iMechanica - micro-cracks
https://imechanica.org/taxonomy/term/13917
enJournal Club for November 2023: Top-down multi scale damage mechanics
https://imechanica.org/node/26948
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/866">damage mechanics</a></div><div class="field-item odd"><a href="/taxonomy/term/13916">internal length</a></div><div class="field-item even"><a href="/taxonomy/term/1048">microstructure</a></div><div class="field-item odd"><a href="/taxonomy/term/13917">micro-cracks</a></div><div class="field-item even"><a href="/taxonomy/term/1968">Multi-scale modeling</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p> </p>
<p>Chloé Arson, Ph.D.</p>
<p>Professor, Cornell University School of Civil and Environmental Engineering</p>
<p> </p>
<p> <strong>Introduction</strong><br />
</p><p class="p1">Continuum Damage Mechanics (CDM) initially aimed to predict the reduction of stiffness and strength with the propagation of defects in solids. Kachanov’s pioneering micromechanical contributions to CDM (Kachanov, 1980, 1982a,b, 1992) inspired many researchers to use a bottom-up approach to understand the relationship between statistical microstructure descriptors (such as micro-crack distributions) and mechanical properties (such as stiffness) that can be defined at the scale of a Representative Elementary Volume (REV). In bottom-up approaches, the REV-scale field variables and properties are calculated by statistical averaging (Bazant and Oh, 1985; Bazant and Prat, 1988; Krajcinovic et al., 1991; Gambarotta and Lagomarsino, 1993; Pensee et al., 2002; Pensee and Kondo, 2003; Paliwal and Ramesh, 2008) or homogenization (Krajcinovic and Sumarac, 1989; Ju, 1991; Dormieux et al., 2006; Pouya et al., 2016; Shen et al., 2019; Shen and Arson, 2019; Shen et al., 2021a; Xu et al., 2022; Xu and Arson, 2022, 2023), from microscopic field variables that are calculated for every single type of crack or defect present in the REV. Homogenization schemes are attractive because they are explanatory, but they are notorious for being difficult to calibrate. The self-consistent homogenization method is particularly computationally intensive, because it requires an iterative calculation of the local field variables. The use of the theory of Thermodynamic of Irreversible Processes (TIP) with continuum damage dissipation variables is an alternative to the formulation of micromechanical models of crack propagation. The second order damage tensor is typically defined as follows:</p>
<p><img src="https://imechanica.org/files/Equation1.jpeg" alt="Equation 1" width="519" height="45" /></p>
<p>in which ρ(<strong>n</strong>) is the distribution of crack planes of normal direction <strong>n</strong> on the unit sphere. It can be shown mathematically (see (Lubarda et al., 1994; Yang et al., 1999) for details) that the crack density function ρ(<strong>n</strong>) is related to the damage tensor as follows:</p>
<p><img src="https://imechanica.org/files/Equation2.jpeg" alt="Equation 2" width="218" height="36" /></p>
<p class="p1">The free enthalpy of the REV can then be expressed as a function of the second order damage tensor <strong>D</strong>, as follows:</p>
<p class="p1"><img src="https://imechanica.org/files/Equation3.jpeg" alt="Equation 3" width="436" height="62" /></p>
<p class="p1">In this post, I will explore the strategies that have been explored to date to enrich the theory of TIP with microstructure variables, mainly, fabric enrichment, non-local formulations, the theory of micromorphic media and the framework of stochastic gradient plasticity. Such top-down approaches are sometimes viewed as too phenomenological and not physical enough. But they present the advantage of depending on fewer constitutive parameters than bottom-up models such as homogenization methods. Moreover, rigorous calibration procedures have been developed for several classes of top-down TIP-based models. More details on this topic are available in the review paper that I published under a Creative Commons License in <em>Open Geomechanics</em> (Arson, 2020), from which I used some of the text presented below.</p>
<p class="p1"> </p>
<p><strong>Fabric enrichment</strong>
</p><p>Fabric-enriched models are essentially phenomenological models in which the damage tensor is defined as a convolution of moments of probability of microstructure descriptors. The free energy and the dissipation potential are postulated at REV scale and constitutive equations are obtained from the thermodynamic conjugation relationships. For example, consider a second-order damage tensor <strong>D</strong> = k<strong>F</strong>, in which k = Tr (<strong>D</strong>) and <strong>F</strong> is a trace-less second-order fabric tensor. The most general form of the REV Helmholtz free energy density ψ that ensures that the undamaged stress/strain relationship is linear is given by:</p>
<p><img src="https://imechanica.org/files/Equation4.jpeg" alt="Equation 4" width="416" height="117" /></p>
<p>where the coefficients c<span class="s1">i </span>are functions of k and of the two first invariants of <strong>F</strong>, the expressions of which are given in (Zysset and Curnier, 1995). The expression of <strong>F</strong> can be customized to capture the evolution of measurable microstructure features that are relevant to the phenomena observed at REV scale. In material sciences, poly-dispersed media are often characterized by (Reid, 1955; Philleo, 1983; Torquato et al., 1990; Lu and Torquato, 1992b,a; Torquato and Lu, 1993; Blum and Eisenlohr, 2009):</p>
