Download e-book for kindle: Finite Rotation Shells: Basic Equations and Finite Elements by K. Wisniewski

By K. Wisniewski

ISBN-10: 9048187605

ISBN-13: 9789048187607

This book treats the formula and finite components for shells, and its topic is outlined by means of the next topics:

  1. Computational mechanics of non-linear shells,
  2. Shell equations: Reissner kinematics, finite rotations, finite strains,
  3. Shell finite parts: 4-node, better or combined or mixed/enhanced,
  4. Drilling rotation: drilling Rotation Constraint or Allman form functions,
  5. Normal pressure: recovered or parameterized,
  6. Constitutive equations: incremental, aircraft tension or 3D.

 A wide selection of purposes of shell components signifies that they need to be flexible, i.e. account for finite rotations and lines, admit the incorporation of varied constitutive legislation and let handy linking with different components. it is a severe problem, which calls for a number of elements of the element's formula to be very complicated.

The simple details on linear shell components might be present in a few textbooks on FEs, yet this publication includes numerous complicated themes with regards to non-linear shells, equivalent to e.g. the parametrization of finite rotations, the tools of together with the drilling rotation, quite a few tools of treating the conventional pressure, and the improved in addition to combined and mixed/enhanced finite components.

Some of those issues were a subject matter of my learn for years, and all of the defined equipment were carried out in my very own components, and confirmed. as a result, i think that my figuring out of this complex topic is true and mature adequate to be canned, and offered to others.

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Additional resources for Finite Rotation Shells: Basic Equations and Finite Elements for Reissner Kinematics

Example text

47) Note that the AMB, eq. 5), was exploited in the derivation of eq. 28), and earlier of eq. 24), but its weak form of eq. 12) is not present in the integrand of eq. 46). Three-field potential. On the basis of eq. 47), by using the CL of eq. 32), we can define the three-field potential . 48) where Fext is defined in eq. 42). This also proves that the use of δskew(QT P) in eq. 45) was indeed correct. Remark 1. The right stretch U can also be eliminated from the fourfield formulation in another way.

Besides, we must take det F > 0 to exclude annihilation of line elements and negative volumes, see [159], pp. 85 and 87. Hence, (det F)/(det U) > 0 and, therefore, det R = +1. 2 Rotation Constraint equation Instead of calculating R as FU−1 , we can find a tensor Q ∈ SO(3), by solving the RC equation . C = skew(QT F) = 0. 8) This it permitted because the equations QT F = U and skew(QT F) = 0, are equivalent, which is shown below. 1. skew(QT F) = 0 ⇒ QT F = U. e. QT F is symmetric. Using this symmetry, we have (QT F)2 = (QT F)(QT F) = (QT F)T (QT F) = FT QQT F = U2 .

If we use in eq. 47) the CL sym(QT P) = ∂U˜ W of eq. 31), then the nominal stress P remains only in the term . skew(QT P). Hence, we can define a skew-symmetric tensor Ta = skew(QT P) with only three components and abandon using P with nine components. That is the basic motivation behind using the Biot stress in the next section. 4 3-F and 2-F formulations for Biot stress In this section, we describe a three-field formulation in terms of {χ, Q, Ta }, developed in [42, 191]. This formulation can be obtained from the threefield formulation for the nominal stress tensor, which is described in the previous section, just by introducing the definition of the Biot stress.

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Finite Rotation Shells: Basic Equations and Finite Elements for Reissner Kinematics by K. Wisniewski

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