CZMInterfaceKernelTotalLagrangian

Description

This class assembles the integrated first Piola-Kirchoff traction computed by a cohesive zone model (CZM) to the system residual vector, which ensures traction equilibrium across an interface. A CZMInterfaceKernelTotalLagrangian acts only on one displacement component and therefore the user must set up a separate instance of this kernel for for each dimension of the problem. The CZMInterfaceKernelTotalLagrangian uses the PK1 traction and its derivatives provided by a CZM Compute Global Traction Total Lagrangian to impose the appropriate residual and to provide the appropriate Jacobian. This kernel accounts for both interface area changes and rotations.

Residual

The strong form of the force equilibrium equation in vector form can be written as: $var element = document.getElementById("moose-equation-4d5ec764-63d4-4733-ad34-a20f2542eb3f");katex.render(" F^- -F^+ = \\int_{A^-}{T^- dA^-} - \\int_{A^+}{T^+ dA^+} = 0", element, {displayMode:true,throwOnError:false});$ where superscripts $var element = document.getElementById("moose-equation-77d34f55-d7d2-4d4e-b035-f48afc0d474a");katex.render("+", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-d1e30f9c-3b3b-4786-94c3-094897050227");katex.render("-", element, {displayMode:false,throwOnError:false});$ identify the primary and secondary surfaces of the cohesive zone, respectively. Furthermore, $var element = document.getElementById("moose-equation-99737485-bfea-4dff-ab22-989ca7cf530a");katex.render("F", element, {displayMode:false,throwOnError:false});$ represents the force, $var element = document.getElementById("moose-equation-68b70689-0b5a-4c2d-aad1-88dffe9df711");katex.render("T", element, {displayMode:false,throwOnError:false});$ the first Piola-Kirchoff traction, and $var element = document.getElementById("moose-equation-823d8f0f-9599-4004-b117-92c2350c1fb5");katex.render("A", element, {displayMode:false,throwOnError:false});$ the undeformed area. The primary surface is the one where the interface normal is computed.

By utilizing the principle of virtual work and recognizing that forces are work conjugate of displacements, the weak form of the equilibrium equation can be written as $var element = document.getElementById("moose-equation-e8a96391-9ca3-47f6-b18e-e91e14df794e");katex.render(" \\int_{A^-}{T^- \\psi^- dA^-} - \\int_{A^+}{T^+ \\psi^+ dA^+} = 0", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-3ae528b2-5bd3-4dcb-b63f-dfc244d843ce");katex.render("\\psi", element, {displayMode:false,throwOnError:false});$ is a vector of test functions. Each of the test function in $var element = document.getElementById("moose-equation-200f4ea1-057c-4659-9ad4-a253f2ab1d03");katex.render("\\psi", element, {displayMode:false,throwOnError:false});$ is associated to a specific displacement component.

In the undeformed configuration, $var element = document.getElementById("moose-equation-ce05d72e-8d82-43b6-9dd9-579148fcb8c8");katex.render("A^-=A^+=A", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-10e9145c-8a29-4030-a1d8-e14df2d0dbe9");katex.render("T^+=T^-=T", element, {displayMode:false,throwOnError:false});$ . Therefore, the equilibrium equation for the displacement component $var element = document.getElementById("moose-equation-400cf193-e088-45e3-abac-5f85155469c3");katex.render("i", element, {displayMode:false,throwOnError:false});$ can be rewritten as $var element = document.getElementById("moose-equation-dff6b73d-449f-4f02-bc0f-01a721c7f80d");katex.render(" T_i (\\psi^- - \\psi^+) = 0", element, {displayMode:true,throwOnError:false});$ Therefore the residual for the primary and secondary surfaces, for a specific test function $var element = document.getElementById("moose-equation-316f006a-94a1-482a-971e-bcd940d216f4");katex.render("j", element, {displayMode:false,throwOnError:false});$ can be rewritten as $var element = document.getElementById("moose-equation-125be435-6565-4066-9d0a-4155b3d6282a");katex.render("\\begin{aligned} R_i^+ & = & - T_i \\psi^{+}_{j} \\\\ R_i^- & = & T_i \\psi^{-}_{j} \\end{aligned}", element, {displayMode:true,throwOnError:false});$

