Microscale equivalent modeling and experimental validation of CCF/PLA using a 100-to-1 downscaling strategy

Geometric modeling

A cuboid RVE aligned with the global coordinate system was constructed in ABAQUS to represent the mesoscale filament arrangement. The RVE contains 100 filaments arranged in a regular 10×10 array. Each circular filament was represented by an equal-area square surrogate to simplify meshing while strictly preserving the local fiber area and fiber volume fraction. Equal-area mapping yields s = π 2 d , where d is the circular filament diameter and s is the side length of the equal-area square. With d = 7 μm, the mapped side length is s = 6.2 μm.

A rectangular unit cell was selected over an ideal hexagonal close-packed array as an explicit engineering trade-off rather than a numerical convenience. First, straight raster regions in continuous-fiber FDM frequently exhibit laterally compressed, grid-like bundle morphologies; accordingly, a rectangular array is more compatible with subsequent path-scale modeling and coupling. ³⁰ Second, a rectangular unit cell substantially simplifies node matching on opposing faces, edges, and vertices, thereby improving numerical robustness when enforcing node-to-node PBCs. This choice is conservative with respect to transverse responses: micromechanics indicates that inclusion cross-sectional shape strongly affects matrix stress fields, and square inclusions introduce corner stress concentrations; in addition, a regular square array imposes stronger transverse geometric constraints than a hexagonal or randomly packed array. Prior work suggests that, when the geometric discrepancy is small, the difference between square and hexagonal arrays remains within engineering tolerance. ³¹ Nevertheless, the present study anticipates that the square-surrogate and regular-array simplification may bias transverse stiffness toward an upper-bound estimation, and the magnitude of this bias is explicitly tracked in later sections through theoretical comparison and fiber-volume-fraction-dependent homogenization.

It should be noted that the equal-area square representation is not intended to reconstruct the full local morphology of real circular filaments. In an actual CCF tow, adjacent filaments may undergo local contact, slight misalignment, lateral compression, resin-rich separation, and nonuniform matrix infiltration during deposition. These effects can modify fiber–fiber interaction and redistribute local matrix stress, particularly in the transverse and shear directions. By contrast, the present RVE regularizes the filament arrangement into a 10×10 square array and therefore suppresses stochastic packing disorder and local filament rearrangement. This simplification improves mesh controllability and enables stable PBC enforcement, but it may reduce the model’s ability to capture local stress transfer around individual circular fibers. Therefore, the square inclusion should be interpreted as an equal-area mechanical surrogate for homogenized stiffness extraction rather than as a direct geometric reconstruction of the original tow microstructure.

Three geometric configurations were defined to isolate the influence of matrix filling state and gap-related defects (Figure 1). The fully filled configuration assumes complete PLA filling between adjacent equivalent filaments, shared fiber-matrix nodes, and perfect bonding. The gapped configuration contains prescribed inter-filament voids to represent under-impregnation or porosity and its expected weakening effect on transverse and shear-related responses. The gap-free configuration removes the inter-filament gaps and serves as a fiber-dominated limiting reference without explicitly resolving interfacial failure or contact behavior.OPEN IN VIEWER

In the fully filled configuration, shared nodes enforce displacement compatibility between the fiber and matrix phases. This treatment is computationally robust and suitable for extracting effective elastic constants, but it neglects local interfacial slip, imperfect bonding, debonding initiation, and frictional contact. Consequently, the local stress-transfer path is more constrained than that in a real printed CCF/PLA tow, which may further contribute to the conservative stiffness tendency, especially in transverse deformation modes. Because the objective of this section is to construct a reduced-order RVE for macroscopic effective-constant extraction, rather than to simulate interfacial failure or microscale damage initiation, no dedicated interfacial constitutive law was introduced for the fully filled case. ³² The influence of this idealization is therefore evaluated indirectly through the comparison among geometric configurations and through the subsequent comparison with the Tandon–Weng model and Hashin–Shtrikman bounds.

