BOUNDARY LAYER EFFECTS IN A FINITE LINEARLY ELASTIC PERIDYNAMIC BAR

  • ADAIR R AGUIAR Departamento de Engenharia Estrutural, Universidade de São Paulo, São Carlos, SP, Brasil http://orcid.org/0000-0001-9816-6129
  • TÚLIO V.B. PATRIOTA Dipartamento de Meccanica, Politecnico di Milano, Italia
  • GIANNI ROYER-CARFAGNI University of Parma, Parma, Italy
  • ALAN B SEITENFUSS Dipartamento de Ingegneria e Architettura, Università degli Studi di Parma, Parma, Italia

Abstract

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.

Author Biographies

ADAIR R AGUIAR, Departamento de Engenharia Estrutural, Universidade de São Paulo, São Carlos, SP, Brasil

Department of Structural Engineering

Associate Professor

TÚLIO V.B. PATRIOTA, Dipartamento de Meccanica, Politecnico di Milano, Italia

Student also at ...

Department of Mechanical Engineering

University of São Paulo

São Carlos School of Engineering 

São Carlos, SP, Brazil

GIANNI ROYER-CARFAGNI, University of Parma, Parma, Italy

Department of Engineering and Architecture 

Professor

ALAN B SEITENFUSS, Dipartamento de Ingegneria e Architettura, Università degli Studi di Parma, Parma, Italia

Department of Mechanical Engineering

Doctoral Student

Published
2018-03-19
Section
MecSol 2017 Joinville