CCM/PD Alternating Solution

• CCM/PD Alternating Solution is a coupled computational method integrating classical continuum mechanics for thermal analysis with peridynamic modeling for displacement and fracture.
• The approach alternates between solving the heat conduction equation and peridynamic equilibrium on a unified finite element mesh to capture thermo-mechanical behavior.
• A novel directional damage tensor refines the prediction of anisotropic degradation in thermal conductivity and fracture propagation.

The CCM/PD Alternating Solution refers to a coupled computational methodology that integrates classical continuum mechanics (CCM) for thermal field computation with peridynamic (PD) modeling for displacement and fracture evolution, specifically in the context of thermo-mechanical fracture problems. Implemented on a unified finite element discretization, the scheme alternates between solving the thermal diffusion equation and the peridynamic equilibrium equation—each leveraging its respective strengths: CCM for accurate, mesh-based thermomechanical transport, and PD for robust prediction of discontinuities such as cracks. A central innovation of the approach is a new, directionally sensitive definition of peridynamic damage, enabling refined modeling of anisotropic crack-induced degradation in thermal conductivity and fracture response.

1. Framework of the CCM/PD Alternating Solution

The core of the methodology is an operator splitting (alternating) strategy applied on a common finite element mesh. Each time step or iteration comprises two main computational phases:

• CCM Phase (Thermal Field): The temperature field $T$ is computed using a finite element discretization of the heat conduction equation, governed by the energy conservation law and Fourier’s law:

$$\mathbf{C} \cdot \frac{dT}{dt} + \mathbf{K} \cdot T = P$$

where $\mathbf{C}$ is the heat capacity matrix, $\mathbf{K}$ is the (potentially damage-modified) conductivity matrix, and $P$ denotes external heat input. For time integration, a generalized midpoint scheme is used:

$$T_{n+\theta \Delta t} = (1-\theta) T_n + \theta T_{n+1}\;, \quad \dot{T}_{n+\theta \Delta t} = (T_{n+1} - T_n) / \Delta t$$

• PD Phase (Deformation and Damage): The updated temperature field is supplied to the PD model, which solves the nonlocal equilibrium equation for the displacement:

$$\int_{H_\delta(x)} f(x', x, \hat{T})\,dV_{x'} + b(x) = 0, \quad \forall\, x \in \Omega$$

Here, $f(x', x, \hat{T})$ is the thermally augmented bond force, typically realized as:

$$\hat{f}(x', x, \hat{T}) = \frac{1}{2} c(x, |\xi|) \left[u_\xi(x') - u_\xi(x) - a(x)\hat{T}(x, \xi)\right] e_\xi,$$

where $u_\xi$ is the projected displacement in the bond direction $\xi$, $c$ is the bond stiffness, $a(x)$ is the micro-expansivity, and $e_\xi$ is the unit bond direction.

2. Traditional and Directional Peridynamic Damage Definitions

2.1 Classical Scalar Damage Indicator

Standard PD models quantify the local damage at point $x$ as a scalar ratio of broken bonds:

$$d(x,t) = 1 - \frac{\int_{H_\delta(x)} \mu(x', x, t) \,dV_{x'}}{\int_{H_\delta(x)} dV_{x'}}$$

$\mu(x',x, t) = 0$ if bond is broken at $t$ (bond stretch exceeds threshold $s_0$), $1$ otherwise.

Limitation: This scalar damage captures only the overall density of failed bonds and cannot distinguish the orientation or spatial distribution of bond failures—a critical limitation for modeling anisotropic damage and directional property degradation.

2.2 Enhanced Directional Damage Tensor

To address anisotropy, the solution introduces a tensorial (or vector-valued in 2D) damage metric:

Three-Dimensional Case

Damage tensor:

$$\hat{d} = \begin{bmatrix} d_1 & 0 & 0 \ 0 & d_2 & 0 \ 0 & 0 & d_3 \ \end{bmatrix}$$

with directionally resolved components:

$$d_i^*(x,t) = 1 - \frac{\int_{H_\delta(x)} \nu_i^*(x', x)\, \mu(x', x, t) dV_{x'}}{\int_{H_\delta(x)} \nu_i^*(x', x)\, dV_{x'}}$$

$\nu_i^*(x', x) = 1$ if $e_i^* \cdot \xi > 0$, $0$ otherwise, with $e_i^*$ denoting the principal (typically orthonormal) axes.

