# DO WE STILL NEED IMAGENET PRE-TRAINING IN REMOTE SENSING SCENE
CLASSIFICATION?

Vladimir Risojević Corresponding author Vladan Stojnić  Faculty of Electrical
Engineering
University of Banja Luka
Banja Luka, Bosnia and Herzegovina
{vladimir.risojevic<EMAIL_ADDRESS>
###### Abstract

Due to the scarcity of labeled data, using supervised models pre-trained on
ImageNet is a de facto standard in remote sensing scene classification.
Recently, the availability of larger high resolution remote sensing (HRRS)
image datasets and progress in self-supervised learning have brought up the
questions of whether supervised ImageNet pre-training is still necessary for
remote sensing scene classification and would supervised pre-training on HRRS
image datasets or self-supervised pre-training on ImageNet achieve better
results on target remote sensing scene classification tasks. To answer these
questions, in this paper we both train models from scratch and fine-tune
supervised and self-supervised ImageNet models on several HRRS image datasets.
We also evaluate the transferability of learned representations to HRRS scene
classification tasks and show that self-supervised pre-training outperforms
the supervised one, while the performance of HRRS pre-training is similar to
self-supervised pre-training or slightly lower. Finally, we propose using an
ImageNet pre-trained model combined with a second round of pre-training using
in-domain HRRS images, i.e. domain-adaptive pre-training. The experimental
results show that domain-adaptive pre-training results in models that achieve
state-of-the-art results on HRRS scene classification benchmarks. The source
code and pre-trained models are available at
https://github.com/risojevicv/RSSC-transfer.

###### keywords:

Convolutional neural networks, Transfer learning, Domain-adaptive pre-
training.

##  1 INTRODUCTION

Transfer learning opened up the possibility of applying deep learning to
domains in which labeled data is scarce, difficult or expensive to acquire,
such as remote sensing (Ball et al.,, 2018). A standard approach for applying
transfer learning in high resolution remote sensing (HRRS) scene
classification has been to start with a supervised model trained on ImageNet
and either use it for feature extraction or fine-tune it to the target task
(Hu et al.,, 2015; Nogueira et al.,, 2017). We refer to the first round of
training as pre-training. In the case of feature extraction the pre-trained
model is used to compute the features of the target images, which can
subsequently be used e.g., for training a classifier or for image retrieval.
On the other hand, in fine-tuning the entire network is optimized for the
target classification task.

The most prominent benchmark in HRRS scene classification for a long time has
been UCM dataset with 21 classes and only 100 images per class (Yang and
Newsam,, 2010), which is not enough for training a model from scratch.
However, in the previous years, several larger datasets of HRRS images have
appeared with a goal of establishing new benchmarks for HRRS scene
classification (Xia et al.,, 2017; Cheng et al.,, 2017; Zhou et al.,, 2018; Li
et al.,, 2020; Qi et al.,, 2020). These datasets contain more classes and more
images per class than UCM and, having in mind the domain difference between
everyday objects in ImageNet and remote sensing scenes, a question arises
whether we still benefit from ImageNet pre-training in HRRS scene
classification.

On the other hand, ImageNet has been successfully used for self-supervised
pre-training and it was recently shown in (Ericsson et al.,, 2021) that self-
supervised models transfer better than supervised models to a number of
downstream tasks. Therefore, to obtain a complete picture of usefulness of
ImageNet pre-training in remote sensing scene classification self-supervised
pre-training should also be included in the analysis.

To answer the titular question we both trained from scratch and fine-tuned
ImageNet pre-trained supervised and self-supervised convolutional neural
networks on multiple HRRS image datasets and compared the resulting
classification accuracies. Furthermore, we examined whether representations
learned on HRRS image datasets transfer better to other HRRS scene
classification tasks compared to the representations learned on ImageNet. We
used multiple source and target HRRS datasets of various sizes and class
distributions with a goal of finding the factors that influence the model
transferability.

In addition, it has recently been observed that transfer performance of large
unsupervised pre-trained language models can be improved by additional
unsupervised pre-training on data from the target domain, an approach known as
domain-adaptive pre-training (Gururangan et al.,, 2020). Similar results with
self-supervised and supervised learning for several computer vision tasks were
presented in (Reed et al.,, 2021). Motivated by these results, we also
investigated if domain-adaptive pre-training can improve upon pre-training on
only ImageNet or HRRS images. In contrast to (Gururangan et al.,, 2020), we
experiment with supervised domain-adaptive pre-training and show that it can
improve the classification accuracies on the remote sensing test datasets
without any additional model complexity. The resulting performances are
similar or better than state-of-the-art obtained using architectural
modifications. Therefore, our results can be regarded as the new baselines and
models reused as backbones in all applications where ImageNet pre-trained
supervised networks have been used.

The main contributions of this paper are:

  1. 1.

Comparison of training from scratch and fine-tuning ImageNet pre-trained
supervised and self-supervised convolutional neural networks on HRRS image
datasets.

  2. 2.

Evaluation of transferability of representations learned from scratch on HRRS
image datasets to other HRRS scene classification tasks.

  3. 3.

Introduction of supervised domain-adaptive pre-training to HRRS scene
classification and empirical analysis of the factors contributing to its good
performance.

  4. 4.

Publishing the source code and pre-trained models to facilitate new research
and applications of transfer learning.

The rest of the paper is organized as follows. In Section 2 related work is
reviewed. The datasets and training details are described in Section 3. The
experimental results are presented in Section 4. We discuss the results in
Section 5 and Section 6 concludes the paper.

##  2 RELATED WORK

It has long been known that networks pre-trained on ImageNet can produce
representations suitable for different target tasks (Sharif Razavian et al.,,
2014; Donahue et al.,, 2014). Consequently, the question of factors
influencing the transferability of the learned features had been investigated
in (Yosinski et al.,, 2014; Azizpour et al.,, 2015; Huh et al.,, 2016). It was
shown that similarity of the source and target tasks, as well as diversity and
size of the source dataset influence the performance on a target task.
However, these results are not consistent across the source and target tasks,
and recently in (He et al.,, 2019) was observed that ImageNet pre-training did
not improve object detection and instance segmentation performance.
Additionaly, it was found in (Kornblith et al.,, 2019) that the networks pre-
trained on ImageNet did not improve performance on target tasks from
significantly different domains, requiring fine-grained classification or
having more training data. These results spurred the interest in examining the
effects of ImageNet pre-training in medical imaging (Raghu et al.,, 2019; Ke
et al.,, 2021; Hosseinzadeh Taher et al.,, 2021), which shares some of the
problems, such as the lack of large labeled datasets and need for domain
experts for labeling, with remote sensing. However, to the best of our
knowledge, a systematic investigation of ImageNet pre-training in HRRS scene
classification is still missing.

Models pre-trained on ImageNet have quickly gained popularity for remote
sensing image classification (Penatti et al.,, 2015; Hu et al.,, 2015;
Marmanis et al.,, 2015; Liang et al.,, 2016; Nogueira et al.,, 2017; Zhao et
al.,, 2017; Tong et al.,, 2020). Although pre-training on ImageNet is a de
facto standard, several papers also reported experiments with pre-training on
remote sensing image datasets. In (Chen et al.,, 2019) a HRRS scene
classification model pre-trained on a dataset constructed taking a union of
RESISC45, PatternNet, and RSI-CB showed slightly improved results on UCM
classification compared to ImageNet pre-training. In contrast, an analysis in
(Pires de Lima and Marfurt,, 2020) showed that the networks pre-trained on
ImageNet outperform the ones pre-trained on PatternNet in transfer learning to
AID and UCM datasets. Concurrently with this work, Million-AID, the largest
HRRS image dataset to date, has been published along with the experiments
showing that pre-training on Million-AID results in better transfer learning
performance on RESISC45 and AID than pre-training on ImageNet (Long et al.,,
2022).

The work in (Neumann et al.,, 2020) is similar to ours in the sense that
transfer from both ImageNet and remote sensing image datasets is explored. The
experiments had been performed on three medium-resolution (BigEarthNet (Sumbul
et al.,, 2019), EuroSAT (Helber et al.,, 2019), and So2Sat (Zhu et al.,,
2019)) as well as two high-resolution datasets (RESISC45 and UCM) and the
results showed that fine-tuning the models pre-trained on remote sensing
datasets resulted in better classification accuracies than fine-tuning the
models pre-trained on ImageNet. Furthemore, multiresolution datasets led to
more transferable representations and medium-resolution datasets did not yield
good generalization to high-resolution datasets. However, since only two HRRS
image datasets were used, the question of the factors that influence
transferability for HRRS scene classification remained unanswered. In this
paper we perform the experiments on multiple HRRS image datasets of various
sizes and with different numbers of classes with a goal of identifying the
factors that influence the transferability of the obtained representations the
most.

Self-supervised learning holds a promise to reduce the need for large labeled
datasets in training deep learning models (Jing and Tian,, 2020). Recently, it
has been shown that the best self-supervised ImageNet models can outperform
supervised ImageNet models in transferring to various downstream tasks
(Ericsson et al.,, 2021). Although in most cases ImageNet or larger datasets
of images of everyday objects and scenes are used for self-supervised pre-
training, there are also attempts to use remote sensing images for that
purpose (Mañas et al.,, 2021; Ayush et al.,, 2021; Stojnic and Risojevic,,
2021). However, since the preliminary experiments with pre-trained models from
(Mañas et al.,, 2021) resulted in poor performances, we did not use remote
sensing images for self-supervised pre-training.

