Example of what I need to make work

tests/stokes/homogenized.i

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+[Mesh]
+  second_order = true
+  [mesh]
+    type = CartesianMeshGenerator
+    dim = 3
+    dx = '1 1'
+    dy = '1 1'
+    dz = '1 1'
+    ix = '8 8'
+    iy = '8 8'
+    iz = '8 8'
+    subdomain_id = '1 2 3 4 5 6 7 8'
+  []
+[]
+
+[Variables]
+  [p]
+    order = FIRST
+    family = LAGRANGE
+  []
+  [u]
+    order = SECOND
+    family = LAGRANGE_VEC
+  []
+[]
+
+[AuxVariables]
+    [deviatoric_stress_xx]
+        family = MONOMIAL
+        order = CONSTANT
+    []
+    [deviatoric_stress_yy]
+        family = MONOMIAL
+        order = CONSTANT
+    []
+    [deviatoric_stress_zz]
+        family = MONOMIAL
+        order = CONSTANT
+    []
+    [deviatoric_stress_xy]
+        family = MONOMIAL
+        order = CONSTANT
+    []
+    [deviatoric_stress_yz]
+        family = MONOMIAL
+        order = CONSTANT
+    []
+    [deviatoric_stress_xz]
+        family = MONOMIAL
+        order = CONSTANT
+    []
+    [stress_xx]
+        family = MONOMIAL
+        order = CONSTANT
+    []
+    [stress_yy]
+        family = MONOMIAL
+        order = CONSTANT
+    []
+    [stress_zz]
+        family = MONOMIAL
+        order = CONSTANT
+    []
+    [stress_xy]
+        family = MONOMIAL
+        order = CONSTANT
+    []
+    [stress_yz]
+        family = MONOMIAL
+        order = CONSTANT
+    []
+    [stress_xz]
+        family = MONOMIAL
+        order = CONSTANT
+    []
+[]
+
+[AuxKernels]
+    [deviatoric_stress_xx]
+        type = ADMaterialRankTwoTensorAux
+        variable = deviatoric_stress_xx
+        property = 'deviatoric_stress'
+        i = 0
+        j = 0
+    []
+    [deviatoric_stress_yy]
+        type = ADMaterialRankTwoTensorAux
+        variable = deviatoric_stress_yy
+        property = 'deviatoric_stress'
+        i = 1
+        j = 1
+    []
+    [deviatoric_stress_zz]
+        type = ADMaterialRankTwoTensorAux
+        variable = deviatoric_stress_zz
+        property = 'deviatoric_stress'
+        i = 2
+        j = 2
+    []
+    [deviatoric_stress_xy]
+        type = ADMaterialRankTwoTensorAux
+        variable = deviatoric_stress_xy
+        property = 'deviatoric_stress'
+        i = 0
+        j = 1
+    []
+    [deviatoric_stress_yz]
+        type = ADMaterialRankTwoTensorAux
+        variable = deviatoric_stress_yz
+        property = 'deviatoric_stress'
+        i = 1
+        j = 2
+    []
+    [deviatoric_stress_xz]
+        type = ADMaterialRankTwoTensorAux
+        variable = deviatoric_stress_xz
+        property = 'deviatoric_stress'
+        i = 0
+        j = 2
+    []
+    [stress_xx]
+      type = ParsedAux
+      variable = stress_xx
+      coupled_variables = 'p deviatoric_stress_xx'
+      expression = 'deviatoric_stress_xx - p'
+    []
+    [stress_yy]
+      type = ParsedAux
+      variable = stress_yy
+      coupled_variables = 'p deviatoric_stress_yy'
+      expression = 'deviatoric_stress_yy - p'
+    []
+    [stress_zz]
+      type = ParsedAux
+      variable = stress_zz
+      coupled_variables = 'p deviatoric_stress_zz'
+      expression = 'deviatoric_stress_zz - p'
+    []
+    [stress_xy]
+      type = ParsedAux
+      variable = stress_xy
+      coupled_variables = 'deviatoric_stress_xy'
+      expression = 'deviatoric_stress_xy'
+    []
+    [stress_xz]
+      type = ParsedAux
+      variable = stress_xz
+      coupled_variables = 'deviatoric_stress_xz'
+      expression = 'deviatoric_stress_xz'
+    []
+    [stress_yz]
+      type = ParsedAux
+      variable = stress_yz
+      coupled_variables = 'deviatoric_stress_yz'
+      expression = 'deviatoric_stress_yz'
+    []
+[]
+
+[ICs]
+  [u]
+    type = VectorFunctionIC
+    variable = u
+    function_x = '1.0 * x'
+    function_y = '1.0 * y'
+    function_z = '1.0 * z'
+  []
+[]
+
+[Kernels]
+  [equil]
+    type = ADStokesStressDivergence
+    variable = u
+  []
+  [pressure]
+    type = ADStokesPressure
+    variable = u
+    pressure = p
+  []
+  [incompressible]
+    type = ADStokesIncompressibility
+    variable = p
+    velocity = u
+  []
+[]
+
+[Materials]
+  [strain]
+    type = StokesStrainRate
+    velocity = u
+  []
+  [stress1]
+    type = StokesLinearViscous
+    mu = 1
+    block = '1 4 6 7'
+  []
+  [stress2]
+    type = StokesLinearViscous
+    mu = 2
+    block = '2 3 5 8'
+  []
+[]
+
+[BCs]
+  [Periodic]
+    [p]
+      variable = p
+      auto_direction = 'x y z'
+    []
+    [u]
+      variable = u
+      auto_direction = 'x y z'
+    []
+  []
+[]
+
+[Postprocessors]
+  [avg_stress_xx]
+     type = ElementAverageValue
+     variable = stress_xx
+  []
+  [avg_stress_yy]
+     type = ElementAverageValue
+     variable = stress_yy
+  []
+  [avg_stress_zz]
+     type = ElementAverageValue
+     variable = stress_zz
+  []
+  [avg_stress_xy]
+     type = ElementAverageValue
+     variable = stress_xy
+  []
+  [avg_stress_yz]
+     type = ElementAverageValue
+     variable = stress_yz
+  []
+  [avg_stress_xz]
+     type = ElementAverageValue
+     variable = stress_xz
+  []
+[]
+
+[Executioner]
+  type = Steady
+
+  solve_type = 'newton'
+  petsc_options_iname = '-pc_type -pc_factor_shift_type -pc_mat_solver_package'
+  petsc_options_value = 'lu NONZERO superlu_dist'
+  line_search = none
+
+  l_max_its = 10
+  l_tol = 1e-8
+  nl_max_its = 15
+  nl_rel_tol = 1e-10
+  nl_abs_tol = 1e-12
+[]
+
+[Outputs]
+  exodus = true
+[]