VARIOUS ISSUES IN THE LARGE STRAIN THEORY OF TRUSSES

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VARIOUS ISSUES IN THE LARGE STRAIN THEORY OF
TRUSSES
C. A. Kaklamanis and K.V. Spiliopoulos Department
of Civil Engineering National Technical
University of Athens
STAMM 2006 Symposium on Trends in Applications of
Mathematics to Mechanics Vienna, Austria, 1014
July 2006
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OUTLINE

• INTRODUCTION
• Manifold View of the Truss and Embeddings
• KINEMATICS
• Notation and Point Transformation
• The Deformation Gradient and its (Diagonal) Polar
  Decomposition
• Various Strain Measures
• The Velocity Gradient and the Deformation and
  Spin Tensors
• Various Strain Rates
• CONSTITUTIVE MODELING
• Strain Energy Rate and Strain Energy Density Rate
• Conjugate Stress and Strain Measures
• A Linear, Hyperelastic Model and Constitutive
  Transformations
• The Strain Energy and the Finite Element
  Methodology
• AN EXAMPLE
• CONCLUDING REMARKS

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INTRODUCTION (Manifold View of the Truss and
Embeddings)

• Mathematically, view the truss as a 3D
  differentiable manifold.
• Use three parameters for particle identification,
  (s1, s2, s3).
• The truss manifold consists of the s1 and the s2
  s3 submanifolds.
• To make observations, we consider embeddings of
  the truss submanifolds.
• An acceptable embedding is denoted by mC, with m
  being some possible configuration labeling, such
  that m 0 is the natural reference
  configuration. All quantities associated with a
  given configuration obtain the corresponding left
  superscript.
• We postulate that
• Acceptable embeddings of the s1 submanifold
  correspond to straight line segments with length
  mL.
• Acceptable embeddings of the s2 s3 submanifold
  correspond to bounded plane areas being normal to
  the embedded s1 line.
• To describe this area, we need two linearly
  independent vectors (not necessarily orthogonal)
  that span it, while being normal to the unit
  vector that lies along the s1 line.

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KINEMATICS (Notation and Point Transformation)

• To develop the truss kinematics, consider the
  truss in two different configurations as shown in
  the Figure
• Note that and are
  not necessarily orthogonal pairs.
• We will use n y or m z for these pairs when they
  are not orthogonal and x otherwise.
• F represents the transformation relating mC with
  nC. This is the transformation that we want to
  find.

X3
nx1
F
j
nC
mx2
nX(j)
nx2
mx3
nx3
mC
j
mx1 Local Frames




i
mX(i)

mX(j)
i
nX(i)
X2 Global Frame
mXs(nx1)
nXs(nx1)
X1
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KINEMATICS (Notation and Point Transformation)

• Consider a free vector a. Its representation in
  some frame ? is denoted as a?. We have that
• a? R? aX
• where R? is the transformation relating the ?
  and X components of a.
• With reference to the previous figure, we find
  that
• (1)
• where
• nL and mL are the truss lengths in each
  configuration. n?y and m?z are the norms of the
  n y ,m z vector pairs used to span the trusss
  cross section in each configuration.
• Bold subscripts denote the vector pairs used to
  describe the trusss cross section.

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KINEMATICS (The Deformation Gradient)

• Equation (1) is a point mapping, telling us how
  points in nC transform to point in mC.
• To describe truss deformations, we need to know
  how line elements transform.
• A line element is a vector, bound to some point
  of the embedded manifold, that connects this
  point with some point that is infinitesimally
  close to it.
• Mathematically, a line element is a vector
  belonging to the manifolds tangent space.
• By (1), we get by differentiation, that
• 
  
  (2)
• where and are line elements
  of the truss in nC and mC respectively.
• is known as the deformation gradient.
• By (2) and (1), we see that points and line
  elements transform from one configuration to
  another in exactly the same way, i.e. via the
  deformation gradient.

