Porous Plasticity
Use the Porous Plasticity subnode to define the properties of a plasticity model for a porous material.
The Porous Plasticity node is only available with some COMSOL products (see www.comsol.com/products/specifications/). The material model is available for 3D, 2D, and 2D axisymmetry.
When a Porous Plasticity node is present, a Volumetric Plastic Strain plot and a Current Void Volume Fraction plot are available under Result Templates.
Porous Plasticity Model
Use this section to define the plastic properties of the porous material.
Material Model
Select the Material model for the porous plasticity criterion — Shima–Oyane, Gurson, Gurson–Tvergaard–Needleman, Fleck–Kuhn–McMeeking, FKM–GTN, or Capped Drucker–Prager.
For FKM–GTN, select the Plastic potential — Associated, Gurson–Tvergaard–Needleman, or Fleck–Kuhn–McMeeking.
Shima–Oyane
For Shima–Oyane enter the following data:
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Initial yield stress σys0.
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Shima–Oyane parameter α.
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Shima–Oyane parameter γ.
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Shima–Oyane parameter m.
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Initial void volume fraction f0.
Gurson
For Gurson enter the following data:
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Initial yield stress σys0.
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Initial void volume fraction f0.
Gurson–Tvergaard–Needleman
For Gurson–Tvergaard–Needleman enter the following data:
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Initial yield stress σys0.
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Tvergaard correction coefficient q1.
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Tvergaard correction coefficient q2.
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Tvergaard correction coefficient q3.
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Initial void volume fraction f0.
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Select an Effective void volume fraction — Bilinear, Asymptotic, or User defined.
For Bilinear and Asymptotic, enter the following data:
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Critical void volume fraction fc.
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Failure void volume fraction ff.
For User defined, enter the effective void volume fraction as a function of for example the void volume fraction, variable <phys>.fcv.
Fleck–Kuhn–McMeeking
For Fleck–Kuhn–McMeeking enter the following data:
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Initial yield stress σys0.
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Initial void volume fraction f0.
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Maximum void volume fraction fmax.
FKM–GTN
For FKM–GTN enter the following data:
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Initial yield stress σys0.
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Tvergaard correction coefficient q1.
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Tvergaard correction coefficient q2.
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Tvergaard correction coefficient q3.
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Initial void volume fraction f0.
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Maximum void volume fraction fmax.
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Void volume fraction cutoff for GTN model fGTN.
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Void volume fraction cutoff for FKM model fFKM.
Capped Drucker–Prager
For Capped Drucker–Prager enter the following data:
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Select the Plastic potential Qp related to the flow rule — Associated or Nonassociated.
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Select how the Equivalent plastic strain εpe is computed — Associated, von Mises, or User defined. Enter a User defined expression in the hp field as needed. See Hardening Rule for details.
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Initial yield stress σys0.
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Yield function parameter a1.
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When Plastic potential is set to Nonassociated, the Potential parameter a1Q.
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Initial void volume fraction f0.
The material properties use values From material (default) or User defined.
Enter the data for the compression cap in the Cap Model section.
See also Drucker–Prager Criterion and Porous Plasticity in the Structural Mechanics Theory chapter.
Octahedral Section
The shape of the yield function in the Octahedral Plane can be controlled by setting the Octahedral section Γ(θ) — Circular, Gudehus, or Lagioia–Panteghini.
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For Gudehus, enter the Strength ratio c.
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For Lagioia–Panteghini, enter the parameters c1, c2, c3.
The octahedral section for the Capped Drucker–Prager model is always circular.
See also Void Nucleation and Growth in the Structural Mechanics Theory chapter.
Void Growth
It is possible to Include void nucleation in tension or Include void growth in shear by selecting the corresponding checkbox.
When Include void nucleation in tension is selected, enter the Void volume fraction of nucleating voids, the Standard deviation for void nucleation, and the Mean strain for void nucleation. For each property use the value From material or enter a User defined value or expression.
When Include void growth in shear is selected, enter the Void growth rate parameter. Use the value From material or enter a User defined value or expression.
See also Void Nucleation and Growth in the Structural Mechanics Theory chapter.
Isotropic Hardening
Select the type of linear or nonlinear isotropic hardening model from the Isotropic hardening model list. See Isotropic Hardening for details.
