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4: General Hook’s law4.1: Constitutive models and their importanceModern soil mechanics was founded on the day when Terzaghi wrote (1936),σ′ = σ – u (30)The effective stress principle. With the principle of effective stress we can make a sensible
link between the deformation of soil and the stress acting on it. Hereafter, principles of
continuum mechanics are found to be applicable to soils.
By the statement that the stress carried by soil skeleton (the effective stress) determines the
deformation of soil, Terzaghi became the founder of modern soil mechanics. The work of
constitutive modelling is to find out mathematical relations between the deformation of soil
and the stress acting on it. Constitutive models of soils must be very important!Definition of a constitutive model:
A constitutive model describes the change in the strain state of an element
of material to the change in the stress state acting on the element. It
includes the strength and deformation of a material.For an element of soil (infinitesimal in size), we study stress σ and strain ε of the element.
Generally, constitutive equations can be expressed as the following form23( ) χξ=ε γ,,, σ′ dfd σ′ (31)Strain increment
Current stress state
Material propertiesThe deformation of soil resulting from a stress increment is dependent on factors such as the
current stress state, stress history, strain history, and material properties. Constitutive relations
are stress and strain relationship.Important rules of soil constitutive models:
(1) In all our engineering designs and safety check, we are only concerned about two facts:
(a) the strength of a structure, and (b) the deformation of a structure. Only through the
knowledge of the strength and deformation of an element of the material (a constitutive model
of the material) can we compute the strength and deformation of an engineering structure. Noother way!!!
The link between an element of material and a structure of the material is by numerical
analysis. FEM is one of the methods.
(2) Since we have good mathematical models and advanced numerical analysis packages, the
results of engineering computation are essentially controlled by the accuracy of our
constitutive models and reliable determination of model parameters.
Currently about 70% of the failures of civil engineering structures are attributed to problems
in the ground foundations. By far the dominating cause for the foundation problems is
because the mechanical properties of geomaterials are not well understood and not well
modlled.4.2: General Hook’s lawWhen we can make a sensible link between the deformation of soil and the stress acting on it,
the next thing we do is to calculate the deformation induced the stress. Naturally we start from
the simplest case: elasticity.
The law for elastic deformation was firstly
proposed by Hook (1678) as “deformation is
proportional to force”. For an element of
material, Hook’s law can be written as: Strain is
proportional to stress. Mathematically, it is,1
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