Mechanical properties assessment of specimens subjected to random non-uniform general corrosion and tensile load

Maritime Engineering and Technology – Guedes Soares et al. © 2012 Taylor & Francis Group, London, ISBN 978-0-415-62146-5 Mechanical properties assessment of specimens subjected to random non-uniform general corrosion and tensile load B.Q. Chen, Y. Garbatov & C. Guedes Soares Centre for Marine Technology and Engineering (CENTEC), Instituto Superior Técnico, Technical University of Lisbon, Portugal ABSTRACT: The objective of this work is to analyze the strength and deformability of steel specimens subjected to random non-uniform general corrosion and tensile load. The real shape of corrosion surface has been generated by a probabilistic model of corrosion deterioration. Non-linear, large deformation and threedimensional finite element analyses have been employed to analyze the mechanical properties of deteriorated specimens. The strength of corroded specimens is estimated as a function of the average corrosion depth and the minimum cross sectional area. The deformability and energy absorption of the corroded plates are estimated as a function of the surface roughness represented by the difference of averaged plate thickness and that at the section of minimum cross sectional area. 1 INTRODUCTION Steel structures such as ships, offshore platforms and land-based structures operate in a very complex sea water environment. The corrosion in steel structures is often very severe, not only under sustained immersed condition in ballast tanks, but also under general exposure to atmospheric conditions. To assess the structural performance of steel structures such as aged ships, it is important to analyze the residual strength and energy absorption of corroded plates. The corrosion attacking on metals could be divided into general corrosion, galvanic cells, under-deposit corrosion, CO2 corrosion, top-of-line corrosion, weld attack, erosion corrosion, corrosion fatigue, pitting corrosion, microbiological corrosion and stress corrosion cracking.Among these various types of corrosion, general wastage that results in a generalized decrease of plate thickness and pitting which consists of much localized corrosion with very deep holes appearing in the plate are two main corrosion mechanisms generally presented in the steel plates. Most of the work referred in this paper relate to the former, general corrosion. Nowadays, numerical simulation is being used to replacing time-consuming and expensive experimental work. A great number of researchers have studied structural performance of steel structures subjected to corrosions. Paik et al. (2003) investigated the ultimate strength characteristics of steel plate elements with pit corrosion wastage and under axial compressive load. While the similar case but under in-plane shear loads had been investigated as well. Garbatov and Guedes Soares (2008) analyzed the non-linear corrosion wastage of main deck plate of tanker ships and bulk carriers based on the model proposed by Guedes Soares and Garbatov (1999). The corrosion wastage of deck plates was examined with respect to how the corrosion wastage varies in time as a result of generalized corrosion. Nakai et al. (2004a, 2004b, 2006) analyzed circular cone-shaped pits in the hold frames of bulk carriers, and ellipsoidal-shaped pits in the bottom shell plates of a tanker. They investigated actual pitting corrosion observed on hold frames of bulk carriers in different studies. A total of 256 nonlinear FEA which was the full combination of two cases of transverse pitting locations, five cases of plate thicknesses, four cases of pitting breadths, four cases of pitting lengths and two cases of pitting depths had been carried out by Ok et al. (2007). They had proved that the length, breadth and depth of pit corrosion have weakening effects on the ultimate strength of the plates while plate slenderness has only marginal effect on strength reduction. Empirical formulae to predict strength reduction due to pitting corrosion had been derived by applying the multi-variable regression method. Jiang and Guedes Soares (2010) studied the influence of scattered pitted plates on the collapse strength by using the mathematical model proposed by Daidola et al. (1997), who developed a method to estimate the residual thickness of pitted plates. Silva et al. (2011) studied the effect of non-linear randomly distributed non-uniform corrosion on the ultimate strength of unstiffened rectangular plate subjected to axial compressive loading has been studied. A series of 570 plate surface geometries are generated by Monte Carlo simulation for different degree of corrosion, location and ages and nonlinear finite element analyses were carried out, using a commercial finite element code and based on a regression 293 analysis, empirical formulae to predict strength reduction because of corrosion have been derived demonstrating a good accuracy. The objective of the present work is to apply a complete true stress–strain relationship