<ul>
<li class="p1">the pore size distribution: probability density function (pdf) of a pore size (typically, largest segment that can be fit in a virtual ellipse);</li>
<li class="p1">the lineal-path function L(z): probability of finding a line segment of length z wholly in one of the phases (e.g., grain, pore) of the sample;</li>
<li class="p1">the chord length pdf: pdf of line segments contained fully within a phase (e.g., grain or pore) and extending between two points on the surface of the object in that phase;</li>
<li class="p1">the void nearest-surface distribution function h<span class="s1">j</span>(r): probability that at an arbitrary point, the nearest surface of a particle or pore of phase j lies at a distance between r and r + dr;</li>
<li class="p1">the void exclusion probability E<span class="s1">V</span>(r): probability of finding a spherical region of radius r that is empty of solid particle center.</li>
</ul>
<p>In structural geology, fabric is typically described with the distributions of crack length, aperture and orientation (Long et al., 1982; Wilson et al., 2003; Chester et al., 2004). In particulate mechanics, the fabric tensor is usually a measure of the orientation of the vectors that are normal to the plane tangent to two grains in contact (called contact vectors). Figure 1 shows two fabric tensors defined by Li and Li (Li and Li, 2009) to explain the internal variable used in the Anisotropic Critical State Theory (ACST) initially proposed by Dafalias’s group (Dafalias and Manzari, 2004; Dafalias et al., 2004; Li and Dafalias, 2011; Fu and Dafalias, 2011; Gao et al., 2014). In the ACST, any deviatoric second-order tensor can be used as a measure of fabric. Both tensors defined by Li and Li require a Voronoi-Delaunay tessellation of the granular medium. The contact-based fabric tensor depends on the contact vector from void centroid to grain contacts, and characterizes the arrangement of so-called “solid cells”, which each contains a grain and its assigned void space. The assigned void space is calculated by dividing each void element into void tetrahedra whose vertices are the three vertices of a Delaunay boundary surface and the center of the void element, and by joining these tetrahedra to the solid elements that share the same Delaunay boundary surfaces. The void-based fabric tensor depends on the contact vector from grain centroid to grain contacts, and characterizes the arrangement of so-called “void cells”, which each contains a pore (or void) and its assigned solid space. The latter is calculated by dividing each grain into solid tetrahedra whose vertices are the three vertices of a Delaunay boundary surface and the center of the grain, and by joining these tetrahedra to the void elements that share the same Delaunay boundary surfaces.</p>
<p class="p1"><img title="Figure 1. Fabric descriptors defined by Li and Li (2009). Figure 6 from (Arson, 2020), article licensed under the Creative Commons Attribution Non-Commercial Share Alike 4.0 License." src="https://imechanica.org/files/Figure1_0.jpg" alt="Figure 1" width="675" height="435" /> </p>
<p class="p1"><strong>Figure 1.</strong> Fabric descriptors defined by Li and Li (2009). f(<strong>n</strong>): norm of contact vector from void centroid to grain contacts; v(<strong>n</strong>) norm of contact vector from grain centroid to grain contacts; H<span class="s1">0</span>, G<span class="s1">0</span>: normalization coefficients. Figure 6 from (Arson, 2020), article licensed under the Creative Commons Attribution Non-Commercial Share Alike 4.0 License.</p>
<p class="p1">The contact-based fabric tensor and the void-based fabric tensor can be used to characterize damage in rocks made of polycrystals of cemented aggregates. Cracks affect the geometry of the grains (crystals or aggregates) and of the pores, which makes the contact-based and void-based fabric tensors particularly efficient to track the evolution of intra- and inter-granular crack sizes, shapes, orientations and connectivity. A fabric tensor <strong>F</strong> can be defined for each descriptor, by integrating the corresponding pdf. In 2D, fabric tensors are 2 x 2 and take the following expression:</p>
<p class="p1"><img src="https://imechanica.org/files/Equation5.jpeg" alt="Equation 5" width="274" height="41" /></p>
<p class="p1">where <strong>F</strong> is a symmetric second-rank tensor; Γ is the whole solid angle, and equals to 2π for 2D images; E(Γ)is a pdf. The components of the fabric tensors are calculated as follows:</p>
<p class="p1"><img src="https://imechanica.org/files/Equation6list_0.jpeg" alt="Equation 6" width="211" height="168" /></p>
<p class="p1">N is the total number of measures (or angles) considered in the image.</p>