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KINEMATICS (The Polar Decomposition of the
Deformation Gradient)

• The deformation gradient contains all the
  kinematic information necessary to describe the
  motion of the truss.
• In order to uncover the building blocks of this
  motion, we polar decompose it, getting
• (3)
• where R is an orthogonal matrix representing a
  3D rotation of the truss and U and V are
  symmetric, positive definite matrices.
• There are three types of second order tensors
  Eulerian, Lagrangian and mixed type (or two
  point) for which the mC only, the nC only, or
  both configurations are related.
• Clearly, F and R are two point tensors, whereas
  U is Lagrangian and V Eulerian.
• We would like to find the conditions under which,
  U or V are intrinsically diagonal, since this is
  a great simplification.
• These conditions have to do with the vector pairs
  used to describe the trusss cross section, as
  well as with the frames used in the various
  representations.

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KINEMATICS (Conditions for Intrinsically Diagonal
U and V Matrices)

• We find that
• where
• (Note two eigenvalues only)
• Note that the z y subscript, indicates that the
  non orthogonal mz pair used to span the trusss
  cross section in mC, is related to the ny pair
  used to span the trusss cross section in nC, via
  the following
• Note that nyo and mzo used in the
  representations, are the orthogonal frames
  corresponding to the orthogonal rotation part of
  the polar decomposition of the non orthogonal
  matrix relating the ny and mz frames with the X
  frame.

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KINEMATICS (Conditions for Intrinsically Diagonal
U and V Matrices)

• Similarly, when orthogonal pairs are used for the
  description of the trusss cross section (denoted
  by use of x subscript), as well as the
  corresponding frames for the representations, we
  get
• where
• Note that here we have three distinct eigenvalues
  as opposed to the non orthogonal pairs case
  where we had only two.
• It can be shown that the conditions for diagonal
  U or V amount to succeeding in having the
  Eulerian counterparts of U equaling the
  Lagrangian counterparts of V where the Eulerian
  counterpart of U is given by
• By these relations, we find that the trusss
  motion consists of a series of transformations
  that can take place in any order a rigid body
  rotation given by R and a deformation involving
  stretching in three directions, described by ?.

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KINEMATICS (Various Strain Measures)

• Our concern lies in describing the deformation
  portion of the trusss motion. Hence, we must use
  ? to do that. (In the following we assume
  orthogonal vector pairs only).
• A quite general way of doing this, is by means of
  Seths strain measures
• (4)
• Where k , -2, -1, 0, 1, 2, . E is a
  Lagrangian tensor, whereas e is the Eulerian
  counterpart of E.
• The strain measures in (4) (all equally
  plausible), measure the deformation experienced
  by the truss in going from the reference nC to
  the current mC configuration.
• For k 1, we get the Biot strain, or corrotated
  engineering. For k 2, we get the Green
  Lagrange strain, for k -2 we get the Almansi
  strain, whereas for k -1, we get the Hyperbolic
  strain.
• For k 0, we get , known as the Lagrangian and
  Eulerian Logarithmic (or Hencky) strains
  respectively.

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KINEMATICS (The Velocity Gradient and the
Deformation and Spin Tensor)

• The rate of change of change of the trusss
  velocity with respect to position, is the
  velocity gradient mL.
• We let the following
• where
• We find that
• where
• are the deformation (symmetric part of mL) and
  spin tensors (antisymmetric part of mL)
• The Lagrangian counterpart of md is
• We are now able to calculate the strain rates

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KINEMATICS (Various Strain Rates)

• The material rates of the various strain measures
  are found to be
• For k 0, note that we get the material rates of
  the Logarithmic strains to be identical with the
  corresponding rates of deformation, a result that
  is not true in general, but only in cases where
  the principal directions of U and V remain fixed.
• The material rates are not objective. An
  objective rate is Jaumanns rate, given by
• When k 0 and a is the corrotating or the
  reference local frame, we get that
• This result is a special case of a more general
  one given by Xiao in 1997, where it is shown that
  the Logarithmic rate of the Eulerian Logarithmic
  strain is identical to the Eulerian rate of
  deformation. For the truss, we find that it is
  the Jaumanns rate is identical to Xiaos
  Logarithmic rate.

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CONSTITUTIVE MODELING (Strain Energy Strain
Energy Density Rate )