Question 1
An Anisotropic material does not have any planes of symmetry. Therefore it is the type of material that has the largest stiffness and compliance matrixes composed of 21 independent constants. Therefore it is difficult to predict the physical behaviors of this material analytically because many inputs are needed.
A Monoclinic material is one that has one plane of symmetry. At this plane of symmetry there are only
13 independent constants which still makes it difficult to analytically predict material behavior.
Most composite materials are Orthotropic. An Orthotropic material has 3 orthogonal planes of symmetry which reduces the number of independent constants in the stiffness matrix to 9, meaning the material properties are symmetric about these planes. Therefore only 9 material properties are needed to analytically determine the physical behavior of the material. In addition, using translation equations one can analyze the solid even for stresses oriented at a certain angle away from the principal directions. Note that for orthotropic material there is no interaction between the principal normal stresses and principal shear strains. Also there is no interaction between principal shear stresses and principal shear strains in different planes.
Transversely Isotropic materials are those for which there is a plane of symmetry for every point within the material. In this plane the material properties are equal in all directions and thus the number of independent constants in the stress-strain relations reduces to 5.
The simplest form is an Isotropic material which has material properties equal in all directions at any point. Therefore there is infinite planes of symmetry at any point. Examples of these material as metals.
These materials are very easy to analyze because only 3 material properties are needed to obtain a prediction of physical behavior.
Question 5
Maximum Stress Theory
The criterion for the maximum stress theory is for the stresses in the principal material directions to be less than the respective strengths of the material in that direction. If the principal stress is larger than the material strength in that direction then the material is assumed to have failed. The criterion is for both tensile and compressive strengths in the normal direction, and for the shear strength. This criterion does not take into account interaction between modes of failure.
Maximum Strain Theory
For the maximum strain theory the material is assumed to fail if one of the strains experienced in the principal directions is larger than then the maximum strain allowable in that respective principal strain direction for that material. The criterion is for both tensile and compressive strengths in the normal direction, and for the shear strength. This criterion does not take into account interaction between modes of failure.

Tsai-Hill Theory
The Tsai-Hill theory is an anisotropic yield criterion which is used as an anisotropic strength criterion. It is an extension of von Mises’ isotropic yield criterion which is related to the amount of energy used to distort the body. However distortion and dilatation cannot be separated for composite materials. This assumption is made on the basis that both yield and strength are limits of linear elastic behavior which is failure for composite materials since they do not yield like metals. This yield criterion for anisotropic materials is:
(G+H)σ1^2 + (F+H)σ2^2 + (F+G)σ3^2 - 2Hσ1σ2 - 2Gσ1σ3 - 2Fσ2σ3 + 2Lτ23^2 + 2Mτ13^2 + 2Nτ12^2 = 1
Where F, G, H, L, M, and N are yield strengths that are related to the failure strengths by Tsai.