CPINet: Parameter identification of path-dependent constitutive model with automatic denoising based on CNN-LSTM

Highlights

• CPINet with automatic denoising to identify constitutive parameters from strain field sequence and non-temporal features.

• Improved LSTMs effectively associate temporal features with non-temporal features.

• CNN enables CPINet to remove Gaussian noise and extract strain features.

• RPEs were 0.4419%, 0.4699%, 0.6218%, and 0.8259% for SNR levels of 0%, 1%, 3% and 5%.

• Identified results of CPINet and FEMU with the consumed time of 0.158s and 51387s are in good agreement.

Abstract

Based on convolutional neural network (CNN) and improved long short-term memory (LSTM) neural network, a deep learning model CPINet is proposed for instant and accurate identification of path-dependent constitutive model parameters with excellent denoising performance. The elastic-plastic constitutive model with isotropic hardening is taken as an example for illustration. The results show that the CPINet can capture the intricate relationship between the strain field sequence and the non-temporal features (loading sequence and geometry dimensions) to identify constitutive parameters instantly and accurately. The denoising analysis revealed that the denoising processing and strain feature extraction of CNN provides excellent denoising ability to CPINet. Finally, the CPINet is validated by comparing the identified constitutive parameters of 6061 aluminum alloy with CPINet and finite element model updating method. To our knowledge, this is the first study that demonstrates the feasibility and considerable potential of using a deep learning technique to instantly and accurately identify constitutive parameters.

Introduction

The material properties can be used to predict the mechanical response of a material under various loading conditions that differ from those in the tests, order materials regarding their suitability for a particular application, and be applied as input parameters in finite element analysis of mechanical system, which are crucial in structural design, safety assessment (Prates et al., 2019; Sessa et al., 2020). Thus, identification of constitutive parameters is a very active and important area (Avril et al., 2008). Identification of constructive parameters is to solve the inverse problem of solid mechanics with known boundary conditions (displacement on S_u and load distribution on S_f) and mechanical response (displacement u and strain ε measured from experiments) to obtain the parameters of constitutive model (the constitutive model is pre-selected). At present, there are mainly two methods: one is the classical uniform stress method (USM), and the other is the method based on the full-field measurement.

The USM often relies on simple tests to fit constitutive parameters by assuming homogeneous strain and stress states over the area of interest. Based on the USM and least square method, Bai et al. (2018a) obtained the constitutive parameters of the Ramberg–Osgood model near the laser-welded joint of 6061 aluminum alloy. Saranath et al. (Saranath and Ramji, 2015) extracted the elastic and plastic properties of an electron beam-welded Ti–6Al–4V titanium alloy from the full-field strain measured by digital image correlation (DIC) using the USM and the virtual field method (VFM), respectively. The results show that the distribution tendencies of the constitutive parameters obtained by the two methods along the direction perpendicular to the weld are in good agreement. Sutton et al. (2008) used USM and VFM to estimate specific heterogeneous material properties throughout the weld zone. The research results show that when the strain ϵ_xx < 0.0025, the local stress–strain results based on the uniform stress assumption and simplified uniaxial virtual field model have good consistency; however, when ϵ_xx > 0.0025, strain localization occurs in the heat-affected zone of the specimen, resulting in necking and structural effects, which hinders the extraction of the local stress–strain behavior using a relatively simple model. Although the USM has the advantages of simplicity and universality, it also has some limitations, such as the homogeneous stress assumption, which is not valid after the onset of necking. When the complex constitutive model is calibrated, a large number of tests are required.

With the development of full-field measurement technology, such as DIC technology (Su et al., 2016a, 2016b), electronic speckle pattern interferometry (Chen et al., 2003; Aswendt et al., 2003), finite element model updating (FEMU) method (Bresolin and Vassoler, 2020; Spranghers et al., 2014), constitutive equation gap method (CEGM) (Florentin and Lubineau, 2010), equilibrium gap method (EGM) (Claire et al., 2004), and VFM (Rahmani et al., 2014), have attracted increasing attention. Martins et al. (2018) introduced the implementation and validation of the above methods in detail and summarized their advantages and disadvantages. In addition, for the identification results of these methods for the elastic-plastic model with isotropic hardening, the FEMU method achieves the most accurate results in the presence of data polluted with noise, but it requires significantly more time. Haddadi et al. (Haddadi and Belhabib, 2012) applied the FEMU method to identify the work-hardening law of non-standard specimens based on the total force and strain fields. Latourte et al. (2008) presented the CEGM to identify the elastic-plastic constitutive parameters of an elastic-plastic model with linear kinematic hardening, and obtained promising numerical and experimental results. Périé et al. (2009) proposed a new method to identify the anisotropic damage law based on EGM. Bai et al. (2018b) employed the VFM to identify the elastic-plastic constitutive parameters near the laser-welded joint of 6061 aluminum alloy from the strain fields measured by the three-dimensional (3D)-DIC measurement system from the uniaxial tensile test, and the results showed that the mechanical properties near the laser-welded joint were significantly reduced. The VFM is also employed to identify the viscoplastic constitutive parameters of 304 stainless steels from room temperature to 900 °C (Valeri et al., 2017). Roux et al. (Roux and Hild, 2020) indicated that these methods all revert to the minimization of a quadratic norm (or semi-norm) between measured and computed displacement/strain fields. They generally require many iterations to solve the least squares problem using the simplex method, particle swarm optimization (PSO), or other optimization algorithms, which may, on occasion, be CPU-intensive, and tend to become sensitive to measurement noise if based on scarce experimental data.