<p class="p1">For example, my group proposed a fabric tensor based on the statistical analysis of 2D halite micrographs extracted at key stages of oedometer tests and triaxial tests (Shen et al., 2021b). Contact branches, defined as segments linking the centers of two grains in contact, were plotted. A map of polygons was obtained, as illustrated in Figure 2(a). The solid volume fraction of of each polygon was then calculated. At grain scale, the magnitude of the Local Solid Volume Fraction (LSVF) was calculated as the total solid volume fraction of the polygons that intersect with the grain. The LSVF is plotted grain by grain in Figure 2(b). The LSVF orientation of a grain was calculated as an average of the grain branch orientations, in which the weights were proportional to the solid volume fraction of the polygons intersected by the grain. Mathematically, the local solid volume fraction fabric tensor is defined by the three equations above giving the expressions of F11, F22 and F12, in which p<span class="s1">k </span>is the local solid volume fraction of each polygon and θ<span class="s1">k </span>is the angle between the horizontal and the line connecting a polygon’s center with the image’s center.</p>
<p class="p1"><img src="https://imechanica.org/files/Figure2.jpeg" alt="Figure 2" width="675" height="349" /></p>
<p class="p1"><strong>Figure 2.</strong> Local solid volume fraction of grains. <strong>a.</strong> The polygons map is constructed by plotting branches from grain centroid to grain centroid; the local solid volume fraction is then calculated for each polygon. <strong>b.</strong> The area of the surface covered by the polygons that overlap with a grain is called the domain of this grain; the solid volume fraction of a grain is the averaged solid volume fraction of the domain of that grain. Figure 5 from the article (Shen et al., 2021b) published in <a href="https://www.springer.com/journal/11440">Acta Geotechnica</a>, reproduced under Springer License agreement n. 501857376.</p>
<p class="p1">With very little error, all the fabric tensors that we calculated from image analysis were found to be diagonal and orthogonal, and for each image i, we defined a normalized fabric tensor <strong>H<span class="s1">i </span></strong>as follows:</p>
<p class="p1"><img src="https://imechanica.org/files/equation10.jpeg" alt="Equation 10" width="125" height="18" /></p>
<p class="p1">where <strong>G<span class="s1">i </span></strong>and <strong>B<span class="s1">i </span></strong>are the grain orientation and branch orientation fabric tensors, <strong>L<span class="s1">i </span></strong>and <strong>S<span class="s1">i </span></strong>are the traceless components of the fabric tensors of LSVF and grain solidity, and γ<span class="s1">i </span>is a normalizing coefficient used to make Tr(<strong>H<span class="s1">i</span></strong>) equal to 1. Grain solidity is defined as the ratio between the area of a grain image by the area of the convex hull of the grain. In the expression of the grain solidity fabric tensor, p<span class="s1">k </span>is the grain solidity of each grain and θ<span class="s1">k </span>is the angle between the horizontal and the line connecting a grain’s center with the image’s center. For the tensors of grain orientation and branch orientation, θ<span class="s1">k </span>is the orientation angle of the vector n in reference to a set axis in the image. Figure 3 shows the evolution of the descriptors used in the definition of the fabric tensor during odometer tests conducted on specimens with different initial porosities.</p>
<p><img src="https://imechanica.org/files/Figure3.jpeg" alt="Figure 3" width="675" height="573" /></p>
<p><strong>Figure 3.</strong> Probability density functions of four microstructure descriptors obtained in oedometer tests for samples with different porosities. Figure 7 from (Arson, 2020), article licensed under the Creative Commons Attribution Non-Commercial Share Alike 4.0 License.</p>
<p>The second-rank fabric tensor <strong>H<span class="s1">i </span></strong>can be written as k<span class="s1">i</span><strong>I</strong>+<strong>K<span class="s1">i</span></strong>. k<span class="s1">i </span>is a scalar, and <strong>K<span class="s1">i </span></strong>is a traceless second-rank tensor. Calculating <strong>H<span class="s1">i </span></strong>for each image i allows tracking the evolution of fabric with deformation. Using the expression of the free energy of an elastic medium enriched with microstructure (expression of Ψ above), we expressed the stiffness tensor of salt rock as a function of k, <strong>K</strong> and material properties. The coefficients a<span class="s1">i </span>can be understood as functions of k and Lame-like constants, μ<span class="s1">c </span>and λ<span class="s1">c</span>. Considering that the salt rock Young’s modulus is an exponential function of porosity (Turner and Cowin, 1987), we proposed to relate μ<span class="s1">c </span>and λ<span class="s1">c </span>to the average local solid fraction α<span class="s1">l </span>and to average grain solidity α<span class="s1">s </span>as follows:</p>
<p><img src="https://imechanica.org/files/equation11.jpeg" alt="Equation 11" width="292" height="15" /></p>
<p>We calibrated the parameters m and n against the experimental oedometer modulus (i.e., the ratio of axial stress by axial strain). After calibration, the model was used to calculate the fabric tensor<strong> H<span class="s1">i </span></strong>for each image and to deduce the damaged stiffness tensor, according to the expression of Ψ above. Figure 4 shows how the oedometer modulus calculated numerically (red dots) fits the damaged oedometer modulus measured experimentally (solid line).</p>
<p><img src="https://imechanica.org/files/Figure4_0.jpg" alt="Figure 4" width="421" height="272" /></p>
<p><strong>Figure 4.</strong> Calibration of the Lamé-like parameters in the fabric-enriched CDM model, against oedometer tests. Figure 8 from (Arson, 2020), article licensed under the Creative Commons Attribution Non- Commercial Share Alike 4.0 License.</p>
