4.17 Incompressible material

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The constitutive model in linear elasticity depends on two parameters, most

often the modulus of elasticity E and Poisson’s ratio ν. In soil mechanics the

bulk modulus and the shear modulus are preferred instead:

K = E

3(1 − 2 ν), G= E

2(1 + ν) . (4.123)

The stresses are split into deviatoric shear stresses and a uniform pressure

σ = (GEG + K EK) ε, or

⎡

⎢⎢⎢⎢⎢⎢⎣

σxx

σyy

σzz

σxy

σxz

σyz

⎤

⎥⎥⎥⎥⎥⎥⎦

=

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

G

⎡

⎢⎢⎢⎢⎢⎢⎣

4/3 −2/3 −2/3 0 0 0

−2/3 4/3 −2/3 0 0 0

−2/3 −2/3 4/3 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

⎤

⎥⎥⎥⎥⎥⎥⎦

+ K

⎡

⎢⎢⎢⎢⎢⎢⎣

1 1 1 0 0 0

1 1 1 0 0 0

1 1 1 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

⎤

⎥⎥⎥⎥⎥⎥⎦

⎫⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎭

⎡

⎢⎢⎢⎢⎢⎢⎣

εxx

εyy

εzz

εxy

εxz

εyz

⎤

⎥⎥⎥⎥⎥⎥⎦

.

(4.124)

Near ν = 0.5, the bulk modulus exceeds the modulus of elasticity and the

shear modulus by a large margin. Theoretically the bulk modulus K becomes

infinite (see Fig. 4.60). This is why the material is said to be incompressible

even though water, for example, retains a bulk modulus of about 2,000 MPa.

Close to ν = 0.5, special techniques are necessary to avoid locking, as for

example by using so-called enhanced strain elements, or by introducing threefield

mixed formulations [258].

In a Lagrangian approach of the dynamic analysis of nearly incompressible

fluids the rotational derivatives of the deformations have to be suppressed with

a penalty function to avoid spurious modes (see Fig. 4.59) because the mass

matrix has still modes which are suppressed in the stiffness matrix.

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