**Concentration of Lipase from** *Aspergillus oryzae* **Expressing** *Fusarium heterosporum* **by Nanofiltration to Enhance Transesterification**

**Hans Wijaya 1,2, Kengo Sasaki 3, Prihardi Kahar 3, Emmanuel Quayson 1, Nova Rachmadona 1, Jerome Amoah 3, Shinji Hama 4, Chiaki Ogino 1,\* and Akihiko Kondo 1,3**

Received: 12 March 2020; Accepted: 7 April 2020; Published: 11 April 2020

**Abstract:** Nanofiltration membrane separation is an energy-saving technology that was used in this study to concentrate extracellular lipase and increase its total activity for biodiesel production. Lipase was produced by recombinant *Aspergillus oryzae* expressing *Fusarium heterosporum* lipase (FHL). A sulfonated polyethersulfone nanofiltration membrane, NTR-7410, with a molecular weight cut-off of 3 kDa was used for the separation, because recombinant lipase has a molecular weight of approximately 20 kDa, which differs from commercial lipase at around 30 kDa for CalleraTM Trans L (CalT). After concentration via nanofiltration, recombinant lipase achieved a 96.8% yield of fatty acid methyl ester (FAME) from unrefined palm oil, compared to 50.2% for CalT in 24 h. Meanwhile, the initial lipase activity (32.6 U/mL) of recombinant lipase was similar to that of CalT. The composition of FAME produced from recombinant concentrated lipase, i.e., C14:1, C16:0, C18:0, C18:1 cis, and C18:2 cis were 0.79%, 34.46%, 5.41%, 45.90%, and 12.46%, respectively, after transesterification. This FAME composition, even after being subjected to nanofiltration, was not significantly different from that produced from CalT. This study reveals the applicability of a simple and scalable nanofiltration membrane technology that can enhance enzymatic biodiesel production.

**Keywords:** nanofiltration; lipase; *Fusarium heterosporum*; fatty acid methyl ester

#### **1. Introduction**

Over the past decade, interest in biodiesel as an alternative to diesel fuel continues to increase throughout the world due to concerns about global climate change. Increasingly, there is a desire for renewable/sustainable energy sources, and an interest in developing domestic supplies of fuel that are more secure [1]. Biodiesel (fatty acid alkyl esters) is produced from renewable natural sources such as vegetable oils (e.g., palm oil), animal fats, and microalgal oil [2,3]. Biodiesel can be used directly in existing diesel engines without major modifications, or as a mixture with petroleum diesel, and the burning of biodiesel produces gas emissions such as sulfur oxide, which are less harmful than those emitted by the burning of petroleum-based fuels [4].

The viscosity of vegetable oils is improved via a transesterification pathway which involves triglycerides and alcohols of lower molecular weights and homogenous or heterogenous substances that are used as catalysts to yield biodiesel and glycerol [4]. Transesterification via enzymatic catalysis has attracted much attention because it is an eco-friendly process that produces no by-products, features easy product recovery, and requires a low reaction temperature [5]. However, the process is expensive, and has a relatively slow reaction rate [6,7]. A variety of lipases (EC <IP_ADDRESS>) from various microorganisms (*Candida antarctica, Rhizopus oryzae*, *Pseudomonas cepacia*, *Thermomyces lanuginosus*, etc.) have been used to accomplish both transesterification and esterification [8–10]. Many researchers have attempted to solve the limitations of lipase-catalyzed biodiesel production by immobilizing the enzymes or cells on a suitable matrix [8,11] or via the use of a lipase cocktail [12]. In contrast, at least one previous study has successfully conducted biodiesel production using recombinant *Aspergillus oryzae* that expresses *Fusarium heterosporum* lipase (FHL), which has demonstrated a high level of tolerance to water [13]. With the use of that particular enzyme, however, the conversion of oil to fatty acid methyl ester (FAME) remained low. To increase the conversion rate, the total activity of this enzyme was increased using a simple process such as a concentrating method. In general, the conversion increases proportionally with the increase of lipase concentration [14]. Concentration methods such as precipitation [15,16] require costly chemicals such as ammonium sulfate. Another concentration method is membrane separation technology, which has advantages that include energy savings, selectivity, no chemical requirement, and simplicity of operation and scale-up [17].

Reinehr et al. [17] previously reported a membrane concentration of lipase that could be produced from *Aspergillus niger* by using microfiltration and ultrafiltration separation processes. The present study is the first to apply a nanofiltration membrane to simply concentrate lipase produced from recombinant *A. oryzae* (expressing FHL) and enable a high level of transesterification compared to a commercially available lipase, CalleraTM Trans L (CalT) (Novozymes, Bagsvaerd, Denmark). In this study, we use unrefined palm oil as a model substrate for transesterification. Palm oil is well known as one of the most suitable sources for biodiesel production. Indonesia and Malaysia produce approximately 85% of global crude palm oil, which is likely to increase in the future [3,18]. The aim of the present study was to efficiently produce FAME from unrefined palm oil using membrane-concentrated lipase.

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

#### *2.1. Materials and Microorganisms*

Unrefined palm oil was purchased as a substrate from Malang, East Java, Indonesia. *Aspergillus oryzae* expressing FHL used in this study was obtained as described previously [19,20].

#### *2.2. Lipase Production*

Sakaguchi flasks (500 mL) containing 100 mL of DP medium (2% glucose, 2% polypeptone, 1% KH2PO4, 0.2% NaNO3, 0.05% MgSO4·7H2O) were aseptically inoculated with spores from *A. oryzae* expressing FHL in Czapek-Dox (CD)-NO2-methionine selection plate agar [20]. The flasks were incubated at 30 ◦C and shaken at 150 rpm for 96 h on a reciprocal shaker. The culture broth was collected and then centrifuged at 6000× *g* for 15 min at 4 ◦C to recover the supernatant. The culture supernatant was dialyzed in MEMBRA-CEL® dialysis tubing with a molecular weight cut-off (MWCO) of 3500 Da (RC, SERVA Electrophoresis GmbH, Heidelberg, Germany), followed by filtrations through different filter papers in the following order: (1) a polycarbonate filter (0.8 μm pore size); (2) a polycarbonate filter (0.5 μm pore size); and (3) a polystyrene filter (0.22 μm pore size). The supernatant was then subjected to nanofiltration-membrane concentration. Lipase from CalleraTM Trans L, a liquid lipase from *Thermomyces lanuginosus* lipase (CalT) (Novozymes, Bagsvaerd, Denmark), was used as a control.

#### *2.3. Nanofiltration Membrane Separation Processes*

A sulfonated polyethersulfone nanofiltration membrane, NTR-7410, with a 3000 Da MWCO was obtained from the Nitto Denko Corporation (Osaka, Japan). The membrane was cut into a circle (diameter: 7.5 cm; effective area: 32 cm2). The nanofiltration (NF) process was carried out at room temperature using a flat membrane test cell (model C40-B, Nitto Denko Corporation, Osaka, Japan) [21]. The lipase supernatant was then subjected to the test cell. The inside of the test cell was stirred at 300 rpm at a pressure of 2.5 MPa under nitrogen gas for one hour.

