Numerical analysis of the wrinkling behavior of thin membranes

Abstract

Membranes are widely applied in the large-span buildings and spatial deployable structures. They are prone to wrinkle under compression due to their small bending stiffness. The wrinkling deformation may affect the surface precision of a membrane structure and its static and dynamic behaviors. Research on the wrinkling behavior of a membrane and its variation with the wrinkle-influencing factors would shed light on the evolution of wrinkles and contribute significantly to the effective control over the wrinkling deformation. In this paper, a rectangular membrane under shear is numerically studied based on the stability theory of plates and shells to explore the wrinkle-influencing factors, such as boundary conditions, pre-stress, thickness and material constants, and their effects on the characteristic parameters of wrinkles. Besides, some of the results are also compared with those derived from the membrane element previously proposed by the authors for a further exploration. Some new and interesting findings are obtained.

Appendix

1.1 A.1. Wrinkling model based on the stability of plates and shells

The wrinkling model based on the stability of plates and shells regards the wrinkling deformation of a membrane as a kind of local buckling of a thin plate or shell with very small bending stiffness and obtains its out-of-plane deformation through the nonlinear buckling analysis.

An arbitrary point P on the mid-surface of an undeformed thin shell (its configuration is denoted by $\Omega _{0})$ can be determined by the position vector ${\overline{{{\varvec{R}}}}}$ as:

$$\begin{aligned} {\overline{{{\varvec{R}}}}}(\xi ^{1},\xi ^{2})=\xi ^{\alpha }{\overline{{{\varvec{G}}}}}_\alpha \end{aligned}$$

(A-1)

where the Greek alphabet $\alpha = 1,2$; ${\overline{{{\varvec{G}}}}}_\alpha$ and $\xi ^{\alpha }$ are, respectively, the covariant base vector and the contravariant coordinate on the mid-surface of the shell with ${\overline{{{\varvec{G}}}}}_\alpha ={\partial {\overline{{{\varvec{R}}}}}}/{\partial \xi ^{\alpha }}$; According to the Einstein’s summation convention, repetition of the index $\alpha$ represents the summation over its range, i.e., $\xi ^{\alpha }{\overline{{{\varvec{G}}}}}_\alpha =\sum _{\alpha =1}^2 {\xi ^{\alpha }{\overline{{{\varvec{G}}}}}_\alpha }$. The unit vector ${\overline{{{\varvec{N}}}}}$ of the normal on this point can be represented by the base vectors as:

$$\begin{aligned} {\overline{{{\varvec{N}}}}}=\frac{{\overline{{{\varvec{G}}}}}_1 \times {\overline{{{\varvec{G}}}}}_2 }{\left| {{\overline{{{\varvec{G}}}}}_1 \times {\overline{{{\varvec{G}}}}}_2 } \right| }=\frac{e^{\alpha \beta }e_{ijk} \overline{{R}}_{,\alpha }^j \overline{{R}}_{,\beta }^k {{\varvec{e}}}_i }{2\sqrt{\overline{{G}}}} \end{aligned}$$

(A-2)

where $\overline{{G}}=\det (\overline{{G}}_{\alpha \beta } )$ and $\overline{{G}}_{\alpha \beta }$ are the component of the metric tensor in the initial configuration $\Omega _{0}$ with $\overline{{G}}_{\alpha \beta } ={\overline{{{\varvec{G}}}}}_\alpha \cdot {\overline{{\varvec{G}}}}_\beta$ ($\alpha$,$\beta = 1,2$); $e^{\alpha \beta }$ and $e_{ijk}$ (i, j, $k = 1,2,3$) are the second and third permutation symbols, respectively; $(\;)_{,\alpha } ={\partial (\;)}/{\partial \xi ^{\alpha }}$; ${{\varvec{e}}}_{i}$ and $\overline{{R}}_{,\alpha }^j$ are, respectively, the base vector and the jth component of the vector ${\overline{{{\varvec{R}}}}}_{,\alpha }$ in the Cartesian coordinate system. A point P’ is located at a distance of $\zeta$ from point P along the normal ${\overline{{{\varvec{N}}}}}$ and its position vector can be formulated as:

$$\begin{aligned} {{\varvec{R}}}={\overline{{{\varvec{R}}}}}(\xi ^{1},\xi ^{2})+\zeta {\overline{{{\varvec{N}}}}} \end{aligned}$$