These are the residual equations implemented in the CZMInterfaceKernelTotalLagrangian. The traction vector $var element = document.getElementById("moose-equation-cfa47f00-ea53-4025-b0a2-ffc2fce9f4d9");katex.render("T", element, {displayMode:false,throwOnError:false});$ by the CZM Equilibrium Traction Calculator Total Lagrangian

Jacobian

The Jacobian for this model is exact. The Jacobian requires calculating the derivative of the residual with respect to the discrete displacements $var element = document.getElementById("moose-equation-7a31182e-6dc6-4a06-b06e-8fed65c6f038");katex.render("u^{\\pm,k}", element, {displayMode:false,throwOnError:false});$ . $var element = document.getElementById("moose-equation-98500a64-f1ab-4193-826b-389045866d18");katex.render("\\begin{aligned} \\frac{\\partial R_i^+}{\\partial u^{+,k}_s} & = & - \\frac{\\partial T_{i}}{\\partial u^{+,k}_s} \\psi^{+}_{j} \\\\ \\frac{\\partial R_i^+}{\\partial u^{-,k}_s} & = & - \\frac{\\partial T_{i}}{\\partial u^{-,k}_s} \\psi^{+}_{j} \\\\ \\frac{\\partial R_i^-}{\\partial u^{+,k}_s} & = & \\frac{\\partial T_{i}}{\\partial u^{+,k}_s} \\psi^{-}_{j} \\\\ \\frac{\\partial R_i^-}{\\partial u^{-,k}_s} & = & \\frac{\\partial T_{i}}{\\partial u^{-,k}_s} \\psi^{-}_{j} \\\\ \\end{aligned}", element, {displayMode:true,throwOnError:false});$

Assuming the traction is only a function of the the midplane deformation gradient, $var element = document.getElementById("moose-equation-e7cb9ad3-46c4-4e1d-909f-e4dde84db389");katex.render("\\hat{F}", element, {displayMode:false,throwOnError:false});$ , and of the displacement jump in global coordinates, $var element = document.getElementById("moose-equation-035cd49c-db36-4d55-9db9-e7918e29c259");katex.render("\\llbracket u \\rrbracket", element, {displayMode:false,throwOnError:false});$ , the partial derivatives of the traction can be rewritten using the chain rule as: $var element = document.getElementById("moose-equation-78a1b87a-4326-4ab5-8096-076c5e95c9fd");katex.render("\\begin{aligned} \\frac{\\partial T_{i}}{\\partial u^{+,k}_s} & = &\\frac{\\partial T_{i} }{\\partial \\hat{F}_{pq}} \\frac{\\partial \\hat{F}_{pq}}{\\partial u^{+,k}_s} + \\frac{\\partial T_{i}}{\\partial \\llbracket u \\rrbracket_{p}} \\frac{\\partial \\llbracket u \\rrbracket_{p}}{\\partial u^{+,k}_s}\\\\ \\frac{\\partial T_{i}}{\\partial u^{-,k}_s} & = & \\frac{\\partial T_{i} }{\\partial \\hat{F}_{pq}} \\frac{\\partial \\hat{F}_{pq}}{\\partial u^{-,k}_s} + \\frac{\\partial T_{i}}{\\partial \\llbracket u \\rrbracket_{p}} \\frac{\\partial \\llbracket u \\rrbracket_{p}}{\\partial u^{-,k}_s}\\\\ \\end{aligned}", element, {displayMode:true,throwOnError:false});$ Substituting the last two equations in the Jacobian definition one obtains the equation implemented in this kernel.

The CZM Equilibrium Traction Calculator Total Lagrangian provides $var element = document.getElementById("moose-equation-dd91d4be-e524-4d9f-846b-d12cc7300f8b");katex.render("\\partial T_{i}/\\partial \\hat{F}_{pq}", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-3f9f9fda-c94c-45c8-a3fd-26850010172f");katex.render("\\partial T_{i} / \\partial \\llbracket u \\rrbracket_{p}", element, {displayMode:false,throwOnError:false});$ . This kernel is responsible for computing $var element = document.getElementById("moose-equation-b961a1db-c0c6-4781-b442-8adfff886701");katex.render("\\partial \\llbracket u \\rrbracket_{p} / \\partial u^{\\pm,k}_{s}", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-a75fa694-1875-40ea-9a8c-1589d4ffa264");katex.render("\\partial \\hat{F}_{pq} / \\partial u^{\\pm,k}_{s}", element, {displayMode:false,throwOnError:false});$ .

