A fracture theory for a heterogenous aggregate material which exhibits a gradual strain-softening due to microcracking and contains aggregate pieces that are not necessarily small compared to structural dimensions is developed. Only Mode I is considered. The fracture is modeled as a blunt smeard crack band, which is justified by the random nature of the microstructure. Simple triaxial stress-strain relations which model the strain-softening and describe the effect of gradual microcracking in the crack band are derived. It is shown that it is easier to use compliance rather than stiffness matrices and that it suffices to adjust a single diagonal term of the complicance matrix. The limiting case of this matrix for complete (continuous) cracking is shown to be identical to the inverse of the well-known stiffness matrix for a perfectly cracked material. The material fracture properties are characterized by only three parameters—fracture energy, uniaxial strength limit and width of the crack band (fracture process zone), while the strain-softening modulus is a function of these parameters. A method of determining the fracture energy from measured complete stres-strain relations is also given. Triaxial stress effects on fracture can be taken into account. The theory is verified by comparisons with numerous experimental data from the literature. Satisfactory fits of maximum load data as well as resistance curves are achieved and values of the three material parameters involved, namely the fracture energy, the strength, and the width of crack band front, are determined from test data. The optimum value of the latter width is found to be about 3 aggregate sizes, which is also justified as the minimum acceptable for a homogeneous continuum modeling. The method of implementing the theory in a finite element code is also indicated, and rules for achieving objectivity of results with regard to the analyst's choice of element size are given. Finally, a simple formula is derived to predict from the tensile strength and aggregate size the fracture energy, as well as the strain-softening modulus. A statistical analysis of the errors reveals a drastic improvement compared to the linear fracture theory as well as the strength theory. The applicability of fracture mechanics to concrete is thus solidly established.

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Crack band theory for fracture of concrete

The fracture front in concrete, as well as rock, is blunted by a zone of microcracking, and in ductile metals by a zone of yielding. This blunting causes deviations from the structural size effect known from linear elastic fracture mechanics (LEFM). The size effect is studied first for concrete or rock structures, using dimensional analysis and illustrative examples. Fracture is considered to be caused by propagation of a crack band that has a fixed width at its front relative to the aggregate size. The analysis rests on the hypothesis that the energy release caused by fracture depends on both the length and the area of the crack band. The size effect is shown to consist in a smooth transition from the strength criterion for small sizes to LEFM for large sizes, and the nominal stress σN at failure is found to decline as (1+λ/λ0)-1/2 in which λ0=constant and λ=relative structure size. This function is verified by Walsh's test data. If reinforcement is present at the fracture front and behaves elastically, ...

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Size Effect in Blunt Fracture: Concrete, Rock, Metal

A COHESIVE zone type interface model, taking full account of finite geometry changes, is used to study the decohesion of a viscoplastic block from a rigid substrate. Dimensional considerations introduce a characteristic length into the formulation. The specific boundary value problem analysed is one of plane strain tension with a superposed hydrostatic stress. For a perfect interface, if the maximum traction that the viscoplastic block can support is greater than the interfacial strength, decohesion takes place in a primarily tensile mode. If this maximum traction is lower than the interfacial strength, a shear dominated decohesion initiates at the block edge. Imperfections in the form of a non-bonded portion of the interface are considered. The effects of imposed stress triaxiality, size scale, loading rate and interfacial properties on the course of defect dominated decohesion are illustrated. The characterization of decohesion initiation and propagation in terms of rice's (J. appl. Mech. 35, 379, 1968) J-integral is investigated for a variety of interface descriptions and values of the superposed hydrostatic stress.

An analysis of tensile decohesion along an interface

Concrete fracture models: testing and practice

Nacre, also known as mother-of-pearl, is a hard biological composite found in the inside layer of many shells such as oyster or abalone. It is composed of microscopic ceramic tablets arranged in layers and tightly stacked to form a three-dimensional brick wall structure, where the mortar is a thin layer of biopolymers (20–30 nm). Although mostly made of a brittle ceramic, the structure of nacre is so well designed that its toughness is several order of magnitudes larger that the ceramic it is made of. How the microstructure of nacre controls its mechanical performance has been the focus of numerous studies over the past two decades, because such understanding may inspire novel composite designs though biomimetics. This paper presents in detail uniaxial tension experiment performed on miniature nacre specimens. Large inelastic deformations were observed in hydrated condition, which were explained by sliding of the tablets on one another and progressive locking generated by their microscopic waviness. Fracture experiments were also performed, and for the first time the full crack resistance curve was established for nacre. A rising resistance curve is an indication of the robustness and damage tolerance of that material. These measurements are then discussed and correlated with toughening extrinsic mechanisms operating at the microscale. Moreover, specific features of the microstructure and their relevance to associated toughening mechanisms were identified. These features and mechanisms, critical to the robustness of the shell, were finely tuned over millions of years of evolution. Hence, they are expected to serve as a basis to establish guidelines for the design of novel man-made composites.

