PHYSICAL REVIEW E 91, 032134 (2015)
Failure criterion for materials with spatially correlated mechanical properties J. Faillettaz'
3G, University of Zurich, 8057 Zurich, Switzerland D. Or
STEP, ETH Zurich, 8092 Zurich, Switzerland (Received 22 October 2013; revised manuscript received 15 April 2014; published 24 March 2015) The role of spatially correlated mechanical elements in the failure behavior of heterogeneous materials represented by fiber bundle models (FBMs) was evaluated systematically for different load redistribution rules. Increasing the range of spatial correlation for FBMs with local load sharing is marked by a transition from ductilelike failure characteristics into brittlelike failure. The study identified a global failure criterion based on macroscopic properties (external load and cumulative damage) that is independent of spatial correlation or load redistribution rules. This general metric could be applied to assess the mechanical stability of complex and hetero geneous systems and thus provide an important component for early warning of a class of geophysical ruptures. DOl: 10.1103/PhysRevE.91.032134
PACS number(s): 64.60.av, 62.20.M -, 46.50.+a, 89.75.Fb
Introduction. Various geom orphological processes and natural hazards are triggered by gravity-driven instabilities (landslides, rockfalls, snow avalanches, and glacier failure). C om m on to such phenom ena are the abrupt and often extensive failures w ithin naturally occurring heterogeneous m aterials. In the context o f natural hazards, the prediction o f such catas trophic events is required for developing effective m itigation m easures. U nfortunately, the predictability o f such events is hindered by the com plex and highly nonlinear precursory dam age and failure processes within heterogeneous media. Certain brittle geologic m edia may fail by the propagation o f a single extensive crack, and considerable efforts have been devoted to the understanding, detection, and prevention o f crack nucleation [1]. For many heterogeneous materials the failure m ode is often m ore gradual and ductile in its nature [2], C atastrophic failure o f heterogeneous materials occurs as the culm ination o f progressive dam age involving com plex interactions betw een m ultiple defects and growing m icrocracks [2]. These precursor internal dam age events that could be detected through m icroseism ic or acoustic em ission [3] as has been show n for rockfall [4] or soil slope failure [5-7], M odels for m echanical interactions am ong many elem ents provide a m eans for studying progression towards failure, including the branching model, the therm al fuse model, the spring-block m odel, and the fiber bundle m odel (see [2,8] for a review). The sim plicity and versatility of the fiber bundle model (FBM ) offer a useful fram ew ork for system atically studying precursory processes preceding global failure [9-13]. A dditionally, the discrete nature o f failure events offers a direct link with acoustic em issions suggested for continuous m onitoring o f such progressive failures [3]. The m echanical properties o f natural m aterials are characterized by structural spatial correlation resulting from the com bined action of several physical, chem ical, or biological processes and from com plex interactions betw een geology, topography, clim ate, and soil use [14], However, studies o f the effect of such spatial correlations on the rupture m aturation process appear to be lacking.
’Also at STEP, ETH Zurich, 8092 Zurich, Switzerland. 1539-3755/2015/91 (3)/032134(6)
This study system atically evaluates the influence o f spatial organization o f m echanical properties on rupture behavior and its im plications for precursory events linked w ith early w arning for natural hazard prediction. Follow ing a b rief description o f the different num erical m odels developed in this work, the rupture behavior and associated m acroscopic properties at rupture o f sim ulated FBM s are presented for different spatial arrangem ents. Precursory activity preceding com plete breakdow n is also exam ined in a statistical fram ew ork aim ed at developing m etrics for failure im m inence. Finally, a general rupture criterion based on m acroscopic properties is presented, and its potential application to gravity-driven geophysical instabilities is discussed. Failure events in correlated fiber bundle model. We consider a set o f parallel elastobrittle fibers assem bled on a square regular lattice w ith periodic boundary conditions. The fibers exhibit a linear-elastic behavior [with a Y oung’s m odulus E (Pa) arbitrarily set to £ = 1] follow ed by an abrupt failure at a prescribed critical threshold load term ed •
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FIG. 1. (Color online) Illustration of the redistribution rule for the LFBM4 (left) and LFBM8 (right) on a 16 x 16 lattice with approximately the same number of failed fibers. Empty gray circles represent the fibers. Filled gray circles indicate failed fibers. The large thick cross in magenta indicates the failed fiber. The thick crosses represent the fiber connected to the failed fiber, forming the “crack” according to the four- (left) or eight- (right) neighbor scheme. The blue thick circle represents the fibers to which load will be redistributed in the next step. Note that periodic boundary conditions connect opposite borders, enabling a large crack to form. As evidenced here, long-range load redistribution is possible, depending only on the geometrical characteristics of the cluster formed by failed fibers.