Displacement Jump and deformation gradient derivatives

Recalling $var element = document.getElementById("moose-equation-05b42bcd-595c-4ef6-8a67-05c948ad697f");katex.render("\\llbracket u \\rrbracket = u^{-} - u^{+}", element, {displayMode:false,throwOnError:false});$ and that $var element = document.getElementById("moose-equation-8e4192fe-f957-4fc6-b7b5-9bea1dd50b4f");katex.render("u_i =\\sum_z \\phi_{i} u^z_i", element, {displayMode:false,throwOnError:false});$ we can write, $var element = document.getElementById("moose-equation-e155cf5f-9261-42a7-9b7b-13c86abfdfd5");katex.render("\\begin{aligned} \\frac{\\partial \\llbracket u \\rrbracket_i}{\\partial u^{+,z}_j} &=- \\begin{bmatrix} \\phi_{1}^{+,z} & 0 & 0 \\\\ 0 & \\phi_{2}^{+,z} & 0 \\\\ 0 & 0 & \\phi_{3}^{+,z} \\\\ \\end{bmatrix} \\\\ \\frac{\\partial \\llbracket u \\rrbracket_i}{\\partial u^{-,z}_i} & = \\begin{bmatrix} \\phi_{1}^{-,z} & 0 & 0 \\\\ 0 & \\phi_{2}^{-,z} & 0 \\\\ 0 & 0 & \\phi_{3}^{-,z} \\\\ \\end{bmatrix} \\end{aligned}", element, {displayMode:true,throwOnError:false});$

Recalling $var element = document.getElementById("moose-equation-7fa41712-0c60-4e2c-a9e5-a4d8dac8170c");katex.render("\\hat{F} = 0.5 \\left( F^+ + F^-\\right)", element, {displayMode:false,throwOnError:false});$ and that $var element = document.getElementById("moose-equation-1b9ca82a-1030-413d-8b82-36cac46fa8c0");katex.render("F_{ij}=\\delta_{ij}\\sum_z \\left(\\phi_{i,j} u^z \\right)", element, {displayMode:false,throwOnError:false});$ we can write $var element = document.getElementById("moose-equation-e0e269db-8df8-453c-ab29-21a2fffe0bab");katex.render("\\begin{aligned} \\frac{\\partial F_{ij}}{\\partial u^{+,z}_k} = \\frac{1}{2} \\begin{bmatrix} \\begin{bmatrix} \\nabla \\phi_{1,1}^{+,z} & \\nabla \\phi_{1,2}^{+,z} & \\nabla \\phi_{1,3}^{+,z} \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ \\end{bmatrix}_{k=1} \\\\ \\\\ \\begin{bmatrix} 0 & 0 & 0 \\\\ \\nabla \\phi_{2,1}^{+,z} & \\nabla \\phi_{2,2}^{+,z} & \\nabla \\phi_{2,3}^{+,z} \\\\ 0 & 0 & 0 \\\\ \\end{bmatrix}_{k=2} \\\\ \\\\ \\begin{bmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ \\nabla \\phi_{3,1}^{+,z} & \\nabla \\phi_{3,2}^{+,z} & \\nabla \\phi_{3,3}^{+,z} \\\\ \\end{bmatrix}_{k=3} \\end{bmatrix} \\\\ \\frac{\\partial F_{ij}}{\\partial u^{-,z}_k} = \\frac{1}{2} \\begin{bmatrix} \\begin{bmatrix} \\nabla \\phi_{1,1}^{-,z} & \\nabla \\phi_{1,2}^{-,z} & \\nabla \\phi_{1,3}^{-,z} \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ \\end{bmatrix}_{k=1} \\\\ \\\\ \\begin{bmatrix} 0 & 0 & 0 \\\\ \\nabla \\phi_{2,1}^{-,z} & \\nabla \\phi_{2,2}^{-,z} & \\nabla \\phi_{2,3}^{-,z} \\\\ 0 & 0 & 0 \\\\ \\end{bmatrix}_{k=2} \\\\ \\\\ \\begin{bmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ \\nabla \\phi_{3,1}^{-,z} & \\nabla \\phi_{3,2}^{-,z} & \\nabla \\phi_{3,3}^{-,z} \\\\ \\end{bmatrix}_{k=3} \\end{bmatrix} \\end{aligned}", element, {displayMode:true,throwOnError:false});$