(A-3)

The corresponding curvilinear coordinate frame in the configuration $\Omega _{0}$ can then be derived from Eq. (A-3):

$$\begin{aligned} {{\varvec{G}}}_\alpha =\frac{\partial {{\varvec{R}}}}{\partial \xi ^{\alpha }}=\frac{\partial {\overline{{{\varvec{R}}}}}}{\partial \xi ^{\alpha }}+\zeta \frac{\partial {\overline{{{\varvec{N}}}}}}{\partial \xi ^{\alpha }},\quad {{\varvec{N}}}={\overline{{{\varvec{N}}}}} \end{aligned}$$

(A-4)

According to the Weingarten’s equation in Differential Geometry [43], we obtain:

$$\begin{aligned} \frac{\partial {\overline{{{\varvec{N}}}}}}{\partial \xi ^{\alpha }}=\hbox {-}{K}_\alpha ^{\cdot \beta } {\overline{{{\varvec{G}}}}}_\beta =\hbox {-}{K}_{\alpha \beta } (\overline{{G}}_{\beta \gamma } )^{-1}{\overline{{{\varvec{G}}}}}_\gamma \end{aligned}$$

(A-5)

where ${K}_{\alpha \beta }$ is the coefficient of the second fundamental form of the undeformed surface with ${K}_{\alpha \beta } =\frac{\partial ^{2}{\overline{{{\varvec{R}}}}}}{\partial \xi ^{\alpha }\partial \xi ^{\beta }}\cdot {\overline{{{\varvec{N}}}}}$. Thus, Eq. (A-4)$_{1}$ can then be rewritten as:

$$\begin{aligned} {{\varvec{G}}}_\alpha =\frac{\partial {{\varvec{R}}}}{\partial \xi ^{\alpha }}={\overline{{{\varvec{G}}}}}_\alpha \hbox {-}\zeta \,{K}_{\alpha \beta } (\overline{{G}}_{\beta \gamma } )^{-1}{\overline{{{\varvec{G}}}}}_\gamma \end{aligned}$$

(A-6)

When the shell buckles under compression (its corresponding configuration is denoted by $\Omega$), the position of point P changes and it can be determined by:

$$\begin{aligned} {\overline{{{\varvec{r}}}}}(\xi ^{1},\xi ^{2})=\xi ^{\alpha }{\overline{{{\varvec{g}}}}}_\alpha \end{aligned}$$

(A-7)

where ${\overline{{{\varvec{r}}}}}$ and ${\overline{{{\varvec{g}}}}}_\alpha$ are, respectively, the position vector and the covariant base vector in the current configuration $\Omega$ with ${\overline{{{\varvec{g}}}}}_\alpha ={\partial {\overline{{{\varvec{r}}}}}}/{\partial \xi ^{\alpha }}$. The normal direction at point P also varies with the deformation and it is given by Eq. (A-8):

$$\begin{aligned} {\overline{{\varvec{n}}}}=\frac{{\overline{{\varvec{g}}}}_1 \times {\overline{{\varvec{g}}}}_2 }{\left| {{\overline{{\varvec{g}}}}_1 \times {\overline{{\varvec{g}}}}_2 } \right| }=\frac{e^{\alpha \beta }e_{ijk} \overline{{r}}_{,\alpha }^j \overline{{r}}_{,\beta }^k {{\varvec{e}}}_i }{2\sqrt{\overline{{g}}}} \end{aligned}$$

