Chapter 3. Mechanism and Mathematical Representation of Energy Dissipation

3.1 Introduction

In the previous chapter, a brief review of existing energy-based liquefaction evaluation

procedures was presented. Expressions were given quantifying the energy Demand

imparted to the soil by the earthquake, but physical interpretations of these expressions

were limited. Also, various terms were used without full description (e.g., material

damping and dissipated energy). Although the diverse formulations of the liquefaction

evaluation procedures necessitated the presentation style of Chapter 2, the purpose of this

chapter is to provide better physical insight into the energy imparted to the soil and the

dissipation mechanisms. A brief treatise is also given to the equivalent linear technique

for mathematical modeling of energy dissipation. Finally, a consistent set of expressions

is derived for computing the energy dissipated in cyclic triaxial, simple shear, and hollow

cylinder triaxial-torsional shear tests.

3.2 Mechanisms of Energy Dissipation in Sands

As seismic waves propagate through soil, a portion of their energy dissipates, resulting in

a reduction in the amplitude of the waves. With the exceptions of the Arias intensity

approaches, all of the energy-based Demand expressions presented in the previous

chapter attempt to quantify the portion of energy that dissipates in the soil as some

fraction of the seismic wave energy arriving at the site. For cohesionless soils, the

dominant mechanism of energy dissipation is the frictional sliding at grain-to-grain

contact surfaces (Whitman and Dobry 1993). Additionally, if the soil is saturated, energy

also dissipates from the viscous drag of the pore fluid moving relative to the soil skeleton.

The contributions from other mechanisms, such as particle breakage, are relatively

insignificant for most soils and are not discussed further. The conceptual relationship

between the energy imparted to the soil and that dissipated by friction and viscous drag is

shown in Figure 3-1, where the fraction attributed to viscous drag is exaggerated for

illustrative purposes. The stored energy represents that portion of the input energy that

continues to propagate through the soil column, and accordingly, at the end of the

78

shaking, is equal to zero. Each of the energy components shown in Figure 3-1 is

examined in more detail.

12

Energy Dissipatedby Viscous Drag

Stored Energy

Energy Dissipatedby Friction

Total Dissipated Energy

504030 600 10 20

10

8

Ener

gy (

×10)

6

4

2

0

Time (sec)

Figure 3-1. Conceptualization of the cumulative energy imparted to the soil by an earthquake and the portions dissipated by frictional and viscous mechanisms. (Loosely adapted from Hall and McCabe 1989).

3.2.1 Frictional Dissipation Mechanism

In relation to liquefaction, the portion of the energy dissipated by friction is of

considerable importance. This is because liquefaction requires the complete breakdown

of the soil structure, which inherently involves slippage of contact surfaces as the

particles rearrange. The physics of energy dissipation by friction can be understood by

the interaction of two elastic spheres under the action of normal and shear forces, which

has been study at depth by several investigators: Mindlin (1949); Mindlin et al. (1951);

Mindlin (1954); Duffy and Mindlin (1956); Johnson (1961); Goodman and Brown

(1962); Deresiewicz (1974); Dobry et al. (1982). The following discussion on the

interaction of two elastic spheres is based largely on this work, unless otherwise noted.

79

The contact forces and corresponding stresses between two spheres are shown in Figure

3-2. The radius of the contact area (a) is a function of both the applied normal force and

the elastic properties of the spheres. This is shown in Figure 3-3, along with the variation

of the normal stress (σc) across the contact area. As the tangential force T increases, there

is a proportional increase in the lateral displacement (δ) between the centers of the

spheres. However, gross (or complete) sliding across the entire contact area does not

occur until T = f⋅N, where N is the normal force and f is the coefficient of friction of the

contact surfaces. The progression of slippage across the contact area as T increases from

0 to f⋅N is shown in Figure 3-4. As illustrated in this figure, sliding starts at the outer

radius of the contact area and progresses inward, thus forming an annulus of slippage

surrounding a zone of no slippage for 0 < T < f⋅N. A series of laboratory experiments

verified this theoretical behavior (e.g., Deresiewicz 1974). A photograph from one of the

experiments is shown in Figure 3-5 in which two spheres were in contact and subjected to

an oscillating tangential force 0 < T < f⋅N. In this figure, the wear marks formed by the

annulus of slippage can be clearly identified.

