### Elasticity of Transversely Isotropic Materials

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This approach enables the estimation of a complete set of parameters for an incompressible, transversely isotropic, linear elastic material.

Mechanical anisotropy is a feature of many soft tissues [ 1 — 5 ]. The dependence of the mechanical response on the direction of loading arises from microstructural features such as collagen fiber bundles [ 6 — 9 ]. The mechanical characterization of anisotropic materials is a fundamental challenge because of the requirement that the responses to multiple loadings must be combined to develop even a linear elastic material characterization see, e. Our specific interest is brain tissue, which presents additional experimental challenges because it is delicate and highly compliant moduli lie in the 0.

Brain tissue contains both white matter myelinated axonal fibers , which is structurally anisotropic, and gray matter, which has no apparent structural anisotropy [ 6 , 13 ]. Brain tissue mechanics are central to mathematical models of brain biomechanics and might be an important determinant of injury susceptibility [ 14 ]. Such models would ideally include the complete characterization of the anisotropic mechanics and structure-function relationships in brain tissue.

However, techniques involving stretching, such as biaxial stretch plus indentation [ 15 ], are not feasible for brain tissue, because of the difficulty of gripping specimens. Cyanoacrylate adhesives have been used to hold samples in tension [ 11 ], however, the use of adhesives preclude testing a single sample in more than one direction. The requirement for multiple loading scenarios to characterize anisotropic materials restricts test procedures to those that do not permanently alter the mechanics of a specimen.

Furthermore, fibrous anisotropic materials may exhibit different properties when loaded in tension and compression, because fibers stretch in tension, but may buckle in compression. A long-term objective of this work is the identification of an appropriate form and all of the parameters for an anisotropic constitutive model of brain tissue. Proposed models include the hyperelastic white matter constitutive model of Meaney [ 16 ] or the nonlinear transversely isotropic viscoelastic model of Ning et al. As an example, the model of Ning et al. As a first step towards this long-term objective, we develop and demonstrate a procedure for finding the complete set of parameters of a transversely isotropic linear elastic model for a soft gel undergoing small strain.

The proposed procedure involves the combination of dynamic shear and asymmetric indentation tests, which are promising methods for probing mechanical anisotropy in brain tissue because they require only simple fixtures to hold the sample, and they are nondestructive at small strains.

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We show that the combination of these two tests can be used to determine all of the parameters of an incompressible transversely isotropic linear elastic material. Shear tests, performed in the plane of isotropy and in a plane perpendicular to the plane of isotropy, uniquely identify two distinct shear moduli. Indentation with a rectangular tip, as proposed by Bischoff [ 19 ], applies different stresses to the material in directions parallel and perpendicular to the long axis of the tip.

Thus, a different force-displacement curve will be obtained depending on whether the long axis is aligned with the predominant fiber direction. Several groups have measured the mechanical properties of brain tissue either by symmetric indentation [ 20 ] or by dynamic shear testing DST alone [ 1 , 21 ]. Dynamic shear testing can characterize anisotropy in a shear modulus, if the plane in which the shear is applied is either parallel or normal to the dominant fiber direction.

It is very difficult, however, to use DST to illuminate the contribution of fiber stretch to the mechanical response. Studies using symmetric indentation or unconfined compression [ 20 , 22 , 23 ] alone do not detect anisotropy. Cox et al. However, the principal stretches are difficult to determine reliably, and require significant additional instrumentation.

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In contrast, the proposed asymmetric indentation method requires only the force-displacement curves, interpreted in the context of corresponding numerical simulations. In this paper, we demonstrate the combined shear-indentation approach by applying it to characterize the linear elastic properties of an anisotropic fibrin gel. Fibrin gel can be made anisotropic by allowing the gel to polymerize in a high magnetic field, which leads to a network with a preferred fiber axis aligned with the magnetic field [ 25 , 26 ].

The mechanical properties of this network depend on fiber bending and rotation; hence, they are related to the orientation of fibrils [ 27 ]. The following sections describe the theory and methods behind the use of combined shear-indentation procedures to measure the mechanical parameters of soft transversely isotropic materials.

Transversely isotropic materials are symmetric about an axis perpendicular to the plane of isotropy. We assume a is known, and the reference coordinate system is aligned with the e 1 axis parallel to a. The stress-strain relationship for the compressible material is derived from the partial derivatives of W c in Eq.

Since a unique stiffness matrix cannot be defined for an incompressible material, we derive a compliance matrix S I by inverting the stiffness matrix for the compressible material Eq. For an incompressible, linearly elastic, transversely isotropic material with the strain energy function of Eq.

The simulation of the response of a transversely isotropic material to asymmetric indentation provides the necessary relationships between experimental data stiffness measurements and intrinsic material parameters moduli. The idealizations of the indenter shape, isotropy, plane strain, and an infinite sample width precludes the use of this solution to obtain numerical estimates of transversely isotropic elastic moduli directly from the experimental indentation data.

A 3-D finite element FE model of an asymmetric rigid tip indenting a transversely isotropic elastic layer of material see Fig. The FE model geometry consisted of a layer of elastic material 3. The corners of the rectangular indenter were rounded; hence, the initial contact width was 1. The model included geometric nonlinearity.

### 1. Introduction

To reduce the number of elements required, only one quarter of the gel was modeled and symmetry boundary conditions were applied to the straight edges of the quarter gel model Fig. The mesh was graded in the e x and e z directions to increase the number of elements in the indented region. The smallest elements under the edge of the indenter were approximately 0. The quarter gel model contained , eight node brick elements C3D8 and the rigid rectangular indenter was discretized into rigid elements R3D3.

Contact between the indenter and the gel was approximated as frictionless sliding. The displacement u z of all nodes on was set to zero to approximate frictionless contact between the gel and rigid substrate. All other surfaces had traction-free boundary conditions. The plane of isotropy is the 2—3 plane, and the vector normal to the plane of isotropy a is aligned with the e 1 unit vector. Only one quarter of the circular gel is modeled. The model predicted force-displacement curves for indentation depths of 0 to 0. From the parametric FE simulations, f and g were chosen as functions of the material parameters.

The choice for the specific form of functions f and g is discussed in Sec. The values of the unknown coefficients a o , b o , and c o were obtained by linear least squares fits. Fibrin gel samples were fabricated in a two-step process: a the preparation of separate fibrinogen and thrombin solutions, and b mixing the two solutions and pouring the mixture into circular dishes. Aligned gels were created by immediately placing the filled dishes in a high field strength magnet during gelation.

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## How to determine elastic constants of transversely isotropic material by tension experiment?

Thrombin Sigma-Aldrich, St. Louis, MO, Product No. T was diluted to 0. The direction of the magnetic field during polymerization, denoted by the unit vector e 1 , was recorded and marked on each sample.

## Elasticity of Transversely Isotropic Materials

Circular samples were cut using an The approximate thickness and weight w of each sample were measured before testing. The samples were placed on a custom-built dynamic shear testing system see Fig. The sample is deformed in simple shear by the harmonic displacement of the base, while the force on the stationary upper surface is measured. The vertical and horizontal lines indicate the dominant fiber directions of the aligned gel. When the imposed displacement is parallel to the dominant fiber axis, shear is imposed in a plane normal to the plane of isotropy.

We have a dedicated site for Germany. This book aims to provide a comprehensive introduction to the theory and applications of the mechanics of transversely isotropic elastic materials. There are many reasons why it should be written. First, the theory of transversely isotropic elastic materials is an important branch of applied mathematics and engineering science; but because of the difficulties caused by anisotropy, the mathematical treatments and descriptions of individual problems have been scattered throughout the technical literature.