5. General issues and future developments
5.6. Constitutive relations
Continuum mechanics provides a good description of the behaviour of solids and fluids and yet it is only an approximation to the reality because it does not take into account the full details of the microscopic structure of the concrete material. The basic equations of continuum mechanics are an expression of the fundamental conservation laws of physics and are universally applicable irrespective of the specific properties of the material under study. This universality means, for instance, that the dynamical equations of motion are not sufficient for determining the evolution of the physical state of a deformable solid under prescribed loading and additional information has to be supplied about the specific way in which the material responds to changes in the state variables (such the strain tensor invariants, the temperature, and other relevant degrees of freedom).
The physical state of certain volume of rock at a given moment of time is specified by the values of all relevant state variables. One can say that the accessible states of a physical system correspond to points in an abstract space of dimension equal to the number of state variables (in the case of a mechanical system of material particles this is the phase space of coordinates and momenta).
One of the fundamental assumptions of physics is that the dynamics of a physical system is determined by some function of the state variables. In the case of a system of material points this function is called the action. For a deformable solid the elastic potential energy (or, more generally the internal energy) as well as the entropy are of importance and the role of the state function is given to the so-called Helmholtz free energy defined as the total internal energy of the system minus the product of the entropy and the temperature.
The way in which a physical body responds to small changes in the state variables is determined by the details of the interactions between its microscopic constituents and is manifested at the macroscopic level by a set of functional relationships between the state variables and some suitably defined macroscopic (averaged) quantities. The stress tensor is one example of such macroscopic quantity which is the averaged result of the local
intermolecular interactions for the material under study. A typical constitutive relation will have the form of an expression for some component of the stress tensor as a function of the components (one or more) of the strain tensor. The Hooke’s law is a classical example for a constitutive relation in the case of an elastic solid and within a limited interval of small deformations.
For complex and nonlinear materials the constitutive relations are also non-linear and can include a dependence on the derivatives of the state variables and on the loading history.
When the constitutive relations contain a dependence on the loading history it is said that they correspond to a material with memory.
5.6.1. Theoretical derivation of constitutive relations
Theoretically the constitutive relations are generated by the partial derivatives of the Helmholtz free energy with respect to the components of the strain tensor. Therefore a numerical model based on a postulated form of the Helmholtz free energy is already equipped with constitutive relations and any additional input of stress-strain relations could lead to internal inconsistencies. The generic link between the free energy and the way in which the material responds to variations of the deformation is rooted in the physical meaning of the Helmholtz free energy, namely: it is the amount of internal energy stored in the material which is available for conversion into mechanical work.
The damage-rheology model IDRM is an example of a numerical procedure in which the constitutive relations are derived from an explicit expression for the Helmholtz free energy.
Many numerical models of practical importance do not include a postulate about the free energy and are based on explicit constitutive relations which are usually derived from the analysis of experiments and observations.
5.6.2. Empirical constitutive relations
Empirical constitutive relations are in the form of stress-strain curves for the material under study and for different loading ranges and patterns. The data for the stress-strain curves is obtained either from laboratory experiments or from field observations. There is a
considerable contamination with both systematic and statistical errors in empirical
constitutive relations which may lead to significant distortion of the modelled data relative to the corresponding real data.
The constitutive description of failure in large scale volumes of rock surrounding a mining excavation can be approached in two fundamentally different ways. In the first case, it can be assumed that the medium is partitioned into representative volume elements (RVE’s) and that the overall deformation is controlled by the local deformation of each RVE and the inertial interaction between these elements. This class of model embodies the theories of plasticity or damage mechanics to provide descriptions of the element strain as a function of the imposed loading and loading history. An alternative approach is to assume that the medium deformation is controlled by failure between constituent particles or blocks. In this case, constitutive descriptions are expressed in terms of the interface forces such as bond strength and frictional resistance. These two classes can be considered to be dual and exclusive descriptions of material failure. It is important to observe that the surface separation philosophy depends also on the divisibility of the material that is controlled, ultimately, by molecular separation forces. In practical terms, this scale is many orders of magnitude smaller than the scale of engineering interest and the possibility exists that the defining fracture surfaces may possess a fractal character with dimension falling between 2 and 3.
In terms of the specific numerical models that are considered to be potential candidates for the integration of seismic activity with modelling, the boundary integral codes, MINSIM and DIGS assume that material failure occurs on representative surfaces, whereas the ISR code assumes that failure occurs in volumetric elements. The finite element code ELFEN,
boundary integral code MAP3Di and distinct element code UDEC allow failure on both separating surfaces and in volumetric elements. It is important to note that the different models demand different classes of constitutive relationships. The surface controlled models require a description of bond cohesion and friction resistance whereas the volumetric models require parameters to describe the effective “flow” of the material. The calibration of each model class is difficult. Detailed frictional behaviour may depend on the velocity and velocity
history of the sliding interfaces. In assigning point properties to the volumetric elements, it may be necessary to assume that the outcome of a laboratory experiment, in which detailed localisation mechanisms occur, is representative of the average behaviour of a different sized zone.