ORIGINAL PAPER

Experimental study on cracking damage characteristicsof a soil and rock mixture by UPV testing

Y. Wang • X. Li

Received: 24 January 2014 / Accepted: 4 September 2014 / Published online: 30 September 2014

� Springer-Verlag Berlin Heidelberg 2014

Abstract This paper investigates the ultrasonic pulse

velocity (UPV), mechanical properties and cracking char-

acteristics of a soil and rock mixture (SRM) with varying

rock percentages under uniaxial compression. Cylindrical

SRM specimens (50 mm diameter and 100 mm height)

with rock percentages of 20, 30, 40 and 50 % were pro-

duced to perform a series of uniaxial compressive strength

(UCS) tests. A P-wave transducer (500 kHz) and associ-

ated equipment were employed for all the testing to record

the ultrasonic parameters during the whole deformation

process. Test results indicates the UCS and UPV decreased

with increasing rock percentages for all specimens. The

failure mechanism of all specimens showed a splitting-

sliding mixed pattern; macro-cracks have a direction of 0�–10� parallel or sub-parallel to the normal stress. In addition,

an equation was proposed for the relationship between

UPV and crack width. Crack initiation stress was lower for

specimens with a high rock percentage. The crack initiation

stress level was about 0.2–0.5 times of peak-strength, and

the total width of cracks was about 2–5 mm at peak-

strength. Based on the width of cracks and UPV, the total

stress–strain curve was divided into three stages: the linear-

elastic stage; the damage initiation and stable development

stage; and the damage acceleration stage. Moreover, a

three-stage damage evolution equation and constitutive

model were established and compared with the testing data.

These results confirm that the UPV and mechanical

properties of SRMs are closely related to the rock per-

centage. In this regard, the UPV test can be suitably

exploited for determing the cracking evolution character-

istics for SRM.

Keywords Soil and rock mixture (SRM) � UPV testing �Mechanical properties � Cracking characteristics

Introduction

Soil and rock mixture (SRM) is a unique type of compli-

cated inhomogeneous geomaterial and widely encountered

in geotechnical engineering projects (Medley and Lindquist

1995; Goodman and Ahlgren 2000; Lindquist and Good-

man 1994). Ancient landslides, debris-flow and rock-filling

dams are usually comprised of SRM (Li et al. 2004; Chen

et al. 2003; Zhang et al. 2004). Also, with the development

of many kinds of large-scale engineering projects, the sta-

bility of an engineering geological body (such as a slope, a

foundation and adjoining rock in tunnels, etc.) are con-

trolled mostly by the mechanical properties of the SRM.

SRMs, as a special engineering geological body, consist of

many components such as stiff rock blocks, comparatively

soft soils or particles, mixed blocks ranging in shape and

size. The individual components of SRMs usually have

different mechanical and physical properties and different

responses under internal and external loadings. Further-

more, complicated relationship exists among those indi-

vidual components. Thus, different mechanical behaviors

(such as cracking characteristics, translation of stress, stress

propagation, carrying capacity and fracture mode, etc.) exist

between SRMs and other homogeneous geomaterials. The

physical and mechanical properties of SRMs are more

complicated than those of general soil and rock mechanics

Y. Wang � X. Li (&)

Key Laboratory of Shale Gas and Geoengineering, Institute

of Geology and Geophysics, Chinese Academy of Science,

Beijing 100029, China

e-mail: lixiao@mail.iggcas.ac.cn

Y. Wang

e-mail: wangyu@mail.iggcas.ac.cn

123

Bull Eng Geol Environ (2015) 74:775–788

DOI 10.1007/s10064-014-0673-x

due to a complicated composition and interior structure. To

study the physical and mechanical properties of SRMs,

many scholars have conducted significant studies from

various points of view. For example, Lindquist (1994),

Lindquist and Goodman (1994) and Medley (2001) studied

the strength and deformation characteristics of SRMs via

multistage triaxial tests and field investigations; results

show that block percentages influence the mechanical

behavior of SRMs to a large extent. Also, block size dis-

tributions based on chords was proposed for studying the

field distribution characteristics of SRMs. You and Tang

(2002) and Xu and Hu (2007) studied the physical and

mechanical properties of SRMs and the influential factors

via in situ tests. Chen et al. (2005) studied spatial factors

such as the configuration, structure, environment and evo-

lution of SRMs in the context of the spatial effect of an

SRM slope on a large scale. Vallejo and Mawby (2000)

studied the influence of porosity on the shear strength of

granular material/clay mixtures. Xu et al. (2008) studied the

mesostructure and meso-mechanical properties of SRMs

using a digital image processing-based finite element

method. Unfortunately, one of the challenges in investi-

gating the mechanical properties and meso-structure is

inspection in real-time, which includes the detection of

damage zones, cracks and defects.

