Discrete Crack Modelling in Concrete Structures. Thesis, University of Colorado, Boulder.
- A unified theory of elastic degradation and damage based on a loading surface.
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- Constitutive model for bimodular elastic damage of concrete.
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Elsevier, Amsterdam. Fracture energy based bi-dissipative damage model for concrete. Yielding of steel sheets containing slits. Solids, 8: Matrix computations, third edition page D, Kay, G. Anisotropic modeling and numerical simulation of brittle damage in concrete.. Engng, General theory of uniqueness and stability of elasto-plastic solids. Solids, 6: — Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements.
Cement Concrete Res. Tensor algebra and tensor analysis for engineers: With applications to continuum mechanics.
Springer, Berlin. Embedded crack model. I: Basic formulation. Methods Engrg. Failure analysis of elasto-plastic material models on different levels of observation. Behavior of concrete under biaxial stresses. ACI J. Biaxial behavior of plain concrete of nuclear containment building. Experimental research for strength and deformation of concrete under biaxial tension-compression loading in Chinese. Journal of Hydraulic Engineering, 8: Introduction to the mechanics of a continuous medium.
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Prentice-Hall International, New Jersey. An anisotropic elastoplastic damage model for plain concrete. On advanced solution strategies to overcome locking effects in strong discontinuity approaches. Energy-based simulation of concrete cracking using an improved mixed-mode cohesive crack model within a meshless discretization. Finite element analysis of reinforced concrete beams, ACI J. Mixed-mode fracture and anchor bolts in concrete analysis with inner softening bands.
Multple hardening plasticity for concrete material. A consistent characteristic length for smeared cracking models. Modeling strong discontinuities in solid mechanics via strain softening contitutive equations. On the use of strain-softening models for the simulation of strong discontinuities in solids. Material instabilities in solids: , Wiley.
Cervera, M. Strong discontinuities and continuum plasticity models: the strong discontinuity approach. On the discrete constitutive models induced by strong discontinuity kinematics and continuum constitutive equations. From continuum mechanics to fracture mechanics: the strong discontinuity approach. When the rate of growth becomes large enough, fatigue striations can be seen on the fracture surface. Striations mark the position of the crack tip and the width of each striation represents the growth from one loading cycle.
Striations are a result of plasticity at the crack tip. When the stress intensity exceeds a critical value known as the fracture toughness, unsustainable fracture will occur, usually by a process of micro-void coalescence. Prior to final fracture, the fracture surface may contain a mixture of fatigue and fast fracture.
The American Society for Testing and Materials defines fatigue life , N f , as the number of stress cycles of a specified character that a specimen sustains before failure of a specified nature occurs.
Engineers have used a number of methods to determine the fatigue life of a material: . Historically, fatigue has been separated in two regions of high cycle fatigue that require more than 10 4 cycles to failure where stress is low and primarily elastic and low cycle fatigue where there is significant plasticity. Experiments have shown that low cycle fatigue is also crack growth. A mechanical part is often exposed to a complex, often random , sequence of loads, large and small.
Since S-N curves are typically generated for uniaxial loading and therefore some equivalence rule is needed whenever the loading is multiaxial. For simple, proportional loading histories lateral load in a constant ratio with the axial , Sines rule may be applied. For more complex situations, such as non-proportional loading, critical plane analysis must be applied. In , M. Miner popularised a rule that had first been proposed by A. Palmgren in Usually, for design purposes, C is assumed to be 1. This can be thought of as assessing what proportion of life is consumed by a linear combination of stress reversals at varying magnitudes.
Although Miner's rule may be a useful approximation in many circumstances, it has several major limitations:. This is a graph of the magnitude of a cyclic stress S against the logarithmic scale of cycles to failure N. S-N curves are derived from tests on samples of the material to be characterized often called coupons where a regular sinusoidal stress is applied by a testing machine which also counts the number of cycles to failure.
