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Potential-based versus non potential-based cohesive models accounting for loading and unloading with application to sliding elastic laminates

Francesco Freddi, Filippo Riva

TL;DR

The paper develops a rigorous, unified framework to compare potential-based (variational) and non potential-based (equilibrium) cohesive-zone models in a planar, quasistatic setting of two elastic laminates with a mixed-mode anisotropic cohesive interface. It introduces an intrinsic construction method for cohesive energies $\Phi$ and tensions $\mathcal{T}$ that incorporate loading and unloading by starting from a purely loading description $\Psi$ or tension $\mathcal S$, and demonstrates that variational models are limited to special cases (e.g., equal directional energies $\Phi_1=\Phi_2$) or restrictive unloading paths, while non-variational models accommodate general loading/unloading paths. Representative 2D examples show non-trivial loading/unloading paths where non-variational models remain physically consistent, whereas variational formulations can fail or produce non-physical responses. The authors establish well-posedness by proving existence of energetic solutions via a minimizing-movements scheme and of equilibrium solutions via Kakutani fixed points, thus providing a rigorous foundation for comparing these two modeling paradigms in anisotropic cohesive interfaces. The results have practical implications for modeling sliding laminates and similar cohesive interfaces, enabling robust descriptions across loading, unloading, and mixed-mode regimes in small-strain elasticity.

Abstract

A rigorous unified perspective of cohesive zone models is presented, including and comparing potential-based and non potential-based formulations, and encompassing known examples studied in literature. The main novelty of the work consists in the natural inclusion of loading and unloading effects in a general mixed-mode framework, incorporated through an intrinsic construction of energy densities or tensions. The proposed mathematical investigation identifies and proves the limitations of variational models with respect to non-variational ones, the latter yielding a feasible description of real instances in all relevant situations and regimes. This validates existing empirical and numerical observations. An application to a mechanical process of two elastic laminates sliding one on each other along their cohesive interface is finally analyzed, and existence results in both potential-based and non potential-based versions are obtained, extending previous contributions.

Potential-based versus non potential-based cohesive models accounting for loading and unloading with application to sliding elastic laminates

TL;DR

The paper develops a rigorous, unified framework to compare potential-based (variational) and non potential-based (equilibrium) cohesive-zone models in a planar, quasistatic setting of two elastic laminates with a mixed-mode anisotropic cohesive interface. It introduces an intrinsic construction method for cohesive energies and tensions that incorporate loading and unloading by starting from a purely loading description or tension , and demonstrates that variational models are limited to special cases (e.g., equal directional energies ) or restrictive unloading paths, while non-variational models accommodate general loading/unloading paths. Representative 2D examples show non-trivial loading/unloading paths where non-variational models remain physically consistent, whereas variational formulations can fail or produce non-physical responses. The authors establish well-posedness by proving existence of energetic solutions via a minimizing-movements scheme and of equilibrium solutions via Kakutani fixed points, thus providing a rigorous foundation for comparing these two modeling paradigms in anisotropic cohesive interfaces. The results have practical implications for modeling sliding laminates and similar cohesive interfaces, enabling robust descriptions across loading, unloading, and mixed-mode regimes in small-strain elasticity.

Abstract

A rigorous unified perspective of cohesive zone models is presented, including and comparing potential-based and non potential-based formulations, and encompassing known examples studied in literature. The main novelty of the work consists in the natural inclusion of loading and unloading effects in a general mixed-mode framework, incorporated through an intrinsic construction of energy densities or tensions. The proposed mathematical investigation identifies and proves the limitations of variational models with respect to non-variational ones, the latter yielding a feasible description of real instances in all relevant situations and regimes. This validates existing empirical and numerical observations. An application to a mechanical process of two elastic laminates sliding one on each other along their cohesive interface is finally analyzed, and existence results in both potential-based and non potential-based versions are obtained, extending previous contributions.

Paper Structure

This paper contains 12 sections, 14 theorems, 89 equations, 7 figures.

Table of Contents

  1. Setting
  2. Potential-based model
  3. Non potential-based model
  4. Construction of cohesive energy densities and tensions
  5. Construction of the potential $\Phi$
  6. Construction of the non-potential tension $\mathcal{T}$
  7. Examples of loading density $\Psi$
  8. Examples of loading tension $\mathcal{S}$
  9. Representative instances: non trivial loading/unloading paths
  10. Proof of the existence results
  11. Energetic solutions
  12. Equilibrium solutions

Key Result

Theorem 1.5

Let the stiffness tensors $\mathbb{C}_i$ satisfy hyp:C1-hyp:C5 and let the cohesive energy density $\Phi$ and the cohesive variables $\mathfrak g$ satisfy hyp:phi1-hyp:phi5 and hyp:g1-hyp:g2. Then, given a prescribed displacement $\ell$ and initial data $(\bm u^0,\gamma^0)$ fulfilling eq:externalloa

Figures (7)

  • Figure 1: CASE 1: potential-based model. a) Evolution of displacement slip values $y_1, y_2$ and history variables $z_1, z_2$, b) Energy evolution, c) Traction-displacement slip relation $\mathcal{T}_1 - y_1$, d) Traction-displacement slip relation $\mathcal{T}_2 - y_2$.
  • Figure 2: CASE 1: non potential-based model. a) Traction-displacement slip relation $\mathcal{T}_1 - y_1$, b) Traction-displacement slip relation $\mathcal{T}_2 - y_2$.
  • Figure 3: CASE 2: potential-based model. a) Evolution of displacement slip values $y_1, y_2$ and history variables $z_1, z_2$, b) Energy evolution, c) Traction-displacement slip relation $\mathcal{T}_1 - y_1$, d) Traction-displacement slip relation $\mathcal{T}_2 - y_2$.
  • Figure 4: CASE 2: non potential-based model. a) Traction-displacement slip relation $\mathcal{T}_1 - y_1$, b) Traction-displacement slip relation $\mathcal{T}_2 - y_2$.
  • Figure 5: CASE 3: potential-based model. a) Evolution of displacement slip values $y_1, y_2$ and history variables $z_1, z_2$, b) Energy evolution, c) Traction-displacement slip relation $\mathcal{T}_1 - y_1$, d) Traction-displacement slip relation $\mathcal{T}_2 - y_2$.
  • ...and 2 more figures

Theorems & Definitions (32)

  • Remark 1.1
  • Example 1.2
  • Remark 1.3
  • Definition 1.4
  • Theorem 1.5
  • Remark 1.6
  • Definition 1.7
  • Remark 1.8
  • Theorem 1.9
  • Remark 2.1
  • ...and 22 more