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On the effect of isotropic and anisotropic dissipative response functions with associated and non-associated flow on the inelastic behaviour of polymeric composites

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Pith's one-line read Pressure-dependent yield functions with an associated flow rule predict physically wrong tensile transverse strains under shear-dominated combined loading of fibre composites; replacing the flow rule with a non-associative deviatoric…

desk verdict A useful, honest side-by-side comparison of three Drucker-Prager-type composite plasticity models with associated vs. deviatoric non-associated flow; the main conclusion is real but conditional on the assumed zero-dilatancy flow potential. read the letter →

arxiv 2412.08656 v1 pith:XEPVXNZR submitted 2024-11-27 cond-mat.soft cond-mat.mtrl-sci

Swaroop Gaddikere Nagaraja , Thomas Antretter , Clara Schuecker This is my paper · ORCID
classification cond-mat.softcond-mat.mtrl-sci
keywords anisotropicplasticityfibre-reinforcedcompositesDrucker-Prageryieldfunctionassociatedflowrulenon-associativepressure-dependenttransverseisotropyAS4/PEEK
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks whether the choice of flow rule inside pressure-dependent plasticity models of unidirectional fibre composites changes the predicted inelastic response, and shows that it does. Three Drucker-Prager-type models—an isotropic one, a mapped-tensor one, and an anisotropically extended one—are calibrated against pure-shear and transverse-compression data for a carbon-fibre/PEEK composite and then tested on biaxial load paths. With the associated (normality) flow rule, the models produce physically unrealistic transverse strains under shear-dominated combined stress states: tensile strains for Models I and III, and a spurious compressive strain for Model II under pure shear. Switching to a non-associative flow rule with a purely deviatoric plastic potential corrects the flow direction, but the shear response under transverse compression remains overestimated. The conclusion is that pressure sensitivity of the yield function is necessary while the associated flow rule is not usable for shear-dominated loads, and that choosing the right flow rule needs additional biaxial yield-surface data.

What carries the argument

The load-bearing object is the transversely isotropic Drucker-Prager-type yield function $\chi = \kappa p + \|\Sigma\|_{\mathbb{C}_8} - \sqrt{2/3}[y_0 - \beta] + \frac{b}{2}\|\beta\|^2_{\mathbb{C}_9}$, where $p$ is the hydrostatic pressure in the effective stress, $\mathbb{C}_8$ is a fourth-order deviatoric projection tensor that selects shear modes while leaving the fibre direction elastic, and $m = a \otimes a$ encodes the fibre orientation. The associated flow rule generates plastic strain rates from $\partial_\sigma \chi$ and therefore carries the pressure term $\frac{\kappa}{3}(1-m)$ into the flow direction; this is the term that produces tensile transverse strain under shear. The non-associative machinery is a second plastic potential $\phi = \chi|_{\kappa=0}$ that is purely deviatoric, so the plastic strain rate is governed only by $\mathbb{C}_8 : \Sigma / \|\Sigma\|_{\mathbb{C}_8}$. Comparing Models I, II, and III isolates what the choice of isotropic versus anisotropic dissipative response adds on top of the flow-rule choice.

What would settle it

Measure the plastic transverse strain increment during pure in-plane shear of the AS4/PEEK composite: the non-associated models predict zero transverse plastic strain because the plastic potential is purely deviatoric, so any measured transverse plastic strain or measurable dilatancy under shear with compression would refute the assumed pressure-independent flow potential.

