On study of nonclassical problems of fracture and failure mechanics and related mechanisms

Transcription

1 On study of nonclassical problems of fracture and failure mechanics and related mechanisms Alexander N. Guz Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev, Ukraine Abstract Nonclassical problems of fracture and failure mechanics that have been analysed by the author and his collaborators at the SPTimoshenko Institute of Mechanics (Kiev, National Academy of Sciences of Ukraine) during the past thirty-five years are considered in brief. Results of analysis are presented in form that would be quite informative for the majority of experts interested in various fundamental and applied aspects of the fracture and failure problems including the identification of the related mechanisms. This paper was prepared on invitation of the Editorial Board of the journal "Annals - " and may be considered as the Extended Pascal Medal Lecture (the 2007 Blaise Pascal Medal in Materials Sciences of the EAS). Keywords Nonclassical problems; Fracture and failure mechanics; Fracture of composites; Cracked materials; Dynamic brittle fracture mechanics Foreword On the borderline between IInd and IIIrd millennia, all humanity analyzes the situation (successes achieved and progress perspectives) established in various directions of human activity. It is quite naturally that representatives of different scientific directions analyze also at that time the achieved scientific results, since the analysis and generalization of established scientific results were and remain the one of basic and crucial problems of scientific activity. One is necessary to note that the forms of analysis and generalization of established scientific results can be very diverse for different scientific directions what is linked with objects in the study, considered time durations, and specificity of concrete scientific directions. While preparing the generalizing publications in mechanics, it is to all appearance expediently to take into account the specifity and place of mechanics in the existing modern system of scientific investigations. It can be assumed that mechanics specificity as the science consists in that mechanics is one of the major sciences of a fundamental character, and at the same time mechanics actuality is defined by the importance for engineering of studied nowadays problems. At all stages of the human progress, beginning from ancient times, the importance of mechanics for engineering can scarcely be exaggerated: in many cases, mechanics and engineering have been considered as a single whole. Mechanics is transformed to present time into the very ramified area of knowledge. Some idea about diversity of problems considered in mechanics can be received from the written below information on separate scientific directions. By the form of physical models used in phenomena studies, mechanics is divided on rigid body mechanics, fluid, gas, and plasma mechanics, and solid mechanics. If two or three of mentioned above models are used together for exploring the sufficiently complicated phenomena, Ann. - Eur. Acad. Sci. ( ) Liège: EAS Publishing House,

2 Alexander N. Guz then these investigations are referred conditionally to general mechanics or to the its separate directions such as, for example, aerohydroelasticity. By the applied methods used in studies of phenomena, mechanics is divided on analytical mechanics, computational mechanics, and experimental mechanics. And besides, even international congresses are often organized separately according to mentioned above directions of mechanics. Let us note that general problems, including the formulations of a closed statement of problems, uniqueness theorems, variational principles and other related problems, are referred to analytical mechanics. At present time, analytical mechanics is treated in the broader sense, when corresponding problems of fluid, gas, and plasma mechanics are referred to the analytical one. Of course, methods and approaches corresponding to two or three mentioned above directions are applied together for a number of problem studying. Sufficiently often, mechanics is divided on the separate scientific directions corresponding to the separate specific directions of practical human activity. When this approach is realized, then one can say, for example, about space flight mechanics, civil engineering or structural mechanics, composite materials mechanics, rock mechanics, structural mechanics of planes or ships, biomechanics, mechanics of a man, celestial mechanics, and a number of other directions. Solid mechanics is divided on theory of elasticity, theory of plasticity, theory of viscoelasticity, creep theory, and fatigue theory by the form of mechanical models used in the phenomena studied. Within the framework of this approach, one is also expediently to distinguish mechanics of coupled fields in materials and structure members, which studies the behaviour of materials and members of structure under the combined action of force, temperature, and electromagnetic fields. Solid mechanics can be also divided on statics, dynamics, stability, and fracture by the character of mechanical phenomena considered. And besides, fracture mechanics investigates the mentioned phenomenon both under static and under dynamic loads. It is need emphasize that in the general case the fracture phenomenon has more complicate character and includes mechanical, physical, and chemical aspects. Therefore, the fracture phenomenon is investigated in the general case with taking into account mentioned above aspects. Above, a series of directions in mechanics are shown as examples of a classification. At present time, the detailed classifiers of scientific directions in mechanics are elaborated in a number of states and specialized editions on mechanics. References on these classifiers are widely used in scientific results publications. It is necessary to note that the next scientific directions in mechanics: computational mechanics, mechanics of a space flight, biomechanics, mechanics of composite materials, fracture mechanics, mechanics of coupled fields in materials and structure members, and also a number of other directions were actively developed in the second half of the XXth century. In general modern problems of fracture and failure mechanics may be divided into classical and nonclassical problems. This paper is devoted to modern nonclassical problems of fracture and failure mechanics including the identification of the related mechanisms. Introduction Griffith defined in his classical work [7] a new scientific area in natural sciences fracture mechanics. It was one of the most actively developed fundamental and applied areas in mechanics in the latter half of the XXth century. Presently, this area in its topicality, fundamentality, and applicability to engineering may be compared only with the mechanics of composites. This is supported by the fact that the two eight-volume collective encyclopedic monographs [3, 5] 36

3 On study of nonclassical problems of fracture and failure mechanics and related mechanisms devoted to the two above-mentioned scientific areas were published in the midsecond half of the XXth century. The author together with his collaborators became engaged in nonclassical problems of fracture mechanics more than 30 years ago. The author's paper [8] was the first to be devoted exclusively to one of the nonclassical problems of fracture mechanics. In succeeding years, extensive studies addressing a number of nonclassical problems of fracture mechanics were made. The results of these studies were included in nine doctoral theses (second doctorate degree), several PhD theses, and numerous publications in various periodicals. The results obtained for each nonclassical problem of fracture mechanics were analyzed and reviewed in [8, 4, 6, 8, 9, 20-23, 25-29, 36] and others publications. These findings, along with other general results, were also included in the monographs [2, 9-2, 7], which are completely or partially devoted to nonclassical problems of fracture mechanics. Information in English on these monographs is presented in monograph [24] also thus, we may say that the results in nonclassical fracture mechanics obtained by the author and his collaborators were comparatively adequately described in the Russian and non-russian scientific literature. Nevertheless, it is the author's opinion that the above mentioned findings in nonclassical fracture mechanics, due to the traditional ideas of scientific exposition of fundamental results on mechanics, are stated in a form that is perceived well by representatives of certain scientific areas but is not always adequately informative for many experts in the mechanics of a deformable body and fracture mechanics. It should be noted that fracture problems attract the attention of representatives of various scientific and engineering areas, including mechanics, physics, the science of materials, etc., who are engaged in various aspects of engineering. In this connection, apparently, a rather topical problem is to represent findings in nonclassical fracture mechanics in a form that would be quite informative for the majority of experts interested in various fundamental and applied aspects of the fracture problem. At the same time, such a representation should include the basic aspects of the problems under consideration (the nonclassical nature of the fracture mechanism, rigorous description, analysis of the basic approaches and results, description of new mechanical effects, etc.). It is quite obvious that in representing results in nonclassical fracture mechanics as described above, we should abandon detailed information on especially mathematical methods of solution and satisfaction with specific graphical and tabular results. Thus, along with the reviews [4, 6, 22, 25, 27-29, 36] and the monographs [2, 9-2, 7], the necessity arose of reviewing results pertaining to nonclassical fracture mechanics provided that they are reported as mentioned above. It should be noted that the intention of writing a review using the above stated approach arose in preparing the invited paper [20] for the IUTAM Symposium (Cambridge, UK, 995) and the paper [23] for the encyclopedia [6]. The author already attempted several times to write a similar review [8, 9, 2]; nevertheless, these papers do not give a sufficiently complete and informative representation of the results obtained in nonclassical fracture mechanics by the author and his followers. Lecturing at the Yildiz Technical University (Istanbul, March, 998), the Institute of Mechanics (Hanoi, December, 998), and the University of Technology (Vienna, December, 999), the author finally became convinced that writing a review using the above-stated approach is quite expedient. This has been partially implemented in the review [26], which may be considered the time the author started on the present review. Let us first clarify the terminology used in mechanics as applied to the fracture of materials and structure elements. By fracture is meant breaking characterized by propagation of one or several cracks. Fracture mechanics studies the fracture of materials or structure elements. By failure is meant breaking characterized by exhaustion of the load carrying capacity of a material or a structure element and is mainly manifested not only as propagation of single cracks. Failure 37

