all models with heterogeneities in the mechanical properties show large - scale patterns of stress maxima and minima when they are compressed .
1 . the nature of the patterns in stress and how the patterns are impacted by elastic heterogeneity
2 , which shows the equivalent von mises stress for four of the models . in all models illustrated here
although this degree of variation in elastic properties is unrealistic , i use this model as an extreme end - member . in fig . 2d
the young s moduli of the grains varies from 100 to 120  gpa , which is similar to the degree of variation seen in high - symmetry minerals like beta quartz or halite .
the other models have variations in elastic properties that are intermediate ( table 1 ) .
although the patterns occur on a scale much larger than the grain - scale , they do not reflect the pattern of grain boundaries because individual grains contain significant stress gradients . in fig .
2d the pattern can be described as isolated islands of low equivalent stress , but in fig .
the change in patterning from the most heterogeneous model to the least is in the spacing of the  streams and the intensity of the stress concentration in the 
streams. the vertically oriented patterns originate from variations in the magnitude and direction of the maximum compressive stress ( 3 ) . a subordinate horizontal pattern ( best seen in fig .
figure 3 shows a portion of a model with the full stress tensor plotted . with the exception of the softest grains ,
the orientation of the maximum compressive stress can deviate substantially away from the macroscopic compression direction and the stress state can even become tensile .
these variations are produced by the difference in elastic properties of neighboring grains even before plastic flow in the weakest grains initiates .
the patterns in stress are established in the first displacement increment and change only in numerical value until the first yielding occurs . as yielding initiates
, the patterning in 3 deteriorates , while the patterning in 1 intensifies ( fig .
4 ) . all of the models produce strain localization in the form of shear bands ( fig .
5 ) , and the spacing of the shear bands is coupled to the stress patterning ; denser patterning produces more closely spaced shear bands concentrating smaller degrees of stain .
therefore , the location of shear bands is dictated by the spatial distribution of locations where yielding first occurs ( fig .
the degree to which shear bands grow and intensify or not as a function of global strain depends on the level of work hardening included in the plasticity model .
all models with heterogeneities in the mechanical properties show large - scale patterns of stress maxima and minima when they are compressed .
1 . the nature of the patterns in stress and how the patterns are impacted by elastic heterogeneity
2 , which shows the equivalent von mises stress for four of the models . in all models illustrated here
although this degree of variation in elastic properties is unrealistic , i use this model as an extreme end - member . in fig . 2d
the young s moduli of the grains varies from 100 to 120  gpa , which is similar to the degree of variation seen in high - symmetry minerals like beta quartz or halite .
the other models have variations in elastic properties that are intermediate ( table 1 ) .
although the patterns occur on a scale much larger than the grain - scale , they do not reflect the pattern of grain boundaries because individual grains contain significant stress gradients . in fig .
2d the pattern can be described as isolated islands of low equivalent stress , but in fig .
the change in patterning from the most heterogeneous model to the least is in the spacing of the  streams and the intensity of the stress concentration in the 
streams. the vertically oriented patterns originate from variations in the magnitude and direction of the maximum compressive stress ( 3 ) . a subordinate horizontal pattern ( best seen in fig .
figure 3 shows a portion of a model with the full stress tensor plotted . with the exception of the softest grains ,
the orientation of the maximum compressive stress can deviate substantially away from the macroscopic compression direction and the stress state can even become tensile .
these variations are produced by the difference in elastic properties of neighboring grains even before plastic flow in the weakest grains initiates .
the patterns in stress are established in the first displacement increment and change only in numerical value until the first yielding occurs . as yielding initiates
, the patterning in 3 deteriorates , while the patterning in 1 intensifies ( fig .
4 ) . all of the models produce strain localization in the form of shear bands ( fig .
