**3. Metabolic improvements with the intragastric baloon**

The intragastric baloon is a spherical silicone elastomer balloon that is resistant to degradation by gastric acid for approximately 6 months. It can be placed endoscopicallyand filled with 400 to 700 ml of saline and methylene blue dye, which changes the color of the urine in the event of balloon rupture.

Balloon insertion and removal are performed under conscious sedation or general anesthesia. Before the insertion, an upper gastrointestinal endoscopy is performed to detect possible contraindications to the procedure. The baloon placement device is inserted through the mouth into the stomach. Then the balloon is positioned in the fundus under endoscopic control, and inflated by injecting saline mixed with 10-ml methylene blue into the catheter. Finally, once the desired volume has been injected, the balloon is released by a short pull on the catheter. The baloon should be removed after a maximum of 6 months because beyond this period, the risk of spontaneous balloon deflation significantly increases.

A meta-analysis by Imaz et al. [21] of 15 studies comprising 3698 patients estimated 14.7 kg weight loss, 32.1% excess weight loss (EWL), and 5.7 kg/m2 decrease in BMI after 6 months.

In a review including 22 studies with a total of 4371 patients implanted with the intragastric baloon, demonstrated a mean weight loss of 17.8 Kg, with extremes of the means of 4.9– 28.5 kg and higher absolute values observed in higher BMI categories. [22].

A prospective study, evaluating the effect of the baloon on weight, insulin resistance, and liver steatosis in obese patients showed that 76% of the patients had a BMI decrease of 3.5 Kg/m2 or more. The mean (SD) weight loss with respect to baseline values was 16.4 (8.2) kg with a corresponding mean (SD) BMI reduction of 6.4 (3.2) kg/m2. The absolute percentage of participants with glycemia levels of 100 mg/dL or higher decreased from 50% to 12%, those with triglyceridemia 150 mg/dL or higher from 58% to 19%, and those with abnormal ALT level from 38% to 7% [23].

Two studies (one randomized, one uncontrolled) totaling 143 patients have reported that, one year after BIB removal, patients had regained 41% and 28% (mean values, respectively) of the absolute weight loss observed at BIB removal [24,25]. Another study following 88 patients for a median of 22 month after baloon withdrawal, observed that (50%) regained some weight, 34 (39%) maintained their weight, and the remaining 10 (11%) continued to lose weight [23].

It is also important to consider that 20–40% of patients fail to achieve a significant weight loss (often defined as ≥10% baseline weight or ≥25% excess weight). Such failures may be related to the request of early baloon removal by patients who present a digestive or psychological intolerance to the baloon, to the early vanishing of anticipated effects on hunger and early satiety, or to patient's adaptation of food intake [23].

In conclusion, the BIB strategy may be an alternative to current management of obesity focused on lifestyle changes, drug therapy, and treating associated metabolic complications.

Although the baloon has not yet proved to be a convincing means of primary long term weight loss, it holds some promise for improving co-morbidities and quality of life in nonmorbidly obese patients or those who are unwilling to undergo bariatric surgery. New perspectives are also beginning to show its potential value in specific patient groups especially, for example, those preparing for surgery.


**2.3 Chile**

The best laboratories for diagnosis of PID in Chile are located in Santiago, Temuco, Valparaiso and Concepcion. Initial screening tests for PID can be performed in large hospitals, although it does not receive government financial support (Goic & Armas, 2010). As in most underdeveloped countries in Latin America, the diagnosis of PID is often performed after numerous episodes of infection and treatment, and patient referral to specialist in infectious diseases, and finally to immunology center. Patients with PID, who need IVIG, are not reimbursed by the public health system for PID, burdening the costs to patients, which makes the treatment is not performed or interrupted. Chile has a three-year residency in immunology at the University of Santiago of Chile, providing training care of adult and pediatric patients (Condino-Neto *et al.,* 2011).


**Pathology and Repair**

**References**

76 Histology

Elsevier; 2012. pp. 253-277

Cell. 2014;**30**:449-462

1992;**172**:305

2005;**33**:223-233

2011;**17**:345-354

Wolters Kluwer; 2006. pp. 251-258

[1] Nanci A. Ten Cate's Oral Histology Development, Structure and Function. 8th ed.

[2] Kumar GS. Orban's Oral Histology and Embryology. 13th ed. Elsevier; 2014. pp. 291-312 [3] Eroschenko VP. DiFiore's Atlas of Histology with Functional Correlations. 11th ed.

[4] Chaurasia's BD. Human Anatomy Regional and Applied Dissection and Clinical: Head,

[5] Cutler LS, Gremski W. Epithelial-mesenchymal interaction in the development of the

[7] Nedvetsky PI, Emmerson E, Finley JK, Ettinger A, Cruz-Pacheo N, et al. Parasympathetic innervation regulates tubulogenesis in the developing salivary gland. Developmental

[8] Melvin JE, Yule D, Shuttleworth T, Begenisich T. Regulation of fluid and electrolyte secretion in salivary gland acinar cells. Annual Review of Physiology. 2005;**67**:445-469 [9] Egdar WM. Saliva: Its secretion, composition and function. British Dental Journal.

[10] Dodds MW, DA J, Yeh CK. Health benefits of saliva: A review. Journal of Dentistry.

[11] Spielmann N, Wong DT. Diagnostic and therapeutic perspectives. Oral Diseases.

[6] Avery JK. Oral Development and Histology. 3rd ed. BC Decker Inc; 1988. pp. 316-319

Neck and Brain. 4th ed. Satish Kumar Jain; 2006. pp. 133-138; 157-162

salivary gland. Critical Reviews in Oral Biology and Medicine. 1991;**2**(1):1

**Chapter 5**

**Provisional chapter**

**A Study of the Correlation between Bacterial Culture**

**A Study of the Correlation between Bacterial Culture**

*Helicobacter pylori* (*H. pylori*) is one of the most common chronic bacterial infections in the world, and it is currently estimated that approximately half of the world's population is infected with the bacterium. The correct diagnosis and effective treatment of *H. pylori* gastric infection are essential in controlling this condition. The available diagnostic methods have advantages and limitations related to factors such as age of patients, technical difficulty level, costs and extensive accessibility in hospitals. The eradication therapy of *H. pylori* infection is still a challenge for gastroenterologists. One of the main causes of failure in *H. pylori* eradication is antibiotic resistance. Biopsy cultures are the most widely used methods among the antimicrobial susceptibility tests. In case of a negative culture, *H. pylori* can be clearly recognised in histological sections. The sensitivity and specificity of histology for the diagnosis depend on clinical settings, density of colonisation and the experience of the histopathologist. A prospective study was performed in order to analyse patients with *H. pylori* gastric infection with positive histology and positive culture versus positive histology and negative culture.

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

*Helicobacter pylori* (*H. pylori*) is a Gram-negative spiral-shaped bacterium which colonises the gastrointestinal tract of nearly half of the world's population, causing local inflammation in

DOI: 10.5772/intechopen.80257

**and Histological Examination in Children with**

**and Histological Examination in Children with**

*Helicobacter pylori* **Gastritis**

*Helicobacter pylori* **Gastritis**

Mihaela Bălgrădean

**Abstract**

**1. Introduction**

Mihaela Bălgrădean

Felicia Galoș, Gabriela Năstase, Cătălin Boboc, Cristina Coldea, Mălina Anghel, Anca Orzan and

Felicia Galoș, Gabriela Năstase, Cătălin Boboc, Cristina Coldea, Mălina Anghel, Anca Orzan and

Additional information is available at the end of the chapter

**Keywords:** *Helicobacter pylori* culture, histology, children

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.80257

#### **A Study of the Correlation between Bacterial Culture and Histological Examination in Children with** *Helicobacter pylori* **Gastritis A Study of the Correlation between Bacterial Culture and Histological Examination in Children with**  *Helicobacter pylori* **Gastritis**

DOI: 10.5772/intechopen.80257

Felicia Galoș, Gabriela Năstase, Cătălin Boboc, Cristina Coldea, Mălina Anghel, Anca Orzan and Mihaela Bălgrădean Felicia Galoș, Gabriela Năstase, Cătălin Boboc, Cristina Coldea, Mălina Anghel, Anca Orzan and Mihaela Bălgrădean

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.80257

#### **Abstract**

*Helicobacter pylori* (*H. pylori*) is one of the most common chronic bacterial infections in the world, and it is currently estimated that approximately half of the world's population is infected with the bacterium. The correct diagnosis and effective treatment of *H. pylori* gastric infection are essential in controlling this condition. The available diagnostic methods have advantages and limitations related to factors such as age of patients, technical difficulty level, costs and extensive accessibility in hospitals. The eradication therapy of *H. pylori* infection is still a challenge for gastroenterologists. One of the main causes of failure in *H. pylori* eradication is antibiotic resistance. Biopsy cultures are the most widely used methods among the antimicrobial susceptibility tests. In case of a negative culture, *H. pylori* can be clearly recognised in histological sections. The sensitivity and specificity of histology for the diagnosis depend on clinical settings, density of colonisation and the experience of the histopathologist. A prospective study was performed in order to analyse patients with *H. pylori* gastric infection with positive histology and positive culture versus positive histology and negative culture.

