In a hy- perelastic material, the stress is obtained from a stored energy function that depends on the local deformation. By isotropy at a point X, we mean that the material is insensitive to superposed rigid body motions on the reference configuration. We see that the left Cauchy-Green deformation tensor b satis- fies the definition of isotropy but not necessarily that of objectivity. Thus, with frame invariance and isotropy, the number of independent deformation variables is reduced further to three. The concept of multiplicative split was motivated by crystal plasticity since F p may be viewed as the deformation induced when single crystals slip via dislocation motion, while F e rotates and distorts the crystal lattice Lee Fortunately, in the multiplicative plasticity theory, we do not need to know the intermediate configuration, so the deformation is not essentially modified by any discontinuous mapping.
Motivated by the result shown in equation 5. This results in complete uncoupling of the elastic and plastic responses. The dissipation inequality of thermodynamics then takes the following form analogous to 3. Thus, 5. In view of the requirement of objectivity, we shall note at the outset that some of the expressions will contain so-called initial stress terms, in addition to the moduli terms.
We shall also invoke the prop- erty of isotropy so that we can use all of the nice results we developed from the preceding section. Inserting expres- sion 5. Note that d can replace l in 5. A hyperelastic material is subjected to combined tension and shear in plane strain. This makes it more convenient to impose relevant conservation laws on the reference configuration than on the current configuration.
Throughout this section, we shall thus adopt a Lagrangian description for convenience. The variational equation then develops following standard lines i. The volume integral on the left of 5. Thus, the volume integral on the left of 5. The hyperelastic constitutive equa- tion 5. The relevant rate equation to integrate is 5. Note that the plastic corrector is nothing else but the negative of the Lie derivative of be. More specifically, the predictor-corrector format is facilitated by a so- called product-formula algorithm Simo , Simo and Hughes In a nutshell, the algorithm consists of two steps.
Note in this case that the Lie derivative of be has no spin, which means that the plastic correction takes place at fixed principal directions. From the coaxiality of the relevant tensors, the principal values of the left- and right-hand sides of 5. Therefore, we can simply work on the principal values, which are scalar quantities.
The above predictor-corrector equa- tion, first presented by Simo , preserves the additive return-mapping format of the infinitesimal theory. Box 5. Alternatively, m A can be constructed from available computer pack- ages Borja et al. Note that the spectral directions of be tr and be are the same.
We are interested in taking the first variation of this volume integral with respect to the displacement field u for purposes of develop- ing a consistent algorithmic tangent operator for nonlinear finite element calculations. We need two preliminary results. Combining the above equation with 5. The first term on the right-hand side of 5. Show that c has minor symmetries with respect to its first two and last two indices. It exploits numerous elements of nonlinear continuum mechanics, including various def- initions of stresses and deformation.
Under the assumption of isotropy, the spectral decomposition technique is once again exploited in both the theoretical developments and numerical implementation of the theory. The definition of an elastic material is quite broad: if it stores but does not dissipate energy, and if it returns to its undeformed shape when the loads are removed, then the material is elastic.
Nonlinear elastic behavior could be due to the elastic parameters being intrinsically dependent on the state of stress material non- linearity , or to the large deformation that developed in the specimen during testing geometric nonlinearity. The elastic properties of geomaterials, such as soil, rock, and concrete, are known to depend on the confining pressure and density, but these materials are also known to have a very small elastic range.
On the other hand, rubber can sustain a very large deformation and still behave elastically. Figure 6. In both figures, the strains experienced by the material during testing are too small to alter the geometry of the specimen. Variable elastic moduli typically are incorporated into the constitutive for- mulation using the framework of hypoelasticity Borja and Lee This approach has a serious drawback in that on a closed elastic loop, the incremen- tal constitutive equation could either generate or dissipate energy Zytynski et al.
An overarching rule on any elastic model is that it should be conservative in the sense that no energy may be generated or lost during a closed elastic cycle. This is fulfilled by a hyperelastic formulation, a constitutive framework that relies on the existence of a stored energy function. Nonlinear elasticity is used in the formulation of critical state plasticity models.
Dynamic values are from wave velocities. Data from Brace and Simmons and Brace Figure reproduced from Jaeger and Cook Data from Hardin and Richart To describe yielding in compression, as well as the accompanying inelastic compaction, the yield surface is provided with a compression cap that can expand or shrink depending on the mechanism of deformation.
In this chapter we present a sampling of plasticity models that employ a compression cap to capture plastic compaction in geomaterials. We combine these cap plasticity models with hyperelasticity and present an elegant numerical stress-point integration technique based on re- turn mapping in strain space that accommodates the intricate coupling of the elastic and plastic deformation responses. It possesses the major symmetry and two minor symmetries with respect to the first two and last two indices.
Inelastic volume change, whether by compaction or dilation, is inter- preted as shear-induced and not a result of simple isotropic compression or extension. However, a material yielding at critical state neither compacts nor dilates when sheared. The CSL separates the compression cap from the dilation side of the yield surface.
By comparison, the Cam-Clay yield surface has the shape of a bullet with vertices at the intersection points with the p-axis. Therefore, the CSL may be considered as a limiting failure envelope for frictional materials. Thus, we can develop a hardening law for pc by performing loading-unloading tests in isotropic compression.
These two hardening laws are accurate up to moderately high pressures. At great depths i. The mechanism for expansion and contraction of the yield surface is shown in Fig. At the critical state the size of the yield surface is fixed perfect plasticity. It may be worthwhile to compare the above hardening law with the one we used for J2 plasticity: The hardening law for the CC and MCC models is driven by the volumetric component of plastic strain, whereas the hardening law for J2 plasticity is determined by the deviatoric component of plastic strain.
This is motivated by Figs. The coupling term e is represented by D12 , which is not zero, resulting in coupled volumetric and deviatoric responses. We should emphasize at the outset that the presented exponential form for the stored energy function was motivated by the experimentally observed dependence of the elastic moduli on the confinement for some materials, and is valid only over a certain range of elastic strains.