##  3 MATERIALS AND METHODS

We used six HRRS image datasets: MLRSNet (Qi et al.,, 2020), RESISC45 (Cheng
et al.,, 2017), PatternNet (Zhou et al.,, 2018), RSI-CB (Li et al.,, 2020),
AID (Xia et al.,, 2017), and UCM (Yang and Newsam,, 2010). The main properties
of these datasets are listed in Table 1. The first five datasets were used as
both source and target datasets while UCM was used only as the target dataset
because it is too small for training a network from scratch. MLRSNet can be
used for training both single-label and multi-label classifiers so we included
both scenarios in our experiments. We also used ImageNet-1k and ImageNet-100
as source datasets. ImageNet-100 is a subset of ImageNet-1k with 100 classes
and total of 131,689 images (Tian et al.,, 2020). Having the size similar to
MLRSNet, ImageNet-100 enabled us to assess how the source domain influences
representation transferability. With ImageNet-1k as the source dataset we
experimented with both supervised and self-supervised pre-training. For self-
supervised pre-training we chose SwAV (Caron et al.,, 2020), which had shown a
good transfer performance in (Ericsson et al.,, 2021). We did not train
networks on ImageNet-1k ourselves and rather used the pretrained supervised
model available in Keras as well as the PyTorch implementation and weights of
a self-supervised model pre-trained using SwAV and provided by the authors of
(Caron et al.,, 2020).

Dataset | Size | Classes | Image size | Resolution (m) | Annotations
---|---|---|---|---|---
MLRSNet (Qi et al.,, 2020) | 109,161 | 46/60 | $256\times 256$ | 0.1 - 10 | single/multi-label
RESISC45 (Cheng et al.,, 2017) | 31,500 | 45 | $256\times 256$ | 0.2 - 30 | single-label
PatternNet (Zhou et al.,, 2018) | 30,400 | 38 | $256\times 256$ | 0.062 - 4.693 | single-label
RSI-CB (Li et al.,, 2020) | 24,000 | 35 | $256\times 256$ | 0.22 - 3 | single-label
AID (Xia et al.,, 2017) | 10,000 | 30 | $600\times 600$ | 0.5 - 8 | single-label
UCM (Yang and Newsam,, 2010) | 2,100 | 21 | $256\times 256$ | 0.3 | single-label
Table 1: Details of HRRS image datasets used in the experiments.

For each source dataset, we used 80% of images from each class for
training/fine-tuning and the rest for testing. We either trained from scratch
or fine-tuned a ResNet-50 (He et al.,, 2016) model pre-trained on ImageNet-1k
for 100 epochs using Adam with batch size 100. We linearly increased the
learning rate for the first 5 epochs to $3\cdot 10^{-3}$, in the case of
training from scratch, or $3\cdot 10^{-4}$, in the case of fine-tuning, and
reduced it with the factor of 0.2 in the 50th, 70th, and 90th epochs. In both
cases, we applied the following augmentations: resize to $292\times 292$
pixels and random crop of a $256\times 256$ pixels block, random flip left-
right and up-down, and random rotation for $\left\\{90,180,270,360\right\\}$
degrees. At test time, the images were resized to $292\times 292$ pixels and a
$256\times 256$ pixels block was cropped from the center.

In the experiments with transfer learning, we used the pre-trained source
models as either fixed feature extractors and trained a softmax classifier, or
replaced the classification layer and fine-tuned the whole network on a target
dataset. For target datasets, we used 20% of images from each class for
training and the remainder for testing. Following the usual protocol (Ericsson
et al.,, 2021), we did not perform data augmentation when training the softmax
and used the same augmentations as for the source datasets when fine-tuning
the models. In the case of feature extraction data augmentation is omitted in
order to assess the quality of the extracted features because they can also be
used in different downstream tasks, such as image retrieval. It should be
noted that a single dataset was not used as both the source and target dataset
in the same experiment.

For domain-adaptive pre-training, we fine-tuned ImageNet-1k pre-trained
supervised and self-supervised ResNet-50 models on MLRSNet, as described
previously, and used the resulting model as the pre-trained source model for
transfer learning. In these experiments we did not use MLRSNet as a target
dataset.

On the target datasets, we trained the softmax classifiers or fine-tuned the
whole network for 100 epochs using Adam with batch size 100. For training the
softmax we used a fixed learning rate $10^{-3}$ and for fine-tuning we used
the same learning rate schedule as for fine-tuning the network pre-trained on
ImageNet-1k. We report the classification accuracies on the test set for
single-label tasks and F1-measures, with threshold 0.5, for multi-label tasks.
All the networks were trained or fine-tuned on two Nvidia GTX 1080Ti GPUs with
CUDA 11.0 and Intel Core i7-8700K CPU running Ubuntu 18.04.

We used the nonparametric bootstrap to estimate 95% confidence intervals for
each performance metric. We drew 1,000 replicates from the test set, and
computed the performance metric on each replicate. This procedure produced a
distribution for each metric, and we reported the 2.5 and 97.5 percentiles as
a confidence interval.

##  4 EXPERIMENTAL RESULTS

###  4.1 Training from scratch vs. fine-tuning

In the first experiment we compare training from scratch and fine-tuning the
ImageNet-1k pre-trained network on remote sensing image datasets. In Table 2
the test accuracies/F1-measures of the models trained or fine-tuned on 80% of
the images from the used source datasets are reported. From these results we
can observe that fine-tuning both supervised and self-supervised models pre-
trained on ImageNet outperforms training from scratch in all the cases.
However, for both variants of MLRSNet, as well as for PatternNet and RSI-CB,
the differences are very small, and for RESISC45 the difference is around 2%,
indicating that, even for medium-sized datasets, ImageNet pre-training plays a
diminishing role in HRRS scene classification. The differences between fine-
tuned supervised and self-supervised models are not statistically significant.

Dataset | Training
---|---
Scratch | Fine-tuning (supervised) | Fine-tuning (SwAV)
MLRSNet (multi-label) | 91.83 (91.69, 91.97) | 92.41 (92.27, 92.54) |  92.58 (92.45, 92.71)
MLRSNet (single-label) | 97.74 (97.55, 97.95) | 98.61 (98.46, 98.76) |  98.85 (98.70, 99.01)
RESISC45 | 95.11 (94.56, 95.65) |  97.04 (96.57, 97.48) | 96.87 (96.44, 97.30)
PatternNet | 99.49 (99.31, 99.65) |  99.84 (99.74, 99.93) | 99.82 (99.70, 99.92)
RSI-CB | 99.39 (99.17, 99.60) | 99.55 (99.35, 99.72) |  99.64 (99.47, 99.80)
AID | 93.92 (92.85, 95.00) | 97.30 (96.55, 98.00) |  97.85 (97.20, 98.45)
Table 2: Test accuracies/F1-measures (%) with 95% confidence intervals of
models trained/fine-tuned on 80% of the images from the HRRS datasets.

###  4.2 Transfer learning

####  4.2.1 Feature extraction

We now turn our attention to the transferability of representations learned on
ImageNet-1k and HRRS image datasets. The results obtained by training a
softmax classifier on the features extracted from each of the target datasets
using the pre-trained models are shown in Table 3. We can see that feature
extractors pre-trained on HRRS image datasets outperform supervised feature
extractors pre-trained on both ImageNet-100 and ImageNet-1k in all the cases.
Interestingly, self-supervised pre-training considerably outperforms
supervised pre-training on ImageNet-1k although in both cases domain-specific
images are not used. Moreover, self-supervised pre-training is comparable to
pre-training on remote sensing datasets, except in the case when RESISC45 is
the target dataset.

When HRRS image datasets are used for pre-training, the best results are
obtained using feature extractors pre-trained on MLRSNet, both single-label
and multi-label, with multi-label pre-training winning in 3 out of 5 cases. In
the tests on PatternNet and RSI-CB, single-label pre-training on MLRSNet
marginally outperforms multi-label pre-training but the difference is very
small and the performance is already saturated. Interestingly, pre-training on
RESISC45 results in only slightly worse classification accuracies than pre-
training on MLRSNet, despite being around three times smaller. Furthermore,
pre-training on RESISC45 is comparable to supervised pre-training on
ImageNet-1k and better than pre-training on ImageNet-100. Surprisingly, pre-
training on PatternNet and RSI-CB yields much worse representations compared
to all the other datasets. This was not expected having in mind that these
datasets are comparable in sizes with RESISC45, significantly larger than AID,
and with the classes similar to the classes in the target datasets.