For the fully filled configuration, fiber volume fraction V_f was controlled by adjusting the filament pitch p. Because s denotes the side length of the equal-area square surrogate, the fiber volume fraction satisfies V_f = (s/p)². Three representative levels were evaluated (35%, 65%, and 95%), covering resin-rich to near-close-packed regimes (Figure 2). This design also provides an indirect assessment of fiber-spacing and interaction effects: as V_f increases, the matrix ligament between adjacent equivalent filaments becomes thinner, and the influence of the imposed square packing and shared-node constraint becomes more pronounced. Therefore, the V_f -dependent results are used not only to obtain engineering constants, but also to examine the applicability boundary of the simplified geometry under dense packing.OPEN IN VIEWER

The circular-to-square mapping necessarily induces local stress concentration at inclusion corners. This local field distortion can affect damage-initiation fidelity in the matrix and may exaggerate stress gradients near the fiber-matrix boundary. However, the present work targets effective elastic constants of the RVE. Under homogenization theory, effective stiffness is a volume-averaged quantity and is therefore less sensitive to localized stress singularities than local failure prediction. This consideration supports the use of the circular-to-square simplification for macroscopic stiffness prediction while enabling substantial computational savings. At the same time, the resulting transverse and shear responses should be interpreted with this geometric limitation in mind, especially when the model is applied to failure-sensitive or interface-dominated problems.

Constituent material properties

The composite system consists of continuous carbon fiber (TC33 1.5 K, TAIRYFIL; filament diameter ≈7 μm) and a PLA matrix (PolySonic™ PLA, Polymaker). Key properties are summarized in Table 1. In the finite-element model, the carbon filament was treated as transversely isotropic, with the filament axis defined as the 1-direction and the transverse directions defined as 2–3. Representative transversely isotropic parameters for high-strength PAN-based carbon fibers reported by Duan et al. ³³ were adopted to ensure a physically reasonable stiffness contrast, and the full parameter set is listed in Table 2. The PLA matrix was modeled as isotropic linear elastic with E = 3.5×10³ MPa and ν = 0.36, consistent with commonly used literature values. ³⁴

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Table 1.

Key constituent properties.

Material	Grade	Supplier	Elastic modulus, E (MPa)	Density, ρ (g/cm3)	Filament diameter, d (μm)
CCF	TC33 1.5 K	TAIRYFIL	2.3×105	1.8	7
PLA	PolySonic™	Polymaker	3.5×103	1.23	1.75×103

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Table 2.

Transversely isotropic filament parameters used in simulation.

E1 (MPa)	E2 (MPa)	E3 (MPa)	ν 12	ν 13	ν 23	G12 (MPa)	G13 (MPa)	G23 (MPa)
2.3 × 105	2.18 × 104	2.18 × 104	0.223	0.223	0.25	1.25 × 104	1.25 × 104	6 × 103

The above material treatment should be interpreted as an effective small-strain elastic approximation rather than a complete constitutive description of printed PLA. In practice, PLA may exhibit viscoelasticity, rate sensitivity, and temperature-dependent relaxation, especially during deposition or long-term loading. However, the present RVE analysis focuses on quasi-static extraction of effective elastic constants under infinitesimal deformation, where the objective is to quantify the stiffness contribution of the equivalent fiber arrangement rather than to reproduce time-dependent matrix relaxation. Therefore, the linear-elastic assumption provides a controlled baseline for comparing different V_f levels and geometric configurations. At the same time, neglecting viscoelastic compliance may make the matrix phase slightly stiffer than the real printed material, which can contribute to a conservative stiffness tendency in transverse and some shear-related responses.