Ultrasonic techniques, known as being non-destructive

and easy to apply for in situ and laboratory conditions, are

commonly used for establishing the strength of concrete or

rock via UPV measurement (Kahraman 2001; Yasar and

Erdogan 2004). These techniques have been used for mea-

suring various concrete properties. The UPV method has also

been suggested as a being useful for estimating elastic and

strength properties of rock; some empirical correlations

between the UPV and compressive strength and modulus of

elasticity have been established (Saka and Uchikawa 1995).

UPV can also be used to evaluate the cracks or defects in a

material (Akkaya et al. 2003) or to analyze the concrete

microstructure development and strength (Ercikdi et al.

2014; Zhang et al. 2010; Su et al. 2012). When associated to

tomography, UPV can give good qualitative information on

the changes in a material properties as well as on its micro-

cracking state (Grinzato et al. 2004; Meglis et al. 2005).

Kahraman (2004) also studied the influence of the fracture

roughness of granites on UPV and provided a correlation

between both parameters. Although acoustic emission seems

to be more appropriate in the evaluation of the crack damage

in concrete, and especially in rocks under uniaxial com-

pression (Farmer 1983; Eberhardt et al. 1999), UPV appears

also to provide some indication about the damage in concrete

(Selleck et al. 1998; Mirmiram and Wei 2001).

The basic objective of the present work is to analyze and

develop the usefulness of UPV testing for exploring the

mechanical properties and crack evolution in an SRM

specimen, and also to establish a damage evolution equa-

tion and constitutive model. To the authors’ knowledge, so

far no experimental results have been published using UPV

testing to research the mechanical and deformation char-

acteristics of SRMs. The SRM specimens are cylinder-

shaped with a 50 mm diameter and a height of 100 mm

with varying rock percentage (20, 30, 40, and 50 %, mass

proportion). All three produced specimens were tested.

Correlation between UPV and UCS and width of cracks are

established. Based on the width of cracks and UPV, the

total stress–strain curve of SRMs was divided into three

stages. Moreover, the three-phase damage evolution

equation and constitutive model were established, and also

compared with the testing data.

Experimental procedure

The testing material

Remolded SRM specimens were used for the experiments.

The soil was obtained from a pit in the Chinese Academy

of Sciences Institute of Atmospheric Physics at a depth of

15 m. According to the geotechnical testing standard for

soil test method (GB/T 50123-1999), some physical and

mechanical parameters are shown in Table 1. The soil

contained a notable amount of strongly hydrophilic clay

minerals. The liquid limit of the hard clay can reach 40 %

and the plastic limit can reach 36 %; the plasticity index

was about 48 and the liquidity index was about 0.05–0.127.

These indices indicated this soil is a typical hard plastic

and high plastic clay. To identify the mineral composition

and mineral contentwe conducted Scanning Electron

Microscope (SEM) and X-Ray diffraction (XRD) tests on

the soil. By XRD analysis, the main clay minerals can be

identified. The clay minerals were identified from their

basal reflections determined from the XRD pattern (Moore

and Reynolds 1997) after: (1) air drying (normal); (2)

glycolation for 48 h; and (3) heating to 550�. The net peak

areas of the basal reflection of the clay minerals were

calculated above the background and considered as a

Table 1 Basic properties of soil material and rock blocks from geo-

technical testing

Property Soil Rock blocks

Natural density (g/cm3) 1.66 2.67

Dry density (g/cm3) 2.03 –

Water content ( %) 9.5 –

Relative density (GS) 2.72 –

Compressive strength wet (MPa) 0.56 2.727

Compressive strength dry (MPa) 50.65 100.74

776 Y. Wang, X. Li

123

proporation of each mineral in the mixture. The total

association was taken to by equal to 100 %. Then the rel-

ative properties were deduced semi-quantitatively. SEM

tests, as shown in Fig. 1, revealed rodlike and irregular

quartz grains with a grain size of about 0.01–0.03 mm and

probably surrounded by clay minerals. The XRD tests

revealed the main clay mineralogical composition, as

shown in Table 2. According to Table 2, it is clear that the

soil has a higher percentage of clay mineral, similar to

kaolinite, montmorillonite, and illite.