4 Alexander N. Guz mechanics studies the failure of materials and structure elements. By damage is meant breaking manifested as accumulation of dispersed developing or incipient cracks or other breaks. Damage mechanics studies the laws (kinetics) of damage accumulation, mainly within the framework of various continual representations, by invoking a definitely chosen damage indicator. The above classification is certainly a matter of convention; however, to the author's opinion, it is quite useful and informative for the analysis of various results in fracture mechanics, in the broad sense of this term. For example, the results of nonclassical problems that are analyzed here pertain to fracture mechanics and failure mechanics. This paper includes the results of papers [23, 26] and books [] (Foreword pp. 3-6, the article pp ) and [39] (the article pp. -62). New results on mechanics of nanocomposites were obtained during [30-34] taking into account the above mentioned studies on the nonclassical problems of fracture and failure mechanics.. On nonclassical problems of fracture mechanics Nowadays, we may apparently consider that fracture mechanics (in a broad sense) includes well-defined basic concepts and approaches to the formulation of fracture criteria. The issues below may be classed as the basic concepts and approaches in fracture mechanics (in a broad sense).. The fundamental Griffith theory of brittle fracture. 2. The concept of quasi-brittle fracture (Irwin, Orowan, and others). 3. The Griffith energy fracture criterion or the equivalent (more easily implemented) Irwin force criterion. 4. The concept of an integral ( J -integral,! -integral, Eshelby, Cherepanov, Rice) independent of the path of integration. 5. The critical crack opening criterion. The above stated concepts and approaches assume that the following conditions are satisfied. Condition. Tension or shear arises near cracks, compression being excluded. Condition 2. During deformation, a cracked body does not change sharply its configuration (for example, buckling does not precede fracture). Condition 3. During deformation of a cracked body, the prefracture deformation pattern is not changed sharply (for example, subcritical crack growth and changes in the boundary conditions during deformation are absent). It should be noted that Condition is principal one, since the above concepts and approaches do not work upon compression along cracks. Under Condition 2, all the above-mentioned concepts and approaches may work; however, a preliminary study should be made of the stress-strain state of a body sharply changing its configuration during deformation. Presently, such an analysis has not been performed in the overwhelming majority of studies in fracture mechanics (in a broad sense). Under Condition 3, all the concepts and approaches may work; however, a preliminary study should be made of the stress-strain state of the body when the prefracture deformation pattern is sharply changed (for example, when the boundary conditions change during deformation). Presently, such an analysis has not been performed in the overwhelming majority of studies in fracture mechanics. 38

5 On study of nonclassical problems of fracture and failure mechanics and related mechanisms Based on the foregoing considerations, results and problems corresponding to the above five concepts or approaches and obtained under Conditions -3 may be considered as classical problems of fracture mechanics, among which are the following studies.. Determination of the stress intensity factors for complex cracked bodies under various mechanical, thermal, and electromagnetic actions. To this end, analytical, numerical (computeraided), experimental, and experimental-theoretical methods are applied. The results of these studies (stress intensity factors) together with the fracture criteria mentioned provide the necessary information on the fracture of materials and structure elements in cases where these fracture criteria are applicable. 2. Experimental investigation of the complex fracture of materials and structural elements. The results are mostly descriptive and obtained without due analysis and attempts to formulate new fracture criteria corresponding to the phenomena under consideration. It should be noted that currently the overwhelming majority of publications pertains to classical fracture mechanics in the above mentioned sense. Because of this, apparently, many scientists conclude that an idea crisis exists in fracture mechanics at the current stage of its development. Note also that the second area in classical mechanics may serve as the first stage in the study of nonclassical fracture mechanics. The following studies may be conditionally classed among nonclassical problems of fracture mechanics.. Study of new mechanisms of fracture that are not described within the framework of the above five concepts and approaches (under Conditions -3) with appropriate analysis and attempts to formulate new fracture criteria corresponding to the phenomena under consideration. 2. Study of certain classes of problems related to new mechanisms of fracture of materials and structural elements by invoking specially formulated fracture criteria. As already mentioned, the above classification (into classical and nonclassical problems) is rather conventional and not always unambiguous. Nevertheless, this classification determines fairly well the orientation of studies and their novelty, which seems to be rather important for the end-point analysis. It should be also noted that the number of fracture mechanisms increases considerably if the material microstructure, which is described differently, is taken into account. This feature pertains to the fracture mechanics of composites whose microstructure is considered at various levels. Scientists who study nonclassical problems and fracture mechanisms apply rather approximate design models. Such approximate design models are applied to the analysis of fracture in the microstructure of composites. The application of approximate design models leads to significant errors and, in many cases, to qualitative differences. Hence, it is rather difficult to carry out a reliable analysis of nonclassical problems and fracture mechanisms using approximate models. Therefore, results obtained in studying nonclassical problems and fracture mechanisms on the basis of quite strict design models are considerable value. Some approximate schemes and models and associated results are considered as examples in Section 6 of the present paper. The results on nonclassical problems and fracture mechanisms that have been obtained by the author and his followers at the S. P. Timoshenko Institute of Mechanics of the National Academy of Sciences of Ukraine within the past thirty years are briefly described (with annotations) here. Primary attention is drawn to a qualitative analysis of the basic aspects of the problems in question according to the idea stated in the Introduction. The following seven problems and associated fracture mechanisms are considered quite regularly: 39

6 Alexander N. Guz. Fracture of composites compressed along reinforcing elements. 2. End-crush fracture of composites subject to compression. 3. Shredding fracture of composites stretched or compressed along the reinforcing elements. 4. Brittle fracture of cracked materials with initial (residual) stresses acting along the cracks. 5. Fracture under compression along parallel cracks. 6. Brittle fracture of cracked materials under dynamic loads (with contact interaction of the crack faces). 7. Fracture of thin-walled cracked bodies under tension with prebuckling. It should be noted that results for Problems -3 pertain only to the fracture mechanics of composites, whose microstructure, as already mentioned, is described at various levels, and results for Problems 4-7 equally pertain to the fracture mechanics of composites and to the fracture mechanics of metals and alloys. Based on the terminology accepted here (see the Introduction), Problems -3 are attributed to failure mechanics and Problems 4-7 to fracture mechanics. Analyzing the results mentioned above, we will not refer to the periodicals where these results were first published, since they are already referred to in the reviews [4, 6, 8, 9, 20-23, 25-29, 36] and the monographs [2, 9-2, 7]. Note that the following two features are characteristic of the studies made by the author and his disciples on the above mentioned seven nonclassical problems of fracture mechanics (as compared with the studies of other authors).. The studies were carried out by invoking the most strict and exact formulations within the framework of the mechanics of deformable bodies. For example, the three-dimensional linearized theory of stability of deformable bodies (for example [24]) and the three-dimensional equations of statics of deformable bodies were applied to the study of, respectively, buckling and the stressstrain state. This remark addresses Problems -6. As for Problem 7, the stringency of the formulation and methods of study is ensured by applying the two-dimensional linearized theory of stability of thin-walled structural elements. 2. Problems -3 were studied within the framework of both the continual approximation (the model of a homogeneous orthotropic body with reduced constants a three-dimensional formulation) and the model of a piecewise-homogeneous medium (the three-dimensional equations for a filler and binder and continuity conditions at interfaces). It should be noted that the indicated model of a piecewise-homogeneous medium (in a three-dimensional formulation) is the most strict and exact within the framework of the mechanics of deformable bodies as applied to composites. We can introduce some corrections (specification of results) only by considering other boundary conditions at interfaces and other constitutive equations for the filler and binder. Nevertheless, note that the above stated nonclassical problems and associated mechanisms of fracture were not adequately elucidated in the generalizing monographic encyclopedic literature on fracture mechanics in a broad sense. For example, they were not included in the eight-volume collective generalizing encyclopaedic monograph on fracture [5]. The encyclopedia of fracture mechanics [6] published in 998 contains brief information (as an individual article [23]) on these problems and the corresponding results. It should also be noted that other results in nonclassical fracture mechanics and associated fracture mechanisms (along with the results of the author and his followers) obtained at the S. P. Timoshenko Institute of Mechanics of NANU are reported in the collective four-volume (five-book) monograph [37]. 40

7 On study of nonclassical problems of fracture and failure mechanics and related mechanisms 2. Fracture of composites compressed along the reinforcing elements In the scientific literature on the fracture mechanics of composites, it is currently agreed that the paper [4] was the first to describe the microbuckling of fibers as a form of fracture of a unidirectional composite under compression. Nevertheless, some other authors described quantitatively this phenomenon (the theory of fracture) using various approximate design models. The following experiment illustrates the feasibility of the fracture mechanism associated with the buckling of fibers (fillers) in the matrix. Glass fibers are placed in epoxy (or other) resin, which is then polymerized at certain temperature. After that, the resultant block cools down to room temperature. In cooling, compressive loads act upon the fibers fixed by the resin (matrix) due to the difference between the coefficients of thermal expansion. As a result, buckling occurs and the fibers take on a well-defined sinusoidal form. Fig. illustrates this experiment. Note that before (Fig. 2a), hardening, glass fibers 0.0 mm in diameter freely floated in the binder. From Fig. (50! magnification), it is seen that after hardening, the whole strand and individual fibers take on a well-defined sinusoidal form, which is indicative of microbuckling. Fig.. This fracture mechanism is realized in a pure form in unidirectional fibrous composites laminated composites (Fig. 2b), orthogonally reinforced composites, and composites with a slightly curved structure. This mechanism is the first stage of Chinese-lantern fracture, fracture of the internal layers in thick-walled cylinders, and some other types of fracture. Note also that this fracture mechanism is realized in compressed zones of a composite subject to bending. In all the above cases, the beginning of fracture under compression is identified with the buckling of the microstructure of the material, when the critical load (the theoretical ultimate strength for the fracture mechanism being considered) is determined by parameters characterizing the microstructure of the composite (the concentration of the filler and binder, the relative rigidity of the filler and binder, etc.) rather than by the dimensions and shape of the specimen or structure element. There exists a simple criterion of realizability of the considered fracture mechanism the critical load corresponding to the buckling in the microstructure of the composite must be less than the critical load corresponding to the buckling of the entire specimen or structure element. Fig. 2. 4