Boundary conditions and extraction of effective constants

Node-to-node PBCs were applied to all RVEs, and effective engineering constants were obtained using a reaction force–displacement method. The EasyPBC plugin was used to generate node pairing and constraint equations, thereby reducing uncertainty associated with manual pairing on complex boundaries. The cuboid nominal dimensions Lx, Ly, and Lz were defined by the coordinate extrema of mesh nodes in each direction (Figure 3). Nodes on the six opposing faces were identified by EasyPBC and paired one-to-one based on non-normal coordinate differences. A tolerance of 1×10⁻⁵ mm was used to ensure unique pairing and geometric consistency. Furthermore, the regular rectangular geometry provided robust pairing behavior and mitigated failure modes common in irregular meshes (e.g., one-to-many pairing), while preventing spurious rigid-body motion or artificial constraints under periodicity enforcement.OPEN IN VIEWER

After PBCs were imposed (Figure 4), six external reference points (RP₁–RP₆) were introduced and coupled to the nodes of the corresponding loading faces via multi-point constraints. Independent base-displacement load cases were applied to these reference points to realize three uniaxial tension states (x, y, z) and three shear states (xy, xz, yz). Reaction forces were extracted and, via Hooke’s law, were used to back-calculate the effective stiffness matrix and engineering constants of the RVE.OPEN IN VIEWER

Together with the shared-node interface, the PBC formulation defines a fully bonded and periodically repeated equivalent microstructure. This treatment is required for stable homogenization and reproducible extraction of effective constants, but it also suppresses local interfacial slip, partial debonding, and microscale mismatch between adjacent filaments and matrix ligaments. Its influence is expected to be limited for E₁, which is dominated by the continuous carbon fibers, whereas transverse and shear-related constants are more sensitive to this idealized constraint. The resulting stiffness tendency is therefore interpreted together with the theoretical comparison presented in the section on theoretical analysis.

Meshing strategy

Mesh design was treated as a primary factor controlling both numerical stability and accuracy, particularly at V_f = 95%, where inter-fiber gaps become far smaller than typical element sizes. To maintain consistency and reproducibility across the full parameter space, the entire RVE was discretized using C3D6 (6-node linear wedge elements) with a global mesh size of 0.003 mm. Although C3D8R or C3D10 elements are often preferred to mitigate shear locking, C3D6 offers a decisive advantage under the present rectangular-array topology: the wedge geometry naturally accommodates the diagonal partitioning implied by the square packing and can generate topologically continuous, face-to-face node-corresponding meshes even in extremely narrow gaps, which is critical for successful PBC enforcement.

While C3D6 can exhibit shear-locking tendencies, the extraction of E₁ is dominated by normal stress under axial tensile loading; consequently, the influence of shear locking is comparatively limited for the axial modulus under PBCs and a near-pure normal-stress state. However, this numerical effect should not be completely neglected for transverse and shear loading cases. In particular, when the matrix ligament becomes narrow at high V_f, the combination of C3D6 discretization, regular square packing, and fully bonded displacement compatibility may restrict local deformation and thereby elevate the predicted transverse stiffness, with a component-dependent influence on the shear response. Therefore, the selected meshing strategy is treated as part of the same engineering compromise as the geometric simplification: it ensures convergence and PBC compatibility across the full V_f range, while the resulting transverse properties, together with some shear-related components, should be interpreted as conservative stiffness estimates rather than exact local-field predictions.

Theoretical prediction and error assessment

To quantify the systematic bias introduced by geometric, material, interface, and mesh simplifications, finite-element predictions were benchmarked against classical micromechanics. The Tandon–Weng model, based on Mori–Tanaka mean-field theory, was selected as the primary reference; it assumes non-interacting ellipsoidal inclusions randomly distributed in the matrix and is widely used to provide reasonable modulus estimates, sometimes tending toward a lower-bound trend. Relative error was evaluated by comparing FEM-predicted constants X_FEM against theoretical values X_TW. Given the square inclusion geometry, shared-node perfect bonding, linear-elastic PLA matrix, regular periodic packing, and C3D6 mesh topology, the transverse predictions, and in some cases shear-related predictions, may tend toward a stiffer response than those based on random or circular-inclusion assumptions.