As stated above, the mechanics of SRMs are restricted

by the shape, distribution, size and percentage of rock

blocks. The rock percentage is the most important index

influencing the mechanical properties of SRMs (Lindquist

and Goodman 1994; Xu et al. 2008). As such, we ignored

the other factors influencing the mechanical properties. The

rock blocks used corundum balls with 8-mm diameters;

properties of the corundum material are listed in Table 1.

Specimen preparation

A total of 60 SRM specimens were prepared for UPV

testing. A compact test was used to produce the specimen

(Donaghe and Torrey 1994). According to the relationship

between the density and the compaction number, the

optimal hammer count was determined to be 20. During the

preparation of SRM specimens, an extra amount of free

water was added to the mixture; the optimal water content

was determined by compaction test to be 9.5 %. The rock

blocks used for specimen preparation and SRM specimens

with different percentages after air-drying are shown in

Fig. 2.

The required amount of rock blocks and soil material for

each specimen (Table 3) were mixed and homogenized in a

mixer. Then, the mixtures were poured into cast iron cyl-

inders 50 mm in diameter by 100 mm in height. The

compaction apparatus was used to compact the mixture

with 20 counts with three layers. The specimens were then

sealed with plastic warp and allowed to air-dry.

Experimental system

The testing system utilized for the UPV tests includes a

rigid loading device, an ultrasonic detector and an ultra-

sonic transducer (500 kHz) specially designed for this

test. The overall setup of the test system is shown in

Fig. 3.

During the test, axial load is applied by the hydraulic

jack, which can provide a maximum axial force of 100 kN.

The axial force is measured by stress sensors. The load

controller can record the axial force at every stress level.

The precision of the load controller is 0.01 kN. The axial

deformation is measured by a micrometer installed on the

platform; its precision is to 0.001 mm. The ultrasonic

detector is a common detector utilized in concrete detec-

tion (Model ZBL-520) which can provide a 1,000 V spike

for a duration of 20 ls to 20 ms for the transducer and also

can accurately record wave signals with good precision. In

the UPV testing, the sampling interval was 0.1 ls and the

arrival time of each pulse could be read to 0.05 ls; sam-

pling length is 1,024. Before the measurements, the mid-

dle-lateral surfaces of SRM specimens were made smooth

and flat. A thin film of Vaseline was applied to the surface

of the transducers (transmitter and receiver) in order to

Fig. 1 SEM images of: a soil sample #1; b soil sample #2

Table 2 Main mineralogical composition of a soil specimen obtained

from XRD

Mineral Soil specimen #1 (%) Soil specimen #2 (%)

Montmorillonite 61.52 60.28

Kaolinite 26.73 24.86

Illite 6.25 9.58

chlorite 3.5 3.28

Cracking damage characteristics of soil and rock 777

123

ensure ful contact and to eliminate the air pocket between

transducers and the specimen.

The specially designed transducer is the core compo-

nent determining the success of the test. The transducer

(500 kHz) is cylindrical and one end can be connected

with the plane at the middle part of specimens. The pie-

zoelectric ceramic disk is equipped in the cylindrical bore

(as shown in the top left corner of Fig. 3) and one end

connects with a tungsten powder mixture filler and spring,

the other end connects with a boss button. The tungsten

powder mixture filler can make the piezoelectric ceramic

disk move forward and emit signals. A thread cover and a

shim are used to constrain the boss button and the spring

makes the head of the boss button extend 1 mm from the

center bore of the shim once no pressure is applied. When

the transducer is subjected to pressure, the boss button

moves backward by compressing the spring and the

pressure can be afforded by the shim. During the tests, the

transducer was fixed using a rubber strip, enabling the

piezoelectric ceramic disk can to be in close contact with

the specimen during the tests.

Testing procedure

All devices were installed as shown in Fig. 3 and were

checked to ensure that they were working normally. The

UPV testing method employed was the ultrasonic trans-

mission method (through-transmission method). Uniaxial

compressive strength tests for the specimens were carried

out at the speed of 0.1 kN/step. Complete information

regarding the stress value, displacement value and ultra-

sonic parameters were recorded. Every three specimens of

varying rock percentages were tested.

As is known, the first cycle wave is stable and renewable

under the same transducer and same contact between

transducers and specimens. The first cycle wave is scarcely

contaminated by other waves arriving late and is easy to

identify. Therefore, the first cycle wave was selected as the

initial wave. The waveforms collected by the receiving

transducer consist of an initial transmitted pulse, followed

by later reflections at various interfaces, such as the rock–

soil interfaces, and the transducers and the specimen. Due

to absorption attenuation, scattering attenuation and diffu-

sion attenuation into an ultrasonic wave, the received

ultrasonic frequency was reduced to some extent. Figure 4a

shows the received waveform of sample SRM20-1 before

loading by ultrasonic detector. The initial wave was

selected to obtain the travel time t at each loading step

(Fig. 4b). After measuring the path length L the velocities

were calculated as UPV = L/t.

