Abstract
This chapter covers the mechanical modeling of a single layer with unidirectionally aligned reinforcing fibers embedded in a homogeneous matrix, a so-called lamina. It is shown that a lamina can be treated as a combination of a plane elasticity element and a classical plate element. For both classical structural elements and their combination, the continuum mechanical modeling based on the three basic equations, i.e., the kinematics relationship, the constitutive law, and the equilibrium equation, is presented. Combining these three questions results in the describing partial differential equations. The chapter closes with different orthotropic failure criteria, i.e., the maximum stress, the maximum strain, the Tsai–Hill, and the Tsai–Wu criterion.
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Notes
- 1.
Note that according to the assumptions of the classical thin plate theory the lengths \(\overline{\textit{0A}}\) and \(\overline{\textit{0}'{} \textit{A}'}\) remain unchanged.
- 2.
Three-dimensional isotropy requires as well only two independent elastic constants.
- 3.
Three-dimensional orthotropy requires nine independent elastic constants.
- 4.
In the case of a shear force \(\sigma _{ij}\), the first index i indicates that the stress acts on a plane normal to the i-axis and the second index j denotes the direction in which the stress acts.
- 5.
If gravity is acting, the body force f results as the product of density times standard gravity: \(f=\tfrac{F}{V}=\tfrac{mg}{V}=\tfrac{m}{V}g=\varrho g\). The units can be checked by consideration of \(1\,\text {N}=1\tfrac{\text {m}\text {kg}}{\text {s}^2}\).
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Öchsner, A. (2023). Macromechanics of a Lamina. In: Composite Mechanics. Advanced Structured Materials, vol 184. Springer, Cham. https://doi.org/10.1007/978-3-031-32390-4_3
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DOI: https://doi.org/10.1007/978-3-031-32390-4_3
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