to an elastoplastic material model of LS-DYNA (Livermore Software Technology, Livermore, CA, USA) to assess the strength and deformability of steel specimen subjected to random non-uniform general corrosion and tensile load. 2 2.1 SIMULATION OF CORRODED SURFACES Non-linear corrosion wastage model Guedes Soares and Garbatov (1999) proposed a model that describes the growth of corrosion wastage by a non-linear function of time in three phases. In the first phase, it is assumed that there is no corrosion because a corrosion protection system is effective. Failure of the protection system will occur at a random point of time and the corrosion wastage will start a non-linear process of growth with time, which levels off asymptotically at a long-term value of corrosion wastage. Since then, several authors have proposed some variants of this model or have compared it with other modifications. The time dependent model of corrosion degradation is illustrated in Figure 1, separated into four phases, where in the first one there is no corrosion (O′ O). The second phase corresponds to the initiation of failure of the corrosion protection system, which leads to corrosion with the decrease of thickness of the plate (OB, fast growing corrosion). The third phase BC corresponds slowly growing corrosion and the last one (t > C) corresponds to a stop in the corrosion process when the corrosion rate becomes zero. The model is based on the solution of a differential equation of the corrosion wastage: Figure 1. Thickness of corrosion wastage as a function of time. where d∞ is the long-term corrosion wastage, d(t) is the corrosion wastage at time t, τc is the time without corrosion which corresponds to the start of failure of the corrosion protection coating (when there is one), and τt is the transition time duration, which may be calculated as d∞ /tgα, where α is the angle defined by OA and OB. 2.2 Non-uniform general corroded surface model Based on 1168 measurements of deck plates from ballast tanks with original nominal thicknesses varying from 13.5 to 35 mm on ships with lengths between perpendiculars in the range of 163.5 to 401 m, Garbatov et al. (2007) determined the parameters of the mean value of the corrosion depth as a function of time under the assumption that it is approximated by the exponential function given in Eqn. (1), where the long-term corrosion wastage is equal to 1.85 mm, the time without corrosion is equal to 10.54 years and the transition period is equal to 11.14 years. Another important statistical descriptor of the data set was the standard deviation for each yearly subset of data. The standard deviation as a function of time was fitted to a logarithmic function: where the coefficients a1 and b1 are defined as 0.384 and 0.710 respectively, for the deck plates of ballast tanks. Several distributions were evaluated by Garbatov et al. (2007) and it was concluded that corrosion wastage depth is the best fit by the Log-normal distribution, which was as well adopted in the present work. Based on their work, a reformative non-uniform general corroded surface model is proposed here. Figure 2c displays the non-uniform general corroded surfaces models of 8 mm thickness tensile specimens with three different levels of corrosion defined by controlling the minimum plate thickness. The corroded plate surface is modelled as random plate thickness that results in the random vertical position of the coordinates of corroded surface for equally spaced reference points positioned along the x and y direction of the plate. These reference points are defined in a Monte Carlo simulation as being the nodes of the finite element mesh on the plate. The plate thickness, Zijcorroded , at any reference point with coordinates x, y for the corroded plate surface, is defined by the random thickness of the intact plate surface, Zijintact affected by the random vertical reduction resulting from the corrosion depth, corrosion depth Zij as defined by Silva et al. (2011): where Z are the matrixes of the corroded and intact surface and corrosion depth. This convention is used to derive the formulation that describes the vertical position of the surface of the 294 Figure 3. Geometry of tensile specimen. Figure 2a. Corroded surfaces, Case 1. Figure 2b. Corroded surfaces, Case 2. Figure 4a. True stress – true strain relation (Ahmmad and Sumi, 2010), full range curve. Figure 2c. Corroded surfaces, Case 3. non-linear corroded plate in the Monte Carlo simulation resulting in randomly distributed plate thicknesses for randomly defined reference nodes at a specific year based on Eqn. (3) applying the corrosion degradation levels as defined by Eqn. (1) and (2). The vertical random coordinates (corrosion depth) of the corroded and intact plate surfaces and corrosion depths are modelled by a log-normal distribution. The intact plate surface coordinates and corresponding corrosion depths are considered as not correlated. The average thicknesses of plates subjected to these levels of corrosion titled as case 1, case 2 and case 3, are 7.380 mm, 6.554 mm and 5.852 mm, respectively. 