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Extended reading notes

Core claim

The central claim is that the negative slope of Drucker-Prager-type yield surfaces in the pressure–shear plane breaks the normality assumption: an associated flow rule turns pure shear and shear-dominated combined stress states into non-physical transverse strain, with Models I and III reporting tensile transverse strain where experiments show compression and Model II reporting a small spurious compressive strain under pure shear. The authors demonstrate this defect in all three model formulations and then show that a non-associative flow rule built from the same yield function with the pressure coefficient set to zero in the plastic potential ($\kappa=0$ in Eqs. (23), (33), (41)) restores the experimentally observed flow direction under pure shear. However, on the compression-preload biaxial path the non-associated models still overestimate the shear response, which the authors attribute to the pressure-independent flow rule and possibly to matrix dilatation due to crazing. The discovery is therefore a separation of effects: the flow-rule choice controls the direction of plastic strain, while the pressure coupling controls the magnitude of the shear–compression interaction, and neither the associative nor the non-associative form alone reproduces the full biaxial response.

Load-bearing premise

The non-associated models remove pressure effects from the plastic flow direction entirely, assuming the polymer matrix never changes volume while it flows; if real PEEK expands under shear with compression, for instance through crazing, the remaining overprediction of the shear response may be an artifact of that choice.

Editorial extensions

If this is right

  • Whenever a Drucker-Prager-type yield function with a negative pressure slope is used, an associated flow rule is unreliable for shear-dominated combined loads because it generates non-physical tensile transverse strains.
  • The non-associative deviatoric plastic potential corrects the plastic strain direction under pure shear and shear-preload paths while preserving the calibrated pure-shear and transverse-compression responses.
  • The remaining overprediction of shear response under transverse compression is shared by all three model structures, so a pressure-independent plastic flow alone cannot close the gap.
  • Anisotropic plastic response functions track the experimental biaxial response more closely than isotropic ones, but they still overpredict at the highest preloads.
  • Additional experimental data on the $\tau_{12}\to-\sigma_{22}$ biaxial path, which maps the evolution of the yield surface and plastic flow potential, are needed to decide between associative and non-associative formulations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural extension the paper leaves implicit is a family of plastic potentials with a pressure coefficient between zero and the associative $\kappa$; the observed overprediction brackets the admissible value from above, so a small positive pressure coupling in the flow potential may preserve the corrected flow direction while reducing the shear-under-compression error.
  • The associated-flow inconsistency is a property of the negative yield-surface slope in the pressure–shear plane, so the same problem should appear in other cohesive pressure-dependent materials such as glassy polymers, foams, and granular solids under combined shear and compression.
  • A testable rate-dependent extension is to replace the rate-independent setting with a Perzyna-type viscous regularization using a low pressure slope, as the paper suggests, and check whether the overestimated shear-under-compression response relaxes towards the experiments at finite strain rates.
  • The paper's comparison implies that a purely isotropic pressure-dependent yield function cannot reproduce the biaxial response; an anisotropic plastic potential built from the same projection tensors as the yield function is the most direct cure, and its pressure coefficient should be fit to the missing $\tau_{12}\to-\sigma_{22}$ data.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper develops three Drucker-Prager-type elastoplastic constitutive models for unidirectional fibre composites — a modified isotropic Drucker-Prager model (Model-I), a mapped-tensor model following Car, Oller and Oñate (Model-II), and an anisotropic representation-theorem extension (Model-III) — and implements each in both associated and non-associated flow variants. The models are calibrated against pure in-plane shear and transverse compression of AS4/PEEK and then compared with experimental biaxial shear–compression load paths. The central finding is that associated flow produces physically inconsistent tensile transverse strains under pure shear and shear-dominated combined loads, whereas non-associated flow with a pressure-independent plastic potential removes this inconsistency but still overpredicts the shear response under transverse compression preloads. The authors acknowledge the residual deviation and attribute it to the assumed pressure-independent flow potential and to the lack of dilatancy data.

