i(n)},$$ (117)

where the square bracket refers to the forward jump as a corner of the boundary is traversed. Thus, [⋅] = (⋅)₊ − (⋅)₋, where the subscripts _± identify the limits as a corner located at arclength station s is approached through larger and smaller values of arclength, respectively. The sum accounts for the contributions from all corners. Here we assume the boundary to be piecewise smooth in the sense that its tangent τ is piecewise continuous.

It follows from (101) that admissible virtual powers have the form

$$P={\int }_{\Omega }{g}_i{v}_i\mathit{da}+{\int }_{\partial {\Omega }_t}{t}_i{v}_i\mathit{ds}+{\int }_{\partial {\Omega }_m}{m}_i{v}_{i,\nu }\mathit{ds}+\sum _{*}{f}_{i(n)}{v}_{i(n)},$$ (118)

where g_i, t_i, m_i and f_i(n) respectively are the distributed force, the edge traction, the double force, and the corner force. The notion of double force figures prominently in 2nd-grade elasticity theory (Toupin, 1962, 1964). Here, ∂Ω_t and ∂Ω_m respectively are parts of ∂Ω where r_i and r_i,ν are not assigned, and the starred sum includes only the corners where position is not assigned. We suppose that r_i and r_i,ν are assigned, and hence that v_i and v_i,ν vanish, on ∂Ω \ ∂Ω_t and ∂Ω \ ∂Ω_m, respectively, and that position is assigned at the corners not included in the starred sum, so that v_i(n) vanishes there. For our present purposes, these restrictions on the virtual velocity constitute the definition of kinematic admissibility. The fundamental lemma then yields the Euler-Lagrange equations

$$T_{i\alpha ,\alpha }+g_i=0\text{in}\Omega ,$$ (119)

and

$$t_i=T_{i\alpha }{\nu }_{\alpha }-{(M_{i\alpha \beta }{\nu }_{\alpha }{\tau }_{\beta })}^{\prime },m_i=M_{i\alpha \beta }{\nu }_{\alpha }{\nu }_{\beta }\text{and}{f}_{i(n)}=-{[M_{i\alpha \beta }{\nu }_{\alpha }{\tau }_{\beta }]}_{(n)}$$ (120)

on ∂Ω_t, ∂Ω_m and at the n^th corner, respectively.

In conventional plate theory, it is customary to specify a couple c on part of the boundary. This is related to the double force by c = r_,ν × m (Steigmann, 2018), where r_,ν = r_i,νe_i and m = m_ie_i. The couple and double force are not equivalent, however, because the former is sensitive only to the part of the latter that is perpendicular to r_,ν. It follows that, in general, specification of the couple is not consistent with the variational statement (101).

We use (82) to deduce the structure of the response functions N_iα and M_iαβ for a single lamina. Thus,

$$\dot{W}=h{Q}_{\epsilon }^{*}\cdot \dot{\epsilon }\text{+}\frac{1}{12}{h}^3{Q}_{\kappa }^{*}\cdot \dot{\kappa }.$$ (121)

Equivalently,

$$\dot{W}={\sum }_{\alpha \beta }{\dot{\epsilon }}_{\alpha \beta }+M_{\alpha \beta }{\dot{b}}_{\alpha \beta },$$ (122)

where

$${\sum }_{\alpha \beta }=h{Q}_{\epsilon }^{*}\cdot {\mathbf{e}}_{\alpha }\otimes {\mathbf{e}}_{\beta }\text{and}{M}_{\alpha \beta }=-\frac{1}{12}{h}^3{Q}_{\kappa }^{*}\cdot {\mathbf{e}}_{\alpha }\otimes {\mathbf{e}}_{\beta }.$$ (123)

and the derivatives $Q_{\epsilon }^{\ast }$, $Q_{\kappa }^{\ast }$ are given by (85) or (87). Here we use (104)₂ to compute the variation

$${\dot{\epsilon }}_{\alpha \beta }=\frac{1}{2}(r_{i,\alpha }{v}_{i,\beta }+v_{i,\alpha }{r}_{i,\beta }).$$ (124)