<p>It remains challenging to capture the full range of possible rock fabrics and textures encountered in nature (Vernon, 2018). For instance, connections between cracks imply enhanced hydraulic crack interaction, but not necessarily mechanical crack interaction, especially if cracks are randomly oriented and if the distribution of cracks is dense (Schubnel et al., 2006). Two different damage variables are necessary to model permeability enhancement and stiffness degradation (Maleki and Pouya, 2010).</p>
<p class="p1"> </p>
<p><strong>Non-local damage formulations</strong>
</p><p>Integral formulations</p>
<p>Non-local formulations consist in replacing one or more local field variables by a non-local counterpart, which translates the influence of a field variable defined at a position x on that field variable in a neighborhood around x. In integral formulations (Bazant and Ozbolt, 1990; Bazant and Jirasek, 2002), space averages are weighted by attenuation functions. If η(x) is a local field variable in a solid body occupying a domain V, the corresponding non-local field variable ηV(x) is defined as:</p>
<p><img src="https://imechanica.org/files/equation12.jpeg" alt="Equation 12" width="196" height="39" /> </p>
<p>where α<span class="s1">′</span>(x, ξ) is a weight function that satisfies the partition of unity and depends on an internal characteristic length l<span class="s1">c</span>. The length l<span class="s1">c </span>can be determined experimentally by comparing the responses of specimens in which the damage remains distributed with the response of fractured specimens in which damage localizes (Bazant and Pijaudier-Cabot, 1989). It can also be determined by comparing the simulation results for various values of l<span class="s1">c </span>with the experimental response (Geers et al., 1999). Integral non-local regularization was performed on the averaged energy release rate (Pijaudier-Cabot and Bazant, 1987), the damage variable (Bazant and Pijaudier-Cabot, 1988), the equivalent strain (Bazant and Lin, 1988), the specific fracture strain (Pijaudier-Cabot and Bazant, 1987), the inelastic stress (Jirasek, 1998), the inelastic stress rate (Jirasek, 1998) and the inelastic stress calculated from the non-local strain (Bazant et al., 1996). Jirasek (Jirasek, 1998) demonstrated that only averaging the equivalent strain, the energy release rate or the specific fracturing strain can correctly reproduce large post-peak deformation or complete fracture. Other integral non-local models lead to spurious residual stresses and to a dilation of the softening zone. Of note, phase-field fracture propagation models are numerically equivalent to integral non-local isotropic damage models and mathematically similar to gradient-enhanced damage models (Planas et al., 1993; Voyiadjis and Mozaffari, 2013; Ambati et al., 2015). Phase-field methods are computationally efficient but remain challenging to use for predicting fracture surfaces explicitly.</p>
<p>In my group, we used an integral non-local formulation to model geomaterials that exhibit tensile softening (Jin and Arson, 2018). The equation of the free energy was similar to the expression of Ψ above, with different coefficients for open and closed cracks. In order to account for the non-local nature of damage, the equivalent strains that control damage evolution was replaced by their weighted average defined on an influence domain V, as follows:</p>
<p><img src="https://imechanica.org/files/equation13.jpeg" alt="Equation 13" width="316" height="39" /></p>
<p>where x is the position vector of the material point considered, and ξ is the position vector of points in the influence domain of x. α<span class="s1">′</span>(x, ξ) is the normalized Bell-shaped function. Figure 5 shows the results of FEM simulations of three-point bending tests, with the local and the non-local formulations of the damage model and for two mesh refinements. The internal length parameter in these simulations was l<span class="s1">c </span>= 0.01 m. The element size was 0.065 m for the coarse mesh and about 0.002 m for the fine mesh, so that the ratio characteristic length to element size was 0.15 for the coarse mesh and 5 for the fine mesh. Results show that non-local enhancement avoids mesh dependency during crack development: the width of the process zone is the same for both mesh refinements (marked with a rectangle in Figure 5).</p>
<p><img src="https://imechanica.org/files/Figure5_0.jpg" alt="Figure 5" width="675" height="506" /></p>
<p><strong>Figure 5.</strong> Horizontal damage component (i.e. vertical crack density) obtained by simulating a three-point bending test without and with non-local enhancement, for various mesh densities. Only one half of the beam is shown. Figure 8 from (Arson, 2020), article licensed under the Creative Commons Attribution Non-Commercial Share Alike 4.0 License.</p>
<p> </p>
<p>Differential formulations</p>
<p>In differential formulations (de Borst et al., 1999; Bazant and Jirasek, 2002), local field variables are developed in Taylor series (De Vree et al., 1995; Peerlings et al., 1996; Askes et al., 2000; Askes and Sluys, 2002):</p>
<p><img src="https://imechanica.org/files/equation14.jpeg" alt="Equation 14" width="387" height="19" /></p>