This process is sometimes known as coupon testing. For greater accuracy but lower generality component testing is used. Analysis of fatigue data requires techniques from statistics , especially survival analysis and linear regression. The progression of the S-N curve can be influenced by many factors such as stress ratio mean stress , loading frequency, temperature , corrosion , residual stresses, and the presence of notches.
A constant fatigue life CFL diagram  is useful for the study of stress ratio effect. The Goodman-Line is a method used to estimate the influence of the mean stress on the fatigue strength. It plots stress amplitude against mean stress with the fatigue limit and the ultimate tensile strength of the material as the two extremes.
Strain localization and failure mechanics for elastoplastic damage solids
Alternative failure criteria include Soderberg and Gerber. As coupons sampled from a homogeneous frame will display a variation in their number of cycles to failure, the S-N curve should more properly be a Stress-Cycle-Probability S-N-P curve to capture the probability of failure after a given number of cycles of a certain stress. Due to the proportionality between stress and strain, high cycle fatigue can also be expressed as strain amplitude vs. High cycle fatigue can be approximated by equating the total strain to just the elastic strain. Using this approximation,.
The figure below shows high cycle fatigue as the right-most linear portion. Any test performed in the bottom left region i. As shown in the figure above the left-most linear section and as described in the next section, the total strain is approximated to be equal to just the plastic strain. For regions between high and low cycle fatigue, an unweighted sum of the high cycle and low cycle expressions gives a reasonable approximation with a built-in safety factor. An estimate of the fatigue life of a component can be made using a crack growth equation by summing up the width of each increment of crack growth for each loading cycle.
Safety or scatter factors are applied to the calculated life to account for any uncertainty and variability associated with fatigue. The rate of growth used in crack growth predictions is typically measured by applying thousands of constant amplitude cycles to a coupon and measuring the rate of growth from the change in compliance of the coupon or by measuring the growth of the crack on the surface of the coupon. For normal manufacturing finishes this may cover the most of the fatigue life of a component where growth can start from the first cycle.
All these techniques aim to match the crack tip conditions on the component to that of test coupons which give the rate of crack growth. Additional models may be necessary to include retardation and acceleration effects associated with overloads or underloads in the loading sequence. In addition, small crack growth data may be needed to match the increased rate of growth seen with small cracks. Typically, a cycle counting technique such as rainflow-cycle counting is used to extract the cycles from a complex sequence.
This technique, along with others, has been shown to work with crack growth methods. Crack growth methods have the advantage that they can predict the intermediate size of cracks. Sources:  . Dependable design against fatigue-failure requires thorough education and supervised experience in structural engineering , mechanical engineering , or materials science. There are at least five principal approaches to life assurance for mechanical parts that display increasing degrees of sophistication: .
Fatigue testing can be used for components such as a coupon or a full-scale test article to determine:. These tests may form part of the certification process such as for airworthiness certification. Following the King Louis-Philippe I 's celebrations at the Palace of Versailles , a train returning to Paris crashed in May at Meudon after the leading locomotive broke an axle. The carriages behind piled into the wrecked engines and caught fire. At least 55 passengers were killed trapped in the carriages, including the explorer Jules Dumont d'Urville. This accident is known in France as the "Catastrophe ferroviaire de Meudon".
The accident was witnessed by the British locomotive engineer Joseph Locke and widely reported in Britain. It was discussed extensively by engineers, who sought an explanation.
The derailment had been the result of a broken locomotive axle. Rankine's investigation of broken axles in Britain highlighted the importance of stress concentration, and the mechanism of crack growth with repeated loading. His and other papers suggesting a crack growth mechanism through repeated stressing, however, were ignored, and fatigue failures occurred at an ever-increasing rate on the expanding railway system.
Other spurious theories seemed to be more acceptable, such as the idea that the metal had somehow "crystallized". To determine the parameters, the following tests. With one uniaxial fatigue test and knowing. The constant A. Figure 5. Comparison between softening function and ex-. Figure 6. Figure 7. With one biax-. Figure 8 shows the prediction results of biaxial limit. The theo-. The following. Figure 9 shows the comparison of the model predic-. The experimental data are of an equal biaxial. Lastly, the predicted.