Physical admissibility was further assessed using the Hashin–Shtrikman (HS) bounds, which provide theoretical limits for effective properties of two-phase composites. Predictions falling within the HS lower and upper bounds, or marginally approaching the upper-bound neighborhood when anisotropic packing and ordered microstructural constraints are considered, were treated as consistent with admissible effective behavior. In this framework, the finite-element RVE is not regarded as a fully resolved reconstruction of all local physical mechanisms, but as an idealized equivalent system for efficient stiffness extraction. The combination of transversely isotropic fibers, linear-elastic PLA, perfect bonding, and PBC-imposed periodicity is expected to preserve the dominant axial load-bearing response, while shifting the transverse response, and some shear-related responses, toward a stiffer estimation range. This treatment further defines the applicable range of the “100-to-1” downscaling strategy by relating the finite-element results to theoretical references and physical bounds, while making clear that the improved computational efficiency is achieved at the cost of certain idealized assumptions.

Validation of PBCs and solver stability

Under a consistent primary loading path (Figure 5), the fully filled configuration exhibited a smooth, strictly linear reaction force-displacement response over the entire loading range, with no nonphysical artifacts observed. The gapped configuration transferred transverse load less efficiently, producing a more compliant response with a reduced linear regime. The gap-free configuration displayed an anomalously high initial stiffness and localized deformation discontinuities because the inter-filament gaps were removed without explicitly resolving interfacial failure or contact behavior. These comparisons identify the fully filled configuration as the most mechanically consistent RVE for homogenization-based property extraction.OPEN IN VIEWER

Deformation compatibility under PBCs was confirmed for the fully filled RVE. Axial displacement followed a strictly linear gradient in the loading direction, transverse displacement showed a regular layered pattern, and strain transitioned smoothly between the fiber and matrix. Stress remained concentrated in the fiber phase while transmitting continuously through the matrix, with no spurious jumps or numerical oscillations indicative of incorrect node pairing (Figure 6).OPEN IN VIEWER

Across the six independent loading cases (Figure 7), reaction force-displacement curves remained highly linear and converged in a few solver iterations, with no stiffness-matrix singularities or unintended activation of contact-driven nonlinearity. This behavior supports the use of the fully filled RVE with PBCs for stable homogenization.OPEN IN VIEWER

Mesh convergence and element sensitivity

Mesh generation feasibility depended strongly on fiber volume fraction (V_f) in the square-surrogate and square-packing topology (Figure 8). At V_f = 35%, C3D8R, C3D10, and C3D6 all produced valid meshes. At V_f = 65%, C3D8R suffered severe element distortion and node-matching failure within narrow matrix channels, while C3D10 frequently triggered node-pairing “Mismatch” errors on periodic opposing faces. At V_f = 95%, the theoretical inter-fiber gap became far smaller than conventional element sizes; only C3D6, leveraging its wedge geometry, generated a topologically continuous mesh with strictly corresponding nodes along the diagonal regions of the square packing.OPEN IN VIEWER

These observations explain the use of C3D6 as the uniform meshing strategy introduced in Section 2.4. This choice represents an explicit engineering compromise: linear wedge elements can exhibit shear-locking-related stiffness elevation, but they provide the most stable node correspondence and PBC compatibility under dense square packing. In the present context, that risk is accepted to secure numerical convergence for the “100-to-1” downscaling strategy under dense packing. Mesh-size convergence further supported this selection (Figure 9): once the global mesh size reached 0.003 mm, the extracted engineering constants stabilized, and further refinement produced changes within an engineering-acceptable range.OPEN IN VIEWER

Extracted effective constants and hierarchical model comparison

Using the fully filled RVE with the selected meshing strategy, effective engineering constants were extracted at V_f = 35%, 65%, and 95% (Table 3) and compared with Tandon–Weng predictions based on Mori–Tanaka mean-field theory (Table 4). The calculated Hashin–Shtrikman lower and upper bounds were used as admissibility references for the FEM results; all extracted effective constants fell within the corresponding admissible intervals, although the detailed bound values are not tabulated here for brevity.

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Table 3.

Effective engineering constants from FEM homogenization (fully filled RVE).