8 Alexander N. Guz In studying the fracture mechanism being considered, it is expedient to examine individually two types of fracture. The first type is fracture of the whole specimen throughout the thickness or internal fracture due to the internal instability of the microstructure of the composite; in this case, an infinite composite may be considered. The second type is near-surface fracture (only the nearsurface layers of the specimen or structure element collapse) due to surface instability; in this case, a semi-infinite composite should be considered and phenomena subsiding with distance from the boundary should be analyzed. The near-surface fracture should be also analyzed near stress concentrators; in this case, it is necessary to calculate the compressive load in view of the stress concentration phenomenon. The above mentioned aspects of the problem being considered are detailed in the monograph [7]. The above stated approach leads to eigenvalue problems, which, when solved, produces a dependence of the load on the wave formation parameter. Typical dependences for four various laminated composites are shown in Fig. 3. It is usually assumed that the composite buckles on the wave formation in the microstructure if the curve of the dependence (Fig. 3) of the load Fig. 3. # parameter p on the parameter! (! = " h l, where 2h a a a a is the thickness of the filler layer and l is the buckling half-wave length) has a minimum for! " 0 for the given material. In this case, a p and! correspond to this minimum. In Fig. 3, the values of p and! are given for cr cr cr cr material 4. For such an approach, p corresponds to the theoretical ultimate compression cr strength. Many authors used various approximate design models based on applied two-dimensional theories of stability of rods, plates, and shells to study the above considered fracture mechanism (for example, [40]). It is certainly difficult to reliably analyze this phenomenon in the microstructure of a composite with the help of the above mentioned approximate models due to its complexity. The author was the first to analyze the fracture mechanism by invoking the threedimensional linearized theory of stability of deformable bodies (for example, the monographs [9, 0, 5, 24]) for elastic (brittle fracture) and plastic (the fracture of a composites with a metal matrix) models. Such an approach excludes occurrence of errors characteristic of various approximate models. The major results obtained by the author and his followers were reported in the monograph [7]. Two approaches were developed within the framework of the above mentioned analysis by using the three-dimensional linearized theory of stability of deformable bodies. The first approach is based on the model of a homogeneous anisotropic body with reduced constants (the continual approximation and homogenization of a medium). In this case, the parameters characterizing the structure of the material are incorporated into the reduced constants of the homogeneous anisotropic body. The second approach is based on the model of a piecewise-homogeneous medium, where the equations of the three-dimensional linearized theory of stability of deformable bodies are used separately for the filler and the binder and the continuity conditions for the stress and displacement vectors are satisfied at the filler-binder interfaces. 42

9 On study of nonclassical problems of fracture and failure mechanics and related mechanisms Let us consider, as examples, some results obtained by the first approach as applied to brittle fracture. These results pertain to a plane problem solved in the plane x 0x for uniaxial 2 compression along the axis 0x, though similar results were also obtained for a three-dimensional! " (the theoretical ultimate strength under uniaxial problem. To determine the quantity ( ) T compression along the axis 0x in the case of internal fracture fracture of a specimen or structure element throughout the thickness), the following expression was derived:! ( ) " = (2.) G. T 2 " To determine the quantity ( )!! (the theoretical ultimate strength under uniaxial compression T along the axis 0x in the case of near-surface fracture fracture near the surface x = const of a 2 specimen or structure element), the following expression was derived: 2 ' & # 2 () ) G ( G $ ' ( ' * * )( ' * * ). 2 $ ! (2.2) T % EE2 "! In addition, to determine the quantity ( ) K " (the theoretical ultimate strength under uniaxial T compression along the axis 0x in the case of near-surface fracture in the presence of stress concentration), the following expression was derived for a specimen or structural element with a stress concentrator (a cavity or an opening) with contour to which the plane x = const is tangent: 2! K! ( ) ( ( )) ( ) " = K A, A, A, G, R! " ". (2.3) T T In (2.)-(2.3): G is the reduced shear modulus, E and E are the reduced Young's moduli, 2 2!,...,! are the reduced Poisson's ratios, A, A and A are the reduced elastic constants, K A, A, A, G, R is the stress concentration factor calculated for an orthotropic body with ( ) reduced constants and a stress concentrator, and R is a geometrical parameter characterizing the 0 curvature of the contour of the opening or cavity. It should be noted that similar results were also obtained for plastic fracture. In this case, all the reduced constants are calculated at the instant of buckling. For example, the role of the reduced shear modulus is played by the tangent reduced shear modulus calculated at the instant of buckling. The theory of delayed fracture of composites was developed based on such an approach (within the framework of the continual approach). The above results address only the case of uniaxial compression. Similar results were also obtained for biaxial and triaxial compression. The considered continual theory constructed within the framework of the above-mentioned first approach enables us, apart from determination of the theoretical ultimate strengths (results (2.)-(2.3)), to explain the nature of fracture for the case under consideration. To determine the nature (mechanism) of fracture within the framework of the indicated theory, the propagation of small perturbations caused by the initial stage of fracture is considered. Let us first present some experimental results that indicate the specific nature of fracture in the case being considered. As examples Fig. 4 shows the fracture of a fibreglass plastic specimen compressed perpendicularly to the reinforcing elements (perpendicularly to the reinforcement plane) and Fig. 5 shows the fracture of specimens made of unidirectional fibreglass and subject to uniaxial compression. The monograph [7] refers to experimental results similar to Figs. 4 and 5. Some other publications indicate the same behavior of composites collapsing under compression. A common feature of the 43

10 Alexander N. Guz above studies is that fracture occurs in planes that are almost perpendicular to the direction of the compressive load. The classical nature of the fracture mechanism for metals and alloys under uniaxial compression manifests itself in the fact that fracture occurs at an angle of about 45 to the direction of the compressive load. It should be noted that Figs. 4 and 5 exclude phenomena associated with fracture near the ends. For example, the cross-sectional area was increased near the ends of the specimens in Fig. 4 and the ends of the specimens in Fig. 5 were fixed in rigid rings. The above mentioned experimentally observable fracture mechanism was also predicted by the continual theory for both brittle fracture (composites with a polymeric matrix) and plastic fracture (composites with a metal matrix). The basic results were detailed in the monograph [7]. According to the continual theory of fracture, the fracture of composites under compression can be described as follows. Near-surface fracture first occurs near stress concentrators for values of the external load considerably lower than the minimum reduced shear modulus, which is determined as for a homogeneous anisotropic body. With further increase in the load (up to values slightly smaller than the minimum reduced shear modulus), near-surface occurs near the free surfaces of the specimen or structural element. Once the value of the external load becomes equal to the minimum reduced shear modulus, avalanche like (instantaneous) internal fracture (of the entire specimen or structuel element) extends over surfaces, which are almost perpendicular to the direction of the external load. The above stated laws are determined by expressions (2.)-(2.3) and are concerned with the continual theory of brittle and plastic fracture. Fig. 4. Fig. 5. The mechanisms of internal and near-surface fracture of unidirectional fibrous (Fig. 2a) and laminated (Fig. 2b) composites were investigated within the framework of the second approach (the model of a piecewise-homogeneous medium). As an example, Fig. 3 presents results on internal fracture of laminated materials obtained by using the second approach. The second approach was also used to solve problems on the internal fracture of unidirectional fibrous composites under uniaxial compression. The cross sections of the materials that where examined are shown in Figs. 6a-e: one fiber (a composite with a small concentration of fibers) (Fig. 6a), two fibers (a composite with a small concentration of fibers, two isolated fibers interact with each other) (Fig. 6b), an infinite periodic row of fibers (no interaction between separate rows of fibers) 44

