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Publisher: Springer-Verlag (Total: 2352 journals)

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Showing 1 - 200 of 2352 Journals sorted alphabetically
3D Printing in Medicine     Open Access   (Followers: 2)
3D Research     Hybrid Journal   (Followers: 21, SJR: 0.222, CiteScore: 1)
4OR: A Quarterly J. of Operations Research     Hybrid Journal   (Followers: 10, SJR: 0.825, CiteScore: 1)
AAPS J.     Hybrid Journal   (Followers: 23, SJR: 1.118, CiteScore: 4)
AAPS PharmSciTech     Hybrid Journal   (Followers: 7, SJR: 0.752, CiteScore: 3)
Abdominal Imaging     Hybrid Journal   (Followers: 17, SJR: 0.866, CiteScore: 2)
Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg     Hybrid Journal   (Followers: 4, SJR: 0.439, CiteScore: 0)
Academic Psychiatry     Full-text available via subscription   (Followers: 27, SJR: 0.53, CiteScore: 1)
Academic Questions     Hybrid Journal   (Followers: 8, SJR: 0.106, CiteScore: 0)
Accreditation and Quality Assurance: J. for Quality, Comparability and Reliability in Chemical Measurement     Hybrid Journal   (Followers: 30, SJR: 0.316, CiteScore: 1)
Acoustical Physics     Hybrid Journal   (Followers: 11, SJR: 0.359, CiteScore: 1)
Acoustics Australia     Hybrid Journal   (SJR: 0.232, CiteScore: 1)
Acta Analytica     Hybrid Journal   (Followers: 7, SJR: 0.367, CiteScore: 0)
Acta Applicandae Mathematicae     Hybrid Journal   (Followers: 1, SJR: 0.675, CiteScore: 1)
Acta Biotheoretica     Hybrid Journal   (Followers: 4, SJR: 0.284, CiteScore: 1)
Acta Diabetologica     Hybrid Journal   (Followers: 18, SJR: 1.587, CiteScore: 3)
Acta Endoscopica     Hybrid Journal   (Followers: 1)
acta ethologica     Hybrid Journal   (Followers: 4, SJR: 0.769, CiteScore: 1)
Acta Geochimica     Hybrid Journal   (Followers: 7, SJR: 0.24, CiteScore: 1)
Acta Geodaetica et Geophysica     Hybrid Journal   (Followers: 3, SJR: 0.305, CiteScore: 1)
Acta Geophysica     Hybrid Journal   (Followers: 11, SJR: 0.312, CiteScore: 1)
Acta Geotechnica     Hybrid Journal   (Followers: 7, SJR: 1.588, CiteScore: 3)
Acta Informatica     Hybrid Journal   (Followers: 5, SJR: 0.517, CiteScore: 1)
Acta Mathematica     Hybrid Journal   (Followers: 13, SJR: 7.066, CiteScore: 3)
Acta Mathematica Hungarica     Hybrid Journal   (Followers: 2, SJR: 0.452, CiteScore: 1)
Acta Mathematica Sinica, English Series     Hybrid Journal   (Followers: 6, SJR: 0.379, CiteScore: 1)
Acta Mathematica Vietnamica     Hybrid Journal   (SJR: 0.27, CiteScore: 0)
Acta Mathematicae Applicatae Sinica, English Series     Hybrid Journal   (SJR: 0.208, CiteScore: 0)
Acta Mechanica     Hybrid Journal   (Followers: 21, SJR: 1.04, CiteScore: 2)
Acta Mechanica Sinica     Hybrid Journal   (Followers: 5, SJR: 0.607, CiteScore: 2)
Acta Metallurgica Sinica (English Letters)     Hybrid Journal   (Followers: 7, SJR: 0.576, CiteScore: 2)
Acta Meteorologica Sinica     Hybrid Journal   (Followers: 3, SJR: 0.638, CiteScore: 1)
Acta Neurochirurgica     Hybrid Journal   (Followers: 7, SJR: 0.822, CiteScore: 2)
Acta Neurologica Belgica     Hybrid Journal   (Followers: 2, SJR: 0.376, CiteScore: 1)
Acta Neuropathologica     Hybrid Journal   (Followers: 4, SJR: 7.589, CiteScore: 12)
Acta Oceanologica Sinica     Hybrid Journal   (Followers: 3, SJR: 0.334, CiteScore: 1)
Acta Physiologiae Plantarum     Hybrid Journal   (Followers: 3, SJR: 0.574, CiteScore: 2)
Acta Politica     Hybrid Journal   (Followers: 15, SJR: 0.605, CiteScore: 1)
Activitas Nervosa Superior     Hybrid Journal   (SJR: 0.147, CiteScore: 0)
adhäsion KLEBEN & DICHTEN     Hybrid Journal   (Followers: 8, SJR: 0.103, CiteScore: 0)
ADHD Attention Deficit and Hyperactivity Disorders     Hybrid Journal   (Followers: 25, SJR: 0.72, CiteScore: 2)
Adhesion Adhesives & Sealants     Hybrid Journal   (Followers: 9)
Administration and Policy in Mental Health and Mental Health Services Research     Partially Free   (Followers: 18, SJR: 1.005, CiteScore: 2)
Adsorption     Hybrid Journal   (Followers: 5, SJR: 0.703, CiteScore: 2)
Advances in Applied Clifford Algebras     Hybrid Journal   (Followers: 4, SJR: 0.698, CiteScore: 1)
Advances in Atmospheric Sciences     Hybrid Journal   (Followers: 37, SJR: 0.956, CiteScore: 2)
Advances in Computational Mathematics     Hybrid Journal   (Followers: 19, SJR: 0.812, CiteScore: 1)
Advances in Contraception     Hybrid Journal   (Followers: 3)
Advances in Data Analysis and Classification     Hybrid Journal   (Followers: 57, SJR: 1.09, CiteScore: 1)
Advances in Gerontology     Partially Free   (Followers: 8, SJR: 0.144, CiteScore: 0)
Advances in Health Sciences Education     Hybrid Journal   (Followers: 32, SJR: 1.64, CiteScore: 2)
Advances in Manufacturing     Hybrid Journal   (Followers: 4, SJR: 0.475, CiteScore: 2)
Advances in Polymer Science     Hybrid Journal   (Followers: 45, SJR: 1.04, CiteScore: 3)
Advances in Therapy     Hybrid Journal   (Followers: 5, SJR: 1.075, CiteScore: 3)
Aegean Review of the Law of the Sea and Maritime Law     Hybrid Journal   (Followers: 7)
Aequationes Mathematicae     Hybrid Journal   (Followers: 2, SJR: 0.517, CiteScore: 1)
Aerobiologia     Hybrid Journal   (Followers: 3, SJR: 0.673, CiteScore: 2)
Aesthetic Plastic Surgery     Hybrid Journal   (Followers: 11, SJR: 0.825, CiteScore: 1)
African Archaeological Review     Hybrid Journal   (Followers: 21, SJR: 0.862, CiteScore: 1)
Afrika Matematika     Hybrid Journal   (Followers: 1, SJR: 0.235, CiteScore: 0)
AGE     Hybrid Journal   (Followers: 7)
Ageing Intl.     Hybrid Journal   (Followers: 7, SJR: 0.39, CiteScore: 1)
Aggiornamenti CIO     Hybrid Journal   (Followers: 1)
Aging Clinical and Experimental Research     Hybrid Journal   (Followers: 3, SJR: 0.67, CiteScore: 2)
Agricultural Research     Hybrid Journal   (Followers: 6, SJR: 0.276, CiteScore: 1)
Agriculture and Human Values     Hybrid Journal   (Followers: 14, SJR: 1.173, CiteScore: 3)
