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Configurational Forces in Plates and Shells

oliver oreilly's picture

Dear Colleague, 


I'm pleased to announce our latest paper on configurational mechanics has just appeared in Acta Mechanica:


Nathaniel N. Goldberg and Oliver M. O’Reilly. A Material Momentum Balance Law for Shells and Plates with Application to Phase Transformations and Adhesion. Acta Mechanica, 2022.


Open Access is kindly provided by UC Berkeley Library.


In the paper, a balance law for material momentum (configurational forces) in shells and plates is proposed along with a concomitant energy momentum tensor C. The balance law reduces to a local form,


dp/dt = ρ0b + Cβ|β


that is identically satisfied by a prescription for the assigned material force ρ0b and a jump condition. The jump condition has the compact form:


[[c + p vν ]]γ + Bγ = 0.


Here, the contact material force c = 0, where υ0 is the unit surface normal vector to the interface. 


As with rods and 3-dimensional continuua, an identity for the power of sources of material momentum Bγ, linear momentum, director momentum, angular momentum, and energy can be established.


The implications of the jump condition for material momentum, [[c + p vν ]]γ + Bγ = 0, for the propagation of defects, phase transformations, and shocks are also explored. The developments are presented in the context of a purely mechanical theory of a Cosserat (or directed) shell [4,5] and specialized to the cases of a Kirchhoff–Love shell theory and a linearly elastic plate theory. 


Our development of Kirchhoff-Love shell theory from the more elaborate Cosserat theory is based on the work of Steigmann [8]. Our work on the balance law for material momentum is based on our earlier work with various rod theory [6] and is inspired by the work of Pietraskiewicz and co-workers on jump conditions and phase transformations in shells.



To explore connections to the literature [2,9,10] on delamination and adhesion, the balance law is applied to the problem of blistering. For these problems, the jump condition 

[[c + p vν ]]γ + Bγ = 0.


reduces to a familiar adhesion moment boundary condition. 




  1. Eremeyev,V.A.,Pietraszkiewicz,W.: The nonlinear theory of elastic shells with phase transitions. J.Elast. 74,67–86(2004).  
  2. Gioia, G., Ortiz, M.: Delamination of compressed thin films. In: Hutchinson, J. W., Wu, T.Y. (eds.) Advances in Applied Mechanics, 33, 119–192. Elsevier (1997). 
  3. Makowski,J.,Pietraszkiewicz,W.,Stumpf,H.:Jump conditions in the non linear theory of thin irregular shells. J.Elast. 54, 1–26 (1999). 
  4. Naghdi, P.M.: The theory of shells and plates. In: Truesdell, C. (ed.) Linear Theories of Elasticity and Thermoelasticity: Linear and Nonlinear Theories of Rods, Plates, and Shells, pp. 425–640. Springer-Verlag, Berlin, Heidelberg (1973).  
  5. Naghdi,P.M.: Finite deformation of elastic rods and shells. In:Carlson,D.E.,Shield,R.T.(eds.) Proceedings of the IUTAM Symposium on Finite Elasticity, Bethlehem PA 1980, pp. 47–104. Martinus Nijhoff, The Hague (1982). 
  6. O’Reilly, O.M.: Modeling nonlinear problems in the mechanics of strings and rods: the role of the balance laws. Interact. Mech. Math. (2017). 
  7. Pietraszkiewicz,W.,Eremeyev,V.,Konopin ́ska,V.: Extended non-linear relations of elastic shells undergoing phase transitions. Z. Angew. Math. Mech. 87(2), 150–159 (2007). 
  8. Steigmann,D.J.: On the relationship between the Cosserat and Kirchhoff–Love theories of elastic shells. Math.Mech.Solids 4(3), 275–288 (1999). 
  9. Storåkers, B., Andersson, B.: Nonlinear plate theory applied to delamination in composites. J. Mech. Phys. Solids 36(6), 689–718 (1988). 
  10. 10. Williams,M.L.:The fracture threshold for an adhesive interlayer. J.Appl.Polym.Sci.14(5),1121–1126(1970).
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