**Keywords:** *Helicobacter pylori* culture, histology, children

## **1. Introduction**

*Helicobacter pylori* (*H. pylori*) is a Gram-negative spiral-shaped bacterium which colonises the gastrointestinal tract of nearly half of the world's population, causing local inflammation in

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

the stomach and duodenum and inducing a systemic humoral immune response [1]. *H. pylori* survival in the acidic gastric environment is mediated by mechanisms such as activity of the urease enzyme, which catalyses and hydrolyses urea to form carbon dioxide and ammonia, producing a neutral environment that is essential for its survival. The primary routes of transmission are considered to be faecal-oral and oral-oral, but some indirect evidences report that the infection can also be acquired by drinking water and by other environmental sources [2, 3].

before the beginning of the study, previous intestinal surgery (except for polypectomy and appendectomy), concomitant severe disease (heart, lungs, kidney and endocrine diseases)

A Study of the Correlation between Bacterial Culture and Histological Examination in Children…

http://dx.doi.org/10.5772/intechopen.80257

81

All patients underwent endoscopy with biopsy specimens for histology (one for the antrum, one for the corpus). One sample from the antrum was used for rapid urease test. Two additional biopsies were taken from the antrum for bacterial culture. The samples were placed into separate vials, previously identified, containing the appropriate medium for each test.

This procedure was performed in patients with a minimum of 10 hours of fasting, under general anaesthesia or conscious sedation. Vital signs were continuously monitored for the

Written informed consent was obtained from the parent or guardian of each child included

The biopsy specimens collected for bacterial culture were transported in commercial selective transport *H. pylori* medium, Portagerm pylori (BioMérieux SA, Marcy l'Etoile, France), and were inoculated after a few hours onto selective medium pylori agar (BioMérieux Italia). The plates were incubated under microaerobic condition at 37° for 72 hours. Once incubated, the colonies resembling *H. pylori* were identified by Gram stain and by oxidase, catalase and urease tests. Suspensions from the primary plates were prepared in sterile solution to perform an E-test on pylori agar. An agar plate was streaked in three directions with a swab dipped into each bacterial suspension to produce a lawn of growth; an E-test strip (E-test; AB Biodisk, Solna, Sweden) was placed each onto separate plates, which was immediately incubated in a microaerobic atmosphere at 37° for 72 hours. Isolated strains were tested for amoxicillin, clarithromycin, metronidazole and levofloxacin resistance following the recommendations of

A biopsy of the gastric body and antrum was fixed in a solution of formaldehyde 10%. Subsequently, the gastric mucosa samples were processed, following the usual steps of dehy-

Two stains were used for histological study: haematoxylin-eosin and Giemsa. Haematoxylin-eosin stain was used to evaluate inflammatory cells and *H. pylori* (**Figure 1**). Giemsa stain was needed when haematoxylin-eosin stain failed to identify the bacterium (**Figure 2**). Giemsa stain is the preferred stain for detecting *H. pylori* because of its technical simplicity, high sensitivity and low cost.

the European Committee on Antimicrobial Susceptibility Testing.

and smoking or alcohol consumption among adolescents.

The study was approved by the ethics committee.

The first sample was used for bacterial culture.

**2.2. Endoscopy**

entire procedure.

**2.3. Bacterial culture**

in the study.

**2.4. Histology**

dration and paraffin embedding.

*H. pylori* represents a key factor in the aetiology of various gastrointestinal diseases, ranging from asymptomatic chronic active gastritis to peptic ulceration, gastric adenocarcinoma and gastric mucosa-associated lymphoid tissue (MALT) lymphoma. Other diseases caused by the pathogen are iron deficiency anaemia, chronic idiopathic thrombocytopenic purpura and growth retardation. *H. pylori* was classified as a class I human carcinogen by the World Health Organization in 1994.

Numerous diagnostic tests are available for detecting *H. pylori* infection: invasive techniques, which means endoscopy with biopsies for a rapid urease test (RUT), histology and culture and noninvasive techniques, such as serology, 13C-Urea breath test (13C-UBT) and stool antigen test. There is no single method to detect *H. pylori* infection reliably and accurately. The choice of the diagnostic method depends on patients' age and complaints, technical difficulty level, costs and extensive accessibility in hospitals.

Two tests are recommended to define *H. pylori* status in children: bacterial culture of gastric biopsy and histology [4]. Bacterial culture of gastric biopsy has 100% specificity, but its sensitivity is low. *H. pylori* can be cultured from gastric biopsies, although this method often presents some difficulties. *H. pylori* soon loses viability when exposed to the environment, and biopsies should be cultured quickly. If it is not possible, a transport media may be used. Histology provides an excellent diagnostic accuracy, allowing for the detection of the bacteria as well as for the grading of gastritis. The sensitivity and specificity of histology for the diagnosis depend on clinical settings, density of colonisation and experience of the histopathologist [5].

The aims of our study were to assess the histological findings and to compare them with the results of bacterial cultures, obtained through gastric biopsy, in children with *H. pylori* gastritis. We also wanted to find out the possible factors that may influence bacterial culture outcomes.


*Edited by Yueh-Hsin Lo,*

*Juan A. Blanco and Shovonlal Roy*

The term biodiversity has become a mainstream concept that can be found in any newspaper at any given time. Concerns on biodiversity protection are usually linked to species protection and extinction risks for iconic species, such as whales, pandas and so on. However, conserving biodiversity has much deeper implications than preserving a few (although important) species. Biodiversity in ecosystems is tightly linked to ecosystem functions such as biomass production, organic matter decomposition, ecosystem resilience, and others. Many of these ecological processes are also directly implied in services that the humankind obtains from ecosystems. The first part of this book will introduce different concepts and theories important to understand the links between ecosystem function and ecosystem biodiversity. The second part of the book provides a wide range of different studies showcasing the evidence and practical implications of such relationships.

ISBN 978-953-51-2028-5

Biodiversity in Ecosystems - Linking Structure and Function

Photo by cturtletrax / iStock


**2. Data**

As for crude oil prices, we used the nominal daily data derived from the US Energy Information Administration (EIA, https://www.eia.gov/dnav/pet/hist/RWTCD.htm (accessed on 1 May 2021)) for West Texas Intermediate (WTI). After computing the daily log-returns, we obtained quarterly overall, upside ("good"), and downside ("bad") realized variances by taking the sum of the daily squared returns, positive returns only, and negative returns only over a specific quarter. As a robustness check, we also analyzed the quarterly *RV* of the Brent crude oil returns, which, in turn, was also sourced from the US EIA (see https://www.eia.gov/dnav/pet/hist/RBRTED.htm (accessed on 1 May 2021)).

Figure 1 plots the *RV* (and its "good" and "bad" counterparts, as defined in Section 3) of both the WTI and Brent crude oil returns. During the GFC, sharp fluctuations in *RV*s were observed over 2020 associated with the COVID-19 outbreak, thus, highlighting the importance of our question.

Uncertainty is a latent variable and, hence, requires methods to measure it. As documented by Gupta et al. [27], there are three broad approaches to quantify uncertainty, apart from the various ones associated with financial markets (such as implied-volatility indices, like the popular VIX, realized volatility, idiosyncratic volatility of equity returns, and corporate spreads): (1) a text-based approach, with the main idea to construct indices from searches of key words or terms related to (economic and policy) uncertainty in major newspapers or country-reports; (2) using stochastic-volatility estimates from various small and large-scale structural models (related to macroeconomics and finance) to derive measures of uncertainty; and (3) using the dispersion of professional forecaster disagreements to obtain uncertainty estimates.

For our metric of uncertainty, we used the first approach outlined by Ahir et al. [28], mainly because it is not model-specific, as it does not require any complicated estimation of a large-scale model to generate it in the first place. In addition to the uncertainty data, the associated spillover of the G7 economies and China to other economies in the world,

are available publicly for download (https://worlduncertaintyindex.com/data/ (accessed on 1 May 2021)).

**Figure 1.** Realized volatility. For better readability, the vertical axes of the panels are not the same for all time series.

From frequency counts of "uncertainty" (and its variants) in the quarterly Economist Intelligence Unit (EIU) country reports from 1996 for 8 of the 143 countries, Ahir et al. [28] constructed quarterly indices of economic uncertainty for 37 countries in Africa, 22 in Asia and the Pacific, 35 in Europe, 27 in the Middle East and Central Asia, and 22 in the Western Hemisphere. The EIU reports provide an analysis and forecasts of political, policy, and economic conditions, as well as a discussion of significant political and economic developments in each country. These reports are compiled by a central EIU editorial team from work done by country-specific teams of analysts.

In order to make the uncertainty indexes comparable across countries, the raw counts were scaled by the total number of words in each report. In addition to the uncertainty indexes of each of the 143 countries, the dataset of Ahir et al. [28] also provides the uncertainty spillover metrics for the G7 and China, which, in turn, determine the choice of the countries in our paper, and the quarterly sample period of 1996Q1 to 2020Q4, which was the latest available data at the time of writing this paper. Specifically, the eight (G7 plus China) uncertainty spillover indexes of one of these particular countries to the remaining 142 countries was computed by counting the percent of word "uncertain" (or its variant) mentioned within a proximity to a word related to a particular G7 country or China in the EIU country reports.

The spillover index was then rescaled by multiplying 1,000,000 with a higher number suggesting higher uncertainty related to the specific country involving the G7 or China and vice versa. For further details regarding the words related to the G7 and China that were used, the reader is referred to Ahir et al. [28]. We used the cross-sectional sum over time to obtain the total uncertainty spillover (on to the remaining 142 economies) indexes of each of these eight countries.

Understandably, since we aimed to contribute to the oil *RV* forecasting literature by analyzing whether accounting for spillovers of uncertainty of the G7 countries and China to the rest of the world mattered over and above the uncertainty of these economies, we relied on the method of Ahir et al. [28] for a matter of consistency and similarity in how both these indexes are derived, even though alternative ways of constructing country-level uncertainty indexes, though not spillovers, are available in the public domain (see, for example, the indexes avilable at: http://policyuncertainty.com/ (accessed on 9 July 2021) based on the work of Baker et al. [29]).

