BOUNDARY LAYER EFFECTS IN A FINITE LINEARLY ELASTIC PERIDYNAMIC BAR
THE PERIDYNAMIC THEORY IS AN EXTENSION OF THE CLASSICAL CONTINUUM MECHANICS THEORY. THE PERIDYNAMIC GOVERNING EQUATIONS INVOLVE INTEGRALS OF INTERACTION FORCES BETWEEN NEAR PARTICLES SEPARATED BY FINITE DISTANCES. THESE FORCES DEPEND UPON THE RELATIVE DISPLACE-MENTS BETWEEN MATERIAL POINTS WITHIN A BODY. ON THE OTHER HAND, THE CLASSICAL GOVERNING EQUATIONS INVOLVE THE DIVERGENCE OF A TENSOR FIELD, WHICH DEPENDS UPON THE SPATIAL DERIVATIVES OF DISPLACEMENTS. THUS, THE PERIDYNAMIC GOVERNING EQUATIONS ARE VALID NOT ONLY IN THE INTERIOR OF A BODY, BUT ALSO ON ITS BOUNDARY, WHICH MAY INCLUDE A GRIFFITH CRACK, AND ON INTERFACES BETWEEN TWO BODIES WITH DIFFERENT MECHANICAL PROPERTIES. NEAR THE BOUNDARY, THE SOLUTION OF A PERI-DYNAMIC PROBLEM MAY BE VERY DIFFERENT FROM THE CLASSICAL SOLUTION. IN THIS WORK, WE INVESTIGATE THE BEHAVIOR OF THE DISPLACEMENT FIELD OF A UNIDIMENSIONAL LINEARLY ELASTIC BAR OF LENGTH L NEAR ITS ENDS IN THE CONTEXT OF THE PERIDYNAMIC THEORY. THE BAR IS IN EQUILIBRIUM WITHOUT BODY FORCE, IS FIXED AT ONE END, AND IS SUBJECTED TO AN IM-POSED DISPLACEMENT AT THE OTHER END. THE BAR HAS MICROMODULUS C, WHICH IS RELATED TO THE YOUNG'S MODULUS E IN THE CLASSICAL THEORY AND IS GIVEN BY DIFFERENT EXPRESSIONS FOUND IN THE LITERATURE. WE FIND THAT, DEPENDING ON THE EXPRESSION OF C, THE DISPLACEMENT FIELD MAY BE SINGULAR NEAR THE ENDS, WHICH IS IN CONTRAST TO THE LINEAR BEHAVIOR OF THE DISPLACEMENT FIELD OBSERVED IN THE CLASSICAL LINEAR ELASTICITY. IN SPITE OF THE ABOVE, WE SHOW THAT THE PERIDYNAMIC DIS-PLACEMENT FIELD CONVERGES TO ITS CLASSICAL COUNTERPART AS A LENGTH SCALE, CALLED PERIDYNAMIC HORIZON, TENDS TO ZERO.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).