Modeling of Textile Composites
Theory and Physics
What are Woven Composites?
Professor, how are "woven composites" different from UD materials (unidirectional reinforced materials)?
In UD materials, all fibers are aligned in one direction, but woven composites have fibers that are woven (interlaced). They have weave structures like plain weave, twill weave, and satin weave.
What are the advantages of weaving?
Types of Weaves
| Type | Structure | Characteristics |
|---|---|---|
| Plain Weave | One-over-one interlacing | Stable. Low drapability |
| Twill Weave | 2/1, 2/2, etc. patterns | High drapability |
| Satin Weave | 5HS, 8HS (long float) | Highest drapability. Fibers are nearly straight |
| NCF (Non-Crimp Fabric) | Fibers fixed by stitching without weaving | No crimp. Best mechanical properties |
What is "crimp"?
The waviness of fibers going over and under each other in a weave structure. Crimp causes fibers to be bent, resulting in a 10-20% reduction in tensile stiffness and strength compared to UD materials. NCF has no crimp, so it has performance close to UD materials.
FEM Modeling of Woven Composites
FEM modeling of woven composites has three levels:
| Level | Approach | Accuracy |
|---|---|---|
| Macro | Equivalent homogeneous shell (CLT-based) | Low (global behavior) |
| Meso | RVE (fiber tow + matrix) modeling | High (local stress) |
| Micro | Modeling individual fibers | Highest (for research) |
Is the mesoscale RVE model practical?
At the mesoscale, one unit of the weave pattern (Unit Cell) is modeled with solid elements, and equivalent properties are calculated using periodic boundary conditions. Specialized tools like TexGen and WiseTex automatically generate the Unit Cell geometry.
Summary
Let me organize the theory of woven composites.
Key points:
- Structure with interlaced fibers — Plain weave, twill weave, satin weave, NCF
- Stiffness and strength reduced by crimp — 10-20% lower than UD materials
- Excellent drapability and damage resistance — Suitable for forming on curved surfaces
- Three levels of modeling — Macro (CLT), Meso (RVE), Micro (individual fibers)
- Mesoscale RVE analysis is practical — Unit Cell generation with TexGen/WiseTex
Elastic Properties of Woven Composites and Unit Cell Theory
Woven composites are composed of repeating unit cells where warp and weft yarns intersect. Elastic properties can be calculated more accurately by FEM homogenization analysis of the unit cell, which better reflects the fiber bridging effect (crimp), compared to simple test values from ISO 527. A mere 5% crimp can reduce strength by 10-20%, and they offer better in-plane isotropy than unidirectional prepreg.
Physical Meaning of Each Term
- Inertia term (mass term): $\rho \ddot{u}$, i.e., "mass × acceleration". Have you ever experienced being thrown forward when slamming on the brakes? That "feeling of being pulled" is precisely the inertial force. Heavier objects are harder to set in motion and harder to stop once moving. Buildings shake during earthquakes because the ground moves suddenly while the building's mass "gets left behind". In static analysis, this term is set to zero, assuming "forces are applied slowly enough to ignore acceleration". It cannot be omitted for impact loads or vibration problems.
- Stiffness term (elastic restoring force): $Ku$ or $\nabla \cdot \sigma$. When you stretch a spring, you feel a "force trying to return it", right? That's Hooke's law $F=kx$, the essence of the stiffness term. Now a question — an iron rod and a rubber band, which stretches more under the same force? Obviously the rubber. This "resistance to stretching" is the Young's modulus $E$, which determines stiffness. A common misconception: "high stiffness = strong" is incorrect. Stiffness is "resistance to deformation", strength is "resistance to failure" — they are different concepts.
- External force term (load term): Body forces $f_b$ (gravity, etc.) and surface forces $f_s$ (pressure, contact forces, etc.). Think of it this way — the weight of a truck on a bridge is a "force acting on the entire volume" (body force), while the force of the tires pushing on the road is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A common mistake here: getting the load direction wrong. Intending "tension" but applying "compression" — sounds like a joke, but it actually happens when coordinate systems rotate in 3D space.
- Damping term: Rayleigh damping $C\dot{u} = (\alpha M + \beta K)\dot{u}$. Try plucking a guitar string. Does the sound continue forever? No, it gradually fades. That's because vibrational energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle — intentionally absorbing vibrational energy for a smoother ride. What if damping were zero? Buildings would keep swaying forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is crucial.
Assumptions and Applicability Limits
- Continuum assumption: Treats material as a continuous medium, ignoring microscopic heterogeneity
- Small deformation assumption (for linear analysis): Deformation is sufficiently small compared to initial dimensions, and stress-strain relationship is linear
- Isotropic material (unless otherwise specified): Material properties are independent of direction (anisotropic materials require separate tensor definitions)
- Quasi-static assumption (for static analysis): Ignores inertial and damping forces, considering only equilibrium between external and internal forces
- Non-applicable cases: Large deformation/large rotation problems require geometric nonlinearity. Nonlinear material behavior like plasticity or creep requires constitutive law extensions
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify load and elastic modulus to MPa/N system |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Note unit system inconsistency when comparing with yield stress |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note distinction between engineering strain and logarithmic strain (for large deformation) |
| Elastic modulus $E$ | Pa | Steel: ~210 GPa, Aluminum: ~70 GPa. Note temperature dependence |
| Density $\rho$ | kg/m³ | In mm system: tonne/mm³ (= 10⁻⁹ tonne/mm³ for steel) |
| Force $F$ | N (Newton) | Unify as N in mm system, N in m system |
Numerical Methods and Implementation
Mesoscale RVE Analysis
Please explain the procedure for RVE (Representative Volume Element) analysis.