11 On study of nonclassical problems of fracture and failure mechanics and related mechanisms (Fig. 6c), a finite number of periodic infinite rows of fibers (Fig. 6d), and a doubly periodic system of fibers (Fig. 6e). It should be noted that the studies for the cases in Figs. 6a-e were made for brittle and plastic fracture. The second approach was employed to study the near-surface fracture of unidirectional fibrous composites under uniaxial compression. The cross sections of the materials that were examined are shown in Figs. 7a-e: one fiber near the surface (a composite with a small concentration of fibers) (Fig. 7a), two fibers along the surface (Fig. 7b) and two fibers perpendicular to the surface (a composite with a small concentration of fibers, two isolated fibers interact with each other) (Fig. 7c), an infinite periodic row of fibers (no interaction between separate rows of fibers) (Fig. 7d), and a finite number of periodic infinite rows of fibers (Fig. 7e). It should be noted that the studies for the cases in Figs. 7a-e were made for brittle and plastic fracture. In the cases shown in Figs. 6a-e and Figs. 7a-e, the exact solutions were derived by invoking a unified general method. This method involves the general solutions of the threedimensional linearized theory of stability of deformable bodies, representation of the solutions for fibers and matrices in local circular cylindrical coordinates, application of the theorem on summing cylindrical functions, derivation of characteristic equations in the form of infinite determinants, proof of the fact that the infinite determinants obtained are normal type, which substantiates the application of the reduction method to their numerical analysis, and numerical analysis of the finite characteristic determinants obtained by the reduction method. It should be noted that the above stated exact method was realized for brittle and plastic fracture. Fig. 6. Fig. 7. Various buckling modes should be examined in solving various problems on unidirectional fibrous composites subject to uniaxial compression along fibers and having cross sections shown in Figs. 6a-e and Figs. 7a-e. Let us consider, as an example, internal fracture in the case of two fibers (Fig. 6b). To this end, it is necessary to consider, in the cross-section plane, the buckling modes shown in Figs. 8a-d and arbitrarily called as follows: in phase buckling from the plane of two fibers (Fig. 8a), antiphase buckling from the plane of two fibers (Fig. 8b), antiphase buckling in the plane of two fibers (Fig. 8c), and in phase buckling in the plane of two fibers (Fig. 8d). For the other cases (Figs. 6a-e and Figs. 7a-e), many more buckling modes exist. 45

12 Alexander N. Guz Fig. 8. In addition, it should be noted that it is considerably more difficult to solve problems of fracture mechanics (within the framework of the second approach) for unidirectional fibrous composites under uniaxial compression for the cases in Figs. 6a-e and Figs. 7a-e than problems of the dynamics and stability of crystal lattices (for the corresponding cases) within the framework of solid-state physics. The point is that solid-state physics investigates the behavior of perfectly rigid balls interacting with each other via mechanical springs linking them. In the above problems of fracture mechanics (solved within the framework of the second approach), fibers interact via the matrix (via a deformable body) and, moreover, a fiber itself is a deformable body. Thus, the difference indicated corresponds to that between the mechanics of discrete systems and the mechanics of distributed-parameter systems. In a mathematical sense, this distinction corresponds to that between problems for ordinary differential equations and problems for partial differential equations. The results obtained in the above formulations by the author and his followers were reported in the monographs [9, 0, 7], the third and fourth reviews in the special issue [36] being devoted to the problem under consideration. The same results were included in the doctoral theses (second doctorate degree) of I. Yu. Babich, Vik. N. Chekhov, and Yu. N. Lapusta. In closing the present section, we should summarize that results pertaining to the first of the seven problems under consideration have been analyzed rather carefully. Moreover, some general issues of fracture mechanics for composites have been stated. In this connection, we shall use below the above information to analyze other problems in a reduced form. 3. End-crush fracture of compressed composites Crush of the ends of specimens and structural elements made of composites is a rather frequently encountered fracture mechanism. In general terms, this phenomenon lies in the fact that, for example, uniaxial compression of specimens and structure elements made of composites results in local fracture of the material near the ends, this fracture not spreading far from the ends and decreasing with distance from them. As an example, Fig. 9 shows how the ends of a unidirectional boron-aluminum composite (50% volume content of boron fibers) crush under uniaxial compression. Additional data are presented in the monograph [7], which also refers to publications describing the results corresponding to Fig. 9. Figure 0 shows the cross section of the above mentioned unidirectional boron-aluminum composite. It should be noted that the design model depicted in Fig. 6e represents quite well the structure of the unidirectional boron-aluminum composite in the cross section. 46

13 On study of nonclassical problems of fracture and failure mechanics and related mechanisms Fig. 9. Fig. 0. It should be noted that experiments on composite specimens are usually based on various constructive and technological techniques that exclude crush of the ends. Among such methods are winding a string on the specimen near its ends, increase in the cross-sectional area of the specimen near its ends (for example. Fig. 4), and fixation of the ends of the specimen in clips made of a more rigid material (for example. Fig. 5). These techniques have been partially discussed in the previous section in considering the conditions for experimental investigation of internal fracture (fracture throughout the thickness). Thus, in experimentally investigating endcrush fracture, one should not use techniques that exclude this phenomenon. Note also that the investigation of end-crush fracture is rather complicated and sometimes yields ambiguous results. This is because of the fact that the experiment produces a result (a picture of a specimen destroyed near the ends and values of the ultimate strength) corresponding to a fracture process that has already been and whose initial stage was characterized by crush of the ends. To examine and describe the phenomenon in question, data on the initial stage of fracture must be available (in order to reveal the causes and mechanism of this phenomenon). It is difficult to identify processes at the initial stage of fracture from the complex end-crush pattern (corresponding to the final stage of fracture). The above information and considerations address experimental investigations of end-crush fracture. A theoretical analysis of end-crush fracture addresses a unidirectional or orthogonally reinforced composite loaded with a compressive force applied to the end surface and directed along the reinforcing elements (fibers). Thus, the composite is compressed along the axis of symmetry of the material properties. In the above described situation at the initial stage of fracture with crush of the ends, the most likely fracture mechanism is near-surface buckling of the microstructure of the composite near the loaded end with buckling modes subsiding with distance from the end. With such an interpretation of the phenomenon, a number of results were obtained 47

14 Alexander N. Guz by using various approximate models based on one and two-dimensional models of rods and plates. The author was the first to analyze end-crush fracture by using the three-dimensional linearized theory of stability of deformable bodies (for example [9, 0, 5, 24]) for elastic (brittle fracture) and plastic (fracture of composites with a metal matrix) models. Such an approach excludes occurrence of errors characteristic of various approximate models. The major results obtained by the author and his followers within the framework of this approach were reported in the monograph [7]. Two approaches were developed within the framework of the abovementioned analysis and the three-dimensional linearized theory of stability of deformable bodies, as for the fracture mechanisms analyzed in the previous section. The first approach is based on the model of a homogeneous anisotropic body with reduced constants (the continual theory). The second approach is based on the model of a piecewise-homogeneous medium, the equations of the three-dimensional linearized theory of stability of deformable bodies used separately for the filler and binder, and the conditions of continuity of the stress and displacement vectors at the filler-binder interfaces. Let us consider some results obtained by the first approach as applied to brittle fracture with! crush of the ends. We will present, as an example, the theoretical ultimate strength ( ) " for end-crush brittle fracture of a unidirectional composite under uniaxial compression along the axis 0x, 3 2 " $ G# 2 2 E % # $!! # %. 3 T ' ' (( SM ( ) G ( ) & = " " " ) EE# ) E# ** In this case, the composite material is simulated by a homogeneous transversally isotropic body with reduced elastic constants for which the planes x = const are isotropy planes. In (3.) 3 and below, the following notation is used: E! is Young's modulus along the axis 0x, E is 3 Young's modulus in the plane x = const, 3! " and! are Poisson's ratios, and G! = G = G is the 23 3! " shear modulus. In terms of the above notation and (2.), the theoretical ultimate strength ( ) for the case of internal fracture (fracture throughout the thickness far from the ends) under compression along the axis 0x is expressed as follows: 3 ( )! 3 G". T SM 3 T (3.) # = (3.2) The following inequality holds for a fibrous unidirectional (along the axis 0x ) composite: 3 E! >> G!. (3.3) From (3.)-(3.3), for the composite being considered, we obtain!! ( ) ( ) SM. 3 T 3 T " > " (3.4) From inequality (3.4), it follows that, according to the continual theory, end-crush fracture begins somewhat earlier than internal fracture (massive fracture of the specimen). This conclusion fully complies with experiments and explains the necessity of adopting additional constructive and technological techniques, partially considered above, to obtain experimental results on the massive (internal) fracture of the specimen. It should be noted that similar results were also obtained for plastic fracture (composites with a metal matrix). Note also that some results on brittle fracture were obtained on the basis of the 3 T 48