From the above, it can be seen that energy is dissipated through friction, even before

gross sliding across the entire contact surface occurs. Accordingly, scenarios can be

hypothesized where the applied tangential force is 0 < T < f⋅N, and an infinite amount of

energy could be dissipated without the occurrence of liquefaction. The strain-based

liquefaction evaluation procedure presented in Chapter 2 may be used to screen for these

scenarios, which was the purpose for including it in the parameter study. This scenario

can only exist if the induced strain (γ) is less than the threshold strain (i.e., γ < γth), where

the threshold strain is that corresponding to T = f⋅N. It can be further hypothesized that,

for a transient earthquake type loading, energy is dissipated prior to the arrival of a pulse

of sufficient amplitude to induce a shear strain that exceeds the threshold strain (i.e., γ >

γth). From the examination of typical earthquake acceleration time histories, such as

shown in Figure 3-6, it can be seen that the large amplitude shear waves arrive early in

the record. From this, it is assumed that if the threshold strain is exceeded, it will occur

early in the shaking, and little energy will be dissipated prior to its exceedance.

80

Figures 3-3 and 3-4 provide a further breakdown of the normal and shear stresses across the contact surfaces.

τc (no slip)

τc (slip)f⋅σc

σc

N

N

T T

c a

δ/2

δ/2

R

R

N

N

T

T

a

δ/2

δ/2

R = radius of spheres σc = normal stress between the spheres N = normal force τc = shear stress between the spheres T = tangential force a = radius of the contact area δ = lateral displacement of spheres c = radius of the non-slip contact area f = coefficient of friction between the spheres

Figure 3-2. Contact forces and stresses between two equal sized spheres of radius R.

81

( ) 5.0232

3 ρπ

σ −= aaN

c

31

2

4)1(3

−=

ENRa ν

contact surfaces

ρ

a

σc

N

N

R = radius of spheres σc = normal stress between the spheres N = normal force ρ = distance from the center of the contact area a = radius of the contact area E = Young’s modulus ν = Poisson’s ratio

Figure 3-3. Radius of the contact area (a) and the normal stress (σc) across the contact area of the two spheres.

82

NfT

ac

⋅−=

1

3

−−⋅

−−=

32

114

)1()2(3Nf

TaE

Nfννδ

a

Gross slippage

T ≥ f⋅N

τc (slip) f⋅σc N

N

T T

a c=0

δ/2

δ/2

Zone of no slippage

Zone of slippage 0 < T < f⋅N

ac

τc (slip) f⋅σc N T

Tc

N a

δ/2

δ/2

Plan view of the contact T = 0

a, c

No slippage

τc (no slip)

N T

T

aN

, c

δ/2

δ/2

R = radius of spheres σc = normal stress between the spheres N = normal force τc = shear stress between the spheres T = tangential force a = radius of the contact area δ = lateral displacement of spheres c = radius of the non-slip contact area f = coefficient of friction of the spheres ν = Poisson’s ratio E = Young’s modulus

Figure 3-4. Relative slippage of the spheres: no slippage at T = 0; gross slippage at T = f⋅N. At intermediate values 0 < T < f⋅N, there is an annulus of slippage surrounding a zone of no slippage.

83

Outer boundary of contact

area Zone of slippage

ac

Zone of no slippage

Figure 3-5. Photograph of an actual sphere, which in contact with another sphere was subjected to an oscillating tangential force 0 < T < f⋅N. Wear marks formed by the annulus of slippage between the spheres can be clearly identified. (Adapted from Deresiewicz 1974).

Approximate arrival time of shear waves

Time (Seconds)

-0.1

Acc

eler

atio

n (g

) 0.1

30 2515 20105 0

Figure 3-6. Typical acceleration time history with the arrival of the large amplitude shear waves occurring early in the record.

