# Rectangular and rhombohedral inclusion reinforcements

Results show that the singular stress field predicted by the linear elastic solution for the rigid inclusion model can be generated in reality, with great accuracy, within a material. In particular, photoelastic experiments:

- agree with the fact that the singularity is lower for obtuse than for acute inclusion angles;
- show that the singularity is stronger in Mode II than in Mode I (differently from a notch);
- validate the model of rigid quadrilateral inclusion;
- for thin inclusions, show the presence of boundary layers deeply influencing the stress field, so that the limit case of rigid line inclusion is obtained in strong dependence on the inclusion’s shape.

# Thin inclusion reinforcement (stiffener)

The presence of a second phase in a matrix material leads to inhomogeneity in the mechanical fields.The theoretical solution of a zero-thickness, infinitely rigid line inclusion embedded in an elastic material has the following features:

- 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.

We have also shown the possibility of the emergence of shear bands at the tip of the stiffener.

**Related papers:**

- Misseroni, D., Dal Corso, F., Shahzad, S. and Bigoni, D. (2014).

Stress concentration near stiff inclusions: validation of rigid inclusion model and boundary layers by means of photoelasticity.*Engineering Fracture Mechanics*, 121-122, 87-97; - D. Bigoni, Nonlinear Solid Mechanics Bifurcation Theory and Material Instability. Cambridge University Press, 2012, ISBN:9781107025417.
- F. Dal Corso, D. Bigoni and M. Gei,

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*, 2008, 56, 815–838; - D. Bigoni, F. Dal Corso and M. Gei,

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*, 2008, 56, 839–857; - F. Dal Corso and D. Bigoni,

The interactions between shear bands and rigid lamellar inclusions in a ductile metal matrix.

*Proceedings of the Royal Society A*, 2009, 465, 143-163; - G. Noselli, F. Dal Corso and D. Bigoni,

The stress intensity near a stiffener disclosed by photoelasticity.

*International Journal of Fracture*, 2010, 166, 91–103.

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