Jet J, Sofar SrraeTees Vol 34. No. 20. pp. 2563 2 81 2768397517+f PI:S0020-7683(96)0017-1 BRITTLE SOLID UNDER COMPRESSION PART I: GRADIENT MECHANISMS OF MICROCRACKING L BLECHMAN National Building Research Institute. Faculty of Civil Engineering Technion-Israe ute of Technology. Technion City Haifa 32000 Israel r Receired 25 October 1995: in rerisell form 28 March 19961 Abstract-The venerable problem of the origins of cracking and failure of a brittle heterogeneous solid(heterogen) under compression is analyzed here from a new point of view. with the cause of its microcracking and atrophy (degeneration) under load. found. first of all. in the differences in Poissons ratio of its components na. found in experiments. are the basis of the new ch to the nder compression(1) The non-linear part of the ascend Inch of SSe is due to formation and accumulation of stab/e microcracks. (2) The intrin modulus remains constant up to the peak point. (3] Concrete and rock fail in splitting. (4) The pattern of SSc is the same for different types of concrete and rock under diflerent types of loading It is found that some kinds af 'gradient mechanisms can induce local transerse strains of tension n. The first creates local strain gradient nong the components due to difference in their Poisson extension. when the components with ower Poisson s ratio are l d in the lateral direction there is also n internal thrust due to gradient in the elastic moduli of the component It is shown that in a brittle solid built from randomly oriented crystals. a population of lateral tensioned crystals, called"acron", are created due to gradients in Poisson's ratio of a single crystal along its three axes. The modeis ident strain in the acron are given including the equatior of critical strains. The problem of crystals acting as"pistons"due to a process of sliding is also sed nd their stochas instead of growing into macrocrack s, they are stable. in good accordance with a vast number of experiments. Gradient mechanisms. especially that of Poisson are universal, descriptir'e and base They suffice to exhaust the bearing capa creasing compression without recourse to shear stresses. They affect every heterogen pression: rock materials, concrete. ceramics -and do not need initial microcracking to an realize the process of atrophy (degradation) of the brittle solid. ( 1997 Elsevier Science DEFINITIONS t first a brittle heterogeneous solid will be referred to as a heterogen. A atrogen is a spe without cry close the particles, but it does not create a matrix. The case of interest is a cre stalon, a kind of p of random/n oriented crystals almost without glue components. NOMENCLATURE Heterogen brittle heterogeneous solid longitudinal strain ateral strain induced by load nd their gradients, respectivcly lateral strain induced by Poisson extensi gitudinal and lateral stress respectivel elastic modulus gradient fact f the
Pergamon BRITTLE SOLID UNDER COMPRESSION. PART I: GRADIENT MECHANISMS OF MICROCRACKING I.BLECHMAN National Building Research Institute. Faculty of Civil Engineering. Technion-Israel Institute of Technology. Technion City. Haifa 32000. Israel Abstract---The venerable problem of the origins of cracking and failure of a brittle heterogeneous solid (heterogen) under compression is analyzed here from a new point of view. with the cause 01 its microcracking and atrophy (degeneration) under load. found, first of all. in the differences in Poisson’s ratio of its components. Four fundamental phenomena, found in experiments, are the basis of the new approach to the behavior and failure of a het~rogen under compression. (I) The non-linear part of the ascending branch of SSc is due to formation and accumulation of stahk microcracks. (2) The intrinsic elastic modulus re~1~u111.s UVZS~U~~ up to the peak point. (3) Concrete and rock fail ill sp/itrb!y. (4) The pattern of SSc i.s rile suww for different types of concrete and rock under different types of loading. It is found that some kinds of’gradien/ rnecham.ws can induce loctil IIZ~.YL’W.W .struin.c of tensiott and cause microcracking in a heterogen under cornprrssion. The firsr creates local strain gradients among the components due to difference in their Poisson extension, when the components with lower Poisson’s ratio are tensioned in the lateral direction. There is also a mechanism which creates an internal thrust due to gradir~it ilz /he elastic moduuli of the component. It is shown that. in a brittle solid built from randomly oriented crystals. a population of lute&/J, tc~vioned cr~,stals, called “acrons”. are created due to &radients in Poisson’s ratio of a single crystal along its three axes. The models of gradient strain in the acronq are given. Including the equation of critical strains. The problem of crystals acting as “pistons” due to a process of sliding is also discussed. The gradient models explain the appearance of microcracks and their stochasticity and why. instead of grooving, into n~~rocraclis. they are stable. in good accordance with a vast number of experiments, Gradient mechanisms. especially that of Poisson, are rrnircrsol, descriptiw und bawd on measurable parameters. They suffice to exhaust the bearing capacity of a heterogen under inci-easing compression without recourse to shear stresses. They affect every heterogen under compression : rock materials, concrete. ceramics -and do not need initial microcracking to initiate and realize the process of atrophy (degradation) of the brittle solid. ( 1997 Elsevier Science Ltd. DEFINITIONS At first a brittle heterogeneous solid will be referred to as a l~r~oyrn. A tizcrtro,qen is a special important class 01 heterogens. which consist of a continuous ntatri.u and of prrrticles (aggregate) that “float” in the matrix without mutual contact, In an artificial matrogen, like concrete. the granulometry is given and kept under very close control. A heterogen built of contacted particles or grains can be called a pa~rugrri. There can be a ‘glue’ between the particles. but it does not create a matrix. The case of interest is a cr~:?tnlon. a kind of patrogen which is built of random/r oriented crJTttds almost without glue components. Heterogen ssc i:J i:. i:* 0. 02 E 6 Lt. m NOMENCLATURE brittle heterogeneous solid the curve of stress-strain relationship longitudinal strain lateral strain induced by load transverse strains and their gradients, respectively lateral strain induced by Poisson extension longitudinal and lateral stress. respectively elastic modulus Poisson’s ratio gradient factor indices of the components