84

3.2.2 Viscous Dissipation Mechanism

Viscosity is the measure of a fluid’s resistance to flow. Viscous drag is the force

resisting the relative movement of a fluid and a solid and is analogous to the frictional

force between two solids. The theory by Biot (1956) may be used for the theoretical

evaluation of energy dissipation by viscous mechanisms in soils. Hall (1962) and Hall

and Richart (1963) outline the results of a laboratory study examining the influence of

various parameters on the total energy dissipated in granular materials, including the

viscosity of the pore fluid. By comparing the energy dissipated in dry samples to similar

saturated samples, the relative contributions from friction and viscous drag can be

discerned.

In their study, Hall (1962) and Hall and Richart (1963) performed a series of resonant

column tests where the specimens were excited at their first mode of vibration and then

set in free vibration, while the amplitude of the rotations being recorded. The decay in

the rotational amplitudes results from energy dissipation. Accordingly, by comparing the

rotational amplitude decay in similar saturated and dry specimens, the relative

contributions of the viscous and frictional energy dissipation mechanisms can be

examined. Such comparisons are shown in Figures 3-7 and 3-8 for Ottawa sand and glass

beads, respectively. In these figures, the vertical axes are the logarithmic decrement:

ln(ui/ui+1), where ui is the peak rotational displacement of ith cycle, and the horizontal

axes are the double amplitudes of the rotations. As may be observed from these figures,

the logarithmic decrement for the saturated specimens shows less variation with

rotational amplitude than the dry specimens (i.e., the curves for the saturated samples

have a flatter slope than the curves for the dry samples). Therefore, the portion of energy

dissipated by viscous mechanisms increases as the amplitude of the rotations decreases

(Hall 1962).

The specimens tested were 1.59in diameter and 10.8in long and were subjected to

torsional oscillations. Accordingly, the amplitude of the induced shear strains varied

across the diameter of the samples. Assuming the deformations in the samples were

linearly distributed, the outer surfaces of the specimens were subjected to shear strains of

85

approximately 0.015%, for the largest amplitude oscillations. This is slightly larger than

the threshold strain determined by Dobry et al. (1982), which was conservatively

estimated as 0.01%. However, pre- and post-test measurements showed little-to-no

differences in the void ratios, implying little-to-no change in density and effective

confining stress in the dry and saturated samples, respectively. This confirms that the

results shown in Figures 3-7 and 3-8 for dry and saturated samples are comparable.

Unfortunately, similar comparisons between dry and saturated samples subjected to large

shear strains do not exist due to the tendency of the samples to densify, resulting in

increased density of dry samples and elevated pore pressures in saturated samples.

However, from extrapolation of the trends shown in Figures 3-7 and 3-8, Whitman and

Dobry (1993) pose a corollary interpretation to that of Hall (1962) stated above: the

portion of energy dissipated by frictional mechanisms increases with increasing rotational

amplitude (or increasing strain) and becomes the dominant mechanism in both saturated

and dry specimens subjected to large strains, such as those of interest in earthquake

engineering (Whitman and Dobry 1993).

677psf1152psf2320psf5010psf

Dr = 37%, dry

Dr = 48%, saturated

691psf 1410psf

Loga

rithm

ic D

ecre

men

t

0.2

0.1

0.05

0.02

10-3

Double Amplitude, Radians10-4 10-2

Figure 3-7. Comparison of the variation of logarithmic decrement with amplitude for dry and saturated Ottawa sand in torsional oscillation. (Adapted from Hall and Richart 1963).

86

dry

Dr = 89%

3600psf7340psf

1469psf734psf

saturated

0.2

0.1

0.05

10-5 0.02

Lo

garit

hmic

Dec

rem

ent

10-310-4

Double Amplitude, Radians

Figure 3-8. Comparison of the variation of logarithmic decrement with amplitude for dry and saturated glass beads in torsional oscillation. (Adapted from Hall and Richart 1963).

3.3 Modeling of Energy Dissipation

3.3.1 Hysteresis loops

As discussed above, if two spheres are in contact acting under a normal force (N) and

tangential force (T), partial or total slippage of the contact area will occur. As a result, if

the tangential force is applied, removed, and then re-applied, a plot of the resulting force-

displacement relation scribes a hysteretic loop, such as shown in Figure 3-9a. The

mathematical expressions defining the shape of the loop are given by the expressions in

Figures 3-3 and 3-4. Similar to the interaction of two spheres, the force-displacement

response of an assemblage of particles also scribes a hysteresis loop, which is often

represented by bi-linear, hyperbolic, or Ramberg-Osgood models (e.g., bi-linear: Idriss

and Seed 1968; hyperbolic: Lee and Finn 1978; Ramberg-Osgood: Streeter et al. 1973).