Source dataset | Target dataset
---|---
MLRSNet (multi-label) |  MLRSNet (single-label) | RESISC45 | AID | PatternNet | RSI-CB | UCM
ImageNet-1k |  83.77 |  91.69 | 86.94 | 90.81 | 98.63 | 98.50 | 92.86
ImageNet-100 |  81.23 |  88.22 | 82.10 | 87.21 | 98.16 | 97.81 | 90.12
ImageNet-1k (SwAV) |  85.83 |  93.22 | 89.21 | 92.98 | 99.07 | 98.82 | 93.27
MLRSNet (multi-label) |  - |  - | 93.21 | 92.68 | 98.96 | 98.57 | 93.45
MLRSNet (single-label) |  - |  - | 92.57 | 91.09 | 99.03 | 98.87 | 92.32
RESISC45 |  85.19 |  91.96 | - | 90.47 | 98.56 | 98.08 | 92.14
AID |  80.83 |  85.78 | 79.81 | - | 96.99 | 96.49 | 86.49
PatternNet |  79.19 |  84.19 | 76.56 | 78.37 | - | 97.10 | 83.57
RSI-CB |  77.66 |  80.04 | 69.20 | 72.09 | 95.72 | - | 74.82
Table 3: Classification accuracies/F1-measures (%) on 20% of the images from
the target datasets obtained by training softmax classifiers on the features
extracted using the pre-trained networks. The best result for each target
dataset is given in bold, and the second best is underlined.

An intuitively plausible reason for good performance of the representations
learned on HRRS datasets is that the source and target datasets contain the
same or similar classes. To investigate this assumption we split MLRSNet into
two subsets, one containing the classes present in UCM and the other with the
rest of the classes. The sizes of these subsets are 50,197 and 58,964 images,
respectively. We also made a third MLRSNet subset containing all the classes
and half (54,573) the images. We train networks on all three subsets and use
them to extract features from UCM. The resulting classification accuracies are
given in Table 4.

Subset | Accuracy (%)
---|---
Same classes | 91.79
Different classes | 89.46
All classes | 92.26
Table 4: Classification accuracies on UCM when different subsets of MLRSNet
are used for training the feature extractor.

We observe that pre-training the feature extractor on the subset with the
classes present in UCM results in better classification accuracy compared to
pre-training on the subset with different classes. Moreover, pre-training on
the subset with the same classes results in higher accuracy compared to
training on, significantly larger, ImageNet-100. On the other hand, when the
subset with different classes is used, the performance drop is less than 2%
compared to the subset with the same classes and around half percent compared
to ImageNet-100. These results suggest that the performance indeed benefits
from pre-training on the same or similar classes as in the target task.

When the subset with all the classes is used, the classification accuracy is
additionally improved and is similar to the accuracies obtained by pre-
training on RESISC45, full single-label MLRSNet, and supervised ImageNet-1k
showing that in addition to domain similarity and overlap between the classes,
class diversity also plays a role in training good feature extractors.

####  4.2.2 Fine-tuning

The classification accuracies obtained by fine-tuning the pre-trained networks
on 20% of the images from the target datasets are given in Table 5. For
comparison, we also report the classification accuracies obtained by training
ResNet-50 on the target datasets from scratch. We see that both supervised and
self-supervised pre-training on ImageNet-1k outperform training from scratch
and pre-training on HRRS datasets in all the cases, with pre-training on
multi-label MLRSNet being worse by around 1%. Supervised and self-supervised
pre-training on ImageNet-1k result in comparable performances, with self-
supervised pre-training having a slight advantage. These results suggest that
both the number of images and class diversity in ImageNet-1k contribute to
obtaining a good initialization for fine-tuning in spite of the domain gap
between ImageNet-1k and remote sensing images. Nevertheless, it should be
noted that the differences are small challenging again the role of ImageNet-1k
as an ubiquitous pre-training dataset.

Source dataset | Target dataset
---|---
MLRSNet (multi-label) |  MLRSNet (single-label) | RESISC45 | AID | PatternNet | RSI-CB | UCM
Scratch |  89.19 |  93.87 | 85.44 | 79.14 | 98.04 | 97.29 | 58.93
ImageNet-1k |  90.53 |  96.62 | 93.85 | 94.40 | 99.51 | 99.15 | 94.64
ImageNet-100 |  88.35 |  93.79 | 88.99 | 90.95 | 99.03 | 98.78 | 87.86
ImageNet-1k (SwAV) |  90.81 |  97.27 | 94.48 | 95.37 | 99.65 | 99.25 | 94.29
MLRSNet (multi-label) |  - |  - | 93.75 | 93.60 | 99.19 | 99.00 | 93.81
MLRSNet (single-label) |  - |  - | 92.17 | 92.16 | 99.11 | 98.90 | 92.50
RESISC45 |  89.08 |  94.53 | - | 91.45 | 99.01 | 98.66 | 92.14
AID |  88.46 |  93.60 | 88.52 | - | 98.93 | 98.42 | 88.27
PatternNet |  87.71 |  91.83 | 83.76 | 83.28 | - | 98.13 | 86.73
RSI-CB |  86.57 |  90.26 | 80.29 | 79.19 | 98.02 | - | 78.39
Table 5: Classification accuracies/F1-measures (%) on 20% of the images from
the target datasets obtained by fine-tuning the pre-trained networks. The best
result for each target dataset is given in bold, and the second best is
underlined.

Pre-training on both variants of MLRSNet outperforms training from scratch in
all the cases, and pre-training on RESISC45 is worse than training from
scratch only in the case when the target task is multi-label MLRSNet.
Furthermore, pre-training on both variants of MLRSNet, as well as on RESISC45
and AID, outperforms pre-training on ImageNet-100 on all the target datasets
except PatternNet and RSI-CB, where the differences are very small.

Similarly to the feature extraction case, when PatternNet and RSI-CB are used
for pre-training the obtained results are worse than in all the other cases.
Moreover, for these two source datasets, the classification accuracies on most
of the target datasets are not improved compared to training from scratch. A
notable exception is UCM with too few training images for training a
classifier from scratch. This result is in line with (Pires de Lima and
Marfurt,, 2020).

By comparing the results in Table 3. and 5. we see that fine-tuning the
networks pre-trained on MLRSNet and RESISC45 for smaller target datasets only
slightly improves classification performances in comparison with training only
a classifier on the features extracted using the pre-trained network. On the
other hand, in both experiments with MLRSNet as the target dataset, fine-
tuning the network pre-trained on other HRRS source datasets results in
improved classification performance compared to training only a softmax
classifier. However, it should be noted that these fine-tuned networks exhibit
similar performances as the networks trained from scratch on both MLRSNet
tasks. Interestingly, fine-tuning the networks pre-trained on both
ImageNet-100 and ImageNet-1k in all the cases considerably improves
classification accuracies compared to the feature extraction case.

To examine the impact of the class overlap between the source and target
datasets on the fine-tuning performance we fine-tune the networks pre-trained
on the subsets of MLRSNet on UCM. The obtained results are shown in Table 6.
We can see that the obtained classification accuracies do not differ much
compared to the feature extraction case. The improvements are at most half
percent and the classification accuracies are comparable with those obtained
by pre-training on RESISC45 and full single-label MLRSNet. Similarly to the
feature extraction case, pre-training on the subset with the same classes
outperforms pre-training on the subset with different classes. However, more
diversity in the subset with all MLRSNet classes in this case does not bring
additional improvement.

Subset | Accuracy (%)
---|---
Same classes | 92.38
Different classes | 89.88
All classes | 92.38
Table 6: Fine-tuning accuracies on UCM when different subsets of MLRSNet are
used for pre-training.

###  4.3 Domain-adaptive pre-training

The results in Section 4.2.1 show that, when pre-trained models are used as
feature extractors, in-domain pre-training is comparable to self-supervised
ImageNet pre-training and both approaches outperform supervised ImageNet pre-
training. Furthermore, all the approaches result in similarly performing fine-
tuned models. Therefore, in order to leverage the advantages of both ImageNet
and in-domain pre-training we investigated the quality of the representations
obtained by domain adaptive (DA) pre-training, i.e. fine-tuning the network
pre-trained on ImageNet-1k using an in-domain dataset different from the
target dataset and using the resulting model as a feature extractor or
additionally fine-tuning it for the target task. In this section, we used
MLRSNet (single and multi-label variants) as the source dataset and RESISC45,
AID, and UCM as the target datasets. We excluded PatternNet and RSI-CB from
these experiments because their performances had already been saturated in the
previous experiments and pre-training on these datasets did not bring
improvements on the target tasks compared to training from scratch. In all the
cases we used supervised fine-tuning on MLRSNet. To get better insight into
the impact of class overlap between the dataset used for domain adaptation and
the target dataset we also performed DA pre-training using the subsets of
MLRSNet containing the same and different classes as UCM and evaluated the
domain-adapted models on UCM classification.

####  4.3.1 Feature extraction

The results obtained by training softmax classifiers on the features extracted
using domain-adapted models are given in the upper half of Table 7 (marked
with FE). In comparison with the results in Table 3 we can see that DA pre-
training improves the classification accuracies on the target datasets
compared to using only ImageNet-1k or MLRSNet pre-training. The improvement is
present when both supervised and self-supervised ImageNet pre-training are
used, with self-supervised pre-training outperforming supervised. In contrast
to pre-training only on MLRSNet, in this case the multi-label variant of
MLRSNet does not show any advantages over the single-label variant.