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Select Perfectly plastic (ideal plasticity) if the material can undergo plastic deformation without any increase in yield stress.
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For Linear the default Isotropic tangent modulus ETiso uses values From material (if it exists) or User defined. The flow stress (yield level) σys is modified as hardening occurs, and it is related to the equivalent plastic strain in the porous matrix εpe as
with
For the linear isotropic hardening model, the flow stress (yield stress) increases proportionally to the equivalent plastic strain in the porous matrix εpe. The Young’s modulus E is taken from the elastic material properties.
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Select Ludwik from the list to model nonlinear isotropic hardening. The flow stress (yield level) σfm is modified by the power law
The Strength coefficient k and the Hardening exponent n use values From material (if it exists) or User defined.
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Select Johnson–Cook or Modified Johnson–Cook from the list to model strain rate dependent hardening. The Strength coefficient k, Hardening exponent n, Reference strain rate , and Strain rate strength coefficient C use values From material (if it exists) or User defined.
Select a Thermal softening model — No thermal softening, Power law, or User defined.
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For Power law, enter the Reference temperature Tref, the Melting temperature Tm, and the Temperature exponent, m.
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For User defined, enter the Thermal Softening function f(Th), the Reference temperature Tref, and the Melting temperature Tm. The softening function f(Th) typically depends on the built-in variable for the normalized homologous temperature Th and have the properties f(0) = 0 and f(1) = 1. The variable is named using the scheme <physics>.<elasticTag>.<plasticTag>.Th, for example, solid.lemm1.popl1.Th.
The yield stress and hardening function for the Johnson–Cook model is given by
For the modified Johnson–Cook model, the yield stress is instead given by
In the case of power law softening, .
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For Swift nonlinear isotropic hardening, the Reference strain ε0 and the Hardening exponent n use values From material (if it exists) or User defined. The yield stress depends on the power law
Typically, the reference strain equals the onset of plasticity, ε0 = σys0/E.
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Select Voce from the list to model nonlinear isotropic hardening. The yield level is modified by the exponential law
The Saturation flow stress σsat and the Saturation exponent β use values From material (if it exists) or User defined.
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For Hockett–Sherby nonlinear isotropic hardening, the Steady-state flow stress σ∝, the Saturation coefficient m, and the Saturation exponent n use values From material (if it exists) or User defined. The yield level is increased by the exponential law
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For Hardening function, the isotropic Hardening function σh(εpe) uses values From material or User defined. The flow stress (yield level) is modified as
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This definition implies that the hardening function σh(εpe) in the Material node must be zero at zero plastic strain. In other words, σys = σys0 when εpe = 0. With this option it is possible to enter any nonlinear isotropic hardening curve. The hardening function can depend on more variables than the equivalent plastic strain in the porous matrix, for example the temperature. Select User defined to enter any function of the equivalent plastic strain εpe. The variable is named using the scheme <physics>.<elasticTag>.<plasticTag>.epe, for example, solid.lemm1.popl1.epe.
See also Isotropic Hardening in the Structural Mechanics Theory chapter.
Cap Model
Use this section the enter data for the compression cap when the Capped Drucker–Prager material model is selected.
Select the Cap model — Elliptic, Planar, or Cam-clay.
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For Elliptic, enter the Initial pressure limit pc0 and the Initial ellipse centroid pcc0. The ellipse aspect ratio is then derived from the intersection of the ellipse with Drucker–Prager criterion.
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For Planar, enter the Initial pressure limit pc0.
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For Cam-clay, enter the Initial pressure limit pc0.
Select the Hardening model — Perfectly plastic (no hardening), Linear, Exponential, or Hardening function.
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When Linear is selected from the list, the default Hardening modulus Kc is taken From material.
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When Exponential is selected from the list, the default Hardening modulus Kc and the Maximum volumetric plastic strain εpvol,max are taken From material.
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When Hardening function is selected from the list, the default Hardening modulus pch and the Maximum volumetric plastic strain εpvol,max are taken From material.
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Select the Hardening variable — Volumetric plastic strain, Equivalent plastic strain, or User defined.
See also Compression Cap in the Structural Mechanics Theory chapter.
