Back to Compression and shear  Based on part of the GeotechniCAL
reference package by Prof. John Atkinson, City University, London 
Stiffness  Back to Basic mechanics of soils 
As stresses are increased or decreased a material body will tend to change size and shape as strains occur: stiffness is the relationship between changes of stress and changes of strain.
The stiffness E' is the gradient of the stressstrain curve. Stiffnesses may be described by
a tangent modulus E'_{tan} = ds' / de
or by a secant modulus E'_{sec} = Ds' / De
Note: If the material is linearly elastic the stressstrain curve is a straight line and E'_{tan} = E'_{sec}.
Change of size: bulk modulus  Back to Stiffness 
As the mean stress increases materials compress (reduce in volume). The bulk modulus K' relates the change in stress to the volumetric strain.
K'= ds'_{mean} de'_{v}
 where
 s'_{mean} = (s'_{x} + s'_{y} + s'_{z}) / 3
 Note:
 In soils volumetric strains are due to changes of effective stress.
Change of shape: shear modulus  Back to Stiffness 
As the shear stress increases materials distort (change shape). The shear modulus G' relates the change in shear stress to the shear strain.
G =  dt 
dg 
Sinse water has no shear strength, the value of the shar modulus, G, remains the same, independant of whether the loading process is drained or undrained.
Uniaxial loading: Young's modulus and Poisson's ratio 
Back to Stiffness 
Young's modulus and
Poisson's ratio are measured directly in uniaxial compression or extension
tests, i.e. tests with constant (or zero) stress on the vertical surfaces.
i.e. ds'_{r} = 0
Young's modulus
 ds'_{a} 
de'_{a} 
Poisson's ratio n' =  de_{r} / de_{a}
Note:
If the material is incompressible, e_{v} = 0 and Poisson’s ratio, n = 0.5.
Uniaxial compression is the only test in
which it is possible to measure Poisson's ratio with any degree of
simplicity.
Typical values of E  Back to Uniaxial loading 
These are a function of the stress level, and the loading history, however a range is given below.
Typical E  
Unweathered overconsolidated clays  20 ~ 50 MPa 
Boulder clay  10 ~ 20 MPa 
Keuper Marl (unweathered)  >150 MPa 
Keuper Marl (moderately weathered)  30 ~ 150 MPa 
Weathered overconsolidated clays  3 ~ 10 MPa 
Organic alluvial clays and peats  0.1 ~ 0.6 MPa 
Normally consolidated clays  0.2 ~ 4 MPa 
Steel  205 MPa 
Concrete  30 MPa 
Relationships between stiffness moduli  Back to Stiffness 
In bodies of elastic material the three stiffness moduli (E', K' and G') are related to each other and to Poisson’s ratio (n'). It assumed that the material is elastic and isotropic (i.e. linear stiffness is equal in all directions). The following relationships can be demonstrated (for proofs refer to a text on the strength of materials).
G' = E' / 2(1 + n')
K' = E' / 3(1  2n')
Material behaviour  Back to Basic mechanics of soils 
The stressstrain curve has features which are characteristic of different aspects of material behaviour and which are represented by different theories.
Elasticity  Back to Material behaviour
Forward to Perfect plasticity 
Shear modulus G = dt / dg
Bulk modulus K' = ds'_{mean} / de_{v } or Young’s modulus E' = ds'_{a} / de_{a}
(where ds'_{r} = 0) 
Perfect plasticity  Back to Material behaviour
Forward to Elastoplasticity 
OA  rigid
AB  plastic
During perfectly plastic straining, plastic strains continue indefinitely at constant stress. The ratio of plastic strains is related to the yield stress, which also represents the failure stress.
Yield  Back to Perfect plasticity 
Yield stress is the stress at the end of elastic behaviour
(in a perfectly plastic material this is the same as the failure stress).
In a 2dimensional stress system the combination of yield stresses forms a yield curve, inside which failure can not occur.

Normality  Back to Perfect plasticity 
The relationship between the ratio of plastic strains is such that the plastic strain vector is normal to the yield curve.
Normality is also known as associated flow.
Elastoplasticity  Back to Material behaviour 
There are simultaneous elastic and plastic strains and the plastic strains cause the yield stress to change. There are two cases which are typically found:
Strain hardening
The plastic strain de_{p} causes an increase in the yield stress.
Soils with loosely packed grains are strain hardening because the disturbance during sharing causes the grains to move closer together.  
Strain softening The plastic strain de_{p} causes a decrease in the yield stress. Soils with densely packed grains are strain softening because disturbance during sharing causes the grains to move apart causing dilation. 
Elasticperfectly plasticity  Back to >Material behaviour 
For relatively low ~ medium risk projects, an assumption may be made that the stress  strain response can be represented by two straight lines, to describe an initial linear elastic stiffness (OA) and the yield stress or strength at failure during plastic straining (AB).