Characterizing the deformation behavior of weak sandstone from micro-vision

Abstract

This study explores the mechanism associated with the characteristics of the weak sandstone from micro-vision. Firstly, compression tests with tiny specimens are conducted and the photographs of deformation could be obtained simultaneously using microscope. Then, the digital image correlation (DIC) technique is used to effectively measure the whole strain field in micro-vision. According to the analysis results, the deformation of weak sandstone possesses the following characteristics: (1) weak sandstone exhibits non-uniform strain pattern under loading, and the matrix portion is relatively softer than the grain; (2) the plastic deformation mostly accumulates in the matrix portion; and (3) deducing from stress-strain curves, the modulus of each element can be obtained, and the modulus of grain are higher than that of matrix.

Keywords: weak sandstone, digital image correlation, plastic strain, microscopic

mechanism.

1 Introduction

The term “sandstone” denotes a large class of sedimentary rocks with different mineral compositions, diagenetic process and ages or degree of diagenesis. This diverseness in rock-forming origins results in a great variety of rock properties, including the strength, deformability and resistance to weathering. In western Taiwan, sandstones have a digenetic age of no more than 70 million years and such relatively short rock forming period is insufficient to classify them as hard rocks. The typical strength of these sandstones in Taiwan is often below 50 MPa and their mechanical behavior differs from that of many hard rocks [1].

In the past, while tunneling through the weak rock strata, several unsuccessful cases have been reported [2]. Difficulties, including severe squeezing and collapse,

Paper 165

Characterizing the Deformation Behavior of

Weak Sandstone from Micro-Vision

M.C. Weng1, S.H. Tung1, M.H. Shih2, C.C. Yu1 and Y.T. Huang1

1 _{Department of Civil and Environmental Engineering}

National University of Kaohsiung, Taiwan

2 _{Department of Construction Engineering}

National Kaohsiung First University of Science and Technology, Taiwan

©Civil-Comp Press, 2008

Proceedings of the Sixth International Conference on Engineering Computational Technology, M. Papadrakakis and B.H.V. Topping, (Editors), Civil-Comp Press, Stirlingshire, Scotland

were encountered during construction of these tunnels. For instance, a crown settlement of 180 cm of a 12.4 m wide highway tunnel has been reported. Therefore, the deformational characteristics of weak sandstones should be further studied.

In order to identify the deformational behavior of weak sandstones, several research projects were conducted by the authors. Based on the previous research results [3], the deformation characteristics of weak sandstones include: (1) lower stiffness in bulk modulus and shear modulus; (2) significant shear dilation and distortion during shearing; and (3) substantial plastic deformation occurs prior to the failure state during shearing.

Furthermore, from micro point of view, how the weak sandstones deform or fracture, when subjected to external loading, has been an interesting and important issue. The plastic behavior may be related to complex micro- and meso-crack coupled growth, and accumulation of these cracks. Therefore, how to effectively measure the strain field from micro-vision becomes increasingly necessary to gain more insight into the deformation behavior of weak sandstones.

Digital image correlation method (DIC) is a latest optical measuring technique, which offers effective and high precision strain distribution for a field. Chu et al. [4] developed a measuring technology by combining deformation theory and digital image correlation method, and strived to expand the field of applications using interpolation theory. Recently, the DIC method has been successfully applied to determine strains in various materials, including resin films, polymer fibers, concrete, wood fiber and so on [5-9].

By digital image correlation method, this paper firstly observes the deformation patterns and the fracture propagation of weak sandstone under uniaxial loading. Afterward, the microscopic mechanism that accounts for the significant deformation is further explored.

2 Theoretical basis of stain analysis by digital image

correlation method

2.1 Two-dimensional digital image correlation method

Digital image correlation method is used to analyze so-called “structural speckle” on the surface, which results in grey scale distribution on image surface. Based on the distribution feature of grey scale, the characteristics of undeformed and deformed images are compared, and relative position of images inferred accordingly. Thus, displacement vectors of various image points are calculated, while other physical values, such as normal strain ε_x and ε_y , shear strain ε_xy and v. Mises strain

' 2

2 J

γ = (where ' 2

J is the second deviatoric strain invariant 2

1 '

2 ij ij

J = e e ; eij =

second deviatoric strain tensor), can also be inferred.

Recently, digital image correlation method is widely applied to the field of image identification technique. By comparing local correlation of two images, the relationship (under assumption of parameter’s functional relationship) of two undeformed and deformed images could be identified. As shown in Figure 1, central point prior to deformation is point P, and then changed to point P* after deformation, with the functional relationship expressed below:

* _{( , )} x = +x u x y (1a) * _{( , )} y = +y v x y (1b) y, y x, x △ y △ y △ x △ x P Q Q P △ y y y y y x x x x ▽ Area of Scanning Undeformed Subimage Deformed Subimage Pixel Location Sampling Grid Q Q Q Q P P P P * * * * * * * * * * *

Figure 1 Schematic diagram of relative location of deformed and undeformed images [4].