2564 L. RUPTURE OF BRITTLE SOLID 1. State of the In traditional description. the limiting strength of brittle solids is considered with the failure inducing peak load, whereby the failure concept is associated with development of a major crack. Today this definition is obsolete. With more precise specimen-testing techniques. the descending branch in the stress-strain curve (SSc)is well recognized. and an ever-growing olume of findings regarding its fcatures and design applications is being published. The upshot is that in spite of the inherent brittleness of concrete and rock the maximum stress not the endpoint of the loading curve: although it still represents the peak response of the specimen. it is no longer associated with total failure. In other words, it can no longer be identified with the moment of rapid collapse of the material following the onset and development of a major crack. Hence the need has arisen to bring out the"cushioned, soft effects whereby the resistance of the material reduces gradually as the stress level increases Thus the strength problem is a part of a general one that of describing the state changes in the material which accompany the increase in the strains and stresses This problem is under intensive study in a number of directions, which may be classified according to approach and to the type of the considered physical process, namely Models of macrofracture( fracture mechanics Meso-models(damage analysis) Local (micro )mechanical models of constituent interactions Models on the molecular level Abstract models(purely mathematical treatment In fracture mechanics, Mindess(1983). failure is attributed to an initially present major crack. Today it is clear that this scheme (referred to as "Mode I'") cannot by itself account for the physical aspects of damage and strength of brittle solids To quote NMAB Report (1983), ". The committee identified two separate behavior patterns in compressive fracture. The first is an extrinsic behavior pattern resulting from large cracks, comparable with the size of the part, amenable to normal approaches of fracture mechanics The sccond is an intrinsic behavior pattcrn, resulting from the accumulation of dormant microcracks and culminating in terminal shear faulting that has been analyzed previously onstitutive instability, not amenable to treatment br normal procedures of frac Naturally, in fracture mechanics, priority is given to the domain from the point of peak stress down the descending branch of the SSc. The fracture approach is applied in a considerable number of works on fracture energy determination--as a basic characteristic for prediction of concrete failure based on measurements of the fracture energy [ Swamy (1971). Shah (1985), van Mier (1986), Malvar(1987). Bazant (1987). The difficulties inherent in the fracture approach are properly illustrated in Mindess(1983) The molecular approach is mostly concerned with the structure of hard cement paste Feldman(1968), and existing data are qualitative rather than quantitativ As regards the mathematical approach, considerable effort has been invested in searches for abstract stress-strain relationships(of which more than twenty have been proposed to date). without reference to the mechanism of the microcracking process Popovics, 1971) With this lack of physical meaning, none of them can claim superiority he (relatively recent)so-called damage model approach deals with the nonlinear part of the ascending branch--in terms of the microcracking effect(Kzajcionovic, 1986: Lemaitre 1987) Here the material is considered at the level of"unit cells "containing a statistically valid sample of weak spots and microcracks which exists before the loading and is treat by micromechanical models. Yet it was established experimentally long ago, that the main phenomenon of this stage is accumulation of stable microcracks, which begin to merge near the strength point, forming macrocracks (with consequent division of the solid in parts (Berg(1950.1961,1971). Slate(1963.1981a,1981b.1986). Li and nordlund(1993)
2564 I. Blechman 1. RUPTURE OF BRITTLE SOLID 1.1. Stute qf the urt In traditional description, the limiting strength of brittle solids is considered with the failureinducing peak load, whereby the failure concept is associated with development of a major crack. Today this definition is obsolete. With more precise specimen-testing techniques, the descending branch in the stress-strain curve (SSc) is well recognized. and an ever-growing volume of findings regarding its features and design applications is being published. The upshot is that, in spite of the inherent brittleness of concrete and rock the maximum stress is not the endpoint of the loading curve; although it still represents the peak response of the specimen. it is no longer associated with total failure. In other words, it can no longer be identified with the moment of rapid collapse of the material following the onset and development of a major crack. Hence the need has arisen to bring out the “cushioned, soft” effects whereby the resistance of the material reduces gradually as the stress level increases. Thus the strength problem is a part of a general one--that of describing the state changes in the material which accompany the increase in the strains and stresses. This problem is under intensive study in a number of directions, which may be classified according to approach and to the type of the considered physical process, namely : Models of macrofracture (fracture mechanics) : Meso-models (damage analysis) ; Local (micro)mechanical models of constituent interactions ; Models on the molecular level : Abstract models (purely mathematical treatment). Injracture mechanics, Mindess (1983). failure is attributed to an initially present major crack. Today it is clear that this scheme (referred to as “Mode I”) cannot by itself account for the physical aspects of damage and strength of brittle solids. To quote NMAB Report (1983), ” The