Figure 10 shows the. The experimental data are from the work. An anisotropic damage model is established to predict. A class of damage mechanics is utilized recognizing that. Figure 8. Comparison between experimental data  and. Figure 9. Comparison of increment of compliance of equal.
Figure Predictions of the stress strain relationship of. The experimental da ta are from the work of. The experimental data are from the work of. In this work a bounding surface theory is. The limit strength state. The changes of the material properties and the ine-. The forms of response tensors allow the. Strength anisotropy is also studied and addressed.
Smith and K. Glass-Fiber Reinforced Composite. Fatigue and Frac-. Biaxial and Multiaxial Fatigue, , pp. Khan, et al. Stinchcomb and K. Fiber Reinforced Composite, II. Failure Criteria for Fa-. Composite Materials , Vol. Natarajan , H. Gangarao and V. Journal of Composite Materials , Vol. Boutman and S. Mandell, D. Cairns, D. Samborsky, R. Morehead and D. Grande, T. Tsiang and F. Highsmith and K.
Composite Materials : Basic Mechanisms , Accumulation ,. Tolerance , and Characterization , American Society for. Testing and Materials, Phil adelphia, , pp.
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Degrieck and W. Yoshioka and J. Chou and F. Elsevier Science Publis hers, Amsterdam, Wen and S. Yazdani and H. Yazdani and S. Budiansky and R. Horri and S. All rights reserved. It is argued that the limit surface is only a special case in such a formulation when the fatigue cycle is set to one and that under fatigue environment the limit surface contracts to a failure residual strength state based on the number of cycle, stress path, and stress magnitude. Within the formula- tion, specific kinetic relations for microcrack growth are postulated for woven fabric composites and a new direction function is specified to capture strength anisotropy of th e material.
Anisotropic stiffness degradations and inelastic strain propagation due to damage processes are also obtained utilizing damage mechanics formulation. The paper con- cludes with comparing theoretical predictions against experimental records showing a good agreement. The majority of the published works have ad- dressed various topics associated with the uniaxial stress path loading.
By comparison, the amount of work on the multiaxial modeling has been small. For one, the ex- perimental testing under multiaxial stress state is difficult to conduct requiring special instrumentation and appara- tus. This has lead to a small amount of experimental data to be available to develop and validate constitutive and failure models. However, the increasing use of woven fabric composites in struct ures subjected to complex loadings has required engineers to enhance the modeling and the predictive tools for a more reliable design. Com- pounding the difficulties is the complexity of micro- structures of the woven composites itself and the pres- ence of various defects and interfaces within the material.
There are three types of interfaces in woven compos- ites : resin rich area to longitudinal fiber group, resin rich area to transverse fiber group, and the interface between longitudinal fiber group and transverse fiber group.
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When loaded in the longitudinal direction, while the second kind of interface has little effects on the direc- tions of crack propagation, the other two kinds of inter- faces tend to stop the development of cracks perpendicu- lar to the direction of the load. Due to the strength and stiffness of the longitudinal fiber group, cracks propa- gating in the perpendicular direction stop at the longitu- dinal fiber group.
The resultant stress concentration would then redirect cracks to the weak interface areas around the longitudinal fiber group and initiate breaking of interfaces between adjacent layers. After a number of the weak interfaces are broken down and resultant sepa- rate interface areas join together, delamination emerges. Under these complex phenomena, several different dam- age modes are present: micro-cracking, cracking, debond- ing, delamination, and fiber fracture .
Many resear- chers report that the fatigue process can be divided into three stages [7,8]. In the first stage, the main damage modes are matrix micro-cracking and cracking; the sec- ond stage is controlled by a combination of matrix crack- ing, interfacial cracking and delamination; while the fiber failure dominates the last stage.