# Stress Concentration Layers in Finite Deformation of Fibre-Reinforced Elastic Materials

## Abstract

We consider finite plane strain of incompressible elastic materials reinforced by inextensible fibres. Deformations of inextensible materials often give rise to stress concentration layers, which are sheets of fibres which carry infinite direct stress but finite force, and across which the shear stress may be discontinuous. For linear elastic materials reinforced by straight parallel fibres these layers are well understood, and asymptotic methods of analysis have been developed for linear elastic materials with small but finite extensibility. When finite deformations occur, a qualitatively new feature arises because, in general, the fibres become curved and then the normal stress across them may also be discontinuous. We consider two examples, namely simple shear and shear bending of a rectangular block. In both of these examples the solution for inextensible material involves surface stress concentration layers. For ‘almost inextensible’ material we obtain approximate solutions for the displacement and stress in the neighbourhood of these surfaces, and show that the solution for an ideally inextensible material can be interpreted as the limit of the solution for material with small but finite extensibility as the ratio of shear modulus to fibre extension modulus tends to zero.

## Keywords

Simple Shear Fibre Direction Finite Deformation Rectangular Block Fibre Tension## Preview

Unable to display preview. Download preview PDF.

## References

- 1.J. E. Adkins and R. S. Rivlin,
*Large elastic deformations of isotropic materials X. Reinforcement by inextensible cords*, Phil. Trans. R. Soc.**A248**(1955), 201–223.MathSciNetADSGoogle Scholar - 2.A. C. Pipkin and T. G. Rogers,
*Plane deformations of incompressible fibre-reinforced materials*, J. Appl. Mech.**38**(1971), 634–640.ADSMATHCrossRefGoogle Scholar - 3.A. J. M. Spencer,
*Deformations of Fibre-Reinforced Materials*, Clarendon Press, Oxford, 1972.MATHGoogle Scholar - 4.G. C. Everstine and A. C. Pipkin,
*Stress channelling in transversely isotropic elastic composites*, Z. angew. Math. Phys.**22**(1971), 825–834.MATHCrossRefGoogle Scholar - 5.G. C. Everstine and A. C. Pipkin,
*Boundary layers in fiber-reinforced materials*, J. Appl. Mech.**40**(1973), 518–522.ADSMATHCrossRefGoogle Scholar - 6.A. J. M. Spencer,
*Boundary layers in highly anisotropic plane elasticity*, Int. J. Solid Structures**10**(1974), 1103–1123.MATHCrossRefGoogle Scholar - 7.A. J. M. Spencer,
*Theory of invariants*in*Continuum Physics*, Vol. 1 (A. C. Eringen, ed.), Academic Press, New York, 1971.Google Scholar - 8.J. E. Ericksen and R. S. Rivlin,
*Large elastic deformations of homogeneous anisotropic materials*, J. Rat. Mech. Anal.**3**(1954), 281–301.MathSciNetMATHGoogle Scholar - 9.A. E. Green, R. S. Rivlin and R. T. Shield,
*General theory of small elastic deformations superposed on finite elastic deformations*, Proc. R. Soc.**A211**(1951), 128–154.MathSciNetADSGoogle Scholar - 10.A. E. Green and W. Zerna,
*Theoretical Elasticity*, Clarendon Press, Oxford, 1954.MATHGoogle Scholar - 11.J. G. Oldroyd,
*Finite strains in an anisotropic elastic continuum*, Proc. R. Soc.**A202**(1950), 345–358.MathSciNetADSGoogle Scholar - 12.A. J. M. Spencer,
*Finite deformations of an almost incompressible elastic solid*, in*Proceedings of the International Symposium on Second-Order Effects in Elasticity, Plasticity and Fluid Mechanics*(M. Reiner and D. Abir, eds.), Pergamon Press, Oxford and Jerusalem Academic Press, Jerusalem, 19??.Google Scholar - 13.A. J. M. Spencer,
*The static theory of finite elasticity*, J. Inst. Maths. Applics.**6**(197?), 164–200.Google Scholar - 14.D. F. Parker, Discussion of
*Some research directions in finite elasticity theory*by R. S. Rivlin, Rheologica Acta**16**(1977), 112.CrossRefGoogle Scholar - 15.D. F. Parker,
*Finite-amplitude oscillations of a spherical cavity in a nearly-incompressible elastic material*, Archives of Mechanics**30**(1978), 777–790.MATHGoogle Scholar