87

b) τ

γ

Dissipated energy per

unit volume

a) Dissipated

energy

T

δ

Figure 3-9. Hysteresis loop resulting from: a) the application and removal of a tangential force T. b) the application and removal of a shear stress τ.

The area bound by the hysteresis loop quantifies the energy dissipated in the system of

particles. Similar to Figure 3-9a, a corresponding plot can be made in terms of stress (τ)

and strain (γ), as shown in Figure 3-9b. For this case, the area bound by the hysteresis

loop quantifies the dissipated energy per unit volume of material (∆W). Laboratory

studies have shown that the shape of the hysteresis loop is independent of the load rate

for dry sands. This implies that for a given amplitude load, the quantity of energy

dissipated by the frictional mechanism is independent of the frequency of the applied

loading (Hardin 1965). On the contrary, energy dissipated by viscous mechanisms is

directly proportional to the frequency of the applied loading. The frequency dependency

of energy dissipated by viscous mechanisms can be understood from the viscous dampers

often used on screen doors: if you close a screen door with a viscous damper quickly, it

takes much more effort than if you close the same door slowly. For saturated, undrained

samples, the area bound by the hysteresis loops represents the energy dissipated by all

mechanisms, and the contributions from friction and viscous drag cannot be discerned.

Hysteresis loops from a typical stress-controlled cyclic triaxial test conducted on a

saturated, undrained sample are shown in Figure 3-10.

88

80

-2 0 2 4Axial Strain (%)

-46

60

Dev

iato

r Stre

ss (k

Pa)

40 20 0

-20 -40

--60 86

Figure 3-10. Hysteresis loops from a stress-controlled cyclic triaxial test.

Referring back to Figure 3-1, “stored energy” and the “total dissipated energy” are related

to the hysteresis loop as shown in Figure 3-11.

τ

Stored energy

Total dissipated

energy (∆W)

γ

Figure 3-11. Graphical definitions of stored energy and total dissipated energy.

3.3.2 Equivalent Lineariztion and Damping Ratios

As mentioned above, the hysteretic response of an assemblage of particles is often

approximated by bi-linear, hyperbolic, or Ramberg-Osgood models. However, even with

these relations, a second order, non-linear partial differential equation is needed to

describe the phenomena associated with wave propagation. Depending on the response

89

quantity of interest, a further simplification referred to as equivalent linearization may be

employed. This technique is based on the idea of replacing a non-linear system by a

related linear system in such a way that the difference between the two is minimized in

some statistical sense (Jacobsen 1930, Iwan and Yang 1971, Dobry 1970, and Dobry et

al. 1971). The equivalent linear model is used in the site response computer program

SHAKE (Schnabel et al. 1972).

The rheological models for the non-linear hysteretic and linearized hysteretic systems,

and the corresponding hysteresis loops, are shown in Figure 3-12. Although the

linearized model is based on a visco-elastic material, the viscous damping coefficient (η)

can be set inversely proportional to the circular frequency of the applied loading (ω) to

remove the frequency dependence of the hysteresis loop (Hardin 1965). In selecting a

linearized hysteretic system, it is typical to equate the secant modulus (G) and the area

bound by the hysteresis loop (∆W) to those of the non-linear hysteretic system. The

secant modulus is commonly defined as the slope of a line drawn through the origin and

the point of load reversal. However, the hysteresis loop scribed by the linearized

hysteresis model is elliptical with no clear point of load reversal. For this case, the secant

modulus is drawn through the origin and the point of maximum shear strain.

90

γmax

G τ

γ

Point of maximum

shear strain

Linearized Hysteretic Model

2 G Dω η =G

time τ

γmax

Point of tangency

Gtan

Gτ

γ

Non-linear Hysteretic Model

Gtan = f (γ)

timeτ

Figure 3-12. Rheological models and corresponding hysteresis loops for hysteretic and equivalent linear materials.