Significance. The paper's main claim is practically relevant: if confirmed, it shows that associated Drucker-Prager-type plasticity should not be used for shear-dominated combined loading of polymeric composites, and that a non-associated formulation is a simple remedy. A clear strength is that the biaxial load paths used for evaluation were not used for calibration; only pure shear and transverse compression data enter the parameter identification, so the comparisons in Figs. 5 and 6 are genuine predictions. The unified implementation and comparison of three model structures in one numerical framework is also useful. The authors are transparent about the residual deviations and about the need for additional experimental data. However, the quantitative claim is weakened by the absence of error metrics and by the untested zero-dilatancy assumption in the flow potentials, so the significance is conditional on strengthening those points.

major comments (2)
  1. [Section 5.2 and Figs. 5-6] The central conclusion that non-associated flow circumvents the inconsistencies of associated flow rests on the specific choice of a pressure-independent plastic potential, κ̃ = 0 in Eqs. (23), (33) and (41). This is a constitutive assumption, not a measured property; the experiments of Ref. [3] are not used to determine the plastic strain direction or the inelastic volume change. Section 5.2.4 itself concedes that crazing or other dilatancy in the polymer matrix would require a pressure-dependent flow potential, and the Conclusions state that the choice of flow rule is unclear without additional biaxial data. The residual overprediction of shear-with-compression (Figs. 5b and 6a) may therefore be an artifact of the assumed potential rather than an intrinsic limitation of non-associated flow. I ask the authors to either determine κ̃ (or, equivalently, the plastic strain direction) from an independent measurement, or to report a sensitivity study over κ̃ in the range from 0 to κ and show that the qualitative correction in Fig. 4 and the deviations in Figs. 5 and 6 are robust within that range.
  2. [Section 5.2 and Figs. 5-6] The manuscript repeatedly describes the comparison with experiments as 'quantitative' (abstract, Section 5) but nowhere reports error metrics, scatter bands, or uncertainty quantification. Judgments such as 'significant deviations' (Section 5.2.3) and 'slightly overpredicted' (Section 5.2.3) are based on visual inspection. Because the main conclusion distinguishes between acceptable and unacceptable model behavior, the absence of quantitative agreement measures is load-bearing. Please include a deviation measure per load path (e.g., RMSE or maximum relative error in stress and strain) and, if available, experimental scatter from Ref. [3]. This would also make it possible to judge whether the residual overprediction is small model misspecification or evidence against the non-associated structure.
minor comments (4)
  1. [Figure 4 caption] The caption states that the corrected response is shown in panels (b), (d) and (e), but panel (e) already displays Model-III-a; the panel for the corrected Model-III-b response is presumably (f). Please correct the panel references.
  2. [Section 5.2.4] The opening sentence of the Discussion, 'the predictions are in excellent agreement with the experimental results whilst using an associative flow rule', conflicts with the earlier statements of 'erroneous predictions' and 'not satisfactory' in Sections 5.2.1 and 5.2.2. Please clarify which models and load paths are meant.
  3. [Section 4.2.3] The convexity condition 'a1−2 ≥ 0' appears to be a typographical error or is at least unclear, since a1 has units of inverse stress squared from Eq. (39). Please state the intended condition on a1 and a2 in a dimensionally consistent form.
  4. [Section 5.2.3] For Model-II, the text notes that the non-associative predictions are 'largely similar' to the associative ones. Since a change of flow potential would normally affect the plastic strain direction significantly, a brief explanation of why the transformation mapping suppresses this difference would help the reader reproduce and trust the result.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: biaxial predictions are external to the calibration, and the κ=0 non-associated flow potential is an openly stated constitutive choice rather than a fitted or concealed input.

full rationale

The paper calibrates each model only against the standard in-plane shear and transverse compression responses (Fig. 1; Tables 1–3) and then tests the calibrated models against biaxial τ12→−ε22 and −σ22→γ12 load paths not used in the fit, so the central quantitative comparison is independent of the calibration inputs. The non-associated variants are constructed by setting κ=0 in the flow potential (Eqs. 23, 33, 41), which forces a deviatoric plastic-flow direction; the resulting vertical flow direction under pure shear (Fig. 4) is therefore a direct consequence of the chosen ansatz, not an empirical prediction. The paper does not hide this: it labels the potential as 'chosen following [31]' and, in Section 5.2.4, attributes the residual overprediction of shear-with-compression to 'the choice of a pressure-independent plastic flow potential'. Material parameters inherited from the authors' previous work [16] supply constitutive coefficients and the same external experimental data [3] are used for validation; no load-bearing uniqueness theorem or self-citation is invoked. The derivation chain is thus self-contained against the external benchmark data, with the κ=0 assumption as a clearly flagged limitation rather than a circular step.

