A Two-Dimensional Gradient-Elasticity Theory for Woven Fabrics

Abstract

A gradient elasticity theory is proposed for the mechanics of woven fabrics. This is motivated by a series of recent experiments in which strongly localized deformation features are observed in the so-called bias test. Such features are reminiscent of solutions to problems posed in the setting of gradient elasticity. In turn, gradient effects in woven fabrics may be motivated by the presence of a local length scale in the pattern of the weave. The presumed influence of this scale on macroscopic constitutive response leads naturally to a special gradient theory.

Appendix: Euler Equations and Boundary Conditions

The derivation of the Euler equations and boundary conditions in second gradient elasticity is well known [24–29]. It is reproduced here for the sake of completeness, and for the purpose of constitutively relating admissible loads to the deformation. To this end, we adopt the framework of the virtual-work statement

$$ \dot{E}=P, $$

(58)

where the superposed dot refers to the variational—or Gateâux—derivative,

$$ E=\int_{\varOmega}W(\mathbf{F},\mathbf{G})da $$

(59)

is the strain energy and P is the virtual power of the edge loads, the form of which is made explicit below. We note that conservative loads are those for which there exists a potential L such that $P=\dot{L}$, and in this case the problem is to minimize the potential energy E−L.

We have

$$ \dot{E}=\int_{\varOmega}\dot{W}(\mathbf{F},\mathbf{G})da, $$

(60)

where

$$ \dot{W}=\partial W/\partial F_{iA}u_{i,A}+\partial W/\partial G_{iAB}u_{i,AB} $$

(61)

and $\mathbf{u}=\dot{\boldsymbol{\chi}}$ is the variation of the position field. Writing

$$\begin{aligned} \partial W/\partial F_{iA}u_{i,A} =&(\partial W/F_{iA}u_{i})_{,A}-u_{i}(\partial W/ \partial F_{iA})_{,A}\quad\text{and} \\ \partial W/\partial G_{iAB}u_{i,AB} =&(\partial W/\partial G_{iAB}u_{i,A})_{,B}-u_{i,A}(\partial W/ \partial G_{iAB})_{,B}, \end{aligned}$$

(62)

we then have

$$ \dot{E}=\int_{\varOmega}P_{iA}u_{i,A}da+\int _{\partial\varOmega}(\partial W/\partial G_{iAB})u_{i,A}N_{B}ds, $$

(63)

where N is the rightward unit normal to the boundary curve ∂Ω in the sense of the Green–Stokes theorem, and

$$ P_{iA}=\partial W/\partial F_{iA}-(\partial W/\partial G_{iAB})_{,B}. $$

(64)

Decomposing this as in (62)₁ furnishes

$$ \dot{E}=\int_{\partial\varOmega}\bigl[u_{i}P_{iA}N_{A}+( \partial W/\partial G_{iAB})u_{i,A}N_{B}\bigr]ds-\int _{\varOmega}u_{i}P_{iA,A}da, $$

(65)

and hence the Euler equation

$$ P_{iA,A}=0, $$

(66)

which holds in Ω. With this satisfied we then have

$$ \dot{E}=\int_{\partial\varOmega}\bigl[u_{i}P_{iA}N_{A}+ \partial W/\partial G_{iAB}\bigl(T_{A}N_{B}u_{i}^{\prime}+N_{A}N_{B}u_{i,N} \bigr)\bigr]ds, $$

(67)

where use has been made of the normal-tangential decomposition

$$ \nabla\mathbf{u}=\mathbf{u}^{\prime}\otimes\mathbf{T}+\mathbf{u}_{,N} \otimes \mathbf{N}, $$

(68)

in which T=X′(s)=k×N is the unit tangent to ∂Ω; and u′=d u(X(s))/ds and u _,N respectively are the tangential and normal derivatives of u on ∂Ω. We now substitute

$$ (\partial W/\partial G_{iAB})T_{A}N_{B}u_{i}^{\prime}=( \partial W/\partial G_{iAB}T_{A}N_{B}u_{i})^{\prime}-( \partial W/\partial G_{iAB}T_{A}N_{B})^{\prime}u_{i} $$

(69)

and thereby arrive at

$$\begin{aligned} \dot{E} =&\int_{\partial\varOmega}u_{i}\bigl[P_{iA}N_{A}-( \partial W/\partial G_{iAB}T_{A}N_{B})^{\prime} \bigr]ds+\int_{\partial\varOmega}u_{i,N}(\partial W/\partial G_{iAB}N_{A}N_{B})ds \\ &{}-\sum u_{i}[\partial W/\partial G_{iAB}T_{A}N_{B}], \end{aligned}$$

(70)

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

We conclude from (58) that admissible powers are of the form

$$ P=\int_{\partial\varOmega_{t}}t_{i}u_{i}ds+\int _{\partial\varOmega _{m}}m_{i}u_{i,N}ds+\sum _{\ast}f_{i}u_{i}, $$

(71)

where

$$ \begin{aligned} t_{i}&=P_{iA}N_{A}-(\partial W/\partial G_{iAB}T_{A}N_{B})^{\prime },\qquad m_{i}=\partial W/\partial G_{iAB}N_{A}N_{B} \quad\text{and}\\ f_{i}&=-[\partial W/\partial G_{iAB}T_{A}N_{B}] \end{aligned} $$

(72)

are the edge traction, edge double force and the corner force, respectively. Here, ∂Ω _t and ∂Ω _m respectively are parts of ∂Ω where χ _i and χ _i,N are not assigned, and the starred sum refers to corners where position is not assigned. We suppose that χ _i and χ _i,N are assigned on ∂Ω∖∂Ω _t and ∂Ω∖∂Ω _m , respectively, and that position is assigned at the corners not included in the starred sum.

A simple example of conservative loading is furnished by the potential

$$ L=\int_{\partial\varOmega_{t}}t_{i}\chi_{i}ds+\int _{\partial\varOmega _{m}}m_{i}\chi_{i,N}ds+\sum _{\ast}f_{i}\chi_{i}, $$

(73)

in which t _i , m _i and f _i are all independent of the deformation.

To interpret the double force in mechanical terms, we consider the special case in which no kinematical data are assigned anywhere on ∂Ω, so that rigid-body deformations are kinematically admissible. The variational derivative of such a deformation is expressible in the form u=ω×χ+a, where a and ω are spatially uniform vectors. Because the strain-energy function is invariant under such deformations, we have $\dot{E}=0$ and (58) reduces to P=0; i.e.,

$$ \mathbf{a}\cdot\biggl(\int_{\partial\varOmega}\mathbf{t}ds+\sum \mathbf{f}_{i}\biggr)+\boldsymbol{\omega}\cdot\biggl[\int _{\partial\varOmega}(\boldsymbol{\chi}\times \mathbf{t}+\boldsymbol{\chi}_{,N} \times\mathbf{m})ds+\sum\boldsymbol{\chi}_{i}\times \mathbf{f}_{i}\biggr]=0, $$

(74)

where χ _i =χ(X _i ). Because the rigid motion may be arbitrary we then have

$$ \int_{\partial\varOmega}\mathbf{t}ds+\sum\mathbf{f}_{i}= \mathbf{0}\quad\text{and}\quad\int_{\partial\varOmega}(\boldsymbol{\chi} \times \mathbf{t}+\boldsymbol{\chi}_{,N}\times\mathbf{m})ds+\sum \boldsymbol{\chi}_{i}\times\mathbf{f}_{i}=\mathbf{0}. $$

(75)

It follows that χ _,N ×m is a distribution of edge couples. It is interesting that these couples are configuration dependent in the example of conservative loading described by (73).