Fractal Continuum Maxwell Creep Model
In this work, the fractal continuum Maxwell law for the creep phenomenon is introduced. By mapping standard integer space-time into fractal continuum space-time using the well-known Balankin’s approach to variable-order fractal calculus, the fractal version of Maxwell model is developed. This method...
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| Main Authors: | , , , , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-01-01
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| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/14/1/33 |
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| Summary: | In this work, the fractal continuum Maxwell law for the creep phenomenon is introduced. By mapping standard integer space-time into fractal continuum space-time using the well-known Balankin’s approach to variable-order fractal calculus, the fractal version of Maxwell model is developed. This methodology employs local fractional differential operators on discontinuous properties of fractal sets embedded in the integer space-time so that they behave as analytic envelopes of non-analytic functions in the fractal continuum space-time. Then, creep strain <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ε</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula>, creep modulus <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>J</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula>, and relaxation compliance <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> in materials with fractal linear viscoelasticity can be described by their generalized forms, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>ε</mi><mi>β</mi></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>,</mo><mspace width="0.166667em"></mspace><msup><mi>J</mi><mi>β</mi></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mrow><mi> </mi><mi>and</mi><mi> </mi></mrow><msup><mi>G</mi><mi>β</mi></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo>=</mo><mi>d</mi><mi>i</mi><mi>m</mi><mi>S</mi><mo>/</mo><mi>d</mi><mi>i</mi><mi>m</mi><mi>H</mi></mrow></semantics></math></inline-formula> represents the time fractal dimension, and it implies the variable-order of fractality of the self-similar domain under study, which are <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mi>i</mi><mi>m</mi><mi>S</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mi>i</mi><mi>m</mi><mi>H</mi></mrow></semantics></math></inline-formula> for their spectral and Hausdorff dimensions, respectively. The creep behavior depends on beta, which is characterized by its geometry and fractal topology: as beta approaches one, the fractal creep behavior approaches its standard behavior. To illustrate some physical implications of the suggested fractal Maxwell creep model, graphs that showcase the specific details and outcomes of our results are included in this study. |
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| ISSN: | 2075-1680 |