To compute the variation ${\stackrel{.}{b}}_{\alpha \beta }$, we first take the variation of (106), reaching

$$v_{i,\alpha \beta }={\Gamma }_{\alpha \beta }^{\mu }{v}_{i,\mu }+{\dot{\Gamma }}_{\alpha \beta }^{\mu }{r}_{i,\mu }+b_{\alpha \beta }{\dot{n}}_i+{\dot{b}}_{\alpha \beta }{n}_i,$$ (125)

and then contract with n_i to obtain

$${\dot{b}}_{\alpha \beta }=n_i(v_{i,\alpha \beta }-{\Gamma }_{\alpha \beta }^{\mu }{v}_{i,\mu }).$$ (126)

Because the energy (82) is valid to leading order in the strain, we adopt the consistent-order approximation (a_αβ)⁻¹ = δ_αβ + O(|ε|). From (108) we then have

$${\Gamma }_{\alpha \beta }^{\mu }={\overline{\Gamma }}_{\mu \alpha \beta }+O(\left|\epsilon \right|),$$ (127)

where

$${\overline{\Gamma }}_{\mu \alpha \beta }={\epsilon }_{\alpha \mu \text{,}\beta }+{\epsilon }_{\beta \mu \text{,}\alpha }-{\epsilon }_{\alpha \beta ,\mu }.$$ (128)

To the same order of approximation, the unit normal n = n_ke_k to the deformed surface is given by

$$e_{\alpha \beta }{n}_k=e_{ijk}{r}_{i,\alpha }{r}_{j,\beta },$$ (129)

where e_αβ and e_ijk respectively are the two- and three-dimensional permutation symbols. Combining these results with (111) and (122), we arrive at

$$M_{i\alpha \beta }=n_i{M}_{\alpha \beta }\text{and}{N}_{i\alpha }=r_{i,\beta }{\sum }_{\beta \alpha }-M_{i\beta \mu }{\overline{\Gamma }}_{\alpha \beta \mu },$$ (130)

in which the non-standard term $M_{i\beta \mu }{\stackrel{‒}{\Gamma }}_{\alpha \beta \mu }$ arises from the use of partial derivatives rather than the customary covariant derivatives (see (107)) on the deformed surface (Ciarlet, 2005). We note that it is a purely nonlinear term and hence absent in linear plate theory.

The reduction of laminate theory proceeds along entirely similar lines and is therefore not made explicit. The only adjustments to the foregoing are that the definitions of Σ_αβ and M_αβ are to be based on (100) rather than (82).

Eqs. (103) may be substituted into (113)₂, (119) and (120) to derive expressions for the distributed force, the traction, the double force and the corner forces. In particular, the double force assumes the form

$$m_i=M{n}_i,\text{with}M=M_{\alpha \beta }{\nu }_{\alpha }{\nu }_{\beta }.$$ (131)

This implies that the edge couple, c = r_,ν × m, is tangential to the deformed surface, and that it vanishes if and only if the double force vanishes.

5. Gradient flow and dynamic relaxation

We apply the foregoing to the analysis of problems in which there is no distributed load over the surface of the plate (g_i = 0) and parts ∂Ω_c and ∂Ω_f of the boundary are either clamped (r_i and r_i,ν assigned) or free (t_i = 0 and m_i = 0), respectively. In all cases the corners of the plate are the endpoints of clamped segments, so that r_i are assigned there. The normal-tangential decomposition (116) implies that the full gradient r_i,α, is effectively assigned on a clamped boundary. This follows from the fact that the specification of r_i(s) yields the tangential derivatives $r_i^{\prime }(s)$. The specification of the normal derivatives r_i,ν(s) may be interpreted in terms of an assigned position field in a narrow seam of width w, say, welded to the boundary. If ν is the normal coordinate on the edge, then r_i(s, ν) = r_i(s) + νr_i,ν(s) + o(ν) for 0 ≤ ν ≤ w. Dividing by ν and passing to the limit, we conclude that specification of the normal derivative furnishes the leading-order model for the specification of position r_i(s,ν) in the adjoining seam.

With the foregoing conditions in effect, the virtual-power statement (101) reduces simply to $\stackrel{.}{E}=0$. Moreover, the problem is trivially conservative, with E playing the role of the potential energy. According to the energy criterion for conservative problems, stable equilibria r(u) may be identified with minimizers of this energy.