Research idea

SRMs are characterized by complex ingredients and an

anomalistic structure distribution. Failure characteristics of

SRMs are complicated under internal or external loadings.

Different mechanical properties exist among soils and

rocks. Determining the level of failure may be difficult and

unreliable without using complicated methods and proce-

dures, such as in situ experiments, indoor experiments,

numerical simulation and so on (Lindquist and Goodman

1994; Goodman and Ahlgren 2000; You and Tang 2002; Li

et al. 2004; Xu and Hu 2007). Sometimes special proce-

dures and methods have to be designed, tried and then

applied to the element under consideration. Such methods

are usually slow and costly.

However, UPV testing as a useful and reliable non-

destructive tool for assessing the mechanical characteristics

of rock and concrete material demonstrates a strong

advantage (Su et al. 2012; Grinzato et al. 2004). The

Fig. 2 Specimens with UPV testing. a Rock blocks in the specimens; b specimens after air-drying

Table 3 The required amount of blocks and soil for each specimen

Rock

percentage (%)

Dry soil

(g)

Dry soil and

water (g)

Block mass

(g)

Ball

no.

20 337.4751 371.2226 364.4731 110

30 303.5799 333.9379 327.8663 170

40 267.7268 294.4995 289.1449 240

50 229.7410 252.7151 248.1203 300

778 Y. Wang, X. Li

123

method present here is a technique that can be applied to

structurally cracked elements in order to explore the

cracking characteristics of SRMs using UPV testing.

Due to the elastic mismatch between a soil matrix and

rock blocks, soil and blocks are considerd to be in a weak

cementation state. Under loading, differential deformation

occurs at the interface between the rock blocks and soil,

which causes differential sliding, moving and rotation of

the rock blocks. As such, the local concentration of stress

causes tensile damage around the rock/soil interface.

Afterwards, a series of non-linear behavior appears,

including crack initiation, propagation and coalescence and

movement of blocks.

The research idea is to measure the velocity through the

SRM specimens under uniaxial compressive test in real-

time. When cracks appear in the specimen, it is obvious

Fig. 3 Testing system (1. Upper cross beam; 2. Rigid column; 3.

Platform; 4. Guide bar; 5. Pedestal; 6. Transmission line; 7. Force

sensor; 8. Load controller; 9. Hydraulic jack; 10. Micrometer gauge;

11. Rigid cushion; 12. SRM specimen; 13. Rubber strip; 14.

Transmitter; 15. Receiver; 16. Ultrasonic detector)

Fig. 4 Received wave form for sample SRM20-1 in its initial state (a) and the principle to obtain UPV (b)

Cracking damage characteristics of soil and rock 779

123

that the velocity is reduced (see Fig. 5). The velocity

through an SRM is higher than the velocity through air or

water; the crack is either filled with air or water. Hence, a

reduction in the measured velocity can be noticed when the

specimen cracks. However, when the cracks are wide, the

sound waves are wholly reflected and no signal is received.

Furthermore, a relation between the UPV and total crack

width was deduced. The basic idea was that the reduction

in the velocity through an SRM is basically due to the

formation of cracks, as shown in Fig. 5. According to the

principle that states when in the state of weak deformation,

travel time in the medium is almost constant, the final

relationship was as follows:

w ¼ L1

V� 1

V0

� ��1

Va

� 1

V0

� �ð1Þ

, where w is the total crack width, V is the velocity in the

SRM at any stress level, V0 is the velocity in the SRM at

zero stress level, Va is the wave velocity in air, taken as

340 m/s and L is the side length of the specimen. In terms

of physical interpretation of Eq. (1), during compressive

loading the travel time increment of the ultrasonic wave

through a specimen is equal to the time increment when the

cracks are filled with air. (Note: Eq. (1) is deduced when

cracks appear in the specimen. So, if ‘‘w’’ is positive, it

indicates that the SRM specimens are cracking and if the

‘‘w’’ is negative, it implies that the SRM specimens are in a

consolidation stage.)

Results and discussion

Peak strength variation against rock percentage

Axial stress–strain curves for typical specimens (SRM20-1,

SRM30-4, SRM40-7 and SRM50-10) are shown in Fig. 6.

