PSI - Issue 66

Umberto De Maio et al. / Procedia Structural Integrity 66 (2024) 502–510 Author name / Structural Integrity Procedia 00 (2025) 000–000

504

3

2. Underlying theory and methodologies This section briefly examines the theoretical background for evaluating the homogenized properties of microstructured materials (Section 2.1). Genetic algorithms and neural networks for structural optimization are discussed in Sections 2.2 and 2.3, respectively. 2.1. Homogenized properties in microstructured periodic media We will consider a representative volume element (RVE) of a nonlinear composite microstructure based on a deep sea hexactinellid sponge. This porous skeletal structure has a repeating pattern, a square grid crosshatched with a diagonal struts pattern, as reported in Section 3. We consider a finitely deformed composite material made of nearly incompressible hyperelastic constituents, characterized by a strain energy density ( ) W F corresponding to a neo Hookean material:

1 2

1 2

,

(1)

2

( ) F

(tr( ) 3)  C

ln( ) J

ln( ) J

W



 







where  denotes the initial shear modulus, C represents the right elastic Cauchy-Green tensor, J is the Jacobian determinant, and λ governs the material's compressibility, set to 1000 μ to simulate the incompressible nature of the material phases. The first and second derivatives of ( ) W F correspond to the stress tensor R T and the tangent moduli tensor R C . In the context of a quasi-static loading process, where body forces are absent, the equation of motion for the undeformed configuration is expressed as Div( ) 0 R  T . The macroscopic gradient deformation tensor F and the macroscopic first Piola-Kirchhoff stress tensor R T are defined through the boundary values of the microscopic deformation field and the nominal traction vector:

1

1

1







( )

,

F

 x n

F X

 x n

dS

dV

dS







( ) i

( ) i

( ) i

( ) i

( ) i

V

V

V

( ) i

( ) i

( ) i

V

B

H





( ) i

( ) i

( ) i

(2)

1

1





( )

,

T

t

X

R T X

dS

dV







( ) i

( ) i

R

R

V

V

( ) i

( ) i

V

B



( ) i

( ) i

where ( ) i B and ( ) i H denote the solid and void portions of the volume ( ) i V occupied by the RVE, respectively. Additional details about the evaluation of the homogenized properties can be found in (Greco et al., 2016). 2.2. Convolutional neural network for instability recognition Convolutional Neural Networks (CNNs) belong to the deep learning class of tools designed to process structured data, such as images or signals. They are characterized by three neural layers: convolutional, pooling, and fully connected. The first one may identify local patterns and produce feature maps by applying filters or kernels to the input images. After that, the pooling layers manipulate the feature maps to minimize their spatial dimensions and overfitting. Finally, the fully connected layers combine all the extracted features, allowing the final stage before the classification of the image to proceed. The softmax layer is often used at the output to convert raw class scores into probabilities, making the final prediction more interpretable. As depicted in Fig.1, the CNN architecture starts with inputting the first mode shape obtained from a linear buckling analysis under compression along the vertical direction. A kernel is applied to the input, and the data moves through several convolution layers with Rectified Linear Unit (ReLU) activation, which introduces non-linearity, followed by pooling operations that progressively downsample the feature maps. After passing through multiple convolution and pooling layers, the data is flattened into a 1D vector, ready to be processed by fully connected layers. These perform