<p>Δ<span class="s1">2 </span>denotes the Laplacian operator and the coefficients c<span class="s1">i </span>depend on the weight function α and on the averaging volume V. The coefficients c<span class="s1">i </span>can be expressed explicitly in terms of l<span class="s1">c </span>if the weight function is a Gaussian distribution. The implicit second-order scheme, expressed as:</p>
<p><img src="https://imechanica.org/files/equation15.jpeg" alt="Equation 15" width="174" height="19" /></p>
<p>is widely invoked to simulate softening in problems of dynamics (De Borst et al., 1995; Askes et al., 2000), brittle damage (De Borst et al., 1995; De Vree et al., 1995; Askes and Sluys, 2002) and plasticity (De Borst et al., 1995; de Borst et al., 1999). The parameter ζ with the dimension of a length squared is related to the internal length. As indicated by Geers et al. (Geers et al., 1998), applying the equation above with a constant ζ provides inconsistent predictions in mode I crack propagation, because the damage zone becomes wider and wider in a direction perpendicular to the crack, where the material should unload. It is possible to overcome this limitation by accounting for the transient behavior of the gradient parameter during damage evolution.</p>
<p>An important limitation of the non-local damage models is the lack or relationship between the internal length parameter and characteristic dimensions of the material components. Bazant and Pijaudier-Cabot (Bazant and Pijaudier-Cabot, 1989) set an internal length equal to 3 times the maximum aggregate size in concrete while using an integral non local approach. However, this number was chosen based on experiments conducted on only one type of concrete material. Giry et al. (Giry et al., 2011) and Vandoren and Simone (Vandoren and Simone, 2018) proposed to relate a variable internal length to both the characteristic size of the material and the stress field. A fiber bundle model (Villette et al., 2020) recently explored the effect of material characteristic length on the size of the fracture process zone. In a fiber bundle model, the material is seen as an assembly of fibers that can break upon traction. When Δ fibers get broken at constant displacement and only through the redistribution of forces, this is called an avalanche of size Δ. Material damage is quantified based on the number of broken fibers. The statistical damage due to the propagation of a cohesive crack embedded in the fiber was studied by introducing a characteristic length by means of a spatially correlated field of fiber tensile strength, represented by an auto-correlated random process. It was found that the avalanche distribution is sensitive to the auto-correlation length, and that a transitional regime of avalanche distributions appears if the characteristic size of the material heterogeneities is small enough compared to the (finite) characteristic size of the shape of the stress field.</p>
<p>Non-local formulations not only avoid non-physical localization of damage and strains, but also capture the dependence of mechanical behavior on specimen size. The key is the introduction of an internal length parameter, which scales the size of the zone of influence of local field variables. The next sections discuss two alternatives to the non-local formulations: the theory of micromorphic media and the stochastic gradient plasticity framework.</p>
<p class="p1"> </p>
<p><strong>Theory of micromorphic media</strong>
</p><p>The theory of micromorphic media introduces additional degrees of freedom for microscopic translations and rotations. Toupin (Toupin, 1964), Mindlin (Mindlin, 1964; Mindlin and Eshel, 1968) and Germain (Germain, 1973) developed the theory of microstructure enriched elasticity. Vernerey and collaborators (Vernerey et al., 2007) extended the formulation to microstructure enriched elastoplasticity for hierarchical materials. The microscopic displacement, noted u<span class="s1">′</span>i<span class="s2">(x</span>′<span class="s1">i</span>), is defined in reference to the macroscopic displacement, noted u<span class="s1">i</span>(x<span class="s1">i</span>). Expanding u<span class="s1">′ </span>i<span class="s2">(x</span>′<span class="s1">i</span>) in Taylor’s series and truncating the higher order terms, one gets the microscopic displacement field of a so-called micromorphic continuum of degree one: <span class="s2">u</span>′i <span class="s2">= u</span>i <span class="s2">+ χ</span>ij<span class="s2">x</span>′<span class="s1">j </span>. The kinematic description of a micromorphic continuum of degree one only depends on the macroscopic displacement field u<span class="s1">i </span>and on the gradient of micro-displacement ∂ju<span class="s1">′</span><span class="s1">i </span>= γ<span class="s1">ij </span>. By definition, γ<span class="s1">ij </span>is a second order tensor, the symmetric part of which, (γ<span class="s1">ij </span>+ γ<span class="s1">ji</span>)/2, is called the micro-strain rate tensor, and the anti-symmetric part of which, (γ<span class="s1">ij </span>− γ<span class="s1">ji</span>)/2, is named the micro-rotation rate tensor. Noting the macroscopic strain ϵ<span class="s1">ij </span>= (∂<span class="s1">i</span>u<span class="s1">j </span>+ ∂<span class="s1">j</span>u<span class="s1">i</span>)/2, the relative deformation η<span class="s1">ij </span>= ∂<span class="s1">i</span>u<span class="s1">j </span>− γ<span class="s1">ij </span>and the microdeformation gradient κ<span class="s1">ijk </span>= ∂<span class="s1">i</span>γ<span class="s1">jk </span>= γ<span class="s1">ij,k</span>, the volumetric Helmholtz free energy ψ can be expressed in terms of the deformation tensors ϵ<span class="s1">ij </span>, η<span class="s1">ij </span>, κ<span class="s1">ijk</span>. The work-conjugated stress variables are respectively the Cauchy symmetric stress σ<span class="s1">ij </span>, the micro-structure relative stress tensor s<span class="s1">ij </span>, and the third order double stress tensor ν<span class="s1">ijk</span>.</p>
