Rigid line inclusion explained
A rigid line inclusion, also called stiffener, is a mathematical model used in solid mechanics to describe a narrow hard phase, dispersed within a matrix material. This inclusion is idealised as an infinitely rigid and thin reinforcement, so that it represents a sort of ‘inverse’ crack, from which the nomenclature ‘anticrack’ derives.
From the mechanical point of view, a stiffener introduces a kinematical constraint, imposing that it may only suffer a rigid body motion along its line.
Theoretical model
The stiffener model has been used to investigate different mechanical problems in classical elasticity (load diffusion,[1] inclusion at bi material interface [2]).
The main characteristics of the theoretical solutions are basically the following.
- Similarly to a fracture, a square-root singularity in the stress/strain fields is present at the tip of the inclusion.
- In a homogeneous matrix subject to uniform stress at infinity, such singularity only arises when a normal stress acts parallel or orthogonal to the inclusion line, while a stiffener parallel to a simple shear does not disturb the ambient field.
Experimental validation
The characteristics of the elastic solution have been experimentally confirmed through photoelastic transmission experiments.[3]
Interaction of rigid line inclusions
The interaction of rigid line inclusions in parallel, collinear and radial configurations have been studied using the boundary element method (BEM) and validated using photoelasticity.[4]
Shear bands emerging at the stiffener tip
Analytical solutions obtained in prestressed elasticity show the possibility of the emergence of shear bands at the tip of the stiffener.[5] [6] [7] [8]
References
- Koiter, W.T., On the diffusion of load from a stiffener into a sheet. Q. J. Mech. Appl. Math. 1955, VIII, 164–178.
- Ballarini, R., A rigid line inclusion at a bimaterial interface. Eng. Fract. Mech., 1990, 37, 1–5.
- Noselli . G. . Dal Corso . F. . Bigoni . D. . 2010 . The stress intensity near a stiffener disclosed by photoelasticity . International Journal of Fracture . 166 . 1–2 . 91–103 . 10.1007/s10704-010-9502-9 . 0376-9429.
- Numerical evaluation of the interaction of rigid line inclusions using strain intensity factors . 10.1016/j.ijmecsci.2019.01.017 . 2019 . Jobin . T.M. . Ramji . M. . Khaderi . S.N. . International Journal of Mechanical Sciences . 153-154 . 10–20 .
- Bigoni, D. Nonlinear Solid Mechanics: Bifurcation Theory and Material Instability. Cambridge University Press, 2012 . .
- Dal Corso . Francesco . Bigoni . Davide . Gei . Massimiliano . 2008 . The stress concentration near a rigid line inclusion in a prestressed, elastic material. Part I.Full-field solution and asymptotics . Journal of the Mechanics and Physics of Solids . 56 . 3 . 815–838 . 10.1016/j.jmps.2007.07.002 . 2008JMPSo..56..815D . 0022-5096.
- Bigoni . Davide . Dal Corso . Francesco . Gei . Massimiliano . 2008 . The stress concentration near a rigid line inclusion in a prestressed, elastic material. Part II.Implications on shear band nucleation, growth and energy release rate . Journal of the Mechanics and Physics of Solids . 56 . 3 . 839–857 . 10.1016/j.jmps.2007.07.003 . 2008JMPSo..56..839B . 0022-5096.
- Dal Corso . Francesco . Bigoni . Davide . 2008 . The interactions between shear bands and rigid lamellar inclusions in a ductile metal matrix . Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences . 465 . 2101 . 143–163 . 10.1098/rspa.2008.0242 . 1364-5021.
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