Figures 2 and 3 plot the uncertainty time series and the international spillover effects. While the uncertainty series tended to fluctuate consistently over time, the spillovers had sudden massive spikes from the country(ies) of origin of the GFC, the European sovereign debt crisis, "Brexit", and the outbreak of the Coronavirus pandemic.

**Figure 2.** Time series of uncertainty. For better readability, the vertical axes of the panels are not the same for all time series.

**Figure 3.** Time series of international spillovers. For better readability, the vertical axes of the panels are not the same for all time series.


**6.6 Shock absorber spring**

The material of the spring should have high fatigue strength, high ductility, high resilience and it should be creep resistant, hence carbon steel was chosen. The free length of 100 mm and the spring wire diameter was determined to be 5 mm.


**Author details**

Tor Erik Jørgensen and Steinar Daae Johansen\*

\*Address all correspondence to<EMAIL_ADDRESS>
Faculty of Biosciences and Aquaculture, Genomics Group, Nord University, Bodø, Norway


*Article* **Numerical Homogenization of Multi-Layered Corrugated Cardboard with Creasing or Perforation**

**Tomasz Garbowski 1 , Anna Knitter-Pi ˛atkowska 2, \* and Damian Mrówczy ´nski 3**

**Abstract:** The corrugated board packaging industry is increasingly using advanced numerical tools to design and estimate the load capacity of its products. This is why numerical analyses are becoming a common standard in this branch of manufacturing. Such trends cause either the use of advanced computational models that take into account the full 3D geometry of the flat and wavy layers of corrugated board, or the use of homogenization techniques to simplify the numerical model. The article presents theoretical considerations that extend the numerical homogenization technique already presented in our previous work. The proposed here homogenization procedure also takes into account the creasing and/or perforation of corrugated board (i.e., processes that undoubtedly weaken the stiffness and strength of the corrugated board locally). However, it is not always easy to estimate how exactly these processes affect the bending or torsional stiffness. What is known for sure is that the degradation of stiffness depends, among other things, on the type of cut, its shape, the depth of creasing as well as their position or direction in relation to the corrugation direction. The method proposed here can be successfully applied to model smeared degradation in a finite element or to define degraded interface stiffnesses on a crease line or a perforation line.

**Keywords:** corrugated cardboard; numerical homogenization; strain energy equivalence; perforation; creasing; flexural stiffness; torsional stiffness

## **1. Introduction**

Colorful boxes and packaging are designed to attract the customers' attention and, as a consequence, to drive the sales of various goods ranging from bulky products, through food, children's toys, cosmetics, and many others. A growing awareness of concern for the natural environment has led many companies to opt for packaging that can be easily recycled or disposed of, biodegradable, and space-saving after manufacturing. A corrugated cardboard undoubtedly has all of these qualities. Moreover, it is easy to print on, for example, the brand name. Corrugated cardboard is easy to shape via creasing along the suitable lines and, furthermore, creating openings, ventilation holes, or perforations does not cause much difficulty. The latter is essential with regard to shelf-ready packaging (SRP) or retail-ready packaging (RRP) when the product, after transportation to the site, is placed on the shelves and after tearing off the flap along the appropriately designed perforation, is ready for sale. Thus, a lot of time is saved, which nowadays leads to significant profits for large companies.

Of course, one cannot only focus on the aesthetic values because the packaging, in fact, plays a much more important role such as securing the goods during storage or safe transport to the destination place. The load-bearing capacity of the corrugated cardboard boxes and the influence of humidity, openings and perforation arrangement, or the location of flaps is under constant investigation. Therefore, scientific research has become an integral

**Citation:** Garbowski, T.; Knitter-Pi ˛atkowska, A.; Mrówczy ´nski, D. Numerical Homogenization of Multi-Layered Corrugated Cardboard with Creasing or Perforation. *Materials* **2021**, *14*, 3786. https:// doi.org/10.3390/ma14143786

Academic Editor: Michele Bacciocchi

Received: 31 May 2021 Accepted: 5 July 2021 Published: 6 July 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

part of a distinct branch of industry (i.e., cardboard packages production). Manufacturers of these packaging types strive for effective, economical, and easy-to-use solutions, which results in the continuous, lasting over many years, development of research on cardboard strength while using various analytical, numerical, and experimental methods.

Compressive, tensile, or bursting strength tests are routinely executed to assess the load-bearing capacity of corrugated cardboard boxes. The box compression test (BCT) and the edge crush test (ECT) are the best known. Inextricably related to the mechanical strength of the paperboard or corrugated cardboard boxes are two characteristic in-plane directions of orthotropy (i.e., perpendicular to the main axis of the fluting and parallel to the paperboard fiber alignment—machine direction (MD) as well as parallel to the fluting—cross direction (CD)).

Another option for estimating the compressive strength of the boxes is the application of analytical formulae in which, in general, three groups of parameters such as paper, board, and box parameters are present [1]. Ring crush test (RCT), Concora liner test (CLT), liner type, weights of liner and fluting, corrugation ratio, and a constant related to fluting belong to the first group. Thickness, flexural stiffnesses in MD and CD, ECT, and moisture content are affiliated with the second group whereas dimensions and perimeter of the box, applied load ratio, stacking time, buckling ratio, and printed ratio are in the third one. Already in 1952, Kellicutt and Landt [2] proposed the calculations of box compressive strength while employing the formula with parameters introduced in the paper (RCT, flute constant) and box (perimeter, box constant). In 1956, Maltenfort [3] indicated the relation between the critical force and paper parameters (CLT, type of liner) and cardboard box dimensions in the BCT. In the approach proposed by McKee, Gander, and Wachuta [4] in 1963, the parameters of the paperboard (ECT, flexural stiffnesses) and the box perimeter were applied. Even though this formula is commonly used in the packaging industry due to its simplicity, which leads to quick and easy solutions for practical implementations, it is applicable only to simple standard boxes. Therefore, scientists have been making attempts to extend the implementation of McKee's analytical approach. Allerby et al. [5] modified the constants and exponents, whilst Schrampfer et al. [6] improved McKee's method by expanding the range of cutting methods and equipment. Batelka et al. [7] augmented the relationship by introducing the dimensions of the box and Urbanik et al. [8] included the Poisson's ratio. Further modification of the above-mentioned McKee's formula for solving more complex problems has been proposed by Aviles et al. [9] and later, by Garbowski et al. [10–12].

Over recent decades, meshless and meshfree methods (e.g., the collocation method) have become popular numerical techniques for solving partial differential equations and have been beneficial while considering corrugated cardboard problems. Wang and Qian [13] proposed the meshfree stabilized collocation method (SCM) and introduced the reproducing kernel function as the approximation. Wang et al. [14] employed the meshfree radial basis collocation method (RBCM), which utilizes infinitely continuous radial basis functions (RBFs), as the approximation for the static and dynamic eigenvalue analysis of the thin functionally graded shells (FGSs) with in-plane material inhomogeneity. The buckling analysis of thin FG plates, also with in-plane material inhomogeneity, while applying radial basis collocation method (RBCM) and Hermite radial basis function collocation method (HRBCM) was discussed by Chu et al. [15]. The main advantages of the above-mentioned approaches are high accuracy and exponential convergence.

Unquestionably, many determinants affect the compression strength of the corrugated paperboard boxes [16] including the moisture content of the box [17,18], openings, ventilation holes and perforations [11,12,19], storage time and conditions [20], stacking load [21], or a very significant one—creasing. As a result of such a process, fold and perforation lines are performed and through this, the mechanical strength of the manufactured corrugated paperboard boxes is diminished.

A very effective, commonly applied in engineering, technique to determine the strength of the boxes is the finite element method (FEM). Thakkar et al. [22] compared the experimental and FEM numerical results to investigate the creasing impact on the local

strength of corrugated paperboard; Beex and Peerlings [23], in turn, conducted physical and numerical experiments to examine the influence of creasing and subsequent folding on the mechanical properties of the laminated paperboard. A constitutive model was implemented by Giampieri et al. [24] in order to obtain the mechanical response of creased paperboard after folding. FEM simulations of paperboard creasing, which appeared to be significant from a practical standpoint, have been proposed by Domaneschi et al. [25] and Awais et al. [26]. Leminena et al. [27] performed experimental and numerical analyses to examine the influence of the creasing process during the press forming on the paperboard mechanical properties. FEM has also been involved in research raising the issue of numerical analysis in relation to transverse shear stiffness of the corrugated cardboards [28–32] or buckling and post-buckling phenomena [33].

The examined models can be facilitated to one single layer described by the effective properties of the composite instead of building layers composed of different materials. Such a method, called homogenization, has been used extensively over the last years by Garbowski et al. [32,34–37]. A clear advantage of this technique is the significant saving in calculation time while preserving the precision of the results. Hohe [38] proposed a representative element of the heterogeneous and homogenized elements based on strain energy to analyze sandwich panels. A periodic homogenization method presented by Buannic et al. [39] enabled them to obtain an equivalent membrane and pure bending characteristics of period plates and, in a modified version, to incorporate the transfer shear effect in the analysis. Biancolini [40] engaged FEM to study a micromechanical part of the considered plate. Thanks to the energy equivalence between the model and the homogenized plate, the stiffness properties of the sandwich plate were received. Decomposition of the plate into two beams in directions of the plate allowed Abbès and Guo [41] to define the torsion rigidity of the orthotropic sandwich plates. An interesting approach based on empirical observation can also be found in the recent work of Gallo et al. [21]. A multiple scales asymptotic homogenization approach was presented by Ramírez-Torres et al. [42] where the effective properties of hierarchical composites with periodic structure at different length scales has been studied, whereas in [43], the authors used the asymptotic homogenization technique to the equations describing the dynamics of a heterogeneous material with evolving micro-structure, obtaining a set of upscaled, effective equations.

