Materials & Properties: Mechanical Behaviour

Proceedings of the 2017 CERN–Accelerator–School course on Vacuum for Particle Accelerators, Glumslöv, (Sweden) Materials & Properties: Mechanical Behaviour C. Garion CERN, Geneva, Switzerland Abstract All systems are expected to be designed to fulfil their functions over their requested lifetime. Nevertheless, failure of a system may occur, and this is unfortunately, true also for vacuum systems. From a mechanical point of view, buckling, leak by fatigue crack propagation, rupture of a fixed support under unbalanced vacuum force… may happen. To well understand and anticipate the behaviour of a structure, it is mandatory to first understand the mechanical behaviour of materials as well as their failure modes. This is presented in a first part of the document. Then, the structural behaviour of a vacuum system and its relationship with the material properties is discussed. Finally, guidelines for the material selection for a given application are roughed out. 1 Introduction Used as vacuum envelope, the material is of primary importance to ensure the mechanical reliability of the vacuum systems. In general, vacuum chambers have to be more and more optimized for mechanical or physics considerations, or to fulfil space requirements which are more and more stringent, or basically for economic reasons. To optimize the chambers and therefore, minimize margin to a reasonable value while ensuring the integrity of the system, the material properties, their behaviour and their influence on the mechanical behaviour of the structure have to be well understood. The first chapter is devoted to different models of materials based on continuum mechanics. After simple linear isotropic elasticity, other notions from plasticity to fracture mechanics are presented. These models can be wisely used for applications where material failure might be expected (fatigue, initial defect propagation, etc.). The basic parameters of isotropic linear thermal elasticity to be used in most of the applications are introduced. The second chapter deals with the design of the vacuum chambers as a structural component but also of the whole vacuum system as a mechanical system. A general approach is presented covering the main loads applied on a vacuum chamber. Two key points shall be kept in mind of all designers of vacuum systems: First, the stability of structures subjected to pressure and external pressure in particular for vacuum applications; second the pressure thrust force induced by bellows on the adjacent chambers and consequently on the supports. Despite these two phenomena are well known, numerous examples of failure with dramatic consequences can be found in accelerators. The third chapter aims at giving some information about the material selection. This process uses the material properties but introduces also the figures of merit of materials for a given applications. The chapter is focused on the material selection based on their physical or mechanical properties. Other points, not covered in this paper, shall be considered as well for the material selection: the availability, the transformation process (machining, forming, etc.), the integration/joining to the system (welding or brazing), radiation compatibility (resistance and activation), cost, etc. This paper is not intended to give an exhaustive view of materials and mechanics but aims at giving a comprehensive overview of both fields and how both are connected together and to the mechanical design of vacuum systems. Available online at https://cas.web.cern.ch/previous-schools 1 2 Material modelling 2.1 Basic notions To characterize the mechanical properties and behaviour of materials, intrinsic variables are introduced. The stress in a section is defined as the ratio between the force applied on a section and its surface area. Usually denoted as s, it is commonly expressed in MPa (N·mm-2). Similarly, the strain is given by the relative variation of a given length. This variable is dimensionless. Basic behaviour and mechanical properties are assessed from tensile tests carried out on a sample of probing length, L and section S0. The force is measured as a function of the elongation. Then, the typical curve giving the nominal or engineering stress (F/S0) as a function of the nominal strain (DL/L0=(L-L0)/L) is derived. Three main material properties are assessed: The Young’s modulus, denoted E, corresponding to the stiffness of the material in the elastic domain; the yield stress sy or Rp (elastic limit) and the tensile strength sm or Rm (Fig. 1). The Poisson ratio, n, defined by the ratio between the transversal and axial strain can be determined as well by the tensile tests. Fig. 1: Typical tensile curve of metals Different categories of material behaviours can be observed. Among them: - Elastic brittle material: the deformation is reversible, and failure occurs without noticeable overall plastic deformation (Fig. 2(a)). - Elastic plastic behaviour: after elastic reversible deformation, slips occurs in crystals and lead to residual deformation after unloading (Fig. 2(b)). - Elastic plastic damageable materials: for large deformation, microcracks and/or microvoids appear in the materials leading to noticeable decrease of the stiffness (Fig. 2(c)). - Elastic viscoplastic behaviour: material response in the plastic domain depends on the strain rate (Fig. 2(d)). 