Upon reaching peak strength, the specimens remained

complete, but cracks parallel or sub-parallel to axial

direction appeared (Fig. 8). The average peak strengths

with different rock percentages are shown in Table 4.

As Fig. 7 depicts, increasing the rock percentage results

in a reduction in peak strength. Also, peak strains decease

Fig. 5 Test procedure in non-cracked and cracked specimens

Fig. 6 Stress-strain curve for typical SRM specimens

Table 4 Peak strength value and peak strain value for specimens

with different rock percentages

Rock percentage

( %)

Peak strength (MPa) Peak strain ( %)

Mean Standard

deviation

Mean Standard

deviation

20 5.897 0.449 1.081 0.014

30 4.676 0.385 1.037 0.033

40 4.549 0.087 0.885 0.027

50 3.212 0.185 0.798 0.081

780 Y. Wang, X. Li

123

with an increasing rock percentage. Our data is in aggre-

ment with the results of Medley (2001) and some of the

results of Xu and Hu (2007).

According to Table 4, peak strength and strain are not

uniform for the specimens with the same rock percentage.

This is due to the difference of block distribution in the

specimen.

Failure mechanism

Because the rock–soil interface was in the state of weak

cementation, differential deformation occurred at the inter-

face, causing differential sliding between the soil and rock

blocks under axial loading androck block movement and

rotation. As such, locally concentrated stress causes tensile

damage around the rock/soil interface. Figure 8 is the failure

morphology of SRM specimens with different rock per-

centages. As shown in Fig. 8, the cracks are almost parallel

to the axial direction. The observed failure patterns for

specimens with different rock percentage revealed that all

specimen failures followed the same mechanism. Figure 8

shows typical failure patterns observed for tested specimens

under uniaxial compressive loading. The combination of

splitting and sliding failire patterns lead to the formation of

two kinds of cracks, the rock/soil main cracks and secondary

cracks surrounding the rock blocks. With an increasing rock

percentage, the number of cracks increased. Splitting failure

occurred in the soil matrix with crack propagation and coa-

lescence and the main cracks formed in the soil. Sliding

occurred along the interfaces between blocks and soil and

this caused formation of secondary cracks. The sliding fail-

ure is simply a result of the relative movement and rotation of

the blocks in the SRM. These failure mechanisms at different

rock percentage were consistent with previously reported

results (Xu et al. 2008) (Fig. 9).

Ultrasonic pulse velocity

As in the analysis above, the failure mode of SRM speci-

mens are a mixed pattern (a combination of splitting and

sliding). Cracks in specimens propagated and coalesced

with an increasing axial loading. Cracks are filled with air

and when ultrasonic waves pass through the specimens, the

UPV deceases gradually. Figure 10 is the relationship

between the UPV and axial stress of typical specimens

SRM20-1, SRM30-4, SRM40-7 and SRM50-10. As is

shown, rock percentage is the main factor influencing the

UPV; at the same stress level the UPV is lower in speci-

mens with more rock blocks than those with less blocks.

Fig. 7 Peak strength and peak strain versus rock percentage for

tested specimens

Fig. 8 Failure morphology for SRM specimens under uniaxial compressive test

Fig. 9 Sketch maps of the failure morphology for SRM specimens with different rock percentages

Cracking damage characteristics of soil and rock 781

123

Table 5 is the UPV of specimen unloading and failure. We

can see that the UPV decreased with an increasing rock

percentage. Also, the UPV of the same rock percentage is

different, due to the different distribution of rock blocks in

the specimens.

Cracking evolution analysis

The total crack width is calculated according to Eq. (1);

they are caused by micro cracking and plastic deformation

during compressive processes. The width of cracks was

plotted against relative stress (i.e., the ratio of axial stress

to the maximum axial stress) for typical specimens

SRM20-1, SRM30-4, SRM40-7 and SRM50-10. Figure 11

depicts a clear crack evolution pattern and the cracking

evolution can be divided into three stages:

1. Linear-elastic deformation: in this stage, the SRM

specimen was consolidated, pores and opening cracks

were closed by being subjected to axial loading, little

elastic energy was released, although some elements

were damaged;

2. Rock/soil jointed crack initiation and stable crack

growth: These actives can be attributed to local

degradation such as interconnection of the large pores,

rock–soil jointed interfaces cracking and multiple

branching cracks. It is clear that in fact the cracking

processes operate at this stage, which seems to control

the lifetime of the SRM;

3. Crack acceleration and coalescence: when the speci-

men approaches its ultimate strength it is assumed that

unstable cracking occurs by interconnection between

the defects created in the second stage.