3 GEOMETRIC AND MATERIAL PROPERTIES OF STEEL SPECIMEN 3.1 Tensile specimen A typical geometry of the specimen of tensile test, which has enlarged ends or shoulders for gripping, is shown in Figure 3. The cross-sectional area of the gage section that is the important part of the specimen is reduced relative to that of the remainder of the specimen so that deformation and failure will be localized in this region. The gauge length is the region over Figure 4b. True stress – true strain relation (Ahmmad and Sumi, 2010), detailed curve of 0 – 0.1 true strain. which measurements are made and is centered within the reduced section. Ahmmad and Sumi (2010) estimated the strength and deformability of steel plates with various pit sizes, degrees of pitting intensity, and general corrosion both experimentally and numerically. The vision-sensor technology was used in their study to obtain the relationship between the true stress and the true strain beyond the onset of localized necking. The true stress– strain relationship, including the material response in both pre- and post-plastic localization phases, is necessary as input for numerical analyses (see Figure 4). The mechanical properties of the material used in their work are listed in Table 1. 3.2 Stress-strain relationship In the conventional engineering tension test, the engineering measures of stress and strain, denoted in this module as σe and εe respectively, are determined from 295 Table 1. Material properties. Setting Value Unit Material Yield strength Mass density Young’s modulus Tensile strength SM490A 365.0 7853.6 206.9 509.4 – MPa kg/m3 GPa MPa Figure 5. Finite element model. the measured the load and deflection using the original specimen cross-sectional area A0 and length L0 as: where δ is the controlled displacement applied in one end of the specimen in a tensile test, while P is the load corresponding to the displacement. When the stress (σe ) is plotted against the strain (εe ), an engineering stress-strain curve is obtained. However, the engineering stress-strain curve must be interpreted with caution beyond the elastic limit, since the specimen dimensions experience substantial change from their original values during the test. If the true stress, based on the actual cross-sectional area of the specimen (A), is used, the stress-strain curve increases continuously to fracture. If the strain measurement is also based on instantaneous measurement, the curve that is obtained such as that shown in Figure 4 is known as true stress-strain curve. Any point on the true stress-strain curve can be considered the yield stress for a metal strained in tension by the amount shown on the curve. Thus, if the load is removed at this point and then reapplied, the material will behave elastically throughout the entire range of reloading. The true stress σt is expressed in terms of engineering stress σe and engineering strain εe by: While the true strain εt , is determined from the engineering strain εe by: 4 4.1 STRESS-STARAIN ANALYSIS Finite element model Numerical analyses were carried out by using a nonlinear implicit finite element code, LS-DYNA, applying a quasi-static type of load. The constitutive material model is an elastoplastic material where an arbitrary stress versus strain curve can be defined. This material model is based on the J2 flow theory with isotropic hardening. The geometric model of the tensile specimen has been shown in Figure 3, in which the total and gauge length are 308 and 140 mm respectively, the width is 50 mm and the reduced width in the gauge section is 40 mm. The thickness of the intact specimen is 8 mm, while three different levels of corrosion are used using the reformative non-uniform general corroded surface model. Figure 5 demonstrates the finite element model of the tensile specimen. Four different element sizes, 10 mm, 5 mm, 2.5 mm and 1.25 mm were used in different regions. In the left side of specimen, the degrees of freedom of all nodes are constrained in all directions, while a constant displacement of 3 mm/min was applied in all nodes of the right side, in the loading direction. Fracture is introduced by allowing elimination of elements when strain to failure is achieved. The material selected from the library of LS-DYNA is ‘Mat.024 – Piecewise linear plasticity’, which allows the definition of a true stress–strain curve as an offset table. The true stress-strain curve used in the study of Ahmmad and Sumi (2010) (see Figure) are used as an input for the calculations. 