#### *2.4. Sodium Dodecyl Sulfate-Polyacrylamide Gel Electrophoresis (SDS-PAGE) and Zymography*

The *A. oryzae* supernatant expressing FHL was either directly applied to polyacrylamide gel electrophoresis in the presence of SDS-PAGE, or applied after filtration through a 5K MWCO Spin-X Ultrafiltration Concentrator (Corning, UK) to concentrate the lipase. CalT was applied to SDS-PAGE after dilution. The proteins were then stained with Coomassie brilliant blue R-250. Zymography analysis was carried out to detect lipase activity, as described previously [22]. The SDS-PAGE gels were incubated for one hour at room temperature in developing solution consisting of 3 mM α-naphthyl acetate, 1 mM Fast Red TR (Sigma, St. Louis, MO, USA), and 100 mM sodium phosphate buffer, pH 8.0. Precision Plus Protein™ Dual Color Standards (Bio-Rad, Hercules, CA, USA) were used as a standard marker.

#### *2.5. Measurement of Lipase Activity and Protein Assay*

The hydrolytic activities of lipase were tested using *p*-nitrophenyl butyrate (pNPB) as a chromogenic substrate. A stock solution was prepared by dissolving 5 μL of pNPB in 250 μL of ethanol, with a further dilution to 50 mL using distilled water. The stock solution was then incubated in a Bioshaker (Taitec, Saitama, Japan) for 10 min at 30 ◦C to allow the development of lipase-hydrolytic activity. After incubation, 5% trichloroacetate was added to terminate the reaction. The absorbance of para nitrophenol (pNP) that was produced was measured at 400 nm (UV-Vis spectrophotometer, Shimadzu, Kyoto, Japan). One unit (U2) of lipase activity was defined as the amount of lipase that liberates 1 μmol of pNP from pNPB per minute [12]. Protein concentrations were measured using a Pierce™ BCA Protein Assay Kit (Thermo Scientific™, Rockford, IL, USA).

#### *2.6. FAME Production by Enzyme*

FAME production was carried out in triplicate using a 10 mL glass tube with a silicon cap equipped with a stirrer (Thermo Scientific VARIOMAG Magnetic Stirrers, Waltham, MA, USA) for circulation at 800 rpm. The glass tube was then immersed in a heated water bath Thermo Minder EX TAITEC (Taitec, Saitama, Japan) that was set to 30 ◦C. The reaction mixture consisted of 4 g of unrefined palm oil and 1.2 mL of lipase (unconcentrated and concentrated lipases from recombinant *A. oryzae* and CalT diluted 245-fold). The transesterifications were carried out at 32.6 U/mL lipase activity for concentrated lipase from recombinant *A. oryzae* and for that from CalT diluted 245-fold (Table 1). To avoid deactivation of the lipase, 186 μL of methanol (corresponding to a 1:1 molar ratio of unrefined palm oil to methanol) was added step-wise at 0, 2, 4, and 6 h. Samples were taken at 0, 2, 4, 6, 9, and 24 h [13].

#### **Table 1.** Comparison of lipase activity and protein content.

#### *2.7. Analytical Methods*

Fatty acid methyl ester (FAME) produced during the course of the transesterification reaction was measured via gas chromatography. Samples taken at specified times were centrifuged at 12,000× *g* for 5 min at 5 ◦C, and the upper layer was analyzed via a GC-17A (Shimadzu, Kyoto, Japan) equipped with a ZB-5HT capillary column (0.25 mm × 15 m) (Phenomenex, Torrance, CA, USA), an auto-sampler, and a flame ionization detector, as previously described [12]. During the analysis, the temperatures of the injector and detector were set at 320 ◦C and 380 ◦C, respectively, using helium as a carrier gas at a flow rate of 58 mL/min. The column was configured with an initial temperature of 130 ◦C for 2 min, which was raised to 350 ◦C at 10 ◦C/min, and then to 370 ◦C at 7 ◦C/min. The FAME composition in each reaction mixture was reported as the percentage of the oil in the reaction mixture using tricaprylin as an internal standard [11].

Transesterification was conducted by following the protocol from the fatty acid methylation kit (Nacalai Tesque Inc., Kyoto, Japan). The FAME composition was analyzed using a gas chromatography-mass spectrometer (GC-MS) (Shimadzu, Kyoto, Japan). The GC-MS was equipped with a 0.25 mm × 30 m DB-23 capillary column (J&W Scientific, Folsom, CA, USA). The carrier was helium gas with a flow rate of 0.8 mL/min at 1:5 split ratios. The initial column temperature was 250 ◦C, which was increased to 50 ◦C for 1 min and then increased 25 ◦C/min to 190 ◦C and 5 ◦C/min to 235 ◦C for 4 min. An internal standard C8:0 (octanoic acid) was included in each sample and FAME was detected at the provided retention time (Supplementary Table S1). The amount of FAME (%) was calculated as the percentage of each fatty acid to the total weight of fatty acids produced [23].

#### **3. Results and Discussion**

#### *3.1. Characterization of Lipases before and after Membrane Concentration*

As described previously, the molecular weights of lipases ranged from 20 to 80 kDa [17,24] or up to 150 kDa [25]. Thus, NTR-7410 with a MWCO of 3 kDa [26] was selected as the membrane that would best concentrate lipase. Nanofiltration concentration was performed for one hour at 2.5 MPa.

Both concentrated and unconcentrated lipases produced from recombinant *A. oryzae* were characterized and compared with commercial lipase, CalT. At first, the molecular weight of the lipase produced by recombinant *A. oryzae* was determined via SDS-PAGE. Then, lipase enzyme activity was detected using the Zymography technique. As a result, the supernatant of *A. oryzae* contained plural proteins with molecular weights that ranged from 20 to 50 kDa (Figure 1A). However, lipase enzyme was detected as a single band at around 20 kDa (Figure 1B). As expected, the concentration of lipase had definitely increased. By comparison, the CalT contained a major protein at around 30 kDa along with some minor proteins that also showed lipase activity (Figure 1C,D). Therefore, we assumed that the observed bands other than that at around 30 kDa also represented small amounts of lipase (Figure 1D). These results suggest that the FHL produced in *A. oryzae* was a smaller molecule compared with the lipases produced in CalT. In addition, recombinant *A. oryzae* produced some other unknown proteins that were not lipase. Due to this contamination by other enzymes, the lipase produced from recombinant *A. oryzae* showed low specific activity (Table 1) compared with lipase reported elsewhere (more than 66 U/mg) [25]. However, the nanofiltration concentration of lipase produced by recombinant *A. oryzae* successfully increased its total activity from 6.4 U/mL to 32.6 U/mL (five-fold concentration factor) in the short time of one hour. Enzyme activity losses in this study were low (5.5%, Table 2), and may have occurred due to adsorption on the membrane surface as a function of fouling [17]. The activity loss was generally lower, because the use of NTR-7410 (MWCO of 3 kDa) could retain more lipase protein than that produced by recombinant *A. oryzae* at around 20 kDa. The denaturation did not occur due to the pressure applied lower than 400 MPa as used in [27].