Agroforestry Systems     Hybrid Journal   (Followers: 20, SJR: 0.663, CiteScore: 1)
Agronomy for Sustainable Development     Hybrid Journal   (Followers: 13, SJR: 1.864, CiteScore: 6)
AI & Society     Hybrid Journal   (Followers: 9, SJR: 0.227, CiteScore: 1)
AIDS and Behavior     Hybrid Journal   (Followers: 14, SJR: 1.792, CiteScore: 3)
Air Quality, Atmosphere & Health     Hybrid Journal   (Followers: 4, SJR: 0.862, CiteScore: 3)
Akupunktur & Aurikulomedizin     Full-text available via subscription   (Followers: 1)
Algebra and Logic     Hybrid Journal   (Followers: 6, SJR: 0.531, CiteScore: 0)
Algebra Universalis     Hybrid Journal   (Followers: 2, SJR: 0.583, CiteScore: 1)
Algebras and Representation Theory     Hybrid Journal   (Followers: 1, SJR: 1.095, CiteScore: 1)
Algorithmica     Hybrid Journal   (Followers: 9, SJR: 0.56, CiteScore: 1)
Allergo J.     Full-text available via subscription   (Followers: 1, SJR: 0.234, CiteScore: 0)
Allergo J. Intl.     Hybrid Journal   (Followers: 2)
Alpine Botany     Hybrid Journal   (Followers: 5, SJR: 1.11, CiteScore: 3)
ALTEX : Alternatives to Animal Experimentation     Open Access   (Followers: 3)
AMBIO     Hybrid Journal   (Followers: 10, SJR: 1.569, CiteScore: 4)
American J. of Cardiovascular Drugs     Hybrid Journal   (Followers: 16, SJR: 0.951, CiteScore: 3)
American J. of Community Psychology     Hybrid Journal   (Followers: 29, SJR: 1.329, CiteScore: 2)
American J. of Criminal Justice     Hybrid Journal   (Followers: 9, SJR: 0.772, CiteScore: 1)
American J. of Cultural Sociology     Hybrid Journal   (Followers: 18, SJR: 0.46, CiteScore: 1)
American J. of Dance Therapy     Hybrid Journal   (Followers: 5, SJR: 0.181, CiteScore: 0)
American J. of Potato Research     Hybrid Journal   (Followers: 2, SJR: 0.611, CiteScore: 1)
American J. of Psychoanalysis     Hybrid Journal   (Followers: 21, SJR: 0.314, CiteScore: 0)
American Sociologist     Hybrid Journal   (Followers: 16, SJR: 0.35, CiteScore: 0)
Amino Acids     Hybrid Journal   (Followers: 8, SJR: 1.135, CiteScore: 3)
AMS Review     Partially Free   (Followers: 4)
Analog Integrated Circuits and Signal Processing     Hybrid Journal   (Followers: 7, SJR: 0.211, CiteScore: 1)
Analysis and Mathematical Physics     Hybrid Journal   (Followers: 5, SJR: 0.536, CiteScore: 1)
Analysis in Theory and Applications     Hybrid Journal   (Followers: 1)
Analysis of Verbal Behavior     Hybrid Journal   (Followers: 6)
Analytical and Bioanalytical Chemistry     Hybrid Journal   (Followers: 32, SJR: 0.978, CiteScore: 3)
Anatomical Science Intl.     Hybrid Journal   (Followers: 3, SJR: 0.367, CiteScore: 1)
Angewandte Schmerztherapie und Palliativmedizin     Hybrid Journal  
Angiogenesis     Hybrid Journal   (Followers: 3, SJR: 2.177, CiteScore: 5)
Animal Cognition     Hybrid Journal   (Followers: 20, SJR: 1.389, CiteScore: 3)
Annales françaises de médecine d'urgence     Hybrid Journal   (Followers: 1, SJR: 0.192, CiteScore: 0)
Annales Henri Poincaré     Hybrid Journal   (Followers: 3, SJR: 1.097, CiteScore: 2)
Annales mathématiques du Québec     Hybrid Journal   (Followers: 4, SJR: 0.438, CiteScore: 0)
Annali dell'Universita di Ferrara     Hybrid Journal   (SJR: 0.429, CiteScore: 0)
Annali di Matematica Pura ed Applicata     Hybrid Journal   (Followers: 1, SJR: 1.197, CiteScore: 1)
Annals of Biomedical Engineering     Hybrid Journal   (Followers: 17, SJR: 1.042, CiteScore: 3)
Annals of Combinatorics     Hybrid Journal   (Followers: 4, SJR: 0.932, CiteScore: 1)
Annals of Data Science     Hybrid Journal   (Followers: 12)
Annals of Dyslexia     Hybrid Journal   (Followers: 10, SJR: 0.85, CiteScore: 2)
Annals of Finance     Hybrid Journal   (Followers: 34, SJR: 0.579, CiteScore: 1)
Annals of Forest Science     Hybrid Journal   (Followers: 7, SJR: 0.986, CiteScore: 2)
Annals of Global Analysis and Geometry     Hybrid Journal   (Followers: 1, SJR: 1.228, CiteScore: 1)
Annals of Hematology     Hybrid Journal   (Followers: 15, SJR: 1.043, CiteScore: 2)
Annals of Mathematics and Artificial Intelligence     Hybrid Journal   (Followers: 12, SJR: 0.413, CiteScore: 1)
Annals of Microbiology     Hybrid Journal   (Followers: 11, SJR: 0.479, CiteScore: 2)
Annals of Nuclear Medicine     Hybrid Journal   (Followers: 5, SJR: 0.687, CiteScore: 2)
Annals of Operations Research     Hybrid Journal   (Followers: 10, SJR: 0.943, CiteScore: 2)
Annals of Ophthalmology     Hybrid Journal   (Followers: 12)
Annals of Regional Science     Hybrid Journal   (Followers: 8, SJR: 0.614, CiteScore: 1)
Annals of Software Engineering     Hybrid Journal   (Followers: 13)
Annals of Solid and Structural Mechanics     Hybrid Journal   (Followers: 9, SJR: 0.239, CiteScore: 1)
Annals of Surgical Oncology     Hybrid Journal   (Followers: 15, SJR: 1.986, CiteScore: 4)
Annals of Telecommunications     Hybrid Journal   (Followers: 9, SJR: 0.223, CiteScore: 1)
Annals of the Institute of Statistical Mathematics     Hybrid Journal   (Followers: 1, SJR: 1.495, CiteScore: 1)
Antonie van Leeuwenhoek     Hybrid Journal   (Followers: 5, SJR: 0.834, CiteScore: 2)
Apidologie     Hybrid Journal   (Followers: 4, SJR: 1.22, CiteScore: 3)
APOPTOSIS     Hybrid Journal   (Followers: 9, SJR: 1.424, CiteScore: 4)
Applicable Algebra in Engineering, Communication and Computing     Hybrid Journal   (Followers: 2, SJR: 0.294, CiteScore: 1)
Applications of Mathematics     Hybrid Journal   (Followers: 2, SJR: 0.602, CiteScore: 1)
Applied Biochemistry and Biotechnology     Hybrid Journal   (Followers: 44, SJR: 0.571, CiteScore: 2)
Applied Biochemistry and Microbiology     Hybrid Journal   (Followers: 18, SJR: 0.21, CiteScore: 1)
Applied Categorical Structures     Hybrid Journal   (Followers: 4, SJR: 0.49, CiteScore: 0)
Applied Composite Materials     Hybrid Journal   (Followers: 49, SJR: 0.58, CiteScore: 2)
Applied Entomology and Zoology     Partially Free   (Followers: 6, SJR: 0.422, CiteScore: 1)