The compressive cracking process up to peak load is well

described by the stress markersrci andrcd (see Fig. 12) where

the typical stress–strain curve up to peak strength is dis-

cussed. The onset of microcracking is associated with the

stress level rci and is followed by a nonlinear increase of the

lateral strain. Unstable microcracking occurs for the crack

damage stress level rcd and is associated with the reverse

point in the total volumetric strain curve (Vr). This point is

connected to the maximum compaction of the specimen and

to the onset of dilation, since the increase in volume gener-

ated by the cracking process is larger than the standard vol-

umetric decrease due to the axial load.

The relations between the width and relative velocity

(i.e., the ratio of velocity to the maximum velocity) is

illustrated in Fig. 13. We can see that the sum of crack

widths is closely related to the UPV. It is also clear from

Fig. 14 that the relative velocity stays almost constant in

the first stage until the relative velocity reaches a certain

level, and then a slower and severe reduction in the relative

velocity is obtained. Also, from Fig. 9, a clear pattern of

crack evolution can be obtained. The point corresponding

to crack initiation and crack acceleration is associated to rci

and rcd. Like rock material, this is the first that we obtain

the rci and rcd for SRM specimens using UPV tests.

Results of rci and rcd related to the corresponding stress

level are shown in Table 6. With an increasing rock per-

centage, the relative stress for rci and rcd is reduced.

Damage constitutive model

In recent years, continuum damage mechanics was applied

to study the initiation and growth of cracks in rock and soil.

Great achievements were made (Chaboche 1981; Kacha-

nov 1986; Lemaitre and Chaboche 1990), but no experi-

mental results about damage characteristics of SRMs under

loading were republished. Under loading conditions,

internal structure, strength, and deformation characteristics

would change accordingly for SRMs. The UPV testing is a

reliable and non-destructive tool, having been used in

concrete and rock widely; some mechanical properties

related to damage characteristics have been conducted. In

Fig. 10 Relationship between the UPV and axial stress of typical

specimens

Table 5 UPV of initial specimens and of failure with different rock

percentages

Rock

percentage

(%)

Initial UPV value Failure UPV value

Mean

(m/s)

Standard

deviation

(m/s)

Mean

(m/s)

Standard

deviation

(m/s)

20 4859.946 26.380 3536.056 26.258

30 4563.342 35.957 1795.870 9.921

40 4332.505 39.399 1678.357 20.694

50 3769.232 44.642 1533.664 17.408

782 Y. Wang, X. Li

123

Fig. 11 Typical plots of total crack width against relative stress for specimens SRM20-1, SRM30-4, SRM40-7 and SRM50-10

Fig. 12 Typical stress–strain curve for rocks under uniaxial com-

pressive loading up to peak stress

Fig. 13 A typical plot showing the relationship between crack width

and relative velocity

Cracking damage characteristics of soil and rock 783

123

this paper we try to establish a damage evolution equation

and a constitutive model for SRM specimens by UPV

testing.

According to classic damage mechanics, the damage

factor can be defined as follows:

D ¼ 1 � ~E=E ð2Þ

where D is the damage factor, ~E and E are the undamaged

and damaged Elastic modulus.

When the longitudinal ultrasonic wave with a certain

frequency goes through the SRM specimens, the velocity v,

bulk density q, modulus of elastic E and Poisson’s ratio mexist in the following relationship:

V2 ¼ E

q1 � m

1 þ mð Þ 1 � 2mð Þ ð3Þ

If the change of the Poisson’s ratio and density are

ignored during loading for damage specimens,

~V2 ¼~E

_

q1 � m

1 þ mð Þ 1 � 2mð Þ ð4Þ

Combining with Eqs. (2) and (3), the damage factor

defined using UPV, is as follows:

D ¼ 1 � ~V2=V2 ð5Þ

where V and ~V are the velocity of undamaged and damaged

material. This definition is based on the assumption that the

initial damage factor of SRM is 0, and the damage factor is

1 when the specimen has failed. Because the changes in

ultrasonic wave velocity can comprehensively reflect

changes in the internal structure of SRM specimens, phe-

nomenon such as crack initiation, propagation and coa-

lescence can be reflected by UPV. So the damage factor

definition based on UPV could completely reflect the

macro mechanics of micro cracks for SRM specimens.

There will be a problem when the damage factor

obtained by Eq. (5); in linear-elastic stage the UPV can

increase due to the compaction effect. In this case, D\ 0;

however, this is impossible, so we specify it as D = 0.