The following article, as the next one in the series, provides theoretical considerations that develop and extend the numerical homogenization technique already presented in the prior works of the authors. The proposed homogenization procedure also takes into consideration the creasing and/or perforation of corrugated board (i.e., processes that evidently weaken the stiffness and strength of the corrugated board locally). However, it is not always easy to estimate how exactly these processes affect the bending or torsional stiffness. The fact is that the decrease in stiffness depends, among others, on the type of cut, its shape, and the depth of creasing as well as their position or direction in relation to the corrugation orientation. The method proposed here can be successfully implemented to model smeared degradation in a finite element or to define degraded interface stiffnesses on a crease line or a notch line.

#### **2. Materials and Methods**

#### *2.1. Corrugated Board—Material Definition*

Corrugated board, as a fibrous material, is characterized by strong orthotropy. The mechanical properties of its components (i.e., cardboard) depend on the direction of the fibers in the individual layers of the composite. Paper and paperboard are more than twice as stiff in the machine direction (MD) than in the cross direction (CD). This is related to the fibers which, due to the production process, arrange along the MD. In this direction, the material is more resistant to tearing and crushing, although it has lower ductility than in CD (see Figure 1).

ۏ ێ ێ ێ ۍ ଵଵߝ ଶଶߝ ଵଶߝ2 ଵଷߝ2 ےଶଷߝ2 ۑ ۑ ۑ ې =

ۏ ێ ێ ێ ۍ

ଶܧ ଵܧ

ଵଶߥ ଵܧ = ଶଵߥ ଶܧ

**Figure 1.** Paperboard mechanical behavior. The stress–strain relationships in different material directions.

000 ଶܧ⁄ଶଵߥ− ଵ ⁄ܧ 1 000 ଶ ⁄ܧ 1 ଵܧ⁄ଵଶߥ− 0 0 ଵଶܩ01⁄ 0 0 ଵଷܩ1⁄ 00 0 ےଶଷܩ1⁄ 00 0 0

ଶଵߥ ଵଶߥ ଵଶܩ

ۑ ۑ ۑ ې ۏ ێ ێ ێ ۍ ଵଵߪ ଶଶߪ ଵଶߪ ଵଷߪ ےଶଷߪ ۑ ۑ ۑ ې

The linear elastic orthotropic material can be described by the following stress– strain relationships:

$$
\begin{bmatrix}
\varepsilon\_{11} \\
\varepsilon\_{22} \\
2\varepsilon\_{12} \\
2\varepsilon\_{13} \\
2\varepsilon\_{23}
\end{bmatrix} = \begin{bmatrix}
1/E\_1 & -\nu\_{21}/E\_2 & 0 & 0 & 0 \\
0 & 0 & 1/G\_{12} & 0 & 0 \\
0 & 0 & 0 & 1/G\_{13} & 0 \\
0 & 0 & 0 & 0 & 1/G\_{23}
\end{bmatrix} \begin{bmatrix}
\sigma\_{11} \\
\sigma\_{22} \\
\sigma\_{12} \\
\sigma\_{13} \\
\sigma\_{23}
\end{bmatrix} \tag{1}
$$

where *E*1 is the Young's modulus in the machine direction (MD); *E*2 is the Young's modulus in the cross direction (CD); *G*12 is the Kirchhoff's modulus, *ν*12; *ν*21 is the Poisson's coefficients. Due to the symmetry of the material compliance/stiffness matrix, the relationship between the Poisson's coefficients is as follows:

$$\frac{\nu\_{12}}{E\_1} = \frac{\nu\_{21}}{E\_2} \tag{2}$$

(MPa) (MPa) (MPa)

The material orientation was always the same in all layers (see Figure 2). This is related to the corrugated board production process in which the paper (for the production of both flat and corrugated layers) is rolled on a corrugator machine from multi-tone bales.

**Figure 2.** Material orientation.

ଶଷܩ ଵଷܩ ଵଶܩ ଵଶݒ ଶܧ ଵܧ The paperboard, as already mentioned, was modeled here using classical linear elastic orthotropy (see Equation (1)). The material data were taken from the literature [40,44,45]. All material data are presented in Table 1 (i.e., *E*1, *E*2, *v*12, *G*12, *G*13 and *G*23, which represents Young's moduli in both directions, Poisson's ratio, in-plane shear modulus and two transverse shear moduli, respectively).

ࣇ

**Table 1.** Material data of intact double wall corrugated cardboard used for modeling the paper layers according to orthotropic constitutive relation.

The thickness of all flat layers (liners) in both single- and double-walled corrugated boards was assumed to be 0.30 mm; for all corrugated layers (flutes) in both models, the thickness was also taken as 0.30 mm.

ࣇ

(MPa) (MPa) (MPa)

#### *2.2. Creases and Perforations—Numerical Study*

The main goal of this work was to numerically analyze many cases of perforation with possible creasing and its effect on the stiffness reduction of corrugated board. The variants include not only different types of perforation (e.g., 4/4—4 mm cut, 4 mm gap; 2/6—2 mm cut, 6 mm gap; and 6/2—6 mm cut, 2 mm gap), but also different orientations of the cuts in the sample (from 0 to 90 deg. every 15 degrees). All cases are compiled in Table 2 and are shown in Figure 3.

**Table 2.** Sample symbols.

**Figure 3.** Perforation types: (**a**) Type 2/6—model SW; (**b**) Type 4/4—model SW; (**c**) Type 6/2—model SW; (**d**) Type 2/6—model DW; (**e**) Type 4/4—model DW; (**f**) Type 6/2—model DW.

Two hypothetical corrugated boards were analyzed here, namely single-walled (SW) with 8 mm flute period, 4 mm height and double-walled (DW) with 4 mm flute period, 2 mm flute height (for lower layer) and 8 mm flute period, 4 mm flute height (for higher layer). Figure 4 shows the visualizations of the geometry of both examples.

**Figure 4.** Geometry of the sample: (**a**) single layer; (**b**) double layer.

Both the influence of the flute orientation and the cutting orientation on the decrease in the stiffness of the corrugated board were examined. In case C, the cutting orientation changed to 00, 15, 30, 45, 60, 75, 90 degrees (see Figure 5) while the flute orientation remained constant.

**Figure 5.** Perforation orientation in sample SW-44-C: (**a**) rotation by 15 degrees; (**b**) rotation by 30 degrees; (**c**) rotation by 45 degrees; (**d**) rotation by 60 degrees; (**e**) rotation by 75 degrees; (**f**) rotation by 90 degrees.

In case F, the flute orientation were changed to 00, 15, 30, 45, 60, 75, 90 degrees (see Figures 6 and 7) while the cut orientation remained constant. All cases are summarized in Table 2.

Both single-walled and double-walled models with perforations of 4/4 mm, 2/6 mm, and 6/2 mm in the variant 00 deg. of cut and flute rotation were crushed by 10, 20, and 30%. This consideration results from the observation of the serial production of packaging in which crushing is an element built into the entire cutting and perforation process. The additional crushing during cutting is the result of using rubber in the area of perforation knives that additionally crush the cross-section. The crushed geometry of both kinds of samples is shown in Figure 8.

All crushed samples were marked with an additional symbol R-xx, where xx means the amount of crush (i.e., 10, 20, or 30). Therefore, for example, a single-walled specimen with a cut/flute rotated by 0 degrees with a cut version of 44 and crushed by 10% has the symbol SW-44-C-00-R-10.

**Figure 6.** Perforation orientation in sample SW-44-F: (**a**) rotation by 15 degrees; (**b**) rotation by 30 degrees; (**c**) rotation by 45 degrees; (**d**) rotation by 60 degrees; (**e**) rotation by 75 degrees.

**Figure 7.** Perforation orientation in sample DW-44-F: (**a**) rotation by 15 degrees; (**b**) rotation by 30 degrees; (**c**) rotation by 45 degrees; (**d**) rotation by 60 degrees; (**e**) rotation by 75 degrees.

Additionally, what was verified during this research was the influence of the position of the cut in the corrugated boards' cross-section along the wave on the stiffness reduction. For this purpose, four additional representative volumetric element (RVE) models were created in two variants of the SW and DW samples, in which the flute was shifted by 1/16 of the period (P) from 1/16 P to 4/16 P (see Figure 9).

**Figure 8.** Crushed samples: (**a**–**c**) Single-walled sample crushed by 10%, 20%, and 30%, respectively; (**d**–**f**) Double-walled sample crushed by 10%, 20%, and 30%, respectively.

**Figure 9.** Cross section of the corrugated board along the wave: (**a**) the reference SW sample—no offset; (**b**) SW sample offset equal to 1/16 P; (**c**) SW sample—offset equal to 2/16 P; (**d**) SW sample—offset equal to 3/16 P; (**e**) SW sample—offset equal to 4/16 P; (**f**) the reference DW sample—no offset; (**g**) DW sample—offset equal to 1/16 P; (**h**) DW sample—offset equal to 2/16 P; (**i**) DW sample—offset equal to 3/16 P; (**j**) DW sample—offset equal to 4/16 P.

#### *2.3. Homogenization Technique*

In order to determine the effect of cuts on the stiffness of the corrugated board, the numerical homogenization method was used here. This method, originally proposed by Biancolini [40] and later extended by Garbowski and Gajewski [32], is based on the elastic energy equivalence between the simplified shell model and the full RVE of corrugated cardboard. The RVE is a finite element (FE) representation of a small, periodic section of the full 3D corrugated board structure. The complete derivations of the constitutive model can be found in [32]. In the present study, only the basic assumptions are presented below.