An effective method for treating problems of this kind is the so-called gradient-flow algorithm, in which the stationarity condition $\stackrel{.}{E}=0$ is replaced by the following artificial first-order dynamical problem, with v(u, ϵ) = ∂r(u; ϵ)/∂ϵ:

$$c{v}_i=T_{i\alpha ,\alpha }\text{in}\Omega ,$$ (132)

and

$$-c{v}_i=T_{i\alpha }{\nu }_{\alpha }-{(M_{i\alpha \beta }{\nu }_{\alpha }{\tau }_{\beta })}^{\prime }\text{on}\partial {\Omega }_f,\text{and}-c{v}_{i,\nu }=M_{i\alpha \beta }{\nu }_{\alpha }{\nu }_{\beta }\text{on}\partial {\Omega }_f,$$ (133)

with c a positive constant and with v_i = 0 and v_i,α = 0 on ∂Ω_c = ∂Ω \ ∂Ω_f. With reference to (115) and (117), this yields

$$\frac{\partial }{\partial ϵ}E[\mathbf{r}(\mathbf{u};ϵ)]=-c^{-1}({\int }_{\Omega }{\left|\mathbf{g}\right|}^{2}da+{\int }_{\partial {\Omega }_f}{\left|\mathbf{t}\right|}^{2}ds+{\int }_{\partial {\Omega }_f}{\left|\mathbf{m}\right|}^{2}ds),$$ (134)

where the components of g, t and m are given respectively by (119) and (120)_1,2.

Thus, ∂E[r(u; ϵ)]/∂ϵ ≤ 0, with equality (and hence E stationary) if and only if the equilibrium conditions are satisfied. Forward integration of (132) and (133) in ‘ϵ - time’ with arbitrary initial conditions thus generates an energy-minimizing sequence r_n(u) = r(u; ϵ_n), where ϵ_n are discrete values of ϵ; that is, E[r_n+1(u)] < E[r_n(u)]. This sequence converges to a minimizer provided that the energy is lower semi-continuous. This in turn is guaranteed by the condition of quasiconvexity of the areal strain-energy function W with respect to the 2nd gradient r_i,αβ (Ball et al., 1981). The relevant term is Q*(κ), in which κ depends on r_i,αβ, linearly (cf. (109) and (110)). Thus W is a non-negative, quadratic and hence convex function of r_i,αβ. It is not strictly convex, however, because while Q* is a positive-definite function of κ, the latter is insensitive to the part of r_i,αβ orthogonal to n_i. Nevertheless, convexity is sufficient for quasiconvexity and hence for lower semicontinuity, which in turn ensures the convergence of the gradient-flow algorithm. It is in this sense that plate theory furnishes a regularization of pure membrane theory.

The foregoing statements are not sufficiently precise to be entirely equivalent to what has been proved. We refer to Ball et al. (1981) for detailed statements and proofs of the relevant theorems.

In practice, the rate of convergence of the gradient-flow algorithm is rather slow. To address this, we instead use the dynamic relaxation method (Shugar, 1990; Topping and Khan, 1994; Rezaiee-pajand et al., 2011), which entails the addition of an artificial inertia to (132). This is motivated by the fact that, in the linear theory of damped vibrations, inertia hastens the transient response of the system and its approach to equilibrium. Thus we replace (132) by

$$\rho {\dot{v}}_i+c{v}_i=T_{i\alpha ,\alpha }\text{in}\Omega \text{,}$$ (135)

where $\stackrel{.}{\mathbf{v}}(\mathbf{u},ϵ)={\partial }^2\mathbf{r}(\mathbf{u};ϵ)∕\partial {ϵ}^2$ and ρ is a positive constant. If the latter is interpreted as a referential mass density, and if the artificial viscosity c vanishes, then (135) furnishes an appropriate dynamical extension of the equilibrium equation. A physical viscosity may be included in the constitutive equations for T_iα, but this is of no consequence here as (135) is intended solely to expedite the computation of equilibria.