<p>Micro-polar media, governed by Cosserat models (Cosserat and Cosserat, 1909), are a particular case of micromorphic medium of degree one, in which only micro-rotations are accounted for (i.e. microscopic translations are ignored) (Matsushima et al., 2000; Chambon et al., 2001; Vernerey et al., 2007). Second gradient models are also models of micromorphic media of degree one, in which it is assumed that the gradient of micro deformation is equal to the macro deformation. The micropolar and second gradient theories were successfully applied to rocks (Tamagnini et al., 2001; Sulem and Vardoulakis, 2014) and granular materials (Chambon et al., 2001). To circumvent the scale effects in conventional damage models with softening, continuum damage mechanics models were enhanced with the gradient of strain (Zhou et al., 2002; Zhao et al., 2005). The conventional strain and the strain gradient are split into an undamaged elastic part and a damaged irreversible part, and the increment of work-conjugated Cauchy stress and higher order stress are updated thanks to the damage yield function and the damage potential.</p>
<p>Most gradient plasticity theories rely on a generalized plastic strain expressed as a function of the local plastic strain and its high-order gradients (Voyiadjis and Al-Rub, 2005):</p>
<p><img src="https://imechanica.org/files/equation16.jpeg" alt="Equation 16" width="270" height="24" /></p>
<p>in which l is a length parameter. The interaction coefficients γ<span class="s1">1</span>, γ<span class="s1">2 </span>and γ<span class="s1">3 </span>(which control how ϵ<span class="s1">p </span>and ∇<span class="s1">n</span>ϵ<span class="s1">p </span>are coupled) are typically found by deriving a set of dislocation mechanics-based equations. g (l<span class="s1">n</span>∇<span class="s1">n</span>ϵ<span class="s1">p</span>) is the measure of the effective plastic strain gradient of order n. The equation above should ensure that ϵ<span class="s1">p </span>→ ϵ<span class="s1">p </span>whenever ϵ<span class="s1">p </span>>> l ∇ϵ<span class="s1">p </span>and that ϵ<span class="s1">p </span>→ l ∇ϵ<span class="s1">p </span>whenever ϵ<span class="s1">p </span><< l ∇ϵ<span class="s1">p</span>. In the gradient plasticity theory introduced by Fleck and Hutchinson (Fleck and Hutchinson, 1993; Fleck et al., 1994; Hutchinson and Fleck, 1997; Begley and Hutchinson, 1998; Fleck and Hutchinson, 2001), γ<span class="s1">1 </span>= 2m, γ<span class="s1">2 </span>= γ<span class="s1">3 </span>= 2, f (ϵ<span class="s1">p</span>) = (ϵ<span class="s1">p</span>)^(1/m)<span class="s1"> </span>and g (l<span class="s1">n</span>∇<span class="s1">n</span>ϵ<span class="s1">p</span>) = l η, with l η = c<span class="s1">1 </span>(∇ϵ<span class="s1">p</span>)<span class="s1">iik </span>(∇ϵ<span class="s1">p</span>)<span class="s1">jjk </span>+c<span class="s1">2 </span>(∇ϵ<span class="s1">p</span>)<span class="s1">ijk </span>(∇ϵ<span class="s1">p</span>)<span class="s1">ijk </span>+c<span class="s1">3 </span>(∇ϵ<span class="s1">p</span>)<span class="s1">ijk </span>(∇ϵ<span class="s1">p</span>)<span class="s1">kji</span>, in which the material parameters c<span class="s1">1</span>, c<span class="s1">2 </span>and c<span class="s1">3 </span>are of length-square dimension. The first gradient of the plastic strain ∇ϵ<span class="s1">p </span>is alternatively defined as the second-order gradient of the displacement field, ∇.∇u. The mechanism-based strain gradient plasticity theory and the Taylor-based non-local theory proposed by Gao and co-workers (e.g., (Nix and Gao, 1998; Gao et al., 1999; Huang et al., 2000; Gao and Huang, 2001; Guo et al., 2001; Hwang et al., 2002; Qiu et al., 2003; Huang et al., 2004)) are based on the framework proposed by Fleck and Hutchinson and correspond to the case γ<span class="s1">1 </span>= 2, γ<span class="s1">2 </span>= 1, γ<span class="s1">3 </span>= 2/m, f (ϵ<span class="s1">p</span>) = (ϵ<span class="s1">p</span>)^(1/m)<span class="s1"> </span>and g (l<span class="s1">n</span>∇<span class="s1">n</span>ϵ<span class="s1">p</span>) = lη, where η = √[(∇ϵ<span class="s1">p</span>)<span class="s1">jjk </span>(∇ϵ<span class="s1">p</span>)<span class="s1">jjk </span>/4] and in which l is a length parameter that depends on the shear modulus and Burgers vector of the material.</p>
<p class="p1"> </p>
<p><strong>Stochastic gradient plasticity framework</strong>
</p><p>The main inconvenient of the gradient plasticity models that are formulated within the theory of micromorphic media is that they depend on a third- or higher-order stress tensors that are work conjugate to ∇<span class="s1">n</span>ϵ<span class="s1">p </span>(n being the degree of the micromorphic medium), for which higher-order surface tractions are typically not uniquely defined or difficult to satisfy when solving boundary-value problems (Voyiadjis and Al-Rub, 2005). Furthermore, it was noted by Bazant and Guo (Bazant and Guo, 2002) that the asymptotic behavior at small sizes is too strong due to the presence of third-order stresses in the formulation. That is why stochastic gradient plasticity frameworks are attractive to understand the non-local effects of micro-mechanical processes such as dislocations on the macroscopic behavior of a cracked REV.</p>