The displacement based on finite element formulation for a linear analysis can be represented by an equation:

$$\mathbf{K}\_{\ell} \ \mathbf{u}\_{\ell} = \mathbf{F}\_{\ell \prime} \tag{3}$$

۹ ۴ = ܝ , ܝ ۹ ۴ where **K***e* is a statically condensed global stiffness matrix of the RVE; **u***e* is a displacement vector of external nodes; and **F***e* is a vector of the nodal forces applied to external nodes. In Figure 10, the FE mesh and mesh nodes are shown.

Static condensation relies on the removal of unknown degrees of freedom (DOF) and then the formulation of the stiffness matrix for a smaller number of degrees of freedom, called the primary unknown or principal DOF. In the analyzed cases, the eliminated degrees of freedom is the internal RVE nodes and the external nodes are the primary unknowns. The statically condensed FE stiffness matrix is computed from the equation:

$$\mathbf{K}\_{\ell} = \mathbf{K}\_{\ell\ell} - \mathbf{K}\_{\ell i} \mathbf{K}\_{\text{ii}}^{-1} \mathbf{K}\_{\text{ii}\prime} \tag{4}$$

where the stiffness matrix contains four subarrays related to internal (subscript *i*) and external (subscript *e*) nodes: ۹ ۴ = ܝ ,

**Figure 10.** RVE—external (in red color) and internal nodes and finite elements: (**a**) SW model; (**b**) DW model.

Static condensation reduces the total elastic strain energy to the work of external forces on the corresponding displacements. The total elastic strain energy can be calculated from the equation:

$$E = \frac{1}{2} \mathbf{u}\_{\varepsilon}^{T} \ \mathbf{F}\_{\varepsilon}.\tag{6}$$

The balance of the total energy for the full 3D shell model and the simplified shell model is ensured by an appropriate definition of displacements in the external RVE nodes and by enabling the membrane and bending behavior. More details can be found in Garbowski and Gajewski [32]. The generalized displacements are related to the generalized strains on the RVE edge surfaces, which can be represented by the relationship:

$$\mathbf{u}\_{i} = \mathbf{H}\_{i} \,\,\varepsilon\_{i\prime} \,\,\tag{7}$$

where for a single node (*xi* = *x*, *yi* = *y*, *zi* = *z*) the **H***i* matrix adopted for RVE shell model can be determined:

$$
\begin{bmatrix} u\_x \\ u\_y \\ u\_z \\ \theta\_x \\ \theta\_y \end{bmatrix}\_i = \begin{bmatrix} \mathbf{x} & \mathbf{0} & y/2 & xz & \mathbf{0} & yz/2 & z/2 & \mathbf{0} \\ \mathbf{0} & y & \mathbf{x}/2 & \mathbf{0} & yz & xz/2 & \mathbf{0} & z/2 \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & -\mathbf{x}^2/2 & -y^2/2 & -xy/2 & \mathbf{x}/2 & y/2 \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & -y & -\mathbf{x}/2 & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{x} & \mathbf{0} & y/2 & \mathbf{0} & \mathbf{0} \end{bmatrix} \begin{bmatrix} \varepsilon\_x \\ \varepsilon\_y \\ \gamma\_{xy} \\ \kappa\_x \\ \kappa\_y \\ \kappa\_{xy} \\ \gamma\_{yz} \\ \gamma\_{yz} \end{bmatrix} \tag{8}
$$

While using the definition of the elastic strain energy for a discrete model:

$$E = \frac{1}{2} \mathbf{u}\_{\varepsilon}^{T} \mathbf{K} \,\mathbf{u}\_{\varepsilon} = \frac{1}{2} \boldsymbol{\varepsilon}\_{\varepsilon}^{T} \,\mathbf{H}\_{\varepsilon}^{T} \,\mathbf{K} \,\mathbf{H}\_{\varepsilon} \,\boldsymbol{\varepsilon}\_{\varepsilon} \tag{9}$$

and considering a finite element as subjected to bending, tension, and transverse shear, the elastic internal energy is expressed by:

$$E = \frac{1}{2} \epsilon\_{\varepsilon}^{T} \mathbf{H}\_{\mathbf{k}} \,\varepsilon\_{\varepsilon} \{area\}. \tag{10}$$

For a homogenized composite, the stiffness matrix can be easily determined as:

$$\mathbf{H}\_k = \frac{\mathbf{H}\_\ell^T \mathbf{K} \,\mathbf{H}\_\ell}{area}.\tag{11}$$

The presented homogenization method is based on replacing the full 3D shell model with a simplified shell model and computing the effective stiffness of the RVE. Such a procedure significantly accelerates the computations and maintains a very high accuracy of the results.

The matrix **H***k* is formed by the matrices **A**, **B**, **D**, and **R** as follows:

$$\mathbf{H}\_{k} = \begin{bmatrix} \mathbf{A}\_{3 \times 3} & \mathbf{B}\_{3 \times 3} \\ \mathbf{B}\_{3 \times 3} & \mathbf{D}\_{3 \times 3} \\ & & \mathbf{R}\_{2 \times 2} \end{bmatrix} \tag{12}$$

where **A** represents extensional and shear stiffnesses; **B** represents extension-bending coupling stiffnesses; and **D** represents bending and torsional stiffnesses, while **R** represents transverse shear stiffness.

In general, the stiffness matrix **A** is independent of the position of a neutral axis. For the most symmetrical cross sections, all elements of stiffness matrix **B** are equal to zero. However, for unsymmetrical sections (i.e., double-walled corrugated board samples) matrix **B** is a non-zero, which indicates that there is a coupling between bending/twisting curvatures and extension/shear loads. Traditionally, these couplings have been suppressed for most applications by choosing the position of the neutral axis that minimizes the values of **B**. Alternatively, uncoupled matrix **D** can be computed from the formula:

$$\mathbf{D} = \mathbf{D}\_0 - \mathbf{B} \mathbf{A}^{-1} \mathbf{B},\tag{13}$$

where **D**0 represents the original (coupled) bending and torsional stiffnesses.

Within all analyses, the 3-node triangular general-purpose shell elements, named S3, were used for the computations. In every examined case, approximate global size equal to 0.5 mm was assumed. Due to the analysis of different orientations of flutings or cuts in the sample, the number of elements changed. For example, in the case of the SW-44-C-00 sample—2002 elements, 1099 nodes, and 6594 degrees of freedom were obtained, and for the DW-44-C-00 sample—3972 elements, 2074 nodes, and 12,444 degrees of freedom were obtained.

#### **3. Results**

#### *3.1. Validation of the Proposed Method*

The proposed numerical method was first verified by direct comparison of the obtained results with the existing solutions from the literature. One example concerns an assembled sandwich structure consisting of a corrugated tooth-shaped core enclosed between two sheets. A reference solution is available from Buannic et al. [39]. According to the notation used in the literature, the T2 panel was tested here. The FE models used in this comparison for the T2 sandwich consists of 3-node and 4-node shell elements and are shown in Figure 11. Error estimation was performed and the maximum deviation was less than 2.5%.

ۯ

۰,ଵିۯ۰ − ۲=۲

۰ ۲

۰

۲

۰

**Figure 11.** Representative shell elements of saw tooth geometry with quadrilateral mesh (single period): (**a**) model with a fine 4-node mesh; (**b**) model with a coarse 3-node mesh; (**c**) model geometry.

On the basis of the above validation (see Table 3) carried out on two numerical models: (a) model with a fine mesh (see Figure 11a) and (b) model with a coarse mesh (see Figure 11b), it was found that the solution does not depend on the element type and on the size of the finite element. It is important, however, to correctly represent any curvatures, therefore, in the case of sinus-like fluting, at least 16 segments are required to obtain correct results [32].

**Table 3.** The stiffnesses of representative shell element computed for a different approach of modeling confronted with data from [39] for saw tooth geometry.

#### *3.2. Detailed Results*

This section presents all the results of numerical tests for both single-walled (SW) and double-walled (DW) corrugated board samples. First, Tables 4 and 5 show an example of the **A***k* matrix, calculated while using the SW and DW models, respectively (both unperforated).

**Table 4.** Constitutive stiffness matrix **A***k* for the SW model without perforation.

**Table 5.** Constitutive stiffness matrix **A***k* for the DW model without perforation.

Due to the volume limitations of the data that can be presented in all the following tables, only the values from the main diagonals of the **A***k* matrix are shown. This simplification does not introduce an error in the analyses of the results, mainly because the components (∗)12 are related to the elements (∗)11 and (∗)22 in each matrix. The **B** matrix was also disregarded. However, it has been accounted for using Equation (13) in the **D** matrix, which is presented in all tables below.

Since the DW model is asymmetric, all matrices **A**, **B**, **D**, and **R** are non-zero; in particular, matrix **B** (see Table 5), which combines the bending effects with the membrane stiffness of the plate.

Table 6 shows the selected stiffnesses of all SW models with no perforation and fluting, rotated by an angle of 0 to 90 every 15 degrees. It is worth noting that in the case of models with rotated fluting by 90 degrees SW-0-F-90 and with non-rotating fluting SW-0-F-0, the stiffness values (∗)11 and (∗)22 were swapped (the same holds for (∗)44 and (∗)55).

**Table 6.** Selected stiffnesses in SW samples with no perforation and with different flute orientations.

Table 7 shows the selected stiffnesses of all DW models with no perforation and fluting rotated by an angle of 0 to 90 every 15 degrees (see Figure 7). For the DW-0-F-45 and SW-0F-45 samples, the same values were obtained for all (∗)11 and (∗)22 as well as (∗)44 and (∗)55, which was expected. This is, of course, due to the symmetry in both the geometrical setup and the material orientation.

