An analytical model for cure-induced spring-in of curved composite parts considering tool-part interaction

Technical Summary: An Analytical Model for Cure-Induced Spring-In of Curved Composite Parts Considering Tool-Part Interaction

Problem Statement
Cure-induced deformation is a primary source of dimensional deviation in thin-walled curved composite parts, which are increasingly utilized in aerospace engineering for their high specific stiffness and packaging efficiency. While existing analytical models can describe the basic thermochemical origins of spring-in (anisotropic thermal expansion and chemical shrinkage), they often fail to capture the experimentally observed thickness dependence of the phenomenon. Specifically, simplified relations do not account for the complex coupling between interfacial restraint (tool-part interaction), cure shrinkage, and shear-lag deformation. Although numerical studies have highlighted the significance of tangential constraints and interfacial friction, these factors have not been explicitly incorporated into closed-form analytical solutions for curved laminates, limiting the ability to rapidly and accurately predict spring-in for thin-walled structures.

Methodology
The study develops a new analytical model that extends the generalized plane strain framework previously established by Ding et al. The core innovation is the explicit integration of tool-part interaction into the mechanical equilibrium equation.

• Theoretical Framework: The model assumes a generalized plane strain condition during the rubbery state (between gelation and vitrification). It accounts for through-thickness non-mechanical strain, longitudinal constraints, and the shear-lag effect caused by low interlaminar shear stiffness in the rubbery state.
• Tool-Part Interaction: Unlike previous models, this approach introduces interfacial friction as a bounded frictional shear stress ($\tau_{\theta}^{ru}$) acting on the tool-part interface during the rubbery stage. This stress is defined as the minimum of the Coulomb friction limit ($\mu^{ru}P$) and a maximum interfacial shear stress ($\tau_{max}^{ru}$).
• Derivation: By incorporating this frictional term into the force-equilibrium formulation, the authors derive a non-homogeneous second-order ordinary differential equation. Solving this equation yields a closed-form solution for the spring-in angle that includes terms for shear-lag deformation, thermochemical strain mismatch, and the specific contribution of interfacial friction.
• Validation: The analytical model is validated against:
  1. Experimental Data: Published results for AS4/8552 unidirectional and cross-ply C-shaped parts with varying thicknesses (1–4 mm).
  2. Finite Element Analysis (FEA): A 3D FEA model developed in Abaqus, utilizing a path-dependent constitutive model to simulate the viscous, rubbery, glassy, and demoulding stages. The FEA model explicitly models surface-to-surface contact with Coulomb friction and varying friction limits to isolate the effects of the rubbery versus glassy stages.

Key Results

• Accuracy: The proposed model demonstrates satisfactory accuracy in predicting the thickness dependence of spring-in. For 1 mm thick laminates, the prediction error is 5.9% for cross-ply parts and 11.5% for unidirectional parts.
• Comparison with Existing Models: Compared to the model by Ding et al. (Eq. 4), the inclusion of friction reduces the prediction error by 9.2 percentage points for cross-ply parts and 8.8 percentage points for unidirectional parts at 1 mm thickness. The model outperforms simpler strain-mismatch relations (Eqs. 1 and 2) and shear-lag models without friction (Eq. 3).
• Role of Friction Timing: Finite element analysis reveals that interfacial friction introduced only after vitrification (in the glassy state) has negligible influence on the final spring-in angle. Conversely, friction active during the rubbery stage significantly reduces the spring-in angle.
• Thickness Dependence: The influence of interfacial friction is most pronounced in thinner parts. In the limit of very small thickness, friction-induced reduction accounts for approximately 9.2% (cross-ply) and 10.5% (unidirectional) of the total spring-in.
• Thin-Limit Approximation: A simplified approximation for ultra-thin parts (thickness < 1 mm) is derived, showing that the spring-in angle varies only slightly in this regime, allowing for rapid assessment.

Significance and Claims
The paper claims to provide a physically interpretable and efficient analytical tool for predicting cure-induced spring-in in thin-walled curved composite structures. By explicitly incorporating tool-part interaction, the model bridges the gap between simplified analytical relations and complex numerical simulations.

The authors emphasize that their work improves the physical completeness of spring-in prediction by unifying thermochemical strain mismatch, shear-lag deformation, and tangential tool restraint within a single framework. They assert that this model is particularly valuable for the rapid assessment of thin-walled structures where interfacial friction plays a critical role in dimensional accuracy, offering a more accurate alternative to existing analytical solutions without the computational cost of full finite element analysis. The study does not propose new applications or future experimental campaigns but focuses on validating the theoretical formulation against existing literature and numerical benchmarks.