The full expression for the damping coefficient used in the linearized hysteretic model is:

ωη GD2

= (3-1)

where: η = Damping coefficient (units of stress ⋅ sec).

G = Secant shear modulus (units of stress).

D = Damping ratio (dimensionless).

ω = Circular frequency (rad/sec).

More than any other quantity, reference is made to the damping ratio (D) when

describing the soil’s ability to dissipate energy (i.e., material damping). Using the

91

definitions for W and ∆W1 given in Figure 3-13 (Jacobsen 1960), the damping ratio (D) is

commonly given as Equation (3-2).

maxmax21 γτ=W

τmax

γmax∆W1

Gτ

γ

Figure 3-13. Quantities used in defining damping ratio (D).

WWD 1

41 ∆

⋅=π

(3-2)

where: D = Damping ratio.

∆W1 = Dissipated energy per unit volume in one hysteretic loop.

W = Energy stored in an elastic material having the same G as the visco-elastic material.

From Equation (3-1), it can be observed that the damping coefficient η is a function of

both the shear modulus (G) and the damping ratio (D). In turn, both G and D are

functions of the induced shear strain (γ). Extensive laboratory studies have been

conducted to develop shear modulus and damping degradation curves, which empirically

relate G and D to γ. The shear modulus degradation curves were introduced in Chapter 2

(Section 2.2.2) and are developed from strain controlled cyclic tests on saturated-drained,

or dry specimens. By subjecting soil samples to cyclic shear strains of different

amplitudes, the corresponding secant shear moduli are determined. The shear modulus

degradation curve is a plot of the secant shear modulus as a function of shear strain

amplitude. Typically, the shear moduli are normalized by the secant shear modulus for γ

= 10-4% (i.e., Gmax). This process is illustrated below.

92

0.00.0001

0.2

0.4

0.6

0.8

1.0

γ1 < γ2 < γ3

G3τ

γ3

G2τ

γ2

G1τ

γ1

G3/Gmax

G2/Gmax

G1/Gmax

γ3 γ2 γ1

G/G

max

Shear Strain (%)

0.01 0.1 10.001

Figure 3-14. Development of shear modulus degradation curves.

93

Damping degradation curves are developed in a similar fashion. However, instead of

computing the area bound by the hysteresis loops directly, which is required for

determining the damping ratio per Equation (3-2), the amplitude decay of a sample in

free vibration is often used. In this procedure, the soil sample is oscillated at its resonant

frequency and then the exciting force is removed, while the free vibration motion of the

sample is recorded. The damping ratio (D) is related to the peak displacements of

sequential cycles of free vibrating soil as (Chopra 1995):

⋅≈

+1

ln21

i

i

uu

Dπ

for D ≤ 20% (3-3)

where: D = Damping ratio.

ui = Peak displacement of cyclic i of the soil sample in free vibration.

ui+1 = Peak displacement of cyclic i+1 of the soil sample in free vibration.

The term ln(ui/ui+1) in Equation (3-3) is referred to as the “logarithmic decrement” and

was used as the vertical axis of the data plotted in Figures 3-7 and 3-8.

As stated in Chapter 2, the Ishibashi and Zhang (1993) shear modulus degradation curves

were used throughout this research. These curves were selected because they are

presented in equation form and are expressed as functions of both effective confining

stress and plasticity index (Ip). Expressions for the shear modulus degradation curves

were given previously in Chapter 2 (i.e., Equation (2-16)). The following empirical

expression is for the damping degradation curves:

( )

+

⋅−

⋅⋅

+⋅=

⋅−

1547.1586.02

1333.0),(),(max

2

),(max

0145.0 3.1

pp

p

II

I

p GG

GGeID

γγ

γ (3-4)

Plots of the shear modulus and damping degradation curves for various mean initial

effective confining pressures are shown in Figure 3-15 for a non-plastic soil.