**Figure 1.** SDS-PAGE and Zymogram. (**A**) SDS-PAGE for the culture supernatant of *Aspergillus oryzae* expressing *Fusarium heterosporum* lipase (FHL) (UC: unconcentrated, C 10×: concentrated 10-fold). (**B**) Zymogram for the detection of lipase produced from recombinant *A. oryzae*. (**C**) SDS-PAGE of the lipase from *Thermomyces lanuginosus* (CalT). (**D**) Zymogram of CalT. (X 3.000 at dilution 3.000-fold, X 10.000 at dilution 10.000-fold). SM: standard marker.

**Table 2.** Concentration factors and activity loss obtained in previous processes involving the concentration of proteins using membrane separation technologies.

#### *3.2. E*ffi*cient FAME Production by Concentrated Lipase*

FAME yield was compared for concentrated and unconcentrated lipases produced from recombinant *A. oryzae* and commercial lipase (CalT). In these reactions, the total lipase activity of concentrated lipase produced by recombinant *A. oryzae* was arranged to be nearly the same as that of diluted CalT. The results for the FAME yield are shown in Figure 2. Interestingly, FAME production was enhanced by using concentrated lipase produced by recombinant *A. oryzae* (designated as AC), compared with diluted CalT (designated as CalT), and unconcentrated lipase (designated as BC). Other controls were 0.5 AC + 0.5 CalT, which contained half of the concentrated lipase produced by recombinant *A. oryzae* and half of diluted CalT, and 0.5 BC + 0.5 CalT, which contained half of the unconcentrated lipase produced by recombinant *A. oryzae* and half of diluted CalT. By comparing these data, it was apparent that activity for lipase produced by recombinant *A. oryzae* to convert unrefined palm oil to FAME was significantly enhanced by nanofiltration concentration, compared with that produced by CalT. The reason for this remains unclear. We hypothesized that the FHL produced by recombinant *A. oryzae* would have different characteristics from commercial lipase (CalT).

AC BC CalT 0.5 AC + 0.5 CalT 0.5 BC + 0.5 CalT

**Figure 2.** Fatty acid methyl ester (FAME) production by lipase enzymes. AC: lipase produced from recombinant *A. oryzae* after nanofiltration (NF) concentration; BC: lipase produced from recombinant *A. oryzae* before NF concentration; CalT: lipase from *Thermomyces lanuginosus* showing lipase activity similar to AC (diluted around 245-fold); 0.5 AC + 0.5 CalT: half AC and half CalT; and 0.5 BC + 0.5 CalT: half BC and half CalT. The arrows indicate the time to add methanol.

The FAME compositions after 24 h of enzymatic reaction were analyzed by GC-MS, as shown in Table 3. In general, there were no major differences in the FAME composition converted from unrefined palm oil by both concentrated and unconcentrated lipases, produced either by recombinant *A. oryzae* or CalT. These results suggest that the biodiesel produced using concentrated lipase from recombinant *A. oryzae* has high potential to be used in the same manner as biodiesel produced from CalT. Furthermore, nanofiltration concentration can be used to enhance the quality of lipase used in biodiesel production without a loss of FAME quantity.

Previously, most of the research focusing on enzymatic catalysis has employed lipase immobilized on polymer support as a catalyst. However, the immobilization process is neither simple nor inexpensive [32]. Using the suggested membrane separation technology to concentrate lipase therefore simplifies the process and reduces the cost. In addition, concentrated lipase produced from recombinant *A. oryzae* can be used as a sole lipase or as a supplement to other commercially available lipases to reduce costs and improve biodiesel conversion yields from unconventional feedstock.

**Table 3.** FAME profiles of transesterification results.

AC: lipase produced from *A. oryzae* expressing FHL after NF concentration, BC: lipase produced from *A. oryzae* expressing FHL before NF concentration, CalT: lipase from *Thermomyces lanuginosus* showing lipase activity similar to AC (diluted around 245-fold), 0.5 AC + 0.5 CalT: half AC and half CalT, and 0.5 BC + 0.5 CalT: half BC and half CalT.

#### **4. Conclusions**

In this study, the total activity of lipase produced by recombinant *A. oryzae* was successfully increased at about 5-fold using nanofiltration membrane separation technology. Concentrated lipase produced from recombinant *A. oryzae* showed a higher FAME yield of 96.8% from unrefined palm oil, compared to 50.2% for CalT in a 24 h period, although the lipase activity (32.6 U/mL) was nearly the same between concentrated lipase and CalT. FAME composition using concentrated lipase was unchanged, compared with that using CalT. The FAME, C14:1, C16:0, C18:0, C18:1 cis, and C18:2 cis were produced, respectively, at 0.79%, 34.46%, 5.41%, 45.90%, and 12.46% after transesterification at 30 ◦C for 24 h. In this study, a simple and inexpensive process was developed using a nanofiltration membrane that is expected to improve enzymatic and industrial biodiesel production.

**Supplementary Materials:** The following are available online at http://www.mdpi.com/2227-9717/8/4/450/s1, Table S1: Retention time of target compound of fatty acid methyl ester obtained in Gas Chromatography-Mass Spectrometer analysis.

**Author Contributions:** Conceptualization, H.W., K.S., P.K., J.A., E.Q., N.R.; experiment, H.W., E.Q., N.R.; methodology, K.S., P.K., J.A., S.H., C.O., A.K.; validation, K.S., P.K., J.A.; data curation, H.W., K.S., P.K., J.A.; formal analysis, H.W., K.S., J.A., P.K.; writing—original draft preparation, H.W., K.S., P.K., J.A., E.Q.; writing—review and editing, K.S., P.K., J.A., S.H., C.O., A.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the International Joint Program, Science and Technology Research Partnership for Sustainable Development (SATREPS), Innovative Bioproduction Kobe (iBioK) from the Japan Science and Technology Agency (JST), the Japan International Cooperation Agency (JICA), and The Ministry of Education, Culture, Sports, Science and Technology.