5 ) , and the spacing of the shear bands is coupled to the stress patterning ; denser patterning produces more closely spaced shear bands concentrating smaller degrees of stain .
therefore , the location of shear bands is dictated by the spatial distribution of locations where yielding first occurs ( fig .
the degree to which shear bands grow and intensify or not as a function of global strain depends on the level of work hardening included in the plasticity model .
the fem models presented here make predictions about the heterogeneity and spatial distribution of elastic and plastic strain in polycrystalline materials that are consistent with experimental observations as well as results from other numerical studies .
many microstructural studies of plastically deformed polycrystals observe intragranular heterogeneity of plastic strain ranging from those conducted on highly anisotropic materials like olivine and ice67 to metals , which generally have more isotropic single - crystal rheological properties2891011 .
the degree of strain heterogeneity is thought to be a function of the anisotropy of the material17 .
the development of shear - strain localization at very low degrees of strain has been observed in a number of materials781112 .
the development of tensile elastic strain during compression has been observed in ice7 and zirconium11 .
large variations in stress states within single grains and plastic features that span many grains have been observed in three - dimensional ( 3d ) finite element models of polycrystalline fcc metals1 .
however , consistent with what is observed in synchrotron and neutron diffraction studies of deforming materials , elastically stiffer grains on average strain less than more compliant grains ( see , for example , the study by turner and tome13 , or diffraction data within the study by burnley and zhang14 ) ( fig .
. referred to as force chains , the patterns are produced by stress percolation20 and consist of semi - linear subsets of particles arranged parallel to compression that bear the applied load through the so - called 
granular materials are most obviously distinct from fully dense polycrystalline materials in that the constituent grains have limited or no cohesion . both shear tractions and tensile forces
there are extreme variations in the mechanical properties of the constituent structural components ( grain boundaries and grain interiors ) , whereas in fully dense polycrystalline materials variations between structural components , which may or may not include rheologically distinct grain boundaries21 , are comparatively muted . thus , granular materials and fully dense polycrystalline materials fall on a continuum of variation in component properties .
the value in making this connection is that it suggests new ways of thinking about deformation of polycrystalline materials .
for example , force chains are understood to be the underlying mechanism of shear localization in granular materials ( see discussion in the study by tordesilla and muthuswamy22 and references therein ) .
therefore , examining the role of stress patterning in shear localization in polycrystalline materials may prove fruitful .
the magnitude of stress concentrations in the stress patterns and the regularity with which they occur may well yield to statistical descriptions , as they do in granular materials23 and should lead to a better understanding of phenomena that are tied to stress concentrations , such as the development of stylolites , slatey cleavage and compositional banding in rocks2425 , as well as help explain how these rhythmic features initially arise from unpatterened rocks such as shale and granite .
the recognition of stress patterning will have a number of important implications for experimental characterization and modeling of rheological properties . in experimental studies ,
experimental samples whose size allows only a portion of the stress pattern to be captured , may present difficulties for obtaining reproducible results ; a phenomena that would be most noticeable in high - pressure studies where sample sizes are restricted . changing the grain size or overall size of experimental samples
can be cumbersome , but numerically it should be comparatively painless to predict the appropriate rve by increasing the size of the grain ensemble being modeled until the tails of the stress distribution are fully described .
another important implication of stress patterning is that it provides a framework for modeling aggregate behavior over multiple - length scales based on single - crystal properties .
robust predictive models of the mechanical behavior of rocks are needed for understanding the mechanical behavior of the interior of the earth and other planets because extrapolations in both length scale and strain rate will always be required26 .
the need for reliable predictive rheological models is not confined to the geosciences but extends to engineering and materials science problems that present a variety of challenges for direct observation ; for example , shock loading of nuclear weapon components or detonation of plastic - bonded explosives .
these extrapolations can be made with more confidence if they are generated by models that are based on the physics of deformation : capturing the behavior of the crystal lattice as well as the short- and long - range interaction between grains within the polycrystal .
plastic and viscoplastic self - consistent models describe single - crystal plasticity but assume that a single set of average boundary conditions apply to each grain .
these models could be modified such that the boundary conditions applied to each grain are allowed to vary over a range suitable to the elastic and plastic heterogeneity of the structural components of the material .
perhaps the most important impact that the recognition of stress patterning can have is if it , as a simple and intuitive paradigm , can replace the default assumption of a reuss state ( isostate ) of stress in the minds of those who work on stress - related phenomena .
although it has been long recognized that the reuss state of stress is an oversimplification , the assumption still pervades our thinking on everything from the attenuation of seismic waves to the development of paleopiezometers . by providing a means of leaving the reuss stress state assumption behind ,
the recognition of patterning in stress should lead to new insights in a number of disciplines .
each grain was modeled as a hexagon containing 48 six - noded triangular elements ( fig .