1. Generate Unit Cell geometry — 3D shape of the weave pattern using TexGen, WiseTex, etc.
2. Mesh generation — Mesh the Unit Cell with TET10
3. Apply periodic boundary conditions — Displacements on opposing faces have a linear relationship
4. Six load cases — Apply $\varepsilon_{11}, \varepsilon_{22}, \varepsilon_{33}, \gamma_{12}, \gamma_{23}, \gamma_{13}$ sequentially
5. Homogenization — Calculate equivalent elastic constants from the average stress of each load case
So the nine elastic constants (orthotropic) are determined from the six load cases.
Yes. You obtain $E_1, E_2, E_3, G_{12}, G_{23}, G_{13}, \nu_{12}, \nu_{23}, \nu_{13}$. These are used as material properties for macro-scale shell/solid elements.
Specialized Tools
| Tool | Features |
|---|---|
| TexGen | Unit Cell geometry generation. Open source (University of Nottingham) |
| WiseTex | Weave geometry + mechanics. Developed by KU Leuven |
| DIGIMAT | Multi-scale material modeling. eXstream/Hexagon |
| MicroMechanics | Fiber-matrix RVE analysis. MCT (Multi-Continuum Theory) |
It's great that TexGen is free.
TexGen automatically generates 3D geometry of weave structures and can directly output Abaqus input files. It's practically the standard tool for research on woven RVE analysis.
Handling Woven Composites at the Macro Scale
How are woven composites handled at the macro scale (regular FEM analysis)?
Treat them as equivalent homogeneous materials. Use the equivalent elastic constants obtained from RVE analysis as one layer in CLT and perform regular laminate analysis.
Points to note:
- Crimp effect — Verify if the equivalent properties include the influence of crimp
- Failure criteria — Applying UD material criteria like Tsai-Wu/Hashin directly can be inaccurate. Modifications for woven materials are needed
- Draping — Reflect changes in fiber angle during forming in the FEM
Summary
Let me organize the numerical methods for woven composites.
Key points:
- Calculate equivalent properties via RVE analysis — Nine elastic constants from six load cases
- Generate Unit Cell with TexGen (free) — Standard research tool
- Multi-scale coupling with DIGIMAT (commercial) — For industry
- Treat as equivalent homogeneous material at macro scale — One layer in CLT
- Do not use UD material failure criteria as-is — Modifications for woven materials are needed
Homogenization FEM Analysis of Woven Composites
Homogenization analysis of a unit cell proceeds in the order: ① 3D FEM construction of the cell, ② application of periodic boundary conditions, ③ calculation of equivalent elastic matrix from reaction forces under six-component unit loads. Computational cost is about solving one cell's FEM (~100k elements) six times, within a few hours. Specialized unit cell meshing tools like Software TexComp (KU Leuven), WiseTex, and TexGen are publicly available, automatically generating meshes just by inputting weave parameters (yarn width, thickness, crimp ratio).
Linear Elements (1st-order elements)
Linear interpolation between nodes. Low computational cost but low stress accuracy. Beware of shear locking (mitigated with reduced integration or B-bar method).
Quadratic Elements (with mid-side nodes)
Can represent curved deformation. Stress accuracy improves significantly, but degrees of freedom increase by about 2-3 times. Recommended: when stress evaluation is important.
Full integration vs Reduced integration
Full integration: Risk of over-constraint (locking). Reduced integration: Risk of hourglass modes (zero-energy modes). Choose appropriately.
Adaptive Mesh
Automatic refinement based on error indicators (e.g., ZZ estimator). Efficiently improves accuracy in stress concentration areas. Includes h-method (element subdivision) and p-method (order increase).
Newton-Raphson Method
Standard method for nonlinear analysis. Updates tangent stiffness matrix each iteration. Quadratic convergence within convergence radius, but high computational cost.
Modified Newton-Raphson Method
Updates tangent stiffness matrix at initial value or every few iterations. Cost per iteration is low, but convergence is linear.
Convergence Criteria
Force residual norm: $||R|| / ||F_{ext}|| < \epsilon$ (typically $\epsilon = 10^{-3}$ to $10^{-6}$). Displacement increment norm: $||\Delta u|| / ||u|| < \epsilon$. Energy norm: $\Delta u \cdot R < \epsilon$
Load Increment Method
Apply total load in small increments rather than all at once. The arc-length method (Riks method) can trace beyond limit points on the load-displacement curve.
Analogy: Direct Method vs Iterative Method
The direct method is like "solving simultaneous equations accurately with pen and paper" — reliable but takes too long for large-scale problems. The iterative method is like "repeatedly guessing to approach the correct answer" — starts with a rough answer but improves accuracy with each iteration. It's the same principle as looking up a word in a dictionary: it's more efficient to open it at an estimated location and adjust forward/backward (iterative method) than to search sequentially from the first page (direct method).
Relationship Between Mesh Order and Accuracy
1st-order elements are like "approximating a curve with a ruler" — represented by straight line segments, so accuracy is limited. 2nd-order elements are like a "flexible curve" — can represent curved changes, and with the same mesh density...
Related Topics
なった
詳しく
報告