Exercise 6. Evaluate the elements of the Hessian matrix D e and state whether this matrix is diagonal or fully populated. To capture this feature, two-invariant models such as the MCC must be enhanced with the third stress invariant. For illustration purposes, we enhance the two-invariant MCC yield func- tion to include the third stress invariant. Argyris et al. Extracting the derivatives by brute force is not an attractive option since it entails an enormous amount of work Willam Fortunately, through the spectral representation technique, we can simplify the task of obtaining the derivatives by working in principal stress space.
In this case, a simple perturbation would be necessary to arrive at three distinct eigenvalues. Recall that in Chapter 4 a similar perturbation has been suggested in the spectral decomposition of the Cauchy stress tensor to ensure that the three principal stresses are distinct. Use the result from Exercise 6. Nor-Sand was developed to capture the dilatant behavior of dense sand during shearing. In what follows we present the yield function for the enhanced version of this model. The exponent parameter N determines the curvature of the yield surface on the hydrostatic axis, and typically has a value less than 0.
Cross section on deviatoric plane left and isometric view in principal stress space right. Figure reproduced from Andrade and Borja It is important to choose the parameters of the yield and plastic potential functions appropriately to ensure a nonnegative plastic dissipation. The above hard- ening law for sand should be contrasted with the hardening law for MCC given in 6.
In the MCC model, the size of the yield surface is driven by the volumetric component of plastic strain; in the sand model, the size is driven by the deviatoric component. The next section presents a plasticity model for concrete in which the movement of the compression cap is driven by the cumulative plastic strain as well as by the mean normal stress to capture increased ductility under increased confinement.
In this section, we consider a cap plasticity model proposed by Kang and Willam that fits nicely within the framework of the present three- invariant formulation. The discussion below focuses on the mechanism of yielding that includes hardening, failure, and softening. We elucidate the individual components of the yield function below. Note from the inset that the local cohesion softening is accompanied by a local friction hardening near the vertex. The von Mises stress q1 is selected so that the compressive meridian passes through the point of uniaxial compressive strength.
Hence, the cross-sections of the failure surface on the deviatoric plane are more triangular near the tensile strength region, and more circular at high pressures. These features are in agreement with the experimental data for plain concrete shown in Fig. It may be useful to compare the above hardening law with 6. When the hardening parameter k reaches the limiting value of unity, the compression cap disappears, and the yield surface coincides with the failure surface.
The softening mechanism defines the move- ment of the vertex p0 toward zero, signifying the loss of tensile strength with plastic deformation. The hardening surface is inactive during the softening process.
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The Kang-Willam concrete model uses linear elasticity with constant elas- tic moduli. The surfaces are all smooth except at the vertex. However, in the infinitesimal regime the additive split is still valid in the strain space. In the finite deformation regime, we have seen that the additive split is also valid in the space of principal logarithmic strains.
In general, a return mapping algorithm in the strain space is more robust than the classical return mapping in the stress space because it can readily accommodate nonlinear elasticity. We readily see an additive split in the incremental strains irrespective of the elastic constitutive behavior. Note that if a constant elasticity tensor ce operates on both sides of the above equation, the standard additive predictor-corrector split in stress space is recovered.
In this case, 6. The next step is to solve the local nonlinear problem. Note that a return map on the principal axes reduces the number of local unknowns, as well as ameliorates the treatment of the third stress invariant. The algorithm does not require that the Cauchy stresses be stored, rather, they are simply calculated from the elastic strains. Box 6. Similar formulations can be used for other isotropic constitutive models such as those described in the previous sections. We recall from Section 5. In the finite deformation range, the geometric meanings of the strain in- variants are as follows.
By analogy with equation 6. We note at the outset that the above equation is not an acceptable model of elasticity for extreme strains. Next, we turn to the yield function for the MCC model. Finally, we consider the hardening law for the MCC model. To obtain the plastic part, we subtract the elastic rebound. According to equation 6. Consider a stored energy function of the form cf. Then, show that this function pre- dicts the correct behavior for extreme strains.
In equation 6. Simulations involving variable density are uncommon, first, because density as a state variable is rarely integrated into the constitutive formulation, and, second, because one has to measure and quantify the spatial density variation within the domain of analysis to make the simulation meaningful.
Current advances in digital imaging have made the latter process feasible, since we now have the technology to quantify the spatial variation of density noninvasively. In the example that follows, we use X-Ray Computed Tomography CT and digital image processing to quantify the spatial density variation in a rect- angular specimen of sand. The simulation involves axially compressing this heterogeneous specimen under plane strain loading condition.
The higher the specific volume, the looser the sand. The black and white image is a negative image, i. For reference, similar finite element simulations were conducted by Borja et al. The refined mesh has an equally refined descrip- tion of heterogeneity, making the mesh sensitivity study more challenging.
The bottom side of the finite element mesh is supported on vertical rollers to simulate a sled used during actual testing. The top side is also supported on rollers except the upper left-hand corner is pinned to the support. Stress histories are all stored at the Gauss integration points. For the most part, stresses in the looser regions lie on the compression cap of the yield surface. The specimen is mm tall by 40 mm wide by 80 mm deep out-of-plane. We employ the sand constitutive model described in Section 6. The complete set of material parameters can be found in Borja et al.
We use a finite deformation formulation based on multiplicative plasticity similar to the development presented in the previous section for the MCC model. Movement of the compression cap is driven by the deviatoric component of the plastic strain. This state parameter determines whether the stress point lies below or above the critical state line.
Depending on the sign of this state parameter, the compression cap can either expand or shrink. Both meshes suggest the development of a shear band, a localized deformation style discussed further in Chapters 7 and 9. This suggests a lack of mesh sensi- tivity during the early part of loading up until the peak load. Beyond the peak load, the two curves diverge from each other, suggesting some mesh sensitivity issues afflicting the solution.
