CZMComputeGlobalTractionTotalLagrangian

Compute the equilibrium traction (PK1) and its derivatives for the Total Lagrangian formulation.

Overview

The CZMComputeGlobalTractionTotalLagrangian uses the local traction, $var element = document.getElementById("moose-equation-de086ce9-5e2e-443e-9a2c-13f56faf268a");katex.render("\\hat{t}", element, {displayMode:false,throwOnError:false});$ , and the derivatives w.r.t. to interface displacement jump, $var element = document.getElementById("moose-equation-42f74469-408d-4db9-9d68-2a977b626855");katex.render("\\partial \\hat{t} / \\partial \\llbracket \\hat{u} \\rrbracket", element, {displayMode:false,throwOnError:false});$ , calculated from any cohesive zone constitutive model, to computes the first Piola-Kirchoff traction in global coordinates, $var element = document.getElementById("moose-equation-38c7fa29-11cf-4de4-9605-d8cbed864d57");katex.render("T", element, {displayMode:false,throwOnError:false});$ , and its derivatives. This object computes the following partial derivatives: $var element = document.getElementById("moose-equation-8e4b57e9-17e5-425e-b788-2dc329d045ce");katex.render("\\partial T / \\partial \\llbracket \\hat{u} \\rrbracket", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-68c01fd9-11cc-4924-8915-a6e4e6b990b3");katex.render("\\partial T / \\partial F", element, {displayMode:false,throwOnError:false});$ , assuming the two are independent. This object assumes finite strain and does account for the interface rotation and area changes caused by deformations and/or rigid body motion.

Theory

Kinematic and geometric variables

The total Lagrangian approach always assumes as reference configuration the initial configuration. The interface midplane deformation gradient $var element = document.getElementById("moose-equation-bd0c53c4-019c-41c1-a696-e63ed03b94eb");katex.render("F", element, {displayMode:false,throwOnError:false});$ is assumed to be: $(1)var element = document.getElementById("moose-equation-be245e90-1592-4a99-9355-7465911fc673");katex.render(" F = \\frac{1}{2}\\left(F^+ + F^-\\right)", element, {displayMode:true,throwOnError:false});$ Using the multiplicative decomposition we can define : $(2) var element = document.getElementById("moose-equation-201334fa-0fbc-41b4-ab34-565f207ef274");katex.render(" F = R U", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-8651da89-c077-494a-ac36-cc3fbc7e5005");katex.render("R", element, {displayMode:false,throwOnError:false});$ is the rotation matrix transforming from the undeformed to the current configuration, and $var element = document.getElementById("moose-equation-8d0e55f3-125a-4d22-a6ca-cad171609d7c");katex.render("U", element, {displayMode:false,throwOnError:false});$ is the corresponding stretch. Let's define $var element = document.getElementById("moose-equation-0e111565-53fb-4b94-a58d-548586f3c99d");katex.render("N", element, {displayMode:false,throwOnError:false});$ as the midplane unit normal in global coordinates in the undeforemd configuration and $var element = document.getElementById("moose-equation-319b0e22-fa93-4a4a-b796-98dae491567b");katex.render("n", element, {displayMode:false,throwOnError:false});$ as the midplane normal in global coordinates in the current configuration. The two normals are related by the rotation $var element = document.getElementById("moose-equation-2efdec87-13e8-4d78-ae59-1b916ddad581");katex.render("R", element, {displayMode:false,throwOnError:false});$ as follows: $(3) var element = document.getElementById("moose-equation-80f19fb7-da9e-460d-bc9f-f32108feae58");katex.render(" n = R N", element, {displayMode:true,throwOnError:false});$ Let's now introduce the rotation matrix $var element = document.getElementById("moose-equation-4e4d924b-a88a-479a-a007-8eb513e0aad8");katex.render("Q_0", element, {displayMode:false,throwOnError:false});$ transforming from the interface coordinate system in the undeformed configuration to the global coordinate system also in the undeformed configuration. One can define the total rotation matrix which transform from the interface coordinate system in the undeformed configuration to the global coordinate system in the deformed configuration as: $(4)var element = document.getElementById("moose-equation-701b9b6c-477c-48f1-8a30-7db8044c7931");katex.render(" Q = R Q_0", element, {displayMode:true,throwOnError:false});$

First Piola-Kirchoff Traction

By definition the first Piola-Kirchoff traction is the Cauchy traction, $var element = document.getElementById("moose-equation-99a1a01b-e327-46bf-b2de-acbf09836734");katex.render("t", element, {displayMode:false,throwOnError:false});$ , acting on the reference area $var element = document.getElementById("moose-equation-1520eee5-e4f3-4a2a-b460-b95cc4e15f7f");katex.render("dA", element, {displayMode:false,throwOnError:false});$ . Hence, using force equilibrium we can write: $(5)var element = document.getElementById("moose-equation-bce34424-f445-4256-a4c0-7693ceeb7827");katex.render(" T = \\frac{da}{dA} t", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-44c20b81-aaf2-4eb0-a220-925005597159");katex.render("da", element, {displayMode:false,throwOnError:false});$ is the area in the current configuration. Nanson's formula allows to compute the area ratio between the deformed and undeformed configuration: $(6)var element = document.getElementById("moose-equation-e7ee209c-b378-48fa-bf56-0418cf360d9c");katex.render(" \\frac{da}{dA} = \\textrm{det}\\left(F\\right) \\vert\\vert F^T N \\vert\\vert = J \\vert\\vert F^T N \\vert\\vert", element, {displayMode:true,throwOnError:false});$