Vf (%)	E1 (MPa)	E2 (MPa)	E3 (MPa)	G12 (MPa)	G13 (MPa)	G23 (MPa)	ν 12	ν 13	ν 23
35	8.262e4	7.566e3	7.56e3	2.489e3	2.489e3	1.942e3	0.305	0.305	0.389
65	1.508e5	1.224e4	1.223e4	4.379e3	4.379e3	2.747e3	0.274	0.274	0.315
95	2.23e5	2.0570e4	2.057e4	1.0889e4	1.089e4	5.415e3	0.227	0.227	0.251

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Table 4.

Tandon–Weng predictions for effective engineering constants.

Vf (%)	E1 (MPa)	E2 (MPa)	E3 (MPa)	G12 (MPa)	G13 (MPa)	G23 (MPa)	ν 12	ν 13	ν 23
35	8.279e4	6.198e3	6.198e3	2.311e3	2.311e3	1.959e3	0.307	0.307	0.491
65	1.507e5	9.839e3	9.839e3	4.173e3	4.173e3	3.007e3	0.266	0.266	0.425
95	2.187e5	1.883e4	1.883e4	1.003e4	1.003e4	5.315e3	0.229	0.229	0.291

Figure 10 compares selected FEM-homogenized engineering constants with Tandon–Weng predictions across V_f = 35%–95%. E₁ exhibits close agreement with the theoretical prediction, with deviations remaining within approximately 2%, confirming that the axial response is mainly controlled by the continuous carbon fibers and is only weakly affected by the cross-sectional idealization. By contrast, the transverse moduli show a more evident discrepancy. At V_f = 35%, E₂ increases from 6198 MPa in the Tandon–Weng prediction to 7566 MPa in the FEM result, corresponding to a deviation of approximately +22.1%; E₃ shows a similar deviation of approximately +22.0%. This deviation is mainly related to the different microstructural assumptions of the two methods. The Tandon–Weng model treats fibers as non-interacting ellipsoidal inclusions embedded in a matrix and therefore allows a more relaxed transverse deformation mode. Under PBCs and shared-node compatibility, the matrix ligaments between adjacent equivalent filaments are more strongly constrained, and the corners of the square inclusions further intensify local stress concentration. As a result, the transverse load-transfer path becomes stiffer than that predicted by the mean-field model.OPEN IN VIEWER

The deviation in ν₂₃, as listed in Tables 3 and 4, should be interpreted together with this transverse constraint. At V_f = 35%, ν₂₃ decreases from 0.491 in the Tandon–Weng prediction to 0.389 in the FEM result, giving a difference of approximately −20.8%. This trend is not associated with the axial load-bearing prediction, but reflects the restricted lateral deformation of the 2–3 plane in the square-packed RVE. In a matrix-rich configuration, transverse deformation and Poisson coupling are still strongly governed by the matrix phase. Once the circular fibers are represented by square inclusions and the fiber-matrix interface is tied by shared nodes, local matrix distortion and lateral contraction are suppressed compared with the random or circular-inclusion assumption in the Tandon–Weng model. Therefore, E₂ and E₃ tend to increase, whereas ν₂₃ tends to decrease. This opposite deviation trend is consistent with a more constrained transverse deformation mode rather than with numerical instability. As V_f increases, the transverse response gradually becomes more fiber-dominated, and ν₂₃ moves closer to the intrinsic transverse Poisson behavior of the carbon fiber phase; consequently, the relative discrepancy in ν₂₃ is reduced at V_f = 95%.

The result at V_f = 95% should be regarded as a near-close-packed limiting case rather than a direct reconstruction of an experimentally typical CCF/PLA tow. At this fiber content, the PLA matrix ligament becomes extremely thin, and the assumption of a continuous linear-elastic PLA phase is physically idealized. In real FDM-printed CCF/PLA, such a dense filament arrangement may involve incomplete impregnation, local voids, resin discontinuity, or direct fiber-fiber contact. The fully filled RVE at V_f = 95% is therefore introduced mainly to examine the numerical robustness and applicability boundary of the proposed “100-to-1” downscaling strategy under extreme packing. Its matrix continuity should be understood as a theoretical upper-bound condition for stiffness extraction, not as a claim that a continuous and fully effective PLA network necessarily exists in the real material at this fiber volume fraction.