<p>The theory of stochastic gradient plasticity was initially proposed by Aifantis and collaborators (Aifantis, 1984, 1987; Zbib and Aifantis, 1988; Tsagrakis and Aifantis, 2002; Zbib and Aifantis, 2003) and then used by other authors (e.g., (Yang et al., 2011; Segura Valdivieso, 2017)). In this theory, the evolution of the deformation field depends on elastic deformation and on the production, annihilation and motion of defects (such as cracks or dislocations). The plastic strain is expressed as a function of the densities of these defects, based on geometric or phenomenological considerations. The starting point is to write a conservation equation for the density of each family of defects. For the k<span class="s1">th </span>family of defects of density ρ<span class="s1">k</span>:</p>
<p><img src="https://imechanica.org/files/equation17.jpeg" alt="Equation 17" width="466" height="35" /></p>
<p>in which <strong>J<span class="s1">k </span></strong>is the flux of the defects of the k<span class="s1">th </span>family, F<span class="s1">k </span>is a source term, v<span class="s1">k </span>is the velocity of the k<span class="s1">th </span>family of defects and D<span class="s1">k </span>is a diffusion coefficient that accounts for random influences on the motion of these defects. G<span class="s1">k</span>w˙<span class="s1">k </span>is an additional stochastic term, where each variable w˙<span class="s1">k represents a</span> random processes, for instance large intrinsic fluctuations of the disclocation velocity (or local slip rate). Tsagrakis and Aifantis (Tsagrakis and Aifantis, 2002) showed that for a single family of gliding dislocations, the equation above may be rewritten in terms of plastic strain. In one dimension, noting γ the plastic shear strain:</p>
<p><img src="https://imechanica.org/files/equation18.jpeg" alt="Equation 18" width="329" height="39" /></p>
<p>in which μ and c are a viscoplastic-like coefficient and a gradient coefficient respectively, σ<span class="s1">ext </span>is the external stress, σ<span class="s1">int </span>is the internal stress with mean value σ<span class="s1">int </span>and fluctuations δσ<span class="s1">int</span>. In the absence of internal stress fluctuations (δσ<span class="s1">int </span>= 0) or viscoplastic strains (μ = 0), one gets the simplest stochastic gradient plasticity model, in which σ<span class="s1">ext </span>= σ<span class="s1">int</span>(γ) − c<span class="s1">∂</span><span class="s1">^2</span><span class="s1">γ/</span><span class="s1">∂x^</span><span class="s2">2 </span>. Noting the constitutive relationship between stress and strain as σ<span class="s1">int</span>(γ) = κ(γ) and expressing σ<span class="s1">ext </span>in terms of equivalent (Von Mises) stress τ:</p>
<p><img src="https://imechanica.org/files/equation19.jpeg" alt="Equation 19" width="143" height="42" /></p>
<p>If one assumes that the strain γ is a function of a random microstrain variable Γ, a Taylor expansion of the mean ⟨Γ⟩ yields an expression that depends on the autocorrelation function of the microstrains C(r) that depends on a correlation length l, which plays the role of an internal length parameter in a non-local formulation or a micromorphic continuum mechanics framework. For a Gaussian auto-correlation function, C(r) = exp[−(r/l)^2]. A phenomenological generalization of the equation above was proposed to include the first-order strain gradient, as follows:</p>
<p><img src="https://imechanica.org/files/equation20.jpeg" alt="Equation 20" width="322" height="21" /></p>
<p>in which σ and ϵ are the Von Mises equivalent stress and strain, respectively and c<span class="s1">1 </span>and c<span class="s1">2 </span>are phenomenological functions of the equivalent strain. In the equation above, the gradient ∇ϵ denotes either the second gradient of the displacement field (“deformation version” of gradient plasticity) or the first gradient of the plastic strain tensor (“flow version” of gradient plasticity).</p>
<p>Abu Al Rub and Voyiadjis (Voyiadjis and Al-Rub, 2005) noted that the gradient plasticity framework as laid out in the following equation:</p>
<p><img src="https://imechanica.org/files/equation16.jpeg" alt="Equation 16" width="282" height="25" /></p>
<p>does not provide satisfactory interpretations of size effects when a fixed internal length parameter l is employed and proposed instead to make it a function of microstructure parameters that vary with plastic deformation. Based on the Taylor model in dislocation mechanics, they showed that l is in the order of the average distance between dislocations (noted L<span class="s1">s</span>), as follows:</p>
<p><img src="https://imechanica.org/files/equation21.jpeg" alt="Equation 21" width="234" height="40" /></p>