Figure 15 is the relationship between relative stress and

the damage factor for typical specimens SRM20-1, SRM30-

4, SRM40-7 and SRM50-10. As shown in the figure, during

uniaxial compression, the damage factors for the specimens

are not uniform. The damage factor changed suddenly at

some stress level and this phenomenon is consistent with the

relationship of crack width and relative stress.

Fig. 14 Relationship between relative velocity and relative stress

during loading

Table 6 The relative stress for rci, rcd and the total crack widths at

peak strength

Specimen

No.

Crack initiation

stress level (%)

Crack damage

stress level (%)

Crack width at

peak strength

(mm)

SRM20-1 0.463 0.768 3.054

SRM20-3 0.481 0.794 2.756

SRM20-4 0.443 0.773 2.934

SRM30-1 0.427 0.736 3.728

SRM30-2 0.403 0.772 4.651

SRM30-4 0.435 0.865 4.897

SRM40-5 0.337 0.653 4.327

SRM40-6 0.332 0.711 4.003

SRM40-7 0.261 0.763 4.605

SRM50-8 0.231 0.682 4.751

SRM50-9 0.226 0.645 4.855

SRM50-10 0.208 0.717 4.343

Fig. 15 Typical plots of the damage factor against relative stress for

typical specimens SRM20-1, SRM30-4, SRM40-7 and SRM50-10

784 Y. Wang, X. Li

123

Research of the damage mechanics of rock and soil is a

fundamental and frontier issue in geotechnical engineering.

Experimental research of damage evolution characteristics

and constitutive relations are reliable and practical condi-

tions to ensure numerical simulation results in geotechnical

engineering. The pre-existing damage constitutive model

can be divided into two major categories. One is the con-

tinuous function and the other is the piecewise describing

function. Because information regarding crack evolution

can be reflected via UPV, the damage evolution of an SRM

is a nonlinear process; it is difficult to describe the cracking

evolution with a continuous function. So, the piecewise

describing function is a good choice. Based on results of

the stress–strain curve and the crack evolution character-

istics above, the pre-peak stress–strain curve can be divided

into a quasi linear stage, a stable damage evolution stage

and a damage acceleration stage.

Taking SRM30-4 as an example, the three-phase dam-

age constitutive model is shown in Fig. 16. According to

the damage factor defined by the UPV, the damage factors

during the uniaxial compressive test were calculated.

Firstly, the relationship of D and e1 was analyzed using the

method of least squares regression. Linear (y = ax ? b),

logarithmic (y = a ? lnx), exponential (y = aex) and

power (y = axb) curve fitting approximations were exe-

cuted and the approximation equations that have the

highest correlation coefficient were determined for the

damage evolution equations. Then, based on the equivalent

strain principle, the relevant constitutive model is expres-

sed as

r1 ¼ E 1 � Dð Þe1 ð6Þ

1. Quasi linear phase

The constitutive model is linear. The constitutive model

in this stage by linear curve fitting approximation is

expressed as:

r1 ¼ 0:0808 þ 5:59408e1 0\r1\2:44076 MPa ð7Þ

The corresponding damage evolution equation is:

D ¼ 0 ð8Þ

Fig. 16 Three-stage damage evolution equation and constitutive model for SRM30-4

Cracking damage characteristics of soil and rock 785

123

2. Damage initiation and stable development stage:

r1 ¼ E 1 � Dð Þe1 ð9Þ

According to the damage factor defined by UPV, the

damage factor during uniaxial compressive tests is calculated,

and the relationship between D and e1 is analyzed using least

squares regression. Linear (y = ax ? b), logarithmic

(y = a ? lnx), exponential (y = aex) and power (y = axb)

curve fitting approximations were executed and the approxi-

mation equations that have the highest correlation coefficient

were determined for the damage evolution equation:

D ¼ 2:4852expð�1:43659=e1Þ ð10Þ

Combination with equations (9) and (10), the constitu-

tive model of SRM30-4 in this stage is expressed as:

r1 ¼ E 1 � 2:4852 exp �1:43695=e1ð Þð Þe12:44076\r1\4:13248 MPa

ð11Þ

3. Damage acceleration stage:

In this stage, the relationship of D and e1 was analyzed

using least squares regression. The damage evolution

equation was as follows:

