# Strain energy density function of a Transversely Isotropic Material

Hi all,

I was going through Constitutive Modeling in Continuum Mechanics. I came across the Transversely Isotropic Materials (TIM). I have a couple of doubts, which are listed in the attached pdf file. I request the Continuum Mechanicians to clarify.

Thank you in advance,

Best regards,

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### TIM

1. a0
is vector which describes the orientation of the fiber in the reference
configuration.

2. For isotropic material, the strain
energy function consists of I1,
I2, I3, but not necessary all of them. For example, for
incompressible neo-Hookean material, only I1 needed. The same for TIM, the
strain energy function is made up of I1,
..., I5, even more depending
on how many fiber families you have, but also not necessary all of them. The
simplest strain energy function for TIM may only contain two invariants: ψ=ψ(I1, I4). Specifically, I4
indicates the stretch of the fiber, I4=Fa0∙Fa0=λF2.

3. In Cauchy
stress, the term with a0dyada0
comes from the derivative of I4
with respect to F.

Here is a paper, which might be a good starting point of learn
constitutive modeling of anisotropic hyperelastic materials:

Mechanical response of fiber-reinforced incompressible non-linearly
elastic solids

Bests,

Kevin

### Math is understood...

Hi Kevin,

Thank you. I undertood the math behind I4 and I5 etc. I am looking for physical explanations. Please take a look at the questions I wrote in the attached file.

Best wishes,

### Yes, I4 is the square of

Yes, I4 is the square of the fiber stretch. I5, also related to the fiber stretch but introduces an additional effect that relates to the behavior of the reinforcement under shear deformations.

### Specific questions

Thank you. Following are my specific questions.

1.ao is a vector, but, this is expressed as a second order tensor. Why? Is it to make the function, y, uniform in terms of its arguments?  When do we convert a first order tensor into a second order (or above) tensors?

2.y  is expressed in terms of five invariants. Among five, three are regular invariants as we see in case of isotropic materials. Rest of the two are (I4 and I5) in terms of C and ao.  What is the basis for these two (why only two?) invariants and how do we get the expressions for I4 and I5 in variants.

Thank you.

### Number of invariants in strain energy density function

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TH

MicrosoftInternetExplorer4

For complete definition (to define stress-strain relation)
of a linear elastic isotropic material two independent material constants are
required. The strain energy function of a linear elastic isotropic material can
be expressed in terms of two invariants of strain tensor.

In case of Transversely Isotropic Material (TIM), five independent
material constants are required for complete definition. The strain energy
density function is also expressed in terms of five strain invariants.

From the above shall I infer that the number of invariants
in the strain energy density function is directly proportional number of independent
constants required for defining the stress-strain relation?

Thank you,

Best regards,

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