**Table 7.** Selected stiffnesses in DW samples with no perforation and with different flute orientations.

Figure 12 shows the stiffness reduction of thee perforated models (both SW and DW) depending on the perforation rotation angle. The normalization term in each case is the **A***k* matrix of the corresponding non-perforated sample (i.e., all stiffnesses in the perforated SW models are divided by the corresponding stiffnesses in nonperforated SW model).

**Figure 12.** Stiffness degradation in sample: (**a**) SW-26; (**b**) SW-44; (**c**) SW-62; (**d**) DW-26; (**e**) DW-44; (**f**) DW-62.

ଵଵܣ Tables 8 and 9 summarize the chosen values of stiffness for a selected case of SW sample with fluting rotated by 15 degrees, for four cases of perforation: (i) no perforation; (ii) 2/6 mm (i.e., the normalized cut is 25%); (iii) 4/4 mm (i.e., the normalized cut is 50%); and (iv) 6/2 mm (i.e., the normalized cut is 75%).

ଶଶܣ ଷଷܣ **Table 8.** The selected stiffnesses in SW models for different perforations and flute rotated by 15 degrees.

ଵଵܣ ଵଵܣ−

∗

∗

ଶଶܣ ଶଶܣ− ∗

ଷଷܣ ଷଷܣ− ∗ ଵଵܦ ଵଵܦ−

ଶଶܦ ଶଶܦ− ∗

ଷଷܦ ଷଷܦ− ∗

−ܴସସ ܴସସ

−ܴହହ ܴହହ ∗

∗ − Figure 13 shows the selected values of the stiffness reduction of the SW samples with the flute rotated by 15, 30, 45, 60, and 75 degrees. All stiffnesses were normalized by the **A***k* matrix of the non-perforated sample with the appropriate fluting orientation (see Figure 6). Figure 14 presents the selected values of the stiffness reduction of the DW samples with the flute rotated by 15, 30, 45, 60, and 75 degrees. All stiffnesses were normalized by the **A***k* matrix of the non-perforated sample with the appropriate fluting orientation (see Figure 7).

**Table 9.** Stiffness reduction for both SW and DW samples with flute rotated by 15 degrees for three cases of perforation.

\* denotes the reference value of non-perforated specimen (i.e., SW-0-F-15).

**Figure 13.** Stiffness degradation in sample SW: (**a**) F-15; (**b**) F-30; (**c**) F-45; (**d**) F-60; (**e**) F-75. Three types of perforations were analyzed (2/6 mm, 4/4 mm, or 6/2 mm).

In the process of cutting corrugated board, perforation may occur in various locations relative to the fluting position, therefore the impact of fluting shift on stiffness changes has also been analyzed. Figure 15 presents the values of the stiffness reduction depending on the location of the cut in relation to the fluting position for the SW and DW samples in three perforation varieties: 2/6 mm, 4/4 mm, and 6/2 mm.

Due to noticed increase of *R*44 and *R*55 stiffnesses (negative stiffness reduction values shown in Figure 15), non-perforated samples were also examined. The values of the stiffness reduction depending on the fluting shift for the SW sample are summarized in Table 10, whereas the values of the stiffness reduction depending on the fluting shift for the DW sample are listed in Table 11.

As the perforation process is inseparable from the crushing process, this effect on the reduction of stiffness has also been tested. The influence of additional crushing of 10, 20, and 30% of the initial height of the corrugated board on the stiffness degradation of SW and DW samples is presented in Figure 16. The comprehensive study of the impact of crushing

ۯ

ۯ

on single-walled corrugated board is presented in a recent study of Garbowski et al. [44], while for the double-walled structures, see Gajewski et al. [45].

**Figure 14.** Stiffness degradation in a sample DW: (**a**) F-15; (**b**) F-30; (**c**) F-45; (**d**) F-60; (**e**) F-75. Three types of perforation were analyzed (2/6 mm, 4/4 mm, or 6/2 mm).

**Figure 15.** Stiffness degradation in sample C-0: (**a**) SW-26; (**b**) SW-44; (**c**) SW-62; (**d**) DW-26; (**e**) DW-44; (**f**) DW-62.

ܴସସ ܴହହ

ܴସସ ܴହହ

ଵଵܦ ଵଵܦ−

ଶଶܦ ଶଶܦ−

ଷଷܦ ଷଷܦ− ∗

**Table 10.** Uncut samples SW. Stiffness reduction in terms of flute offset. −ܴସସ ܴସସ ∗

∗ −

∗ −

\* denotes the reference value of non-shifted flute. ଵଵܦ ଵଵܦ− ∗ − −

**Table 11.** Uncut samples DW. Stiffness reduction in terms of flute offset. ଶଶܦ ଶଶܦ− ∗ −

\* denotes the reference value of non-shifted flute.

**Figure 16.** Stiffness degradation in sample: (**a**) SW-26-C-0-R-xx; (**b**) SW-44-C-0-R-xx; (**c**) SW-62-C-0-R-xx; (**d**) DW-26-C-0-R-xx; (**e**) DW-44-C-0-R-xx; (**f**) DW-62-C-0-R-xx. Here xx is a crush level (0%; 10%, 20%, and 30%).

ܦଶଶ ܴହହ

ଷଷܦ ଷଷܣ

ଶଶܦ ଶଶܣ ଵଵܦ ଵଵܣ

ଵଵܦ ଵଵܣ

ଶଶܣ

ଶଶܦ ଶଶܣ

ଵଵܦ ଵଵܣ

۲ ۯ

ܴହହ

ܴସସ

ܣଷଷ ܦଷଷ ܴହହ ܴସସ

ܴସସ ܴହହ

܀

#### **4. Discussion**

On the basis of the conducted analyses and the obtained results, it can be concluded that the perforations to a greater or lesser extent affected the stiffness degradation not only in the **A** sub-matrix (responsible for the tensile/compression stiffness) and in the **D** sub-matrix (responsible for bending/torsion stiffness), but also in the **R** sub-matrix (responsible for the transversal shear stiffness).

For samples with different perforation orientations (see Figure 5), the reduction in stiffness was related to the rotation angle of the perforation. In the samples with a rotation angle below 30 degrees, the greatest reduction occurred for matrix elements with indices 22 and 55. If the rotation angle was greater than 60 degrees, mainly matrix elements with indices 11 and 44 were reduced. This rule applied to both types of samples (i.e., SW and DW). When the perforation was rotated by an angle equal to 45 degrees, the matrix elements with indices 11, 22, 44, and 55 were evenly degraded.

For 2/6 mm perforation in model SW (see Figure 12a), the maximum degradation did not exceed 10% and was applied to *A*22 (for perforation rotation angle < 30 degrees) and *A*11, *D*11 (for perforation rotation angle > 60 degrees). It is worth noting that the decrease in the stiffness *D*22 and *R*55 for the rotation angle of the perforation equal to 0 degrees was relatively high and amounted to 5% for the perforation type 2/6 mm. The remaining stiffnesses degraded less than 3% in this case. A similar observation applied to the DW model (see Figure 12d).

While considering the 4/4 mm type perforation (see Figure 12b), the observations were as follows: reduction of *A*22, *D*22 was about 25% for a perforation rotation of 0 degrees and about 0% for a 90-degree rotation; *R*55 degraded about 25% when the perforation was rotated by 0 degrees and about 10% when the perforation was rotated by 90 degrees; reduction of *A*33 and *D*33 was about 10% regardless of the perforation rotation angle, while the degradation of *A*11 and *D*11 varied from around 0% to 30% for 0 degrees and 90 degrees, respectively; and the degradation of *R*44 did not exceed 5%. In the DW model (see Figure 12e), a similar decrease could be observed. The reductions *R*44 and *R*55 look slightly different; this is related to a different ratio of the sample height to its dimensions in the plan.

The greatest reductions were observed for the sample with the 6/2 mm perforation type (see Figure 12c,f). This is obviously related to the largest cut-to-gap ratio (which amounts to 75% in this case). In the case of the SW model, both the stiffness reductions *A*11 and *D*11 as well as *A*22 and *D*22 reached a maximum value of slightly more than 50%. The reduction of *A*33, *D*33, and *R*55 varied between 15 and 30%. The *R*44 stiffness reduction was approximately 0% for the non-rotated perforation, while for the rotation angle of 90 degrees, it was about 20%. A very similar stiffness degradation could be observed for the DW model (see Figure 12f).

For samples with different fluting orientations (see Figures 13 and 14), the greatest reduction in stiffness always occurred in the direction perpendicular to the perforation (i.e., (∗)22 and (∗)55), regardless of material orientation. Both *A*22 and *D*22 stiffnesses had the greatest reductions and amounted to about 50% in the case of 6/2 mm perforation for all fluting orientations. Slightly smaller reductions in stiffness were observed for *R*44, *A*33, and *D*33 ranging from 15 to 30% (for 6/2 mm perforation type), depending on the orientation of the fluting. The smallest stiffness reductions were observed for *A*11, *D*11, and *R*55.

When analyzing the stiffness reductions for models with shifted fluting (see Figure 9), even in the case without perforation, slight differences in stiffness could be observed (see Tables 10 and 11) and concerned mainly *R*44 and *R*55. Small fluctuations were also observed in models with perforation for both cases of SW and DW (see Figure 15), where again, the *R*44 and *R*55 showed the greatest dependence on fluting shift.

By also adding to the model the crushing of fluting (see Figure 8) that accompanies the perforations during the treatment of corrugated board, the degradation for some stiffnesses can increase several times (see Figure 16). The more perforated the model (i.e., 6/2 mm perforation type), the smaller the further reductions in the stiffness *A*22, *D*22, and *R*55. The remaining stiffnesses were drastically reduced with the increase in the crushing of the cross-section of the corrugated board. It is worth noting that for the DW model, the stiffnesses reduction of *A*11, *A*22, and *A*33 did not depend on the amount of crushing.

