PSI - Issue 61

İmren Uyar et al. / Procedia Structural Integrity 61 (2024) 195 – 205 İ. Uyar, E. Gürses / Structural Integrity Procedia 00 ( 2019) 000 – 000

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aimed at exploring the interconnected behavior of electro-chemo-elastic materials, as discussed by Yang et al. (2010) and Klinsmann et al. (2016). The focus is on including chemical effects in the equation system. A variational principle is derived for a linearly coupled system using the governing equations of ionic diffusion and momentum balance. The extended Gibbs' free energy function is used as a fundamental reference point. The expression for the extended Gibb's free energy per unit volume can be written by adding chemical energy (Murad et al., 2002) and excluding thermal and electrical based on the assumption that these effects are insignificant compared to the chemical and mechanical effects. The free energy function for the isotropic linearly coupled system is defined in terms of = − ℎ (the elastic strain), ℎ (the chemical strain) and concentration , = 12 ( [ ( − ℎ )] 2 +2 [( − ℎ ) 2 ]) − ( − ℎ ) + 0 + 0 2 (1) where , are the Lamé constants and = + 2 3 . is the gas constant, is temperature, 0 reference chemical potential, 0 initial concentration and [ ] = stands for the trace of tensor . In the original problem, one can observe volume swelling resulting from lithium-ion movement, similar to heat expansion where pure volume expansion occurs under thermal load. This analogy is used to construct the electro chemo-elastic formulation summarized in Table 1.

Table 1. The set of mechanical and electrochemical equations Mechanical

∇ ∙ =0 − ̅ =0 = ̅ = 12 [∇ + (∇ ) ]

Electrochemical ̇+∇∙ =0 0 , =− ̅ = ̅ 0 = − 0 ∇

Independent variables Governing Equations

Boundary Conditions

Strain/Displacement and Flux/Concentration relations Constitutive relations of the coupled system

= − ( − ℎ )+ 0 + 0

= ( − ℎ ) +2 ( − ℎ )

2.2. Phase Field Model of Crack Propagation Following the simulation of the coupled behavior in the fiber region of a structural battery, the analysis is integrated with the study of crack formation in the same region. The presence of a high concentration gradient at the fiber surface leads to the development of mechanical stresses, potentially resulting in the initiation and propagation of cracks within the fiber. When the structure develops cracks, it causes the fibers to lose their mechanical strength, and the battery's ability to charge is also affected because it reduces the movement of lithium ions. To this end, a phase field model for fracture is incorporated into the system to model the formation and growth of cracks. The coupled system is solved with the phase field equation demonstrated by Martínez-Pañeda et al. (2022). The finite element procedure is carried out using the open-source software FENICS. The phase field method has evolved into an efficient approach for addressing the challenges posed by earlier numerical methods. In this method, a sharp crack is transformed into a diffuse crack zone governed by a scalar phase field variable, commonly referred to as the order parameter (Miehe et al. (2010) and Ambati et al. (2015)). Within the context of the phase field method, the crack is represented using a diffuse field variable ∈ [0,1] . =0 denotes intact material, while =1 represents the fully broken material state. The size of the regularized crack surface is governed by the choice of , the length scale parameter. A regularized crack density functional Γ ( , ) can be defined that converges to the functional of the discrete crack as →0 , see in Fig 1 (a). Hence, the fracture energy due to the creation of a crack can be approximated as