Two-Dimensional Case

$$\hat{d} = \begin{bmatrix} \max\{d_1^+, d_1^-\} & 0 \ 0 & \max\{d_2^+, d_2^-\} \ \end{bmatrix}$$

where $d_1^\pm$ and $d_2^\pm$ correspond to positive and negative projections along $x_1, x_2$ axes.

Significance: This enhancement enables modeling cracks and damage propagation that are highly anisotropic, such as those caused by directional thermal gradients or complex loading histories.

3. Coupling Thermomechanical Fields via Anisotropic Conductivity Degradation

The computed damage tensor $\hat{d}$ is directly used to update the thermal conductivity in a physically consistent, direction-dependent fashion:

$\mathbf{k} = (\mathbf{I} - \hat{d}) \mathbf{k}_0$

where $\mathbf{k}_0$ is the undamaged thermal conductivity and $\mathbf{I}$ the identity matrix. As such, cracks and high-damage zones produce local reductions in heat transport along directions experiencing concentrated bond failure, mirroring true anisotropic crack-induced property degradation.

4. Numerical Implementation and Validation

• Unified Mesh Scheme: Both CCM and PD models use the same finite element discretization, bypassing the need for remeshing or data transfer between disparate mesh structures.
• Alternating Solution Procedure: At each nonlinear iteration or time step:
  1. Solve heat conduction (CCM, updating $T$).
  2. Use $T$ in the PD solver for displacement and damage update.
  3. Update conductivity tensor $\mathbf{k}$ as a function of the new $\hat{d}$ and return to the thermal phase.

• Example Validations:

  • Reproduction of analytic thermal–deformation solutions for an undamaged square plate.
  • Simulation of plates with horizontal/vertical insulation cracks, showing that anisotropic $\mathbf{k}$ (arising from tensorial damage) yields realistic heat flux fields.
  • Crack evolution in cruciform plates under various thermal loads, with crack trajectories and damage fields closely matching phase-field method (PFM), extended finite element method (XFEM), and experimental data.
  • Simulation of dynamic crack growth in ceramics subjected to thermal shock.

5. Significance in Thermo-Mechanical Fracture Modeling

This integrated alternating framework is capable of:

• Capturing both isotropic and anisotropic fracture-induced degradation in thermomechanical properties.
• Handling quasi-static and dynamic crack initiation/propagation using a robust, mesh-consistent strategy.
• Providing physically justified, mesh-independent transfer of evolving fracture-induced property heterogeneities from the PD model to the CCM-based thermal solver and vice versa.

6. Theoretical and Practical Implications

The alternating CCM/PD solution advances the state of the art in the following ways:

• Initiates a scalable, generalizable framework for coupling continuum descriptions (CCM) of transport phenomena with nonlocal mechanics (PD) for discontinuity modeling.
• Demonstrates that a directionally resolved damage tensor can be systematically embedded in both the mechanical and thermal components, capturing anisotropic crack effects.
• Provides a validation pathway by comparing numerical predictions with analytical, computational, and experimental benchmarks across multiple regimes (pure thermal deformation, static/dynamic fracture, and coupled property evolution).

7. Outlook

This approach lays the foundation for rigorous, physically faithful modeling of complex thermo-mechanical fracture phenomena in engineering materials—particularly in contexts where directional damage and property degradation are essential, such as ceramics, composites, and structures under severe thermal transients. The methodology is amenable to extension toward fully multiphysics coupling, complex three-dimensional fracture scenarios, and further integration with phase-field or variational fracture models.

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Summary Table: Directional PD Damage Formulation

Aspect	Classical Damage	Directional Damage Tensor
Mathematical form	Scalar $d(x,t)$	Diagonal tensor $\hat{d}$ or vector
Captures bond count?	Yes	Yes
Captures bond orientation?	No	Yes (via direction $e_i^*$, $\nu_i^*$)
Updates conductivity?	Isotropic	Anisotropic, $\mathbf{k} = (\mathbf{I}-\hat{d})\mathbf{k}_0$
Resolves directional cracks?	No	Yes
Implementation mesh	Unified FE mesh	Unified FE mesh

This comprehensive alternating solution provides a platform for future advances in coupled fracture simulations, especially where directionally dependent phenomena are critical (Tao et al., 1 Sep 2025).