#### **5. Conclusions**

This article presents the comprehensive numerical analyses of the effect of perforation on reducing stiffness while implementing homogenization techniques. The acquired knowledge can be used for numerical modeling, for example, of corrugated cardboard packaging with perforations. Knowing the specific values of the stiffness reduction, it is possible to correctly model the perforation line and thus accurately estimate the load capacity of the packaging. The reduction in individual stiffnesses depends not only on the type of perforation, but also on the orientation of the perforation and the orientation of the fluting, but does not depend on the location of the perforation along the wavelength. Further development of the launched research is planned related to the validation of the proposed model with experimental models while engaging the non-contact displacement measurements [46].

**Author Contributions:** Conceptualization, T.G.; Methodology, T.G.; Software, T.G.; Validation, T.G. and D.M.; Formal analysis, T.G. and D.M.; Investigation, D.M., T.G. and A.K.-P.; Writing—original draft preparation, A.K.-P., T.G. and D.M.; Writing—review and editing, A.K.-P. and T.G.; Visualization, D.M. and T.G.; Supervision, T.G.; Project administration, T.G.; Funding acquisition, T.G. and A.K.-P. All authors have read and agreed to the published version of the manuscript.

**Funding:** The APC was funded by the Ministry of Science and Higher Education, Poland, the statutory funding at Poznan University of Life Sciences, grant number 506.569.05.00 and the statutory funding at Poznan University of Technology, grant number 0411/SBAD/0004.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.

**Acknowledgments:** Special thanks to the FEMat Sp. z o. o. company (Pozna ´n, Poland) (www.fematsystems. pl— accessed on 21 May 2021) for providing the commercial software.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


*Review* **Magnetorheological Elastomers: Fabrication, Characteristics, and Applications**

### **Sung Soon Kang 1, Kisuk Choi 2, Jae-Do Nam 2 and Hyoung Jin Choi 1,\***

Received: 31 August 2020; Accepted: 13 October 2020; Published: 15 October 2020

**Abstract:** Magnetorheological (MR) elastomers become one of the most powerful smart and advanced materials that can be tuned reversibly, finely, and quickly in terms of their mechanical and viscoelastic properties by an input magnetic field. They are composite materials in which magnetizable particles are dispersed in solid base elastomers. Their distinctive behaviors are relying on the type and size of dispersed magnetic particles, the type of elastomer matrix, and the type of non-magnetic fillers such as plasticizer, carbon black, and crosslink agent. With these controllable characteristics, they can be applied to various applications such as vibration absorber, isolator, magnetoresistor, and electromagnetic wave absorption. This review provides a summary of the fabrication, properties, and applications of MR elastomers made of various elastomeric materials.

**Keywords:** magnetorheological; elastomer; magnetic particle; viscoelastic; rheological


**Predicting the Effects of Insecticide Mixtures on Non-Target Aquatic Communities**

Alexa C. Alexander and Joseph M. Culp

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/53356


*3.1. Water-Borne Infectious Disease Risk*

A variety of techniques have been used to examine the association between weather and various infectious diseases [30,31]. Water-borne infectious disease outbreaks have been associated with climate/weather, such as sporadic cryptosporidiosis [32,33]. A systematic review examined 24 papers on outbreaks of water-borne infectious disease and found associated factors included low rainfall, increasing temperature and heavy rainfall before the outbreak [34]. The impact of weather on drinking water quality is reduced with the provision of modern drinking water supplies, but is more marked in private water supplies [35]. Although modern treatment should still be able to cope with most extreme weather events, drinking water quality and supply may come under increased pressure with changes in climate. Flooding can impact on the infrastructure of water treatment, sewage disposal, and electricity supply. In the Baltic and beyond, seawater temperature may also be playing a role in the increase in diseases such as cholera related to the bacteria, *Vibrio* spp. [36].

The supply of food may be adversely influenced by drought or floods affecting agriculture [37,38]. As part of the HPRU in Environmental Change and Health Theme 1 research, we have explored the health and well-being consequences of flood exposures in the UK [39] and found that the seasonality of a number of foodborne pathogens can be influenced by weather, particularly temperature [40]. We have also reviewed the seasonality of over 2000 distinct infectious diseases reported in the UK [41] and the co-occurrence of weather conditions with human infection cases of individual pathogens provides

some indication that these might be related. However, because such links can be associated with a range of other variables, we have developed techniques that link local weather to individual patients so that the weather components can be partly decoupled from seasonality; methods for extracting and linking these data locally have also been tested, based on linking the geographical location of where diagnosis data were collected and analyzed, with weather data for that location [42].

One of the problems with linking human infections with the weather across large areas is the differences in monitoring (including methods, reporting and completeness), particularly over a sufficiently long timescale [43]. To facilitate a wider analysis of pathogens, we have also reviewed methods used in linking water-associated diseases with climate [30,31], developed methods based on the infectious disease Campylobacter as a case-study (see Box 1), and produced a website for visualising data and generating hypotheses about the relationships between weather parameters and disease (https://www.data-mashup.org.uk/research-projects/climate-weather-infectious-diseases/).

**Box 1.** Campylobacter Case Study.

Within the Health Protection Research Unit (HPRU) in Environmental Change and Health research, the comparison of data based on the weekly incidence of campylobacter infections in the UK at different temperature and rainfall values has been useful in partially separating out the weather component from seasonality [49,50]. One traditional way of examining a time-series analysis is to use an autoregressive integrated moving average (ARIMA) model, and this has been adapted to include seasonality (SARIMA). A range of methods have been used to examine these, including: a novel Comparative Conditional Incidence (CCI) wavelet analysis [51]; hierarchical clustering [52]; generalized additive models for location, scale and shape (GAMLSS); and generalized structural time series (GEST) models [49,50]. This area of work is going to be further developed by future HPRU research so that it can be applied to a wider range of pathogens, weather/climate change and other complex factors.


**1. Introduction**

34 Ion Exchange Technologies

[66] Tsujimoto Y, Kitada A, Uemura Y. J, Goko T, Aczel A. A, Williams T. J, Luke G. M, Narumi Y, Kindo K, Nishi M, Ajiro Y, Yoshimura K, Kageyama H. Two-Dimensional *S*

= 1 Quantum Antiferromagnet (NiCl)Sr2Ta3O10. Chem. Mater. 2010;22 4625-4631.

The unusual electrical, optical, magnetic, and chemical properties of metal colloids (better known in nowadays as metal nanoparticles, MNPs) have attracted increasing interest of scientists and technologists during the last decade. In fact, although Nanoscience and Nanotechnology are quite recent disciplines, there have already been a high number of publications that discuss these topics. [1-11] What is more, there are quite new high impact peer-reviewed journals especially devoted to these research fields and there is also a particular subject category "Nanoscience & Nanotechnology" in the Journal Citation Reports from Thomson Reuters.

MNPs can be obtained by various synthetic routes, such as electrochemical methods, decomposition of organometallic precursors, reduction of metal salts in the presence of suitable (monomeric or polymeric) stabilizers, or vapour deposition methods. Sometimes, the presence of stabilizers is required to prevent the agglomeration of nanoclusters by providing a steric and/or electrostatic barrier between particles and, in addition, the stabilizers play a crucial role in controlling both the size and shape of nanoparticles.

In this sense, the development of polymer-stabilized MNPs (PSMNPs) is considered to be one of the most promising solutions to the MNPs stability problem. For this reason, the incorporation of MNPs into polymeric matrices has drawn a great deal of attention within the last decade as polymer-metal nanocomposites have already demonstrated unusual and valuable properties in many practical applications.

The modification of commercially available ion exchange resins and the development of suitable polymeric membranes with metal nanoparticles (MNPs) having certain functionality, such as for example, biocide or catalytic activity has proved to be a theme of

great interest. The main advantage of the nanocomposite ion exchange materials is the location of metal nanoparticles near the surface of the polymer what substantially enhances the efficiency of their biocide and catalytic application.

## **1.1. Metal nanoparticles**

The most commonly accepted definition for a nanomaterial is "a material that has a structure in which at least one of its phases has a nanometer size in at least one dimension." [12] Regarding this definition, it is possible to classify nanoobjects in three groups:

Such materials include porous materials (with porous sizes in the nanometer range), polycrystalline materials (with nanometer-sized crystallites), materials with surface protrusions separated by nanometric distances, or nanometer-sized metallic clusters. Among all of these materials, metal nanoparticles (MNPs) have attracted increasing interest of scientists and technologists during the last decade, due to their unique electrical, optical, magnetic, and chemical properties.

**Figure 1.** Bibliographic analysis based on the search of "nanoparticles" in Scifinder Scholar. Blue represents scientific publications (e.g. books, articles, reviews) and green represents patents.

Within the last two decades a new focus has been initiated to control and to better understand the nanometer-size objects due to the appearance of a new interdisciplinary field, which is known now as Nanoscience and Nanotechnology. This has stimulated a new wave of intensive and more detailed studies of MNPs and various nanocomposites on their base. Figure 1 shows the tendency in publication about this issue, where it is shown that the number of publications has increased exponentially due to their wide applications in different fields such as Medicine, Chemistry, and Physics and so on. Moreover, not only scientific publications have been growing in the last decades but also a huge number of patents have been issued in the last decade.

The main goal of Nanoscience and Nanotechnology is the creation of useful/functional materials, devices and systems through the control of matter on the nanometer length scale and exploitation of novel phenomena and properties (physical, chemical and biological) at that scale. To achieve such goal it is necessary to use a multidisciplinary approach: inputs from physicists, biologists, chemists and engineers are required for the advancement of the understanding in the preparation, application and impact of new nanotechnologies.