(A-8)

where $\overline{{g}}=\det (\overline{{g}}_{\alpha \beta } )$ and $\overline{{g}}_{\alpha \beta }$ are the component of the metric tensor in the current configuration $\Omega$ with $\overline{{g}}_{\alpha \beta } ={\overline{{\varvec{g}}}}_\alpha \cdot {\overline{{\varvec{g}}}}_\beta$; $\overline{{r}}_{,\alpha }^j$ is the jth component of the vector ${\overline{{\varvec{r}}}}_{,\alpha }$ in the Cartesian coordinate system. The position of point P’ in the configuration $\Omega$ can then be obtained from Eqs. (A-7) and (A-8):

$$\begin{aligned} {{\varvec{r}}}={\overline{{\varvec{r}}}}(\xi ^{1},\xi ^{2})+\zeta {\overline{{\varvec{n}}}} \end{aligned}$$

(A-9)

According to the Weingarten’s equation in Differential Geometry [43], the curvilinear coordinate frame in the configuration $\Omega$ can be formulated as:

$$\begin{aligned} {{\varvec{g}}}_\alpha =\frac{\partial {{\varvec{r}}}}{\partial \xi ^{\alpha }}=\frac{\partial {\overline{{\varvec{r}}}}}{\partial \xi ^{\alpha }}+\zeta \frac{\partial {\overline{{\varvec{n}}}}}{\partial \xi ^{\alpha }}={\overline{{\varvec{g}}}}_\alpha \hbox {-}\zeta \,\kappa _\alpha ^{\cdot \beta } {\overline{{\varvec{g}}}}_\beta ={\overline{{\varvec{g}}}}_\alpha \hbox {-}\zeta \,\kappa _{\alpha \beta } (\overline{{g}}_{\beta \gamma } )^{-1}{\overline{{\varvec{g}}}}_\gamma ,\quad {{\varvec{n}}}={\overline{{\varvec{n}}}} \end{aligned}$$

(A-10)

where $\kappa _{\alpha \beta }$ is the coefficient of the second fundamental form of the deformed surface with $\kappa _{\alpha \beta } =\frac{\partial ^{2}{\overline{{\varvec{r}}}}}{\partial \xi ^{\alpha }\partial \xi ^{\beta }}\cdot {\overline{{\varvec{n}}}}$. The deformation gradient at point P’ can be derived from Eqs. (A-4) and (A-10):

$$\begin{aligned} {{\varvec{F}}}=\frac{\partial {{\varvec{r}}}}{\partial {{\varvec{R}}}}={{\varvec{g}}}_I {{\varvec{G}}}^{I} \end{aligned}$$

(A-11)

where ${{\varvec{G}}}^{I}$ is the contravariant base vector in the configuration $\Omega _{0}$ with ${{\varvec{G}}}_I \cdot {{\varvec{G}}}^{J}=\delta _I^J$ (I, $J = 1, 2, 3; \delta _I^J$ is the Kronecker symbol); ${{\varvec{G}}}^{3} ={{\varvec{N}}}$ and ${{\varvec{g}}}_{3} ={{\varvec{n}}}$. The Green strain tensor can then be expressed as:

$$\begin{aligned} {{\varvec{E}}}=\frac{1}{2}\left( {{{\varvec{F}}}^{\mathrm {T}}{{\varvec{F}}}-{{\varvec{I}}}} \right) =\frac{1}{2}\left( {g_{IJ} -G_{IJ} } \right) {{\varvec{G}}}^{I}{{\varvec{G}}}^{J} \end{aligned}$$

(A-12)

where $G_{IJ} ={{\varvec{G}}}_I \cdot {{\varvec{G}}}_J$ and $g_{IJ} ={{\varvec{g}}}_I \cdot {{\varvec{g}}}_J$. The second Piola–Kirchhoff (PK2) stress tensor is energetically conjugate to the Green strain and is given by the constitutive relationship:

$$\begin{aligned} {{\varvec{S}}}={{\varvec{C}}}:{{\varvec{E}}} \end{aligned}$$

(A-13)

where C is the constitutive tensor.

