Distributed Fiber Models
Composite materials are made out of an isotropic matrix reinforced with fibers. In these reinforced materials, the fibers act as the main load carrier, while the surrounding matrix supports the fibers and transfer loads to them. These materials are either manufactured, or have a natural origin such as biological tissue, leafs or wood.
In COMSOL Multiphysics it is possible to add fibers to Linear Elastic or Nonlinear Elastic materials, as well as to any of the built-in Hyperelastic material models. The fibers are arranged in a specified orientation inside the matrix material. By subsequently adding Fiber nodes as needed it is possible to include fibers in many directions, thus recreating complex anisotropic structures.
Fibers for Linear and Nonlinear Elastic Materials
For Linear Elastic or Nonlinear Elastic materials, each Fiber node adds a contribution to the stress tensor of the type
Here, Efib E is the difference between the stiffness of the isotropic matrix E and the fibers’ stiffness Efib, vfib is the fiber volume fraction, a is the direction vector for the fiber orientation in the undeformed geometry, and is the elastic strain in the fiber direction a.
For a Linear Elastic Material, Hooke’s law (Equation 3-15) is augmented as
For Nonlinear Elastic Materials, the same contribution to the stress tensor is added, although the stress-strain relation in the isotropic matrix could be nonlinear.
Fibers for Hyperelastic Materials
For Hyperelastic Material Models, the fibers are represented by additional contributions to the strain energy density (Ref. 22 and Ref. 23).
The Holzapfel-Gasser-Ogden constitutive model captures the anisotropic mechanical response observed in arteries. The model combines a nearly incompressible material for the isotropic matrix, with additional terms for two fiber families
Here, Wiso is the strain energy density of the isotropic hyperelastic material. In Ref. 22 Wiso is represented by a nearly incompressible Neo-Hookean material; in COMSOL Multiphysics it is possible to select any of the built-in hyperelastic material models.
The second and third terms, Wfib,1 and Wfib,2 are anisotropic additions that describe the mechanical contribution of collagen fiber networks. These terms contribute with a strain energy density of the type
where
where the parameter k1 represents the fiber stiffness (SI unit: Pa), k2 is a dimensionless tuning parameter, and k3 is the fiber dispersion (dimensionless). A value of k3 = 0 recovers the formulation in Ref. 22, where all the fibers are perfectly oriented in the a direction. Setting k3 = 1/3 means that the fibers are completely dispersed, thus recovering a similar formulation as given in the Delfino material model.
The formulation in Ref. 22 and Ref. 23 is based on the Isochoric Elastic Deformation, where the invariants are defined as
The invariant represents the squared value of the isochoric elastic stretch in the fiber direction a. A common modeling assumption for biological tissues is that fibers cannot sustain compression, so the fiber stiffness is added only for tensile stretches such as .
In general, the anisotropic strain energy density for a hyperelastic material containing N fiber families reads
The second Piola-Kirchoff stress tensor is then computed from
where the stress contribution from each fiber family is given by
The anisotropic strain energy density for the fiber family Wfib can also be specified as a user-defined expression of the strain invariants , , , , , and , where
,
, , and
and b and c are orthogonal directions to the main fiber direction a.
The internal variables , , , , , and are named item.IaCIe, item.IbCIe, item.IcCIe, item.IabCIe, item.IacCIe, and item.IbcCIe. Here, item is the name of the Fiber node, for instance solid.hmm1.fib1.
Thermal Expansion in Fibers
Not only the stiffness in the fibers might differ by orders of magnitude to the matrix stiffness, but also the thermal properties, so it is possible to model thermal expansion in fibers independently from the thermal expansion in the surrounding material. The two basic assumptions are that the volume occupied by the fibers is small as compared to the base material, and that thermal expansion (or contraction) occurs in the fiber direction only (there is zero transverse thermal expansion in the fibers).
For Linear Elastic and Nonlinear Elastic materials, the fibers add a contribution to the stress tensor of the type
where Efib E is the difference between the stiffness of the isotropic matrix E and the fibers’ stiffness Efib, vfib is the fiber volume fraction, a is the direction vector for the fiber orientation in the undeformed geometry, and is the elastic strain in the fiber direction a. When considering the thermal expansion, the stress reads
where εth,fib is the thermal strain in the fibers.
For a Linear Elastic Material, Hooke’s law (Equation 3-15) is then augmented as
For Nonlinear Elastic Materials, the same contribution to the stress tensor is added, although the stress-strain relation in the isotropic matrix could be nonlinear.
The thermal strain in the fibers, εth,fib, can be specified by different means. When the secant coefficient of thermal expansion is used, it reads
Here, the secant coefficient of thermal expansion αs can be temperature-dependent. The reference temperature Tref is the temperature at which there are no thermal strains in the fibers, and Tfib is the current fiber temperature.
When the tangent coefficient of thermal expansion is used, αt, the thermal strain is given by
It is also possible to explicitly enter the thermal strain, dL, as a function of the fiber’s temperature
The coefficient of the thermal expansion or the thermal strain are considered in the fiber direction a, since thermal expansion is assumed to occur in the fiber direction only.
In all cases, the contribution to the thermal strain tensor is defined from the fiber volume fraction vfib, the thermal strain in the fibers εth,fib, and the fiber direction a
See also the description of the Fiber node in the Solid Mechanics interface documentation.
An example of an isotropic hyperelastic material reinforced with fibers is shown in Arterial Wall Mechanics: Application Library path Nonlinear_Structural_Materials_Module/Hyperelasticity/arterial_wall_mechanics