D ¼ 0:7895e0:46391 ð12Þ

Combination with Eqs. (5) and (6), the constitutive

model of SRM30-4 in this stage is expressed as:

r1 ¼ E 1 � 0:7895e0:46391

� �e1 4:13248\r1\4:77452MPa

ð13Þ

Correlation coefficients (r) were 0.9994, 0.94524 and

0.9913 for SRM30-4, respectively. The damage evolution

equation and constitutive model for SRM20-1, SRM40-7

and SRM50-10 were obtained using the same method

(Table 7). As shown in Table 7, the correlation coefficient

of all equations are very good, but they do not necessarily

indicate the goodness-of-fit of the equations. Thus, vali-

dation of the equations was checked by a t and F test. The

significance of the r values can be determined by the t test,

which compares the computed t value with the tabulated

t value using a null hypothesis. The significance of the

regressions was determined by analysis of variance (F test).

In these tests, a 95 % condidence interval (p B 0.05) was

chosen. It is well known that if the computed t and F values

are greater than the tabulated t and F values, the null

hypothesis is rejected (Levine et al. 2011). In this regard,

computed t and F values are greater than the tabulated t and

F values, indicating the validity of these equations

(Table 7). Figure 17 illustrate the stress–strain curve using

measured and estimated data. As is shown, the estimated

curve was in agreement with the measured curve.

Conclusions

The present work encompassed a series of UPV tests on

SRM specimens with different rock percentages (i.e., 20,

30, 40 and 50 %), in an effort to contribute to the field

of UPV, a non-destructive technique for researching

mechanical properties and cracking evolution in real-time.

An experimental system was developed to match the UPV

tests. A technique that can be applied to structurally

cracked elements in order to explore the cracking

Table 7 Damage evolution

equation and constitutive model

for SRM specimens

Specimen

no.

Evolution

stage

Damage evolution

equation

Damage constitutive model Correlation

coefficient (r)

SRM20-1 (1) D = 0 r1 = -0.14121 ? 8.97584e1 0.9994

(2) D = 2.27713exp

(-1.62444/e1)

r1 = E(1-2.27713exp

(-1.62444/e1)) e1

0.8865

(3) D = 0.57467e11.57493 r1 = E(1-0.57467e1

1.57493)e1 0.9394

SRM30-4 (1) D = 0 r1 = 0.0808 ? 5.59408e1 0.9994

(2) D = 2.4852exp

(-1.43659/e1)

r1 = E(1-2.4852exp

(-1.43659/e1))e1

0.9452

(3) D = 0.7895e11.57493 r1 = E(1-0.7895e1

1.57493)e1 0.9913

SRM40-7 (1) D = 0 r1 = 0.0291 ? 4.45117e1 0.9855

(2) D = 1.00844exp

(-0.40446/e1)

r1 = E(1-1.00844exp

(-0.40446/e1))e1

0.9265

(3) D = 1.15365exp

(-0.38258/e1)

r1 = E(1-1.15365exp

(-0.38258/e1))e1

0.8928

SRM50-10 (1) D = 0 r1 = 0.04665 ? 2.81125e1 0.9731

(2) D = 1.50392exp

(-1.26212/e1)

r1 = E(1-1.50392exp

(-1.26212/e1))e1

0.9853

(3) D = 1.05582e11.78802 r1 = E(1-1.05582e1

1.78802)e1 0.9175

786 Y. Wang, X. Li

123

characteristics of SRMs using UPV testing has been con-

ducted. Because deformation of the middle part is most

evident under uniaxial compressive tests, the ultrasonic

transducer (500 kHz) was installed in the middle part of

specimens. Test results revealed that UCS and UPV

deceased with an increasing rock percentage for all speci-

mens. Specimens with a different rock percentage failed by

a combination of splitting and sliding failures; the macro-

cracks were 0–10� parallel to the axial of specimen. A

relationship between crack widths and UPV was estab-

lished. Crack evolution is reflected by the change in UPV,

and the non-linear failure process of SRMs was divided

into three stages. Moreover, a damage evolution equation

and constitutive model corresponding to each stage were

established and were compared with the measured data.

These findings suggest that the UPV test is a reliable, low-

cost and practical method can be used to research the

mechanical properties and cracking characteristics of

SRMs. The UPV and mechanical properties of SRMs are

closely related to the rock percentage.

Acknowledgments The authors would like to thank the Editor and

two anonymous reviewers for their helpful and constructive com-

ments. This work was supported by the National Natural Science

Foundation of China (Grants Nos. 41227901) and the Strategic Pri-

ority Research Program of the Chinese Academy of Sciences (Grants

Nos. XDB10030000, XDB10030300, and XDB10050400).

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