Analysis of fatigue damage character of soils under impact load

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1 Article Analysis of fatigue damage character of soils under impact load Journal of Vibration and Control 19(11) ! The Author(s) 2012 Reprints and permissions: sagepub.co.uk/journalspermissions.nav DOI: / jvc.sagepub.com Xinhua Xue 1, Tinghong Ren 2 and Wohua Zhang 3 Abstract Hammers are most typical of impact machines and are widely used in the mechanical manufacture industry. It is important to predict the effects of the damage growth on the fatigue process of the hammer foundation especially when subjected to long-term repeated impact loading. Although there have been many articles about damage model of soils, few of them have studied the problem of soil failure under cyclic loading as the fatigue damage of soils failure. In this paper, a damage fatigue problem of a hammer foundation system considered with fatigue damage growth and vibration on the machine foundation due to the impact of hammer blows is discussed by numerical results using the concept of damage mechanics based on the interaction between hammer foundation and soil ground. It has been found that the influence of damage due to hammer blows on the responses of the ground soil near the foundation is significantly increased with damage development both in the hammer foundation and the surrounding soil. Therefore, in order to protect the environment from damage due to the blows of a hammer, the design for actively vibration absorbing control and damage control in the hammer foundation system needs to be taken into account. Keywords Control, cumulative damage, dynamic response, fatigue damage, impact load Received: 11 April 2012; accepted: 7 May Introduction Fatigue failure is an important phenomenon in many engineering projects. When a structural component is subjected to impact or shock loading, especially when subjected to long-term repeated impact loading, the transient stress waves are generated and propagated. The response due to dynamic loading can cause elevation of stress levels especially in a damaged zone or in local regions surrounding the cracks and defects. In particular, within the damaged zone, the microstructure of damaged material is significantly changed from the undamaged zone due to the onset and growth of damage. The dynamic response of a damaged structural component is considerably different from the corresponding undamaged one due to the change of microstructure in materials, for example, lower frequency, increased damping ratio and amplitude (Zhang and Valliappan, 1998). Due to the structural behavior of soils, many scholars have introduced damage conception into constitutive models of soils. However, literature about the fatigue damage characteristics of soil under cyclic loading is quite rare. So the study of the impact of the fatigue damage process has remarkable theoretical significance and applied value for dealing with engineering problems. Hammers are most typical of impact machines and are widely used in the mechanical manufacture industry. In order to apply the damage mechanics to the problems of machine foundation design, a damage fatigue problem of a hammer foundation system considered with fatigue damage growth and numerical 1 College of Water Resources and Hydropower, Sichuan University, Sichuan, China 2 Department of Water Resources and Electric Power, Guizhou Province, Guiyang, China 3 MOE Key Laboratory of Soft Soils and Geoenvironmental Engineering, Zhejiang University, Hanghou, China Corresponding author: Xinhua Xue, College of Water Resources and Hydropower, Sichuan University, No. 24 South Section 1, Yihuan Road, Chengdu, Sichuan, , China. scuxxh@163.com