Applied Geomatics     Hybrid Journal   (Followers: 3, SJR: 0.733, CiteScore: 3)
Applied Geophysics     Hybrid Journal   (Followers: 9, SJR: 0.488, CiteScore: 1)
Applied Intelligence     Hybrid Journal   (Followers: 13, SJR: 0.6, CiteScore: 2)
Applied Magnetic Resonance     Hybrid Journal   (Followers: 4, SJR: 0.319, CiteScore: 1)
Applied Mathematics & Optimization     Hybrid Journal   (Followers: 8, SJR: 0.886, CiteScore: 1)
Applied Mathematics - A J. of Chinese Universities     Hybrid Journal   (SJR: 0.17, CiteScore: 0)
Applied Mathematics and Mechanics     Hybrid Journal   (Followers: 5, SJR: 0.461, CiteScore: 1)
Applied Microbiology and Biotechnology     Hybrid Journal   (Followers: 66, SJR: 1.182, CiteScore: 4)
Applied Physics A     Hybrid Journal   (Followers: 10, SJR: 0.481, CiteScore: 2)
Applied Physics B: Lasers and Optics     Hybrid Journal   (Followers: 24, SJR: 0.74, CiteScore: 2)
Applied Psychophysiology and Biofeedback     Hybrid Journal   (Followers: 8, SJR: 0.519, CiteScore: 2)
Applied Research in Quality of Life     Hybrid Journal   (Followers: 12, SJR: 0.316, CiteScore: 1)
Applied Solar Energy     Hybrid Journal   (Followers: 22, SJR: 0.225, CiteScore: 0)
Applied Spatial Analysis and Policy     Hybrid Journal   (Followers: 7, SJR: 0.542, CiteScore: 1)
Aquaculture Intl.     Hybrid Journal   (Followers: 26, SJR: 0.591, CiteScore: 2)
Aquarium Sciences and Conservation     Hybrid Journal   (Followers: 2)
Aquatic Ecology     Hybrid Journal   (Followers: 37, SJR: 0.656, CiteScore: 2)
Aquatic Geochemistry     Hybrid Journal   (Followers: 4, SJR: 0.591, CiteScore: 1)
Aquatic Sciences     Hybrid Journal   (Followers: 14, SJR: 1.109, CiteScore: 3)
Arabian J. for Science and Engineering     Hybrid Journal   (Followers: 5, SJR: 0.303, CiteScore: 1)
Arabian J. of Geosciences     Hybrid Journal   (Followers: 2, SJR: 0.319, CiteScore: 1)
Archaeological and Anthropological Sciences     Hybrid Journal   (Followers: 21, SJR: 1.052, CiteScore: 2)
Archaeologies     Hybrid Journal   (Followers: 12, SJR: 0.224, CiteScore: 0)
Archiv der Mathematik     Hybrid Journal   (Followers: 1, SJR: 0.725, CiteScore: 1)
Archival Science     Hybrid Journal   (Followers: 66, SJR: 0.745, CiteScore: 2)
Archive for History of Exact Sciences     Hybrid Journal   (Followers: 7, SJR: 0.186, CiteScore: 1)
Archive for Mathematical Logic     Hybrid Journal   (Followers: 3, SJR: 0.909, CiteScore: 1)
Archive for Rational Mechanics and Analysis     Hybrid Journal   (SJR: 3.93, CiteScore: 3)
Archive of Applied Mechanics     Hybrid Journal   (Followers: 6, SJR: 0.79, CiteScore: 2)
Archives and Museum Informatics     Hybrid Journal   (Followers: 157, SJR: 0.101, CiteScore: 0)
Archives of Computational Methods in Engineering     Hybrid Journal   (Followers: 6, SJR: 1.41, CiteScore: 5)
Archives of Dermatological Research     Hybrid Journal   (Followers: 7, SJR: 1.006, CiteScore: 2)
Archives of Environmental Contamination and Toxicology     Hybrid Journal   (Followers: 14, SJR: 0.773, CiteScore: 2)
Archives of Gynecology and Obstetrics     Hybrid Journal   (Followers: 17, SJR: 0.956, CiteScore: 2)
Archives of Microbiology     Hybrid Journal   (Followers: 9, SJR: 0.644, CiteScore: 2)
Archives of Orthopaedic and Trauma Surgery     Hybrid Journal   (Followers: 9, SJR: 1.146, CiteScore: 2)
Archives of Osteoporosis     Hybrid Journal   (Followers: 2, SJR: 0.71, CiteScore: 2)
Archives of Sexual Behavior     Hybrid Journal   (Followers: 10, SJR: 1.493, CiteScore: 3)
Archives of Toxicology     Hybrid Journal   (Followers: 17, SJR: 1.541, CiteScore: 5)
Archives of Virology     Hybrid Journal   (Followers: 5, SJR: 0.973, CiteScore: 2)
Archives of Women's Mental Health     Hybrid Journal   (Followers: 16, SJR: 1.274, CiteScore: 3)
Archivio di Ortopedia e Reumatologia     Hybrid Journal  
Archivum Immunologiae et Therapiae Experimentalis     Hybrid Journal   (Followers: 2, SJR: 0.946, CiteScore: 3)
ArgoSpine News & J.     Hybrid Journal  
Argumentation     Hybrid Journal   (Followers: 6, SJR: 0.349, CiteScore: 1)
Arid Ecosystems     Hybrid Journal   (Followers: 2, SJR: 0.2, CiteScore: 0)
Arkiv för Matematik     Hybrid Journal   (Followers: 2, SJR: 0.766, CiteScore: 1)
Arnold Mathematical J.     Hybrid Journal   (Followers: 1, SJR: 0.355, CiteScore: 0)
Arthropod-Plant Interactions     Hybrid Journal   (Followers: 2, SJR: 0.839, CiteScore: 2)
Arthroskopie     Hybrid Journal   (Followers: 1, SJR: 0.131, CiteScore: 0)
Artificial Intelligence and Law     Hybrid Journal   (Followers: 11, SJR: 0.937, CiteScore: 2)
Artificial Intelligence Review     Hybrid Journal   (Followers: 18, SJR: 0.833, CiteScore: 4)
Artificial Life and Robotics     Hybrid Journal   (Followers: 9, SJR: 0.226, CiteScore: 0)
Asia Europe J.     Hybrid Journal   (Followers: 5, SJR: 0.504, CiteScore: 1)
Asia Pacific Education Review     Hybrid Journal   (Followers: 13, SJR: 0.479, CiteScore: 1)
Asia Pacific J. of Management     Hybrid Journal   (Followers: 16, SJR: 1.185, CiteScore: 2)
Asia-Pacific Education Researcher     Hybrid Journal   (Followers: 13, SJR: 0.353, CiteScore: 1)
Asia-Pacific Financial Markets     Hybrid Journal   (Followers: 3, SJR: 0.187, CiteScore: 0)
Asia-Pacific J. of Atmospheric Sciences     Hybrid Journal   (Followers: 19, SJR: 0.855, CiteScore: 1)
Asian Business & Management     Hybrid Journal   (Followers: 9, SJR: 0.378, CiteScore: 1)
Asian J. of Business Ethics     Hybrid Journal   (Followers: 10)
Asian J. of Criminology     Hybrid Journal   (Followers: 6, SJR: 0.543, CiteScore: 1)
AStA Advances in Statistical Analysis     Hybrid Journal   (Followers: 4, SJR: 0.548, CiteScore: 1)
AStA Wirtschafts- und Sozialstatistisches Archiv     Hybrid Journal   (Followers: 5, SJR: 0.183, CiteScore: 0)
ästhetische dermatologie & kosmetologie     Full-text available via subscription  
Astronomy and Astrophysics Review     Hybrid Journal   (Followers: 22, SJR: 3.385, CiteScore: 5)