Pre-training | Transfer | Target dataset
---|---|---
RESISC45 | AID | UCM
ImageNet (supervised) $\rightarrow$ MLRSNet (multi-label) | FE | 94.50 | 92.51 | 94.35
ImageNet (supervised) $\rightarrow$ MLRSNet (single-label) | FE | 94.69 | 92.99 | 94.17
ImageNet (SwAV) $\rightarrow$ MLRSNet (multi-label) | FE | 94.54 | 93.29 | 94.96
ImageNet (SwAV) $\rightarrow$ MLRSNet (single-label) | FE | 95.24 | 93.92 | 96.89
ImageNet (supervised) $\rightarrow$ MLRSNet (multi-label) | FT | 94.27 | 94.31 | 93.87
ImageNet (supervised) $\rightarrow$ MLRSNet (single-label) | FT | 95.14 | 95.54 | 95.12
ImageNet (SwAV) $\rightarrow$ MLRSNet (multi-label) | FT | 95.49 | 96.17 | 96.55
ImageNet (SwAV) $\rightarrow$ MLRSNet (single-label) | FT | 95.89 | 96.09 | 97.14
Table 7: Classification accuracies (%) on the target datasets obtained by
transfer learning using the domain-adapted models. FE denotes using domain-
adapted models for feature extraction and FT fine-tuning the domain-adapted
model.

The classification accuracies obtained by using the feature extractors domain-
adapted on different subsets of MLRSNet are given in Table 8. We can see that
DA pre-training on the MLRSNet subset with the same classes as in UCM
outperforms DA pre-training on the subset with the different classes by around
4% and achieves almost the same results as when the MLRSNet subset with all
the classes and half the images is used. However, when the MLRSNet subset with
different classes is used for DA pre-training, the classification accuracies
on UCM are lower compared to ImageNet-1k pre-training only. Apparently, DA
pre-training shifted the features towards the discrimination between the
classes not present in UCM reducing their performance on UCM classification.
These results suggest that feature extractors can benefit from DA pre-training
in those cases when there exists class overlap between the dataset used for DA
pre-training and the target dataset.

Subset | Accuracy (%)
---|---
Same classes | 95.18
Different classes | 91.25
All classes | 95.30
Table 8: Classification accuracies on UCM when different subsets of MLRSNet
are used for DA training of the feature extractor.

####  4.3.2 Fine-tuning

The results obtained by fine-tuning the domain-adapted models on the target
datasets are given in the lower half of Table 7 (marked with FT). In
comparison with Table 5, we can see that DA pre-training improves the
classification accuracies compared to both supervised and self-supervised pre-
training on ImageNet-1k only. Moreover, the self-supervised model makes for a
better basis for DA pre-training than the supervised one.

Fine-tuning the models obtained by DA pre-training on the different subsets of
MLRSNet results in the classification accuracies given in Table 9. From these
results similar conclusions as in the case of feature extraction can be drawn.
Specifically, DA pre-training on the MLRSNet subset with the same classes as
in UCM is considerably better than pre-training on the MLRSNet subset with the
different classes. In this case, it is even better than DA pre-training with
all the classes and half of the training images. Furthermore, it outperforms
pre-training on both ImageNet-1k and MLRSNet (multi-label). Therefore, when
the models are fine-tuned for the target task, DA pre-training using a dataset
with class overlap with the target dataset is beneficial.

Subset | Accuracy (%)
---|---
Same classes | 95.77
Different classes | 92.44
All classes | 94.70
Table 9: Fine-tuning accuracies on UCM when different subsets of MLRSNet are
used for DA pre-training.

##  5 DISCUSSION

In summary, our experimental results show that training from scratch on most
of the used HRRS image datasets results in only slightly lower performance
than fine-tuning the ImageNet pre-trained models on the same datasets. For
example, the differences on both variants of MLRSNet, PatternNet and RSI-CB
are less than 1%, with somewhat larger gaps on RESISC45 and AID. These results
suggest that for larger and some medium-sized HRRS image datasets we might
avoid ImageNet pre-training and still achieve competitive results.

When pre-trained networks are used as fixed feature extractors, pre-training
on HRRS image datasets outperforms supervised pre-training on ImageNet.
Clearly, the features obtained using the networks pre-trained on HRRS images
are better suited for other HRRS scene classification tasks than the features
computed using the ImageNet pre-trained network. However, when a self-
supervised model is used as a feature extractor, the performances are similar
to pre-training on HRRS image datasets. On the other hand, when pre-trained
networks are end-to-end fine-tuned, supervised ImageNet pre-training slightly
outperforms pre-training on HRRS image datasets, while self-supervised
ImageNet models after fine-tuning outperform both supervised ImageNet and in-
domain models.

When PatternNet and RSI-CB are used as source datasets in both transfer
learning scenarios, the obtained classification accuracies are lower compared
to pre-training on other datasets. However, the classification accuracies on
PatternNet and RSI-CB exceed 99%, showing that the performance on a source
task is not always a good predictor of the performance on a target task. It is
possible that classification of PatternNet and RSI-CB is too easy, which
prevents the network from learning useful features for other HRRS scene
classification tasks.

The recent experiments on Million-AID (Long et al.,, 2022) showed that a large
HRRS image dataset can outperform ImageNet as a source dataset in a transfer
learning scenario and our results indicate that the gap between the models
trained from scratch and fine-tuned models pre-trained on ImageNet is very
narrow even for medium-sized HRRS image datasets. Nevertheless, there is
another benefit of ImageNet pre-training. The models pre-trained on ImageNet
and fine-tuned on a HRRS image dataset, i.e. domain-adaptive pre-trained
models, outperform the models pre-trained only on either ImageNet or MLRSNet
in both transfer learning scenarios. Nevertheless, although the improvements
can be readily achieved by domain-adaptive pre-training on medium-sized
datasets, our results suggest that it is important that there exists class
overlap between the dataset used for domain adaptation and the target dataset.

In Table 10 the results on RESISC45 and AID obtained using DA pre-training are
compared to recent state-of-the-art HRRS scene classification methods. Two of
the methods, ResNet50+EAN (Zhao et al.,, 2020) and PCNet(Zhang et al.,, 2021)
use the same ResNet-50 backbone as in our experiments, while GLDBS (Xu et
al.,, 2021) uses ResNet-34. The best results obtained using pre-training on
Million-AID used DenseNet-169 and ResNet-101 for classification of RESISC45
and AID, respectively. We can see that domain-adaptive pre-training without
any additional modifications of the model is able to surpass the
classification accuracies achieved using the recently proposed methods, as
well as pre-training on an order of magnitude larger Million-AID. Therefore,
DA pre-training can serve as a strong baseline for HRRS scene classification.
Furthermore, DA pre-training is orthogonal to other extensions of the ResNet
architecture proposed in the literature and we believe that it is possible
that their combination would lead to further increase in classification
accuracy.

Method | Target dataset
---|---
RESISC45 | AID
ResNet50+EAN (Zhao et al.,, 2020) | 93.51 | 93.64
GLDBS (Xu et al.,, 2021) | 94.46 | 95.45
PCNet (Zhang et al.,, 2021) | 94.59 | 95.53
Million-AID (Long et al.,, 2022) | 94.26 | 95.40
Domain-adaptive pre-training (ours) | 95.89 | 96.17
Table 10: Comparison with the state-of-the-art methods.

##  6 CONCLUSION

In this paper we empirically showed that, although ImageNet pre-training in
its traditional form gradually loses its appeal, we can obtain additional
improvements by using a second round of pre-training using in-domain data
i.e., domain-adaptive pre-training. Therefore, the answer to the question
posed in the title is positive and we can still benefit from ImageNet pre-
training by coupling it with domain-adaptive pre-training. Additionally, since
self-supervised pre-training is on par or better than supervised pre-training,
for pre-training we only need the images from ImageNet and not their labels.
An important consequence of our work is that domain-pretrained networks can be
used as backbones in all applications where supervised ImageNet pre-trained
networks have traditionally been used.

In the future work we plan to further investigate what makes a dataset
suitable for pre-training, as well as why fine-tuning the network pre-trained
on ImageNet outperforms fine-tuning the networks pre-trained on HRRS image
datasets in spite of the better results of the latter as feature extractors.
Another interesting question is the impact of the choice of the network layers
for fine-tuning on the classifier performance. Finally, transferability of
representations to other target tasks, such as object detection and semantic
segmentation is also an interesting direction of research.

## ACKNOWLEDGEMENTS

We thank Mihajlo Savić for insightful discussion. Valuable comments from the
reviewers are gratefully acknowledged.

This research was funded in part by the European Commission under the Horizon
2020 European research infrastructures grant agreement no. 857645 (NI4OS-
Europe project) and in part by the Ministry of Scientific and Technological
Development, Higher Education and Information Society of the Republic of
Srpska under contract 19.032/961-102/19, “Service for Classification of Remote
Sensing Images”.

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# On the effective surface energy in viscoelastic Hertzian contacts

L. Afferrante<EMAIL_ADDRESS>Department of Mechanics,
Mathematics and Management, Polytechnic University of Bari, Via E. Orabona, 4,
70125, Bari, Italy  G. Violano  Department of Mechanics, Mathematics and
Management, Polytechnic University of Bari, Via E. Orabona, 4, 70125, Bari,
Italy

###### Abstract

Viscoelasticity and rate-dependent adhesion of soft matter lead to
difficulties in modeling the ’relatively simple’ problem of a rigid sphere in
contact with a viscoelastic half-space. For this reason, approximations in
describing surface interactions and viscous dissipation processes are usually
adopted in the literature.