Similar simulations presented by Fig. Color bar is norm of deviatoric principal logarithmic stretches in percent. Numerical solutions exhibit mesh sensitivity beyond a vertical compres- sion of around 6 mm. Chapter 9 addresses this aspect from the point of view of bifurcation of the solution typically associated with softening responses. Plasticity models can be embedded easily within the hypere- lastic framework. For the class of hyperelastic-plastic models considered in this chapter, a return-mapping algorithm in the elastic strain space provides a robust framework for stress-point integration.
The technique is compatible with the algorithm employed for stress-point integration in the finite defor- mation regime, where the return mapping is performed in the deformation space rather than in the stress space; more specifically, in the space defined by the elastic logarithmic principal stretches. The mechanisms of deformation within this zone can be very complex, and they occur at multiple scales. As shearing progresses, a fault core, or cataclasite zone, develops between surrounding less damaged zones made up of joints and sheared joints.
As one moves farther away from the fault core, the rock becomes less and less damaged, until one finds the competent host rock that marks the end of the fault zone Aydin et al. Because the fine-grained materials in the cataclasite zone may be considered frictional in nature, fault zones are also considered frictional. Figure 7. On a smaller scale, fractures and cracks are much narrower zones of discontinu- ity where adjacent surfaces either separate or slide past each other. But as the degree of damage increases, they could become part of a thicker fault core composed of discrete elements that do not have any semblance to the original cracks.
Because of nearly overlapping definitions and variations in scale, qualitative descriptions of failure modes are quite artificial. However, we distinguish between two mechanisms of deformation: a continuum local- ized mode where the two sides of damage zone are in direct physical contact; and a separation mode where traction-free surfaces are created. Image reproduced from Aydin et al. In this chapter, we use the finite element method as a tool to capture these two localized deformation modes. If the discontinuity is known right from the beginning, we can align the sides of the finite elements to conform with the discontinuity and employ classic nonlinear contact mechanics techniques Laursen , Wriggers , Sanz et al.
However, cracks can nucleate and propagate in random di- rections, in which case, it would not be possible to specify the sliding surfaces a priori. Adaptive remeshing has been used in the past to allow the element sides to adaptively define the geometry of a randomly propagating disconti- nuity.
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However, this approach is cumbersome to use, and problems arise when transferring information from the old mesh to the new mesh particularly in the presence of bulk plasticity. This chapter highlights two finite element techniques where a jump in the displacement field is embedded into a fixed finite element mesh, called the background grid, to accommodate randomly propagating discontinuities.
The first is a global enrichment technique based on the extended finite element method, where additional degrees of freedom are introduced into the existing nodes to interpolate the slip continuously. The second is a local enrichment called the assumed enhanced strain method, where local degrees of freedom are added to the elements to enhance the displacement interpolation. In this technique, the additional degrees of freedom are elimi- nated on the element level and the global system of equations is unchanged.
A robust contact algorithm is essential for either method, and below we outline important developments in the area of contact mechanics that are relevant to the two enhancement techniques. We distinguish between stick and slip conditions. For seismic faulting, Dc is the critical slip dis- tance over which strength breaks down during earthquake nucleation Marone and Kilgore The reverse is true for a stepping to a slower slip speed, i.
Equation 7. Therefore, constitutive law 7. The Lagrange multipli- ers method imposes the contact condition exactly; however, it is unwieldy to implement in the presence of cohesive-frictional contact. The penalty method accommodates cohesive-frictional contact, but does not impose the con- tact condition exactly.
The augmented Lagrangian method uses the penalty method to mimic the Lagrange multipliers method, but requires an additional layer of iterations. We present the mathematical formulations for each of the three methods below. In the context of the standard finite element formulation, this requires that the element sides be aligned to the contacting faces. In Section 7. To complete the formulation of the problem, we must impose the constraint of no inter- penetration on S. The tangential component of traction remains to be quantified.
In develop- ing an expression for the evolution of tT , we must consider both stick and slip conditions as well as the incremental nature of loading. Here, a negative value for the gap function gN implies that there is some interpenetration between the contacting surfaces due to the finite value of kN. If the inequality is true, we accept the predictor as the final value stick condition. Furthermore, the method can handle the stick-slip con- dition quite easily.
However, there are some shortcomings of the method as well. The augmented Lagrangian method strives to strike a balance between the more accurate Lagrange multipliers method and the simpler penalty method Simo and Laursen We can make the value of k N as large as the regular penalty parameter kN , or some- what smaller. In the latter case, the overlap between the contacting surfaces is expected to be larger, but the algorithm does provide a way to iteratively reduce the amount of this overlap.
Inserting 7. Box 7. Step 1 initializes the estimate of the Lagrange multiplier, which can be any nonnegative number including zero. Step 3 checks to ensure that the interpene- tration is acceptable. The downside of the augmented Lagrange method is the additional layer of iterations necessary to find an acceptable value of Lagrange multiplier. This layer of iterations nests over another layer to solve the global nonlinear problem in Step 2. In short, there is price to pay for the accuracy and simplicity of the augmented Lagrangian technique. Yes, augmentation convergence achieved and exit.
Step 4. Therefore, unless a discontinuity is predefined, we cannot structure the finite element mesh so that the sides of the elements are always aligned to the discontinuity. Conventional finite ele- ments can only interpolate conforming displacement fields, i. If a discontinuity passes through the interior of these elements, the shape functions must be enhanced to accommo- date this shock.
This section deals with the enhancement of the finite element shape functions with a Heaviside function to accommodate the kinematics of a shock in the form of a displacement discontinuity. The goal is to enrich the finite element shape functions within this subregion in order to resolve a discontinuous displacement field on S. The open dots are nodal values of u, and a displacement jump occurs on S between nodes A and B.
In this re-parameterized version, the displacement field uc x is the conform- ing displacement field. Outside the subregion B h in Fig. Note that uc is not an approxima- tion to the displacement jump on S. Line AB is the conforming displacement field uc. Outside the support of MS x , conventional finite element interpolation is used. To further illustrate the geometric meaning of f h , Figs. For a CST element cut by a discontinuity, the positive side of S could contain either one node or two nodes.
