Use structural implicit gradient regularization to regularize structural models that are inherently subject to numerical instability and pathological mesh sensitivity (such as the strain-softening microplane material model). Applications of structural implicit gradient regularization include modeling concrete and similar materials under cyclic loading.
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The implicit gradient regularization scheme is implemented by defining a nonlocal field using a modified Helmholtz equation, which adds extra degrees of freedom on top of the structural degrees of freedom.
The governing equations are given by the linear momentum balance equation, and additionally a modified Helmholtz equation:
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where 
is the Cauchy stress tensor, 
is the body force vector, 
is the divergence, 
is the gradient, and 
is the Laplace operator. 
is a gradient parameter which controls the range of nonlocal
interaction. 
is the local variable to be enhanced, and 
is its nonlocal counterpart.
The homogeneous Neumann boundary condition applies as follows:
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where 
is the normal to the outer boundary of the nonlocal field.
For displacement 
and nonlocal variable 
, linearizing the governing equations gives:
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where 
, 
are the structural and nonlocal field strain-displacement operator
matrices, and 
, 
are the structural and nonlocal field shape functions.