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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

Stress increment

Strain history

Stress history

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

σ′1

proportional to force”. For an element of

material, Hook’s law can be written as: Strain is

proportional to stress. Mathematically, it is,1

2

33D Cartesian

coordinatesσ′1′σ E=ε<br…

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