Using (133) and (135) in (115) and (117), we obtain

$$\frac{\partial }{\partial ϵ}\{E[\mathbf{r}(\mathbf{u};ϵ)]+K[\mathbf{v}(\mathbf{u};ϵ)]\}=-c{\int }_{\Omega }{\left|\mathbf{v}\right|}^2\mathit{da}-c^{-1}({\int }_{\partial {\Omega }_f}{\left|\mathbf{t}\right|}^{2}ds+{\int }_{\partial {\Omega }_f}{\left|\mathbf{m}\right|}^{2}ds),$$ (136)

where

$$K[\mathbf{v}(\mathbf{u};ϵ)]=\frac{1}{2}{\underset{\Omega }{\int }\rho \left|\mathbf{v}\right|}^{2}da$$ (137)

is the kinetic energy of the surrogate dynamical system and t, m respectively are again defined by (120)_1,2. Thus the total mechanical energy E + K decays on solution trajectories of the dynamical system. Numerical integration of (133) and (135) again yields a minimizing sequence r_n(u) = r(u; ϵ_n), whose limit minimizes E.

To implement this method, we discretize (135) and the edge conditions (133) in space using an adaptation of a finite-difference mesh based on a discrete version of the Green-Stokes theorem with hour-glass control (Silling, 1988; Last and Harkness, 1989); and in ϵ - time using conditionally stable explicit central differences with zero initial velocity and initial positions with randomly distributed out-of-plane displacements. Finite difference methods based on a contour integral technique, such as what we use here, utilize approximations similar to those of four-node quadrilateral finite elements with reduced integration (Belytschko et al., 1982). Stable simulations are achieved by adjusting the artificial mass density and viscosity and repeating the computations, as needed, using successively smaller time steps, until the (non-dimensionalized) norms of the right-hand sides of (133) and (135) fall below a specified tolerance. A related procedure involving kinetic damping (Rezaiee-pajand et al., 2011) in lieu of viscous damping was used in Taylor et al. (2014) to simulate the nonlinear response of isotropic plates. The present problem is identical with respect to the implementation of the foregoing algorithm and, accordingly, we refer the interested reader to Section 3 of that work for a detailed exposition of the numerical solution scheme.

6. Numerical examples and discussion

We close with descriptions of a number of examples that highlight the effects of anisotropy on predicted deformations and wrinkle patterns. All simulations pertaining to transverse isotropy were conducted using the moduli β = 5 kPa, μ_L = 1 kPa and μ_T = 0.5 kPa based on the study of fibrin gel by Namani et al. (2012). Those pertaining to orthotropy are based on the moduli (in units of kPa) μ = 1, μ_a = 2, μ_b = 1, β_a = 10, β_b = 5 and β_ab = 2.5, chosen to be on the order of the fibrin gel but with one fiber orientation stiffer than the other. In the case of laminates, we take the laminae to be transversely isotropic with β_± = 5, μ_L± = 1, and μ_T± = 0.5 (in kPa), i.e. two layers of the fibrin gel material. These moduli satisfy inequalities (92) or (94), as appropriate. Figures 1–3 depict the deformed configurations of initially rectangular thin sheets having dimensions 10cm × 25cm with a total thickness of 0.01cm. Converged solutions were obtained using a regular 200 × 500 mesh. The vertical sides are free and the top and bottom edges are clamped at a relative separation of 1.2 times the initial value. The arrows in the figures indicate the orientations of the fibers on the reference plane and the vertical color scales show the distribution of out-of-plane displacement associated with the wrinkling patterns. The crests of the wrinkles are oriented vertically in the case of isotropy (Taylor et al., 2014; Wang et al., 2019), whereas our simulations reveal a deviation from verticality due either to asymmetry in fiber orientation, to differing moduli for the two fiber families in the case of orthotropy, or to lamination with fibers oriented differently in laminae of various thicknesses. We see that the wrinkles orient themselves towards the direction of strongest fiber reinforcement. In In Figure 3, and subsequent figures involving transverse isotropy, a light arrow indicates the fiber direction in the thinner lamina and the heavier arrow the fiber direction in the thicker lamina.