**Acknowledgments:** We appreciate the help provided by Ayami Fujino and Yasunobu Takeshima.

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

#### **References**

© 2020 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 (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Biotechnology and Bioprocesses: Their Contribution to Sustainability**

#### **Alejandro Barragán-Ocaña 1,\*, Paz Silva-Borjas 1, Samuel Olmos-Peña <sup>2</sup> and Mirtza Polanco-Olguín <sup>1</sup>**

Received: 21 March 2020; Accepted: 2 April 2020; Published: 7 April 2020

**Abstract:** Significant advancements in biotechnology have resulted in the development of numerous fundamental bioprocesses, which have consolidated research and development and industrial progress in the field. These bioprocesses are used in medical therapies, diagnostic and immunization procedures, agriculture, food production, biofuel production, and environmental solutions (to address water-, soil-, and air-related problems), among other areas. The present study is a first approach toward the identification of scientific and technological bioprocess trajectories within the framework of sustainability. The method included a literature search (Scopus), a patent search (Patentscope), and a network analysis for the period from 2010 to 2019. Our results highlight the main technological sectors, countries, institutions, and academic publications that carry out work or publish literature related to sustainability and bioprocesses. The network analysis allowed for the identification of thematic clusters associated with sustainability and bioprocesses, revealing different related scientific topics. Our conclusions confirm that biotechnology is firmly positioned as an emerging knowledge area. Its dynamics, development, and outcomes during the study period reflect a substantial number of studies and technologies focused on the creation of knowledge aimed at improving economic development, environmental protection, and social welfare.

**Keywords:** bioeconomy; bioprocesses; applications; policy; social welfare; sustainability

#### **1. Introduction**

Biotechnology and bioprocesses are two important tools for economic progress and social welfare. The industrial, academic, and government sectors are bound to face technical problems as they develop competitive biotechnological products and processes using synthetic biology, genetics, and molecular biology as alternatives to chemical-based applications. In this regard, the biological control of microbial consortia based on synthetic biology solutions and the regulation and optimization of the migration from batch production to continuous production are ongoing tasks [1,2]. In the biopharmaceutical industry, improved bioprocesses are always in demand to address new regulatory requirements, quality control needs, and production problems in biological products, cell culture titration, and the production of biosimilars [3–5].

Applications derived from biotechnology are very diverse, including food design, processing, and optimization to improve nutritional intake [6]; the optimization of processes to purify monoclonal antibodies for the treatment of different conditions, the analysis of host cell proteins (HCPs), and the production of hematopoietic stem cells (HSCs) for therapeutic purposes [7–9]; the development

of microorganisms for the processing and transformation of biomass into fuels [10–12]; the production of raw materials based on fermentation processes, such as ethanol, butanol [13–15], and other products traditionally derived from chemical sources, such as aliphatic, aromatic, and other macromolecules using bioprocesses, such as (a) separate hydrolysis and fermentation (SHF), (b) simultaneous saccharification and fermentation (SSF), and (c) consolidated bioprocessing (CBP) [16]; and the construction of bioelectronic devices for applications in multivariate data analysis, experiment design, mathematical models, sensors, and biosensors whose data are processed by software to monitor and optimize processes [17–22].

Other important bioprocesses involve the large-scale production of secondary metabolites relevant for the food, cosmetics, pharmaceutical [23], wastewater treatment, and bioremediation industries (all of these are high-value processes) using bacteria and plant cells produced in vitro to protect endangered or scarce plants or to obtain metabolites [24,25] and enzymes produced by filamentous fungi, leveraging the advances of genetic engineering and molecular biology [26]; the development of cells that can be used in the production of new drugs [27]; the application of enzymatic processes to treat textiles [28]; the use of bacteria for the production of enzymes and various chemical products [29]; the use of nanotechnology, for instance, the nano-encapsulation of bioactive compounds, intelligent packaging systems in food production, biocatalysts and biosensors, and microbiological identification [30,31]; the collection and commercialization of recyclable and biodegradable biopolymers such as PLA (polylactide) [32]; and the development of regenerative medicine solutions [33].

Culture collections (CCs) and microbial biological resources centers (mBRCs) are two critical elements during the microorganism characterization and preservation process. In the second case, in spite of pending challenges, Europe has achieved substantial progress in the areas of databases, quality, infrastructure, legislation, and project development. This progress contributes to the preservation of biodiversity and ecosystems and, certainly, to stimulating the innovation, research, and development of biotechnology-based applications [34,35]. Molecular and genetic characterizations of living collections of biological resources provide added value to these biorepositories. As a consequence, their development has technical, financial, and regulatory implications to address depending on the type of collection (microbial cultures and animal or plant germplasm) [36].

Acetic acid bacteria are an interesting group of organisms with potential for the generation of diverse metabolites of industrial application based on sustainable processes; however, current processes still have limitations to address large-scale industrial demand [37]. Another example is the development of yeasts that produce high amounts of glutathione to be used in drugs, cosmetics, and foods; in the wine production industry, the antioxidant effect of glutathione and its action against unwanted aromatic compounds are particularly adequate [38].

As can be appreciated, the impact of biotechnology on social welfare is evident and has been widely discussed. Areas of focus such as sustainability, the decrease in CO2 emissions, technological change, and the bioeconomy are associated with this field of study, whose potential is vital for the development of numerous products based on inputs derived from agriculture or other renewable sources of biological origin [39–41]. For example, marine algae can be used to produce biofuels and bioenergy as a substitute for fossil fuels [42,43]. In the case of genetically modified and improved seeds, potential risks and benefits are the subject of heated debate, especially around the ethics of their development and use, and the issues related to economic and environmental impact have yet to be addressed [44,45]. It is also possible to use materials created by biological agents or to use these agents in environmental remediation applications [46]. This new reality underscores the need to analyze the peculiarities of these inventions in order to address their resulting intellectual property rights adequately [47].

Thus, under an full (economic, social, and environmental) approach, concerned with social welfare and the development of the bioeconomy, sustainability is closely oriented toward achieving objectives and promoting economic growth [48,49]. Although all of these biological resources are renewable and the solutions that they provide are socially valuable, it is essential to respect and preserve their biological sources and optimize the use of water and energy to carry out the bioprocesses. As a consequence, the use of environmental indicators (climate change, water, energy, land use, chemical risk) is necessary to manage resources sustainably [50]. However, evaluating the social impact of these advances is one of the most neglected tasks in the field of bioeconomics because the attention has been focused on environmental and techno-economic elements [51]. Additionally, the adverse effects of globalization on economic equality and the preservation of biodiversity must be considered in each case and context and paying attention to social indicators related to health, food, and employment, among others, must be paramount [52–54].

Biotechnology uses bioprocesses as an operating mechanism, and the development and improvement of these processes provide technological alternatives to solve myriad problems in the health, food, energy, agriculture, and many other industrial sectors. As a result, academia, the business sector, governments, non-governmental organizations, and the societies in which all these applications have a positive economic, social, or environmental impact have taken an interest in bioprocesses. Nevertheless, the alternatives brought forth by the bioeconomy to promote economic development should not be limited to technological advancement per se but include other aspects of interest to different actors.

Specifically, the emphasis must be placed on developments that support social welfare and mitigate environmental impact. Although technologically and economically efficient solutions to address environmental and other types of technical issues that represent a significant social impact are already being developed, it is also true that many challenges have to do with the creation of policies, regulations, and ethical guidelines concerning biosecurity, as well as with technical and risk assessments, industrial scale-up, the efficient use of renewable resources, and industry-driven ad hoc mechanisms to address specific problems derived from this area of application (see Figure 1).

