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A Review of Photoelasticimetry in Granular Matter

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A Review of Photoelasticimetry in Granular Matter Granular Matter manuscript No. (will be inserted by the editor) Enlightening force chains: a review of photoelasticimetry in granular matter Aghil Abed Zadeh · Jonathan Bar´es · Theodore A. Brzinski · Karen E. Daniels · Joshua Dijksman · Nicolas Docquier · Henry O. Everitt · Jonathan E. Kollmer · Olivier Lantsoght · Dong Wang · Marcel Workamp · Yiqiu Zhao · Hu Zheng Received: date / Revised version: date Abstract A photoelastic material will reveal its in- awareness about the ubiquitous presence of granular ternal stresses when observed through polarizing fil- matter in our lives, enlighten its puzzling behavior, and ters. This eye-catching property has enlightened our promote conversations about its importance in environ- understanding of granular materials for over half a cen- mental and industrial contexts. To aid in this endeavor, tury, whether in the service of art, education, or scien- this paper also serves as a front door to a detailed tific research. In this review article in honor of Robert wiki containing open, community-curated guidance on Behringer, we highlight both his pioneering use of the putting these methods into practice [1]. method in physics research, and its reach into the pub- lic sphere through museum exhibits and outreach pro- grams. We aim to provide clear protocols for artists, 1 Introduction exhibit-designers, educators, and scientists to use in their own endeavors. It is our hope that this will build Most of the transparent objects we encounter are pho- toelastic: their degree of birefringence depends on the Aghil Abed Zadeh, Department of Physics & Center for Non- local stress at each point in the material [2, 3]. This linear and Complex Systems, Duke University, Durham, NC, property can be used to visualize, and even quantita- USA · Jonathan Bar´es,Laboratoire de M´ecaniqueet G´enie tively measure, what is usually invisible to our naked Civil, Universit´ede Montpellier, CNRS, Montpellier, France, eye: the stress field. When such materials are subjected E-mail: [email protected] · Theodore A. Brzinski, De- partment of Physics, Haverford College, Haverford, PA, USA · to an external load, and placed between crossed polar- Karen E. Daniels, Department of Physics, North Carolina izing filters, each different region of the material rotates State University, Raleigh, NC, USA · Joshua Dijksman, Phys- the light polarization according to the amount of local ical Chemistry and Soft Matter, Wageningen University & stress [4]. This creates a visual pattern of alternating Research, Wageningen, The Netherlands · Nicolas Docquier, Institute of Mechanics, Material and Civil engineering, Uni- colored fringes (see Fig. 1) within the material which, versit´ecatholique de Louvain, Louvain-la-Neuve, Belgium · on top of their aesthetic and pedagogical aspects, per- Henry O. Everitt, Department of Physics and Department of mits us to quantify the stress field within the material. Chemistry, Duke University, Durham, NC, USA · Jonathan E. Photoelastimetry has its roots in engineering prac- Kollmer, Department of Physics, Universit¨atDuisburg-Essen, Duisburg, Germany · Olivier Lantsoght, Institute of Mechan- tice, where it was widely used to design parts before the ics, Material and Civil engineering, Universit´ecatholique de rise of computational finite element methods [3]. It also Louvain, Louvain-la-Neuve, Belgium · Dong Wang, Depart- provided the first glimpse of the internal forces within ment of Physics & Center for Nonlinear and Complex Sys- granular materials, at first qualitatively [6{9] and later tems, Duke University, Durham, NC, USA · Marcel Workamp, Physical Chemistry and Soft Matter, Wageningen University quantitatively [10, 11]. Today, it remains the most well- & Research, Wageningen, The Netherlands · Yiqiu Zhao, De- developed method for quantifying stresses [12], and meth- partment of Physics & Center for Nonlinear and Complex ods for particle making [5, 13] and image post-processing Systems, Duke University, Durham, NC, USA · Hu Zheng, [4, 14, 15] are under active development. Department of Physics & Center for Nonlinear and Com- plex Systems, Duke University, Durham, NC, USA; School After two decades of quantitative efforts, the scien- of Earth Science and Engineering, Hohai University, Nanjing, tific successes of the photoelastimetry method are nu- Jiangsu, 211100, China merous. In Robert Behringer's group alone, it was re- 2 Aghil Abed Zadeh et al. 2 Photoelasticimetry theory Photoelasticity arises from the birefringent properties of most transparent materials, in which the speed of light (via the index of refraction) depends on the polar- ization of the incident light wave. In some cases, such as glass and polymeric