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 stress-strain 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 stress-strain 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

E'=

 ds'a
 de'a

Poisson's ratio   n' = - der / dea

Note:
If the material is incompressible, ev = 0 and Poissonís 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 Poissonís 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 stress-strain curve has features which are characteristic of different aspects of material behaviour and which are represented by different theories.

OA: linear and recoverable
ABC: non-linear and irrecoverable
BCD: recoverable with hysteresis
DE: continuous shearing
The three basic theories which are relevant to soil behaviour are: elasticity, plasticity and viscus flow (often refered to as creep). In addition, the theories of elasticity and plasticity are combined into elasto-plasticity. If strains are zero the behaviour is rigid.


Elasticity Back to Material behaviour
Forward to
Perfect plasticity

In linear-elastic behaviour the stress-strain is a straight line and strains are fully recovered on unloading, i.e. there is no hysteresis. The elastic parameters are the gradients of the appropriate stress-strain curves and are constant.
Shear modulus G = dt / dg

Bulk modulus K' = ds'mean / dev

or

Youngís modulus E' = ds'a / dea (where ds'r = 0)

Poissonís ratio n' = - der / dea   (where ds'r = 0)

 


Perfect plasticity Back to Material behaviour
Forward to
Elasto-plasticity

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 2-dimensional 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.

 

 

 

 


Elasto-plasticity 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 dep causes an increase in the yield stress.
The hardening law is ds'yield / dep

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 dep 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.

 

 


Elastic-perfectly 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).

 

Produced by Dr. Leslie Davison, University of the West of England, Bristol, May 2000
in association with Prof. Sarah Springman, Swiss Federal Technical Institute, Zurich