Computational homogenization of rope-like technical textiles

Abstract

This contribution investigates a computational homogenization method for one-dimensional technical textiles i.e. ropes or cables. On the macroscopic level rope-like textiles are characterized by a large length-to-thickness ratio, such that a discretization with structural elements, i.e. beam elements, is numerically efficient. The material behavior is, however, strongly influenced by the heterogeneous micro structure. Here, the fibers at the micro structure are modeled explicitly via a representative volume element whereby the contact interactions between fibers are captured. To transfer the microscopic response to the macro level a beam specific homogenization scheme is introduced. Theoretical aspects are discussed, e.g. the advocated beam specific power averaging theorem and the corresponding scale transition, and numerical examples for periodic rope-like structures are given.

Appendix: weak form

For the scale transitions the local Euler equations are needed. They result from the first equation of motion

$$\begin{aligned} \int \limits _{\mathcal {B}_0} \rho _0 \ddot{{\varvec{\varphi }}} dV= \int \limits _{\mathcal {B}_0} {\varvec{b}}_0 dV + \int \limits _{\mathcal {\partial B}_0}{\varvec{t}}_0 dA, \end{aligned}$$

(57)

where $\rho _0$ is the material density and ${\varvec{b}}_0$ is the vector of volume forces in the material configuration. The traction vector is defined by ${\varvec{t}}_0= {\varvec{P}}\cdot {\varvec{n}}_0$ with the outward unit normal ${\varvec{n}}_0=n_i {\varvec{G}}^i$ with the covariant components $n_i$ of ${\varvec{n}}$. It appears for an area element $dA_i$ at the coordinate surface $\theta ^i=$const. that

$$\begin{aligned} {\varvec{n}}_0 dA=\frac{{\varvec{G}}^i}{\sqrt{G^{(ii)}}} dA_i, \end{aligned}$$

(58)

and therewith

$$\begin{aligned} n_i\sqrt{G^{(ii)}} dA = dA_i, \end{aligned}$$

(59)

where $\sqrt{G^{(ii)}}$ is the norm of the vector ${\varvec{G}}^i$ [23]. An incremental force $d {\varvec{f}}$ follows as

$$\begin{aligned} d{\varvec{f}} = {\varvec{t}}_0 dA = \frac{{\varvec{t}}_{0}}{n_i \sqrt{G^{(ii)}}} dA_i = {\varvec{t}}_0 \frac{\sqrt{G}}{n_i}d\theta ^j d\theta ^k={\varvec{t}}^i d\theta ^j d\theta ^k,\nonumber \\ \end{aligned}$$

(60)

with $dA_i=\sqrt{G G^{(ii)}}d\theta ^j d\theta ^k$. It can be shown, that the tractions ${\varvec{t}}^i$ correspond to those in (24) by

$$\begin{aligned} d{\varvec{f}} \!=\! {\varvec{t}}_0 dA \!=\! {\varvec{P}} \cdot {\varvec{n}}_0 dA = \sqrt{G} {\varvec{P}} \cdot {\varvec{G}}^i d \theta ^j d\theta ^k={\varvec{t}}^i d\theta ^j d\theta ^k.\nonumber \\ \end{aligned}$$

(61)

Inserting (60) in the last term of (57) and applying the Gaussian theorem and a transformation to the parameter space of the curvilinear coordinates renders eventually

$$\begin{aligned} \int \limits _{\mathcal {\partial B}_0}{\varvec{t}}_0 dA&= \int \limits \limits _{\mathcal {\partial B}_0}{\varvec{t}}^i \frac{ n_{i}}{\sqrt{G}} dA = \int \limits _{\mathcal {B}_0} {\varvec{t}}^i_{,i} \frac{1}{\sqrt{G}} dV\nonumber \\&= \int \!\!\int \!\!\int \limits _{\theta ^k \theta ^j \theta ^i} {\varvec{t}}^i_{,i}d\theta ^i d\theta ^j d\theta ^k. \end{aligned}$$

(62)

Therewith (57) in the parameter space results as

$$\begin{aligned} \int \!\!\int \!\!\int \limits _{\theta ^1 \theta ^2\theta ^3 } {\varvec{t}}^i_{,i}+ \sqrt{G} [\rho _0 \ddot{{\varvec{\varphi }}}-{\varvec{b}}_0 ]d\theta ^3 d\theta ^2 d\theta ^1={\varvec{0}}. \end{aligned}$$

(63)

The first equation of motion in local form in the material configuration is therefore

$$\begin{aligned} {\varvec{t}}^i_{,i}+ \sqrt{G} [\rho _0 \ddot{{\varvec{\varphi }}}-{\varvec{b}}_0 ]={\varvec{0}}. \end{aligned}$$

(64)

If the acceleration is assumed to be zero and in the absence of volume forces (64) simplifies to

$$\begin{aligned} {\varvec{t}}^i_{,i}={\varvec{0}}. \end{aligned}$$

(65)