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Similar Journals
Journal Cover
International Journal of Fracture
Journal Prestige (SJR): 0.916
Citation Impact (citeScore): 2
Number of Followers: 14  
 
  Hybrid Journal Hybrid journal (It can contain Open Access articles)
ISSN (Print) 1573-2673 - ISSN (Online) 0376-9429
Published by Springer-Verlag Homepage  [2352 journals]
  • Fractional calculus derivation of a rate-dependent PPR-based cohesive
           fracture model: theory, implementation, and numerical results
    • Abstract: Rate-dependent fracture has been extensively studied using cohesive zone models (CZMs). Some of them use classical viscoelastic material models based on springs and dashpots. However, such viscoelastic models, characterized by relaxation functions with exponential decay, are inadequate to simulate fracture for a wide range of loading rates. To improve the accuracy of existing models, this work presents a mixed-mode rate-dependent CZM that combines the features of the Park–Paulino–Roesler (PPR) cohesive model and a fractional viscoelastic model. This type of viscoelastic model uses differential operators of non-integer order, leading to power-law-type relaxation functions with algebraic decay. We derive the model in the context of damage mechanics, such that undamaged viscoelastic tractions obtained from a fractional viscoelastic model are scaled using two damage parameters. We obtain these parameters from the PPR cohesive model and enforce them to increase monotonically during the entire loading history, which avoids artificial self-healing. We present three examples, two used for validation purposes and one to elucidate the physical meaning of the fractional differential operators. We show that the model is able to predict rate-dependent fracture process of rubber-like materials for a wide range of loading rates and that it can capture rate-dependent mixed-mode fracture processes accurately. Results from the last example indicate that the order of the fractional differential operators acts as a memory-like parameter that allows for the fracture modeling of long- and short-term memory processes. The ability of fractional viscoelastic models to model this type of process suggests that relaxation functions with algebraic decay lead to accurate fracture modeling of materials for a wide range of loading rates.
      PubDate: 2019-03-28
       