Here, we develop a fully deterministic model in which adhesive interactions
are described by Lennard-Jones potential and the material behaviour with the
standard linear solid model.

Normal loading-unloading cycles are carried out under different driving
conditions. When loading is performed in quasi-static conditions and, hence,
unloading starts from a completely relaxed state of the material, the
effective surface energy is found to monotonically increase with the contact
line velocity up to an asymptotic value reached at high unloading rates. Such
result agrees with existing theories on viscoelastic crack propagation.

If loading and unloading are performed at the same non-zero driving velocity
and, hence, unloading starts from an unrelaxed state of the material, the
trend of the effective surface energy $\Delta\gamma_{\mathrm{eff}}$ with the
contact line velocity is described by a bell-shaped function in a double-
logarithmic plot. The peak of $\Delta\gamma_{\mathrm{eff}}$ is found at a
contact line velocity smaller than that makes maximum the tangent loss of the
viscoelastic modulus. Furthermore, we show Gent&Schultz assumption partly
works in this case as viscous dissipation is no longer localized along the
contact perimeter but it also occurs in the bulk material.

viscoelasticity, adhesion hysteresis, effective surface energy, crack
propagation, finite element modelling

##  I Introduction

The adhesive contact between a rigid sphere and a viscoelastic substrate is a
problem of scientific and technological interest, as adhesion of soft
materials plays a key role in biomedicine Jeong2015 , soft robotics
Shintake2018 , sensors Yao2020 , tapes Villey2015  and printing industries
Meitl2006 . Moreover, the contact mechanics of soft matter is not yet fully
understood, due to its intrinsic viscoelasticity and rate-dependent adhesive
features ViolanoPart1 ; ViolanoPart2 .

In the 1960s, Lee & Radok Lee1960  investigated the adhesiveless contact
between a rigid sphere and a viscoelastic half-space introducing a simple
method to replace the elastic modulus with the viscoelastic creep function.
However, their solution is exact only for the loading phase. The solution for
the unloading phase was formulated by Ting Ting1966 , whose method is based on
the use of the correspondence principle and Laplace transforms. Greenwood
Greenwood2010  defined Ting’s approach for finding the contact area and
applied load an algebraically messy procedure. To avoid the use of Ting’s
repeated integrations, he proposed to exploit the superposition of an assembly
of viscoelastic Boussinesq circular punch indentations.

In the 1970s, two fundamental theories on adhesion of elastic spheres were
formulated: Johnson, Kendall & Roberts (JKR) theory JKR1971  and Derjaguin,
Muller & Toporov (DMT) theory DMT1975 . The first is based on the assumption
of infinitely short-range adhesive interactions acting inside the contact
area. The latter instead neglects interactions inside the contact area (where
a Hertzian contact shape is assumed) and considers long-range adhesive
interactions acting outside it. Tabor Tabor  showed JKR and DMT theories are
the extreme limits of a single theory parametrized by the parameter
$\mu=\left[R\Delta\gamma^{2}/\left(E^{\ast
2}\varepsilon^{3}\right)\right]^{1/3}$ (known as Tabor’s parameter), where $R$
is the radius of the sphere, $\Delta\gamma$ is the surface energy of adhesion,
$E^{\ast}$ is the composite elastic modulus and $\varepsilon$ is the range of
action of van der Waals forces. In fact, DMT theory works in the limit of
$\mu\ll 1$ (i.e., for small and stiff spheres), while JKR theory applies to
$\mu\gg 1$ (i.e., for large and compliant spheres).

More recently, Maugis Maugis1992 , by exploiting the Dugdale approximation
Dugdale , proposed an analytical model based on the assumption to represent
the surface force in terms of a Dugdale cohesive zone approximation. In this
model (known as Maugis-Dugdale (MD) model) the work of adhesion is assumed to
be $\Delta\gamma=\sigma_{0}d_{0}$, where $\sigma_{0}$ is the maximum force
predicted by the Lennard-Jones force law and $d_{0}$ is the separation at
which the areas under the Dugdale and Lennard-Jones curves match. JKR, DMT and
MD theories are derived in quasi-static conditions and, therefore, cannot be
applied to viscoelastic solids exhibiting rate-dependent adhesion ViolanoPart1
. Real soft materials, like natural rubbers and synthetic polymers, seldom
show a linear elastic constituive behaviour. In fact, the contact mechanical
response of soft matter depends on the instantaneous value of its viscoelastic
modulus, which is a function of the temperature and the frequency of
excitation.

Starting from the 1980s, different strategies for studying adhesion of
viscoelastic solids have been derived in the framework of classical fracture
mechanics MB1980 ; GreenJohn1981 ; Muller1999 . Gent & Schultz (GS) GS1972
and Maugis & Barquins (MB) MB1980  showed, with experiments, the apparent
surface energy depends on the contact line velocity. Such dependence was also
predicted by Schapery in his theoretical studies on crack’s propagation in
viscoelastic media Schapery1975a . The solutions developed in Refs. MB1980 ;
GreenJohn1981 ; Muller1999  moved from GS assumption that predicts viscous
dissipation is localized along the contact perimeter, which can be seen as the
tip of a circular crack. However, such assumption requires that the material
behaves elastically, which is not always the case.

In 2000s, Haiat et. al Haiat2003  formulated a generalization of Ting’s
adhesiveless solution by extending the restricted self-consistent adhesive
model by Tabor Tabor  to viscoelastic spheres. This model requires to solve
simultaneously two equations describing the time-dependence of the contact and
adhesive zones, respectively. For complex contact histories, nested time
integration may lead to decreasing feasibility of the method. Moreover,
adhesive interactions are modeled according to a double-Hertz approximation
G-J1998 . An alternative approach has been proposed by Lin & Hui Lin2002 , who
performed Finite Element (FE) simulations of the adhesive contact between
viscoelastic spheres. In their model, adhesion is implemented by means of a
Dugdale cohesive zone model. Furthermore, Barthel & Fretigny BF2009  have
proposed theoretical expressions to estimate the impact of viscoelasticity on
effective surface energy. More recently, Sukhomlinov & Müser have extended the
original Green’s Function Molecular Dynamics approach Campana&Muser  to
investigate the viscous dissipation caused by rough indenters in sliding
motion Sukhomlinov&Muser .

The depicted state of the art highlights the absence of a model that is able
(i) to describe the real physics of the adhesive interactions at the contact
interface, and (ii) to capture rate-dependent adhesion of soft matter, without
any assumptions on the location of viscous dissipation. To achieve these
manifold objectives, we herein develop a fully deterministic Finite Element
model aimed at investigating the viscoelasticity induced adhesion enhancement
in the contact of a spherical indenter with a smooth flat substrate. In our
model, adhesion is described by means of non-linear springs following the
Lennard-Jones traction-gap law. We study the enhancement of the effective
surface energy with the contact line velocity during retraction. Numerical
results are compared with the predictions of the crack propagation models of
Greenwood Greenwood2004 , and Persson & Brenner (PB) PB2005 , which both show
to be accurate. We also discuss the validity of GS assumption and show when
viscous dissipation may occur in the bulk material.

##  II Formulation

Figure 1 gives a graphical representation of the problem under investigation:
a rigid hemisphere of radius $R$ is pressed into a viscoelastic half-space and
then pulled apart from it. The normal driving velocity $V=d\delta/dt$ is
assumed constant during the loading and unloading, being $\delta$ the contact
penetration. Notice $\delta$ is measured at $r=0$ and corresponds to the
normal displacement of the surface of the half-space with respect to its
undeformed configuration (black dashed line). We denoted with $a$ and $F$, the
contact radius and applied load, respectively, which is assumed positive when
compressive.

Figure 1: The problem under investigation: a rigid hemisphere of radius $R$ is
pressed into a viscoelastic half-space and then pulled apart from it.

A standard linear solid is used to describe viscoelasticity in the half-space.
Figure 2 shows the dependence of the viscoelastic modulus $E$ on the frequency
$\omega$. Specifically, Fig. 2a shows the variation with the dimensionless
frequency $\omega\tau$ of the real and imaginary parts of $E$, while Fig. 2b
the variation of the loss tangent $Im[E]/Re[E]$ with $\omega\tau$.

(a)

(b)

Figure 2: (a) The real (solid line) and imaginary (dashed line) parts of the
viscoelastic modulus $E(\omega)$ for a standard linear viscoelastic solid.
$Re[E]$ and $Im[E]$ are shown in terms of the dimensionless frequency
$\omega\tau$. The curves are given for $E(0)/E(\infty)=0.1$. (b) The
dependence of the loss tangent $Im[E]/Re[E]$ on the dimensionless frequency
$\omega\tau$.

At low frequencies, the material is in the ‘rubbery’ region where the real
part $Re[E]$ of the viscoelastic modulus is constant and the viscous
dissipation related to the imaginary part $Im[E]$ is negligible (loss tangent
$Im[E]/Re[E]\approx 0$). At high frequencies, i.e., in the ‘glassy’ region,
viscous dissipation is again negligible and the real part of $E$ is constant
but much larger than in the ‘rubbery’ region. Viscous dissipation occurs at
intermediate frequencies in the ‘transition’ region where the loss tangent
takes non-zero values.