Despite this stiffness tendency, the FEM results remained within the admissible range defined by the Hashin–Shtrikman limits. The transverse moduli E₂ and E₃ were consistently higher than the Tandon–Weng predictions, whereas G₁₂ and G₁₃ showed only moderate positive deviations and G₂₃ remained close to, or slightly below, the theoretical reference at some V_f levels. These results indicate that the proposed strategy provides highly accurate axial predictions while giving a conservative stiffness estimate mainly for transverse deformation under the chosen square inclusion geometry, ordered square packing, shared-node interface, and C3D6 meshing strategy. Therefore, the comparison defines the applicable range of the simplified RVE: it is reliable for axial-dominated effective stiffness extraction, while transverse Poisson coupling, near-close-packed matrix continuity, and component-dependent shear response should be interpreted within the stated idealized assumptions.

Single-track specimen fabrication and tensile testing

CCF/PLA single-track specimens were fabricated on a self-developed desktop 3D-printing platform using a CoreXY motion architecture and a customized brass nozzle (1.0 mm orifice) to preserve tow integrity during deposition (Figure 11(a)). Five specimens (S-01–S-05) were printed with nominal dimensions of 150 mm × 1 mm × 0.2 mm (Figure 11(b)). To reduce the influence of geometric dispersion, the stress calculation did not rely on the nominal cross-section. Instead, each specimen’s width w and thickness t were measured within the gauge section (L₀ = 50 mm), and the effective load-bearing area was defined as A = w × t.OPEN IN VIEWER

The mean cross-sectional area was 0.207 ± 0.012 mm², and all subsequent stress calculations and FE geometry were based on these measured statistics (Table 5). Uniaxial tensile tests were conducted on a computer-controlled universal testing machine (CMT4204) at a crosshead speed of 2 mm/min (Figure 11(e)). Because the single-track specimens were extremely small and prone to brittle fracture, an extensometer was impractical; therefore, deformation was measured using crosshead displacement. Accordingly, the recorded displacement ΔL includes not only specimen elongation but also grip seating and clearance elimination, as well as elastic deformation of the loading train. This measurement definition is essential for interpreting the observed stiffness discrepancy.

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Table 5.

Measured geometry and effective cross-sectional area of printed CCF/PLA single-track specimens.

Specimen	L0 (mm)	w (mm)	t (mm)	A = w × t (mm2)
S-01	50.00	1.02	0.21	0.214
S-02	50.00	0.98	0.20	0.196
S-03	50.00	1.05	0.19	0.199
S-04	50.00	1.01	0.20	0.202
S-05	50.00	1.03	0.22	0.226
Mean	50.00	1.018	0.204	0.207
SD	​	0.026	0.011	0.012

Fiber content measurement by chemical dissolution method

Fiber content is the key physical variable linking the microscale model to macroscopic behavior. To avoid uncertainty from nominal feedstock composition, an ASTM D3171-referenced chemical dissolution method was used to remove the PLA matrix from fractured tensile specimens. Chloroform (Figure 11(d)) was employed as the solvent to dissolve PLA and recover the carbon fibers for quantification of the true fiber volume fraction V_f. Five parallel specimens were weighed before and after dissolution using a precision balance (FA2004, 0.1 mg resolution) (Figure 11(d)), and V_f was calculated using constituent densities. The mean fiber mass fraction was W_f = 42.18%, corresponding to V_f = 33.95% with a standard deviation of 0.24%. This experimentally measured V_f provides a physically grounded input for the subsequent single-track tensile model. In addition, possible porosity or incomplete impregnation may contribute to the lower experimental stiffness even when the fiber content is controlled Table 6.

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Table 6.

Weighing records and calculated fiber content from the chemical dissolution method.