2 Xue et al analysis of the influences of damage and vibration on the machine foundation and the ground soil near the foundation block due to impact of hammer blows will be discussed in this paper by numerical results based on the interaction between hammer foundation damage and soil ground damage, and some valuable conclusions are obtained. 2. Fatigue damage evolutional equations under repeated impact load In the case of anisotropy, the fatigue damage equation under repeated impact load can be expressed as Z D i ¼ O i t ði ¼ 1, 2, 3Þ ð1þ t Where O i is the anisotropic principal damage variable along the i th direction under a single impact load, and accumulates constantly with time; D i is the fatigue damage variable along the i th direction. Assuming that the cumulative damage O i generated by a single impact load is the integration of the rate of damage, the fatigue damage variable D i under repeated impact load can be expressed as Z D i N ¼ t Z _Odt ¼ O ði ¼ 1, 2, 3Þ ð2þ t Where N is the number of blow times. Then, the evolutional equation of fatigue damage under repeated impact load can be expressed as D i ¼ Z N D i N 0 N dn ¼ Z N N 0 AdN ði ¼ 1, 2, 3Þ ð3þ Where A ¼ D i =N; N 0 is the number of blow times before damage increases. It should be noted that in the fatigue failure process only the excess stress beyond the fatigue strength in a material can produce fatigue damage. Generally, the fatigue strength ( i ) is considered to be a fraction of fatigue limit ( f ), that is, i ¼ a f ¼ ð0:5 0:7Þ f. Therefore, in this particular case of fatigue damage, damage evolution equations under a single impact load can be written in the form of power rule as O i t ¼ 8 >< >: A i i ðtþ ni a f i ðþa t ð1 O i Þ f 1 O i 0 Otherwise ð4þ Where and n are material constants and they can be obtained by fatigue tests. The fatigue damage evolution equation corresponding to the number of blows n can be expressed as D i ¼ N n 8 H >< >: t do i dt dt j iðþja t ð1 O i Þ f 0 Otherwise The fatigue damage process under repeated impact load can not only be regarded as the consumption of the fatigue life of materials, but also as a macro evolution process of micro cumulative damage developing towards the allowable damage value. This paper supposes that the allowable damage values of soils are those when the amplitude of the foundation block achieves the allowable amplitude, as shown in Table 1 (Xu, 2002). That is, when the calculated displacements of the foundation block exceed the allowable amplitude, we think that the hammer foundation system cannot work normally. Therefore, once the damage of ground soil near the foundation block achieves the maximum allowable value, the number of hammer blows would be regarded as the damage fatigue life of ground soil. 3. Anisotropic dynamic damage finite element method equations The equation of an anisotropic damaged ground under dynamic loading can be written as (Valliappan et al., 1990) ½MŠ U þ ½C ðoðþ t ÞŠ _U þ K ½ ðoðþ t ÞŠfUg ¼ Pt ðþ ð6þ Where ½MŠ ¼ R V e ½NŠ T ½NŠdv is the mass matrix for a damaged element; ½NŠ is the shape function matrix; fug, _U, U are the displacement, velocity and acceleration vector of nodal points, respectively. ½K ðoðþ t ÞŠ ¼ R V e ½BŠ T ½T Š T ½D Š½T Š½BŠdv is the time dependent stiffness matrix for an anisotropically damaged element; ½D Š is the anisotropic R damaged material matrix; Pt ðþ ¼ s 2 ½NŠ T Qt ðþ dsþ R V e ½NŠ T Ft ðþ dv is known as the general nodal force vector due to the pressure of the foundation block; C ðoðþ t Þ is the time-dependent damping matrix for a damaged element in the ground; ½T Š is the coordinate transformation matrix defined in 2-D as 2 3 cos 2 sin 2 sin 2 ½T Š ¼ 6 4 sin 2 cos 2 sin 2 0:5 sin 2 0:5 sin 2 cos 2 ð5þ 7 5 ð7aþ