Nonlocal Plasticity Model
Nonlocal plasticity can be used to facilitate, for example, the modeling of material softening. Typical examples that involve material softening are finite-strain plasticity and soil plasticity. In these situations, standard (local) plasticity calculations reveal a mesh and topology dependence, where a mesh refinement fails to reproduce a physically sound solution. Nonlocal plasticity adds regularization to the equivalent plastic strain, thereby stabilizing the solution.
The default setting is None. Select Implicit Gradient to add nonlocal regularization to the equivalent plastic strain. Then, enter values or expressions for:
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Length scale, lint — The regularization length scale for the equivalent plastic strain εpe should ideally be greater than the largest mesh element size.
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Nonlocal coupling modulus, Hnl — The value represents the penalization of the difference between the local and nonlocal variables. A larger value enforces the equivalent plastic strain εpe to be closer to the nonlocal equivalent plastic strain εpe,nl. A typical value would be around 10% the equivalent shear modulus of the material.
See also Nonlocal Plasticity in the Structural Mechanics Theory chapter.
It is possible to apply the implicit gradient method to the variables for the equivalent plastic strain or void volume fraction, by selecting the corresponding checkbox.
Select the Void volume fraction checkbox in order to apply the implicit gradient method of regularization to the void volume fraction variable. Enter a value for the Length scale, void volume fraction, lint,f.
The length scale for the equivalent plastic strain, lint, and the length scale for the void volume fraction, lint,f, can be chosen independently. Ideally, both parameters should be larger than the largest mesh element size, while remaining independent of it.
Discretization
To display this section, click the Show More Options button () and select Advanced Physics Options in the Show More Options dialog, and it is available with the Implicit gradient nonlocal plasticity model.
Select the shape function for the Nonlocal equivalent plastic strain εpe,nl — Automatic, Linear, Quadratic Lagrange, Quadratic serendipity, Cubic Lagrange, Cubic serendipity, Quartic Lagrange, Quartic serendipity, or Quintic Lagrange. The available options depend on the order of the displacement field.
When the Void volume fraction checkbox is selected, select the shape function for the Nonlocal void volume fraction fnl — Automatic, Linear, Quadratic Lagrange, Quadratic serendipity, Cubic Lagrange, Cubic serendipity, Quartic Lagrange, Quartic serendipity, or Quintic Lagrange. The available options depend on the order of the displacement field.
Advanced
To display this section, click the Show More Options button () and select Advanced Physics Options in the Show More Options dialog.
Enter the Maximum damage, which defines the residual stiffness of the model. The default value is 0.995.
Select the Local method to solve the plasticity problem — Automatic, Backward Euler, or Backward Euler, damped. When Backward Euler or Backward Euler, damped is selected, it is possible to specify the maximum number of iterations and the relative tolerance used to solve the local plasticity equations. Enter the following settings:
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Maximum number of local iterations. To determine the maximum number of iteration in the Newton loop when solving the local plasticity equations. The default value is 25 iterations.
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Relative tolerance. To check the convergence of the local plasticity equations based on the step size in the Newton loop. The final tolerance is computed based on the current solution of the local variable and the entered value. The default value is 1e-6.
When the Backward Euler, damped method is selected, the Newton’s method is enhanced by line search iterations. Using this method can improve the robustness of the plasticity algorithm when the plastic potential or hardening model are highly nonlinear. When selected, it is possible to specify the Maximum number of line search iterations. The default value is 4 iterations.
For the Capped Drucker–Prager model, see the settings in the Advanced section for Pressure-Dependent Plasticity and Cap and Cutoff.
See also the Numerical Integration Algorithm section in the Structural Mechanics Theory chapter.
To compute the energy dissipation caused by porous compaction, enable the Calculate dissipated energy checkbox in the Energy Dissipation section of the parent material node.
Location in User Interface
Context Menus
Solid Mechanics > Linear Elastic Material > Porous Plasticity
Solid Mechanics > Nonlinear Elastic Material > Porous Plasticity
Solid Mechanics > Hyperelastic Material > Porous Plasticity
Ribbon
Physics tab with Linear Elastic Material, Nonlinear Elastic Material, or Hyperelastic Material node selected in the model tree:
Attributes > Porous Plasticity