For a quadrilateral element, the positive side of S could contain one, two, or three nodes Borja and Regueiro , Regueiro and Borja Irrespective of how the discontinuity is placed within the element, one can always associate f h x with the standard finite element basis shape functions. Exercise 7. Develop expressions for the blending function f h for an enriched quadrilateral element with one node, two nodes, and three nodes on the positive side of S, see Fig.
We recall that in the classic form 7. Furthermore, there is no blending term in the classic formulation because the discontinuities are located on the exterior bound- aries, and so the trial and weighting functions define conforming fields all throughout the domain, i. For a CST element, the slip and contact pressure are integrated on S using a two-point quadrature rule. The negative sign follows from continuum mechanics convention that negative stresses are compressive, so the Lagrange multipliers are all negative numbers. We now develop the matrix equation for the Lagrange multipliers formu- lation.
The next section deals with a stabilization scheme alleviating the contact pressure oscillation in the context of the Lagrange multipliers formulation. Assume that a discontinuity passes through the interior of this element so that node A is on the slave side, and nodes B and C are on the master side.
Repeat the problem if nodes A and B are on the slave side, and node C is on the master side. A CST element with an embedded crack is enriched to accom- modate a jump in displacement field. The coordinates of nodes defining the element are a 0, 0 , b 1, 0 , and c 0, 1. At what point on the crack is the norm of the displacement jump maximum? For the linear pair, Bochev and co-workers Bochev et al. After Liu and Borja In this example, an elastic cube is clamped at both its top and bottom ends and pressed in the vertical direction. The cube is discretized with three-dimensional tetrahedral elements, and a smooth horizontal plane is embedded midway between the top and bottom surfaces of the cube, cutting the interior of the tetrahedral elements.
The Lagrange multipliers method is used to impose the contact condition. The left figure shows a contour of La- grange multipliers exhibiting unphysical oscillation of the contact pressures, in contrast to the right figure which shows the correct, smooth solution for the contact pressure. A stabilization methodology has been advocated by Liu and Borja to correct the deficiency inherent in the linear displacement-linear contact pressure interpolation with the extended finite element method.
The method- ology follows the same polynomial pressure projection stabilization advocated by Bochev and co-workers described above. The idea is to add stabilizing terms to the no-interpenetration condition 7. We expound on this sub- ject below in the context of the Lagrange multipliers formulation.
The readers may wish to consult Liu and Borja for similar discussions pertaining to the penalty and augmented Lagrangian methods. We first assume a slip mode so that by stabilizing the contact pressure we also stabilize the shear component of traction. For the Lagrange multipliers method the stabilized form of 7. The concept can be extended to 3D tetrahedral elements, i. With the above technique, the solution recovers the smooth distribution of contact pressure as shown on the right contour of Fig. Hence, stabilizing the contact pressure does not necessarily stabilize the tangential traction.
In other words, both the contact pressure and tangential traction must now be stabilized. The submatrices are the same as before, i. For a stick mode, this stabilization technique has been shown to circumvent pressure and shear stress oscillations Liu and Borja As shown in Fig. However, the enrichment is purely local to the element, so it does not engender additional global degrees of freedom to solve.
The assumed enhanced strain technique revolves around the idea of a strong discontinuity. We remark that the displacement discontinuities de- scribed in the previous sections are all strong discontinuities, but they are treated in the context of an interface problem. To this end, the Dirac delta function from theory of distributions Stakgold plays a key role in the formulation. The Dirac delta function is a generalized function that is zero everywhere but has an infinitely sharp peak bounding a unit area.
Consider a strong discontinuity embedded in an elastic continuum. An example of an acceptable yield function is the simple cohesion-friction law given in 7. As noted in Section 7. This implies that the continuum response of the material inside the discontinuity must approach perfect plasticity. Substituting 7. From equation 7. The magnitude of the discontinuity is statically condensed on the element level prior to global assembly, and the overall system of equations is unchanged. For constant strain triangular or constant strain tetrahedral el- ements, there is only one Gauss point required for numerical integration, so a static condensation on the element level is equivalent to a static condensation at the Gauss point level Borja In what follows, we adopt the develop- ment used for the classic and extended finite element formulations to derive the relevant expressions leading to the assumed enhanced strain method.
Consider the classic variational form 7. The displacement field u x for any enriched element can be obtained from the re-parameterized form 7. Therefore, we have a Petrov-Galerkin formulation. Alternatively, we can write the displacement field u x for any enriched element in the original form 7.
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Comparing 7. This is in accord with the initial hypothesis that the host medium is elastic. An enriched CST element always contains a one-node side, which could lie either on the slave side Fig. In general, the latter condition results in an optimal resolution of the kinematics of the discontinuity, but of course, it is not possible to predict the orientation of a propagating discontinuity.
This problem can be alleviated by a minor adjustment of the nodal coordinates, see Borja Note that the jump direction does not change since the discontinuity is embedded in the solid unless the solid is undergoing finite rotation, see Chapter 8. We can rewrite 7. Note that the corrector part in the above predictor-corrector equation emanates from slip relaxation on the discontinuity and not a result of bulk plasticity, which does not enter in the present formulation.
However, the expression for this contact integral must be modified for the case of piecewise constant slips due to the following reason. For piecewise constant slips, the contact integral may be broken up into individual element constraints due to the lack of connectivity of the slips, i. In a way, this is equivalent to the element not passing the patch test for constant strain fields Hughes For higher-order elements one needs to calculate the volume average or area average, for 2D of the stress term. Note from the above expression that S e cancels out, so the above equation is true irrespective of the position of the discontinuity within an element.
This means that there are no additional degrees of freedom to solve other than the conforming nodal displacements. For a constant stress field, the required one-point Gauss integration provides an additional simplification. As a final remark, linear jumps, such as the one proposed by Linder and Armero , can also be accommodated by the strong discontinuity frame- work, although they are not covered in this book.
Quasistatic and isothermal plastic deformations in single crys- tals arise from slips on specific crystallographic planes. Forward and backward slips are treated separately. For a face-centered cubic crystal, there are 24 slip systems including forward and backward systems. Assume that a crystal is deforming in single slip i. Thus, it is of interest to compare how the as- sumed enhanced strain, extended finite element method, and classic solu- tions resolve this singularity at the crack tip.