###  II.1 Numerical model

A Finite Element (FE) model (Fig. 3) has been developed to study the problem
sketched in Fig. 1. The substrate has been modeled with axisymmetric plane
elements with linear shape functions. Constraints are applied at the bottom
boundary. For the rigid sphere we need to define only nodes on the surface
where a single master node has been identified. On this node force and
displacement are applied and its degree of freedoms are coupled with those of
all other sphere nodes by simple constraint equations.

Figure 3: Graphical representation of the finite element model.

The adhesive interaction has been modeled by nonlinear spring elements with a
traction-displacement relation according to the traction-gap law of Lennard-
Jones (L-J)

$p\left(r\right)=\frac{8\Delta\gamma}{3\varepsilon}\left[\left(\frac{\varepsilon}{g\left(r\right)}\right)^{3}-\left(\frac{\varepsilon}{g\left(r\right)}\right)^{9}\right]\text{.}$
(1)

Notice the compression (elongation) of the springs is the sum of the total
’rigid’ displacement $\Delta$ of the sphere and the surface displacement $u$
of the deformable substrate

$g_{0}\left(r\right)-g\left(r\right)=\Delta+u\left(r\right)$ (2)

being $g_{0}=h_{0}+r^{2}/\left(2R\right)$ the initial gap function, where
$h_{0}$ is the assumed initial distance between the sphere and the substrate
at $r=0$.

This type of approach was already adopted in previous works (see, for example,
Refs. Kadinetal2008 ; SongKomvo2011 ) to investigate similar problems with
linear elastic and/or elasto-plastic materials.

As a result, the adhesive force exerted by a spring connected to a surface
node at distance $r$ from the contact center is
$dF\left(r\right)=p\left(r\right)2\pi rdr$, except for the central node where
we have $dF\left(r\right)=p\left(r\right)\pi dr^{2}$.

The substrate has been modeled as a linear viscoelastic material.
Specifically, a classical linear standard model was used with a single
relaxation time $\tau$ and a Prony series representation was adopted

$E\left(t\right)=E_{0}+\left(E_{\infty}-E_{0}\right)\exp\left(-t/\tau\right)$
(3)

where $E_{0}$ is the elastic modulus at zero frequency and $E_{\infty}$ is the
elastic modulus at high frequency.

The stress $\mathbf{\sigma}$ is calculated according to the following
constitutive equation

$\mathbf{\sigma=}\int_{0}^{t}E\left(t-t^{\prime}\right)\frac{d\mathbf{\varepsilon}}{dt^{\prime}}dt^{\prime}$
(4)

where $\mathbf{\varepsilon}$ is the strain and $E$ the relaxation function.

To calculate stresses at time $t_{i+1}=t_{i}+\Delta t$, the classical
incremental finite element procedure has been used

$\mathbf{\sigma}\left(t_{i+1}\right)=\mathbf{\sigma}\left(t_{i}\right)\exp\left(-\Delta
t/\tau\right)+\int_{t_{i}}^{t_{i+1}}\left(E_{\infty}-E_{0}\right)\exp\left(-\frac{t_{i+1}-t^{\prime}}{\tau}\right)\frac{d\mathbf{\varepsilon}}{dt^{\prime}}dt^{\prime}.$
(5)

Finally, the dissipated energy in the viscoelastic substrate is calculated as
the energy required to deform the dashpot in the Maxwell element. Therefore,
the increment of energy dissipated over a time increment $\Delta t$ is

$\Delta E_{d}=\mathbf{\sigma}\cdot\Delta\mathbf{\varepsilon}_{dashpot}\Delta
t$ (6)

where $\Delta\mathbf{\varepsilon}_{dashpot}$ is the dashpot strain increment
over time $\Delta t$ given by

$\Delta\mathbf{\varepsilon}_{dashpot}=\Delta\mathbf{\varepsilon}-\alpha\Delta\mathbf{\sigma}$
(7)

being $\alpha$ the Maxwell spring compliance.

The prepared code was solved with the aid of ANSYS solver.

##  III Results and Discussion

All results presented in this section are obtained for $E_{0}/E_{\infty}=0.1$
and
$\mu_{0}=\left[R\Delta\gamma^{2}/\left(E_{0}^{2}\varepsilon^{3}\right)\right]^{1/3}\approx
3.85$ and are given in terms of the following dimensionless quantities:
$\hat{V}=V\tau/\varepsilon$, $\hat{a}=a/R$, $\hat{\delta}=\delta/\varepsilon$,
$\hat{F}=F/\left(1.5\pi R\Delta\gamma\right)$, and
$\hat{p}=p\varepsilon/\Delta\gamma$.

(a)

(b)

Figure 4: (a) The dimensionless force $\hat{F}$ as a function of the
dimensionless imposed approach $\hat{\delta}$. Loading (dashed line) is
performed at vanishing velocity ($V\approx 0$) so one can neglect viscous
effects. Unloading (coloured solid lines) is instead performed at different
driving velocities $\hat{V}$. Dotted lines connect the two branches of the
curves where jump-in (loading phase) and jump-off (unloading phase)
instabilities occur. Results are given for a standard linear solid with
$E_{0}/E_{\infty}=0.1$ and
$\mu_{0}=[R\Delta\gamma^{2}/(E_{0}^{2}\varepsilon^{3})]^{1/3}\approx 3.85$.
(b) The dimensionless contact radius $\hat{a}$ as a function of the
dimensionless force $\hat{F}$. Loading (dashed line) is performed at vanishing
velocity ($V\approx 0$) so one can neglect viscous effects. Unloading
(coloured solid lines) is instead performed at different driving velocities
$\hat{V}$. Dotted lines identify the regions where jump-off instabilities
occur. Notice no perceptible difference between the curves can be identified
for $\hat{V}>33.33$. Results are given for a standard linear solid with
$E_{0}/E_{\infty}=0.1$ and
$\mu_{0}=[R\Delta\gamma^{2}/(E_{0}^{2}\varepsilon^{3})]^{1/3}\approx 3.85$.

A first set of numerical simulations has been run at vanishing approaching
velocity ($V\approx 0$) to neglect time dependent effects during the loading
phase. In this way, we are sure that the material is in a strain relaxed state
at the end of loading and the following unloading phase always starts from the
same initial conditions corresponding to the ‘rubbery’ region of Fig. 2.
Figure 4 shows the resulting curves after a cycle of loading-unloading
performed at different pulling velocities. Specifically, with reference to
Fig. 4a, hysteresis dissipation (given by the difference between the work
spent to bring the bodies in contact and that required to separate them)
occurs even when the pulling velocity is negligible (i.e., in absence of
viscous effects). In such case, energy loss (quantitatively given by the area
between the loading and unloading curves) is exclusively due to the adhesive
hysteresis related to the jumping-on and jumping-off phenomena (dotted lines
identify the points where jump-on and jump-off instabilities occur). In this
respect, Refs. Wangetal2021 ; ViolAff2021  have proposed a comprehensive
investigation of the elastic adhesive hysteresis occurring between approach
and retraction.

As expected, increasing the pulling velocity, dissipation increases and the
unloading curves (coloured solid lines) more and more deviate from the loading
one (dashed line). As a result, the pull-off force $F_{\mathrm{PO}}$ (i.e.,
the maximum tensile force) is observed to monotonically increase. It reaches a
maximum value at about $\hat{V}\approx 33.33$, after which it remains
constant. Moreover, notice that at the highest velocities snap-off occurs at
positive (compressive) penetrations $\delta$ because increasing the pulling
velocity the time for the viscoelastic substrate to ‘relax’ and recovery its
deformation reduces. As a result, the sphere imprint on the substrate can be
clearly observed even when the sphere has been completely detached.

Figure 4b shows the dependence of the contact radius $a$ on the applied force
during the loading and unloading simulations. In the present context, a
rigorous definition of the contact radius can be disputable because Lennard-
Jones force law imposes the gap between contacting surfaces is always nonzero.
As observed by Feng in Ref. Feng2001 , where he investigated the adhesive
contact between a rigid half-space and a deformable elastic sphere, the edge
of the flattened area coincides well with the point where the tensile stress
reaches its peak when the Tabor parameter $\mu>1$. This agrees with previous
observations of Greenwood Greenwood1997  who proposed to identify the edge of
contact area at the point where the tensile stress takes its maximum value.
However, for smaller $\mu$, the edge of the contact area is less easily
identified and seems to coincide with the point where the local Lennard-Jones
pressure becomes zero Feng2001 . In our calculations, we find a perfect
coincidence of the deformed profile of the substrate with that of the Hertzian
rigid indenter up to the point where the tensile stress is maximum. Therefore,
in the present calculations, the contact radius is identified as the radius at
which the pressure takes its maximum tensile value. No perceptible difference
between the curves can be identified for $\hat{V}>33.33$. Also, during the
initial phase of unloading, the contact radius is practically constant. Such
period (identified as stick time in Refs. Haiat2003 ; ViolanoPart1 ; ViolFront
) increases with the speed and reaches a maximum at about $\hat{V}=33.33$,
above which no further variation is observed.

