ABSTRACT
A constitutive equation is expected to represent stress due to a strain (or deformation) history starting from some specified reference state. If we represent stress as a function of strain, this automatically means that the stress does not depend on the deformation history. This special case is called (by definition) elastic behaviour. Soil is not elastic, so we have to find another type of relation. How can we represent strain history? Some researchers introduced integral transformations using appropriate kernels. This approach is not useful for soils. A general way to introduce history (or path) dependence in physics is to use non-integrable differential forms (or PFAFFean forms), i.e. to represent y by the differential equation
dy = aidxi a2dx2 + • • • andxn•
This equation connects increments dxi , dx2, . with dy (or dyi, dye, ., if y is a vector) in such a way that there is no closed-form representation of y(x). I.e. the relation (which is called incremental, as it relates increments) dy = f (dx,) is not integrable. This is the way we proceed in soil mechanics when we represent the stress increment as a non-integrable function of the strain increment:
do-= f (dE) .