For undeformed images, finite element method (FEM) is used to divide the images into several sub-images. According to digital image correlation method, the sum of grey scale value of undeformed and deformed secondary images is assumed to the same. Assuming undeformed image is A and deformed image B, the correlation is called as a functional relationship. The level of correlation is defined below: 2 2 ij ij ij ij g g COF g g = ⋅

∑

∑ ∑

Where, g and _ij g~ is grey scale of image A on coordinate _ij ( ji, ) and image B on coordinate ( ji, ), respectively. Then, coordinate ( ji, ) of image B corresponds to coordinate ( ji, ) of image A.

If optimum function parameter for every sub-image is recognized by an optimization procedure, the corresponding coordinate of every undeformed and deformed sub-image could be obtained. Accordingly, displacement vector and displacement field can be individually computed.

2.2 Calculation of strain field

1 2

T

E= ⎡_⎣F ⊗ −F I⎤_{⎦ (3)}

Where, F is gradient tensor of displacement field, and I is unit matrix. Tensor E is rewritten into the function of displacement field as follows:

(

)

1 1

2 2

ij i j j i k i k j

E = u +u + u u (4)

Where, , ,i j k∈( , )x y , and ui j, =δui/δj. Strain is obtained as: 2 2 1 2 y x x xx u u u x x x ε =∂ + ⎡⎢_⎜⎛∂ ⎞_⎟ +⎛_⎜∂ ⎞_⎟ ⎥⎤ ∂ ⎢_⎣⎝ ∂ ⎠ ⎝ ∂ ⎠ ⎥_⎦ (5) 2 2 1 2 y _x y yy u u u y y y ε =∂ + ⎡⎢_⎜⎛∂ ⎞_⎟ +⎛_⎜∂ ⎞_⎟ ⎥⎤ ∂ _⎢⎣⎝ ∂ _{⎠ ⎝} ∂ _{⎠ ⎥⎦} (6) 1 1 2 2 y y y x x x xy u u u u u u y x x y x y ε = _⎜⎛∂ +∂ ⎞_⎟+ _⎢⎡∂ ∂ +∂ ∂ ⎤_⎥ ∂ ∂ ∂ ∂ ∂ ∂ ⎝ ⎠ ⎣ ⎦ (7)

3 Setup of experimental study

3.1 Specimen preparation

A typical weak rock, the Mushan sandstone, which has caused squeezing ground hazards for tunnels in Taiwan, is of primary interest in this study [1,3]. The Mushan sandstone has a porosity of 14.1 %, dry density of 2.28 g/cm3, and saturated water content of 5.92 %. The average uniaxial compressive strength is 26.3 MPa in dry condition and 12.8 MPa in saturated condition. Based on the petrographic analyses, the percentages of grains, matrix, and voids are 67.5%, 18.5% and 14.1%. The average particle diameter is 0.72 mm. Mineralogically, Mushan sandstone consists

of 90.7% of quartz, 9.0% of rock fragments, and thus is classified as lithic greywacke.

In this study, the sandstone specimens were prepared perpendicular to the bedding plane and the specimen size was taken as 40 mm x 20 mm x 10 mm. Furthermore, the surface was dyed with the red color to enhance the contrast between grains and matrix and then polished by emery powder for clear observation through the microscope. Afterward, the specimens were oven-dried (105 ℃) for at least 24 hours to maintain the dry condition.

3.2 Testing procedures

The deformation characteristics of weak sandstones were explored by means of uniaxial compression tests. The experimental setup shown in Figure 2 consists of two parts: (a) loading system and (b) the image observation system of DIC. The axial load was provided by a loading frame with a maximum capacity of 100 kN, and it provides loading rates ranging from 0.0005 mm/min to 1000 mm/min, with a precision of 0.0005 mm/min. The observation system consists of a microscope with light source, a CCD sensor and a data-recording computer. The microscope is mounted onto the clamping device of the specimen, and it allowed observing the surface in real time for different load levels.

Figure 2 Schematic diagram of micro digital image correlation system.

The specimen was compressed at the rate of 0.05 mm/sec. Meanwhile, two cycles of unloading/reloading were conducted at the applied stress of 1 kN and 2 kN respectively for obtaining the elastic deformation, which will be used to decompose the total deformation into elastic and plastic components.

The digital images of specimen were taken as 2048 × 1534 pixels (as shown in Figure 3), and the loading is parallel to the vertical direction (y-direction). The observation area is 2.31 mm × 1.73 mm, actually. Therefore, the resolution is 1.13 mm/ pixel. Before the strain analysis, it is required to determine the size of sub-image (element), which has influence upon the resolution from strain analysis. A higher level of displacement accuracy is often obtained from bigger sub-images, but local strain change cannot be reflected. In this research, the sub-image is sized in 64 pixels. Figure 3 depicts the analysis area and the size of element in this test process. Next, an analytical software developed by our group is used for analyzing the acquired images, while analytical procedure is prepared according to the principle in Section 2.1. According to Sutton et al. [10], the analytical accuracy of this procedure is about 0.01 pixels.