Specimen	Crucible mass (g)	Crucible + Sample Mass (g)	Crucible + Fiber Mass (g)	Composite material mass (g)	Fiber mass (g)	Fiber mass fraction (wt%)	Fiber volume fraction (vol%)
1	27.6517	28.8223	28.1460	1.1706	0.4943	42.23	33.99
2	33.1705	34.2942	33.6439	1.1237	0.4734	42.13	33.91
3	25.7985	26.6121	26.1426	0.8136	0.3441	42.29	34.05
4	29.5251	30.5121	29.9373	0.9870	0.4122	41.75	33.57
5	26.7328	27.6947	27.1415	0.9619	0.4087	42.49	34.23
Mean	​	​	​	​	​	42.18	33.95
SD	​	​	​	​	​	​	0.24

Single-track tensile FE model and simulation settings

A single-track tensile FE model was built in ABAQUS/Explicit to provide a controlled axial-response comparison with the experiments. The specimen geometry was constructed using the mean dimensions from Table 5. The matrix was discretized with solid elements, while the fiber tow was simplified using the “100-to-1” downscaling strategy as an array of embedded truss elements. The model fiber volume fraction was set to approximately 32%, which is close to the measured V_f = 33.95% within the geometric discretization of the embedded-truss representation. Thus, the simulation input preserves the experimentally measured fiber-content level without relying on the nominal feedstock composition. The physical assumptions were intentionally idealized to isolate the intrinsic material response: a perfect interface (fully compatible fiber–matrix displacement) and ideal clamping (pure axial displacement at the loading end, excluding any external system compliance).

For constitutive behavior, the matrix employed an elastoplastic damage-evolution fracture criterion with element deletion at high strain, whereas the fiber phase used a maximum-stress failure criterion. The explicit dynamic setting targets brittle fracture and post-peak energy release, while quasi-static loading was verified by monitoring kinetic and internal energies (ALLKE and ALLIE) and maintaining a very low kinetic-to-internal energy ratio to suppress inertial influence on the peak prediction (Figure 12).OPEN IN VIEWER

Experimental–simulation force-displacement comparison and discussion

Force-displacement curves from experiments (S-01–S-05) and the FE simulation (S-06) were plotted together in Figure 13 and evaluated in terms of initial stiffness, peak load, and failure mode. The simulation reproduces the characteristic progression of a linear rise, peak load, and abrupt load drop associated with brittle failure.OPEN IN VIEWER

The predicted peak load was F_max = 231.58 N, compared with the experimental mean F_max = 209.26 N, yielding a relative error of +10.67% (Table 7). Considering the specimen-to-specimen scatter inherent to additively manufactured continuous-fiber composites, this level of agreement supports the model’s capability to predict the axial ultimate capacity of the single-track specimen.

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Table 7.

Summary of experimental–simulation comparison metrics for single-track tensile response.

Metric	Experiment (n = 5), mean ± SD	Simulation (S-06)	Relative error
K0 (N/mm)	615.96 ± 25.47	750.74	+21.88%
Fmax (N)	209.26 ± 3.76	231.58	+10.67%
ΔLmax (mm)	0.3457 ± 0.0159	0.3261	−5.68%

The initial stiffness shows a larger discrepancy: the simulated stiffness K₀ = 750.74 N/mm exceeds the measured apparent stiffness K_total = 615.96 N/mm by 21.88%. Because experimental displacement was measured by crosshead motion, K_total includes both specimen deformation and deformation of the testing system, including the frame, load cell, grips, and drive train. To quantify this contribution, a series spring model was introduced to correct the experimental stiffness. Based on the configuration of the CMT4204 system, the effective machine stiffness was approximated as K_machine = 10 kN/mm, corresponding to a machine compliance of C_machine = 1/K_machine. With springs in series, the total compliance satisfies C_total = C_sample + C_machine, which is equivalently expressed in stiffness form as K_sample = (K_total × K_machine)/(K_machine − K_total). Substituting K_total = 615.96 N/mm and K_machine = 10 kN/mm yields a corrected specimen stiffness of K_sample ≈ 656.4 N/mm. This correction partly reduces the apparent stiffness discrepancy and supports a consistent interpretation: the FE prediction approaches an idealized upper estimate of intrinsic axial stiffness under perfect gripping and zero external compliance, whereas the uncorrected experimental stiffness represents an apparent lower estimate measured through a compliant test system.