With the variation of the position $\delta {\varvec{\varphi }}$ the weak form results as

$$\begin{aligned} \int \!\!\int \!\!\int \limits _{\theta ^1 \theta ^2\theta ^3 } \delta {\varvec{\varphi }}\cdot {\varvec{t}}^i_{,i} d\theta ^1 d\theta ^2 d\theta ^3=0, \end{aligned}$$

(66)

and with the beam kinematics (2) it becomes

$$\begin{aligned}&\int \limits _{\theta ^1 } \delta {{\varvec{\varphi }}}_{\mathcal {L}}\cdot \left[ \iint \limits _{\theta ^2\theta ^3 } {\varvec{t}}^i_{,i} d\theta ^3 d\theta ^2 \right] d\theta ^1 \nonumber \\&\quad + \int \limits _{\theta ^1 } \delta {\varvec{d}}_\alpha \cdot \left[ \iint \limits _{ \theta ^2\theta ^3 } \theta ^\alpha {\varvec{t}}^i_{,i} d\theta ^3 d\theta ^2 \right] d\theta ^1=0. \end{aligned}$$

(67)

The integrals in thickness direction can be simplified by using Eq. (50) to render

$$\begin{aligned}&\iint \limits _{\theta ^2\theta ^3 } {\varvec{t}}^i_{,i}d\theta ^3 d\theta ^2 =\iint \limits _{\theta ^2\theta ^3 }{\varvec{t}}^1_{,1}d\theta ^3 d \theta ^2+\iint \limits _{ \theta ^2\theta ^3 }{\varvec{t}}^\alpha _{,\alpha }d\theta ^3 d\theta ^2 \nonumber \\&\quad =\left[ \,\iint \limits _{\theta ^2\theta ^3 }{\varvec{t}}^1 d\theta ^3 d\theta ^2\right] _{,1} + \int \limits _{\theta ^3 }[{\varvec{t}}^2]^{h^+_2}_{h^-_2}d\theta ^3+ \int \limits _{\theta ^2}[{\varvec{t}}^3]^{h^+_3}_{h^-_3}d\theta ^2 \nonumber \\&\,=: {\varvec{N}}^1_{,1}+ {{\varvec{n}}}^\alpha , \end{aligned}$$

(68)

and by introducing $[\theta ^\alpha {\varvec{t}}^\alpha ]_{,\alpha }= {\varvec{t}}^\alpha + \theta ^\alpha {\varvec{t}}^\alpha _{,\alpha }$ it holds that

$$\begin{aligned}&\iint \limits _{\theta ^2\theta ^3 } \theta ^\alpha {\varvec{t}}^i_{,i}d\theta ^3 d\theta ^2 =\iint \limits _{\theta ^2\theta ^3 } \theta ^\alpha {\varvec{t}}^1_{,1}d\theta ^3 d\theta ^2+\iint \theta ^\alpha {\varvec{t}}^\alpha _{,\alpha }d\theta ^3 d\theta ^2 \nonumber \\&\quad = \left[ \iint \limits _{\theta ^2\theta ^3 } \theta ^\alpha {\varvec{t}}^1d\theta ^3 d\theta ^2 \right] _{,1}-\iint \limits _{ \theta ^2\theta ^3 } {\varvec{t}}^\alpha d\theta ^3 d\theta ^2 \nonumber \\&\qquad + \int \limits _{\theta ^3 }[\theta ^2 {\varvec{t}}^2]^{h^+_2}_{h^-_2}d\theta ^3+\int \limits _{\theta ^2}[\theta ^3{\varvec{t}}^3]^{h^+_3}_{h^-_3}d\theta ^2 \nonumber \\&\quad =: {\varvec{M}}^\alpha _{,1} - {\varvec{N}}^\alpha +{{\varvec{m}}}^\alpha \,. \end{aligned}$$

(69)

Therewith (67) reads

$$\begin{aligned}&\int \limits _{\theta ^1 } \delta {{\varvec{\varphi }}}_\mathcal {L}\cdot \left[ {\varvec{N}}^1_{,1}+ {{\varvec{n}}}^\alpha \right] d\theta ^1 \nonumber \\&\quad + \int \limits _{\theta ^1} \delta {\varvec{d}}_\alpha \cdot \left[ {\varvec{M}}^\alpha _{,1} - {\varvec{N}}^\alpha + {{\varvec{m}}}^\alpha \right] d\theta ^1=0. \end{aligned}$$

(70)

The distributed line loads ${{\varvec{n}}}^\alpha$ and ${{\varvec{m}}}^\alpha$ are here considered to be zero and the local Euler equations thus result in

$$\begin{aligned} {\varvec{N}}^1_{,1} = {\varvec{0}} \quad \text {and}\quad {\varvec{M}}^\alpha _{,1}={\varvec{N}}^\alpha . \end{aligned}$$

(71)