94

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

σ’mo ↓

0.001 10.1

G/G

max

Shear Strain (%)0.01

1.0

0.8

0.6

0.4

0.2

0.00.0001

σ’mo = 3533 psf σ’mo = 2947 psf σ’mo = 2413 psf σ’mo = 1933 psf

σ’mo = 1507 psfσ’mo = 1133 psfσ’mo = 813 psfσ’mo = 547 psf

σ’mo = 333 psf σ’mo = 173 psf σ’mo = 67 psf σ’mo = 13 psf

σ’mo ↓

0.0010.0001 10.1

Dam

ping

Shear Strain (%)0.01

Figure 3-15. Plots of the Ishibashi and Zhang (1993) shear modulus degradation curves for various initial mean effective confining stresses.

3.3.3 Final Comments on D

The damping ratio D used above differs from that of the same name used in conjunction

with viscously damped oscillators. D is defined for the linearized hysteretic material

without introducing the mass (m), and therefore is a material property. The definition for

95

the damping ratio (β) for viscously damped oscillators includes m, and is therefore a

system property (Dobry et al. 1971). In relation to the energy dissipation,

ωω

πβ n

WW

⋅∆

⋅= 1

41 (3-5)

where: β = Damping ratio of viscously damped oscillators.

∆W1 = Dissipated energy per unit volume in one hysteretic loop.

W = Energy stored in an elastic material having the same G as the visco-elastic material.

ω = Frequency of the applied loading (rad/sec).

ωn = Natural frequency of the SDOF system (rad/sec).

As may be observed from Equations (3-2) and (3-5), at resonance (i.e., ωn = ω), D = β.

3.4 Use of Dissipated Energy to Quantify Capacity

As defined in this thesis, the energy Capacity of soil is the cumulative energy dissipated

up to the point of liquefaction. In light of the above discussions on energy dissipation

mechanisms, energy Capacity is re-examined. Given that the energy dissipated by the

frictional mechanism is directly related to slippage of particle contact surfaces and is

independent of the frequency of the applied loading (Hardin 1965), it seems an

appropriate measure of soil Capacity. However, because the energy dissipated by the

various mechanisms (e.g., viscous and frictional) cannot be easily discerned from the

laboratory and field data, the energy dissipated by the frictional mechanism is

approximated by the total dissipated energy. This is justified because at the large strain

amplitudes of interest in earthquake engineering, the frictional mechanism is expected to

dominate (Section 3.2.2). Accordingly, the total amount of energy dissipated up to the

point of liquefaction should be relatively independent of frequency for large amplitude

loads. Contrary to this, for small amplitude strains, the viscous mechanism of energy

dissipation may be significant, and the amount of the energy dissipated by this

mechanism is directly proportional to the frequency of the applied loading. For a load of

arbitrary amplitude, it is expected that the total dissipated energy up to the point of

96

liquefaction is either independent or increases proportionally with the frequency of the

applied loading.

Now compare these trends with those expected using Arias intensity to quantify soil

Capacity. As may be recalled from Chapter 2 (Section 2.3.4.2), the Arias intensity of the

load required to induce liquefaction in a sample subjected to stress-controlled sinusoidal

input motion is inversely proportional to the frequency of the applied load. This was

shown by Equation (2-65), which is repeated below.

f

gnIvo

lh

2max

2, 4τ

σπ

⋅= (2-65)

where: Ih,l = Arias intensity required to induce liquefaction.

τmax = Amplitude of the applied loading.

f = Frequency of the applied loading.

g = Acceleration due to gravity.

n = Number of cycles to failure.

σvo = Total vertical stress at depth z.

Both total dissipated energy and Arias intensity are functions of the applied loading.

However, for loads inducing large amplitude strains (e.g., earthquake loading), the total

dissipated energy should be relatively insensitive to the frequency of the loading, while

Arias intensity retains its frequency dependency.

3.5 Computing Dissipated Energy from Laboratory Tests

In the above sections, the relationship between dissipated energy and the hysteresis loop

was presented. However, the presentation was intentionally vague in specifying which

stress-strain hysteresis loops should be used in computing the dissipated energy per unit

volume of material. The hysteretic loop that should be used is a function of load path.

