Assuming LEHI behavior and a thin incompressible material (v = 0.5) in a state of biaxial plane stress (where the material is loaded only in the x and y directions) with [tex]O_{xx}[/tex] and [tex]O_{yy}[/tex] both positive, the strain in the through-plane direction ([tex]E_{zz}[/tex]) must be:

A. Positive, meaning the material gets thicker in the z direction.
B. Negative, meaning the material gets thinner in the z direction.

The given question is asking about the strain in the through-plane direction (Ezz) in a state of biaxial plane stress, assuming LEHI behavior and a thin incompressible material (v=0.5). The stress components Oxx and Oyy are both positive.

In a state of biaxial plane stress, the material is loaded in only the x and y directions. When Oxx and Oyy are both positive, this means the material is experiencing tensile stress in both the x and y directions.

In an incompressible material, the volume does not change under stress. This means that the material cannot expand or contract in any direction.

In the z direction, the material experiences no stress due to the loading in the x and y directions. Therefore, the strain in the through-plane direction (Ezz) must be zero. This implies that the material does not change its thickness in the z direction.

To summarize, in the given scenario, the strain in the through-plane direction (Ezz) is zero, which means the material does not change its thickness in the z direction.

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