The Cauchy traction and the interface traction are related by the total rotation $var element = document.getElementById("moose-equation-900e4bcb-3a79-4f34-abfd-886233cee772");katex.render("Q", element, {displayMode:false,throwOnError:false});$ : $(7)var element = document.getElementById("moose-equation-90d8b2b0-abae-4c94-b3c7-2fdc6911f3a4");katex.render(" t = Q \\hat{t}", element, {displayMode:true,throwOnError:false});$

Substituting Eq. (7) in Eq. (5) we obtain $(8)var element = document.getElementById("moose-equation-a2fde50d-d39b-4d58-a093-80128a7b7531");katex.render(" T = \\frac{da}{dA} Q \\hat{t}", element, {displayMode:true,throwOnError:false});$

The CZM Interface Kernel Total Lagrangian uses the total PK1 traction computed using equation Eq. (8).

First Piola-Kirchoff Traction derivatives

Using the chain rule, we can decompose the derivative of the traction w.r.t. the discrete displacements as $(9)var element = document.getElementById("moose-equation-9eca7fe8-c716-440b-b192-afa49dee2c0a");katex.render(" \\frac{\\partial T_{i}}{\\partial u^{\\pm,k}_s} = \\frac{\\partial T_{i} }{\\partial F_{pq}} \\frac{\\partial F_{pq}}{\\partial u^{\\pm,k}_s} + \\frac{\\partial T_{i} }{\\partial \\llbracket u \\rrbracket_p } \\frac{\\partial \\llbracket u \\rrbracket_p }{\\partial u^{\\pm,k}_s}", element, {displayMode:true,throwOnError:false});$ By expanding the above terms and using the rule we obtain: $(10)var element = document.getElementById("moose-equation-cf75c195-fa2a-43f9-97f1-d19a563eda8b");katex.render(" \\frac{\\partial T_i}{\\partial u^{\\pm,z}_r} = \\left[ \\frac{\\partial \\frac{da}{dA}}{\\partial F_{pq}} \\left( Q \\hat{t} \\right) + \\frac{da}{dA} \\left( \\frac{\\partial Q_{ij}}{\\partial F_{pq} \\hat{t}_j + Q_{ij} \\frac{\\partial \\hat{t}_j }{\\partial \\llbracket \\hat{u} \\rrbracket_v} \\frac{\\partial Q^T_{vw}}{\\partial F_{pq}\\llbracket u \\rrbracket_w \\right) \\right]\\frac{\\partial F_{pq}}{\\partial u^{\\pm,k}_s} + \\frac{da}{dA} Q_ij \\frac{\\partial \\hat{t}_j }{\\partial \\llbracket \\hat{u} \\rrbracket_r } Q^T_{rp} \\frac{\\partial \\llbracket u \\rrbracket_p }{\\partial u^{\\pm,k}_s}", element, {displayMode:true,throwOnError:false});$

The CZM Interface Kernel Total Lagrangian uses the two terms within curly brackets in Eq. (10) to compute the analytic Jacobian.

Example Input File Syntax

This object is automatically added from the Cohesive Master Master Action when strain=FINITE.

Input Parameters

base_nameMaterial property base name
C++ Type:std::string
Controllable:No
Description:Material property base name
blockThe list of blocks (ids or names) that this object will be applied
C++ Type:std::vector<SubdomainName>
Controllable:No
Description:The list of blocks (ids or names) that this object will be applied
boundaryThe list of boundaries (ids or names) from the mesh where this object applies
C++ Type:std::vector<BoundaryName>
Controllable:No
Description:The list of boundaries (ids or names) from the mesh where this object applies
computeTrueWhen false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the MaterialBase via MaterialBasePropertyInterface::getMaterialBase(). Non-computed MaterialBases are not sorted for dependencies.
Default:True
C++ Type:bool
Controllable:No
Description:When false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the MaterialBase via MaterialBasePropertyInterface::getMaterialBase(). Non-computed MaterialBases are not sorted for dependencies.
declare_suffixAn optional suffix parameter that can be appended to any declared properties. The suffix will be prepended with a '_' character.
C++ Type:MaterialPropertyName
Controllable:No
Description:An optional suffix parameter that can be appended to any declared properties. The suffix will be prepended with a '_' character.
prop_getter_suffixAn optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
C++ Type:MaterialPropertyName
Controllable:No
Description:An optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.

Optional Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
Controllable:No
Description:Adds user-defined labels for accessing object parameters via control logic.
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Controllable:Yes
Description:Set the enabled status of the MooseObject.
implicitTrueDetermines whether this object is calculated using an implicit or explicit form
Default:True
C++ Type:bool
Controllable:No
Description:Determines whether this object is calculated using an implicit or explicit form
seed0The seed for the master random number generator
Default:0
C++ Type:unsigned int
Controllable:No
Description:The seed for the master random number generator

Advanced Parameters

output_propertiesList of material properties, from this material, to output (outputs must also be defined to an output type)
C++ Type:std::vector<std::string>
Controllable:No
Description:List of material properties, from this material, to output (outputs must also be defined to an output type)
outputsnone Vector of output names where you would like to restrict the output of variables(s) associated with this object
Default:none
C++ Type:std::vector<OutputName>
Controllable:No
Description:Vector of output names where you would like to restrict the output of variables(s) associated with this object