Furthermore, from a review of the literature, there appears to be no clear consensus on

which loops should be used, even for a given load path. For example, in computing ∆W

from cyclic triaxial test data, it appears that Simcock et al. (1983) used the axial stress

versus axial strain hysteresis loops, Alkhatib (1994) used the deviator stress versus axial

97

strain hysteresis loops, and Ostadan et al. (1996) used 3/4 times the deviator stress versus

axial strain hysteresis loops.

The purpose of this section is to outline a fundamentally sound approach for computing

dissipated energy for arbitrary load paths. Starting with general expressions for

incremental work, a consistent set of equations is derived for computing the dissipated

energy per unit volume for cyclic triaxial, cyclic simple shear, and cyclic torsional shear

tests.

The stress components acting on a cubical element are shown in Figure 3-16, using both

tensor-suffix notation and engineering notation.

b)

τyz

τyx

τxy

σy

τxz

σx

τzy

τzx σz

z

y

x

a)

σ23

σ21

σ12

σ22

σ13

σ11

σ32

σ31 σ33

x3

x2

x1

F b

igure 3-16. Stresses acting on a differential element a) tensor-suffix notation ) engineering notation. (Adapted from Schofield and Wroth 1968).

The increment in the energy dissipated per unit volume of material in the cubical element

using indicial notation is:

ijijddW εσ= (3-6)

In this expression σij and dεij are the stress and incremental strain tensors, respectively,

and are given as:

98

=

333231

232221

131211

σσσσσσσσσ

σ ij (3-7)

=

333231

232221

131211

εεεεεεεεε

εddddddddd

d ij

Expansion of Equation (3-6) yields:

333332323131

232322222121

131312121111

εσεσεσεσεσεσ

εσεσεσ

dddddd

ddddW

++++++

++= (3-8)

Assuming symmetric tensors (i.e., σij = σji and dεij = dεji), this expression reduces to:

232313131212

333322221111

222 εσεσεσεσεσεσ

dddddddW

+++++=

(3-9)

Using the following relations, Equation (3-9) can be written in engineering notation, as

given by Equation (3-11).

11σσ =x 11εε dd x =

22σσ =y 22εε dd y =

33σσ =z 33εε dd z =

12στ =xy 122 εγ dd xy = (3-10)

13στ =xz 132 εγ dd xz =

23στ =yz 232 εγ dd yz =

yzyzxzxzxyxy

zzyyxx

ddd

ddddW

γτγτγτ

εσεσεσ

+++

++= (3-11)

For an arbitrary load path, the cumulative energy dissipated per unit volume of material

(∆W) can be computed by integrating either Equation (3-9) or (3-11):

∫=∆ dWW (3-12)

99

3.5.1 Cyclic Triaxial Test

The following expressions relate the common notation used in reference to cyclic triaxial

tests with those of the tensor-suffix and engineering notations presented above.

xσσσ == 111'

xa ddd εεε == 11

zy σσσσσ ==== 33223' (3-13)

zyh ddddd εεεεε ==== 3322

where: σ’1 = The major principal effective stress.

σ’3 = The minor principal effective stress.

dεa = The increment in axial strain.

dεh = The increment in lateral strain.

For the cyclic triaxial test, the following boundary conditions apply:

0311332231221 ====== σσσσσσ

Substitution of the boundary conditions into Equation (3-9) yields:

ha dddW εσεσ 31 '2' += (3-14)

This expression can be further simplified by using the following relations for deviatoric

stress (σd) and Poisson’s ratio (ν).

31 '' σσσ −=d (3-15a)

a

h

εε

ν −= (3-15b)

Substituting these expressions into Equation (3-14) yields:

aad dddW ενσεσ )21('3 −+= (3-16)

Finally, for a saturated undrained tests,ν = 0.5, and Equation (3-16) reduces to:

ad ddW εσ= (3-17)

Using the trapezoidal rule to integrate Equation (3-17), the dissipated energy per unit

volume of material (∆W) can be determined by:

100

)()(21

,1,

1

1,1, iaia

n

iididW εεσσ −+=∆ +

−

=+∑ (3-18)

where: ∆W = Dissipated energy per unit volume of material up to the nth load increment.

σd,i = The ith increment in deviatoric stress.

εa,i = The ith increment in axial strain.

n = Total number of increments.