**Figure 1.** Bioeconomy, biotechnology, and sustainability. Source: elaborated by the authors.

#### **2. Method**

A literature search and a patent application search were carried out using the Scopus (in the case of Scopus, the search criterion was as follows: (TITLE-ABS-KEY (bioproce\*) AND TITLE-ABS-KEY (sustainab\*)) AND PUBYEAR > 2009) [55] and Patentscope (in the case of Patentscope, the advanced search criteria for English language in all offices was: bioproce\* AND sustainab\*) [56] databases, respectively, for the period from 2010 to 2019, as a first approach to understand the relationship between sustainability and bioprocess. The purpose of the present study was to identify documents and patent applications related to the development and analysis of sustainability involving bioprocesses to approach our object of study along the economic, social, and environmental axes. In addition, the search sought to reveal an initial outline of the scientific and technological trajectories around these terms during the study period.

Thus, the first of these databases identified the 676 most relevant publications by country, knowledge area, institution, and source; the lowest number of published documents (29) corresponded to 2010, whereas the highest (103) corresponded to 2018. Concerning the patent search, the data considered were the number of applications per country, per institution, and technology area; 1233 applications were found, and 2013 was the year with the most significant number of applications (156). The data obtained from Scopus were subjected to network analysis, including co-occurrence, using the authors' keywords as a unit of analysis and full counting. Additionally, a co-authorship study was included in the analysis, considering the country as a unit of analysis and full counting.

#### **3. Results**

Figure 2 shows that the United States was the leader in bioprocess and sustainability-related research during the study period. Among the ten leading countries, India, China, and Germany published more than 50 documents each. The most developed areas, those with more than 100 documents, were biochemistry, genetics, and molecular biology; chemical engineering; immunology and microbiology; environmental science; energy and engineering. These data highlight the need to increase the number of basic research projects in disciplines focused on the development or improvement of new bioprocesses and their industrial scale-up and the creation of technological applications for the medical, food, environmental, and power generation sectors.

**Figure 2.** Documents by area of knowledge and country (2010–2019). Source: elaborated by the authors based on Scopus [55].

According to Journal Citation Reports (JCR) [57] 2018, two important academic journals led the list with more than 30 published documents: (1) Bioresource Technology, with an impact factor of 6.669 and ranked Q1 in the following three areas: (a) Agricultural Engineering, (b) Biotechnology and Applied Microbiology, and (c) Energy and Fuels; and (2) Applied Microbiology and Biotechnology, with an impact factor of 3.670 and ranked Q2 in Biotechnology and Applied Microbiology. They are followed by journals with ten or more published documents: Biotechnology for Biofuels (impact factor: 5.452); Biotechnology and Bioengineering (impact factor: 4.260); Current Opinion in Biotechnology (impact factor: 8.083); Renewable and Sustainable Energy Reviews (impact factor: 10.556); Journal of Chemical Technology and Biotechnology (impact factor: 2.659); and Biotechnology Journal (impact factor: 3.543) (the quartiles for the rest of the journals with at least 10 documents are distributed according to different categories as follows: (1) Biotechnology for Biofuels: Biotechnology and Applied Microbiology (Q1) and Energy and Fuels (Q1); (2) Biotechnology and Bioengineering: Biotechnology and Applied Microbiology (Q1); (3)- Current Opinion in Biotechnology: Biochemical Research Methods (Q1) and Biotechnology and Applied Microbiology (Q1); (4) Renewable and Sustainable Energy Reviews: Green and Sustainable Science and Technology (Q1) and Energy and Fuels (Q1); (5) Journal of Chemical Technology and Biotechnology: Biotechnology and Applied Microbiology (Q2), Chemistry, Multidisciplinary (Q2), Engineering, Environmental (Q3), Engineering, Chemical (Q2); and (6) Biotechnology Journal: Biochemical Research Methods (Q1) and Biotechnology and Applied Microbiology (Q2).) These top ten journals include the participation of institutions from Denmark, the Netherlands, England, France, the United States, Korea, India, and Brazil (see Figure 3).

**Figure 3.** Documents by affiliation and source of publication (2010–2019). Source: elaborated by the authors based on Scopus [55].

In regard to technological development, patent applications were clearly dominated by the United States, where 552 applications were filed. This country was followed by the Patent Cooperation Treaty (PCT), Australia, and the European Patent Office, with more than 100 applications each. By institution, Genomatica, Inc. ranked first with 118 applications, followed by Regents of the University of California with 44, and the Massachusetts Institute of Technology with 35; all of these institutions are based in the United States. The participation of independent inventors and another American company are noteworthy. These data show that the United States, Australia, and European countries are the target market for this type of invention, according to the data obtained by searching in English (see Figure 4).

**Figure 4.** Patent applications by country and institution (2010–2019). Source: elaborated by the authors based on World Intellectual Property Organization (WIPO) [56].

Most of these applications are related to scientific and technological areas of traditional expertise in the development of bioprocesses, fermentation processes, the use of enzymes to obtain various chemical compounds, and other formulations and applications involving microorganisms or enzymes, in addition to compositions of microorganisms and enzymes that are essential in many applications and processes. The rest of the categories are related to the development of new devices; the manufacture of organic compounds and pharmaceutical products; medical appliances; the production of deodorization, disinfection, and sterilization materials; indexing associated with other microorganism subclasses; applications related to water treatment, wastewater, sewage, and sludges; peptide generation processes; separation methods, and applications associated with sugars, sugar derivatives, nucleosides, nucleic acids, and nucleotides, among other examples (see Table 1).

The network analysis based on the literature search showed that sustainability and bioprocesses are central points in this search; both keywords are grouped in the largest cluster, together with smaller nodes such as green chemistry, enzymes, biocatalysis, and industrial biotechnology, among others. Around this cluster, the figure shows three smaller groupings, whose main nodes are as follows: (1) fermentation, biofuels, bioethanol, biorefining, and lignocellulose; (2) microalgae, anaerobic digestion, bioenergy, biodiesel, biohydrogen, and dark fermentation; and (3) consolidated bioprocessing, biofuel, biobutanol, and clostridium. Finally, a smaller cluster, whose main component is associated with synthetic biology, can also be appreciated. These links showcase the vigorous dynamics of bioprocess design research in connection with sustainability. They outline where most of the scientific research is taking place and where the new areas of opportunity originating around this activity can be found (see Figure 5).

**Table 1.** Patent applications by technological field (2010–2019).

Source: elaborated by the authors based on WIPO [56,58].

**Figure 5.** Keyword co-occurrence analysis: (author)-full counting. Source: elaborated by the authors based on Scopus [55].