15 On study of nonclassical problems of fracture and failure mechanics and related mechanisms second approach (the model of a piecewise-homogeneous medium). These results pertain to elementary problems for laminated composites with a filler of small concentration. The above formulated findings of the author and his followers were reported in the monograph [7]. The same results were partially included in Yu. V. Kokhanenko doctoral thesis (second doctorate degree). 4. Shredding fracture of composites under tension or compression along the reinforcing elements This fracture mechanism is not observed in homogeneous materials (to which metals and alloys may be conditionally attributed) and is only characteristic of composites with a welldefined preferred direction of reinforcement. This mechanism consists in separating material into parts along the direction of the compressive load provided that it acts along the reinforcement direction in unidirectional composites and along one of the reinforcement directions in composites with a well-defined preferred direction of reinforcement. Let us consider some experimental results as examples. Figures,2, and 3 show the fracture of a unidirectional boroaluminum composite, unidirectional fiberglass, and fiberglass of longitudinatransversal winding, respectively, each compressed along the reinforcement direction. Additional information is given in the monograph [7], which refers also to the studies where the results shown in Figs. -3 were obtained. It should be noted that the fracture in Fig. was due to the use of the additional techniques excluding the fracture shown in Fig. 9. The fracture mechanism considered in the present section, following [7], will be conditionally called shredding. Note that this mechanism was repeatedly observed in wood. Remark. It is interesting to note that the phenomenon of shredding was also observed in Fig.. tension of a composite along the preferred reinforcement direction. The related references are given in the monograph [7]. This remark is important, since some authors associate shredding under compression with the limiting values of the transverse elongation (at the cost of Poisson's ratio). Fig

16 Alexander N. Guz The approach indicated cannot explain shredding under tension, since, in this case, transverse compression occurs and the material is destroyed along the direction of the tensile force (perpendicularly to transverse shortening). Therefore, we conclude that shredding is apparently determined by a more complex mechanism of fracture in the microstructure. Note also that even with seemingly successful application of the concept of limiting transverse elongation to compressed unidirectional composites, nevertheless, the mechanism of fracture in the microstructure has yet to be revealed, since this concept operates with integral characteristics (from the viewpoint of the mechanics of microstructure) and the force does not act in the transverse direction. Fig. 3. The basic specific features of shredding fracture (examples are in Figs. -3) may be considered as follows: (i) it occurs in compression and, in some cases, in tension of unidirectional fibrous and laminated composites under a load acting along the fibers and layers, (ii) it is manifested as massive (not local) fracture over planes and surfaces located in parallel to the reinforcing elements, (iii) it is realized if additional techniques excluding internal fracture as in Figs. 4 and 5 are used. It should be noted that shredding fracture occurs under a load acting along the reinforcing elements and in planes and surfaces located along the reinforcing elements. Nevertheless, this fracture mechanism occurring both under compression along the reinforcing elements and, in some cases, under tension along the reinforcing elements allows us to exclude various buckling phenomena from the fracture mechanism being considered. Thus, under uniaxial tensioncompression along the reinforcing elements, the following fracture mechanism is the most probable. Shredding fracture may occur only due to internal forces (stresses) that act perpendicularly to the direction of the external load and arise under the influence of the composite microstructure or due to the associated tangential stresses. Thus, it is necessary to determine the mechanism of occurrence of the mentioned stresses in the microstructure of the composite and to find out whether these stresses reach the limiting values corresponding to the shredding fracture, provided that the stresses along the reinforcement direction (along the external loads applied) do not reach their limiting values. It is well-known from numerous studies that the microstructure of a composite contains various distortions (bending). Thus, it is practically always possible for various stresses to occur in the microstructure of the composite. In view of the above stated information and considerations, the author, in the papers referred to in [7], first explained the mechanism of shredding fracture by internal stresses that arise due to 50

17 On study of nonclassical problems of fracture and failure mechanics and related mechanisms distortions (bending, curved interface) of the microstructure and act on areas the normal to which coincides with that to the curved interface. Let us apply the approach proposed to the microelement shown in Fig. 4 and including the curved interface between a reinforcing element and the matrix. The quantities pertaining to the reinforcing element and the matrix are marked with the indices a and m, respectively. In Fig. 4, H is the rise and! is the distortion halfwave length (the distortion as a curved interface is assumed periodic), n and ô are the unit vectors of the normal and the tangent to the curved ( a) ( m) interface,! and! are the stresses directed along and counterbalanced with the external load, and ( a ) ( ) ( ), a, m nn n! nn " " " and ( m) n! " are the stresses in the reinforcing element and the matrix that are applied to the curved interface and self-balanced within each period of distortion. It should be noted that the latter stresses reverse their sign in a half-period along the axis 0x (Fig. 4). Hence, under a stretching or compressing external load, tensile stresses, which may exhaust the adhesion strength, always occur on isolated areas of the curved interface. Similar considerations are applicable to tangential stresses. Fig. 4. Let us consider some quantitative estimates. For example, the following condition is satisfied for not so large distortions: H <<!. (4.) It is obvious that for composites reinforced along the axis 0x (Fig. 4) and subjected to the load being considered, the following inequality holds: ( m) ( a)! <<! (4.2) by virtue of the natural inequality E E, m a where << (4.3) E and a E are Young's moduli of the reinforcing elements and matrix identically m 0x (Fig. 4). In view of loading being considered for elastic models elongated along the axis (brittle fracture), the following relations are also valid: ( a) ( m) # " E! k # " E! k, (4.4) m a, m m where k and k are coefficients. a m By virtue of the continuity of the stress vector on the curved interface, the following relations hold: ( a) ( m) ( a) ( m) nn nn n! n! " = ", " = ". (4.5) 5

18 Alexander N. Guz Because of conditions (4.5) on the interface, we cannot already obtain estimates for the stresses ( m) ( m)! and " in the form of the second expression in (4.4). Hence, we cannot also obtain nn n! estimates (4.2) for these stresses on the interface despite the presence of inequality (4.3). Thus, the ( m) ( m) stresses! and " on the interface may be considerably larger than those obtained from the nn n! second expression in (4.4). Taking into account the above estimates and considerations, we should clarify the following ( a) ( m) situation. We will consider that the stresses! and! (Fig. 4) are lower than the corresponding ultimate strengths separately for the reinforcing elements and the matrix ( a) ± ( a) ( m) ± ( m)! < ",! < ". (4.6) Conditions (4.6) imply that the composite is not destroyed due to the break of the reinforcing elements and the matrix under the stresses acting along the direction of the external load. Note that conditions (4.6) can be satisfied by imposing constraints on the value of the external load. Thus, it is necessary to prove that there exist reasonable limits of change in H and! (Fig. 4) for which conditions (4.) and (4.6) are satisfied and the following conditions for the stresses ( m) ( m)! and " on the interface are also satisfied: nn n! ( m) ( m) A + or A + nn t n! s " = " =, (4.7) where A + and A + are the ultimate adhesion tensile and shear strengths on the interface of the t s reinforcing elements and the matrix. We should also note that the conditions A t + ( m) + <! (4.8) are usually satisfied for the majority of structural composites due to the presence of various defects on the interface. Thus, for the proposed mechanism of shredding fracture, it is necessary to prove that there exist reasonable limits of change in H and! (Fig. 4) for which, under conditions (4.) and ( m) ( m) ( m) (4.6), stresses! and " (on the interface) no less than the stresses! may occur (Fig. 4). nn n! Note that in solving the above problem, it is necessary to construct a continual theory such that self-balanced (within each distortion) stresses can be determined. Usually, continual theories allow us to determine stresses on areas exceeding the sizes of distortions. Therefore, the author developed a variant of the three-dimensional continual theory that meets the requirements formulated above. The basic results on the construction of this theory were reported in the monograph [7]. The problem formulated above (after expressions (4.8)) was solved positively for elastic models (brittle fracture) within the framework of the above mentioned continual theory. The author and S. D. Akbarov used the exact three-dimensional model of piecewisehomogeneous bodies to develop the fundamentals of the mechanics of laminated and fibrous composites with curved structures for elastic models. The basic results on the construction of the mechanics of composites with curved structures were reported in the monographs [2, 7]. The problem formulated above (after expressions (4.8)) was also solved positively for elastic bodies (brittle fracture) based on the above mentioned mechanics of composites with curved 52

19 On study of nonclassical problems of fracture and failure mechanics and related mechanisms structures and the exact and rigorous three-dimensional model of a piecewise-homogeneous medium. Thus, the specially developed continual theory and the model of a piecewise-homogeneous medium prove the proposed explanation of the mechanism of shredding fracture. The results obtained in the above formulations by the author and his followers arc reported in the monographs [2, 7], the first review in the special issue [36] being devoted to this problem. The same results were included in S. D. Akbarov doctoral thesis (second doctorate degree). 5. Brittle fracture of cracked materials with initial (residual) stresses near cracks Initial or residual stresses occur in materials or structural elements due to various processes and affect significantly the fracture mechanisms. When initial stresses act along cracks (on Fig.5 and 0 below, initial stresses are marked with the index 0,! are initial stresses, and! are 22 additional (acting or service) stresses), it is impossible to take into account the effect of these stresses on the behavior of fracture within the framework of the linear mechanics of brittle fracture. The point is that within the framework of the linear mechanics of brittle fracture and based on the solution of a linear plane problem of elastic theory (the Inglis-Muskhelishvili solution), the initial stresses! do not affect the stress intensity factors and the crack opening displacement 0 (vertical displacements on a horizontal axis); hence, these stresses do not affect the commonly accepted fracture criteria of linear fracture mechanics. 0 Note that the indirect influence of the initial stresses! (Fig. 5) can be realized through the specific surface energy!. However, this is very difficult, since, with such an approach, the quantity! must depend on both the initial stresses and the type of problems, the latter being investigated. Nevertheless, it seems that the initial stresses must exert some effect on brittle fracture. To prove this idea, let us consider two mental experiments. Figure 6a shows two strings located in parallel and close to each other and subjected to a tensile force acting along their axes. To separate the strings, forces are applied to the middle of their length perpendicularly to their axes. It is obvious to spread apart the strings by a given value of the deflection, it is necessary to apply forces with values depending on the tension of the strings, i.e., on the value of the tensile force. Figure 6b shows two rods located in parallel and close to each other and subjected to a compressive load acting along their axes. To separate the rods, forces are applied to the middle of their lengths perpendicularly to the rod axes. It is that to spread apart the rods by a given value of the deflection, it is necessary to apply forces with values depending on the compressive load. When the compressive load reaches the Euler critical value, the system is in neutral equilibrium, and it is sufficient to apply insignificant forces perpendicularly to the axes of the rods to separate them. Fig