4.2 Effect of failure criterion The failure criterion used for the numerical simulation of the tensile test is an important issue. In the present analyses, the failure criterion is defined as the plastic strain tends to fail. The element where necking occurs is deleted from the model when the plastic strain reaches the setting value predefined as a failure criterion. Normally, the failure criterion in terms of the plastic strain behavior in an element can be determined in such a way that the total elongation in FE analysis reaches the experimental elongation. For the sake of studying the effect of the strain to fail on the tensile properties, three different failure criteria for strains, 0.25, 0.55 and 0.85, are applied for the Case 1 in the numerical simulations. Three resultant curves shown in Figure 6 indicate that the initial linear portions of the curves are exactly identical, which means that the strain to fail has no effect on the modulus of elasticity which can be calculated as the slope of the initial linear portion of the curve. However, it is found from this figure that the failure strain has a significant effect on the elongation. With the increase of strain to failure from 0.25 to 0.55, the elongation increases up to twice, and it increases to more than 15% when the strain to fail reaches the value 0.85. There is a small increase in elongation from 296 Table 2. Tensile strength as a function of corrosion level. Case Intact 1 2 3 Corrosion rate (%) Tensile strength (MPa) Strength loss (%) 0.0 548.3 0.0 7.8 504.1 8.1 18.1 444.3 19.0 26.9 438.6 20.0 Figure 6. Stress elongation curve for different failure criteria. Figure 8. Liner part of the nominal stress – nominal strain curve as a function corrosion levels. Figure 7. Nominal stress versus percent elongations curve. strain to fail of 0.85 to that of 0.932. However, for the present study, the strain failure criterion assumed is 0.932. 4.3 Effect of corrosion level Figure 7 shows the nominal stress versus percent elongations curves, of the non-corroded specimen and three different cases of different corrosion levels defined in Section 2.2, in which the strain to fail value is assumed as 0.932. The Case 1, Case 2 and Case 3 refer to the specimens subjected to three different levels of general non-uniformly distributed corrosion with the average plate thickness of 7.380 mm (7.8%), 6.554 (18.1%) mm and 5.852 (26.9%) mm, respectively. Tensile strength is one of the most important parameters that describes the mechanical properties of a steel plate under tensile load. As can be seen from Figure 7, the intact specimen has the highest tensile strength, while the strength decreases with the increase of the corrosion rate. Table 2 lists the tensile strength of each case, in which 8.1% of the strength is lost when the intact specimen suffers to 7.8% general corrosion, after that the strength loss still increases to 19% as the corrosion rate reaches 18.1%. From Case 2 to Case 3, the changing rate of the strength loss becomes lower, from 19.0% in the case of the 18.1% general corrosion to 20.0% in that of 26.9% general corrosion. And it implies that the severer corrosion results in less elongation. Figure 9. The average thicknesses of all cross sections along specimen length. Another important mechanical property of a steel plate is the modulus of elasticity. Figure 8 shows the initial liner portion of the nominal stress – nominal strain curve with various corrosion levels. According to Hooke’s Law, the slope of the stress – strain curve stands for the modulus of elasticity which is also known as the Young’s modulus. It is evident that the plate modulus of elasticity is decreased as the corrosion deterioration increases. 4.4 Effect of minimum cross section Since the corroded surfaces are defined from random variants, it is also interesting to pay special attention on the cross section, where the minimum thickness is located. Figure 9 shows the average thickness of each cross section along the specimen length in all the three cases. The locations, where the failures occurred are indicated. The marked failures occurring locations show a small correlation with the locations of minimum thicknesses of the cross sections, given that they are all 297 identical to one of the minimum section average thickness, but not the least value. It may be concluded that the determination of the failures occurring locations is complicated and depends on various factors including the minimum section average thickness. From the FE analysis it was observed that the failure criteria affect the locations where the failures occur as well. 4.5 Stress-strain analysis The true stress-strain curve of many metals in the region of uniform plastic deformation can be expressed by the simple power law (Ludwik, 1909): Figure 10. Log–log plot of the stress–strain curve. where n is the strain hardening exponent, K is the strength coefficient, and εp is the plastic strain. A log-log plot of the true stress-strain curve from yield point up to the maximum load will result in a straight line if Eqn. (7) is fitted to the observed/ experimental data. Another common variation on the simple power law is the Ramberg-Osgood equation (Ramberg and Osgood, 1943): Table 3. Case n value K value Intact Case 1 Case 2 Case 3 0.251 0.249 0.250 0.251 1054 MPa 971 MPa 859 MPa 843 MPa Table 4. levels. where α and m are dimensionless constants, E is the Young’s modulus and σR is a reference stress. If m is very large, then εp remains small until σ approaches σR , and increases rapidly when σ exceeds σR , so that σR may be regarded as an approximate yield stress. In the limit as m becomes infinite, the plastic strain is zero when σ < σR , and is indeterminate when σ = σR , while σ > σR would produce an infinite plastic strain and is therefore impossible. This limiting case accordingly describes a perfectly plastic solid with yield stress σR . If the deformation is sufficiently large for the elastic strain to be neglected, then Ramberg-Osgood equation can be solved for σ in terms of ε: which is equivalent to the power law, Eqn. (8) if