As mentioned before, the United States was determined to be the primary originator of publications focused on sustainability and bioprocesses, and the same was valid for institutional collaboration. However, although with lower frequency and proximity, the presence of other economies such as India, Germany, China, England, South Africa, Mexico, Chile, Malaysia, Australia, Turkey, Italy,

and Singapore, among others, could also be observed in the network; individually, these countries connected to different clusters. Therefore, it is necessary to increase our efforts to generate new research focused on the sustainability of bioprocesses in local environments in collaboration with scientists from institutions in different parts of the world, since biotechnological solutions cannot always be applied globally, hence the need for ad hoc solutions to specific problems, especially in developing countries (see Figure 6).

**Figure 6.** Co-authorship analysis: country-full counting. Source: elaborated by the authors based on Scopus [55].

#### **4. Conclusions**

Biotechnology has provided society with thousands of bioprocesses to address diseases and food demands, to develop petroleum product substitutes, to provide alternatives for energy production, and to solve agricultural problems, among other benefits. Applications and products based on biological sources are the framework of a bioeconomy that contributes to the economic development of regions and countries. However, social welfare and care for the environment must be inherent to these applications, which is why the generation of ad hoc indicators of these two areas is necessary to monitor these areas.

The scientific and technological trajectory shows how sustainability and bioprocesses are topics of great interest and constantly growing, although further efforts are still needed to move toward an integrated sustainability framework. Microbiology and enzymology are often prevalent in this technological field, although new areas of opportunity are emerging around the new demand for sustainable solutions to support economic growth and industrial development, especially basic science projects, which need to be explored and exploited in greater depth. New bioprocesses based on biorefining, bioethanol, consolidated bioprocessing, microalgae, lignocellulose, biocatalysis, and biohydrogen are among the products and technologies of the future.

In the following decades, actors from the academic, business, and government sectors, in addition to non-governmental organizations and society in general, will have to intensify their collaboration mechanisms, especially in developing economies, where the challenge to move forward sustainably is harder and problems associated with poverty and inequality tend to be more serious. Although biotechnology has shown unprecedented progress, it is also true that the design of bioprocesses must be geared to sustainability criteria, in which social impact must be a priority. Additional aspects to take into account to guarantee the successful development of future biotechnological applications for the benefit of economic development, environmental protection, and social welfare are legislative, normative, and ethical considerations; the optimization of resources and the conservation of biological sources; technical and risk assessment; biosecurity, and intellectual property.

**Author Contributions:** Conceptualization and methodology: A.B.-O. and P.S.-B.; Investigation: A.B.-O., P.S.-B., S.O.-P. and M.P.-O.; Formal Analysis: A.B.-O., P.S.-B. and S.O.-P.; Data curation: P.S.-B., S.O.-P. and M.P.-O. All authors have read and agreed to the published version of the manuscript.

**Funding:** SIP grant numbers 20195587 and 20200773.

**Acknowledgments:** We wish to acknowledge the support provided by the National Polytechnic Institute (Instituto Politécnico Nacional—IPN) and the Secretariat for Research and Postgraduate Studies (Secretaría de Investigación y Posgrado—SIP).

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

#### **References**

© 2020 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 (http://creativecommons.org/licenses/by/4.0/).


**3.1 The Modified Local Green's Function Method - MLGFM**

The Modified Local Green's Function Method (MLGFM) is an integral technique that associates the Finite Element Method and the Boundary Element Method, solving the problem through an integral equations system. Unlike to the BEM and the Trefitz Methods, the MLGFM does not use a fundamental solution and/or a Green's function. The term "Local" indicates that the calculation of the GF projections can be done locally, that is, for each element.

Essentially, the MLGFM uses a transverse integration technique and reciprocity relations to determine, at a local level, the Green's Function, transforming the partial differential operator in an ordinary partial operator (Barcelos & Silva, 1987). The MLGFM uses finite elements at the domain with the purpose to create discrete projections of the Green's Function, corresponding to fundamental solutions, that are used later in the integral equations system associated to the boundary approximation. To analyze a continuum mechanics problem through the MLGFM, such as the plate indicated in figure 4, two meshes are necessary, one for the domain and other for the boundary. With domain elements, the method generates a set of domain equations, which are used to generate automatically the domain Green's projections. Later, a set of boundary equations are also generated and the boundary Green's projections can be determined with the domain projections developed before. At the end, the system is solved only for boundary equations, where the main variables are calculated. Domain values may be obtained once the boundary values are known after the solution of the boundary equation system.

Considering a conventional structural approximation by conventional Finite Element Method, the problem can be expressed by a system of algebraic equations, representing a

> 1 1 11 12 16 21 22 26 2 2 61 62 66 6 6

where *Kij* are the element stiffness matrix, (*dx*, *dy*, *dz*) are the components of the element displacement vector and {*Fa*} and {*Fd*} are the applied force vector and the element damage

The procedure used in this paper to obtain the expected results is a little different because it uses a different computational method known as Modified Local Green's Function Method (MLGFM), in witch the system defined in expression (22) is not directly applied in a conventional FEM. A detailed explanation of this procedure can be found in Barbieri *et al* (1998a,b) and Machado *et al* (2008). The MLGFM is an integral method that determines the unknowns on the boundary, similarly to the Boundary Element Method, but the fundamental solutions are generated automatically by projections of the Green's Functions developed from de field, as in the Finite Element Method. The matrix and the vectors indicated in (22) will be

The Modified Local Green's Function Method (MLGFM) is an integral technique that associates the Finite Element Method and the Boundary Element Method, solving the problem through an integral equations system. Unlike to the BEM and the Trefitz Methods, the MLGFM does not use a fundamental solution and/or a Green's function. The term "Local" indicates that the calculation of the GF projections can be done locally, that is, for

Essentially, the MLGFM uses a transverse integration technique and reciprocity relations to determine, at a local level, the Green's Function, transforming the partial differential operator in an ordinary partial operator (Barcelos & Silva, 1987). The MLGFM uses finite elements at the domain with the purpose to create discrete projections of the Green's Function, corresponding to fundamental solutions, that are used later in the integral equations system associated to the boundary approximation. To analyze a continuum mechanics problem through the MLGFM, such as the plate indicated in figure 4, two meshes are necessary, one for the domain and other for the boundary. With domain elements, the method generates a set of domain equations, which are used to generate automatically the domain Green's projections. Later, a set of boundary equations are also generated and the boundary Green's projections can be determined with the domain projections developed before. At the end, the system is solved only for boundary equations, where the main variables are calculated. Domain values may be obtained once the boundary values are

used to produce values at the boundary, as explained in the next topic.

**3.1 The Modified Local Green's Function Method - MLGFM**

known after the solution of the boundary equation system.

Fig. 4. Symmetric boundary conditions and plate for a 2x2 mesh (4 finite elements and 8 contour elements) by the MLGFM.