20 Alexander N. Guz Fig The above mental experiments clearly demonstrate that the initial stresses! (Fig. 5) must affect brittle fracture. Apparently, the mechanical effects resulting from the mental experiments must also be manifested. Thus, we can consider that the necessity of developing the mechanics of brittle fracture of initially stressed materials is proved to a certain measure. The nonclassical nature of the problem being considered is determined by the fact that the linear mechanics of brittle fracture cannot, as shown in the beginning of the present section, take into account the effect of initial stresses. Apparently, the necessary information on the problem being considered could be obtained by using the basic relations of the general nonlinear theory of elasticity. Nevertheless, with the rather general approach indicated, there is no hope to obtain specific information on cracked materials within the framework of the physically and geometrically nonlinear theory of elasticity for bodies with rather general elastic relations, though such studies deserve appropriate support. Nevertheless, there exists a certain class of problems of the mechanics of brittle fracture for cracked materials with initial stresses acting along the cracks, some success was achieved in solving these problems. This is the case where the initial stresses! are much greater than the acting stresses! (Fig. 5), and, therefore, studies can be made within the framework of the 22 linearized theory of elasticity. The above formulation of problems under the linearized theory of elasticity is quite obvious and logical for composites with cracks located along the reinforcing elements. Note that, in the case being considered (cracks are located along the reinforcing elements), the breaking stresses! in composites with a preferred reinforcement direction (Fig ) can be an order of magnitude less than the initial stresses! (Fig. 5), which are the stresses along the reinforcing elements. In view of the foregoing considerations, the author developed the fundamentals of the mechanics of brittle fracture for cracked materials with initial stresses acting along cracks using the three-dimensional linearized theory of elasticity for compressible and incompressible bodies with an elastic potential of arbitrary structure. The major results were reported in the monographs [2] and vol. 2 [37]. The basic tenets of the above mentioned mechanics of brittle fracture for initially stressed materials are presented below.. In the initial stress-strain state, a cracked body is loaded so that initial stresses do not occur on the crack planes. 2. When additional (with respect to the initial stress-strain state) loads are applied to the body, the perturbations of the stress state are considerably less than the initial stress state. 3. The initial stress-strain state has a structure such that it can be considered (to sufficient accuracy) locally homogeneous near the cracks. 4. The solutions of linearized problems of elastic theory obtained within the framework of the linearized mechanics of cracked materials with initial stresses acting along the cracks are unique. 0 54

21 On study of nonclassical problems of fracture and failure mechanics and related mechanisms The basic results obtained within the framework of the mechanics of brittle fracture for materials with initial stresses [2] and vol. 2 [37] are given below.. The fundamentals of the mechanics of brittle fracture for materials with initial stresses were developed, including formulation of problems, methods for solution of plane and spatial problems, and the formulation of the fracture criterion that takes into account the effect of initial stresses. 2. The studies were made by using the three-dimensional linearized theory of elasticity, the theory of functions of complex variables for plane problems, and the theory of potential for spatial problems. 3. Specific problems were solved; among them are static and dynamic plane problems for one rectilinear crack and static axisymmetric and nonaxisymmetric spatial problems for one circular or elliptic crack. 4. New mechanical (quantitative and qualitative) effects of initial stresses were revealed in studying phenomena under static and dynamic loading. 5. It was proved that as the initial stresses tend to zero, all the results obtained [2] and vol. 2 [37] change over to the analogous results of the classical linear mechanics of brittle fracture. Of the various mechanical effects revealed during the studies being reviewed, we should mention the primary mechanical effect: as the initial stresses approach the values corresponding to the surface instability of the halfspace, the primary quantities change sharply. In the case of free cracks, the ultimate loads in the indicated situation tend to zero, which complies with the inferences of the mental experiments made at the beginning of the present section. The phenomena described are of resonant nature and fit the properties of linear mechanical systems. If we applied the general nonlinear formulation, then the peaks of the resonant phenomena would apparently be smoothed out. As a result, when the initial stresses reach values corresponding to the surface instability of the half-space, the ultimate load is not equal to zero, but the tendency for its sharp decrease must remain. As an example, let us consider a compressible isotropic body with a crack in the form of a circular disk of radius a located in the plane x = 0. Identical initial stresses characterized by the 3 coefficient of elongation! act along the axes 0x and 0x. Thus, in the initial state, the following 2 relations hold:! =! = const,! = 0, " = ". (5.) For the initial stresses (5.), let us consider the general shear problem where uniform shear stresses of intensity Q are applied to the upper and lower crack surfaces (Fig. 7) at an angle! to the axis 0x. According to the exact solution [2] and vol. 2 [37], the longitudinal and lateral shear intensity factors K and K are nonzero in the case being considered. To determine these III II quantities, the following expressions are used: K = K! K, K = K! K. (5.2) III 0 II 0 III III II II In (5.2) and below, the following notation is used: K and K are the longitudinal and lateral III II shear stress intensity factors in an initially stressed material (5.) for a given external load, and K are the longitudinal and lateral shear stress intensity factors in the material without initial 0 II 0 K III 55

22 Alexander N. Guz III II stresses for the same external load, and K and K are the dimensionless longitudinal and lateral shear factors characterizing the effect of initial stresses (5.). Figure 8 shows the factors III II K and K as functions of the elongation coefficient! for compressible isotropic materials with harmonic elastic potentials. Note that! = corresponds to a material without initial stresses,! > to the case of tension, and! < to the case of compression. In Fig. 8, the curves with III numbers,2!!, and 3! pertain to the quantity K and correspond to the values of Poisson's ratio II! = 0.20, 0.30, and 0.50 and the curves with numbers, 2, and 3 pertain to the quantity K and correspond to the values of Poisson's ratio! = 0.20,0.30, and From the results in Fig. 8, it follows that the stress intensity factors (for the problem being considered) depend significantly on the initial stresses. The results obtained in the above formulations by the author and his followers were reported in the monographs [2] and vol. 2 [37] and the review [3]. Fig. 7. Fig Fracture under compression along parallel cracks The fracture mechanics for materials compressed along parallel cracks is a special division of fracture mechanics, which cannot describe the fracture mechanisms within the framework of classical fracture mechanics. This fact follows from the following elementary problem shown in Fig. 9a: an infinite isotropic body compressed along the axis 0x and having in the plane 0x = 0 2 a flat crack infinite along the axis 0x. A more complicated problem is presented in Fig. 9b: an 3 infinite isotropic body containing a system of flat cracks, which are located in parallel planes, and compressed along the cracks. In the cases shown in Figs. 9a, b, a homogeneous stress-strain state occurs for arbitrary nonlinear models of isotropic deformable bodies. A similar situation also arises for arbitrary nonlinear models of orthotropic deformable bodies when the cracks are located 56