  • Gradient damage models applied to dynamic fragmentation of brittle
           materials
    • Abstract: This paper is devoted to the use of gradient damage models in a dynamical context. After the setting of the general dynamical problem using a variational approach, one focuses on its application to the fragmentation of a brittle ring under expansion. Although the 1D problem admits a solution where the damage field remains uniform in space, numerical simulations show that the damage field localizes in space at a certain time and then a fragmentation of the ring rapidly occurs. To understand this phenomenon from a theoretical point of view, one develops a stability analysis of the homogeneous response by studying the growth of small perturbations. A dimensional analysis shows that the problem essentially depends on two dimensionless parameters \(\tilde{\ell }\) and \(\tilde{\dot{\varepsilon }}_0\) , \(\tilde{\ell }\) being related to the characteristic length present in the damage model and \(\tilde{\dot{\varepsilon }}_0\) to the applied expansion rate. Then, since the product \(\tilde{\ell }\tilde{\dot{\varepsilon }}_0\) is small in practice, the problem of stability is solved in a closed form by using asymptotic expansions. The comparison with the numerical results allows us to conclude that the time at which the damage localizes and the number of fragments are really governed by the growth of the imperfections. To conclude, a numerical simulation of the fragmentation of a 2D ring is presented.
      PubDate: 2019-03-27
       
  • Evolution of damage and failure in an additively manufactured 316L SS
           structure: experimental reinvestigation of the third Sandia fracture
           challenge
    • Abstract: The third Sandia Fracture Challenge (SFC3) was a benchmark problem for comparing experimental and simulated ductile deformation and failure in an additively manufactured (AM) 316L stainless steel structure. One surprising observation from the SFC3 was the Challenge-geometry specimens had low variability in global load versus displacement behavior, attributed to the large stress-concentrating geometric features dominating the global behavior, rather than the AM voids that tend to significantly influence geometries with uniform cross-sections. This current study reinvestigates the damage and failure evolution of the Challenge-geometry specimens, utilizing interrupted tensile testing with micro-computed tomography (micro-CT) scans to monitor AM void and crack growth from a virgin state through complete failure. This study did not find a correlation between global load versus displacement behavior and AM void attributes, such as void volume, location, quantity, and relative size, which incidentally corroborates the observation from the SFC3. However, this study does show that the voids affect the local behavior of damage and failure. Surface defects (i.e. large voids located on the surface, far exceeding the nominal surface roughness) that were near the primary stress concentration affected the location of crack initiation in some cases, but they did not noticeably affect the global response. The fracture surfaces were a combination of classic ductile dimples and crack deviation from a more direct path favoring intersection with AM voids. Even though the AM voids promoted crack deviation, pre-test micro-CT scan statistics of the voids did not allow for conclusive predictions of preferred crack paths. This study is a first step towards investigating the importance of voids on the ductile failure of AM structures with stress concentrations.
      PubDate: 2019-03-27
       
  • The third Sandia Fracture Challenge: deterministic and probabilistic
           modeling of ductile fracture of additively-manufactured material
    • Abstract: Within the scope of the third Sandia Fracture Challenge the plasticity and ductile fracture behavior of an additively manufactured 316L stainless steel tensile specimen containing through holes and internal cavities is predicted in a blind round robin format. Only a limited number of experimental results, including flat dogbone-shaped and double-notch tension specimens, as well as EBSD maps of the Challenge geometry are provided by Sandia National Laboratory. A non-associated Hill’48 plasticity model with Swift-Voce strain hardening and Johnson–Cook strain rate hardening is used to accurately describe the large deformation response of the material. A special case of the recently developed Hosford–Coulomb model is used to predict fracture initiation and propagation by crack re-initiation. Very good qualitative and quantitative agreement of the blind prediction with the experimental results is obtained for both global force-displacement responses as well as the local surface strain evolution throughout the test. In a post challenge follow-up study, the role of the plasticity model is evaluated, focusing on the effect of the anisotropy and the strain-rate on the material response. Aside from considering the deterministic model, the statistical material properties of the additively manufactured structure are analyzed by defining a heterogeneous random media model. Probabilistic material properties for both plasticity and fracture are assigned to each element of the Challenge specimen. As an alternative, the role of intrinsic porosities is analyzed by randomly deleting 1% of the pristine geometry. The results of both approaches show that the presence of homogeneities follows a more realistic description of the material behavior, especially in the crack propagation regime post maximum force and when looking at local strains.
      PubDate: 2019-03-27
       
  • Towards objective simulation of quasi-static failure using a bounded rate
           local model with damage
    • Abstract: Today, the realistic simulation of complex industrial problems requires using industrial codes. For the simulation of failure in quasi-static, and if making use of implicit schemes, the convergence is often problematic. In order to ensure convergence and robustness, explicit algorithms are often used. In this case, mass and time scalings are used to allow for quasi-static simulations. How these techniques affect the failure prediction is nevertheless unclear. Moreover, in the case of damage model another difficulty arises, the one of spurious mesh dependency. In order to avoid this problem, the use of non-local models, as for example gradient ones, is the dominant approach. The implementation of such models in industrial software is cumbersome. A simpler possibility is to rely on bounded rate approaches (Allix in Int J Damage Mech 22:808–828, 2013). In fact, these approaches require only local modifications of the constitutive relation. Nevertheless, they have been much less studied and require dynamic analyses to ensure adapted regularisation effects. Considering these two issues, we study, in this paper, the possibility of combining explicit simulations with bounded rate models with damage. The aim is to enable relevant quasi-static damage simulations to perform up to failure. In this context, one main issue concerns the proposal of adapted scaling techniques. This problem is addressed through examples concerning the simulation of failure and the computation of the burst rotating speed of an axisymmetric disk.
      PubDate: 2019-03-14
       
  • Effect of high-rate dynamic comminution on penetration of projectiles of
           various velocities and impact angles into concrete
    • Abstract: The dynamic ‘overstress’, i.e., the apparent increase of strength of concrete at very high strain rates (10– \(10^{6}\) /s) experienced in projectile impact and penetration, has recently been explained by a new theory with partial analogy to turbulence. The increase is attributed to comminution of concrete driven by the release of local kinetic energy of the shear strain rate field of forming fragments which, in projectile impact problems, exceeds the strain energy of the fragments by orders of magnitude. This theory gives the particle size distribution and the additional local kinetic energy density, \(\Delta K\) , proportional to the deviatoric strain rate square. To match test results, \(\Delta K\) must be dissipated during finite element simulations of impact. In previous simulations, \(\Delta K\) was, at first, dissipated by an artificial equivalent viscosity (not empirical but predicted by the theory). Later it was found that dissipation by upscaling material tensile strength is equally effective. This theoretically justified upscaling is adopted here since it is more realistic when microplane constitutive model M7 for fracturing damage in concrete is used. All artificial damping is eliminated from the computer program. While previous simulations with the comminution theory were limited to orthogonal impacts, and only the cases of penetration of slabs of various thickness by projectile of one velocity and penetration depths for different velocities, the present study also analyzes further test data on oblique impacts at various impact angles up to \(35^\circ \) , and on the exit velocities and penetration depths of projectiles of different velocities. For each test series on one and the same concrete, the material parameters are calibrated on one test and then, using the same parameters, all the other tests are predicted. All the predictions of exit velocities and penetration depths of projectiles, as well as entry and exit craters, are quite accurate.
      PubDate: 2019-03-05
       
