Continuum homogenization and RVEs

For now (to help with a conversation that I’m having with a few collaborators) this post provides only the following “infographic” to illustrate the concept of approximating a periodic discrete system with an effective continuum over a sufficiently large scale. (More information will be added about this topic as needed and/or as requested).

Below is shown a five-link chain (in red-blue-green-orange-black). Immediately this colorful chain is a dark-gray plot of the exact (mesocale) lineal density, which is defined at a location “x” to be the mass within an infinitesimal segment dx at that location divided by the segment’s length dx. This local density is shown as the dark-gray shaded plot in the upper-left corner, and it is the slope of the black line in the graph of the lower-left corner.

The continuum concept. Homogenization and identification of an RVE is based on the value of x for which perturbations fall below numerical round-off error for a given application.

The continuum concept. Homogenization and identification of an RVE is based on the value of x for which perturbations fall below numerical round-off error for a given application.

The exact homogenized (macroscale) lineal density at a location “x” is defined as the exact total mass falling inside the span from zero to x,  divided by the chain’s length (x itself).  While the mesoscale density is the local slope at location “x” of the black line in the graph, the macroscale density is the secant slope at location “x” of the same black line.   The continuum (red-dashed) approximation of the local mass distribution ignores local fluctuations from the fact that the chain is actually heterogeneous. For short chain lengths, the exact macroscale density is significantly different from the continuum density, but this discrepancy asymptotes toward zero as the chain length is increased.

 

The theoretical representative volume element (RVE) size corresponds to the size for which the discrepancy (like the plot in the lower-right corner of the infographic) falls below some tolerable threshold, which is determined by considering the tolerable error in an engineering simulation.

These concepts apply to other properties besides density. For example, the macroscale elastic stiffness would be defined as the force applied to the chain divided by the corresponding induced displacement. Like density, this macroscale property varies with the number of links in the chain, but it asymptotes to a steady value as the chain length increases.

Density has a nice asymptotic continuum limit that isn’t sensitive to dilutely distributed statistical perturbations in the local (microscopic) density.  If, for example, 1 in 10000 links is made of light aluminum while the others are made of heavy steel, then the continuum density will be nevertheless close to that of a chain that is made entirely of steel links. The continuum elastic stiffness is likewise not highly sensitive to slight variations in local constituent (link) stiffness.  A chain’s failure strength, on the other hand, is profoundly affected by existence of even a miniscule fraction of weaker links. A mostly steel chain that contains relatively few aluminum links would have a continuum strength equal to the strength of the weaker (aluminum) link. That’s because (in the limit) an infinitely long chain would contain at least one aluminum link. For short chains that are made of, say, 10 links (each of which has a 1 in 10000 chance of being made of aluminum), the average macroscale strength would be higher on average than the strength of longer chains. The strength data for short chains would also be more variable.

These observations give insight into what a modeler must pay attention to when using continuum macroscale properties in simulations of engineering structures.  To design for the structure’s daily (i.e., normal and therefore usually elastic) usage conditions, homogenized continuum properties would be fine. However, continuum strength properties would need to be appropriately perturbed based on the size of the finite elements. This explicit incorporation of statistical variability in continuum properties is required when those perturbations strongly influence the engineering objective of the analysis (such as computing failure risk). In fact, it can be argued that such revisions are crucial to predict fracture and fragmentation whenever the finite-element size is smaller than a few kilometers. For more details on scale-dependent and statistically variable macroscale properties, see Publication: Aleatory quantile surfaces in damage mechanics and the more recent 2015 IJNME article, “Aleatory uncertainty and scale effects in computational damage models for failure and fragmentation” by Strack, Leavy and Brannon.

Publication: Verification tests in solid mechanics

ABSTRACT: Code verification against analytical solutions is a prerequisite to code validation against experimental data. Though solid-mechanics codes have established basic verification standards such as patch tests and convergence tests, few (if any) similar standards exist for testing solid-mechanics constitutive models under nontrivial massive deformations. Increasingly complicated verification tests for solid mechanics are presented, starting with simple patch tests of frame-indifference and traction boundary conditions under affine deformations, followed by two large-deformation problems that might serve as standardized verification tests suitable to quantify accuracy, robustness, and convergence of momentum solvers used in solid-mechanics codes. These problems use an accepted standard of verification testing, the method of manufactured solutions (MMS), which is rarely applied in solid mechanics. Body forces inducing a specified deformation are found analytically by treating the constitutive model abstractly, with a specific model introduced only at the last step in examples. One nonaffine MMS problem subjects the momentum solver and constitutive model to large shears comparable to those in penetration, while ensuring natural boundary conditions to accommodate codes lacking support for applied tractions. Two additional MMS problems, one affine and one nonaffine, include nontrivial traction boundary conditions.

Some eye-catching graphics (see the paper itself for details):

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Publication: Aleatory quantile surfaces in damage mechanics

ABSTRACT: In statistical damage mechanics, a deterministic failure limit surface is replaced with a scale-dependent family of quantile surfaces. An idealized homogeneous isotropic matrix material containing cracks of random size and orientation is used to elucidate expected mathematical character
of aleatory uncertainty and scale effects for initiation of damage in a brittle material. Scope is limited to statistics and scale dependence for the ONSET (not subsequent progression) of shear-driven failure. Exact analytical solutions for probability of such failure (with an interesting pole-point visualization) are derived for axisymmetric extension or compression of a single-crack sample. A semi-analytical bound on the failure CDF is found for a multi-crack specimen by integrating the single-crack probability over an exponential crack size distribution for which the majority of flaws are small enough to be safe from failure at any orientation. Resulting tails of the predicted failure distribution differ from Weibull theory,
especially in the third invariant.

Selected cool pictures (see the article for more images):

2014AleatoryQuantileSurfacesPic1

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F-tables for prescribed deformation

Motion without superimposed rotation

Motion without superimposed rotation

Same deformation with superimposed rotation

Same deformation with superimposed rotation

When developing constitutive models, it is crucial to run the model under a variety of standard (and some nonstandard) homogeneous deformations. To do this, you must first describe the motion mathematically. As indicated in http://csm.mech.utah.edu/content/wp-content/uploads/2011/03/GoBagDeformation.pdf, a good way to do that is to give the deformation gradient tensor, F. The component matrix [F] contains the deformed edge vectors of an initially unit cube, making this a very easy to way to prescribe deformations.

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Tangent mapping


These images show the initial configuration of a body (square) and a nonlinear deformation of that body into a curvy shape (to the right of the square).  Overlaid on the actual deformed shape is the so-called tangent mapping at the indicated point.  It coincides with the nonlinear mapping to first-order accuracy.

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Illustration of polar decomposition

This posting explains the meaning of a polar decomposition, and it gives two numerical methods for computing it.

Below is shown simple shear of a unit square.  The inscribed circle and the lines from corner to corner should be regarded as painted on the material, so they flow with deformation.  The green and red dashed lines show the principal directions of stretch, which are aligned with the major axes of the deformed ellipse and hence move relative to the material as the deformation proceeds.  In the deformed state (far right), the red and green dashed lines are defined to be aligned with the major axes of the deformed ellipse (far right). The red and green dashed lines in the other states show the material points covered by those green and red lines in the deformed state.

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