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# Configurational Forces in Plates and Shells

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,

d**p**/dt = ρ0**b** + **C**β|β

that is identically satisfied by a prescription for the assigned material force ρ0**b** and a jump condition. The jump condition has the compact form:

[[**c** + **p** vν ]]γ + **B**γ = **0**.

Here, the contact material force **c** = **Cυ**0, where **υ**0 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 **B**γ, linear momentum, director momentum, angular momentum, and energy can be established.

The implications of the jump condition for material momentum, [[**c** + **p** vν ]]γ + **B**γ = **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 + **B**γ = **0**.

reduces to a familiar adhesion moment boundary condition.

References

- Eremeyev,V.A.,Pietraszkiewicz,W.: The nonlinear theory of elastic shells with phase transitions. J.Elast.
**74**,67–86(2004). - 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). - Makowski,J.,Pietraszkiewicz,W.,Stumpf,H.:Jump conditions in the non linear theory of thin irregular shells. J.Elast.
**54**, 1–26 (1999). - 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).
- 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).
- O’Reilly, O.M.: Modeling nonlinear problems in the mechanics of strings and rods: the role of the balance laws. Interact. Mech. Math. (2017).
- 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). - Steigmann,D.J.: On the relationship between the Cosserat and Kirchhoff–Love theories of elastic shells. Math.Mech.Solids
**4**(3), 275–288 (1999). - Storåkers, B., Andersson, B.: Nonlinear plate theory applied to delamination in composites. J. Mech. Phys. Solids
**36**(6), 689–718 (1988). - 10. Williams,M.L.:The fracture threshold for an adhesive interlayer. J.Appl.Polym.Sci.
**14**(5),1121–1126(1970).

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