3 1730 Journal of Vibration and Control 19(11) Table 1. The allowable amplitude of hammer foundation Weight of hammer/ton Category and allowable bearing capacity of foundation soil [R]/(ton/m 2 ) <2 25 >5 Amplitude/mm Amplitude/mm Amplitude/mm Crushed stone soil [R] > 40 <1.4 <1.2 <0.96 Cohesive soil [R] > 25 Crushed stone soil [R] ¼ Sand soil [R] ¼ 3040 Cohesive soil [R] ¼ Crushed stone soil [R] ¼ Sand soil [R] ¼ Cohesive soil [R] ¼ Sand soil [R] ¼ <0.46 <0.4 <0.32 Cohesive soil [R] ¼ 8 3 in which is the two-dimensional anisotropic orientation angle. The general transformation matrix of coordinates in 3D space is ½T 2 l 2 1 m 2 1 n m 1 n 1 n 1 l 1 l 1 m 1 l 2 2 m 2 2 n 2 2 m 2 n 2 n 2 l 2 l 2 m 2 l 2 3 m 2 3 n 2 3 m 3 n 3 n 3 l 3 l 3 m 3 Š¼ 2l 2 l 3 2m 2 m 3 2n 2 n 3 m 2 n 3 þm 3 n 2 n 2 l 3 þn 3 l 2 l 2 m 3 þl 3 m l 3 l 1 2m 3 m 1 2n 3 n 1 m 3 n 1 þm 1 n 3 n 3 l 1 þn 1 l 3 l 3 m 1 þl 1 m l 1 l 2 2m 1 m 2 2n 1 n 2 m 1 n 2 þm 2 n 1 n 1 l 2 þn 2 l 1 l 1 m 2 þl 2 m 1 ð7bþ Where fl i, m i, n i g T are direction cosines of the normal unit vectors. Since the microstructure within a material has changed due to damage (Vestroni and Capecchi, 1996; Wang et al., 2006), the material constants and the internal energy dissipation (internal damping) also change (Fahrenthold, 1991; Zhang et al., 2000). Therefore, the stiffness matrix and the damping matrix of a damaged element must be considered as a function of the damage variable fog that varies with time. Strictly speaking, the mass density will also change due to the damage. However, from the point of view of mass conservation, the global mass is unchanged, and a mass matrix independent of the damage has to be assumed. On the other hand, damage causes stiffness degradation, and the frequency spectrum of the structure is downshifted significantly. Hence, the damage has inevitable influences on the internal damping. So far, the influence of damage on material damping has not been the subject of any major experimental or analytical investigation. In order to discuss this problem from the point of view of numerical analysis, it is convenient to assume a Rayleigh-type damping and the equivalent viscous damping. For the Rayleigh damping matrix, one can adopt the usual formulation: ½C ðoðtþþš ¼ ½MŠþ ½K ðoðtþþš ð8þ For the equivalent viscous damping matrix, we have Z ½C ðoðþ t ÞŠ ¼ r ðþn t ½ Š T ½NŠdv ð9þ V e Where, are the Rayleigh damping parameters of the damaged material; r ðþ t is the time-dependent equivalent viscous damping coefficient. For the Rayleigh damping defined in equation (8), the damaged damping ratio corresponding to the i th order vibration mode of a damaged structure can be written in a similar manner to that for the undamaged case as i ¼ 1 2! i þ! i ð10þ Where! i is the i th circular frequency of a damaged structure. The contribution of higher order modes to the dynamic response of a structure is less significant than the contribution of the first and second modes. Hence, the dynamic response can be approximated using only the first-order and second-order damping ratios. In the case of isotropic damage, a simple relationship can be found from equation (10) and the

4 Xue et al relation! i ¼ ð1 OÞ! i if the Rayleigh damping parameters and are assumed to be constant. i ¼ 1 þ ð1 OÞ! i ð11aþ 2 ð1 OÞ! i i ¼ 1 þ! i ð11bþ 2! i The Rayleigh damping parameters and can be approximately measured by a specified frequency of a mode such as by the first or the second frequency as ( 1! 1, 1 =! 1 ) (Zhang, 1992). The ratio of = can be estimated more accurately by the geometry mean from both modes and formulated independently to the damping ratio as = 1=! 1! 2. A simple relationship of damaged and undamaged damping ratio ¼ has been found as a function of the damage variable O and the ratio of the undamaged natural frequency! 1 =! 2 1 ¼ ¼ 1 O þ ð1 OÞ! 1! 2 1 þ! ð12þ 1! 2 Since the ratio of the natural frequency! 1 =! 2 is known when the geometrical and physical parameters of a structure are given, equation (12) can be used to describe the influence of damage on the material damping of damaged structures. Therefore, equation (12) combined with equation (10) can be used to evaluate the damping ratio of damaged materials. Thus, the damaged Rayleigh damping parameters and can be approximately determined using obtained damaged frequencies! 1 =! 2 ¼ 2! 1! 2! 1 þ,! 2 2 ¼! 1 þ! 2 ð13þ Introducing the equivalent viscous damping for damaged and undamaged materials by (Zhang and Valliappan, 1998) ¼ 2 m!, ¼ 2 m! ð14þ Where m is equivalent mass. The ratio of damaged to undamaged equivalent viscous damping can be written as ¼ ¼ ð1 OÞ ¼ 4. Numerical simulations 1 þ ð1 OÞ 2! 1! 2 1 þ! ð15þ 1! 2 The hammer foundation system consists of a frame, a falling weight known as tup, the anvil and the foundation block. To reduce the transmission of impact stresses in the concrete block and the frame, an elastic pad consisting of rubber, felt, cork or timber is generally provided between the anvil and the foundation block.the foundation block is mostly designed to rest directly on soil. In the design procedure of hammer foundation, the dynamic response is computed by Tup χ 2 Spring of elastic pad K2 Damping in absorber C2 Anvil χ 1 χ 2 m 2 C2 K2 m 1 χ 1 Foundation block Foundation soil C1 K1 Damping in Soil C 1 Soil Spring K1 Figure 1. Typical arrangement of a hammer foundation resting on ground soil and analyzed mechanical model.

