Invariance Under Superposed Rigid Body Motions with Constraints
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Invariance Under Superposed Rigid Body Motions with Constraints M.B. Rubin1
Received: 19 January 2020 © Springer Nature B.V. 2020
Abstract The notion of invariance under Superposed Rigid Body Motions (SRBM) is enhanced by a restriction that explicitly states in what sense the responses of a material under SRBM are equivalent. This new restriction is used to develop expressions for the superposed values of the strain energy and the Cauchy stress, instead of assuming their forms. Moreover, it clarifies invariance of the constraint response for kinematic constraints. Keywords Constraints · Form-invariance · Invariance conditions · Superposed rigid body motion Mathematics Subject Classification 74A20
1 Introduction The main use of invariance under Superposed Rigid Body Motions (SRBM) in continuum mechanics is to place restrictions on the constitutive equations. In this paper attention is limited to simple materials which are characterized by local measures like the velocity gradient. Throughout this text a superscript (+ ) is used to denote the value of a variable in the superposed configuration. It is clear from [3, p. 486] that the notion of invariance under SRBM is based on the restriction: (R-1): The balance laws must be form-invariant under SRBM. This means that the balance laws in the superposed configuration with a superscript (+ ) added to each variable are valid for all SRBM. Since the expressions for the superposed values of the strain energy function and the Cauchy stress cannot be deduced from (R-1), an additional restriction is needed. Specifically, it is necessary to specify in what sense the responses of a material under SRBM are equivalent. To this end, the additional restriction is proposed as
B M.B. Rubin
[email protected]
1
Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, 32000 Haifa, Israel
M.B. Rubin
(R-2): The constitutive response of the material relative to its orientation is the same for all SRBM. In this regard it is noted that the Principle of Material Frame Indifference has been characterized by two assumptions in [2]. The first assumption in [2] states that: “The physical quantities which characterize the behaviour of a given material are intrinsic”. The term intrinsic means objective, which essentially assumes the transformation relation for the stress. Here, use is made of the second restriction (R-2) to develop expressions for the superposed values of the strain energy and the stress, instead of assuming that they transform as objective quantities.
2 Basic Equations of SRBM Within the context of SRBM a material point located by the position vector x at time t in the current configuration is transformed to the location x+ at time t + in the superposed configuration, such that x+ = c(t) + Q(t)x ,
t+ = t + c ,
(1)
where c(t) is an arbitrary vector function of time only characterizing superposed translation, Q(t) is an arbitrary proper orthogonal tensor function of time only characterizing superposed rotation QQT = I ,
detQ = +1 ,
(2)
c is
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