Figure 3 The reference image of weak sandstone used in DIC analysis.

4 Results and discussion

4.1 Variations of strain field

Figure 4 illustrates the full-field strain patterns, including ε_x, ε_y, ε_xy and γ , at different loading phases from analytical results. In the uniaxial test, the axial strain

y

ε exhibits more significant than the lateral strain ε_x and the shear strain ε_xy. The observed characteristics of strain field are described below.

Owing to the complicated constituents of sandstones, non-uniform strain field is clearly observed under loading. At the earlier loading stage (as shown in Figure 4a), apparent compressive strain ε_y occurs around the elements of coordinate (3,10) and (2,7). The compressive strain increases as more loading is applied (as shown in Figure 4b). When the stress approaches to the failure state, a shear band develops

and significant increases of lateral strain ε_x can be observed (as shown in Figure 4c). This tendency of deformation can be also found on the variations of v. Mises strain under different loading stages.

(a) Strain pattern under loading of 1 kN (5.3 MPa)

(b) Strain pattern under loading of 2 kN (10.6 MPa)

(c) Strain pattern at the failure loading of 2.6 kN (13.77 MPa)

Figure 4 (cont.) Full-field strain patterns of weak sandstone under axial loading. Furthermore, Figure 5 illustrates the distributions of plastic strain from the unloading procedure. The plastic strains unloading from 1 and 2 kN are shown in Figure 5a and 5b respectively, and the patterns of the two figures are similar. Apparent plastic strain can be observed around the elements of coordinate (3,10) and (2,7).

(a) Plastic strain pattern unloading from 1 kN (5.3 MPa) Figure 5 Full-field plastic strain patterns of weak sandstone.

(b) Plastic strain pattern unloading from 2 kN (10.6 MPa)

Figure 5 (continued) Full-field plastic strain patterns of weak sandstone.

Figure 6 The distribution of grains in the analysis image.

In order to understand the correlations between the local strain and the locations of grains and matrix, the boundaries of grains are identified and further marked shown in Figure 6. Compared with Figure 4, 5 and 6, the matrix portion, around the elements of coordinate (3,10) and (2,7), exhibits more v. Mises strain at the earlier

loading stage, and most of strain is plastic strain. Therefore, the matrix is relatively softer than the grain, and the deformation of matrix is mostly unrecoverable. However, when fracture develops, it reveals that fracture surface tracks through not only the matrix but also the grains. That is, the fracture surface is a result of coalescence of intra-granular micro-cracks for dry sandstones. This phenomenon concurs with the finding by Lin et al. [11].

4.2 Stress-strain relationship

Following, the variations of axial strain induced by apparent axial stress are presented in Figure 7. These stress-strain curves are obtained from four elements at coordinate (3,4), (3,9), (3,10) and (2,11) respectively, and the first two curves represent the deformations of grains and the others reflect the deformations of matrix under compression. In Figure 7, the tendencies of all strain curves are similar, but the magnitudes are quite different. The matrix portion exhibits more ductile in compressive behavior, especially in the early stage. Furthermore, deducing from the experimental results, the variations of secant modulus of each element under loading are plotted in Figure 8. It reveals that all the moduli increase as axial stress arises. When the stress approaches the ultimate strength, softening occurs and the moduli degrade. In addition, the secant moduli of grain are higher than that of matrix. According to theses deformation trends of the two portions, it is concluded that the matrix portion plays an important role in the significant deformation of the weak sandstones. 0 2 4 6 8 10 12 14 16 0.0 0.5 1.0 1.5 2.0 Axial strain (%) A xia l s tr es s ( M Pa ) grain at (3,4) grain at (3,9) matrix at (3,10) matrix at (2,11)

0.0 0.5 1.0 1.5 2.0 2.5 0 2 4 6 8 10 12 14 16

Axial stress (MPa)

Mo du lu s ( G P a) grain at (3,4) grain at (3,9) matrix at (3,10) matrix at (2,11)

Figure 8 The variations of modulus corresponding to increases of stress.

5 Conclusion

Compression tests with tiny specimen of weak sandstone are conducted in this research. In order to effectively measure the whole strain field in micro-vision, the digital image correlation method is used. According to the analysis results, the deformation of weak sandstone possesses the following characteristics:

(1) Weak sandstone exhibits non-uniform strain pattern under loading, and the matrix portion is relatively softer than the grain.

(2) The plastic deformation mostly accumulates in the matrix portion.

(3) When the stress approaches the ultimate strength, the fracture propagation can be identified through digital image correlation method.

(4) Deducing from stress-strain curves, the modulus of each element can be obtained, and the modulus of grain are higher than that of matrix.

Acknowledgements

The research is supported by the National Science Council of Taiwan, Grant no. NSC 96-2221-E-390-021.

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