committee identified two separate behavior patterns in compressive fracture. The first is an extrinsic behavior pattern resulting from large cracks, comparable with the size of the part, amenable to normal approaches of fracture mechanics. The second is an intrinsic behavior pattern, resulting from the accumulation of dormant microcracks and culminating in terminal shear faulting that has been analyzed previously as a constitutive instability, not amenuhle to treutment by normal procedures qf’,flacture mechanics”. Naturally, in fracture mechanics, priority is given to the domain from the point of peak stress down the descending branch of the SSc. The fracture approach is applied in a considerable number of works on fracture energy determination-as a basic characteristic for prediction of concrete failure based on measurements of the fracture energy [Swamy (1971), Shah (1985), van Mier (1986). Malvar (1987), Bazant (1987)]. The difficulties inherent in the fracture approach are properly illustrated in Mindess (1983). The molecular upproach is mostly concerned with the structure of hard cement paste, Feldman (I 968), and existing data are qualitative rather than quantitative. As regards the muthemntirrrl approach, considerable effort has been invested in searches for abstract stress-strain relationships (of which more than twenty have been proposed to date). without reference to the mechanism of the microcracking process (Popovics, 197 I). With this lack of physical meaning, none of them can claim superiority. The (relatively recent) so-called dumcqe model approach deals with the nonlinear part of the useending branch-in terms of the microcracking effect (Kzajcionovic, 1986; Lemaitre, 1987). Here the material is considered at the level of “unit cells” containing a statistically valid sample of weak spots and microcracks which exists before the loading and is treated by micromechanical models. Yet it was established experimentally long ago, that the main phenomenon of this stage is accumulation of stable microcracks. which begin to merge near the strength point, forming macrocracks (with consequent division of the solid in parts), (Berg (1950, 1961. 1971), Slate (1963. 1981a, 1981b. 1986). Li andNordlund (1993))
Brittle solid under compression--I A number of difficulties can be associated with the damage models a. What are the cause and the rules of the onset of microcracking? Where does the in a fully compress b. Why does the tensile effect invariably"pick" new sound spots in concrete and rock creating new microcracks with increasing stress--instead of opening up existing ones? Or why do microcracks, once induced, remain dormant? c. Why are the planes of new microcracks induced under increasing load, oriented Possible answers to these questions are given below in terms of transverse gradient strains. 1. 2. Griffith's point of tieu In view of Griffith's role in fracture mechanics, it may be of interest to begin with a quote from his second paper, (1924, page 62). which states his view on the difficulties in transition from a highly homogeneous amorphous solid. without even any scratches on its surface, to a more or less rough heterogen. Griffith conducted his study on very carefully prepared glass specimens! )"The rupture strain energy of the strongest silica rods is so great enquire. therefore what becomes of this energy when fracture occurs. It is not converted ito heat: Indeed. measurements have shown that the detect of elasticity of the strong material is so inappreciably small that such an operation would take several minutes at ast. What actually happens is that the energy is used up almost entirely in forming new surfaces, that is. surfaces of fracture, in the material. The disintegration of a strong drawn down part is sometimes so complete that recognizable portions of this part are difficult to find. The sick ends also are usually broken, owing to the propagation of an elastic wave from the original fracture In concluding, I wish to refer to some of the difficulties which impede the further development of the theory of rupture. On attempting to pass from isotropic(amorphous) solids to brittle crystals we at once meet the difficulty that the surface tension is not a constant. but is a function of the position of the surface in the space lattice so that the thcorctical strcngth is diffcrcnt for diffcrcnt faccs. Thc anisotropic naturc of the clasticity is the further obstacle As we can see. Griffith saw very clearly the: complexity and difficulties of transition from an isotropic material to a heterogen. It can also be said that these difficulties are not over even now. as shown by Mindess(1983) 1.3. Tension under compression It is now clear enough that destruction in uniaxially compressed concrete is a result of local transverse tension[Berg( 1950, 1971). Slate(1981, 1986). Delibes(1987) F Slate et al. in(1986) drew the following conclusion A tensile (or tensile-shear mechanism is the most relevant crack mechanism controlling failure of concrete in uniaxial compression. This failure occurs in a direction perpendicular to applied load for all the concretes tested Normal strength concretes develop highly irregular failure surfaces including a large amount of bond failure. Medium strength concrete develop a similar mechanism, but at higher strain. The failure mode of high strength concretes is typical of nearly homogenical material. Failure occurs sudden/y in a vertical nearly fat plane passing through the aggregat and the orte This fact is t of the approach, developed in Blechma the nonlinearity of concrete behavior on the ascending branch of SSc is explained b microcracking Yet theoretically, in a continuous elastic solid under uniaxial compression transverse tensile stresses cannot be induced. An attempt to lay the responsibility on the existing oblique cracks is limited by the simple fact that they do not change, Slate(1981,1986 under loading. but stay dormant"up to stresses near the peak point of SSc. Instead of