load redistribution, in a similar fashion to the DFBM. Note also that such a geometrical mode of local load sharing is different from the methods of other studies that redistribute stress also according to the distance to the failed fiber [16-18]. Figure 2 shows for the three considered redistribution schemes the variation of effective stress and the fraction of unbroken fibers with applied stress for a bundle with uniform distribution of fiber strength. As expected, the rupture of the same bundle
appears for lower applied stress and lower number of failed fibers in the case of LFBM4: As such a redistribution rule promotes load concentration on the neighbors (the load is redistributed on fewer neighbors than in the LFBM8 or DFBM), the probability of triggering a cascade of failures of near neighbors is enhanced, favoring the appearance of global rupture. On the contrary, the LFBM8 appears to behave in a similar way to the DFBM as the redistributed load is “spread” on a large number of neighbors for a sufficiently highly damaged lattice (Fig. 1). The effects of heterogeneity patterns on the global rupture behavior of the bundle were studied using the values of fiber strengths a,h chosen randomly from a uniform distribution in the range from 0 to 1. For robust statistical representation, we generated 200 different initial configurations of each FBM parameter combination studied. Moreover, to account for natural spatial correlations (such as induced by a tree root system with large lateral extent), we reorganized the initially uniformly distributed configuration of fiber strengths to obtain different spatial correlation lengths (see Fig. 3, insets). The process involved three steps: (1) An optimized correlated lattice with a given correlation length CL (expressed in terms of lattice units) was computed by convolving in the Fourier space a Gaussian filter with a lattice composed of uniformly distributed fiber strengths. (2) Each fiber of the initial lattice was spatially migrated to the position on the optimized lattice with the nearest strength value. (3) Finally, a test on the reorganized lattice was performed to verify that the resulting spatial distribution of the fiber strengths ath followed the prescribed correlation length (Fig. 3). It is important to note
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Failure criterion for materials with spatially correlated mechanical properties J. Faillettaz'
3G, University of Zurich, 8057 Zurich, Switzerland D. Or
STEP, ETH Zurich, 8092 Zurich, Switzerland (Received 22 October 2013; revised manuscript received 15 April 2014; published 24 March 2015) The role of spatially correlated mechanical elements in the failure behavior of heterogeneous materials represented by fiber bundle models (FBMs) was evaluated systematically for different load redistribution rules. Increasing the range of spatial correlation for FBMs with local load sharing is marked by a transition from ductilelike failure characteristics into brittlelike failure. The study identified a global failure criterion based on macroscopic properties (external load and cumulative damage) that is independent of spatial correlation or load redistribution rules. This general metric could be applied to assess the mechanical stability of complex and hetero geneous systems and thus provide an important component for early warning of a class of geophysical ruptures. DOl: 10.1103/PhysRevE.91.032134
PACS number(s): 64.60.av, 62.20.M -, 46.50.+a, 89.75.Fb
Introduction. Various geom orphological processes and natural hazards are triggered by gravity-driven instabilities (landslides, rockfalls, snow avalanches, and glacier failure). C om m on to such phenom ena are the abrupt and often extensive failures w ithin naturally occurring heterogeneous m aterials. In the context o f natural hazards, the prediction o f such catas trophic events is required for developing effective m itigation m easures. U nfortunately, the predictability o f such events is hindered by the com plex and highly nonlinear precursory dam age and failure processes within heterogeneous media. Certain brittle geologic m edia may fail by the propagation o f a single extensive crack, and considerable efforts have been devoted to the understanding, detection, and prevention o f crack nucleation [1]. For many heterogeneous materials the failure m ode is often m ore gradual and ductile in its nature [2], C atastrophic failure o f heterogeneous materials occurs as the culm ination o f progressive dam age involving com plex interactions betw een m ultiple defects and growing m icrocracks [2]. These precursor internal dam age events that could be detected through m icroseism ic or acoustic em ission [3] as has been show n for rockfall [4] or soil slope failure [5-7], M odels for m echanical interactions am ong many elem ents provide a m eans for studying progression towards failure, including the branching model, the therm al fuse model, the spring-block m odel, and the fiber bundle m odel (see [2,8] for a review). The sim plicity and versatility of the fiber bundle model (FBM ) offer a useful fram ew ork for system atically studying precursory processes preceding global failure [9-13]. A dditionally, the discrete nature o f failure events offers a direct link with acoustic em issions suggested for continuous m onitoring o f such progressive failures [3]. The m echanical properties o f natural m aterials are characterized by structural spatial correlation resulting from the com bined action of several physical, chem ical, or biological processes and from com plex interactions betw een geology, topography, clim ate, and soil use [14], However, studies o f the effect of such spatial correlations on the rupture m aturation process appear to be lacking.