In view of the above issues, a possible solution is the machine learning approach. At present, the application of machine learning in the field of mechanics is rapidly increasing (Nguyen-Thanh et al., 2020; Zhuang et al., 2021; Jiang et al., 2020; Yu et al., 2019; Abueidda et al., 2020; Kallioras et al., 2020). Recurrent architectures networks, which have excellent performance in addressing sequential problems, are ideal choices for identifying material parameters of the path-dependent constitutive model. Abueidda et al. (2021) applied recurrent architecture networks (long short-term memory (LSTM) and gated recurrent unit (GRU)) and temporary convolutional networks to predict the history-dependent material response. They showed that the GRU and temporary convolutional network can both instantly and accurately predict the complex phenomena of these materials. Zhang et al. (2021) presented a LSTM deep learning method to reproduce the stress–strain behavior of soil. The results show that the LSTM method has better accuracy and convergence efficiency than feedforward and feedback neural networks. Mozaffara (Mozaffar et al., 2019) applied an improved GRU to predict the path-dependent plasticity, indicating that recurrent architectures networks have excellent performance addressing elastic-plastic problems.

However, when identifying the constitutive parameters from the strain fields measured by full-field measurement technology, inputting them into recurrent architectures results in a large number of weights. The automatic learning function of a convolutional neural network (CNN) enables the extraction of data features. Cha et al. (Cha and WooramChoi, 2017) proposed a CNN method to detect concrete cracks that avoided defect feature calculations. The results indicate that the proposed method has better performance than the traditional Canny and Sobel edge detection methods, and identifies concrete cracks in realistic situations. Zhang et al. (2019) studied on-line monitoring of defects in an aluminum alloy robot arc welding process based on a CNN. Soukup et al. (Soukup and Huber‐Mörk, 2014) trained a CNN in the photometric stereo image database of metal surface defects. The research results indicate that the performance of the CNN-based method is considerably better than that of the model-based method. In addition to feature extraction, CNN also has remarkable advantages in denoising compared to traditional methods. Jain et al. (Jain and Seung, 2008) proposed a natural image denoising method based on CNN, and found that CNN provides comparable and, in some cases, superior performance than state-of-the-art wavelet and Markov random field methods, with significantly less computational expense. Zhang et al., 2017a, 2017b built a Gaussian denoising CNN model that can address unknown noise levels by incorporating ultra-deep structures, learning algorithms, and regularization methods into image denoising. The above research progress inspired the authors to employ CNN to denoise the strain field to extract strain features.

Although the constitutive parameter identification based on deep learning has considerable development potential, research in this area is scarce. In this study, based on CNN and an improved LSTM, a deep learning model, namely CPINet, is established to automatically denoise, and instantly and accurately identify path-dependent constitutive model parameters from the strain field sequence, loads, and geometry dimensions. The outline of this work is as follows: first, a constitutive parameter identification model, namely CPINet, is proposed based on theoretical analysis. Then, taking the elastic-plastic constitutive model with isotropic hardening as an example, the corresponding CPINet is established by capturing intricate interactions between input features (strain field sequence and non-temporal features) and material parameters. Next, the identification accuracy from the strain fields in the presence of Gaussian noise with different signal-to-noise ratio (SNR) levels is compared, and the denoising performance and mechanism are analyzed. Finally, the constitutive parameters of 6061 aluminum alloy are identified by feeding the strain field sequence measured by the DIC system, loads, and geometry dimensions into the trained CPINet, which were compared with the results of the FEMU method based on PSO to validate the proposed method.