When the body force vanishes, the equilibrium differential equations in the configuration $\Omega$ are simplified as:

$$\begin{aligned} \nabla \cdot {\varvec{\sigma }}=0 \end{aligned}$$

(A-14)

where $\nabla =\frac{\partial }{\partial \eta ^{I}}{{\varvec{t}}}_I (I = 1, 2, 3); {{\varvec{t}}}_1$ and ${{\varvec{t}}}_2$ are the normalized base vectors along the principal directions in the configuration $\Omega$ and ${{\varvec{t}}}_3 ={{\varvec{n}}}$; $\eta ^{1}$ and $\eta ^{2}$ are the principal coordinates and $\eta ^{3}=\zeta$; ${\varvec{\sigma }}$ is the Cauchy stress tensor and can be obtained by pushing forward S:

$$\begin{aligned} {\varvec{\sigma }}=J^{-1}{{\varvec{F}}}\cdot {{\varvec{S}}}\cdot {{\varvec{F}}}^{{\mathrm{T}}} \end{aligned}$$

(A-15)

where $J =\hbox { det}({{\varvec{F}}})$. Pulling back Eq. (A-14) to the initial configuration yields:

$$\begin{aligned} \nabla _0 \cdot {{\varvec{P}}}=0 \end{aligned}$$

(A-16)

where $\nabla _0 =\frac{\partial }{\partial \eta ^{I}}{{\varvec{T}}}_I (I$ = 1, 2, 3), ${{\varvec{T}}}_1$ and ${{\varvec{T}}}_2$ are the normalized base vectors along the principal directions in the configuration $\Omega _{0}$ and ${{\varvec{T}}}_3 ={{\varvec{N}}}$; P is the Lagrange stress tensor with:

$$\begin{aligned} {{\varvec{P}}}=J{{{\varvec{F}}}}^{-1}\cdot {\varvec{\sigma }}={{\varvec{S}}}\cdot {{\varvec{F}}}^{{\mathrm{T}}} \end{aligned}$$

(A-17)

Owing to the complexity of the boundary conditions for a general wrinkling problem, Eq. (A-14) or Eq. (A-16) is usually solved with the finite element method (FEM). But the stiffness matrix of a wrinkled membrane may have a large condition number resulting from the vanishing stiffness in the normal direction of wrinkles and it may thus be difficult to obtain the convergent solution if the Newton iteration scheme is used. The dynamic relaxation method combined with the explicit time integration scheme does not require the inverse operation of the stiffness matrix and ensures the numerical stability. This approach is therefore adopted in this paper to analyze the wrinkling behavior of a membrane based on the stability of plates and shells.

1.2 A.2 Dynamic relaxation method

The dynamic relaxation method (DRM) treats the static equilibrium of a wrinkled membrane as the long-time limit state of a damped transient system and obtains the wrinkling deformation by solving for this limit state [19,20,21, 32, 44, 45]. DRM combined with the explicit time integration scheme intends to address such a nonlinear problem that usually induces divergence in the Newton iteration. When DRM is used for the wrinkling problem, mass and damping in the dynamic equations are not required to match the real physical properties of a membrane. Their values are chosen only for the consideration of the speedy decay of the transient response, the efficient convergence and the stability of computation [32].

According to the assumption of DRM, the pseudo dynamic process of the wrinkling problem can be formulated as:

$$\begin{aligned} \nabla _0 \cdot {{\varvec{P}}}=\rho {{\varvec{r}}}_{tt} +c{{\varvec{r}}}_t \end{aligned}$$

(A-18)

where $(\;)_{tt} ={\hbox {d}^{2}(\;)}/{\hbox {d}t^{2}}$, ${{\varvec{r}}}_t ={\hbox {d}(\;)}/{\hbox {d}t}$, $\rho$ and c are, respectively, the density and damping of membrane. Equation (A-18) is solved in this paper with the central difference method [46], a kind of explicit time integration scheme, to overcome the convergence difficulty that confronts us in the wrinkling analysis with an ill-conditioned stiffness matrix. The detailed procedure of the central difference method is presented as follows:

• (1) Discretize the time domain (i.e., $[0, t_E ]$, $t_E$ is the ending time) of Eq. (A-18) into “n” intervals $[t^{k-1},\;t^{k}](k$ = 1$\sim n)$ with:

  $$\begin{aligned} \Delta t^{k+1/2}=t^{k+1}-t^{k},\quad t^{k+1/2}=\frac{1}{2}\left( {t^{k+1}+t^{k}} \right) ,\quad \Delta t^{k}=t^{k+1/2}-t^{k-1/2} \end{aligned}$$

  (A-19)

• (2) Set the initial values including the displacement ${{\varvec{u}}}^{0}$ and velocity ${{\varvec{u}}}_t^0$ at the moment of $t = 0$, the mass matrix M and the damping matrix ${{\varvec{C}}}^{\mathrm{damp}}$.