The most important steps of the MLGFM are detailed in the work sof Barbieri *et al* (1998a) and Machado *et al* (2008). It is possible to show that through the MLGFM two sets of equations are formed, the first one in the domain (equation (23)) and the other one on the boundary (equation (24)):

$$\mathbf{u}(\mathbf{Q}) = \int\_{\Omega} \left[ \mathbf{G}^{\mathrm{T}}(\mathbf{P}, \mathbf{Q}) \, \mathbf{a}(\mathbf{P}) \right] d\Omega + \int\_{\Gamma} \left[ \mathbf{G}^{\mathrm{T}}(\mathbf{p}, \mathbf{Q}) \, \mathbf{f}(\mathbf{p}) \right] d\Gamma \, ; \; \mathbf{P}, \mathbf{Q} \in \Omega \; ; \mathbf{p} \in \Gamma \tag{23}$$

$$\mathbf{u}(\mathbf{q}) = \int\_{\Omega} \left[ \mathbf{G}^{\mathrm{T}} \{ \mathbf{P}, \mathbf{q} \} \, \mathbf{a}(\mathbf{P}) \right] d\Omega + \int\_{\Gamma} \left[ \mathbf{G}^{\mathrm{T}} \{ \mathbf{p}, \mathbf{q} \} \, \mathbf{f}(\mathbf{p}) \right] d\Gamma \, ; \; \mathbf{P} \in \Omega \ ; \; \mathbf{p}, \mathbf{q} \in \Gamma \tag{24}$$

where Q, P are two points in the domain; q, p are other two points on the boundary; **a**(P) is the vector of independent terms for the original problem; f(p) is the vector associated to the fluxes on the boundary; **G**(i,j) are the Green's functions which may be understood as the generalized displacement in the point i in the direction of an unitary vector **ni**, when a generalized force is applied over the point j, in the direction of a unitary vector **n**j.

Equations (23) and (24) describe completely the problem. Since these equations involve domain and boundary integrals, two types of meshes are necessary, one in the domain and the other on the boundary, using FE and BE methods, respectively. The FE domain approximation is also used to develop the Green's functions which are associated to the matrices **G(**P,Q**), G(**p,Q**), G(**P,q**)** and **G(**p,q**).**

Progressive Stiffness Loss Analysis

where

following functional **F**, which one depends on **G** or **G** :

**B** – is a bilinear form, developed to **G** or **G;**  and are constants whose values are: = 1 and = 0 to determine **G** = 0 and = 1 to determine **G**

 **B1(G,[])** =

 **B2(G,[])** =

(26), which are the main system of the MLGFM.

 **B3(G,G)** = 1

Green's projections

**4. Damage evolution**

where *Fa* is the failure criterion,

**B1**, **B2**, **B3** – are bilinear forms which can be written as:

**G** – corresponds to **G** or **G** depending on the case of interest;

of Symmetric Laminated Plates due to Transverse Cracks Using the MLGFM 133

In order to determine the Green's functions automatically, it must be considered the

 **F(G,G) = B(G,G) - B1(G,) - B2(G, ) + B3(G,G)** (37)

As in the variational approach of the FEM, the minimization of functional **F(G,G)** in Equation (37) results in a linear equation system which can be solved to determine the

where [**K**] is the global stiffness matrix, evaluated in the same way of the conventional finite element stiffness matrix; **A** and **D** are the matrices of Equations (27) and (30), respectively. In this way, the Green's projections are determined directly from Equation (41), and can be applied in Equations (28), (29), (31) and (32) to complete the matrices of equations (25) and

With purpose to quantify the damage accumulation due to a monotonic load increment,

*ij a ij F Z X*

and *Z* is the failure value characteristic to each criterion. The rigidity degradation of a component occurs due to a progressive process during its serviceable life. It is important to note that the evolution of rigidity loss in a structure can be characterized by a single crack or by the occurrence of generalized cracks. Combinations of failure modes can act together

some failure criterion will be used. Generically, failure criteria can be considered as:

2  **G(**Q**) (**Q**)** d (38)

**G(**q**) (**q**)** d (39)

(42)

**N#(**q**) G (**q**). G (**q**)** d (40)

**KG G A D Ω Γ** (Q) (Q) (41)

*ij* are the local stress, *Xij* are the principal material strength

The approximation shape functions are the same as in the conventional FE and BE methods. For the present work, quadratic shape functions were employed to construct nine nodes lagrangean finite elements and three nodes boundary elements.

 The matrices **A**, **B**, **C**, **D**, **E** and **F** can be written as:

$$\mathbf{A} = \int\limits\_{\Omega} \Psi(\mathbf{Q})^{\mathrm{T}} \Psi(\mathbf{Q}) \mathrm{d}\Omega \tag{27}$$

$$\mathbf{B} = \int\limits\_{\Omega} \Psi(\mathbf{Q})^{\mathrm{T}} \mathbf{G}\_{\mathbf{T}}(\mathbf{Q}) d\Omega \tag{28}$$

$$\mathbf{C} = \int\limits\_{\Omega} \Psi(\mathbf{Q})^{\mathrm{T}} \, \mathbf{G}\_{\mathbf{D}}(\mathbf{Q}) d\Omega \tag{29}$$

$$\mathbf{D} = \int\_{\Gamma} \Phi(\mathbf{q})^{\mathbf{r}} \Phi(\mathbf{q}) d\Gamma \tag{30}$$

$$\mathbf{E} = \int\_{\Gamma} \Phi(\mathbf{q}) \mathbf{r} \, \mathbf{G}\_{\mathbf{T}}(\mathbf{q}) d\Gamma \tag{31}$$

$$\mathbf{F} = \int\limits\_{\Gamma} \Phi(\mathbf{q})^{\mathbf{T}} \mathbf{G}\_{\mathbf{d}\mathbf{d}}(\mathbf{q}) d\Gamma \tag{32}$$

where **(**Q**)** and **(**q**)** are matrices with the shape functions in the domain and on the boundary, respectively, and **G(**Q**), G(**Q**), G(**q**), G(**q**)** are the Green's function projections over the boundary and the domain , evaluated on the points Q and q. The Green's projections can be written as:

$$\mathbf{G\_{T}(Q) = \int\limits\_{\Gamma} G^{\tau}(p, Q) \,\Phi(p) \,\mathrm{d}\Gamma} \tag{33}$$

$$\mathbf{G\_{D}(Q) = \int\limits\_{\Omega} \mathbf{G^{T}(P,Q)} \,\Psi(P) \,\mathrm{d}\Omega \,\tag{34}$$

$$\mathbf{G\_{T(q)}} = \int\_{\Gamma\_{\parallel}} \mathbf{G\_{T(p,q)}} \, \Phi(\mathbf{p}) \, \mathrm{d}\Gamma \,\tag{35}$$

$$\mathbf{G\_{f2}(q) = \int\_{\Omega} \colon \mathbf{G^{T}(P,q)} \,\Psi(P) \,\mathrm{d}\Omega \,\tag{36}$$