23 On study of nonclassical problems of fracture and failure mechanics and related mechanisms in the planes of symmetry of the material properties. Since here a homogeneous stress-strain state occurs, no singular part will be in the solution of the above mentioned problems for arbitrary nonlinear models of isotropic and orthotropic deformable bodies. Therefore, for the stress intensity factors, we obtain K = 0, K = 0, K = 0. (6.) I II III By virtue of conditions (6.), we see that the Griffith-Irwin fracture criterion [7, 35] and the critical crack opening displacement criterion do not work in the case being considered (Figs. 9a, b) for arbitrary nonlinear (elastic, plastic, and viscous) models of isotropic and orthotropic deformable bodies. The same is true of all the other derived fracture criteria. Thus, we may conclude that when the compression along system of parallel cracks is considered in fracture mechanics, approaches distinct from those of classical fracture mechanics should be invoked. Fig. 9. To define more clearly the classes of problems examined in the present section, let us formulate the following two remarks. Remark. Some authors assign the case where compression is along two cracks that extend from the contour of a circular opening and are located on the continuation of the same diameter of the opening to the class of problems being considered. The compression along cracks lying in one plane is indeed investigated here, but, in this case, the cracks are in a complex stress field caused by the stresses concentrated near the opening. In particular, tensile stresses also occur near the crack tips, which enables us to apply the fracture criteria of classical fracture mechanics. Thus, the above mentioned problems pertain to classical fracture mechanics and have nothing to do with the problems considered in the present section. Remark 2. Some authors believe that fracture mechanics for the case shown in Figs. 9a, b can be constructed by taking into account the microstructure of the material at the tip of an advancing crack. Therefore, it is assumed that the crack for x = ± a (Fig. 9a) propagates not along the plane line x = 0 (Fig. 9a) but along some broken nearly plane curve following the 2 microstructure of the material at the crack tip. When external loads act, as in Fig. 9a, tensile and shear stresses occur on the faces of the crack propagating along the assumed broken curve. Thus, with the tensile and shear stresses thus introduced, we can already apply the criteria of classical fracture mechanics. Casting no doubt on the feasibility of studies based on the above-stated approach, we should nevertheless point out that a cracked material under the action of tensile and shear stresses is examined at the final stage of these studies. Thus, this approach is in no way related to the problems being analyzed in the present section. Analyzing the prospects for the above stated approach, it is expedient, in the author's opinion, to take into account the following two considerations.. Allowance for the effect of the microstructure of the material at the crack tip for the case shown in Figs. 9a, b is apparently the next stage in learning to apprehend fracture mechanics, which is usually developed in terms of phenomenological (continual) notions. 57

24 Alexander N. Guz 2. In attempting to implement the above approach, it is necessary to conduct very complex studies to identify the phenomena occurring at the crack tip at the microstructural level and determining a broken line the path of crack propagation with the model representations used in fracture mechanics. It is the author's opinion that the above considerations (especially, the second one) do not enable us to expect that (in the near future) finished results will be obtained with the help of the above approach. Based on the foregoing information, we may apparently consider that the problem shown in Figs. 9a, b and formulated in terms of continual notions is so to speak special (singular) within the framework of continual fracture mechanics. In the case being considered, it is natural to consider (as in the mechanics of structure elements under compression) that the onset of fracture coincides with local loss of stability of the equilibrium state of the material that surrounds the crack, which is a rather general and logical concept for the situation presented in Figs. 9a, b. It should be noted that as a continuation of die deformation process, two situations may develop after the local buckling near the crack. In the first situation, the local buckling near the crack leads to massive fracture of the material. In this case, the study of the fracture process is completed by the determination of the critical load associated with the local buckling near the crack. In the second situation, the local buckling initiates a passage to an adjacent equilibrium mode near the crack and the massive fracture of the material does not occur. In this case, the fracture process should be studied based on the stress-strain distribution in the adjacent equilibrium near the crack. Currently, the scientific literature addresses, as a rule, only the former situation. Two approaches to the implementation of the above stated general concept have currently evolved in the scientific literature. In the first approach, which is followed by the majority of authors, whose studies were elucidated in numerous publications, the part of the material (the shaded area in Figs. 20a, b) between the parallel cracks (Fig. 9a) or between the crack and the boundary surface (Fig. 20b) is replaced by a beam (for plane problems) or a plate and a shell (for spatial problems). This approach was called the beam approach and was used in [38] for the first time. Such beams, plates, and shells are analyzed within the framework of various applied theories of stability of thin-walled systems (by invoking the Bernoulli, Kirchhoff-Love, Timoshenko, etc. hypotheses). Specific results are usually obtained under specific boundary conditions (rigid fixing or hinged support) at the ends of thin-walled elements fanned as mentioned above, though we actually deal with, so to speak, an elastic restraint. Fig. 20. Casting no doubt on the possible usefulness of the beam approach in the above formulation, we present the following two considerations.. Such an approach, even from the geometrical standpoint, falls far short of being applicable to all the cases presented in Figs. 9 and 20. For example, it is impossible to assign defensibly a thickness to the beam in the case of one crack (Fig. 9a), applied theories of stability of thin- 58

25 On study of nonclassical problems of fracture and failure mechanics and related mechanisms walled systems do not work in the case of a thick-walled beam, and two short cracks (Fig. 20a), etc. 2. In all the cases presented in Figs. 9 and 20, the beam approach introduces an irreducible error (even for thin-walled elements) into the results it produces. The point is that the energy variation at the crack tip is determined by the nature of the singularity at the crack tip, the energy varying significantly at the crack tips of cracked bodies. However, when applied theories are used to describe the deformation of those beams, plates, and shells, it is impossible to obtain the order of singularity at the crack tip that corresponds to the exact (three-dimensional) description. The above considerations (especially the second one) indicate that we should use more strict approaches that adequately describe the phenomena under consideration (at least, with accuracy accepted in the mechanics of deformable bodies). In the second approach, the basic relations and methods of the three-dimensional linearized theory of stability of deformable bodies (for example, [9, 0, 5, 24]) are used to study local buckling near cracks in the cases shown in Figs. 9 and 20 for finite and small subcritical strains and for elastic and elastoplastic models. In this case, the theoretical ultimate strength under compression along parallel cracks coincides with the critical load corresponding to local buckling near cracks and calculated within the framework of the three-dimensional linearized theory of stability of deformable bodies. Note that, in the second approach, by plastic fracture is meant the case where the entire (not only near cracks) material in a subcritical state deforms in the plastic domain. The second approach does not introduce major errors, which are characteristic of the first approach, and allows us to obtain results with accuracy accepted in the mechanics of deformable bodies. The author proposed to apply this approach to the brittle and plastic fracture of materials (elastic and plastic bodies with rather general constitutive equations). Exact solutions to plane and spatial problems for the arbitrary number of plane cracks located in one plane were obtained with the help of the second approach. The theory of functions of complex variables was applied to plane problems and the theory of potential was applied to spatial problems. In succeeding years, the second approach made it possible to solve problems for the cases shown in Figs. 2a-c: two equal parallel cracks (Fig. 2la), one near surface crack parallel to the free surface (Fig. 2b), and an infinite periodic row of equal parallel cracks (Fig. 2c). Plane problems (for infinite (from a plane) cracks) and spatial problems (for disk-like cracks) were investigated for the cases shown in Figs. 2a-c. In solving the above stated problems, specific numerical results were obtained for hyperelastic materials, structural composite materials, and composite materials with a metal matrix (the model of a homogeneous orthotropic material with reduced constants was applied to composite materials). The method of integral transforms was applied with subsequent reduction to pair integral equations, which in turn were reduced to integral Fredholm equations. To illustrate the above results, let us present two specific examples. Fig

26 Alexander N. Guz Fig. 22. Example. A plane problem on an isotropic hyperelastic material axially compressed along infinite periodic rows of equal cracks (Fig. 22a). The following notation is used in Fig. 22: 2a is the crack length, 2h is the distance between two neighboring cracks in an infinite periodic row of 0 cracks,! is the compressive load,! is the critical shortening along the axis 0x (Fig.22a), "! = # ", where! is the coefficient of elongation along the axis 0x, and! = ha is a 2 dimensionless parameter. Figure 22b shows the critical shortening! as a function of the dimensionless parameter!. The same dependence but in the vicinity of zero is shown in the right portion of Fig. 22b. The dashed line (Fig. 22b) stands for the critical shortening! for one isolated crack (as! " # ). Based on the results in Fig. 22b, we may conclude that the interaction between the cracks in an infinite periodic row of equal cracks results in a decrease in the critical shortening (! ) by an order of magnitude and more. It should be noted that here (Fig. 22a) the local buckling near cracks coincides with the massive fracture of the material, since the local buckling (for the infinite row of cracks) forms so to speak a plastic joint over the entire thickness and, as a result, the structure element loses carrying capacity. Example 2. A plane problem on a structural composite axially compressed along two equal parallel cracks (Fig. 23a). The following notation is used in Fig. 23: 2a is the crack length, 2h is the distance between the cracks,! is the compressive load,! is the critical load (the second 0 ap approach and the exact solution for this approach),! is the critical load (the first approach the beam approximation), and! is the relative error in the critical load determined on the basis of the beam approximation, ap! = " #" $ " # $ 00%. (6.2) Fig