• (3) Compute the force ${{\varvec{f}}}^{k}$. ${{\varvec{f}}}^{k}={{\varvec{f}}}^{\mathrm {ext}}({{\varvec{u}}}^{k},t^{k})-{{\varvec{f}}}^{\mathrm{int}}({{\varvec{u}}}^{k},t^{k})$ in which ${{\varvec{f}}}^{\mathrm {ext}},{{\varvec{f}}}^{\mathrm{int}}$ are the exterior and interior nodal force vectors.

• (4) Compute the acceleration ${{\varvec{u}}}_{tt}^k$:

  $$\begin{aligned} {{\varvec{u}}}_{tt}^k ={{\varvec{M}}}^{-1}\left( {{{\varvec{f}}}^{k}-{{\varvec{C}}}^{\mathrm {damp}}{{\varvec{u}}}_t^{k-1/2} } \right) \end{aligned}$$

  (A-20)

  When k = 0, ${{\varvec{u}}}_t^{-1/2}$ is determined by:

  $$\begin{aligned} {{\varvec{u}}}_t^{-1/2} =\left( {{{\varvec{I}}}-(\Delta t^{0}/2){{\varvec{M}}}^{-1}{{\varvec{C}}}^{\mathrm {damp}}} \right) ^{-1}\left( {{{\varvec{u}}}_t^0 -(\Delta t^{0}/2){{\varvec{M}}}^{-1}{{\varvec{f}}}^{0}} \right) \end{aligned}$$

  (A-21)

  in which I is the unit matrix $\circ$

• (5) Update the time: $t^{k+1}=t^{k}+\Delta t^{k+1/2},\quad t^{k+1/2}=\frac{1}{2}\left( {t^{k+1}+t^{k}} \right)$

• (6) Update the velocity ${{\varvec{u}}}_t^{k+1/2}$:

  $$\begin{aligned} {{\varvec{u}}}_t^{k+1/2} ={{\varvec{u}}}_t^k +(t^{k+1/2}-t^{k}){{\varvec{u}}}_{tt}^k \end{aligned}$$

  (A-22)

• (7) Update the displacement ${{\varvec{u}}}^{k+1}$:

  $$\begin{aligned} {{\varvec{u}}}^{k+1}={{\varvec{u}}}^{k}+\Delta t^{k+1/2}{{\varvec{u}}}_t^{k+1/2} \end{aligned}$$

  (A-23)

• (8) Calculate ${{\varvec{f}}}^{k+1}$.

• (9) Determine ${{\varvec{u}}}_{tt}^{k+1}$ by Eq. (A-20).

• (10) Update the velocity ${{\varvec{u}}}_t^{k+1}$:

  $$\begin{aligned} {{\varvec{u}}}_t^{k+1} ={{\varvec{u}}}_t^{k+1/2} +(t^{k+1}-t^{k+1/2}){{\varvec{u}}}_{tt}^{k+1} \end{aligned}$$

  (A-24)

• (11) Check the energy balance.

• (12) Update the time step: $k\rightarrow k+1$ and return to step (5) till the solution is completed.

It should be noted that the explicit time integration scheme is conditionally stable, and it can obtain the convergent solution only when $\Delta t\le \Delta t_{cr}$. $\Delta t_{cr}$ is the critical time increment (also called the stable time increment), and it can be determined by [44]:

$$\begin{aligned} \Delta t_{{\mathrm{cr}}} =\frac{2}{\lambda _{\max } } \end{aligned}$$

(A-25)

where $\lambda _{max}$ is the maximum eigenvalue of the system.