on 29-Sep-2020 (Tue)

In this study, a damage model by Linde et al. [23] for fibre reinforced composites was used for demonstrating the proposed method, other than focusing on the accuracy and applicability of the damage model itself.

Not being constrained to only elastic properties, in this study, the scope of material characterisation is enriched to form a surrogate model representing a general material property database. It covers nonlinear effective stress/strain relations (macroscale) accounting for damage initiation and propagation with regard to any loading condition, along with material damage states. In addition, the damage information also covers possible failure modes for composites.

The trained ANN is a general material property database for a specific material, and is designated to provide immediate information for future analysis of structures made of the same material and layup.

The inputs of the surrogate model are the effective strain components, as required by the Abaqus user defined material subroutine (UMAT), and the outputs are the corresponding effective stress components along with damage information. Depending on the specific application, the inputs can be three dimensional for plane and shell element usage or six dimensional for solid element usage.

For applications in composite materials, ANNs have been used to create surrogate models for the given input and output data, which are usually related to material properties. After proper training, the predictions of new entries are far more fast than running the simulation or experiment, due to the efficient information processing mechanism of the ANNs. Consequently, using ANNs is a method to reduce the high computational costs of numerical simulations, particu- larly for some large scale problems.

For regression analysis and classification problems in engineering, the most commonly used one is the feed-forward neural network. This network contains one input layer, one output layer and one or more hidden layers (Fig. 4). The information propagates in one direction from the input layer directly via any hidden layers to the output layer without loops.

Here we used Adamax [25], an algorithm for first-order gradient-based optimisation of stochastic objective functions, based on adaptive estimates of lower-order moments for the classification problems, and RMSprop [26], a gradient-based optimisation with adaptive learning rate adaption for the regression problem.

Completness: Every true statement which can be expressed in the notation of the system is a theorem.

Consistency (internal): when every theorem, upon interpretation, comes out true (in some imaginable world)

Warning: When we start using the term "provable statements" instead of "theorems", it shows that we are beginning to blur the distinction between formal systems and their in terpretations.

These are not theorems: truths belonging to a theory which are not provable within the theory.

As the proposed surrogate model covers both constitutive relation- ship and damage state information, multiple ANNs are used to represent them respectively as they are for different problems. The regression analysis is carried out for nonlinear constitutive relationship between the obtained strain and stress data, and classification is performed for damage identification between the strain and damage state variable data. In this example, both regression and classification ANN were trained using Keras [28] (...)

Progressive damage behaviour is considered in the proposed method so that the regression for the constitutive law is nonlinear. A deep neural network with two hidden layers (60 and 50 neurons each) were employed for the plane stress case. The numbers of layers and neurons are hyperparameters in neural networks, which are problem-dependent and can not be learned during the training. Manual search is a simple way to determine the hyperparameters and also there are some complex hyperparameter optimisation methods as introduced in Ref. [30]. In this case, a manual search has been performed to identity these numbers through comparing accuracy/loss during validation. The input/output and hyperparameters for this regression ANN are presented in Table 2. It is noted that for 3D stress application the required hidden layers or number of neurons per layer could be more in the regression analysis.

To avoid overfitting in the neural networks, one should monitor if there is any significant in- crease in the validation loss globally

Note that for the macroscale model, there is only one layer of plane stress elements, it is more interesting to tell which element is damaged or failed regardless of the layerwised answer.

The kraft process (also known as kraft pulping or sulfate process)^[1] is a process for conversion of wood into wood pulp, which consists of almost pure cellulose fibers, the main component of paper. The kraft process entails treatment of wood chips with a hot mixture of water, sodium hydroxide (NaOH), and sodium sulfide (Na₂S), known as white liquor, that breaks the bonds that link lignin, hemicellulose, and cellulose