The pull-off process of a sphere from a substrate is analogous to the opening
of a circular crack. In cracks propagation, viscoelastic dissipation is known
to determine an increase in the propagation energy
$\Delta\gamma_{\mathrm{eff}}$ Schapery1975a ; Greenwood2004 ; PB2005 . It
depends on the nature of the processes occurring in the crack-tip process
zone, which is the region close to the crack tip where the classical Irwin
equations Irwin1968  relating stresses to the distance from the crack tip are
no longer valid.

It has been shown experimentally MB1980 ; GS1972  that
$\Delta\gamma_{\mathrm{eff}}$ depends on the crack tip velocity
$V_{\mathrm{c}}$ and the temperature $T$ according to

$\Delta\gamma_{\mathrm{eff}}=\Delta\gamma[1+f(a_{\mathrm{T}}V_{\mathrm{c}})]$
(8)

where $a_{\mathrm{T}}$ is the Williams-Landel-Ferry (WLF) WLF1955  factor
depending on the temperature and $f(a_{\mathrm{T}}V_{\mathrm{c}})$ is the
function of the viscoelastic energy dissipation in front of the crack tip.
This function is characteristic of the viscoelastic material and is
independent of the contact geometry (see MB1980 ).

In the literature, two different approaches have been developed for studying
crack propagation in viscoelastic solids. The first approach is based on the
cohesive-zone model, which assumes a constant stress $\sigma_{0}$ in the
process zone. Greenwood Greenwood2004  exploited a Maugis-Dugdale surface
force law to find the dependence of the surface (propagation) energy on the
crack tip velocity. For the standard linear solid used in our work, one
obtains

$\Delta\gamma_{\mathrm{eff}}=\Delta\gamma\left[\frac{E_{0}}{E_{\infty}}+\frac{1}{2}\left(1-\frac{E_{0}}{E_{\infty}}\right)\frac{d}{V_{\mathrm{c}}\tau}\frac{E_{0}}{E_{\infty}}\int_{0}^{1}d\xi
H(\xi)e^{-\frac{d}{V_{\mathrm{c}}\tau}\frac{E_{0}}{E_{\infty}}(1-\xi)}\right]^{-1}$
(9)

where

$H(\xi)=2\xi^{1/2}-(1-\xi)\ln\left(\frac{1+\xi^{1/2}}{1-\xi^{1/2}}\right)$
(10)

and $d=\pi E_{0}\Delta\gamma/(4\sigma_{0}^{2})$ is the width of the process
zone.

The second approach is instead based on energetic equilibrium. It was
developed by Persson & Brenner (PB) PB2005 , who showed that the effective
energy required to advance the crack tip by one unit area is related to the
viscoelastic modulus and can be calculated theoretically as

$\Delta\gamma_{\mathrm{eff}}=\Delta\gamma\left[1-\frac{2E_{0}}{\pi}\int_{0}^{2\pi
V_{\mathrm{c}}/s}d\omega\frac{F(\omega)}{\omega}Im\left(\frac{1}{E(\omega)}\right)\right]^{-1}$
(11)

where $F(\omega)=[1-(\omega s/(2\pi V_{\mathrm{c}}))^{2}]^{1/2}$. In eq. (11),
$s=s(V_{\mathrm{c}})$ depends on the crack tip velocity and can be determined
if one assumes that the stress at the crack tip takes some critical value
$\sigma_{\mathrm{c}}$ necessary to break the atomic bonds for the tip to
propagate. This gives

$\frac{\Delta\gamma_{\mathrm{eff}}}{\Delta\gamma}=\frac{s}{s_{0}}$ (12)

where $s_{0}$ is the crack tip radius for a very slowly moving crack and takes
values of the order of the nanometer. Specifically, the crack tip cut-off
radius $s_{0}$ in the Persson-Brener model is evaluated as
$s_{0}=E_{0}\Delta\gamma/\left(2\pi\sigma_{c}^{2}\right)$, where $\sigma_{c}$
is the characteristic stress necessary for bond breaking at the crack tip.
Notice $\sigma_{c}$ is not equivalent to $\sigma_{0}$ in the cohesive-zone
approach as instead erroneously assumed in recent papers ciava ;
ciavaMcMeeking .

The mechanism regulating the contact opening occurring during the unloading
phase in our numerical simulations is the same as the crack propagation at the
interface between a viscoelastic solid and a rigid countersurface Persson2021
. Maugis & Barquins MB1980  and Charmet et al.Charmet1999  observed that
viscoelastic losses are localized at the crack tip and bulk displacements are
purely elastic. Therefore, a first estimate of the increase in the effective
surface energy $\Delta\gamma_{\mathrm{eff}}$ due to viscoelastic dissipation
in the vicinity of the front of detachment can be done according to (see
Lorenz2013 )

$\frac{\Delta\gamma_{\mathrm{eff}}}{\Delta\gamma}\approx\frac{F_{\mathrm{PO}}}{F_{\mathrm{PO,}V_{\mathrm{c}}=0}}$
(13)

where $F_{\mathrm{PO,}V_{\mathrm{c}}=0}$ is the pull-off force measured in the
quasi-static limit ($V\approx 0$).

Figure 5: Relative increase of the interfacial viscoelastic energy
$(\Delta\gamma_{\mathrm{eff}}-\Delta\gamma)/\Delta\gamma$ as a function of the
dimensionless contact line velocity $\hat{V}_{\mathrm{c}}$ measured at snap-
off. Results are shown in a double logarithmic plot. The red dashed line is
the Greenwood’s solution (9) obtained with $\sigma_{0}\approx
1.026\Delta\gamma/\varepsilon$. Solid line refers instead to the prediction
given by Persson-Brener (PB) equation (11) with $s_{0}/\varepsilon\approx
3.33$, i.e., $s_{0}=1$ $\mathrm{nm}$ for $\varepsilon=0.3$ $\mathrm{nm}$. Data
are given for a standard linear solid with $E_{0}/E_{\infty}=0.1$ and
$\mu_{0}=[R\Delta\gamma^{2}/(E_{0}^{2}\varepsilon^{3})]^{1/3}\approx 3.85$.

Figure 5 shows, in a double logarithmic plot, the relative increase of the
effective surface energy $\Delta\gamma_{\mathrm{eff}}$ in terms of the
detachment front propagation velocity $V_{\mathrm{c}}=-da/dt$. Specifically,
$V_{\mathrm{c}}$ is estimated at pull-off (i.e., at the instant where the
tensile force is maximum). Greenwood’s solution (red dashed line) and PB’s one
(black solid line) are also shown as comparison. The first is obtained with
$\sigma_{0}$ equal to the maximum stress of Lennard-Jones law
($\sigma_{0}\approx 1.026\Delta\gamma/\varepsilon$), while the latter assuming
$s_{0}/\varepsilon\approx 3.33$ (i.e., $s_{0}=1$ nm for $\varepsilon=0.3$ nm).

As expected (see Refs. GreenJohn1981 ; Greenwood2004 ; PB2005 ),
$\Delta\gamma_{\mathrm{eff}}\approx\Delta\gamma
E_{\infty}/E_{0}\gg\Delta\gamma$ for high pulling velocities.

Figure 6: The dimensionless contact pressure
($\hat{p}=p\varepsilon/\Delta\gamma$) distribution on the contact region for
different values of the dimensionless imposed approach $\hat{\delta}$ and
dimensionless driving velocity $\hat{V}=3.33$. Results are given for a
standard linear solid with $E_{0}/E_{\infty}=0.1$ and
$\mu_{0}=[R\Delta\gamma^{2}/(E_{0}^{2}\varepsilon^{3})]^{1/3}\approx 3.85$.

Figure 6 shows the contact pressure distribution on the contact region for
different values of the approach $\delta$. Dashed lines refer to the loading
phase, while solid lines to the unloading one. The occurrence of positive
tractions is clearly observable at the edges of the contact area when contact
is established.

(a)

(b)

Figure 7: (a) The dimensionless force $\hat{F}$ as a function of the
dimensionless imposed approach $\hat{\delta}$ for different driving velocities
$\hat{V}$. Loading (dashed lines) and unloading (solid lines) are performed at
the same $\hat{V}$. Dotted lines connect the two branches of the curves where
the jump-in (loading phase) and jump-off (unloading phase) instabilities
occur. Results are given for a standard linear solid with
$E_{0}/E_{\infty}=0.1$ and
$\mu_{0}=[R\Delta\gamma^{2}/(E_{0}^{2}\varepsilon^{3})]^{1/3}\approx 3.85$.
(b) The dimensionless contact radius $\hat{a}$ as a function of the
dimensionless force $\hat{F}$ for different driving velocities $\hat{V}$. The
contact radius is identified as the radius at which the contact pressure takes
its maximum tensile value. Loading (dashed line) and unloading (solid lines)
are performed at the same velocity. Dotted lines identify the regions where
jumping instabilities occur. Results are given for a standard linear solid
with $E_{0}/E_{\infty}=0.1$ and
$\mu_{0}=[R\Delta\gamma^{2}/(E_{0}^{2}\varepsilon^{3})]^{1/3}\approx 3.85$.