### **1.2. General properties**

36 Ion Exchange Technologies

**1.1. Metal nanoparticles**

great interest. The main advantage of the nanocomposite ion exchange materials is the location of metal nanoparticles near the surface of the polymer what substantially enhances

The most commonly accepted definition for a nanomaterial is "a material that has a structure in which at least one of its phases has a nanometer size in at least one dimension."

Such materials include porous materials (with porous sizes in the nanometer range), polycrystalline materials (with nanometer-sized crystallites), materials with surface protrusions separated by nanometric distances, or nanometer-sized metallic clusters. Among all of these materials, metal nanoparticles (MNPs) have attracted increasing interest of scientists and technologists during the last decade, due to their unique electrical, optical,

**Figure 1.** Bibliographic analysis based on the search of "nanoparticles" in Scifinder Scholar. Blue represents scientific publications (e.g. books, articles, reviews) and green represents patents.

Within the last two decades a new focus has been initiated to control and to better understand the nanometer-size objects due to the appearance of a new interdisciplinary field, which is known now as Nanoscience and Nanotechnology. This has stimulated a new wave of intensive and more detailed studies of MNPs and various nanocomposites on their base. Figure 1 shows the tendency in publication about this issue, where it is shown that the

[12] Regarding this definition, it is possible to classify nanoobjects in three groups:

ii. 2D nanometer-size objects (e.g., nanowires, nanorods and nanotubes) iii. 3D nanometer-size objects (e.g. nanoparticles and/or nanoclusters)

the efficiency of their biocide and catalytic application.

i. 1D nanometer-size objects (e.g., thin films)

magnetic, and chemical properties.

Because of the decrease in the scale of the materials, their behaviour changes in a remarkable form. In fact, the reduction of the bulk materials to a nanometric size induces sizedependant effects resultant from:

**Figure 2.** Schematic representation of the change in the ratio surface/volume between a bulk microsphere and the same microsphere composed by NPs.

This can be illustrated, for example, by the dependence of gold melting point on the size of gold nanoparticles, or by suspensions of Ag nanoparticles with sizes ranging from 40 to 100 nm which show different colours. [13] In addition, there are physical phenomena that do not exist in materials with larger grain sizes, as the general quantum-size effect for optical transitions in semiconductor nanocrystals which occurs in very small nanoparticles (<10 nm) due to the quantum confinement effects inherent in particles of that size.

Is due to all of these new properties that, indeed, research centred on nanoscopic materials have a large field of application which extends from the semiconductor industry, where the ability to produce nanometer-scale features leads to faster and less expensive transistors [6], to biotechnology, where luminescent nanoparticles are extremely interesting as bioprobes. [14] Some other particular examples are catalyst for fuel cells [15] or electrocatalysts used in sensing devices with enhanced properties. But, as a rule of thumb, nanoparticles, due to the large percentage of surface atoms [12, 16], have already made a major impact on the field of surface science, as Catalysis or Biocide treatment.

## **1.3. MNPs Preparation: Stability challenges and stabilization mechanisms**

In general there are two routes for the preparation of MNPs (see Figure 3):-Down and Bottom-Up. The top-down methods are those that reduce the macroscopic particles to the nanoscale. This route is not very suitable to prepare uniform particles of very small sizes. In contrast, with the bottom-up methods it is possible to obtain uniform particles (usually of different shapes and structures). These routes start from atoms that can be added (either in solution or gas phase) to form larger particles.

**Figure 3.** Scheme of Top-Down and Bottom-Up approaches to the synthesis of MNPs.

Overall, a good method to classify the different methods of synthesis of MNPs is by Physical, Physicochemical and Chemical routes (See Figure 4). [13] Many synthetic pathways can be used but the chemical ones are generally cheaper and do not require equipment or instruments as specific as in the case of physical methods the physical methods.

However, the main drawback which still limits the wide application of MNPs is their insufficient stability dealing with their high tendency to self-aggregate. [17] MNPs are so reactive that when they touch each other, they surfaces fuse, what results in a loss of the nanometric size and in their special properties. These features of nanoparticles, in part determined by the conditions of synthesis, create enormous difficulties in their fabrication and application. [18]

**Figure 4.** Physical, Physicochemical and Chemical routes for the preparation of MNPs.

It is noteworthy that NPs can aggregate not only as a result of a further manipulation but also during their growth. A typical mechanism of aggregation is the Ostwald ripening which is a growth mechanism where small particles dissolve, and are consumed by larger particles. [19] So, the average nanoparticle size increases with time, the particle concentration decreases and their solubility diminishes.

Therefore, the stabilization of MNPs is specifically required to:

38 Ion Exchange Technologies

This can be illustrated, for example, by the dependence of gold melting point on the size of gold nanoparticles, or by suspensions of Ag nanoparticles with sizes ranging from 40 to 100 nm which show different colours. [13] In addition, there are physical phenomena that do not exist in materials with larger grain sizes, as the general quantum-size effect for optical transitions in semiconductor nanocrystals which occurs in very small nanoparticles (<10 nm)

Is due to all of these new properties that, indeed, research centred on nanoscopic materials have a large field of application which extends from the semiconductor industry, where the ability to produce nanometer-scale features leads to faster and less expensive transistors [6], to biotechnology, where luminescent nanoparticles are extremely interesting as bioprobes. [14] Some other particular examples are catalyst for fuel cells [15] or electrocatalysts used in sensing devices with enhanced properties. But, as a rule of thumb, nanoparticles, due to the large percentage of surface atoms [12, 16], have already made a major impact on the field of

**1.3. MNPs Preparation: Stability challenges and stabilization mechanisms**

**Figure 3.** Scheme of Top-Down and Bottom-Up approaches to the synthesis of MNPs.

instruments as specific as in the case of physical methods the physical methods.

Overall, a good method to classify the different methods of synthesis of MNPs is by Physical, Physicochemical and Chemical routes (See Figure 4). [13] Many synthetic pathways can be used but the chemical ones are generally cheaper and do not require equipment or

However, the main drawback which still limits the wide application of MNPs is their insufficient stability dealing with their high tendency to self-aggregate. [17] MNPs are so reactive that when they touch each other, they surfaces fuse, what results in a loss of the nanometric size and in their special properties. These features of nanoparticles, in part

In general there are two routes for the preparation of MNPs (see Figure 3):-Down and Bottom-Up. The top-down methods are those that reduce the macroscopic particles to the nanoscale. This route is not very suitable to prepare uniform particles of very small sizes. In contrast, with the bottom-up methods it is possible to obtain uniform particles (usually of different shapes and structures). These routes start from atoms that can be added (either in

due to the quantum confinement effects inherent in particles of that size.

surface science, as Catalysis or Biocide treatment.

solution or gas phase) to form larger particles.

The successful synthesis of nanoparticles usually involves three steps: nucleation, growth, and termination by a capping agent or ligand (or stabilizing agent) through colloidal forces.[20] These colloidal forces can be classified in three main types as follows: Van der Waals interactions, electrical double-layer interactions, and steric interactions. In addition, hydrophobic and solvation forces may be important. [21]

Some of the mechanisms regarding the stabilization forces have been thoroughly revised in the literature.[13, 22]

Specially the use polymer-assisted fabrication of inorganic nanoparticles is probably one of the most efficient and universal ways to overcome the stability problem of MNPs and to save their properties. Metal nanoparticles synthesized by this approach exhibit long-time stability against aggregation and oxidation while nanoparticles prepared in the absence of polymers are prone to quick aggregation and oxidation.[18, 23]

In this sense, stabilization of MNPs can be done by different strategies. In the ex-situ synthesis, NPs are dispersed after their synthesis in a solid or liquid medium by using different mechanochemical approaches. The problem is that in these cases, the success of the stabilization is limited by the possibility of re-aggregation of the MNPs along the time. On the opposite hand, by the in-situ synthesis, MNPs are grown directly in the stabilizer medium yielding a material that can be directly used for a foreseen purpose. For this reason, in-situ approaches are getting much attention, because of their technological advantages (Figure 5).

**Figure 5.** Schematic comparison between in-situ and ex-situ methodologies.

Despite the methodology employed, it is of crucial importance to understand the processes occurring in polymer interactions with nanoparticles. In this regard, the mechanism of MNP stabilization with polymers can be explained by two approaches which run simultaneously in the system and influence one another: the substantial increase of viscosity of the immobilizing media (the polymer matrix), and the decrease of the energy of particle-particle interaction in PSMNP systems versus non-stabilized MNP dispersions. [24]

In the first approach, the substantial increase of viscosity of the immobilizing media (the polymer matrix), the Coagulation velocity depends on factors as the range of attraction forces, Brownian motion velocity, concentration of colloidal solution, presence of electrolytes… As follows from the Smoluchowsky equation [25], the rate constant of particle coagulation, kc, is inversely proportional to the viscosity of the media, , (here k stands for the Boltzman constant, and T is the temperature):

$$\mathbf{k}\_c = \frac{8\mathbf{k}\,\mathrm{T}}{\eta} \tag{1}$$

The second approach is the decrease of the energy of particle-particle interaction in PSMNP systems versus non-stabilized MNP dispersions. The potential energy of attraction Ur between two spherical particles of radius r and minimum distance lo between their surfaces can be given by the following equation:

Bifunctional Polymer-Metal Nanocomposite Ion Exchange Materials 41

$$\mathbf{U}\_r \approx \frac{\mathbf{A}\mathbf{r}}{12 \ l\_o} \text{ at } \mathbf{r} \gg \mathbf{l}\_o \tag{2}$$

where A is the effective Hamaker's constant with dimensions of energy. The value of A is known to be close to *kT* for polymer particles (e.g 6.310-20J for polystyrene), while for the metal dispersions it is far higher (4010-20 J for silver). [24]