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6/4 Strength and failure of concrete under short-term, cyclic and sustained loading

(ε1, ε2, ε3) acting orthogonal to the principal planes on which the shear stresses (strains)
are zero.

6.1.3 Deformation and failure theories

Since the eighteenth century many theories and models have been proposed to explain or
predict the deformation, fracture and failure of composite systems. These are categorized
in Table 6.1.

Table 6.1 Categories of theories and models for the behaviour of composite materials

Category Theory/model Remarks
1
‘Classical’ theories Maximum principal stress or strain
2 Maximum shear stress
3 Mathematical models Maximum strain energy of distortion
4 Structural models Maximum octahedral shear stress
5 Rheological models Internal friction theory
6 Statistical models Mohr theory, etc.
Physical models
Fundamental theory

‘Mixture’ laws

Comprising elements for elasticity, plasticity and viscosity

Distributions of properties of elements

Simulations of real material (Griffith theory, finite element
models, etc.)

It is beyond the scope of this chapter to discuss all of these in detail but the following is
a summary of the advantages and disadvantages of the various approaches, paticularly
with regard to their use for concrete.

Category 1
These predict failure when a particular function of stress or strain reaches a critical value
and have limited application to concrete.

Category 2
Such models are based on fundamental theories of physics and mechanics and allow the
evaluation of stresses and strains within composite materials and for different geometrical
arrangements of homogeneous materials. Inglis in 1913 considered an elliptical crack in
an ideal elastic solid under uniform uniaxial tension applied at 90° to the major axis of the
crack. For a major axis of 2b and a minor axis of 2c the radius of the crack tip is b2/c and
the maximum stress at the crack tip is σ(1 + 2c/b) where σ is the stress applied to the
boundary of the solid. The relationship between the radius of the crack tip (non-
dimensionalized) and the intensification of stress at the crack tip (1 + 2c/b) is shown in
Figure 6.1 (Inglis, 1913).

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