The top and bottom boundaries are supported on rollers, and one cor- ner node is pinned to prevent rigid-body horizontal translation. The block is compressed vertically, forcing the crack faces to rub against each other and slide. The finite element mesh consists of nodes and CST elements ar- ranged in a cross-diagonal pattern. For the two enrichment techniques as- sumed enhanced strain and extended finite element methods the crack passes through the element interior, but the elements are oriented so that their sides are nearly aligned to the crack.
For the classic solution, the element sides de- fine the crack itself, where each crack segment is represented by a pair of element faces. Plots show the distribution of plastic slips along the crack: a assumed enhanced strain; b extended finite element; and c classic solutions.
After Borja Figures 7. As expected, slip has a maximum value at the ex- posed face of the block and a minimum value at the crack tip. It is of interest to note that the classic and extended finite element solutions predict nearly the same slip distributions, although they are not one on top of the other in Fig. How- ever, both solutions predict smaller slips overall compared to the assumed enhanced strain solution.
The latter solution exhibits the classical elliptical shape consistent with linear elastic fracture mechanics theory, whereas the other two solutions exhibit bullet-shaped slip distributions.
Clearly, the con- tinuous slip interpolation causes the deformation to lock in the vicinity of the crack tip. This explains why the near-tip deformations calculated by the classic and extended finite element solutions tend to lock. In contrast, slip predicted by the assumed enhanced strain solution shows an infinite gradient at the crack tip. To handle the crack tip singularity, we enrich the description of the near- tip displacement field using a technique proposed by Belytschko and Black With the above crack tip enrichment in place, Fig.
Because of the enriched basis function near the crack tip, the problem of locking has been alleviated, and the solution now accommodates the near-tip singularity of the gradient. The concept was motivated by the cohesive zone models for tensile fracture Barenblatt , Bilby et al. Rocks then fail either by developing a new fault zone or reactivating an old one. The shear strength decays to a lower level on those segments of the fault that slipped.
Various slip weakening laws have been proposed and calibrated in the laboratory Rummel et al. At the point of failure a discontinuity may be embedded into the failed element to simulate the nucleation of a fault. Then, the constitutive description for the element transitions accordingly. The constitutive model for the fault should provide a softening variable to facilitate a smooth tran- sition from the pre-failure to the post-failure states.
For cohesive-frictional materials, the cohesion of the fault is often chosen as this softening variable Borja and Foster Bulk plasticity concentrates near the crack faces as the block slides along the frictional crack. Color bar is cumulative plastic strain in percent. The crack is given an artificial cohesion that decays to zero due to slip weakening. The initial value of this cohesion is determined by equating the yield strength of the pre-failure model, taken as a non-associated Drucker- Prager model in this example, to the yield strength of the post-failure model at the failure stress.
The sides of the finite elements representing the block are not aligned to the crack, so it is not possible to use the classic formulation for this problem. Instead, we can employ either the assumed enhanced strain method Foster et al. Here, we use the latter technique. To allow the frictional resistance on the crack to develop, a pressure of MPa was applied on the two vertical faces of the block. Without this confining stress, the residual strength on the crack would be zero since there would be no contact pressure on it.
The failure stress is reached at a vertical displacement of around 0. The initial rate of vertical compression is 0. After Liu and Borja Figure 7. Bulk plasticity concentrates in the region sur- rounding the crack, suggesting a potential for thickening of the failure zone. The displayed mechanism is in general agreement with the thickening of the fault core shown in the sketches of Fig.
The classic formulation for an interface with negligible thickness may be employed for cohesive-frictional sliding if the geometry of the discontinuity is known a priori. For an evolving interface, such as a crack that nucleates and propagates as the solution progresses, enhanced finite element techniques including the extended finite element and assumed enhanced strain methods may be employed to allow the discontinuity to evolve on a fixed background grid.
Their elastoplastic properties are attributed to the existence of slip planes. The face-centered cubic f. Because of their denser packing, materials with a f. But packing alone does not determine the absolute ductility of a given material. Hexagonal close-packed h. Con- sequently, cadmium, zinc, titanium, beryllium, and other metals with a h. Figures 8. Understanding the elasto-plastic behavior of single crystals is important for the prediction of the overall behavior of crystal aggregates.
Quasistatic and isothermal plastic deformation of single crystals is due to slip on crystal- lographic planes, which occurs when the resolved shear stress on a critical slip system reaches a certain maximum value. The challenge lies in identifying the specific slip systems activated by a given increment of load, since the process typically involves selection from a pool of linearly dependent slip systems. In this chapter, we focus on multi-surface plasticity theory for single crys- tals and present theoretical and computational aspects of imposing linearly independent active constraints.
The name has been coined for this technique because of its two desirable features: it is unconditionally convergent, and it is exact for an imposed crystal deformation varying as a ramp function. Dashed lines denote outline of each Bravais lattice. Moreover, we address both the infinitesimal and finite deformation aspects of crystal plasticity. For the latter aspect, theory of distribution and strong discontinuity concepts are key ingredients of the mathematical formulation leading to the discrete micromechanics of elastoplastic crystals in the finite deformation range.
In keeping with the overall format of the book, we only address the rate-independent aspects of crystal deformation, the reason being that rate-dependent regularization techniques abound in the literature, but a robust rate-independent formulation for elastoplastic crystals remains scarce. There can be no more than N active slip systems, since if a forward slip direction is active then the backward direction must be latent.
For example, a f. Substituting in 8. After Borja and Wren With this law, forward and backward slips gen- erate the same rate of hardening. Crystal orientations can be described by the Euler angles between a fixed reference frame x, y, z and a crystal reference frame xc , yc , zc , as shown in Fig. Figure 8. Some of the yield planes plot one on top of the other, while others cluster in a corner. In light of this, redundant constraints are to be expected when two or more slip systems activate. The algorithm derives its name from the properties that: a it is unconditionally convergent, and b it is exact for imposed crystal deforma- tion varying as a ramp function.
