A Physically-Based Model of Fabric Drape
Figure 1
Nature of the problem
Fabrics are complex mechanisms of interwoven threads that are
themselves mechanisms of twisted fibers. Drape is a property of
fabrics associated with the aesthetic appearance of garments and
other textile structures. Fabric drape can be defined as a description
of the deformation of the fabric produced by gravity when only
part of it is directly supported. This is the case in many applications
where fabrics are used. A garment, for example a skirt, is in
contact with the body only at some places. Most of the rest of
the skirt falls gracefully and forms smooth folds. This characteristic
is perhaps what distinguishes a fabric from other materials. Therefore,
the simulation of fabric drape is an interesting problem and will
have many uses, including computer-aided apparel design.
There have been several attempts since the mid-eighties to model
fabric drape. These include the application of continuum mechanics
and finite element methods, as well as modeling fabric as a mechanism
based on its structure. Since frictional forces between the fibers
and the threads give the fabric mechanism a stable structure,
it may be advantageous to model fabrics at the microstructure
level.
Practical considerations, however, show that such micromodeling
of fabric is not realizable for various reasons. Even though some
work has been done on the micromechanics of fabrics, the properties
of the microstructure are not well known, neither can they be
experimentally determined at this time. Also, to represent fabrics
realistically, a microstructure model would have to contain a
large number of elements or particles. This would make the calculations
reach enormous levels to solve at the present computational capabilities
in any reasonable amount of time.
The problem involved in modeling fabrics is now well understood,
even though proper solutions are not yet available. Fabrics undergo
large deformations for small applied forces. This differentiates
fabrics from other sheet materials. In most sheet materials, computer
simulation is confined to the study of the buckling of the materials.
In fabrics, however, the simulation will have to account for the
shape assumed by the fabric when it attains an equilibrium and
comes to its final shape.
Shell Theory and Finite Element Method
Finite element method is a way of subdividing a system into its
individual components or `elements,' whose behavior is readily
understood, and then rebuilding the original system from such
components to study its behavior. In the study of fabric deformations,
large displacements must be considered. Fabrics also bend about
two normal axes simultaneously. In this respect, fabrics are different
from some other materials which, when bent, undergo stiffening.
For any large deformation problem, the non-linearity due to a
change in the geometry of the body has to be considered in order
to obtain a correct solution. Instead of the one-step solution
found in linear problems, the non-linear problem is usually solved
iteratively. The loads are applied incrementally to the system,
and at each step, the equilibrium equation:
[K] {Delta q} = {Delta f} (1)
is solved by the Newton-Raphson method. The non-linearity is handled
by calculating the stiffness matrix [K] in each step as a function
of the displacement vector {q}. Because during the intermediate
steps, the fabric is no longer a plate, shell elements are used
in the formulation.
Various shell theory and finite element formulations have been
developed during the past several years. The simplest one is to
treat shells as an assembly of plates. This approach has been
used by some earlier researchers. In this approach, a superposition
of stretching and bending of the plates was used to represent
shell behavior, thus avoiding the difficulties associated with
geometry. However, this approach excluded the coupling of stretching
and bending within the elements. The link between in-plane membrane
deformation and out-of-place bending deformation cannot be ignored
in the case of textile materials.
Other researchers took the classical shell theory approach, which
regarded the shell as an inextensible one-director Cosserat surface.
This surface is defined by a middle plane of a shell element and
a unit vector at each point of the surface. The objective is a
precise formulation of the local balance laws, the local constitutive
equations, and the weak form of the momentum equations in a manner
suitable for numerical analysis and finite element implementation.
Because the resultant form is carried out analytically, it is
much more mathematically complicated and computationally intensive.
We use the degenerated solid approach, which has the same underlying
hypothesis as the classical shell theory, but the reduction to
resultant form is carried out numerically. This approach is more
conducive to numerical implementation.
The kinematics of the shell theory consist of measuring membrane
strain on the reference surface from the derivatives of x with
respect to the position on the surface, and the bending strain
from the derivatives of normal n. The strain measures that are
used for this purpose are approximations to Koiter-Sanders Theory
strains. The transverse shear strains are measured as the change
in the projections of n onto the shell's reference surface. The
change in the shell's thickness caused by deformation is neglected.
Input parameters for the model
The input to the model are the Young's modulus, the shear modulus,
and Poisson's ratio. The latter is the ratio of the percentage
reduction in width as compared to the percentage extension. Woven
textile fabrics consist of two sets of threads intersecting each
other at right angles. Thus fabric can be treated as an orthotropic
medium. Accordingly, the Young's modulus values are given in the
model for the two thread directions. Experimentally determined
values of fabric parameters were used in the models.
Simulation
Figure 1 shows the simulation of a 30 centimeter square fabric
drape over a 12 centimeter square surface. A nine node, doubly
curved shell element with five degrees of freedom per node is
used in the simulation. The portion of the fabric in contact with
the table is fully constrained. This means that the elements in
contact with the table have no degrees of freedom. The images
are ray-traced to show the folds. The six stages of draping show
the development of the double curvature, charac-teristic of fabric
drape.
The fabric is a typical suiting fabric made of polyester textured
yarns in warp and weft. Besides the visual similarity exhibited
by the model and the actual fabric, the measured values of deformation
at the edges and at intermediate points on the actual fabric correlated
well with the model values.
Figure 2
The usefulness of the model is further illustrated
in Figure 2. A series of simulations were done with increasing
stiffness of the fabric in one of the thread (warp) directions
to study the effect on drape. The shear modulus and Poisson's
ratio were kept the same for all simulations. The effect of increasing
stiffness is clearly seen. A comparison of the simulations clearly
shows the effect of increasing orthotropy on drape.
The simulations we discussed so far are for fabric drape over
a fixed flat surface. In these cases, the portion of the fabric
that is in contact with the surface is fixed during draping and
no force transfer between the fabric and the surface takes place.
In real draping situations, as in apparel, the fabric is draped
over arbitrary surfaces. No prior knowledge exists of which portion
of the fabric will be in contact with the body after draping.
This amounts to letting a fabric free fall, meet an object, and
drape. There will be some interaction forces between the fabric
and the rigid surface.
The same flexible shell theory and finite element formulation
with appropriate Lagrange multiplier techniques were used to simulate
the drape of fabric over a sphere. We assume the sphere as a rigid
surface with negligible friction between the fabric and the surface.
The simulation of a 30 cm x 30 cm size fabric, draped over a sphere
of 10 cm diameter, is shown in Figure 3. A photograph of an actual
fabric draped similarly is shown in Figure 4. For
an animation of fabric drape click here.
Figure 3 Figure 4
Acknowledgments
This work was supported in part by a grant from the Agriculture
Experiment Station, Ithaca, NY, grant number 329-403. Finite element
modeling was done using ABAQUSreg. on Cornell Theory Center resources.
We would like to acknowledge the support of the visualization
group at the Cornell Theory Center, especially Catherine Devine,
who was responsible for the rendering of the models using IBM's
DataExplorer software.
Publications:
Chen, B., and Govindaraj, M. 'A Parametric Study of Fabric Drape',
Textile Research Journal, 66(1), 16-23, 1996
Chen, B., and Govindaraj, M. 'A Physically-Based Model of Fabric
Drape Using Flexible Shell Elements', Textile Research Journal.
65 (6), 324-330, 1995.
Copyright 1995 by Muthu Govindaraj and Bijian Chen. All
Rights Reserved.
Last updated April 1996 . For info: (mgraj@aol.com
).