Brittle solid under compression- I A number of difficulties can be associated with the damage models : 7565 a. What are the cause and the rules of the onset ~f’microcrucking? Where does the tension in a fully compressed solid come from’? b. Why does the tensile effect invariably “pick” lzebi’ sound spots in concrete and rock, creating new microcracks with increasing stress-instead of opening up existing ones? Or. why do microcracks. once induced, remain dormant? c. Why are the planes of new microcracks, induced under increasing load, oriented longitudinally, in the direction of compression’? Possible answers to these questions are given below in terms of transverse gradient strains. 1.3. Griffith’s poitlt qf’uiew, In view of Griffith’s role in fracture mechanics, it may be of interest to begin with a quote from his second paper, (1924, page 62). which states his view on the difficulties in transition from a highly homogeneous amorphous solid, without even any scratches on its surface. to a more or less rough heterogen. (Griffith conducted his study on very carefully prepared glass specimens!) “The rupture strain energy of the strongest silica rods is so great that if it were all converted into heat it would make the rod red-hot. It is of interest to enquire. therefore, what becomes of this energy when fracture occurs. It is not converted into heat; indeed. measurements have shown that the defect of elasticity of the strong material is so inappreciably small that such an operation would take several minutes at least. What actually happens is that the energy is used up almost entirely in forming new surfaces, that is, surfaces of fracture, in the material. The disintegration of a strong drawndown part is sometimes so complete that recognizable portions of this part are difficult to find. The sick ends also are usually broken, owing to the propagation of an elastic wave from the original fracture. In concluding, I wish to refer to some of the difficulties which impede the further development of the theory of rupture. On attempting to pass from isotropic (amorphous) solids to brittle crystals we at once meet the difficulty that the surface tension is not a constant. but is a function of the position of the surface in the space lattice, so that the theoretical strength is different for different faces. The anisotropic nature of the elasticity is the further obstacle.” As we can see, Griffith saw very clearly the complexity and difficulties of transition from an isotropic material to a heterogen. It can also be said that these difficulties are not over even now. as shown by Mindess (1983). 1.3. Tension under compressiorl It is now clear enough that destruction in uniaxially compressed concrete is a result of local transverse tension [Berg (1950, 1971). Slate (1981, 1986). Delibes (1987)J. F. Slate pt al. in (1986) drew the following conclusion : “A tensile (or tensile-shear) mechanism is the most relevant crack mechanism controlling fhilure qf concrete in uniaxial compression. This failure occurs in a direction perpendicular to applied load for all the concretes tested. Normal strength concretes develop highly irregular failure surfaces including a large amount of bond failure. Medium strength concrete develop a similar mechanism. but at higher strain. The failure mode of high strength concretes is typical of nearly homogenical material. Failure occurs sudden/J) in a vertical, nearly,@& plane passing through the aggwgutr und the mortar.” This fact is the basis of the approach. developed in Blechman (1988,1989,1992), where the nonlinearity of concrete behavior on the ascending branch of SSc is explained by microcracking. Yet theoretically, in a continuous elastic solid under uniaxial compression transverse tensile stresses cannot be induced. An attempt to lay the responsibility on the existing oblique cracks is limited by the simple fact that they do not change, Slate (1981. 1986). under loading. but stay “dormant” up to stresses near the peak point of SSc. Instead of
opening the initial microcracks, new microcracks appear. but they also are stable and do w NMAB Re eport(198 whe the transverse tension also cannot be explained by this part of Poisson exte ch is restricted by press-platens. As commonly known, without restraints, Poisson exter on does not induce stresses. However, when the friction between platens and specimen is eliminated. the pattern of material splitting becomes especially clear. It should be noted that besides concrete and rock. other brittle materials like ceramic and cast iron fail in the ame manner. If so, we should come to the conclusion that the mechanisms of failure of brittle solids are very general and independent from the individual features of concrete rock and so or 2. SPECIFICITY OF HETEROGENS 2.1. General features Heterogens are isotropic in macro, since their behaviour and strength are independent of the direction of loading They are heterogeneous and anisotropic in micro, since they are built of randomly oriented ind randomly combined components, whose properties are different when takenl in the direction of the load The internal order in artificial heterogens like ceramics and concrete is of a heari/1 restricted stochastic structure. Local parameters in it fluctuate only between given limits and mean parameters are kept in line with the technical requirements by rigorous quality control during the production process The intrinsic elastic modulus of heterogen--E, measured under low-cyclic loading. is constant, as long as the integrity of the heterogen is retained, Karsan(1969) To study the brittle solids from nature(rock materials), we have to classify and sort them into groups with the sume siructure. Then the deviations of their features will be restricted. similar to artificial heterogens Under short-term uniaxial load and normal temperature. a heterogen has no plastic strains. which oth out the influence of local gradients. Due to absence of plasticity, the tensile strength of brittle solids in macro is much lower than their ompressive strength, in contrast to heterogen with high plasticity like soft metals. The failure of a heterogen under compression is always preceded by cocracking (see Berg(1950), Slate(1981a), Shah(1968b), Glucklich(1971b)) Microcracks induced in a heterogen during loading are local and stable. Their plane is parallel to the direction of maximal compressive stress. It has also been known for emission. when the previously applied stress state is exceeded( Kaiser effect). Li and Nordlund (1993 Microcracking is the reason for the nonlinearity of the stress-strain curve at the ascending branch under short-term compression in both uniaxial and triaxial comprcssion. It is clear now that there is no plasticity in the non-linear stage of loading. except for triaxial compression with high lateral stress Accumulation of dormant microcracks gradually causes the heterogen to degenerate internally. This process is intrinsical and therefore is called"atrophy". not damage. which can be a result of external mechanical action, Failure sets in when the limiting atrophy is reached, which is the moment when the increment in loading energy absorbed by the heterogen equals the loss of energy due to its atrophy(degeneration). blechman (1992)
