(4) Stress – Stress Behavior (Stiffness) of Soils.

(4) Stress – Stress Behavior (Stiffness) of Soils.

Stress-Strain Behavior

→ depends on the composition, void ratio, stress history of the soil, manner in which the stress applied, and so on.

→ describes based on the theory of elasticity; Nonlinear stress-strain curves of a solid are linearized , i.e., replaced by straight lines.

→ Modulus and Poisson’s ratio are used for describing stress-strain behavior but they are not constant.

→ Concepts from the theory of elasticity.

ⅰ) For uniaxial case,

E

z z

ε

σ

z y

x

ε µε

ε

(If shear stresses τzx are applied, then shear distortion

G

zx zx

γ

τ

) 1 ( 2 +µ

= E

G )

ⅱ) For an elastic material with all stress components acting,

)]

( 1[

z y x

x E σ µ σ σ

ε = − + --- Eq. (1)

)]

( 1[

x z y

y E σ µ σ σ

ε = − + --- Eq. (2)

)]

( 1[

y x z

z E σ µ σ σ

ε = − + --- Eq. (3)

G

xy xy

γ =τ --- Eq. (4)

G

yz yz

γ =τ --- Eq. (5)

G

zx zx

γ =τ --- Eq. (6)

- For isotropic compression (σ_x =σ_y =σ_z =σ0), x y z ^{3 0 (1 2 )}

V

V E

ε ε ε σ µ

∆ = + + = −

⇒ The bulk modulus, ^B= _∆V^σ0_/V ⁼ ₃₍₁₋^E_2µ)

ⅲ) For confined compression (^{εx =εy =0}),

from Eqs.(1) ∼(6), ^{x y} σ^z µ σ µ

σ = = − 1 µ

µ µ

ε σ

−

−

= +

1

) 2 1 )(

1 ( E

z

z

and thus,

) 2 1 )(

1 (

) 1 (

µ µ

µ ε

σ

− +

= −

= E

D

z z

→ Alternate methods of portraying data for confined compression.

- the coefficient of compressibility, a : v

v

v d

a de

−

σ

=

- the compression index, Cc:

) (log _v

c d

C de

− σ

=

- the coefficient of volume change,

m

v v

v d

m d

σ

= ε

Table 3

Relations between various stress-strain parameters for confined compression

Constrained Modulus

Coefficient of volume change

Coefficient of compressibility

Compression index

Constrained

Modulus ^v

D v

ε σ

∆

= ∆

mv

D 1

=

av

D 1+e⁰

=

c va

C D e

435 . 0

) 1 ( + ₀ σ

=

Coefficient of volume

change ^{mv D}

= 1

v v

mv

σ ε

∆

= ∆

1 e0

m_v a^v

= +

va c

v e

m C

σ ) 1 (

435 . 0

+ 0

=

Coefficient of

compressibility ^D

a_v 1+e⁰

= a_v =(1+e₀)m_v

v v

a e

σ

∆

− ∆

=

va c v

a C

σ 435 .

=0

Compression index

D C_c e ^va

435 . 0

) 1 ( + ₀ σ

= 0.435

) 1

( _{0 va v}

c

m C +e σ

= 0.435

va v c

C aσ

=

v c

C e

σ

∆log

− ∆

=

Note. e₀ denotes the initial void ratio. σ_va denotes the average of the initial and final stresses.

Highly non-linear.

( E or G is not constant but depends on strain level.)

Large strain

Small strain Very small

strain

Nonlinear Linear

Shear strain, % (log scale.) G.

10^-3 0.1 1

Very small ( ε < 10^-3 %): dynamics.

Practices according Small (10^-3 %< ε < 0.1 – 1.0 %) : to strain level relevant to many designs under serviceable loads.

Large (>0.1 – 1.0%) : relevant to soil behavior near or correspondingly to failure.

In the very small strain region (<10^-3%), stiffness is approximately constant (Go, Gmax, Eo, Emax).

Influence factors on stress – strain behavior.

i) Soils

- Composition : Grading.

Mineralogy.

Particle shape.

Texture.

- Fabric Particle packing. (including density.) Layering.

Discontinuity (joints, fissures, open cracks).

- Chemical alteration.

ii) In-situ or testing stress or strain condition

- Current stress state.

Mean effective stress level, p’.

Stress difference, q.

Principal stress direction.

- Aging. (Time at current stress state.) - Stress history.

- Stress path imposed by sedimentation and subsequent loading.

- Rate of stress (or strain) change.

- Drainage condition.

G

G

ε log scale ε log scale

ε log scale

Drained cyclic loading

Undrained cyclic loading

ε₁ ε₂ ε₃ ε₁ ε₂ ε₃

ε ε ε

Ageing, cementing

G

Destructuring Faster loading rate Anisotropy

Fig. Effects of confining stress on the strain-dependent shear modulus (Kokusho, 1980)

Fig. Comparison of strain-dependent shear modulus of dense sand from undisturbed samples and from disturbed samples (Katayama ea at. 1986)

Fig. Effects of consolidation histories on strain-dependent modulus and damping ratio (Kokusho et al, 1982)

Test methods.

i) Dynamic methods ( γ < 10^-3%)

- Based on direct measurements of shear wave velocity (vs) G = ρvs2

① Bender Element Test in Triaxial apparatus

two thin piezoceramic plates which were bent by electric excitation, and of which bending changes the electric signal.

② Geophysical methods with elastic wave.

­ Downhole test

­ Crosshole test

­ SASW test

ii) Resonant column test (10^-5% < γ < 10^-1%)

- Measure the resonant frequency of soils with varying amplitude of strain.

⇒ Determine shear wave velocity.

⇒ Give G.

iii) Cyclic TX with internal deformation measurement (inner cell measurements).

10^-3% < ε < 10^-1%

iv) Static with external deformation measurement.

10^-1% < ε

Shear modulus (G^max) for small strain. ( ε < 0.001% ) For sands,

Gmax = AF(e)(σo’)ⁿ

where, A ≡ non-dimensional constant.

σσσσo ’ ≡ mean normal effective stress.

n = 0.5

For Toyoura sand,

5 . 0 2

max ( ')

) 1 (

) 17 . 2

8400( _o

e

G e σ

+

= −

where, Go and σo’ are in terms of kPa.

For Clays (Hardin and Black (1968)),

n o K_s

OCR e

AF

G

= ( )( ) ( σ ' )

where, Ks = 0 for PI < 40 1 for PI > 40.

For remolded kaolinate clay (PI = 21),

5 . 0

max ( ')

1

) 97 . 2

3300( _o

e

G e σ

+

= −

G - γ relationship

- Ramberg – Osgood Model.

R

C G

G ) ( ) (

max max

τ

γ = τ +

C, R ≡ non-dimensional constant.

Tangential Modulus, ¹

max max

1

τ −

⋅

⋅ + γ =

∂ τ

= ∂ _R

t

C G R G G