 m1 = K. consider m1 = n and σR ασER Figure 10 shows the log-log plot of the stress-strain relationships in the present study. All curves of the four cases with different corrosion deteriorations are expressed by a linear regression y = ax + b in which a is equal to the strain hardening exponent (n), and b stands for the logarithmic form of the strength coefficient K. All the values of n and K are listed in Table 3. It can be observed that the strains hardening exponent are around 0.250 in all corrosion rates. While the K value decreases with the increase of the corrosion deterioration. n and K values, with various corrosion levels. Resilience and toughness, with various corrosion Case Unit Ur Ut Intact Case 1 Case 2 Case 3 MPa MPa MPa MPa 0.459 0.380 0.344 0.367 95.9 74.8 62.3 56.5 4.6 Energy absorption analysis Moreover, two parameters with respect to the energy absorption are also analyzed here. One is the modulus of resilience (Ur ) which means the capacity to absorb energy when deformed elastically and recover all energy when unloaded. It can be calculated as the area under the elastic portion of the stress-strain curve. The other is modulus of toughness Ut that stands for the energy to break a unit volume of material, or absorb energy to fracture. It can be calculated as the area under the entire stress-strain curve. From Figure 7 and Figure 8, the resilience and toughness can be easily calculated by integration. The results are shown in Table 4, where one can observe that the modulus of resilience is decreasing with the increase of the corrosion deterioration, except for Case 3. In Case 3, the resilience is even bigger than the one of Case 2, though the Young’s modulus that is the slope of the initial linear portion of the curve in Case 3 is less than that of Case 2. It is because in Case 2 the elongation of the linear portion of the stress-strain curve is bigger. Figure 11 shows the capacity of energy absorption with respect to the general non-uniformly distributed 298 In different cases with various level of corrosion deterioration, the strains hardening exponent n are almost the same. While the strength coefficient K decreases with the increase of the corrosion rate. The capacity of energy absorption decreases from 95.9 MPa to 74.8 MPa as the intact specimen suffers 7.8% general corrosion. After that the energy absorption still decreases with the increase of the corrosion rate, but with a lower rate. REFERENCES Figure 11. Energy absorption with respect to corrosion deterioration. corrosion deterioration. It is concluded that the capacity of the energy absorption decreases from 95.9 MPa to 74.8 MPa as the intact specimen suffers to 7.8% general corrosion. After that the energy absorption still decreases with the increase of the corrosion rate, but with a lower rate. 5 CONCLUSIONS This work presented an analysis of the strength and deformability of steel specimen subjected to random non-uniform general corrosion and tensile load. Based on the results obtained in the present work, several conclusions can be drawn. The strain to failure has no effect on the modulus of elasticity which can be calculated as the slope of the initial linear portion of the stress-strain curve. However, it has a significant effect on the elongation. With the increase of strain to failure from 0.25 to 0.55, the elongation increases to twice as the previous value. And it increases to more than 15% when the strain to fail reaches the value 0.85. There is a small increase in elongation from strain to fail of 0.85 to that of 0.932. The tensile strength decreases with the increase of the corrosion deterioration. 8.1% of the strength is lost when the specimen is subjected to 7.8% general non-uniform corrosion, after that the strength loss still decreases with the increase of the corrosion deterioration, but with a lower changing rate, that it varies from 19.0% to 20.0% as the corrosion changes from 18.1% to 26.9%. The severer corrosion results in less elongation. The modulus of elasticity and the energy absorption capacity of a steel plate is decreased as corrosion deterioration increases. The failures occurring locations show little correlation with the locations of minimum thicknesses of the cross sections. It may be concluded that the failure occurring locations is complicated and depends on various factors including the minimum section average thickness as well as the failure criteria. Ahmmad, M. 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