2566 1. Blechman opening the initial microcracks, new microcracks appear. but they also are stable and do not grow, NMAB Report (1983). The transverse tension also cannot be explained by this part of Poisson extension which is restricted by press-platens. As commonly known, without restraints, Poisson e.yterzsion does not induce stresses. However. when the friction between platens and specimen is eliminated. the pattern of material splitting becomes especially clear. It should be noted that besides concrete and rock. other brittle materials, like ceramic and cast iron, fail in the same manner. If so, we should come to the conclusion that the mechanisms of failure of brittle solids are very general and independent from the individual features of concrete. rock and so on. 2. SPECIFICITY OF HETEROGENS 2.1. General ji7atztre.s ??Heterogens are isotropic irl nmcro, since their behaviour and strength are independem of the direction of loading. ??They are heterogeneous andanisotropic in micro, since they are built of randomly orierlted und randornl~~ combined conlpoilents, whose properties are different when taken in the direction of the load. ??The internal order in artificial heterogens like ceramics and concrete is of a heatli!,, restrictedstochastic structure. Local parameters in it fluctuate only between given limits and mean parameters are kept in line with the technical requirements by rigorous quality control during the production process. a The intrinsic elastic modulus of heterogen-E, measured under low-cyclic loading, is constant, as long as the integrity of the heterogen is retained, Karsan (1969). ??To study the brittle solids from nature (rock materials), we have to classify and sort them into groups with the same structure. Then the deviations of their features will be restricted. similar to artificial heterogens. ??Under short-term uniaxial load and normal temperature. a heterogerz has no plasric struins, which can smooth out the influence of local gradients. Due to absence of plasticity, the tensile strength of brittle solids in macro is much lower than their compressive strength, in contrast to heterogen with high plasticity like soft metals, where the strengths in tension and compression are equal. 2.2. Microcrucking7fundumentals ??The failure of a heterogen under compression is always preceded by microcracking, (see Berg (1950), Slate (1981a), Shah (1968b), Glucklich (1971b)). ??Microcracks induced in a heterogen during loading are local and stable. Their plane is parallel to the direction of maximal compressive stress. It has also been known for a long time, that under repeated load the microcracks are usually detected by acoustic emission, when the previously applied stress state is exceeded (Kaiser effect). Li and Nordlund (1993). ??Microcracking is the reason,/& the nonlinearity of the stress-strain curve at the ascending branch under short-term compression in both uniaxial and triaxial compression. It is clear now that there is no plasticity in the non-linear stage of loading, except for triaxial compression with high lateral stress. ??Accumulution oj’ dornzunt microcracks gradually causes the heterogen to degenerate internally. This process is intrinsical and therefore is called “atrophy”. not damage, which can be a result of external mechanical action. Failure sets in when the limiting atrophy is reached, which is the moment when the increment in loading energy absorbed by the heterogen equals the loss of energy due to its atrophy (degeneration), Blechman (1992)
Brittle solid under compression- I 2.3. Integrity of heterogen The above mentioned features are effective up to the point of peak stress on the stress- strain curve. As is seen from the condition: E= const. integrity of a heterogen in the longitudinal direction is retained in the above domain even when microcracks occur After the peak point of SSe the macro cracks usually split the heterogen into parts (blocks). which can still carry a decreasing load. The features of the macrocracked heterogen are essentially different and we have to treat it as a heterogen of a different kind. Therefore the models considered below are built for an integral heterogen existing in the domain before the peak point of the stress-strain curve. Due to this distinction in the features of the heterogen, one can hardly imagine a unified model valid for both- the ascending and descending branches of the SSc 2. 4. Definition of heteroge For the proposed aim the following main features of the material, considered at the scending branch of SSc, define a brittle heterogen: (a) it is isotropic in macro, (b) it heterogeneous in micro, (c)its integrity is retained in the longitudinal direction of loading (d) there are essential differences in Poissons ratio and or in the elastic moduli of its components. (e) its intrinsic elastic modulus is constant in macro. and (f plasticity is absent under uniaxial compression 2.5. Origins and role of gradients The following main factors can induce gradient strains and stresses in a brittle solid Differences in Poissons ratio and in the elastic moduli of the component Residual stresses and local variations in density Local tension around pores and flaws. Zaitsev (1981) e Local shear due to gradients in shear modulus of the components Note: In this paper only the first item is treated the residual stresses are taken under consideration in part 2 When unconfined, Poisson extension (it is not tension )in uniaxially compressed solids does not create stresses. However, in a restricted state gradients