materials, birefringence arises only when the material is subject to anisotropic stress, with the refractive indexes depending on the eigenval- ues of the local stress tensor. Consequently, photoelas- ticity can provide measurements of the internal stress in the material. Using circularly polarized light results in more pre- cise isotropic measurements, compared to linearly po- Fig. 1 Composite view of a granular photoelastic system. On larized light. Unpolarized light can be transformed into the left side, the grains are imaged with white backlighting. In circularly polarized light using a circular polarizer which the middle, the particles are viewed between crossed polariz- is certain combination of linear polarizer and a quarter- ers, revealing the force chains through photoelasticity. On the right side, the system is imaged from above with a UV light, wave (π=2) phase shift plate, as shown in Fig. 2. On revealing inked bars used to track particle rotation. Within the other side of the photoelastic material, another cir- each particle, an embedded cubic magnet is visible as a dark cular polarizer with opposite polarity, called the \an- square [5]. alyzer", blocks any unperturbed light. If the material is unstressed, there is no transmitted light and a dark image results. However, anyplace in the material where sponsible for identifying the erratic stress fluctuations there is anisotropic stress, the wave components which in sheared granular matter [10, 16, 17], Green's func- are polarized along the two principle axes of the lo- tion response [18], particle-scale anisotropy of the con- cal stress tensor will travel with different speeds. This tact force networks [11, 19], shear jamming [20{24], the speed difference results in a relative phase shift for these dynamics of granular matter under impact [25{27], the two components of the wave, converting circularly po- Reynolds pressure, the Reynolds coefficient [22], and larized light to elliptically polarized light. As a conse- more. quence, a portion of the wave is not completely blocked Far beyond the bounds of his laboratory at Duke by the analyzer, and is therefore recorded as a bright University, the method has been used to examine par- region of the image. This property provides a quantita- ticle shape dependence [28], identify interparticle con- tive measure of local stress via an inverse method. tacts [29], observe sound propagation [30{32], test the To quantitatively measure the local stress, we begin validity of statistical ensembles [33, 34], examine sensi- by assuming that the relation between the local stress tivity to initial conditions [35], identify dilatancy soft- and the refractive index is linear. The difference of re- ening [36], measure force chain order parameters [37], fractive indices between the two principle axes: and observe the effects of fluid flow [38]. In interdis- n1 − n2 = C(σ1 − σ2); (1) ciplinary efforts, photoelastimetry permits scientists to evaluate the grain-scale stresses caused by growing plant where σ1; σ2 are the two eigenvalues of the local stress roots [13, 39, 40], and examine situations relevant to tensor, and n1; n2 are the two refractive indices in the faulting and earthquakes [41{44] corresponding directions. The material constant C is known as the stress-optical coefficient. The relative phase The structure of the paper is as follows. In x2, we shift of wave components in the eigendirections of the briefly review the physics of photoelasticity. This sec- local stress tensor is: tion can be skipped for those only interested in qualita- 2πCd tive uses of the method. In x3, we present various ways α = (σ1 − σ2); (2) of fabricating photoelastic particles by cutting, cast- λ ing or printing, followed by imaging-techniques in x4. where λ the wavelength of the incident light and d These two sections can stand alone for the creation of a the distance traveled inside the material (its thickness). demonstration apparatus. Finally, we present quantita- For this phase shift, the intensity of the wave passed tive methods in x5. In all cases, additional information through the analyzer is given by: and technical specifications are provided on a wiki to α πCd I = I sin2 = sin2 (σ − σ ) : (3) which many of the paper authors have contributed [1]. 0 2 λ 1 2 Enlightening force chains: a review of photoelasticimetry in granular matter 3 Fig. 2 Schematic image of the photoelastic technique for a darkfield transmission polariscope: the combination of linear polarizer 1 and quarter-wave plate 1 (with a π=4 difference of principle directions), converts unpolarized light into circularly polarized light. This light passes through a uniaxially loaded photoelastic particle. A second combination of plate 2 and linear polarizer 2 (producing the opposite chirality and called the analyzer) blocks any circularly polarized light passed
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