Assumptions & free parameters 16 free parameters · 9 assumptions · 0 invented entities

The models rely on a standard pressure-dependent plasticity framework, with material parameters calibrated to pure shear and transverse compression data from [3]. The most consequential added assumption is the pressure-independent plastic potential used for non-associated flow, which is chosen to eliminate the reported inconsistency rather than derived from independent measurements. No new physical entities are introduced.

free parameters (16)
  • Coefficient of hydrostatic pressure kappa (Model-I) = 0.9497 (associative), 1.105 (non-associative)
    Calibrated to pure shear and transverse compression experiments [3] following the procedure in [16]; differs between flow-rule variants, so model comparison is partly confounded.
  • Initial yield stress y0 (Model-I) = 10.6 MPa
    Calibrated to experimental shear and compression curves.
  • Hardening modulus h (Model-I) = 237.9 MPa
    Calibrated to the hardening part of the experimental curves.
  • Hardening exponent n (Model-I) = 0.249
    Calibrated to the curvature of the hardening response.
  • Coefficient of hydrostatic pressure kappa (Model-II) = 1.931 (associative), 1.917 (non-associative)
    Calibrated in the fictitious isotropic space to experimental curves [3].
  • Initial yield stress y0 (Model-II) = 20.5 MPa
    Calibrated to experimental shear and compression curves in the fictitious isotropic space.
  • Hardening modulus h (Model-II) = 415.7 MPa
    Calibrated to the hardening response.
  • Hardening exponent n (Model-II) = 0.241
    Calibrated to the curvature of the hardening response.
  • Anisotropic space yield strength Y22 (Model-II) = 158.6 MPa
    Yield strength used to build the space transformation tensor; calibrated to experimental data.
  • Isotropic space yield strength Ybar (Model-II) = 158.6 MPa
    Yield strength of the fictitious isotropic space, chosen from the matrix-dominated transverse response.
  • Transverse compressive yield stress y22c (Model-III) = 24.6 MPa (associative), 27.4 MPa (non-associative)
    Used in Eq. (39) to determine kappa; calibrated to transverse compression data.
  • In-plane shear yield stress y12 (Model-III) = 9.41 MPa
    Used in Eq. (39) to determine a1; calibrated to in-plane shear data.
  • Transverse shear yield stress y23 (Model-III) = 10.66 MPa
    Used in Eq. (39) to determine a2 and kappa; calibrated to transverse shear data.
  • Hardening modulus h (Model-III) = 177.5 MPa
    Calibrated to the hardening response.
  • Hardening exponent n (Model-III) = 0.246
    Calibrated to the curvature of the hardening response.
  • Pre-strain alpha_bar = 1e-12
    Chosen as a numerical regularization in Eq. (1); the paper states it has negligible effect on results.
assumptions (9)
  • domain assumption The total strain admits an additive decomposition into elastic and plastic parts, with symmetric second-order plastic strain and hardening variables as internal state variables.
    Invoked in Section 2, Eq. (1), as the basis of the infinitesimal plasticity framework.
  • domain assumption The yield function is of generalized Drucker-Prager type, linear in hydrostatic pressure and with a Hill-type deviatoric norm.
    Stated in Eq. (4); this pressure dependence is the source of the reported non-physical associated-flow response.
  • standard math The transversely isotropic symmetry of the composite is fully characterized by the structural tensor m = a tensor a, and scalar invariants from the integrity basis in Eq. (11) generate the response functions.
    Used in Section 3 to construct the anisotropic potential and Model-III; these are established representation theorems [42,47-49].
  • domain assumption Flow rule and hardening evolution follow the generalized normality condition, i.e., associated flow, unless replaced by a separate plastic potential.
    Eqs. (5) and (8); the normality assumption is what produces tensile transverse strains under shear-dominated loads.
  • ad hoc to paper For non-associated flow, the plastic potential is assumed to have the same form as the yield function but with different coefficients, and in Models I-b, II-b, III-b it is taken as pressure-independent, with kappa set to zero in phi.
    Eqs. (23), (33), (41); this assumption directly enforces the corrected flow direction and is identified in Section 5.2.4 as possibly omitting dilatancy or crazing.
  • ad hoc to paper Kinematic hardening is neglected: C0=0 and b1=b2=0.
    Stated in Sections 4.1 and 4.3 because relevant experimental data are unavailable; this affects unloading and cyclic response but is less central for the monotonic biaxial paths considered.
  • domain assumption A single C3D8 hexahedral element under homogeneous stress represents the material response.
    Section 5; no structural localization or mesh-size effects are considered.
  • domain assumption The material response is rate-independent.
    Remark 1; rate-dependence is discussed only as a possible extension.
  • domain assumption For Model-II, a fictitious isotropic space exists and is connected to the real anisotropic space by fourth-order transformation tensors based on yield strength tensors.
    Section 4.2.2 after [18,19]; the transformation tensor is modified from the original Y tensor Y inverse form.