in Poisson extension induce transverse strains in accordance with Hooke's law. When this restriction is due to differences gradicnt will be binations of compressive and tensile transverse strains. At the same time, differences in the elastic moduli of the components create heterogeneity in the strain-stress fields of the heterogen. which also induces lateral tension in it. as shown belor 3. LAYERED ELEMENT-SANDWICH Let us begin with the phenomenon of gradient strains in a uniaxially compressed "sandwich", namely a multilayered element, built from two solids. indexed a and m(aggr gatc and"matrix")with Poissons ratio wu>Ina Cutting from the""a part shown in Fig. I. we can write the following equations: eqn(1)of continuity between the two layers, eqn(2)of equality in the increment of the gradient lateral forces--d F (transverse compression and tension)in these two layers and eqn (3)of the increment in the longitudinal strains, due to the isostress state of COmDr ession in the layers dap -de= dens tda do/E (3b)
2.3. Intr~yritl- qf’hrterogm Brittle solid under compression- I 3567 The above mentioned features are effective up to the point of peak stress on the stressstrain curve. As is seen from the condition: E = const. integrity of a heterogen in the longitudinal direction is retained in the above domain, even when microcracks occur. After the peak point of SSc the wmcro cracks usually split the heterogen into parts (blocks), which can still carry a decreasing load. The features of the macrocracked heterogen are essentially different and we have to treat it as a heterogen of a different kind. Therefore the models considered below are built for an integral heterogen existing in the domain before the peak point of the stress-strain curve. Due to this distinction in the features of the heterogen, one can hardly imagine a unified model valid for both-the ascending and descending branches of the SSc. 2.4. Dyfirzition qf’heterogen For the proposed aim the following main features of the material, considered at the ascending branch of SSc, define a brittle heterogen: (a) it is isotropic in macro, (b) it is heterogeneous in micro, (c) its integrity is retained in the longitudinal direction of loading. (d) there are essential differences in Poisson’s ratio and/or in the elastic moduli of its components, (e) its intrinsic elastic modulus is constant in macro. and ( f) plasticity is absent under uniaxial compression. 2.5. Origins und role qj‘grudierlts The following main factors can induce gradient strains and stresses in a brittle solid under compression : ??Differences in Poisson’s ratio and in the elastic moduli of the components ??Residual stresses and local variations in density. ??Local tension around pores and flaws, Zaitsev (198 1). ??Local shear due to gradients in shear modulus of the components. Note: In this paper only the first item is treated, the residual stresses are taken under consideration in part 2. When unconfined. Poisson extension (it is not tension!) in uniaxially compressed solids does not create stresses. However. in a restricted state gradients in Poisson extension induce transverse strains in accordance with Hooke’s law. When this restriction is due to d~ff~rence.s in Poissorl’s ratio qf’ the conlponents, their gradient will be expressed in very local combinations of compressive and tensile transverse strains. At the same time, differences in the elastic moduli of the components create heterogeneity in the strain-stress fields of the heterogen, which also induces lateral tension in it. as shown below. 3. LAYERED ELEMENT~~“SANDWICH” Let us begin with the phenomenon of gradient strains in a uniaxially compressed “sandwich”, namely a multilayered element, built from two solids. indexed a and 111 (“aggregate” and “matrix”) with Poisson’s ratio \lr, > r,,,. Cutting frorn the “sandwich” a part shown in Fig. 1, we can write the following equations : eqn (1) of continuity between the two layers, eqn (2) of equality in the increment of the gradient lateral forces-dF (transverse compression and tension) in these two layers and eqn (3) of the increment in the longitudinal strains, due to the isostress state of compression in the layers. dF,, = dF,,,> (2) dc,, = dg/F,,, dti,,, = dclE ,,,. (3a) (3b)
I Blechman Fig. I. Incremental strains in layered solid drm, de, -longitudinal strain of m-and a-layers: da leo -free Poisson extension of m-and a-layers; dam--gradient strain of tension in m-layer:de* gradient strain of compression in cf-layer: da-full extension. I Initial state, 2-comlpesseu state where c-free Poisson extension and &*transverse strain induced by gradient: do. de- the increments in longitudinal stress and strain. respectively The increment of the gradient forces can be expressed by the parameters of the layers. when their elastic moduli in the lateral direction are taken as equal to those in the longitudinal direction. (The layers are taken as isotropic in macro. dF= h, e, dag (4a) dFn=h, E,r dem where: E--modulus of elasticity. ha. hm-thicknesses of layers As follows from(2)and(4) hE h e The factor he resses the relationship between the stiffnesses of the two layers The increments in free Poisson extension of the layers are (7b Substituting the above equations in(1) yields
I. Blechman Fig. 1. Incremental strains in layered solid dr:,,,. dL,, -longitudinal strain of v- and a-layers : ds;; _ tk{:‘-free Poisson extension of 11~ and u-layers; ds,T, --gradient strain of tension in m-layer : d+ gradient strain of compression in u-layer; df:-+ full estension. I Initial state, 2 -compressed state. where Y--free Poisson extension and c*-transverse strain induced by gradient : do. dc+ the increments in longitudinal stress and strain. respectively. The increment of the gradient forces can be expressed by the parameters of the layers, when their elastic moduli in the lateral direction are taken as equal to those in the longitudinal direction. (The layers are taken as isotropic in macro.) dF<, = h,,E,, dc:. (4a) dF,,i = II,,, E,,, dG. (4b) where : E--modulus of elasticity, h,,, IT,,,--thicknesses of layers. As follows from (2) and (4) : The factor: expresses the relationship between the stiffnesses of the two layers. The increments in free Poisson extension of the layers are : (6) (7a) (7b) Substituting the above equations in ( 1) yields :