When the approximation of quasi-static loading is no longer valid and, hence,
the material is not in a completely relaxed stress state at the beginning of
unloading, the behaviour of the system significantly changes, as shown in Fig.
7. Here, loading and unloading are performed at the same driving velocity.
Also, unloading starts right after the loading finishes so that the substrate
material does not have the time to ‘relax’.

With reference to Fig. 7a, the small dissipation occurring at vanishing
($V\approx 0$) and very high ($\hat{V}>500$) velocities is exclusively due to
the adhesive hysteresis related to jumping instabilities. The pull-off force
$F_{\mathrm{PO}}$ first increases with the unloading speed, reaches a maximum
and then reduces in a similar way to the hysteresis dissipation. This
behaviour agrees with the viscoelastic modulus curves given in Fig. 2. In
fact, only in the transition region we expect energy loss due to viscoelastic
hysteresis. Therefore, at low and high speeds, i.e., low and high frequencies
of excitation, the only dissipation is that related to the adhesion hysteresis
due to the jump-on and jump-off instabilities. In this case, we can clearly
identify the two regions where the material behaves elastically with modulus
$E_{0}$ (low velocities) and $E_{\infty}$ (high velocities).

Figure 7b, instead, shows the contact radius, i.e., the radius where the
contact pressure takes its tensile peak, in terms of the applied force. No
difference can be identified between the loading and unloading curves at
$\hat{V}\approx 0$ and $\hat{V}>500$ as the substrate material behaves
elastically. However, at high velocities the system is much stiffer, and the
same contact area is reached at much larger forces. Moreover, in the regions
where the material is ‘elastic’ (rubbery and glassy regions) the contact
radius significantly decreases right from the early stages of unloading. On
the contrary, at intermediate driving velocities, we observe an initial phase
where the behaviour of the contact radius is more similar to that observed in
Fig. 4b.

Figure 8: The dependence of the relative increase of the effective surface
energy $\Delta\gamma_{\mathrm{eff}}$ on the dimensionless contact line
velocity $\hat{V}_{\mathrm{c}}$ for a standard linear solid with
$E_{0}/E_{\infty}=0.1$ and
$\mu_{0}=[R\Delta\gamma^{2}/(E_{0}^{2}\varepsilon^{3})]^{1/3}\approx 3.85$.
Results are presented in a double logarithmic plot.

Figure 8 shows the relative increase of the effective surface energy
$\Delta\gamma_{\mathrm{eff}}$ in terms of the contact line velocity
$V_{\mathrm{c}}$ as obtained from numerical simulations performed at different
driving velocities $V$. As above explained, $V_{\mathrm{c}}$ is estimated at
pull-off when we reach the maximum tensile force. Results are quite different
with respect to Fig. 4. Indeed, after reaching a maximum value,
$\Delta\gamma_{\mathrm{eff}}$ reduces when moving towards high contact line
velocities A bell-shape curve for the effective surface energy has been also
found in Ref. Lin2002  by numerical simulations on two contacting viscoelastic
spheres and in Ref. Gustavo  with classical JKR-like experiments.

The velocity where $\Delta\gamma_{\mathrm{eff}}$ takes its largest value does
not correspond to the frequency at which the loss tangent is maximum. In fact,
the frequency of excitation can be roughly estimated as $\sim V/a$ and the
peak of $\Delta\gamma_{\mathrm{eff}}$ is reached at about $V\approx
4\varepsilon/\tau$. Therefore, at pull-off being the contact radius $\sim
0.147R$, the frequency of excitation can be estimated as $\omega\tau\approx
0.055$, which falls in the transition region but is not however close to the
value where the loss tangent peak occurs (see Fig. 2b). A similar result is
reported in Ref. francesi , where the fracture energy is found to reach its
maximum value for low speeds ” _even if the ’liquid zone’ has not emerged yet_
”.

Furthermore, in agreement with the considerations done in Ref. Israelachvili
and the calculations performed in Ref. Lin2002 , we observe that the maximum
dissipation occurs at velocity at which the time of observation (or time scale
of the process) $t_{\mathrm{o}}$ is comparable to the characteristic time
$\tau E_{\infty}/E_{0}$. Therefore, the peak of $\Delta\gamma_{\mathrm{eff}}$
is expected to occur close to $De\approx 1$, where

$De=\frac{\tau E_{\infty}/E_{0}}{t_{\mathrm{o}}}$ (14)

is the so-called Deborah number Reiner1964 . Indeed, we find $De\approx 1.35$
when the interaction time $t_{\mathrm{o}}$ is estimated as the time between
jump-on and jump-off phenomena.

Figure 9: The accumulated dissipated energy density at the initial phase of
unloading (a-c) and at pull-off (b-d). In (a-b) loading is carried out under
quasi-static conditions ($V\approx 0$) and unloading at pulling velocity
$\hat{V}=1$. In (c-d) loading and unloading are performed at the same driving
velocity $\hat{V}=1$. Results are obtained for a standard linear solid with
$E_{0}/E_{\infty}=0.1$ and
$\mu_{0}=[R\Delta\gamma^{2}/(E_{0}^{2}\varepsilon^{3})]^{1/3}\approx 3.85$.

The different behaviour observed in Figs. 4 and 7, can find a possible
explanation in results shown in Fig. 9, where the accumulated dissipated
energy density is plotted at different times (at the initial phase of
unloading and at pull-off). Specifically, in Fig. 9a-b, loading is carried out
under quasi-static conditions ($V\approx 0$) and unloading at pulling velocity
$\hat{V}=1$, while in Fig. 9c-d loading and unloading are performed at the
same $\hat{V}=1$. As expected, in the first case, dissipation mainly occurs at
the interface, near the front of detachment. Here, therefore, the classical
Gent&Schultz assumption of considering viscous effects localized exclusively
close to the crack tip is reasonably satisfied. In the latter case, we notice
that this assumption fails as dissipation basically occurs in the bulk.

Figure 10: The dimensionless contact pressure
($\hat{p}=p\varepsilon/\Delta\gamma$) distribution on the contact region for
different values of the dimensionless imposed approach $\hat{\delta}$ and
$\hat{V}=4$. Dashed lines refer to the loading phase, solid lines to the
unloading one. Results are given for a standard linear viscoelastic solid with
$E_{0}/E_{\infty}=0.1$ and
$\mu_{0}=[R\Delta\gamma^{2}/(E_{0}^{2}\varepsilon^{3})]^{1/3}\approx 3.85$.

Finally, Fig. 10 shows the contact pressure distribution in different instants
of the loading-unloading cycle. Again, tractions at the edges of the contact
area arise when contact is established. Moreover, we observe that, at high
penetrations, the pressure on the compressive contact region is roughly
constant.

##  IV Conclusions

In this paper, we have presented a fully deterministic numerical model for
studying the adhesive normal contact between a rigid sphere and a viscoelastic
substrate. Van der Waals adhesive interactions are described by means of non-
linear springs with a traction-gap relation according to the Lennard-Jones
law, while material viscoelasticity is modeled by a standard linear solid with
a spring in parallel with a Maxwell element constituted by a spring in series
with a dashpot.

In a first set of simulations, we have imposed quasi-static loading conditions
($V\approx 0$) to ensure adhesive equilibrium and negligible viscous effects.
In such case, the material behaves as a soft elastic medium (rubbery region).
Unloading has been instead performed for different driving velocities. We
found that, as unloading starts from a completely relaxed state of the
material, a progressive enhancement of the effective surface energy with the
contact line velocity is found up to an asymptotic value. This is reached for
high contact line velocities that correspond to excitation frequencies falling
in the glassy region of the material, where it behaves as a stiff elastic
medium. Numerical data of $\Delta\gamma_{\mathrm{eff}}(V_{\mathrm{c}})$ are
found in good agreement with the theoretical predictions of two crack’s
propagation theories, namely the cohesive-zone model proposed by Greenwood
Greenwood2004  on the basis of Schapery’s findings, and the energetic approach
developed by Persson & Brenner PB2005 .

In a second set of simulations, we have performed loading and unloading at the
same driving velocity. No dwell time has been waited before unloading. In such
case, the trend of the effective surface energy with the contact line velocity
is described by a bell-shaped function. The maximum of the bell is found for
exciting frequency lower than the frequency that maximize the tangent loss of
the viscoelastic modulus in agreement to previous studies. Moreover, the peak
of dissipation is found close to $De=1$. We have also discussed how the
Gent&Schultz assumption works in this case showing that viscous dissipation is
not always localized along the contact perimeter but it may occur in the bulk
material. In such case, adhesion models based on linear elastic fracture
mechanics are expected to be less accurate.

###### Acknowledgements.

L.A. and G.V. acknowledge support from the Italian Ministry of Education,
University and Research (MIUR) under the program ”Departments of Excellence”
(L.232/2016). Moreover, this work was supported by the project ”FASTire (Foam
Airless Spoked Tire): Smart Airless Tyres for Extremely-Low Rolling Resistance
and Superior Passengers Comfort” funded by the Italian MIUR Progetti di
Ricerca di Rilevante Interesse Nazionale (PRIN) call 2017—grant n. 2017948FEN.

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