# Appendix 5: Constitutive modeling of discontinuities and interfaces

## A5.I Multi-response theory

Discontinuities and interfaces are encountered in many engineering problems such as excavations in rock mass, dam constructions, piling and bolting and so on. Relative sliding or separation movements in such localized zones present an extremely difficult problem in mechanical modeling. The discontinuities have been generally regarded as planar bodies that can only sustain and transfer the normal and tangential tractions between two adjacent bodies in geomechanics. The same type of approach is also followed in the mechanical engineering field in dealing with contact problems. Although this approach seems to be appropriate at first glance, it presents extremely difficult problems mathematically, experimentally and analytically. In this type of consideration, such planes will be internal boundaries within the body at which several conditions of constraints will be required to be satisfied. For example, if the condition of non-penetration is required, it will simply require that the so-called normal stiffness must have a value of infinity. The actual geometry of interfaces or discontinuities is never smooth and has asperities of varying amplitude and wavelength (Figure A5.1). Therefore, assigning a thickness to discontinuities related to the asperity height would

*Figure AS. I*** Surface configurations of discontinuities and interfaces.**

*Figure AS.2*** Mechanical model for discontinuities.**

simplify the problem. Aydan et al. (1990a, b, 1996) and Aydan and Kawamoto (1990) suggested that the thickness of the discontinuity should be twice the average height of asperities, which can be easily determined from the surface morphology measurements (e.g. Aydan and Shimizu 1995).

Let us consider a thin two-dimensional tabular body in a Cartesian coordinate system *(s,n)* as shown in Figure A5.2. Within the thin tabular body of discontinuity/ interface, stress components may be contemplated as *O _{mu}G_{ss}* and

*r*in two dimensions. If a discontinuity/interface is regarded as an internal thin body, the components of the stress tensor, which tend to remain on the discontinuity, may be

_{ns}*o,*and

_{m}*T*only. The relations among the strain and displacement components for discontinuities may be assumed to be of the following forms:

_{m }The behavior of discontinuities and interfaces is highly non-linear and they may show an elasto-plastic behavior from the very beginning of loading. Ichikawa (1985) proposed a multi-response theory for the behavior of materials, which separates the deviatoric and hydrostatic responses. This theory was adopted for interfaces and discontinuities by Aydan (1989, 2018) and Aydan et al. (1990a, b). The response functions of the discontinuities or interfaces for shear and normal stresses are given as

where у,,т,- are the shear strain and stress, and e,,cr, are the normal strain and stress. Superscripts *e* and *p* stand for elastic and plastic, respectively.

The incremental constitutive equations for the elasto-plastic behavior are given in the matrix form as:

Denoting

and taking the inverse of the above expressions and after some rearrangements, we have where

In the above expression, parameters *h _{s}* and

*h*are physically interpreted as the hardening moduli for the respective responses. The symbols

_{n}*p*and /3 denote the friction and the dilatancy factors, respectively. These terms are interrelated to each other in conventional plasticity. On the other hand, the present theory evaluates these terms independently from each other.

The elasto-plastic incremental constitutive law can be easily written by assuming that the total strains are a linear sum of elastic and plastic strains as

The yield function is vectorial and is written in the following form:

If the relation between stresses and strains are assumed to be linear elastic, then w'e have the following conventional form:

where £, and *G,* denote the Young’s modulus and shear modulus of the discontinuity/

interface material.

## References

Aydan, O. 1989. The stabilisation of rock engineering structures by rockbolts. Doctorate Thesis, Nagoya University, 204 p.

Aydan, O. 2018. Rock Reinforcement and Rock Support,

Aydan, O. and T. Kawamoto 1990. Discontinuities and their effect on rock masses. *Proceedings of International Conference on Rock Joints, ISRM,* Loen, 149-156.

Aydan, O. and Y. Shimizu 1995. Surface morphology characteristics of rock discontinuities with particular reference to their genesis. Fractography, Geological Society Special Publication No. 92, 11-26.

Aydan, O., Y. Ichikawa, S. Ebisu, S. Komura and A. Watanabe 1990b. Studies on interfaces and discontinuities and an incremental elasto-plastic constitutive law. *International Conference on Rock Joints,* ISRM, 595-602.

Aydan, O., Y. Ichikawa and T. Kawamoto 1990a. Numerical modelling of discontinuities and interfaces in rock mass. *The 4th Symposium on Computational Mechanics of Japan,* 254-261.

Aydan, О., I.H.P. Mamaghani and T. Kawamoto 1996. Application of discrete finite element method (DFEM) to rock engineering structures. *NARMS’96,* 2039-2046.

Ichikawa, Y. 1985. Fundamentals of the incremental elasto-plastic theory for rock-like materials (in Japanese), Dr. Thesis, Faculty of Engineering, Nagoya University, Nagoya, Japan, 132 p.