Equation (3-18) is illustrated in Figure 3-17 for an actual hysteresis loop from a stress-

controlled cyclic triaxial test.

80

3.5.2 Cyclic Simple Shear Test

Assuming the cyclic shear is applied in the 13 direction, the following boundary

conditions apply for a sample subjected to cyclic simple shear.

02312 ==== yzxy ττσσ

0332211 ====== zyx dddddd εεεεεε (3-19)

Using these boundary conditions, the general expressions for the increment in dissipated

energy given in Equations (3-9) and (3-11) reduce to:

0.0-0.2-0.4-0.6-0.8-1.0 -1.2 -60

0.2

(εa,i+1 - εa,i)

σd,i+1

σd,i

2 (σd,i+1 + σd,i)

60

Dev

iato

r Stre

ss (k

Pa)

40

20

0

-20

-40

0.4Axial Strain (%)

Figure 3-17. The dissipated energy per unit volume for a soil sample in cyclic triaxial loading is defined as the area bound by the deviator stress - axial strain hysteresis loops, Equation (3-18).

101

xzxz dddW γτεσ == 13132 (3-20)

Appling the trapezoidal rule to integrate Equation (3-20), and dropping the xz subscripts,

the cumulative energy dissipated per unit volume of material (∆W) can be determined as:

)()(21

1

1

11 ii

n

iiiW γγττ −+=∆ +

−

=+∑ (3-21)

where: τi = The ith increment in shear stress.

γi = The ith increment in shear strain.

n = Total number of increments.

3.5.3 Hollow Cylinder Triaxial - Torsional Shear Test

With the unique geometry of a hollow cylinder sample, the imposed stresses are shown in

Figure 3-18. From examination of an element of soil in the sample, it can be observed

that the stresses imposed on it are very similar to the cubical element shown in Figure 3-

16.

τvh

σ’v

σ’h

σ’h

σ’h

σ’h

τvh

σ’v

σ’h σ’h

Figure 3-18. Stress conditions for a hollow cylinder triaxial-torsional shear test. (Adapted from Towhata and Ishihara 1985).

102

The following expressions relate the notation used in Figure 3-18 to the tensor-suffix and

engineering notations.

xv σσσ == 11'

zyh σσσσσ ==== 3322'

13σττ == xzvh

xa ddd εεε == 11 (3-22)

zyh ddddd εεεεε ==== 3322

132 εγγ ddd xzvh ==

Additionally, the following stress conditions exist on an element for soil in the sample:

02312 ==== yzxy ττσσ (3-23)

3322 σσσσ === zy

Substituting these conditions into the general expressions of the increment in dissipated

energy per unit volume of material, given as Equations (3-9) and (3-11), yields:

131333331111 222 εσεσεσγτεσεσ dddddddW xzxzzzxx ++=++= (3-24)

Using the notation of Figure 3-18, Equation (3-24) can be equivalently written:

vhvhhhav ddddW γτεσεσ ++= '2' (3-25)

For isotropically consolidated samples subjected to torsional shear, the axial and lateral

deformations are approximately zero:

0≈≈ ha dd εε (3-26)

0≈≈≈ zyx ddd εεε

The expression for the increment in dissipated energy per unit volume of material reduces

to:

13132 εσγτ dddW xzxz == (3-27)

or equivalently,

vhvhddW γτ= (3-28)

103

Appling the trapezoidal rule to integrate Equation (3-27) or (3-28), and dropping the

subscripts, the cumulative energy dissipated per unit volume of material (∆W) can be

determined by:

)()(21

1

1

11 ii

n

iiiW γγττ −+=∆ +

−

=+∑ (3-29)

where: τi = The ith increment in shear stress.

γi = The ith increment in shear strain.

n = Total number of increments.

3.5.4 Use of the Derived Equations

Equations (3-18), (3-21), and (3-29) are a set of consistently derived expressions for

computing the cumulative energy dissipated in soil subjected to cyclic triaxial, cyclic

simple shear, and cyclic torsional shear loading, respectively. These expressions will be

used in Chapters 4 and 5 for computing the energy dissipated in both laboratory

specimens and in soil profiles via site response analyses.

104