In a nutshell, the ultimate algorithm divides the imposed crystal deformation in increments and tracks the sequence of slip system activation one by one. Therefore, the set J act can have no more than five elements. To demonstrate how the ultimate algorithm works, consider first how a primary slip system activates. Writing 8.
A similar idea may be employed to generalize the algorithm to the multislip case. Since the slips are linearly independent, the matrix [gij ] must be invertible. Box 8. A simple LDU factorization of this array identifies the redundant constraints from the zero elements of D. Step 9. Otherwise, return to calling program. However, we can always show that the resulting overall crystal stress is unique and does not depend on the selected systems. As noted earlier, this set results from an arbitrary selection of independent slip systems and is not unique.
The redundant systems do not contribute to the plastic strain; instead, they simply undergo neutral loading. As an illustration, we consider a f. The crystal is subjected to two deformation histories: extension and simple shear. The perfectly plastic crystal shows a hor- izontal plateau as expected, whereas the Taylor-hardening crystal shows a constant tangent modulus after activating all independent slip systems. In the extension test, a duplex system is encountered at initial yield. Further stretching activates two more linearly independent systems, and then a fifth one is detected at a higher strain.
In the simple shear test, the five linearly independent systems are detected one by one. The resulting stress-strain re- sponses are the same irrespective of the linearly independent slip systems selected. Smaller- font numbers denote independent slip systems detected. In formulating a crystal plasticity theory within the framework of multi- plicative plasticity, the following aspects must be considered.
First, the lattice deformation is purely elastic, and plastic deformation is due solely to slips on crystallographic planes. Second, as the crystal deforms, the lattice stretches and rotates, so the embedded slip systems also stretch and rotate with the crystal lattice. Third, the lattice rotation is not the same as the crystal rota- tion, because the crystal rotation also includes a portion contributed by the plastic slips.
The figure resembles a finite element with an embedded strong discontinuity, a subject discussed thoroughly in Chapter 7. The similarity between the two problems, that of a finite element with a strong disconti- nuity and a crystal lattice with an embedded slip plane, suggests that the strong discontinuity concept could very well be used for the crystal plasticity problem.
We explore this idea in the following developments. For convenience, we regu- larize the slip plane and represent it as a layer D0 with thickness h0 in the reference configuration. Let M and N define the primary slip system in the reference configuration, where M is a unit vector in the direction of slip. We have shown earlier that the elastic deformation gradient F e performs a covariant transformation on N , so it must perform a contravariant transformation on M , i.
Since the reference configu- ration is fixed, then M is also fixed. Equation 8. The regular component hD plays the role of the Taylor hardening modulus in subsequent discussions. The fine-scale fields must be upscaled to the crystal level to obtain the coarse-scale deformation field, which is a conform- ing field describing the overall deformation of the crystal. Interestingly, the theoretical developments related to the enriched finite elements presented in Chapter 7 apply equally well to the crystal plasticity problem.
A glimpse of the similarity of the two problems is evident from Fig. If one takes the latter point of view, then the domain delineated in Fig. The continuous deformation field delineates the fine-scale deformation of the crystal lattice; the slip is the fine-scale displacement jump. Over the REV range, the fine-scale fields are homogenized to obtain the conforming crystal deformation. In the present case, G X is an arbitrary smooth ramp function that varies from zero to one over a unit thickness of the crystal volume in the direction of N.
The actual form of the ramp function is immaterial to the formulation since we only need to extract a deformation gradient from it, i. We shall refer to F e tr as the coarse-scale deformation gradient for the crystal. From equation 8. The foregoing formulation shows that the elastic relative deformation gra- dient f e is a function of the current slip direction m, which in turn is a function of f e.
The foregoing development suggests that for single-slip system the push-forward of the continuous field on the slip system is the same as the push-forward of the conforming field on this same slip system. However, the push-forward of either deformation gradient on the slip system is the same. The following is a summary of the stress-point integration algorithm for a crystal deforming in single slip.
Some of these systems may be linearly dependent, in which case, a special filtering algorithm, such as the ultimate algorithm, may be employed to eliminate the redundant con- straints. For the small strain case, we recall that tracking the individual slip systems is an explicit process because the elastoplastic tangential moduli are constant. However, for the finite deformation case, tracking the slip systems is an iterative process. In general, an iterative solution is employed to: a detect the next active slip system s , and b track the evolution of the cur- rent slip systems.
The discussions below are specific to a duplex system, but it is quite easy to see how the developments can be extended to more than two slip systems. The coarse-scale relative deformation gradient f e tr induces an exact push-forward on the primary slip system; therefore, it induces the same push-forward on all inactive systems, i.
This motivates the development of a simplified stress- point integration algorithm for multislip systems. The linearized equation for f e has a predictor- corrector format that resembles the incremental form developed earlier for a crystal deforming in single slip, see 8. For a duplex system, the problem re- duces to the solution of two equations in two unknowns, which is a much smaller system to solve than the eleven equations required to obtain the exact solution.
The example below demonstrates the performance of this algorithm through some stress-point simulations. All the other crystal properties are the same as in the example of Section 8. In these simulations, we only consider the first two critical slip systems i. Smaller-font numbers denote independent slip systems detected. The finite element mesh has 5, nodes and 2, eight-node hexahedral elements, all integrated with the B-bar option. The cylinder is clamped at both its top and bottom ends while the inner and outer vertical faces are assumed to be traction-free. The top end is axially stretched by 0.
Similar simulations, but without stretching the cylinder, are presented by Borja and Rahmani An infinitesimal formulation is used in the simulation. In the first simulation where the crystal orientation is uniform, four vertical deformation bands emerge from the imposed deformation, as shown in Fig. These bands did not form randomly, but rather, they are determined by the relative crystal orientation with respect to the direction of stretching and twisting.
In the second simulation, a small imperfection is embedded in the cylinder a b Fig.
Color bar is second invariant of deviatoric plastic strain in percent. This imperfection generates more intense localized deformation and complementary deformation bands propagating away from the imperfection. Table 8. Note that the convergence of the iterations is faster for the case where the crystal is uniformly oriented, and slower when there is an imper- fection.