’Also at STEP, ETH Zurich, 8092 Zurich, Switzerland. 1539-3755/2015/91 (3)/032134(6)
This study system atically evaluates the influence o f spatial organization o f m echanical properties on rupture behavior and its im plications for precursory events linked w ith early w arning for natural hazard prediction. Follow ing a b rief description o f the different num erical m odels developed in this work, the rupture behavior and associated m acroscopic properties at rupture o f sim ulated FBM s are presented for different spatial arrangem ents. Precursory activity preceding com plete breakdow n is also exam ined in a statistical fram ew ork aim ed at developing m etrics for failure im m inence. Finally, a general rupture criterion based on m acroscopic properties is presented, and its potential application to gravity-driven geophysical instabilities is discussed. Failure events in correlated fiber bundle model. We consider a set o f parallel elastobrittle fibers assem bled on a square regular lattice w ith periodic boundary conditions. The fibers exhibit a linear-elastic behavior [with a Y oung’s m odulus E (Pa) arbitrarily set to £ = 1] follow ed by an abrupt failure at a prescribed critical threshold load term ed •
•
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•
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•
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FIG. 1. (Color online) Illustration of the redistribution rule for the LFBM4 (left) and LFBM8 (right) on a 16 x 16 lattice with approximately the same number of failed fibers. Empty gray circles represent the fibers. Filled gray circles indicate failed fibers. The large thick cross in magenta indicates the failed fiber. The thick crosses represent the fiber connected to the failed fiber, forming the “crack” according to the four- (left) or eight- (right) neighbor scheme. The blue thick circle represents the fibers to which load will be redistributed in the next step. Note that periodic boundary conditions connect opposite borders, enabling a large crack to form. As evidenced here, long-range load redistribution is possible, depending only on the geometrical characteristics of the cluster formed by failed fibers.
load redistribution, in a similar fashion to the DFBM. Note also that such a geometrical mode of local load sharing is different from the methods of other studies that redistribute stress also according to the distance to the failed fiber [16-18]. Figure 2 shows for the three considered redistribution schemes the variation of effective stress and the fraction of unbroken fibers with applied stress for a bundle with uniform distribution of fiber strength. As expected, the rupture of the same bundle
appears for lower applied stress and lower number of failed fibers in the case of LFBM4: As such a redistribution rule promotes load concentration on the neighbors (the load is redistributed on fewer neighbors than in the LFBM8 or DFBM), the probability of triggering a cascade of failures of near neighbors is enhanced, favoring the appearance of global rupture. On the contrary, the LFBM8 appears to behave in a similar way to the DFBM as the redistributed load is “spread” on a large number of neighbors for a sufficiently highly damaged lattice (Fig. 1). The effects of heterogeneity patterns on the global rupture behavior of the bundle were studied using the values of fiber strengths a,h chosen randomly from a uniform distribution in the range from 0 to 1. For robust statistical representation, we generated 200 different initial configurations of each FBM parameter combination studied. Moreover, to account for natural spatial correlations (such as induced by a tree root system with large lateral extent), we reorganized the initially uniformly distributed configuration of fiber strengths to obtain different spatial correlation lengths (see Fig. 3, insets). The process involved three steps: (1) An optimized correlated lattice with a given correlation length CL (expressed in terms of lattice units) was computed by convolving in the Fourier space a Gaussian filter with a lattice composed of uniformly distributed fiber strengths. (2) Each fiber of the initial lattice was spatially migrated to the position on the optimized lattice with the nearest strength value. (3) Finally, a test on the reorganized lattice was performed to verify that the resulting spatial distribution of the fiber strengths ath followed the prescribed correlation length (Fig. 3). It is important to note
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