<p>b<span class="s1">S </span>and b<span class="s1">G </span>are the magnitudes of the Burgers vectors associated with the statistically stored dislocations (SSDs) and geometrically necessary dislocations (GNDs), respectively, α<span class="s1">S </span>and α<span class="s1">G </span>are statistical coefficients which account for the deviation from the regular spatial arrangements of the SSD and GND populations, respectively, r is the Nye factor and M is the Schmidt’s orientation factor. It was later argued that L<span class="s1">s </span>depends on plastic deformation, and the authors revised the equation above as follows:</p>
<p><img src="https://imechanica.org/files/equation22.jpeg" alt="Equation 22" width="143" height="45" /></p>
<p>in which d is the mean grain size and D is the characteristic size of the REV. In the absence of plastic strain, l = h d, which implies that the mean dislocation path in the absence of accumulated plastic strain is equivalent to the grain size. The concept of variable internal length parameter was used by other authors (e.g., (Faghihi and Voyiadjis, 2012; Zhao et al., 2015; Zhang and Aifantis, 2015)) and allowed an interpretation of macroscopic behavior of damaged media in terms of dislocations and pile ups.</p>
<p class="p1"> </p>
<p><strong>Concluding remarks</strong>
</p><p>In top-down modeling approaches, REV-scale state equations are derived from thermodynamic conjugation relationships. In fabric-enriched models, the damage variable is a convolution of moments of probability of microstructure descriptors of characteristic sizes that define internal lengths. Non-local continuum models are based on REV-scale evolution laws for strains and damage, and at least one state or internal variable is defined by developing a local variable in a Taylor’s series or by integrating that variable on a control volume. The internal length in that case is the length that intervenes in the differential development or the size of the control volume. Micromorphic media are based on a balance of energy written in terms of the REV-scale state variables and their micro-scale counterparts. The calculation requires expanding the local displacement field into a Taylor’s series which again, involves an internal length parameter. When only the first derivative of the local displacement field is conserved in the series, one obtains a second-order gradient model. Second-order gradient formulations using a non-local plastic strain tensor were often used in the gradient plasticity theory, in which the internal length parameter typically depends on the shear modulus and Burgers vector of the material. In the stochastic gradient plasticity framework, the plastic strain is expressed as a function of the densities of defects such as cracks or dislocations. The REV-scale strain is a function of a random microstrain variable, the mean of which depends on an auto-correlation function that depends on a correlation length, which plays the role of internal length.</p>
<p>Continuum mechanics models are specific to the REV scale at which are they are formulated, precisely because they depend on internal length parameters that have a physical meaning. Stochastic gradient plasticity approaches do account for variable internal lengths, but the evolution laws of those internal lengths are only valid so long as large localization zones do not exist. Continuum approaches break down when the internal length that characterizes the microstructure changes in order of magnitude, which typically means that the REV needs to be adjusted in size. Crack coalescence has been studied extensively in rock mechanics. Boundary Element simulations highlighted the occurrence of tensile vs. shear driven coalescence from the tips of pre-existing flaws. In metals, void coalescence and subsequent ductile fracture propagation were mostly studied through micromechanical models formulated at the scale of a unit cell containing one or few voids, or through numerical approaches, mainly, the Finite Element Method (FEM) and phase field methods. The interaction between a diffuse crack distribution and a larger fracture, which can be seen as the interaction between coalesced cracks and other cracks, was studied by combining continuum mechanics models of damage in the FEM with cohesive fracture models, either through the extended FEM (XFEM) or by means cohesive Zone Models (CZMs). My group’s work on that topic was published in (Jin et al., 2017; Jin and Arson, 2019, 2020).</p>
<p>Figure 6 summarizes the main analytical, numerical and experimental tools available to date to predict the propagation of a single fracture or a distribution of cracks. Although there exit several methods to couple microstructure to macroscopic damage, there is no method in the current body of literature to predict the response of a microscopic specimen that contains cracks with a length of the order of 10 nanometers (“grey area” in Figure 6). At present, statistical mechanics is too computationally expensive to simulate micro-scale domains from the interaction of atoms or molecules in that domain. Micromechanics relies on Griffith’s theory, which cannot be unconditionally applied at the scale of a few nanometers.</p>
<p><img src="https://imechanica.org/files/Figure6.jpg" alt="Figure 6" width="763" height="429" /></p>
<p><strong>Figure 6.</strong> The main tools to analyze and predict fracture propagation across scales, and the links between them. Legend: SEL: Scaling Effects Laws. MD: Molecular Dynamics. LSO: Local Structure Optimization. SGCMC: Semi-Grand Canonical Monte Carlo.</p>
<p> </p>
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