da (8) The expression in brackets is a combi-gradient 8, which includes the influence of Poisson ratios and elastic moduli We will also denote P Integrating( 8)between the limits 0 and o, with Poissons ratio taken independent of o. we obtain the gradient of transverse strain of tension, accumulated in the m layer 00 The opposite gradient strain of transverse compression in the a-layer is Equation(10a)can be also presented as Eu (10c) n the expression in brackets is a Poisson gradient in the" sandwich", corrected in the second term by the factor Eu Er E With a -aEa the cquation of gradient strain of local tension becomes In the compressed layer the gradient strain is e=P288 2=P,e* As can be seen from eqn(9)with some proportionality between two pairs of Poisson ratio and elastic moduli. the gradient 8, in the"sandwich "can be very slight. For example: if a-0.26、Vm-0.13, when E-40000MPa,Em-20,000 then d,=0 gradient factors at crystals level--dur (described helow in Section 5). will be higher than between the la. described below. not due to interlayer differences
Brittle solid under compressiorl-I 1569 (8) The expression in brackets is a combi-gradient 6,, which includes the influence of Poisson ratios and elastic moduli : We will also denote : I PC, pmi - 1 + PCS ’ and pn = i+Pn Integrating (8) between the limits 0 and C. with Poisson’s ratio taken independent of 0. we obtain the gradient of transverse strain of tension, accumulated in the nz layer: E,T, = p”, (5,G. (1W The opposite gradient strain of transverse compression in the n-layer is Equation (10a) can be also presented as (lob) (IOC) Then the expression in brackets is a Poisson gradient in the “sandwich”, corrected in the second term by the factor EJE,, : With E = a/E,, the equation of gradient strain of local tension becomes (9b) In the compressed layer the gradient strain is : As can be seen from eqn (9) with some proportionality between two pairs of Poisson ratio and elastic moduli. the gradient 6$ in the “sandwich” can be very slight. For example : if v,, = 0.26, v,,, = 0.13, when E, = 40.000 MPa, E,,, = 20,000 then ~5,~ =O! When the “sandwich” is built from crystalline brittle solids, it is quite possible that the gradient factors at crystals level& a,,,, (described below in Section 5). will be higher than between the layers. Then the “sandwich” will be split due to intercrystalline gradients described below. not due to interlayer differences
2570 4. COMPRESSED MATROGEN 41 poi adient i In matrogens. unlike the "sandwich". the aggregate particles are surrounded by a matrix and therefore their longitudinal strains. not the stresses, are almost equal. Then, instead of eqn(3)the increments da dem in their longitudinal strains have to be written da.= de where ku, k, are a deviation of the components from the average increment in the material In an artificial matrogen. like concrete. the granulometry. i.e., the composition of aggregate of different size is kept under very close control. At the same time the dispositions of the large aggregate particles is random enough for assuming the same average value for the stiffness ratio of the components, at every cross-section, as follows k, l mi En (12) k,23E Here la and L are volume fractions of matrix and aggregate, respectively. The matrix comprises the hardened cement paste. sand and voids. For a matrogen, eqn(2)remains in effect and yields The free Poisson extension of the components will be de/ =y dE where I. Im, are the Poisson ratios of the aggregate and matrix, respectively. Equation(I) is also in effect for uhstituting the found expressions in(1)we have let Taking pn= 1/ (1+P, )and integrating(14)from 8=0 to a we obtain the equation for gradient strain of local transverse tension-emf induced in the matrix =pn(k。V-kmCm) The expression 8,=(k, 'a -kum)p can be defined as the gradient factor in matrogen. Then the gradient tensile strain =0 (15b) Then the gradient of the compressive strain will b en=pd,6. Let us attempt a rough estimation of the gradient factor in concrete, with vm=0. 14 and
2.570 I. Blechman 3. COMPRESSED MATROGEN 3.1 . Poissorl ‘s yradim t in matrogrn In matrogens. unlike the “sandwich”. the aggregate particles are surrounded by a matrix and therefore their longitudinal strains, not the stresses, are almost equal. Then, instead of eqn (3) the increments da,,, d;:,,, in their longitudinal strains have to be written as : where k,,, h-,,, are a deviation of the components from the average increment in the material. In an artificial matrogen. like concrete, the granulometry, i.e., the composition of aggregate of different size, is kept under very close control. At the same time the dispositions of the large aggregate particles is random enough for assuming the same average value for the stiffness ratio of the components-p,, at every cross-section, as follows “I = kc,,.; 3 E,, ’ when L’,, + i’,,, = I (12) Here, I’,,, and I’,, are volume fractions of matrix and aggregate, respectively. The matrix comprises the hardened cement paste. sand and voids. For a matrogen, eqn (2) remains in effect and yields : dc,T = p &e. (l5c) Let us attempt a rough estimation of the gradient factor in concrete, with v,, = 0.14 and
Brittle solid under compression v,=0.24, Neville (1971). For E,= 40,000 MPa, En= 15.000 MPa, U=to k=1.05 km=0.95 we have Po=0.375 and Pm, =0.73. Then the gradient factor will bo =0.73*(0.24*0.950.14*1.05)=0.06 4.2. Poisson gradient--alternatire approach In considering the gradient strain, we can try another approach, based on the apparent value of Poissons ratio of the matrogen-In. Measured in tests. Then the average gradient strain-g *induced by Poisson's ratio between the matrix and aggregate can be estimated =(V.一Vm) The condition of continuity is"built in" in(17), as the apparent valuc of Poissons ratio yas taken. Under this approach the gradient factor is defined and calculated as (18) Since poisson's ratio for concrete is about 0.20 and as given above, V,n=0. 