The inelastic response of these materials depends on the crystal orientation relative to the loading directions. Ductility of a material depends on both the atomic packing and the structure of the slip systems. Crystal plasticity theory is de- scribed by multiple yield constraints with significant redundancy, necessitating a robust filtering algorithm to remove the redundant constraints. Figure 9. Both shear bands and compaction bands are understood as examples of a larger class of so-called deformation bands Aydin et al. Deformation bands, typically tabular in form, are generally regarded as a result of material instability, but material instability does not always result in a deformation band.
An opposite mechanism is void growth in metal specimens subjected to a high ratio of isotropic extension- to-deviatoric shear stress, called triaxiality Agarwal et al. From a smaller scale e. Bifurcation theory of continuum mechanics is often used to detect the on- set of instability in the material response. The theory identifies a stress state at which the solution could lose uniqueness. The- ory of plasticity provides a mathematical description of the tangential stress- strain tensor through the continuum elastoplastic tangent operator cep.
A nonlinear elastic solid is an example of a material exhibiting an incrementally linear response. If there is only one yield constraint for the solid, then the tangent operator can have two branches: one for loading and another for unloading. When there are two or more yield criteria to be satisfied, such as in crystal plas- ticity where the slip systems constitute two or more yield constraints, then the tangent operator could have more than two branches. In this section, we establish conditions for the stability of an incrementally linear solid.
Note that the boundary-value problem is stated in incremental, or rate, form, rather than in total form. Stability analysis deals with very small changes in the stress state, necessitating that the problem be stated in rate form. We want to find a point in the solution where more than one local response could emerge. Equation 2. The term on the left-hand side can be written as a surface integral, and also vanishes. This last condition is sometimes called the second-order work for stability Darve et al. In this case, the local uniqueness condition 9. For example, consider a cylindrical specimen of rock compressed axi- ally while maintaining the radial stress constant.
Condition 9. If c has no major symmetry, then condition 9. Further, while severe weakness precludes control, mild to moderate weakness is less well correlated with motor control. For example, the ipsilateral limb after stroke shows abnormal motor control but strength is normal [ ], so motor control requires more than strength. Models such as the Han et al. As such, they are only loosely based on actual data and cannot be used to predict recovery of individual patients. Can the trial-to-trial evolution in movement success during rehabilitation be modeled, and, if so, what are the key learning parameters needed to describe this evolution?
Casadio et al. This model provides insights into the role of assistive force in the recovery process, and the extent to which learned changes in voluntary control decay over time and transfer to subsequent training sessions. The model characterizes the recovery process related to robot-assisted training for improving arm extension in chronic stroke survivors as a linear, discrete-time, shift invariant dynamical system.
The model posits that motor performance is a function of a voluntary control component, an assistive force component, provided either by a robot or a therapist, and a performance noise term. The input signals that drive recovery are movement error or a performance measure. A process noise term accounts for the portion of the recovery not due to these three terms. The most striking result from this model is that the retention parameter predicts the percent change of the Fugl-Meyer score at the 3-month point following the end of the robotic treatment Fig.
This result, which needs to be confirmed with a larger cohort, is potentially important for the individualization of training. After one or few initial training sessions with the robot, the model can be fitted to the data. Then, by examining the retention parameter, one could potentially determine who will benefit from additional robotic motor training.
Note however that this model is based on a set up in which the arm is constrained to move in two dimensions with shoulder and elbow movements. Thus, here again, for example, the issue of restitution versus compensation is difficult to address and the conclusions are limited to simplified movements at only two joints. In the same vein of using computational models for predicting individual recovery, Schweighofer and colleagues developed a first-order dynamical model of stroke recovery with longitudinal data from participants receiving constraint-induced movement therapy in the EXCITE clinical trial [ ].
The goal of the model was to better understand the interactions between arm function and use in human post-stroke following therapy. The time constants in this model were on the order of several weeks. An increase in this parameter, which can be thought of as the confidence to use the arm for a given level of function, led to an increase in self-reported spontaneous use after therapy compared to before therapy. However, as in the previous model of Casadio et al.
It contains only a few parameters and two time constants, one for functional recovery and the other for hand use. The reason for this simplicity is the very sparse data set on which the model was built. Increasing the complexity of the model, and thus the number of parameters was not possible because this would have led to overfitting, i. Rehabilitation therapy often involves interactions between patient and a trainer, be it a rehabilitation therapist, a robotic device, or a computer game.
Modeling the patient-trainer interaction could potentially provide insight into movement recovery. Reinkensmeyer used an adaptive Markov model to examine the role of external mechanical assistance from a robotic device or therapist in promoting movement recovery [ ]. Following research in gait training after spinal cord injury, the model assumes that motor control is characterized by repeated transitions between sensory and motor states; for example, sensing of full hip extension at the end of stance triggers leg flexion.
It further grossly characterizes these states as abnormal and normal sensory states e. The model uses a Hebbian-inspired model of plasticity in which the transitions between specific sensory and specific motor states become more reliable with repetitive activation of that transition. The action of a skilled external trainer robot or human who is assisting in movement is modeled as a mediated increase in the probability of transferring to a state of normal motor output e.
Assistance-as-needed is simulated by mediating this transfer only when the patient is in an abnormal motor state, while assistance-always is simulated by mediating this increase on every movement repetition. While these predictions may sound somewhat intuitive, this is perhaps because they mirror opinions that are often expressed in the current clinical milieu.
The model in this case served to verify that these opinions could be mathematically supported relying on a simple but plausible plasticity rule. Another approach toward modeling patient-trainer dynamics is based on what was first developed as a model of human movement adaptation. As briefly described above, the motor system uses a sunken-v muscle adaption rule to alter force, impedance, and trajectory, explaining a wide variety of experimental findings of reaching in dynamic environments [ , ].