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An Experimental Investigation of Deformation and Fracture of Nacre–Mother of Pearl

Final stretch criterion of failure is applied to the problem of quasi-static extension of a crack embedded in an elastic-plastic or viscoelastic-plastic matrix. The slow growth under subcritical conditions in a rate-sensitive Tresca solid is shown to be a superposition of creep rupture and McClintock's ductile growth. This type of growth occurs at subcritical magnitude of the imposed K-factor and can be accounted for only through a recognition of inelastic properties of solids. In the subcritical range there is no unique value for K sub c independent of geometrical configuration and flaw size. Not only the produced states of stress and strain are dependent on the loading path, but also the material resistance to fracture turns out to be a function of the history of loading that precedes catastrophic failure. A nonlinear integro-differential equation of motion is derived for a crack progressing through a viscoelastic medium with some limited ability to plastic flow. Examples of numerical integration are given incorporating both monotonic and cyclic loading programs.

Quasi-static extension of a tensile crack contained in a viscoelastic-plastic solid

This text is an examination of the nonlinear phenomena created when fractures develop in structural materials upon application of static, cyclic or dynamic external loads. Incorporated in the volume are the mathematical models aimed at the quantative prediction of ductile failure in strain-hardening metallic alloys, or quasi-brittle fracture in strain-softening materials such as rock, concrete and cementitious composites. By scrutinizing the microstructure (mesomechanics) involved, the text incorporates certain recommendations concerning new technologies and procedures required in the manufacture of high performance materials with enhanced resistance to crack propagation and superior mechanical properties.

Nonlinear fracture mechanics

Conditions under which fast fracture may occur in ductile solids are described in terms of a new quantity, named stability index, λ. This index, defined as % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaabs% gacaWGsbGaai4laiaadsgacaWGHbGaeyOeI0IaeyOaIyRaamOuamaa% BaaaleaacaWGbbaabeaakiaac+cacqGHciITcaWGHbGaaiykaiaacI% cacqGHciITcaWGsbWaaSbaaSqaaiaadgeaaeqaaOGaai4laiabgkGi% 2kaadggacaGGPaWaaWbaaSqabeaacqGHsislcaaIXaaaaaaa!4C11!\[({\text{d}}R/da - \partial R_A /\partial a)(\partial R_A /\partial a)^{ - 1}\], encompasses the tearing modulus of Paris % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadw% eacaGGVaGaeq4Wdm3aa0baaSqaaiaaicdaaeaacaaIYaaaaOGaaiyk% aiaacIcacaqGKbGaamOsaiaac+cacaqGKbGaamyyaiaacMcadaWgaa% WcbaGaamyAaaqabaaaaa!42DC!\[(E/\sigma _0^2 )({\text{d}}J/{\text{d}}a)_i\]determined at the point of incipient crack extension, and other geometry dependent quantities. Stability index serves as a measure of the “distance” of any given state of the system considered (specimen containing an extending crack) from the critical state. Instability occurs at λ=0. This work suggests a means of computation of the quantities % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeizaiaabk% facaqGVaGaaeizaiaadggaaaa!3A28!\[{\text{dR/d}}a\]and % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOaIyRaam% OuamaaBaaaleaacaWGbbaabeaakiaac+cacqGHciITcaWGHbaaaa!3C25!\[\partial R_A /\partial a\]necessary for evaluation of the stability index λ when a geometrical configuration of the test-piece is specified and under an assumption that the DBCS model employed here and extended to a moving crack situation, is adequate. Symbols R and RAdenote the extent of the plastic zone developed during a quasi-static crack growth and considered either as a measure of material toughness (R) or as an intensity of the applied field (RA). Symbol “a” denotes the current crack half-length.

Occurrence of catastrophic fracture in fully yielded components. Stability analysis

A local energy criterion of Irwin's type is applied to a problem of a quasi-static extension of a crack embedded in an elastic-plastic or viscoelastic-plastic matrix The derived nonlinear differential equation ℳ(σ, l, dσ/dl) + G(σ, l) = G

 c
 /Ψ(Δ/l) governs subcritical growth up to the point of gross instability The creep rupture under sustained loads and fatigue crack propagation both described by the above equation, are shown to follow almost identical mathematical representations The essential features such as the resistance curves, the dependence of growth rate on the current K-factor and the amount of growth vs time relations are shown to be strikingly similar

Michael P. Wnuk

Papers

Quasi-static extension of a tensile crack contained in a viscoelastic-plastic solid

Nonlinear fracture mechanics

Occurrence of catastrophic fracture in fully yielded components. Stability analysis

Subcritical growth of fracture (inelastic fatigue)

On estimating stress intensity factors and modulus of cohesion for fractal cracks