Example Input File Syntax

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

Input Parameters

componentthe component of the displacement vector this kernel is working on: component == 0, ==> X component == 1, ==> Y component == 2, ==> Z
C++ Type:unsigned int
Controllable:No
Description:the component of the displacement vector this kernel is working on: component == 0, ==> X component == 1, ==> Y component == 2, ==> Z
displacementsthe string containing displacement variables
C++ Type:std::vector<VariableName>
Controllable:No
Description:the string containing displacement variables
neighbor_varThe variable on the other side of the interface.
C++ Type:std::vector<VariableName>
Controllable:No
Description:The variable on the other side of the interface.
variableThe name of the variable that this residual object operates on
C++ Type:NonlinearVariableName
Controllable:No
Description:The name of the variable that this residual object operates on

Required Parameters

base_nameMaterial property base name
C++ Type:std::string
Controllable:No
Description:Material property base name
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
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.
traction_global_namePK1traction
Default:PK1traction
C++ Type:std::string
Controllable:No

Optional Parameters

absolute_value_vector_tagsThe tags for the vectors this residual object should fill with the absolute value of the residual contribution
C++ Type:std::vector<TagName>
Controllable:No
Description:The tags for the vectors this residual object should fill with the absolute value of the residual contribution
extra_matrix_tagsThe extra tags for the matrices this Kernel should fill
C++ Type:std::vector<TagName>
Controllable:No
Description:The extra tags for the matrices this Kernel should fill
extra_vector_tagsThe extra tags for the vectors this Kernel should fill
C++ Type:std::vector<TagName>
Controllable:No
Description:The extra tags for the vectors this Kernel should fill
matrix_tagssystemThe tag for the matrices this Kernel should fill
Default:system
C++ Type:MultiMooseEnum
Options:nontime, system
Controllable:No
Description:The tag for the matrices this Kernel should fill
vector_tagsnontimeThe tag for the vectors this Kernel should fill
Default:nontime
C++ Type:MultiMooseEnum
Options:nontime, time
Controllable:No
Description:The tag for the vectors this Kernel should fill

Tagging 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.
diag_save_inThe name of auxiliary variables to save this Kernel's diagonal Jacobian contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
C++ Type:std::vector<AuxVariableName>
Controllable:No
Description:The name of auxiliary variables to save this Kernel's diagonal Jacobian contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
diag_save_in_var_sideThis parameter must exist if diag_save_in variables are specified and must have the same length as diag_save_in. This vector specifies whether the corresponding aux_var should save-in jacobian contributions from the primary ('p') or secondary side ('s').
C++ Type:MultiMooseEnum
Options:m, s
Controllable:No
Description:This parameter must exist if diag_save_in variables are specified and must have the same length as diag_save_in. This vector specifies whether the corresponding aux_var should save-in jacobian contributions from the primary ('p') or secondary side ('s').
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
save_inThe name of auxiliary variables to save this Kernel's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
C++ Type:std::vector<AuxVariableName>
Controllable:No
Description:The name of auxiliary variables to save this Kernel's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
save_in_var_sideThis parameter must exist if save_in variables are specified and must have the same length as save_in. This vector specifies whether the corresponding aux_var should save-in residual contributions from the primary ('p') or secondary side ('s').
C++ Type:MultiMooseEnum
Options:m, s
Controllable:No
Description:This parameter must exist if save_in variables are specified and must have the same length as save_in. This vector specifies whether the corresponding aux_var should save-in residual contributions from the primary ('p') or secondary side ('s').
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