Penetration of master surface to slave surface in contact analysis

The general contact algorithm in Abaqus/Standard is based upon the interactions and constraint characteristics applied to master and slave surfaces. The user must define a master and a slave surface for each contact pair. Moreover, it is recommended that analytical rigid surfaces and rigid element-based surfaces must always be master surfaces. Master surfaces should be more coarsely meshed, while the slave rough part surface should be more finely meshed and has lower stiffness. The master-slave contact algorithm places no restrictions on the master analytical rigid surface; it may penetrate the slave surface between slave nodes, as illustrated in Figure 2.6. The contact interaction and penetration definition between the surfaces can be specified using one of two approaches; node-to-surface discretization and

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surface-to-surface discretization. A node-to-surface approach is based on each single slave node interacting with a group of master surface nodes. Thus, the slave nodes are assumed not to penetrate the master surface, while the nodes of the master surface can penetrate into the slave surface. Bryant, (2013) found that the node-to-surface discretization method provides less accurate residual stress results than the surface-to-surface discretization as shown in Figure 2.7, because the shape of both master and slave surfaces are not considered in the contact formulation. Therefore, to avoid such problems the surface-to-surface discretization method was chosen for the current research. Additionally, in this method the features of both master and slave surfaces are considered and that minimises the localized penetrations, as well as providing more accurate results for both the residual stresses and residual deflection results.

Figure 2-6 Definition of master and slave surfaces.

Figure 2-7 Contact pressure plots for surface-to-surface and node-to-surface contact discretization method (Bryant, 2013).

55 2.1.7 Strain hardening

To specify the elastic-perfectly plastic or elastic-plastic behaviour, Abaqus requires yield stress and plastic strains to be defined in terms of true strain and the true stress instead of nominal stress and nominal strain. If plasticity behaviour has not been specified for the Abaqus contact analysis, the stress/strain relationship will behave linearly. The elastic-plastic calculation of stress and strain distributions at low strains are based on linear elasticity. The onset of strain hardening is attributed to plastic behaviour and happens at a stress level regarded as the first yield stress (Batdorf and Budiansky, (1949) ; Cook et al.

(1989); Bryantet al., (2012) for example). Any subsequent increase in stress with increased strain occurs according to linear strain hardening. In this approximation, the tangent modulus, ET, characterises the stress-strain relationship post-yield.

This modulus quantifies the level of ‘‘hardening” or ‘‘softening” of the model material that normally happens when it begins to yield, Shankar and Mayuram, (2008). In the model described in section 2.1.2, the elastic-plastic behaviour was defined as elastic perfectly plastic as the initial attempt of finite element simulation for the contact of rough part pressed against the rigid body. However, that analysis aborted and experienced convergence difficulties in obtaining solutions when the model reached a particular deformation. This is related to the stiffness degradation problem of model material associated with elastic-perfectly plastic behaviour. So, the concept of a linear strain hardening behaviour will be adopted for the current research to accommodate the material behaviour and to avoid the numerical convergence problems that occur with the elastic-perfectly plastic approach. In addition, this is helpful in terms of controlling the penetration and contact interaction during the simulation. A value of 50 GPa was used for the tangent modulus (ET) in the current research, giving ET / E ratio of 0.25. However, Kogut and Etsion, ( 2002 ) found that the majority of practical materials have ET / E ≤ 0.05, the strain hardening behaviour was chosen to be a relatively large value in the current simulation to evaluate its effects. This

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gives a clear picture for a judgment of the suitability of the assumption of elastic- plastic behaviour. The strain hardening behaviour can be defined in Abaqus by using the property module, and is specified in the edit materials menu as shown in Figure 2.8 a.

Figure 2-8 Stress-strain relationships; a) Abaqus/CAE 6.13 "Edit Materials" menu for linear strain hardening , b) linear strain hardening (ET / E = 0.25).

a)

b)

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The stress strain behaviour is specified by a table that gives a piecewise linear stress-strain curve. The example of Figure 2.8 specifies a strain of 0.11521 that will be achieved when the von Mises stress is 10.0 GPa in the right and left columns, respectively in the edit materials menu. This high true von Mises stress value of 10 GPa is chosen in order to ensure the required plastic behaviour is monitored throughout the material without any extension of elastic-perfectly plastic behaviour.

The mathematical calculation of initial strain input is calculated by using the following relationship.

𝜀 = 𝜎/𝐸 (2.1)

However, the elastic- plastic approach uses the true stress and strain, instead of the nominal stress and strain, as the latter does not take into account the instantaneous change of cross-sectional area as the model material deforms plastically. The true stress (is the applied load divided by the actual cross-sectional area) and strain takes this factor into account. They can be expressed as follows.

𝜎_{𝑡𝑟𝑢𝑒} = 𝜎_𝑛𝑜𝑚 (1 + 𝜀_𝑛𝑜𝑚) (2.2)

𝜀_{𝑡𝑟𝑢𝑒} = 𝑙𝑛 (1 + 𝜀_𝑛𝑜𝑚) (2.3)

The true strain must then be converted into the true plastic strain in terms of recoverable strain (true stress/E) and true strain as shown in equation below.

ε_{𝑡𝑟𝑢𝑒}^𝑝𝑙 = ε_{𝑡𝑟𝑢𝑒}−𝜎_{𝑡𝑟𝑢𝑒}

𝐸 (2.4)

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Abaqus expects the true strain and stress data to be entered in the properties table in the right and left columns, respectively. Abaqus interpolates linearly between the data points provided.