Hyperplasticity Summary 1

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Finally we provide a brief summary of what has been covered in this course.

Hyperplasticity Summary 2

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First of all, a summary of the hyperplasticity formulation.

A material is defined in terms of strains and internal variables (by using Legendre Transforms itcould be defined instead in terms of stresses and internal variables).

We then define just two scalar functions – a free energy function which is a function of the state,i.e. of strains and internal variables

... and a dissipation which is a function of state and of rate of change of internal variables(homogeneous first order in the latter).

From thermodynamics (augmented by Ziegler’s orthogonality condition) we derive the stressesand an additional equation which effectively allows elimination of the internal variables. Forconvenience we introduce the additional variable chi which is conjugate to the internal variable.

That is all that is needed – the entire constitutive behaviour follows.

In the formulation we can make much use of legendge Transforms to move between alternativeforms.

Hyperplasticity Summary 3

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So what are the main advantages?

First of all the simplicity of specifying an entire model through just two scalar functions – onefor storage of energy, the other for dissipation. Compare this with many models which arespecified through a plethora of equations.

This approach allows development of simple hierarchies of models based on step-by-stepchanges to the defining functions.

We emphasise the importance of the choice of the defining variables – pairs of conjugatevariables such as stress and strain, generalised stress and internal variable. The approach issolidly rooted in the method of thermodynamics with internal variables (TIV)

... hence the models automatically obey thermodynamic principles.

The models are very simple in their basic form (e.g. 1-D elasticity and plasticity), but in theirmore advanced forms become quite sophisticated – see for instance the use of functionals inthe definition of continuous hyperplastic models.

Towards the end of the course we have introduced convex analysis, which we regard as thenatural language for the description of plasticity models.

Hyperplasticity Summary 4

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What are some future directions? The following are in no particular order.

It would be good to expand the work on damage mechanics in a more comprehensive way. Thiswould require more work on regularisation of softening behaviour.

In geotechnics there is a real challenge of unifying small strain theory and critical state – noteasy because there are perhaps too many options as to how the resulting theory could beexpressed.

There is fundamental work to be done on bound theorems – if at all possible bringing them alittle closer together.

Transport phenomena (water and heat flow) can be incorporated more comprehensively

There is much to be done on porous media theory, and specifically on unsaturated soils(including hydraulic hysteresis etc).

Electro-chemical effects can in principle be brought within this framework, but this is an areawhere the author has no particular expertise

There are possible applications in the modelling of composites – especially introducinganisotropy, and possibly adopting homogenisation techniques.

In geotechnics (and other areas) the whole topic of building up macroscopic behaviour frommicromechanical models is important, and hyperplasticity can contribute to this.

.... these are just a taster of some possible future directions: there are many other possibilities.