  • Intergranular fracture prediction and microstructure design
    • Abstract: A model based on discrete unit events coupled with a graph search algorithm is developed to predict intergranular fracture. The model is based on two hypotheses: (i) the key unit event associated with intergranular crack propagation is the interaction of a grain boundary crack with a grain boundary segment located at an angle with the initial crack plane; and (ii) for a given crack path, the overall crack growth resistance can be calculated using the crack growth resistance of a collection of unit events. Next, using a directed graph containing the connectivity of grain boundary junctions and the distances between them, and crack deflection versus crack growth resistance data, a directed graph in the J-resistance space is created. This graph contains information on the crack growth resistance for all possible crack paths in a given grain microstructure. Various crack growth resistance curves are then calculated including those corresponding to: (i) a local resistance minimum; (ii) a global minimum; and (iii) for verification, a path specified by microstructure-based finite element calculations. The results show that the proposed method based on discrete unit events and graph search can predict the crack path and the crack growth resistance for cracks that propagate from one grain boundary junction to another. The proposed computationally inexpensive model can be used to design material microstructures with improved intergranular fracture resistance, and/or to assess the overall crack growth resistance of materials with a known distribution of grain morphology.
      PubDate: 2019-03-04
       
  • Correction to: Multi-scale model of effects of roughness on the cohesive
           strength of self-assembled monolayers
    • Abstract: This article was published with an erroneous version of one of the author’s names. Please find on this page the correct version of the author’s names.
      PubDate: 2019-02-28
       
  • Smooth Generalized/eXtended FEM approximations in the computation of
           configurational forces in linear elastic fracture mechanics
    • Authors: Diego Amadeu F. Torres; Clovis S. de Barcellos; Paulo de Tarso R. Mendonça
      Abstract: The computation of crack severity parameters in the linear elastic fracture mechanics (LEFM) modeling is strongly dependent on the local quality of the approximated stress fields right at the crack tip vicinity. This work investigates the behavior of extrinsically enriched smooth mesh-based approximations, obtained via \(C^{k}\) -GFEM framework (Duarte et al. in Comput Methods Appl Mech Eng 196:33–56, 2006), in the computation of \(\mathcal {J}\) -integral in both pure mode I and mixed-mode loadings for two-dimensional problems of the LEFM. The method of configurational forces is used for this purpose as shown in Steinmann et al. (Int J Solids Struct 38:5509–5526, 2001), for instance, by performing some adaptations according to Häusler et al. (Int J Numer Methods Eng 85:1522–1542, 2011). As such method provides vector quantities, it is also possible to compute the angle \(\theta _{{\mathrm{ADV}}}\) of probable crack advance. The \(C^{k}\) -GFEM is quite versatile and shares similar features with the standard FEM regarding the domain partition and numerical integration (Mendonça et al. in Finite Elem Anal Des 47:698–717, 2011). The tests were conducted using three-noded triangular element meshes and numerical integrations were performed using only global coordinates. The evaluations combined different schemes of polynomial and discontinuous/singular (Moës et al. in Int J Numer Methods Eng 46:131–150, 1999) enrichments. The use of a smooth partition of unity (PoU) can influence the accuracy of computed crack severity parameters. The configurational forces computation is favored by the smoothness, reducing the dependence on the way the crack severity parameters are evaluated.
      PubDate: 2019-02-23
      DOI: 10.1007/s10704-019-00353-1
       
  • The J -contour integral in peridynamics via displacements
    • Authors: Christer Stenström; Kjell Eriksson
      Abstract: Peridynamics is a nonlocal formulation of solid mechanics capable of unguided modelling of crack initiation, propagation and fracture. Peridynamics is based upon integral equations, thereby avoiding spatial derivatives, which are not defined at discontinuities, such as crack surfaces. Rice’s J-contour integral is a firmly established expression in classic continuum solid mechanics, used as a fracture characterizing parameter for both linear and nonlinear elastic materials. A corresponding nonlocal J-integral has previously been derived for peridynamic modelling, which is based on the calculation of a set of displacement derivatives and force interactions associated with the contour of the integral. In this paper, we present an alternative calculation of the classical linear elastic J-integral for use in peridynamics, by writing Rice’s J-integral as a function entirely of displacement derivatives. The accuracy of the proposed J-integral on displacement formulation is investigated by applying it to the exact analytical displacement solution of an infinite specimen with a central crack and comparing the exact analytical expression of its J-integral \(K_I^2/E\) . Further comparison with a well-known peridynamic crack problem shows very good agreement. The suggested method is computationally efficient and further allows testing of the accuracy of a peridynamic model as such.
      PubDate: 2019-02-21
      DOI: 10.1007/s10704-019-00351-3
       
  • Circumventing mesh bias by r- and h-adaptive techniques for variational
           eigenfracture
    • Authors: Aurel Qinami; Eric Cushman Bryant; WaiChing Sun; Michael Kaliske
      Abstract: This article introduces and compares mesh r- and h-adaptivity for the eigenfracture model originally proposed in Schmidt et al. (Multiscale Model Simul 7:1237–1266, 2009), Pandolfi and Ortiz (J Numer Methods Eng 92:694–714, 2012), with the goal of suppressing potential mesh bias due to the element deletion. In the r-adaptive approach, we compute the configurational force at each incremental step and move nodes near the crack tip parallel to the normalized configurational forces field such that the crack propagation direction can be captured more accurately within each incremental step. In the h-adapative approach, we introduce mesh refinement via a quad-tree algorithm to introduce more degrees of freedom within the nonlocal \(\epsilon \) -neighborhood such that a more refined crack path can be reproduced with a higher mesh resolution. Our numerical examples indicate that the r-adaptive approach is able to replicate curved cracks and complex geometrical features, whereas the h-adaptive approach is advantageous in simulating sub-scale fracture when the nonlocal regions are smaller than the un-refined coarse mesh.
      PubDate: 2019-02-20
      DOI: 10.1007/s10704-019-00349-x
       