It should also be noted that the corrected stiffness remains lower than the simulated stiffness of 750.74 N/mm, with a residual difference of approximately 14.37% when normalized by the corrected experimental stiffness. This remaining gap cannot be attributed to system compliance alone. Instead, it is more reasonably associated with the idealized assumptions used in the single-track FE model. First, the embedded-truss representation preserves the axial load-carrying role of the fiber tow, but it does not explicitly reconstruct real filament packing, fiber waviness, resin-rich zones, or microscopic voids in the printed track. Second, the interface between the embedded fiber phase and the matrix is treated as fully compatible, whereas the real printed CCF/PLA specimen may contain incomplete impregnation, local debonding, and nonuniform stress transfer. Third, the boundary condition in the simulation corresponds to ideal axial clamping, while the experiment may still involve residual grip compliance, slight misalignment, and local stress redistribution near the clamped region. Therefore, after system-compliance correction, the remaining stiffness difference should be interpreted as the combined effect of idealized geometry, neglected porosity, simplified interface behavior, and simplified boundary conditions.

The present single-track validation mainly evaluates the axial-dominated response of the “100-to-1” downscaling strategy. Since the experimental displacement was obtained from crosshead motion and no transverse strain measurement or digital image correlation was used, ν₂₃ and lateral contraction of the single-track specimens were not directly extracted from the tensile tests. Accordingly, the single-track comparison should not be overinterpreted as a direct validation of transverse Poisson behavior. The discussion of ν₂₃ and lateral deformation is instead supported by the RVE-level homogenization results in Section 3.3, where the square-packed equivalent geometry was shown to restrict deformation in the 2–3 plane and to produce a conservative transverse response. Thus, the tensile experiment confirms that the equivalent strategy can capture the dominant axial load-bearing capacity and brittle failure signature. By contrast, transverse deformation, ν₂₃, and local lateral constraint cannot be directly validated by the present single-track tensile test and should still be interpreted based on the RVE-level assumptions regarding packing, interface compatibility, and boundary constraints. To assess failure-mechanism consistency and local stress transfer, field variables around the peak load were examined (Figure 14).OPEN IN VIEWER

The LE₁₁ field shows that axial strain localizes within the gauge section rather than at the clamped ends, indicating that the idealized boundary condition does not artificially trigger premature end failure. The von Mises stress field further shows that stress concentration develops along the embedded fiber path and adjacent matrix region before element deletion occurs. This stress pattern is consistent with axial load being primarily carried by the continuous fiber phase, while the surrounding PLA matrix provides lateral constraint and transfers stress between the embedded reinforcement and the external specimen boundary. However, because the fiber tow is represented by embedded truss elements rather than explicitly resolved circular filaments, the model cannot describe the detailed stress distribution around each individual fiber or the local debonding process at a real fiber-matrix interface. Therefore, Figure 14 supports the consistency of the macroscopic failure location and axial stress-transfer path, but it also reflects the limitation of the simplified packing representation for local transverse and interface-dominated stress fields.

Overall, the experimental loop supports the single-track extension of the microscale framework within its intended axial-dominated validation scope. The measured V_f ≈ 34% provides physically grounded simulation inputs, and the ultimate-load prediction error of approximately 10% indicates that the model can reasonably capture the load-bearing capacity of the printed single-track specimen. The initial stiffness discrepancy of 21.88% is partly reduced and physically interpreted after introducing the system-compliance correction, confirming that testing-machine and grip compliance are important contributors to the apparent experimental compliance. Nevertheless, the remaining 14.37% stiffness gap indicates that the FE model still tends toward an idealized upper estimate of the axial stiffness. This residual difference is most plausibly related to the combined influence of idealized fiber geometry, neglected porosity, perfect fiber-matrix compatibility, and simplified axial boundary conditions. Thus, the validation confirms the engineering utility of the “100-to-1” strategy for axial stiffness and ultimate-load prediction, while transverse deformation, ν₂₃, and local fiber-scale stress fields should be interpreted within the limitations of the simplified equivalent model.