1 ) ; this type of element uses a parabolic interpolation function along each edge .
the nodes on the boundaries of each grain are shared with the elements along the edge of the neighboring grain , thus grain boundaries are demarcated only by a change in elastic and plastic properties . as there can be no offset along a grain boundary in the model ,
the model consists of 2,574 hexagonal grains that have been assigned randomly into 25 subsets .
the constitutive equation used for linear elastic solids is : where ij is the stress tensor , ij is the kronecker delta , ij is the strain tensor ,  is the lame constant , and g is the shear modulus .
 can be expressed as : and g can be expressed as : where v is the poisson s ratio and e is the young s modulus .
plastic behavior initiates when the equivalent von mises stress ( vm ) , which is calculated as follows : equals the specified yield strength .
post - yielding behavior was either perfectly plastic or included a work - hardening slope .
the young s modulus and poisson s ratio assigned to each grain set for the models presented in the illustrations are given in table 1 .
for all materials shown in the figures , the yield strength was set to 1/100th of the young s modulus and no work hardening was included .
nodes along the bottom edge were not allowed to move in y ( the vertical axis ) .
nodes along the left edge were not allowed to move in x ( the horizontal axis ) .
depending on the goal of the experiment , either a constant edge load was placed on the top edge of the model or a constant displacement was applied to each node on the top of the model .
each grain was modeled as a hexagon containing 48 six - noded triangular elements ( fig .
1 ) ; this type of element uses a parabolic interpolation function along each edge .
the nodes on the boundaries of each grain are shared with the elements along the edge of the neighboring grain , thus grain boundaries are demarcated only by a change in elastic and plastic properties . as there can be no offset along a grain boundary in the model ,
the model consists of 2,574 hexagonal grains that have been assigned randomly into 25 subsets .
the constitutive equation used for linear elastic solids is : where ij is the stress tensor , ij is the kronecker delta , ij is the strain tensor ,  is the lame constant , and g is the shear modulus .
 can be expressed as : and g can be expressed as : where v is the poisson s ratio and e is the young s modulus .
plastic behavior initiates when the equivalent von mises stress ( vm ) , which is calculated as follows : equals the specified yield strength .
post - yielding behavior was either perfectly plastic or included a work - hardening slope .
the young s modulus and poisson s ratio assigned to each grain set for the models presented in the illustrations are given in table 1 .
for all materials shown in the figures , the yield strength was set to 1/100th of the young s modulus and no work hardening was included .
nodes along the bottom edge were not allowed to move in y ( the vertical axis ) .
nodes along the left edge were not allowed to move in x ( the horizontal axis ) .
depending on the goal of the experiment , either a constant edge load was placed on the top edge of the model or a constant displacement was applied to each node on the top of the model .
how to cite this article : burnley , p. c. the importance of stress percolation patterns in rocks and other polycrystalline materials .
 a new framework for thinking about the deformation behavior of rocks and other heterogeneous polycrystalline materials is proposed , based on understanding the patterns of stress transmission through these materials . here , using finite element models , 
 i show that stress percolates through polycrystalline materials that have heterogeneous elastic and plastic properties of the same order as those found in rocks . 
 the pattern of stress percolation is related to the degree of heterogeneity in and statistical distribution of the elastic and plastic properties of the constituent grains in the aggregate . 
 the development of these stress patterns leads directly to shear localization , and their existence provides insight into the formation of rhythmic features such as compositional banding and foliation in rocks that are reacting or dissolving while being deformed . 
 in addition , this framework provides a foundation for understanding and predicting the macroscopic rheology of polycrystalline materials based on single - crystal elastic and plastic mechanical properties . 
Results
Results of finite element models
Discussion
Methods
Finite element models
Author contributions
Additional information