  • Embrittlement of an elasto-plastic medium by an inclusion
    • Authors: Luca Cimbaro; Adrian P. Sutton; Daniel S. Balint; Anthony T. Paxton; Mark C. Hardy
      Abstract: A mathematical model for the embrittlement of a long elastic-plastic crack by a relatively small, misfitting inclusion is presented. The model makes direct contact with the Dugdale–Bilby–Cottrell–Swinden model as a limiting case. The particular case of an oxide inclusion with a triangular cross-section at the tip of an intergranular crack in the Ni-based superalloy RR1000 at \(650\,^{\circ }\hbox {C}\) is considered. The positive misfit of the intrusion provides an additional tensile load on the crack tip and on the plastic zone, raising the local stress intensity factor \(k_I\) and the crack tip opening displacement \({\varDelta } u\) above those when the inclusion is replaced by a dislocation-free zone of the same length. It is shown that for a given misfit strain and inclusion shape, the enhancement of \(k_I\) and \({\varDelta } u\) is controlled by a dimensionless parameter \(\omega = (\sigma /\sigma _1)\sqrt{c/(2l)}\) where \(\sigma \) is the applied stress, \(\sigma _1\) is the yield stress, c is the crack length and l is the length of the inclusion. The anti-shielding effect of the intrusion is significant only when \(\omega \lesssim 6\) . As a result of the anti-shielding effect of the intrusion, the stress singularity at the crack tip always exceeds the compressive normal stress that exists within the thickest part of the intrusion when it is isolated. It is also shown that the gradient of the hydrostatic stress within the intrusion subjected to different applied stresses drives the oxygen diffusion and, hence, assists the oxidation at the grain boundary. The fracture toughness is considerably greater than that of a bulk sample of the oxide particle, which we attribute to the plastic zone.
      PubDate: 2019-02-19
      DOI: 10.1007/s10704-019-00344-2
       
  • A fracture model for exfoliation of thin silicon films
    • Authors: Martin Ward; Michael Cullinan
      Abstract: The direct exfoliation of thin films from silicon wafers has the potential to significantly lower the cost of flexible electronics while leveraging the performance benefits and established infrastructure of traditional wafer-based fabrication processes. However, controlling the thickness and uniformity of exfoliated silicon thin films has proven difficult due to a lack of understanding and control over the exfoliation process. This paper presents a new silicon exfoliation process and model which enables accurate prediction of the thickness and quality of the exfoliated thin-film based on the exfoliation process parameters. This model uses a parametric, finite element, linear elastic fracture mechanics study with nonlinear loading to determine how each process parameter affects the crack propagation depth. A metamodel is then constructed from the results of numerous simulations to inform the design and operation of a novel exfoliation tool and predict thickness of produced films. In order to manufacture uniform, high-quality films, the tool creates a controlled peeling load that is able to propagate a crack through the silicon in a controlled manner. Finally, exfoliated silicon samples produced with the prototype tool are evaluated and compared to metamodel projections, confirming the ability of the tool to steer crack trajectory within ± 3 microns of the crack depth predictions.
      PubDate: 2019-02-14
      DOI: 10.1007/s10704-019-00350-4
       
  • In-situ microscopy testing of plasticity variation ahead of fatigue crack
           tip in AL2024-T3
    • Authors: Wei Zhang; Liang Cai; Daoqing Zhou; Fuqiang Sun
      Abstract: In this paper, an in-situ experiment is performed under the optical microscopy to investigate the continuous variation of the strain field in the vicinity of the fatigue crack tip in Al2024-T3. The specimen is a small thin plate with an edge-crack, whose surface is speckled with tiny and dense spots with random-shapes. The applied load cycle is divided into a certain number of steps at which high resolution images around the crack tip are taken and recorded by a microscope camera. Once collecting the series of images, the strain distributions at each load step can be evaluated through the digital image correlation technique. Consequently, the continuous variation of the strain field ahead of the crack tip within the applied load cycle can be estimated approximately. Then combining the experimental measurements with the material constitutive relationship, the plastic zone sizes at different load levels can be calculated. In the current study, the continuous variations of the retensile and reversed plastic zones with different stress ratios are measured. Furthermore, the investigation is extended to the simple variable loading case, in which the plastic zone variation is traced before, during and after the single overload. It is shown that the plastic zone experiences an immediate reduction after the overload and gradually restores to the previous level. The crack closure effect and plasticity-induced loading interaction are analyzed. Finally, several conclusions are drawn based on the current investigation.
      PubDate: 2019-02-14
      DOI: 10.1007/s10704-018-00340-y
       
  • Plasticity and ductile fracture modeling of an Al–Si–Mg
           die-cast alloy
    • Authors: Madhav Baral; Jinjin Ha; Yannis P. Korkolis
      Abstract: The plastic anisotropy and ductile fracture behavior of an Al–Si–Mg die-cast alloy (AA365-T7, or Aural-2) is probed using a combination of experiments and analysis. The plastic anisotropy is assessed using uniaxial tension, plane-strain tension and disc compression experiments, which are then used to calibrate the Yld2004-3D anisotropic yield criterion. The fracture behavior is investigated using notched tension, central hole and shear specimens, with the latter employing a geometry that was custom-designed for this material. Digital image correlation is used to assess the full strain fields for these experiments. However, fracture is expected to initiate at the through-thickness mid-plane of the specimens and thus it cannot be measured directly from experiments. Instead, the stresses and strains at the onset of fracture are estimated using finite element modeling. The loading path and the resulting fracture locus were found to be sensitive to the yield criterion employed, which underscores the importance of an adequate modeling of plastic anisotropy in ductile fracture studies. Based on the finite element modeling, the fracture locus is represented with three common criteria (Oyane, Johnson–Cook and Hosford–Coulomb), as well as a newly proposed one as the linear combination of the first two. However, beyond that, it is still questionable if all of these experiments are probing the same fracture locus, since the predicted loading paths of notched tension specimens are highly evolving compared to those of central hole and shear ones.
      PubDate: 2019-02-06
      DOI: 10.1007/s10704-019-00345-1
       
