PSI - Issue 42

Jaynandan Kumar et al. / Procedia Structural Integrity 42 (2022) 806–812 Jaynandan Kumar, Anshul Faye / Structural Integrity Procedia 00 (2019) 000–000

807

2

Monteiro, 2014; Sommer, 2016; Polzer, 2021); reasons of the same remain unclear. (Volokh, 2016) made an attempt to analyse the e ff ect of calcification of AAA by using a High-fidelity generalized method of cells micromechanics analysis. Their results suggest that the strength of the aneurysm decreases under calcification. The matrix (tissue) was modelled as an isotropic hyper-elastic material by (Volokh, 2016). In the present work, we aim to investigate the e ff ect of calcification on the failure envelope of aneurysmatic tissue. E ff ects of calcification will be analyzed by varying the size, shape and fraction of calcium particles in the tissues. A micron-sized RVE is created with calcium particles embedded in aneurysmatic tissue, which will be analyzed using finite element analysis. Tissues are modelled using a two-fibre anisotropic material model; whereas calcium particles are considered to be linear elastic materials. Failure of tissue is modelled using the phase-field method.

2. Methodology

2.1. Constitutive Model

Aneurysmatic tissues are modeled as an anisotropic material. A two-fiber material based model given by Holzapfel (2000) is used. The strain energy function for the material is defined as a sum of the isotropic and anisotropic parts as:

ψ = ψ iso

ani 0 ( I 4 , I 6 ) .

0 ( I 1 , J ) + ψ

(1)

Isotropic part ( ψ iso 0 ) govern the mechanical response tissue at a low strain level when the collagenous structure of 0 ) describes the deformation of tissues at higher stretches when the behaviour is mostly governed by the family of collagen fibres, which become straighter and more resistant to stretch. Isotropic and anisotropic functions are as follows: the tissue is not active. The anisotropic part ( ψ ani

µ 2

ψ iso 0 ( J , I 1 ) = κ ( J − ln J − 1) +

( I 1 − 2 ln J − 3) ,

(2)

k 1 2 k 2 i = 4 , 6

ψ ani

2 ] − 1 } .

0 ( I 4 , I 6 ) =

{ exp[ k 2 ( I i − 1)

(3)

where, I 1 is the first Invariant of right Cauchy-Green tensors, I 4 = m · m ; represent the square of the deformed length of the first fibre. Similarly, I 5 represent the square of the deformed length of the second fibre. Moreover, J defined as the determinant of the deformation gradient which maps the infinitesimal volume element to the spatial volume elements. k 1 and k 2 are a stress like material parameters and a dimensionless parameter, respectively and κ is the penalty parameter used for computational treatment of incompressible materials. Cauchy stress is associated with the strain energy function (1) can be derived as,

µ J

κ J ( J − 1) I + 2 ψ 4 m ⊗ m + 2 ψ 6 m ′ ⊗ m ′ .

( b − I ) +

(4)

σ 0 =

where b is the left Cauchy-Green tensor, µ is the shear modulus, m and m ′ are the directions of the orientation of the first and second fibre respectively. I is second order identity tensor. ψ 4 and ψ 6 are the deformation dependent scalars which are the gradient of strain energy with respect to I 4 and I 6 respectively.