Figure 1:

Axial stretch of a fibrin gel sheet reinforced with a single family of fibers oriented at (a) 0°, (b) 45°, and (c) 90° to the horizontal axis. Non-dimensionalized by sheet height (L = 25cm)

Figure 3:

Axial stretch of a two-layer fibrin gel laminate with upper/lower surface fiber orientations of (a) 0°/90° (η = 0.5), (b) 45°/135° (η = 0.25), (c) 45°/135° (η = 0.5), and (d) 45°/135° (η = 0.75). Non-dimensionalized by sheet height (L = 25cm)

Figures 4–6 show the deformed configurations of a very thin initially square sheet of side 50cm and thickness 0.0001cm in which the corners are displaced diagonally by different asymmetric amounts, indicated in the figure captions. This example is inspired by experiments conducted by Wong and Pellegrino (2006), albeit on isotropic sheets. All sides are free except for short vertical and horizontal segments of length 1.5cm, adjoining the corners, where the sheet is clamped. The upper panel of Figure 4 (a–c) illustrates the significant effect of fiber orientation on the predicted deformation pattern at fixed diagonal displacement; the lower panel (d-f) illustrates the effect of varying diagonal displacements with a given fiber orientation. With a fiber orientation of 45°, the sheets display wrinkle patterns akin to those found by Wong and Pellegrino (2006) in an isotropic polyimide film at various ratios of asymmetric corner stretch. However, the patterns corresponding to orientations of 0° and 90° are quite different. The effects of fiber orientation in orthotropic and laminated plates, with laminae of equal thickness, are illustrated in Figures 5 and 6, respectively. Converged simulations were achieved using a regular 300 × 300 mesh.

Figure 4:

Square transversely isotropic fibrin gel sheet with asymmetric tensile corner displacements. Upper right and bottom left corner stretched diagonally by a factor of 1.01. (a-c) Opposing corners stretched by a factor of 1.1 with fiber orientations of 0°, 45°, and 90°, respectively. (d-f) Fiber orientation fixed at 45° but with opposing corners stretched by factors of 1.05, 1.15, and 1.2, respectively. Non-dimensionalized by sheet height (L = 50cm)

Figure 6:

Square two-layer fibrin gel laminate with asymmetric tensile corner displacements. Upper right and bottom left corner stretched diagonally by a factor of 1.01; opposite corners stretched by a factor of 1.1. Fiber families oriented at 0° (upper layer) and 90° (lower layer) with (a) η = 0.5 and (b) η = 0.25. Non-dimensionalized by sheet height (L = 50cm)

Figure 5:

Square orthotropic gel sheet with asymmetric tensile corner displacements. Upper right and bottom left corner stretched diagonally by a factor of 1.01; opposite corners stretched by a factor of 1.1. (a) Fiber families oriented at 0° and 90°. (b) Fiber families oriented at 45° and 135°. Non-dimensionalized by sheet height (L = 50cm)

Figures 7–9 pertain to an initially rectangular 38cm × 12.8cm sheet of thickness 0.0025cm. The vertical edges are free and the horizontal edges clamped, with the top edge displaced to the right uniformly by 0.256cm, and converged solutions were achieved using a regular 600 × 200 mesh. This example also has an experimental counterpart, described in Wong and Pellegrino (2006), and was simulated in the case of isotropy in Taylor et al. (2014). Figure 7 illustrates the strong effect of fiber orientation in the case of transverse isotropy, in both wrinkle magnitude and orientation, whereas Figures 8 and 9, respectively, indicate that the effect is less pronounced in the case of orthotropy or lamination when the laminae have equal thickness.

Figure 7:

Fibrin gel sheet in shear for fiber families of (a) 0°, (b) 45°, and (c) 90°. Non-dimensionalized by sheet width (L = 38cm)

Figure 9:

Shearing of a two-layer fibrin gel laminate with fiber families of (a) 0° (upper layer) & 90° (lower layer) and (b) 45° (upper layer) & 135° (lower layer). Non-dimensionalized by sheet width (L = 38cm)

Figure 8:

Shearing of orthotropic gel sheet with fibers families of (a) 0° /90° and (b) 45°/135°. Non-dimensionalized by sheet width (L = 38cm)