This update rule can be viewed as the motor system implementing a greedy minimization of a cost function of error and effort [ ]. A gradient descent of a similar cost function of error and effort was shown to provide an efficient assistive controller for rehabilitation [ ]. To model this situation, the trainer would take all of the effort off of the patient by minimizing its cost function:. However, individuals with stroke improve their motor function more when actively attempting to move, rather than relying on the robot to move their arm [ ].
To avoid the patient becoming passive, the trainer can modify its cost function to. If one assumes that the patient behaves according to the sunken-v adaptation rule, if the trainer is slightly more lazy i. Thus, this method and the cost function of Eq. In terms of computational neurorehabilitation, this work suggests that sensorimotor rehabilitation may be able to be modeled in terms of the cost functions that the trainee and the trainer seek to implement, as well as the algorithms they use to implement those cost functions.
We contend that the models reviewed in the previous section are evidence that a fundamental understanding of neurologic recovery will be facilitated by modeling motor learning and plasticity itself in a context specific to rehabilitation. Even the initial, simplified models presented are generating novel ideas concerning the mechanisms of recovery in rehabilitation, and thus are suggesting important directions for future experimental research.
For example, the Han et al. The Reinkensmeyer et al. The Casadio et al. Other interesting ideas generated by the initial models reviewed above are that a key way that rehabilitation therapy may function is by modulating a parameter that controls the effect of arm function on use, and that trainer-trainee interactions can be characterized using parsimonious cost functions. Even given their utility in generating such novel ideas concerning rehabilitation, these initial models clearly have shortcomings. They greatly simplify human movement and the rehabilitation process.
They ignore the fact that every day activity relies on bimanual motor function. They selectively model only one or two learning and plasticity mechanisms. They focus on motor learning rather than plasticity mechanisms associated with restitution, as yet neglecting the rich literature on spontaneous biological recovery, which may ultimately offer greater potential for recovery. They have not yet been used to replicate, let alone extend, the findings achieved with prognostic regression models. None have yet used a cross-validation procedure in order to see how much a model based on the data from a pool of patients can explain the behavior of patients outside of that pool.
In addition, sensorimotor control is just one of many domains addressed by neurorehabilitation. As we surveyed above, the mathematical tools for developing such models are already at least partly available because of the past several decades of work in the field of computational neuroscience and machine learning.
Further, as we also reviewed above, large-scale longitudinal data for each patient and for a large number of patients are now possible to obtain with robotic devices and wearable sensors. This will allow the development of more elaborate, physics-based models that predict recovery at a fine temporal and spatial resolution. Note that conventional motor learning experiments have generally failed to even come close to the number of training trials that gives rise to the high level expertise in sports and work-related skills.
With instrumented rehabilitation and wearable sensors, this situation is now set to change, at least in the context of motor learning during rehabilitation. A key challenge for the future is how to utilize these longitudinal data that are collected on different samples, with different methods of motion capture, and different quantifications of movement into a coherent package such that they can inform models of movement control over the course of recovery after stroke. Following this approach, we predict that soon we will see the emergence of large scale mechanistic models of motor recovery that are driven by the actual movement content of rehabilitation therapy as well as records of daily activity.
Ideally, one would use the same data set from hundreds or even thousands of well-characterized patients, so that computational model output is realistic and appropriate. In addition, the models will increasingly be informed by automatic analysis of MRI scans routinely obtained after stroke, since, as we reviewed above, brain anatomical information is needed because identical behavior can arise through largely different neural processes. A range of novel neuroimaging tools and computational methods, including analysis of grey and white matter structures and structural and functional connectivity [ ], will provide an intermediate level of description with which to bridge the gap between what we know about recovery after stroke from animal models compared to what we know from studies of behavior in humans.
Clearly, initial computational neurorehabilitation models are vast simplifications of a very complex process. While simplification and abstraction are often virtues in modeling, with richer data sets, it will be possible to increase the complexity of computational neurorehabilitation models with less risk of overfitting. Models containing a multiple joint system, such as the whole arm or both arms, will be important for understanding compensation. Upper extremity motor control during real-world, free-living activity involves the movement of all the segments in order to position and orient the limb and to interact with objects [ , ].
If a computational model utilizes only a few of the segments, then the model output will provide only a limited view of the actual solution, essentially ignoring the degrees of freedom problem underscored by Bernstein. Models that can help understand or predict control processes during naturalistic actions will be of high value for the field of neurorehabilitation.
Further, it is currently difficult to experimentally study the interacting effects between different forms of learning, such as supervised, unsupervised, and reinforcement learning, including the role of the timing of rehabilitation on these processes [ 81 , ]. There appear to be methods also to induce beneficial plasticity beyond task repetition, such as the possible enhancing of effects of non-invasive brain stimulation on motor learning [ , ].
Computational neurorehabilitation models can incorporate multiple levels of plasticity and learning, as well as plasticity-enhancing effects of techniques such as electrical stimulation, and even psychological effects important to rehabilitation, helping understand these interactions in computer simulations to guide future experimental work. This is important, as one theoretically could vet hypotheses in an efficient and cost-effective manner, rather than relying solely on randomized, controlled trials, which are costly and time consuming.
A key question, of course, is whether the incorporation of plasticity and learning mechanisms, along with internal physiological states, into models will improve upon the predictive capability now possible with prognostic regression models. We contend they have a good chance to, because they are more likely to isolate the key predictive variables of interest, since these variables likely relate to physiologic function, and computational neurorehabilitation models seek to make just such variables explicit.
Such variables likely vary from patient-to-patient as well, suggesting that their isolation will improve individualized predictions. If so, computational models of neurorehabilitation should ultimately improve rehabilitation for individuals with neurologic injuries. We expect that computational models of recovery, based on early clinical data, kinematic performance, and routine scans, will provide the basis for future clinical software that suggests timing, dosage, and content of therapy.
Such an approach will transform neurorehabilitation by guiding clinicians, patients, and health providers in the optimization of treatments. Global and regional burden of stroke during — findings from the Global Burden of Disease Study Comprehensive overview of nursing and interdisciplinary rehabilitation care of the stroke patient: a scientific statement from the American Heart Association.
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