PSI - Issue 48

Tamás Fekete / Procedia Structural Integrity 48 (2023) 302–309 Fekete / Structural Integrity Procedia 00 (2023) 000 – 000

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A general introduction to the theory Processes in material systems always involve observable motion of the system in ambient space; hence one of the keystones of all physical theories is the motion equations (or equations of motion) following the motion of the material system (or alternatively Material Body, MB ) in external space – see e.g., Bažant, Cedolin (1991), Béda, Kozák, Verhás (1995), Steinmann (2022) – . The description is called Eulerian if the motion is investigated in a coordinate frame fixed to the ambient space; then the coordinate frame is called Eulerian Coordinate System ( ECS ). A description is called a Material – or the Lagrange – description, when looking the motion in a coordinate frame co-moving with the material. This coordinate system is called the Material Coordinate System ( MCS ) – or Substantial or Lagrange coordinate system – . It is well known that since the early days of safety design, DSC s and SIC s for LSPS s have been based on Continuum Field Theories ( CFT ); this aspect is left unchanged. CFT s are formulated to describe the physical behavior of MB s in terms of continuous distributions of the physical fields – in the selected order of approximation – . MB s are considered as manifolds of elementary cells. In each cell, the physical quantities describing the material and its properties can be described by sufficiently smooth distributions. An elementary cell is called the Representative Volume Element ( RVE ). The material point ( P ) – which is characterized by its position ( X ) in the MCS – is that selected point of the RVE , the motion of which – or the value of the physical fields defined at this point – is representative of the averaged motion of the RVE – or the characteristic values of the physical fields of the RVE – with a suitable accuracy. In line with the philosophy of Jet's Manifold Theory ( JMT ), the motion field and the physical fields within an RVE are expressed as their respective k-order truncated Taylor polynomials around P , called the k-Jet of the respective fields. The k - Jet of a field ( N ) around P is denoted by   k J N P . In this paper, the MCS is chosen as primary coordinate system, since the theoretical questions to be answered are the most convenient to be addressed using MCS . The ideas outlined here are based on concepts from JMT , from the Geometric Approach to Continuum Mechanics and from the theory of Mechanics on Material Manifolds , where the MB is considered as a Material Manifold with its own Geometric Structure – see e.g., Saunders (1989), Yavari, Marsden and Ortiz (2006) and Sardanashvily (2014) – . For brevity of space, further technical details are omitted. The motion of P of a given MB in the ambient space is defined by:     0 , , , where d             x X (1) where X stands for the position of P in MCS , but x in ECS ; the gradient is referred to as X  in MCS and  in ECS . For the description used here,  is the instantaneous time coordinate assigned to the fast processes, while  is the time accumulated since the system was started, 0  . The inclusion of the  time coordinate into the description indicates that the motion – and also the physical processes – are followed over the long-term. The form of the equations of motion in (1) is the same as that in point mechanics. The form of the equation of motion in the RVE conjugated to P is shown below. For this, the motion of a point P' – belonging to the RVE , with coordinates d   X X X – is investigated. That is, the transformation behavior of the motion equation under the , d       P P X X X X transformation is considered. It is assumed that on the RVE , the motion field can be described by   2 J  P , – that is the 2-Jet field of the motion over P – with sufficient accuracy. Thus, the motion of P' in ECS is as follows: if it is assumed that the motion field on the RVE – i.e.,   , ,     X – is adequately described by   2 J  P , then the motion of P' may be expressed in ECS as follows:         , , , , , , , : d d d                   x X X X X X X (2)

1 2

X

X X

( , , ) X     X represents the locally linear part of the material

where x ’ denotes position of P ' in the ECS . In eq. (2),

  1 J  P

( , , ) X X      X the locally bilinear part. Restricting the motion model to

deformation gradient over X , and

– i.e., keeping only the locally linear part of the material deformation gradient – , the classical Boltzmann continuum model is obtained; hereafter the description follows the Boltzmann continuum model, and in rest of the paper, ( , , ) X     X is denoted by F . The displacement vector links the positions of P between ECS and MCS as follows:       , , ,         u x X (3)