In Figures 10–12, we display an annular sheet subjected to azimuthal shear. The initial inner and outer diameters are 4cm and 10cm and the thickness is 0.005cm. The circular boundaries are clamped with the outer boundary fixed at its initial configuration and the inner rotated counter-clockwise by 10°. Converged solutions were obtained using a polar mesh consisting of 60 radial nodes and 360 circumferential nodes biased logarithmically towards the central hub. Figure 10, pertaining to transverse isotropy, demonstrates the strong effect of fiber orientation. In the top panel the fibers are oriented horizontally or vertically. The deformation pattern exhibits reflection symmetry with respect to either axis, with one pattern corresponding, as expected, to a 90° rotation relative to the other. The bottom panel indicates that significant differences in the patterns of wrinkling occur in annuli with fibers oriented radially and azimuthally. These, however, exhibit rotational symmetry in the sense that the predicted pattern, though clearly varying with radius, does not vary with azimuth. Figures 11a,b, pertaining to orthotropy, indicate that the deformation pattern reflects the four-fold symmetry of the underlying fiber orientation. Rotation of this orientation is seen to produce a corresponding rotation of the deformation pattern. The situation is naturally different in the case of fibers oriented radially and circumferentially (Fig. 11c), with the deformation pattern again exhibiting rotational symmetry in this case. Similar conclusions apply to laminated annuli consisting of equal-thickness laminae (Fig. 12).

Figure 10:

Annular fibrin gel sheet with inner hub twisted 10° counter-clockwise for fiber families oriented (a) at 0°, (b) at 90°, (c) radially, and (d) circumferentially. Non-dimensionalized by outer diameter (L = 10cm)

Figure 12:

Annular two-layer fibrin gel laminate with inner hub twisted 10° counter-clockwise for fiber families oriented (a) at 0°/90° and (b) radially/circumferentially. Non-dimensionalized by outer diameter (L = 10cm)

Figure 11:

Annular orthotropic gel sheet with inner hub twisted 10° counter-clockwise for fiber families oriented (a) at 0°/90°, (b) at 45°/135°, and (c) radially/circumferentially. Non-dimensionalized by outer diameter (L = 10cm)

Finally, Figures 13–15 depict the response of an initially rectangular 12.5cm × 5cm sheet of thickness 0.001cm with a central circular hole of radius 1.5cm. The vertical edges and the edge of the hole are free, while the horizontal edges are clamped and displaced to 1.10 times their initial separation distance. Converged solutions were obtained using a polar mesh with 100 radial nodes and 400 circumferential nodes. Figure 13 illustrates the symmetric deformation patterns produced when the fibers are oriented parallel to the initial edges of the sheet, whereas asymmetry is observed when the fibers are orientated at 45° to these axes. Wrinkling is widespread, and we observe small triangular floppy zones adjoining the hole boundary where the sheet is lightly stressed. Similar patterns may be observed in desk-top experiments on hand-held sheets of rubber. In Figure 14a we, observe a symmetric deformation pattern in the case of orthotropy with the fibers aligned initially with the edges of the sheet. A slight asymmetry is observed for fibers oriented at 45° and 135°, (Fig. 14b) due to differences in the moduli associated with the two fiber families. Figures 15a,c exhibit symmetric deformation patterns in laminates composed of laminae of equal thickness having the same material properties, with fibers either aligned with the edges of the rectangle or symmetrically disposed with respect to them. The effect of different laminae thicknesses is to produce a slight asymmetry in the deformation pattern, depicted in Figure 15b.

Figure 13:

Axial stretch of a fibrin gel sheet with traction-free central hole reinforced with a single family of fibers oriented at (a) 0°, (b) 45°, and (c) 90° to the horizontal axis. Non-dimensionalized by sheet height (L = 12.5cm)

Figure 15:

Axial stretch of a two-layer fibrin gel laminate with traction-free central hole with upper/lower surface fiber orientations of (a) 0°/90° (η = 0.5), (b) 45°/135° (η = 0.25), and (c) 45°/135° (η = 0.5). Non-dimensionalized by sheet height (L = 12.5cm)

Figure 14:

Axial stretch of an orthotropic gel sheet with traction-free central hole reinforced with two families of fibers oriented at (a) 0°/90° and (b) 45°/135°. Non-dimensionalized by sheet height (L = 12.5cm)

Figure 2:

Axial stretch of an orthotropic gel sheet reinforced with two families of fibers oriented at (a) 0°/90° and (b) 45°/135°. Non-dimensionalized by sheet height (L = 25cm)