  • Multiscale crystal-plasticity phase field and extended finite element
           methods for fatigue crack initiation and propagation modeling
    • Authors: Alireza Sadeghirad; Kasra Momeni; Yanzhou Ji; Xiang Ren; Long-Qing Chen; Jim Lua
      Abstract: This paper presents a physics-based prediction of crack initiation at the microstructure level using the phase field (PF) model without finite element discretization, coupled with an efficient and accurate modeling of crack propagation at macro-scale based on extended finite element method (XFEM). Although the macro-scale model assumes linear elastic material behavior, at micro-scale the behavior of plastically deforming heterogeneous polycrystals is taken into account by coupling the PF model and a crystal plasticity model in the fast Fourier transform computational framework. A sequential coupling has been established for the multiscale modeling where the macro-scale finite element (FE) model determines the hot spots at each cyclic loading increment and passes the associated stress/strain values to the unit-cell phase-field model for accurate physics-based microstructure characterization and prediction of plasticity induced crack initiation. The PF model predicts the number of cycles for the crack initiation and the phenomenological crack growth models are employed to propagate the initiated crack by the appropriate length to be inserted in the FE mesh. Finally, the XFEM solution module is activated to perform mesh independent crack propagation from its initial crack size to the final size for the total life prediction. The effectiveness of the proposed multiscale method is demonstrated through numerical examples.
      PubDate: 2019-02-04
      DOI: 10.1007/s10704-018-00339-5
       
  • Shear–compression loaded interface crack between a rigid substrate and
           an FGM layer—frictional crack closure effects
    • Authors: A. Dorogoy
      Abstract: A typical application of functionally graded materials (FGMs) is the coating of homogeneous or inhomogeneous substrates. This work investigates numerically an interfacial crack between an FGM coating and a rigid substrate that is subjected to shear–compression loading under the effect of friction. Two types of linearly graded coatings and one homogeneous coating exhibiting the same average Young’s modulus were examined. Two different numerical methods were applied for solving the singular receding contact problem: in-house finite difference software and a commercial finite element software. The effect of friction on the crack closure parameters such as tangential shifts and normal gaps of the crack face were studied with both methods and revealed an excellent agreement between the two. The effect of friction on the transition of the crack face from the slip to stick condition was studied as well. An extended J line formulation was used to extract the stress intensity factors (SIFs) for the crack tip for which the adjacent crack face experiences a large frictional contact. It was demonstrated that increasing the coefficient of friction causes a decrease in the tangential shifts and normal gaps until the whole crack face exhibits stick. The linear gradation in which the material is harder on the interface than on the top results in lower crack face displacement and SIFs.
      PubDate: 2019-02-02
      DOI: 10.1007/s10704-019-00348-y
       
  • Effect of thermomechanical coupling on stress-induced martensitic
           transformation around the crack tip of edge cracked shape memory alloy
    • Authors: Yajun You; Xiaojun Gu; Yahui Zhang; Ziad Moumni; Günay Anlaş; Weihong Zhang
      Abstract: In this work, the effect of thermomechanical coupling on stress-induced martensitic phase transformation around the crack tip of a pseudoelastic NiTi SMA compact tension (CT) specimen is studied experimentally and numerically. Six CT specimens are tested under different loading rates and the temperature distributions around the crack tip are measured using an infrared camera. For the numerical evaluation of temperature field and phase transformation, extended thermo-mechanically-coupled ZM (Zaki–Moumni) model is implemented into ABAQUS. The results of temperature field show that experimental and simulation results of temperature show a good agreement, and the maximum temperature at the crack tip increases with loading frequency. Furthermore, numerical results show that the phase transformation region size decreases when the loading frequency increases.
      PubDate: 2019-02-02
      DOI: 10.1007/s10704-019-00346-0
       
  • Characterization of hydraulic fracture geometry in shale rocks through
           physical modeling
    • Authors: Weihong Wang; Kan Wu; Jon Olson
      Abstract: In unconventional shale reservoirs, the presence of natural fractures is common and widespread. Multi-stage hydraulic fracture treatments are used to generate complex fracture geometry and stimulate reservoirs to unlock hydrocarbon in unconventional reservoirs. Natural fractures further complicate the complexity of the fracture geometry. In this paper, the objective is to investigate the impact of natural fractures on multiple hydraulic fracture propagation using an in-house fracture propagation model. We have developed a complex hydraulic fracture model to simulate multiple fracture propagation in naturally fractured reservoirs. The model can calculate partitioning of fluid between multiple fractures, fracture interaction within a stage and between stages, as well as the interaction between hydraulic and natural fractures. For simultaneous multiple fracture propagation, the exterior fractures generally take the majority of fluid and propagate at the expense of the interior fractures. Natural fractures can retard the growth of hydraulic fractures and change the fluid distribution between fractures and propagation trajectories of hydraulic fractures. In a reservoir with a heterogeneous natural fracture distribution, the interruption of natural fractures can complicate fracture geometry. The higher density or smaller spacing of natural fractures can mitigate stress shadow effects and favor more fluid flowing into the inner fractures. Due to the uncertainty of natural fracture distribution, the reliability of the complex fracture geometry provided by numerical models has been challenged. This paper demonstrates the potential of incorporating diagnostic results to constrain numerical results. Through combining diagnostic results with the fracture model, the accuracy of predicted fracture geometry can be improved.
      PubDate: 2019-01-16
      DOI: 10.1007/s10704-019-00343-3
       
  • Critical strain energy release rate for rubbers: single edge notch tension
           versus pure shear tests
    • Authors: David Roucou; Julie Diani; Mathias Brieu; Armel Mbiakop-Ngassa
      Abstract: In order to estimate mode I fracture strain energy release rate of a rubber upon monotonic loadings, the material is submitted to pure shear and single edge notch tension tests. Catastrophic failure happens suddenly for both tests, revealing mirror-like crack surfaces, assessing the fragile fracture. Nonetheless, Griffith failure analysis could be carried out on pure shear tests only. This analysis leads to an energy release rate value that allows challenging approximate expressions existing in the literature for pure shear and single edge notch tension tests. The pure shear approximate expression provides quantities that match the Griffith analysis. Meanwhile, the strain energy release rate values calculated directly from the single edge notch tension tests differ significantly from the values obtained in pure shear. This discrepancy is explored and possible explanations are discussed showing that pure shear tests should be favored.
      PubDate: 2019-01-12
      DOI: 10.1007/s10704-018-00336-8
       
 
 
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