14. the second estimation yields similar result to the above 8,=0.20-0.14-0.06 .3. Significance ng the process of microcracking and predicting the critical loading strains. However. due to stochasticity of the microcracking process, the gradient models cannot be simply used to explain the behavior and strength of concrete. In part 2 of this paper the solution of this problem is given On the other hand models can be used for revealing the resistance of the heterogen to microrupture. For example, if e=1.7*10 and 8=0.06 then the critical lateral gradient strain. which will induce microrupture. can be predicted as eft=1* 10, which falls within the well known limits 0.5-1.5+10-4 millistrain for concrete tensile strain at failure. when we know the critical strain of microcrupture in a heterogen. we can predict the limit of linearity in the sst 4. 4. Thrust in matrogen When the elastic moduli of matrix and aggregate in a matrogen are widely different (as in low-strength concrete), heterogeneity of the stress field will create local domains of thrust between aggregate particles To model the stress gradients induced by thrust in a uniaxially compressed heterogen we will consider the large aggregate particles as spheres of radius--r, arranged as shown in Fig. 2 in layers at distance H in the vertical and L in the horizontal direction. with the particles in a pyramid pattern. Here L> 21/ and bctween every pair of layers of aggregate there is a cushion-layer of matrix. The heterogen is under a longitudinal compressive strain e When the elastic modulus of the matrix is less than that of the aggregate the stresses the aggregate are higher than that in the matrix. Taking the stresses in matrix as references we can find the excess compressive stresses in the large particles for a unit cross-section of the element from the following expression kee(k, e-k, E) where k, is a factor of the disorder for the large particles and of dissipation of the excess stress over the fine- size particles Then the thrust in the base of a pyramid for this scheme will b
Brittle solid under comprrasion- I 2571 v,, = 0.24. Neville (1971). For E,, = 40,000 MPa, E,,, := 15,000 MPa, I’,, = z’,,,, k,, = 1.05. h-,,, = 0.95 we have p. = 0.375 and p,,? = 0.73. Then the gradient factor will be: 6, = 0.73*(0.24*0.95-0.14* 1.05) = 0.06. 4.2. Poisson graclienr-altesilati~e uppimx~i~ In considering the gradient strain we can try another approach, based on the apparent value of Poisson’s ratio of the matrogen---v,,. measured in tests. Then the average gradient strain it:* induced by Poisson’s ratio between the matrix and aggregate can be estimated as 6: = (\‘,, - V,,) )c. (17) The condition of continuity is “built in” in (17), as the apparent value of Poisson’s ratio was taken. Under this approach the gradient factor is defined and calculated as Since Poisson’s ratio for concrete is about 0.20 and, as given above. r,,, = 0.14. the second estimation yields similar result to the above c?, = 0.20-o. 14 = 0.06. 4.3. Sl@1jicancr The importance of the Poisson gradient factor lies, at first, in the possibility of explaining the process of microcracking and predicting the critical loading strains. However, due to stochasticity of the microcracking process, the gradient models cannot be simply used to explain the behavior and strength of concrete. In part 2 of this paper the solution of this problem is given. On the other hand models can be used for revealing the resistance of the heterogen to microrupture. For example, if c,, = 1.7 * 10-j and 6 = 0.06 then the critical lateral gradient strain. which will induce microrupture. can be predicted as E:.,, = 1 * lo-“, which falls within the well known limits 0.5-l .5 * 10pJ millistrain for concrete tensile strain at failure. Or, when we know the critical strain of microcrupture in a heterogen, we can predict the limit of linearity in the SSc. 4.4. Thrust it1 rmlrogm When the elastic moduli of matrix and aggregate in a matrogen are widely different (as in low-strength concrete), heterogeneity of the stress field will create local domains of thrust between aggregate particles. To model the stress gradients induced by thrust in a uniaxially compressed heterogen we will consider the large aggregate particles as spheres of radius--r, arranged as shown in Fig. 2 in layers at distance H in the vertical and L in the horizontal direction, with the particles in a pyramid pattern. Here L > ?H and between every pair of layers of aggregate there is a cushion-layer of matrix. The heterogen is under a longitudinal compressive strain e. When the elastic modulus of the matrix is less than ihat of the aggregate, the stresses in the aggregate are higher than that in the matrix. Taking the stresses in matrix as references we can find the excess compressive stresses in the large particles for a unit cross-section of the element from the following expression : Ao, = kl~3.‘&(li,~~:,,-k,,,E,,,). (19) where k, is a factor of the disorder for the large particles and of dissipation of the excess stress over the fine-size particles. Then. the thrust in the base of a pyramid for this scheme will be
2572 I Blechman Fig. 2. Pyramid scheme of aggregate particles and thrust in matrogen ,=k△a (20a) E- Here, k, is also a factor of disorder, but now related to the geometry of the pyramid Denoting kalkm=k and taking K,= k, k,k,, we obtain o,=K,Ua:(kE-en) In strain terms, dividing(20c) by Em, we have (2la) where the gradient factor of thrust dris E In the bulk of the heterogen the thrust in every"pyramid "is neutralized by the opposite thrust of the"neighbors". The thrust can manifest itself only at the free edges of the specimen and due to space-disorder of aggregate particles Estimation of Kr for concrete. To estimate the Kr we will use the state of local rupture
2512 I. Blechman Fig. 2. Pyramid scheme of aggregate particles and thrust in matrogen cr = k, Arr <- (2Oa) or (20b) Here. k, is also a factor of disorder, but now related to the geometry of the pyramids. Denoting kJk,,, = k, and taking K,, = k,k,k ,,,, we obtain ci, = iY,,c<;“~(kE,, - E,,). WC) In strain terms, dividing (20~) by E,,,, we have where the gradient factor of thrust 6, is (214 (21b) In the bulk of the heterogen the thrust in every “pyramid” is neutralized by the opposite thrust of the “neighbors”. The thrust can manifest itself only at the free edges of the specimen and due to space-disorder of aggregate particles. Estimation of Ktr.fbr concrete. To estimate the K,, we will use the state of local rupture where