5 1732 Journal of Vibration and Control 19(11) modelling these as a lumped-mass-dashpot system based on assumptions given in Meirovitch (1985). This simplified system is the two degree-of-freedom vibration model where the anvil rests on an elastic pad and the foundation block rests directly on the soil, and the weight of the hammer is 2 tons, as shown in Figure 1. To simplify the three dimension problems, the calculation scope of the foundation soil is regarded as a limited range of media in this paper, and the top surface of the foundation block is supposed to be on the same level as the surface of the foundation soil, and a quarter of its volume was calculated due to the symmetry, as shown in Figure 2. Here, the size of foundation block is m 3, the size of foundation soil is m 3. In principle, all boundaries must have one boundary in each direction. In this paper, the bottom boundary of computational model is fixed in the vertical direction; the boundaries of surface a and b are restrained in the horizontal direction. The whole computational model consists of 2904 tetrahedron elements. The parameters used in the model are as follows: 1) Concrete foundation block: E 1 ¼ GPa, E 2 ¼ GPa, E 3 ¼ GPa, v ij ¼ 0.20(i, j ¼ 1 3), ¼ 2700 kg/m 3, A ¼ 2.10e 15, n ¼ 1.04; 2) First layer soil: E 1 ¼ MPa, E 2 ¼ MPa, E 3 ¼ MPa, v ij ¼ 0.42 (i, j ¼ 1 3), ¼ 2000 kg/m 3, A ¼ 3.10e 12, n ¼ 1.05; 3) Second layer soil: E 1 ¼ MPa, E 2 ¼ MPa, E 3 ¼ MPa, v ij ¼ 0.42(i, j ¼ 13), ¼ 2000 kg/m 3, A ¼ 3.10e 13, n ¼ Damage response in the ground soil due to the first hammer blows Response of foundation soil in vertical direction. We assume that the initial damage values and the initial displacements along three principal directions equal to 1.0e-9 and zero, respectively. Figure 3 shows the contour map of damage response in vertical direction at t ¼ 0.108s due to the first hammer blows Damage strain energy release rate. Figure 4 shows the contour map of damage strain energy release rate at t ¼ 0.108s due to the first hammer blows. It can be seen that the maximum value of damage strain energy release rate occurs in the interior of foundation block Response comparison of different positions in foundation block and ground soil due to the first hammer blows Response of different horizontal distances away from the center of foundation block. Figure 5 shows the response of different horizontal distances away from the center of the foundation block due to the first hammer blows. It can be seen that the farther away from the center of the foundation block, the smaller the value of the vibration amplitude. Figure 2. Finite element model of hammer foundation and foundation soil.

6 Xue et al Figure 3. Contour map of damage response in vertical direction at t ¼ 0.108s. Figure 4. Contour map of damage strain energy release rate at t ¼ 0.108s.

7 1734 Journal of Vibration and Control 19(11) Figure 5. Response of different horizontal distances away from the center of foundation block due to the first hammer blows Response of different vertical distances away from the center of the foundation block. Figure 6 shows the response curves of different vertical distances away from the center of the foundation block due to the first hammer blows. It can be seen that the farther away from the center of the foundation block, the smaller the value of the vibration amplitude Vertical displacements of foundation block due to the first hammer blows. The vertical displacements of foundation block due to the first hammer blows were calculated by the fatigue damage model presented in this paper, and are compared with the analytic results (Prakash and Puri, 1988), as shown in Figure 7. It can be seen that the vertical displacement results simulated in this paper are in good agreement with the analytic results Decrement of vibration in ground soil Figure 8 shows the relationship between the amplitude, the depth away from the center of the foundation block and the number of blows. It can be seen that the amplitude of soils increases with the increase of blows, but it reduces with the increase of depth away from the center of the foundation block. However, the influence of blows to the amplitude of soils is not significant in a depth of more than 10 m conditions. Figure 9 shows the relationship among the amplitude, distances away from the center of the foundation block and the number of blows. It can be seen that the amplitude of soils increases with the increase in blows, but it reduces with the increase in the distances away from the center of the foundation block. However, the influence of blows to the amplitude of soils is not significant in a distance of more than 15 m conditions Curves of impact fatigue damage life It is necessary to analyze these fatigue damage problems using various forms of fatigue life curves. In this paper, the following formula is adopted to process the D-N curves: D ¼ a n 1 ð1 D 0 Þ 1 N m ð16þ N f Where a n is a constant corresponding with the damage; D 0 is the initial damage value; N f is the fatigue life; N is

