PSI - Issue 79

Daniel Leidermark et al. / Procedia Structural Integrity 79 (2026) 190–197

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the diversity of the training set by creating artificial copies of existing data. Data augmentation is a tool that is frequently used in image identification training procedures, e.g. Long et al. (2015), Mikołajczyk and Grochowski (2018), Mumuni and Mumuni (2022) and Alomar et al. (2023), where an image can be rotated, scaled, reflected, made black-white, cropped etc. , to easily generate artificial copies of said image. However, such an approach might not be as easily adopted when it comes to training of ML constitutive models that need to describe the mate rial behaviour, for instance monotonic or cyclic load-displacement curves. There are some previous studies on ML constitutive models and data augmentation, however limited, see e.g. Holzapfel et al. (2021) and Cheung et al. (2024).

Hence, this study will define and evaluate two di ff erent data augmentation approaches suitable for ML constitutive models, to enhance a scarce training data set.

2. Machine learning model

To evaluate these smart training strategies, the cyclic constitutive model by Ramberg-Osgood Ramberg and Osgood (1943) was chosen and utilised in amplitude format as follows

+

σ a K ′

1 / n ′

σ a E

(1)

ε a =

where ε a is the strain amplitude, σ a is the stress amplitude, E is the Young’s modulus and K

′ and n ′ are plastic material

constants describing the cyclic hardening behaviour. An ML model of the Ramberg-Osgood constitutive description was constructed based on an artificial neural net work (ANN), taking the three material constants ( E , K ′ , n ′ ) as inputs and σ a at 1% ε a as output. Thus, based on this, a simple ANN structure with one hidden layer consisting of five neurons was set-up and implemented in Matlab. In this context, the training was performed using 80% of the available data and 20% for testing, thus, excluding the usually adopted validation stop-criteria process during training. Sigmoid activation functions were used network-wide, ex cept in the output layer where a linear function was adopted. Normalised Xavier weight initialisation, constant biases equal to unity and input normalisation to [0 . 1 , 0 . 9] were adopted. The backward propagation training was done by a stochastic gradient descent approach with a constant learning rate of 0 . 05 and a momentum rate of 0 . 4 over 10000 epochs.

3. Scarce data

The training of the above ML model was done based on material data of 96 di ff erent low-alloys steels with similar properties, di ff erences reside in grade, heat treatment and test temperature. These were taken from literature, namely Ba¨umel and Seeger (1990), and gathered in a material database, see Table A1. It is to be pointed out that the stress amplitude has been calculated using Eq. (1) at 1% strain amplitude, as it was not explicitly present in the data. Each of the materials in the database are comprised of fitted material parameters based on a number of experimental tests, hence the foundation of the training relies upon a minimum of 96 · 3 = 288 experimental tests (minimum being 3 data points for fitting a non-linear curve). This can be considered many in the perspective of constitutive modelling purposes, but in the context of ML this is low. Now, the usual amount of available experimental tests for a constitutive model fitting lies in the area of 10 − 20, hence a scarce number of data. Based on this, three evaluation technique of the training process was conducted, i) baseline - trained on all 96 available materials, ii) a linear augmentation approach - trained on ten materials and iii) a scatter augmentation approach - trained on ten materials. The ten materials in the augmentation approaches were sampled based on a uniform latin hypercube sampling of all 96 available materials. A one-dimensional sampling in stress space, hence σ a at 1% ε a , was carried out. Thus, generating a sampling in stress amplitude that renders similar global variation as in the available material data. This scarce data set can be seen as a test campaign that will be used to fit (train) the Ramberg-Osgood ML model.