2 a) Elastic brittle behaviour b) Elastic plastic behaviour c) Elastic plastic damageable behaviour d) Elastic viscoplastic behaviour Fig. 2: Some basic material behaviours The geometry of the sample evolves during the loading. Therefore, true variables, strain and stress, are introduced. They are defined from the engineering strain and stress by: $% d� = à � = ln(1 + � . (1) % +,- � � = ≅ � ∙ (1 + � . (2) � +,- +,- A comparison between the engineering stress-strain curve and the true stress-strain curve is shown in Fig. 3. Whereas the difference is not noticeable in the elastic domain, it has to be considered for a large strain to characterize correctly the materials. Fig. 3: Comparison of engineering and true variables 3 2.2 Notions of strain and stress in continuum mechanics The stress vector T defined at a point M and for a given surface represented by its normal n is given by: �� (�, �) = lim (3) $<→> �� It can be written as a function of the stress tensor s, defined by: �(�, �) = �(�) ∙ � (4) The normal and shear stress are called sn and t. They are defined by �, = � ∙ �(�) ∙ � and � = �(�) ∙ � − �, ∙ �, respectively. In tension, sn is positive. It can be shown that s is a symmetric tensor and is therefore represented in the principal base by a diagonal matrix with the principal stresses, sI, sII and sIII. The hydrostatic stress, sh, and deviatoric stress tensor sD are defined by: H � = � ∙ � + �F ; G� = ∙ tr L�MN (5) D D I From a kinematical point of view, considering the function F, that transforms the initial configuration to the actual geometry (M=F(M0)), the deformation gradient F is defined by: �Φ(��) d� = ∙ d�� = �(��) ∙ d�� (6) �� The Green Lagrange deformation is introduced by: 1 �(� ) = ∙ T��(� ) ∙ �(� ) − �W (7) � 2 � � Considering the displacement (M= M0+u(M0)) and small deformations, the strain tensor reads: 1 ��(� ) ��(� ) \ �(�) = ∙ Y � + Z � [ ] (8) 2 �� Finally, the energy variation in the material of a body W is: d� = ∰` � ∶ d�. 2.3 Thermal elasticity 2.3.1 Linear elasticity All materials are composed of atoms which are held together by bonds resulting from electromagnetic interactions. When external load is applied, the lattice is deformed, interatomic distances change leading to interatomic forces that, in a first approach, are proportional to the interatomic distance variation. This deformation is reversible. This behaviour at the atomic level can be extended at the mesoscopic scale under the Hooke’s law: � with the stiffness or elastic tensor � = �: � � (9) For homogeneous and isotropic materials, the elastic tensor is defined by only two independent parameters: Hgh j � = � ∙ tr T��W ∙ � + 2 ∙ � ∙ �� or �� = ∙ � − ∙ tr T�W ∙ � i i (10) 4 � and � stand for the Lamé coefficients. �, sometimes denoted G, is the shear modulus. The relationships between the elastic parameters are given in the set: ji i � = and � = (Hgj)(Hklj) l(Hgj) Imgln m � = � and � = (mgn) l(mgn) (11) 2� � � = � + = 3 3(1 − 2�) The Young’s modulus is a direct consequence of the type of bonds. Typical values are given in Tables 1 and 2 for different bonds and metals, respectively. Table 1: Typical range for Young’s modulus Bond type Typical range of the Young’s modulus [GPa] Covalent 1000 Ionic 50 Metallic 30-400 Van der Waals 2 Table 2: Elastic parameters at room temperature for metals commonly used for UHV applications Aluminium Copper Stainless steel Titanium E [GPa] 70 130 195 110 � 0.34 0.34 0.3 0.32 2.3.2 Thermal linear isotropic elasticity A temperature variation changes the equilibrium state between atoms and introduces a free thermal strain. It is represented by the thermal strain tensor, eth, that is defined for an isotropic material by: rD d� = �(�) ∙ d� ∙ � (12) with a the coefficient of thermal expansion (CTE). For temperature independent dilatation coefficient, the thermal strain reads with respect to a temperature of reference Tref: rD � = � ∙ (� − �t+u. ∙ � (13) The total strain tensor is the sum of the thermal and mechanical contributions, � = �+ + �rD , leading to: � rD � = �: � = � = �: L� − � M (14) In one dimensional case, it reads simply: � = � ∙ (� − �rD. = � ∙ (� − � ∙ ∆�) (15) 5 Therefore, for a free stress state,

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