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Cite this review

Pith. "Pith review of On the effect of isotropic and anisotropic dissipative response functions with associated and non-associated flow on the inelastic behaviour of polymeric composites." pith.science (2026). https://pith.science/paper/XEPVXNZR

@misc{pith2026241208656,
  author       = {Pith},
  title        = {Pith review of: On the effect of isotropic and anisotropic dissipative response functions with associated and non-associated flow on the inelastic behaviour of polymeric composites},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XEPVXNZR}},
  note         = {Machine review of arXiv:2412.08656}
}
read the original abstract

This article investigates the effect of using isotropic and anisotropic plastic response functions in the analysis of the elastic-plastic response of unidirectional fibre composites on the meso-scale. Three model problems that use a Drucker-Prager-type pressure-dependent yield function are considered to simulate the non-linearities exhibited by a composite material. A further core ingredient is the analysis of a canonical and non-conventional constitutive structure, with respect to associated and non-associated flow response, where the use of latter is motivated by the physical inconsistencies induced by the former under shear dominated loads. These models are evaluated quantitatively by comparison to experimental data.

Figures

Figures reproduced from arXiv: 2412.08656 by the authors.

Figure 1
Figure 1. Calibration results. Comparison of the experimental data [3] and the three meso models responses for (a) in-plane shear and (b) transverse compression load. 5. Numerical simulations. The proposed models are implemented as user subroutines (UMAT) in ABAQUS, a gen￾eral purpose non-linear finite element program documented in [58]. The subsequent nu￾merical simulations demonstrate the applicability and predictive capabi… view at source ↗
Figure 2
Figure 2. Predictions for biaxial loads. Comparison of the experimental and associative model responses for the τ12 → −ε22 loading path. an observable over-prediction by the model. In comparison with the previous case, the agreement with the experimental response is not satisfactory, as evident from [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Predictions for biaxial loads. Comparison of the experimental and associative model responses for the −σ22 → γ12 loading path [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Predictions for biaxial loads. Erroneous predictions of the associated flow response and corrected response by the non-associative flow rule for shear dominated loads. for the compressive response. In comparison with Model-I-a, the predicted shear strains are much lowe…
Figure 5
Figure 5. Figure 5: Predictions for biaxial loads. Comparison of the experimental and non￾associative model responses for the τ12 → −ε22 loading path. seen in [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: Predictions for biaxial loads. Comparison of the experimental and non￾associative model responses for the −σ22 → γ12 loading path. predicted for load paths ○05 –○08 [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.