27 On study of nonclassical problems of fracture and failure mechanics and related mechanisms From the results in Fig. 23b, it follows that the beam approximation applied even to comparatively long cracks ( a! 8h ) produces an error of 0%. Thus, it is quite desirable to use the second approach (an exact approach based on the three-dimensional linearized theory of stability of deformable bodies) in studies even for comparatively long cracks. The results obtained by the author and his followers based on the second approach were included in the monographs [7] and vol. 4 book [37]. These results were also included in V. M. Nazarenko doctoral thesis (second doctorate degree). 7. Brittle fracture of cracked materials under dynamic loads (in view of the contact interaction of the crack faces) When loads determined by propagating waves act on a body, the crack faces displace and the displacement reverse sign with time, since, for example, in the specific case of a longitudinal wave, tension and compression phases arise. Thus, the sign reversal of displacements of the crack faces is apparently a typical feature of the phenomena arising under dynamic loads. This feature always appears in dynamic fracture mechanics, at least, in bounded bodies due to the complex diffraction pattern. From the foregoing, it follows that it is necessary (under a correct and strict formulation of problems of dynamic fracture mechanics) to take into account the contact interaction of the crack faces. This interaction occurs irrespective of the level of the acting load and also takes place, naturally, when the linear equations of the mechanics of deformable bodies describing the laws of wave propagation are applied. It is natural that this phenomenon occurs in simulating a crack by a mathematical cut. It should be noted that such simulation is used in all publications on dynamic fracture mechanics. Failure to account for the contact interaction of the crack faces in dynamic fracture mechanics leads to irreducible errors, since the contact interaction is a necessary aspect determined by the physics of phenomena. In this connection, we should apparently consider that all the results on dynamic fracture mechanics obtained earlier do not conform to the physics of the phenomenon under consideration. Certainly, the case in point is the results obtained within the framework of the mechanics of a deformable body. So serious inference will become obvious after the consideration below of an elementary problem of dynamic fracture mechanics, which is, in a certain sense, standard and is presented in practically all monographic issues on dynamic fracture mechanics. Fig

28 Alexander N. Guz Fig. 25. Let us consider the case where a material is simulated by a linearly elastic isotropic body in Cartesian coordinates x ( j =, 2,3 ). Let the body be infinite and have in the plane x = 0 (Fig. 2 j 24) a flat crack infinite along the axis 0x and of constant width 3 2a (the axis 0x is directed 3 perpendicularly to the plane of Fig. 24). Let also a dynamic load independent of the coordinate x 3 be applied. Thus, we can consider a plane problem in the plane x 0x (Fig. 24) for an infinite 2 body with a crack of length 2a along the axis 0x (Fig. 24). Let a plane harmonic expansion wave propagate along the axis 0x (Fig. 24). In this case, in view of the wave diffraction, all 2 quantities (stress and displacements) have the multiplier ( exp i! t ), including near the crack tips. Let us consider the phenomena that occur, for example, near the right tip of the crack (Fig. 25a). Under the action of the load, the following three characteristic phases take place near the right tip of a crack, as well as all over the body: (i) the quiescent state (Fig. 25a), (ii) the phase of tension corresponding to the maximum crack opening (Fig. 25b), and (iii) the phase of compression corresponding to the maximum crack closure (Fig. 25c). It should be noted that phase (iii) occurs a half-period later than phase (ii). In Fig. 25b, A denotes the opening of the crack faces at some point near the right tip in the phase of tension. All publications on dynamic fracture mechanics use crack face boundary conditions that do not take into account the phenomena arising in the phase of compression, the contact interaction being disregarded and so to speak free penetration of the crack faces into each other being allowed. This situation assumed in dynamic fracture mechanics to occur in the phase of compression is shown in Fig. 25c, where the crack faces have as though exchanged places the upper face occupies the position of the lower face (Fig. 25b) and the lower face passes to the position of the upper face (Fig. 25b). In Fig. 25c, B denotes the value of mutual penetration of the crack faces (at the same point as in Fig. 25b) near the right tip of the crack in the phase of compression. The situation specified in Figs. 25a-c is encountered in all publications on dynamic fracture mechanics when the boundary conditions corresponding to an unloaded crack are assigned. The equality A = B (7.) follows from the foregoing and Figs. 25b,c. Thus, in dynamic fracture mechanics, the phenomena described correctly in the phase of tension are of the same order as those described incorrectly in the phase of compression. Hence, the results of dynamic fracture mechanics incorporate reducible error irrespective of the intensity of the load applied. It should be also noted that the approximate design model for the phase of compression shown in Fig. 25c and used in practically all publications on dynamic fracture mechanics indicates that the material receives, so to speak, an additional lag when the contact interaction of crack faces is not taken into account. Therefore, one should expect publications on dynamic fracture mechanics to contain overestimated maximum quantities, including the stress 62

29 On study of nonclassical problems of fracture and failure mechanics and related mechanisms intensity factors. Additional information on the above aspects of dynamic fracture mechanics may be found in [25]. The construction of dynamic brittle-fracture mechanics for cracked materials with regard for the contact interaction of the crack faces was proposed (as doctoral thesis second doctorate degree) to V.V. Zozulya in 989. Today, the main published results on dynamic brittle fracture mechanics in the above formulation belong to the author of the present paper and V.V. Zozulya and their followers. It is natural that problems in the above formulation are nonlinear, since the size of the contact region is determined by the displacements of the crack faces and other quantities. In the elementary case, such a problem consists of linear equations and nonlinear boundary conditions. To solve such problems, step-by-step methods, where a problem with a fixed contact region is solved at each step, are applied. This is a conventional approach to the solution of problems with a contact region whose size is determined from the solution of the same (nonlinear) problem. Specifically, the method of solution based on reduction to boundary integral equations with constraints in the form of inequalities on crack faces, the boundary element method with a time-step scheme, and an iterative process were used in the above-mentioned studies. Specific results were obtained for one and two flat cracks under harmonic dynamic loads. Note that the problem under consideration should be classed among nonclassical problems of fracture mechanics, since it takes into account sharply changing of the configuration of the body at the crack tip, i.e., the contact interaction of the crack faces (Condition 2 in Chapter of this paper). Moreover, the nonclassical nature of this problem is also determined by the fact that it considers the physical phenomenon that was disregarded by other authors and that should be considered, as follows from the considerations in the first part of this section. In summary, let us present one result of dynamic brittle fracture mechanics for cracked materials with regard for the contact interaction of crack faces. This result pertains to the reference problem shown in Fig. 24 (a plane harmonic longitudinal expansion wave in a linearly elastic isotropic body). For example, Fig. 26 presents the dimensionless stress intensity factor K as a function of the dimensionless frequency a! c " (dimensionless wave number). In Fig. 26 and below, the following notation is used: 2a is the crack length,! is the frequency of the harmonic * longitudinal expansion wave, c is the velocity of the expansion wave, K is the ratio of the peak I value of the stress intensity factor K for dynamic loading (with allowance for the contact I interaction of crack faces) to K -the stress intensity factor for static loading. In Fig. 26, curves I and 2 stand for dynamic brittle fracture mechanics, respectively, without and with allowance for the contact interaction of crack faces. Two conclusions follow from the results in Fig. 26: when the contact interaction of crack faces is considered, the maximum value K is (i) decreased and (ii) reached at a different dimensionless frequency. It should be noted that the former conclusion coincides with the physical considerations (stated in the first part of this section) on the increase in the lag effect of the material near the crack tips when the contact interaction of crack faces is not taken into account. * I * I Fig

30 Alexander N. Guz The results obtained on dynamic brittle fracture mechanics in view of the contact interaction of the crack faces (within the framework of a nonclassical problem) were reported in the monograph vol. 4 book 2 [37]. These results were also included in V.V. Zozulya doctoral thesis (second doctorate degree). 8. Fracture of thin-walled cracked bodies under tension with prebuckling In classical fracture mechanics for cracked materials subject to tension and shear, it is tacitly assumed that fracture begins when the body is in the strain-free state. In other words, it is assumed that the body does not change its configuration sharply during deformation prior to fracture, i.e., buckling does not precede fracture (Condition 2 in Chapter of this paper). In actuality, buckling may precede fracture even in tension of thin-walled bodies. To illustrate the foregoing, let us consider the well-known Inglis-Muskhelishvili solution of the problem on the elastic equilibrium of a cracked plate under uniaxial tension (Fig. 27). The distribution of the stresses! acting along the crack periphery is shown in the upper part of Fig. 27. The shaded areas correspond to the compressive stresses! occurring under uniaxial tension at infinity, and the curves with numbers (-; -0.3; -0.) separate the areas with stresses! of different intensity. Since the compressive stresses act along the crack, the local buckling characterized by the normal deflection w may precede fracture for cracks longer than the plate thickness. A similar situation may also arise in thin-walled shells under external loads causing as though only tensile stresses near the cracks. Fig. 27. Thus, the present section has outlined issues associated with the construction of fracture mechanics for thin-walled plates and shells under external loads, which generate mainly tensile stresses near the cracks. The case where local buckling near the cracks may precede fracture has been taken into account. To construct fracture mechanics in this case, it is necessary (i) to study the phenomenon of local buckling and (ii) to study the fracture mechanism based on the shape of the thin-walled element and the stress-strain distribution corresponding to the postbuckling state of the thin-walled element. To date, the majority of publications were devoted, with rare exception, to the former study, including theoretical and experimental approaches. Minor publications containing, with rare exception, experimental results were devoted to the latter study. In study (i), the majority of authors employ approximate design models and schemes, which are as follows. A strip of certain width and length is conditionally selected along the crack (Fig. 28a) (in Fig. 28b, this strip is shaded). After that, the strip is analyzed for stability under loads corresponding to the initial stresses in the cracked plate (Fig. 28c). The approximateness and 64