Patrick Schaedler
SIC - HW 2 – 3-3-13
1.
Refer to PDF folder for MATLAB results for 1.a – 1.e.
a. General Anisotropic Solid – in this case, there are the most non-zero components, 21. This has to do
with the nature of the material in that the properties are different in all directions. The lack of material
symmetry directly relates to this as the stiffness and compliance matrices loose. In this case, only
Independent variables of the same subscript are the same (i.e. Ci,j,= Cj,i). Additionally, the shear stress
and normal strains, as well as the shear stress and shear strains are coupled with a direct effect on one
another. Physically, these material have no planes of symmetry.
b. Monoclinic Solid – non-zero components are further reduced n the monoclinic solid to only 13
components. This is a result of there being one plane of symmetry, in which it is Z = 0. This results in 8
of the shear stress components dropping out of the matrix equations. Physically, there is one plane of
material property symmetry.
c. Orthotropic Solid - In an orthotropic solid, there are two orthogonal planes of material property
symmetry, which results in synetry relative to a third mutually orthogonal plane. This results in the
number of non-zero components being further reduced to 9 components. In this instance, there is no
interaction between the shear stress and normal strains or the shear stress and shear strains in different
planes.
d. Transversely Isotropic Solid – For the transversely isotropic material, every point of the material has one
plane in which the material properties are equal in all directions. This further reduces the number of
non-zero components to 5 components. In this instance, two planes develop a special plane of isotropy,
and their values become interchangeable. For example if the 2-3 planes are the planes of special
isotropy, then the 2-3 subscript terms are interchangeable in the stiffness and compliance matrices.
e. Isotropic Solid – For an isotropic material, there are an infinite number of planes in which the material
property symmetry exists. This further reduces the number of non-zero constants to in the stiffness and
compliance matrix to 2 components. All values in the compliance and stiffness matrix become relative
to the 1-2 planes.
f.
Orthotropic material in plane strain – In an orthotropic material in plane strain, the number of
components is reduced from the 9 that typically are associated with an orthotropic solid. Specifically, if
the component is in plane strain the non-zero components for an entire axis drop out of the matrix, as
all of the strain is in one plane. For example, in a 1-2 system, the 3 plane components will have a value
of zero. Therefore, only 4 non-zero components remain as 5 drop out of the matrix.
2. See maple code and PDF results and excel sheet.
3. See maple code and PDF results.
4. See maple code and PDF results
5.
a. Max Stress Theory – in the maximum stress theory, the stresses in the principal directions must be less
than the maximum stresses for the material, otherwise failure will occur. This applies for compressive
stresses, tensile stresses, and shear stress.
b. Max Strain Theory – The maximum strain theory is similar to the maximum stress theory, with the
exception that the strain in the limiting factor. In other words, the strains in the principal directions
must be less than the maximum strains for the material, otherwise failure will occur. This applies for
compressive strains, tensile strains, and shear stress
c. Tsai-Hill – The Tsai-Hill strength theory is founded on a proposed yield criteria for anisotropic materials.
That theory relates certain failure strengths with the typical failure strengths (X, Y, S), and eventually
leads to one specific failure criterion. This has been seen to be more accurate than the maximum stress
or maximum strain theories, in which 3 different failure criteria are applicable for each theory.
d. Tsai-Wu – The Tsai-Wu strength theory is similar to the Tsai-Hill theory, with the exception that it
increases the number of terms by representing the strength values in tensor form. This increases the
correlation between the experimental and theoretical values and therefore improves on the curve
fitting of the equations.