8 Xue et al Figure 6. Response of different vertical distances away from the center of foundation block due to the first hammer blows. the number of blows; m is a positive constant which is used to describe the cumulative degree of fatigue damage. The parameters used in the model are as follows: a n ¼ 0.47, D 0 ¼ 1.0e 9, N f ¼ 10.8e7. Then the D-N curves can be fitted as! N 0:61 D ¼ 0:47 1 0: ð17þ Figure 7. Vertical displacements of foundation block due to the first hammer blows. Figure 10 shows the D-N curves of the calculated values compared with the fitted values. It can be seen

9 1736 Journal of Vibration and Control 19(11) Amplitude A(m) Depth away from the center of foundation block H(m) Number of blows Figure 8. Relationship among amplitude, depth away from the center of foundation block and number of blows Amplitude A(m) Distances away from the center of foundation block L(m) Number of blows 12 Figure 9. Relationship among amplitude, distances away from the center of foundation block and number of blows.

10 Xue et al numerical investigation of the dynamic properties of damage in the soil ground, it can be seen that the influences of hammer blows on both surface and depth of the soil near the foundation are significant when damage increases. This provides the possibility to work out a method for controlling the damage and its growth in a damaged material, as well as the dynamic response of a damaged structure. Funding This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors. Figure 10. The D-N curves. that the formula of the D-N curves proposed above is reasonable. 5. Conclusions In this paper, a damage fatigue problem of a hammer foundation system considered with fatigue damage growth and numerical analysis of the influences of damage and vibration on the machine foundation and the ground soil near the foundation block due to the impact of hammer blows are discussed by numerical results using the concept of damage mechanics based on the interaction between hammer foundation damage and soil ground damage. From analysis of the simulated results the conclusions can be obtained that when a machine foundation is subjected to strong dynamic loading, the dynamic response increases significantly with the degree of damage and this in turn influences the damage propagation both in the foundation and the soil due to the higher stresses concentrating near the foundation areas. Furthermore, the natural frequencies of the hammer foundation system are reduced significantly with the damage growth and as the damping ratio increases significantly. From the References Fahrenthold EP (1991) A continuum damage model for fracture of brittle solids under dynamic loading. Journal of Applied Mechanics, ASME 58: Meirovitch L (1985) Elements of Vibration Analysis. New York: McGraw-Hill. Prakash S and Puri VK (1988) Foundations for Machine: Analysis and Design. New York: Wiley. Valliappan S, Zhang WH and Murti V (1990) Finite element analysis of anisotropic damage mechanics problems. Engineering Fracture Mechanics 35: Vestroni F and Capecchi D (1996) Damage evaluation in vibrating beams using experimental frequencies and finite element models. Journal of Vibration and Control 2: Wang SS, Ren QW and Qiao PZ (2006) Structural damage detection using local damage factor. Journal of Vibration and Control 12: Xu J (2002) Construction Vibration Engineering Handbook. Beijing: China Architecture & Building Press. Zhang WH (1992) Numerical analysis of continuum damage mechanics. Ph.D. thesis, University of New South Wales, Australia. Zhang WH and Valliappan S (1998) Continuum damage mechanics theory and application Part I: theory; Part II application. International Journal of Damage Mechanics 7: , Zhang WH, Chen YM and Jin Y (2000) A study of dynamic responses of incorporating damage materials and structure. Structural Engineering and Mechanics, An International Journal 12(2):