In order to determine the Green's functions automatically, it must be considered the following functional **F**, which one depends on **G** or **G** :

$$\mathbf{F(G\_{D},G\_{T}) = B(G,G) - \alpha \ B\_{1}(G\_{D}, \mathbf{y}) - \beta \ B\_{2}(G\_{T}, \mathbf{d}) + B\_{3}(G, G)}\tag{37}$$

where

132 Continuum Mechanics – Progress in Fundamentals and Engineering Applications

The approximation shape functions are the same as in the conventional FE and BE methods. For the present work, quadratic shape functions were employed to construct nine nodes

Developing discrete equations from the nodal values, two sets of linear equations are

 **A u = B f + C a** (in the domain) (25)

 **D u = E f + F a** (on the boundary) (26) where **u** and **u** are the domain and the boundary displacements, respectively, **a** and **f** are the independent and the fluxes variables vectors. The matrices **A**, **B**, **C**, **D**, **E** and **F** can be

(Q)T (Q)d (27)

**(**Q**)T G(**Q**)**d (28)

**(**Q**)T G(**Q**)**d (29)

**(**q**)T (**q**)**d (30)

**(**q**)T G(**q**)**d (31)

**(**q**)T G(**q**)**d (32)

**GT(**p,Q**) (**p**)** d (33)

**GT(**P,Q**) (**P**)** d (34)

**GT(**p,q**) (**p**)** d (35)

**GT(**P,q**) (**P**)** d (36)

lagrangean finite elements and three nodes boundary elements.

**A** =

**G(**Q**)** =

where **(**Q**)** and **(**q**)** are matrices with the shape functions in the domain and on the boundary, respectively, and **G(**Q**), G(**Q**), G(**q**), G(**q**)** are the Green's function projections over the boundary and the domain , evaluated on the points Q and q. The Green's

determined:

written as:

 **B =**

 **C =**

 **D =**

 **E =**

 **F =**

 **G(**Q**) =**

 **G(**q**) =**

 **G(**q**) =**

projections can be written as:

**G** – corresponds to **G** or **G** depending on the case of interest;

**B** – is a bilinear form, developed to **G** or **G;**

and are constants whose values are:

= 1 and = 0 to determine **G**

= 0 and = 1 to determine **G**

**B1**, **B2**, **B3** – are bilinear forms which can be written as:

$$\mathbf{B}\_1(\mathbf{G}\_\mathbf{D}[\Psi]) = \int\_{\Omega} \mathbf{G}\_\mathbf{B}(\mathbf{Q}) \, \Psi(\mathbf{Q}) \, \mathrm{d}\Omega \tag{38}$$

$$\mathbf{B\_2(G\_{\sf T}[\Phi]) = \int\_{\sf T} G\_{\sf T}(\mathbf{q}) \, \boldsymbol{\Phi}(\mathbf{q}) \, d\sf{T} \tag{39}$$

$$\mathbf{B}\_3(\mathbf{G}, \mathbf{G}) = \frac{1}{2} \int\limits\_{\Gamma} \mathbf{N}^\bullet(\mathbf{q}) \, \mathbf{G} \text{ (q)}. \, \mathbf{G} \text{ (q)} \, \text{d}\Gamma \tag{40}$$

As in the variational approach of the FEM, the minimization of functional **F(G,G)** in Equation (37) results in a linear equation system which can be solved to determine the Green's projections

$$\begin{bmatrix} \mathbf{K} \end{bmatrix} \begin{bmatrix} \mathbf{G}\_{\Omega}(\mathbf{Q}) \ \vdots \ \mathbf{G}\_{\Gamma}(\mathbf{Q}) \end{bmatrix} = \begin{bmatrix} \mathbf{A} & \vdots & \mathbf{D} \end{bmatrix} \tag{41}$$

where [**K**] is the global stiffness matrix, evaluated in the same way of the conventional finite element stiffness matrix; **A** and **D** are the matrices of Equations (27) and (30), respectively. In this way, the Green's projections are determined directly from Equation (41), and can be applied in Equations (28), (29), (31) and (32) to complete the matrices of equations (25) and (26), which are the main system of the MLGFM.

#### **4. Damage evolution**

With purpose to quantify the damage accumulation due to a monotonic load increment, some failure criterion will be used. Generically, failure criteria can be considered as:

$$F\_a \left(\frac{\sigma\_{ij}}{X\_{ij}}\right) = Z$$

where *Fa* is the failure criterion, *ij* are the local stress, *Xij* are the principal material strength and *Z* is the failure value characteristic to each criterion. The rigidity degradation of a component occurs due to a progressive process during its serviceable life. It is important to note that the evolution of rigidity loss in a structure can be characterized by a single crack or by the occurrence of generalized cracks. Combinations of failure modes can act together

Progressive Stiffness Loss Analysis

plane stress condition, resulted in the expression (43):

**4.5 Strain Energy Release Rate Criterion**

O

Fig. 5. Stiffness loss in composite laminates [0°l/90°m]s – Lim & Tay (1995).

The result is a reduction in the effective stiffness of the laminate in the loading direction, and this is represented by the OB segment in figure 5. Upon further loading, this reduction is verified by the segment BC. This process is repeated until line OF is reached. Note that

**4.3 Tsai-Hill Criterion**

**4.4 Tsai-Wu Criterion**

Tsai-Hill Criterion.

of Symmetric Laminated Plates due to Transverse Cracks Using the MLGFM 135

An adaptation made by Tsai in the Hill Criterion for transverse orthotropic laminate at

(43)

(44)

22 2 1 2 1 2 12 22 2 2 1 *XY X C*

A simple procedure was proposed by Tsai-Wu, changing the Tsai-Hill Criterion in equation (43). When the tensile and compression strength are similar, the expression (44) becomes the

> 22 2 1 2 1 2 12 22 2 1 *XY C XY*

The Strain Energy Release Rate Criterion to describe the damage evolution was applied by Lim and Tay (Lim & Tay, 1996). Consider a symmetric laminated composite of width *b* and length *L*, with the configuration [0°l/90°m]s, where l and m are integers. When the laminated is loaded uniaxially in tension, the stress-strain curve is linear until the failure criterion is reached for the first time, at point A (figure 5). A transverse crack is introduced in 90° layer.

C

E

F

D

H

A

B

G

causing changes in the material properties and in its local stress distribution. In this way, the main difficulty in this kind of analysis is the adoption of a failure criterion, *Fa* , that conveniently describes the damage evolution due to a failure mode.

The theories introduced to prevent the failure of an orthotropic laminate are adaptations of failure criterion for isotropic materials, modified for biaxial stress cases, such as, Maximum Stress Criterion, Maximum Strain Criterion, Tsai-Hill Criterion and Tsai-Wu Criterion (Reddy, 1997; Vasiliev & Morozov, 2001; Mendonça, 2005).

A new criterion is presented, based on the strain energy release rate, to evaluate the formation of a new micro crack (Anderssen et al., 1998; Ji et al., 1998; Kobayashi et al.,2000). The released energy is used because it is practically independent from the crack length (Anderssen et al., 1998). Some of these criterions are presented here.