Since different materials behave differently under the same external load, the constitutive law characterizes the material response of a specific body. In order to describe the material behaviour it is necessary to establish a relationship between the strains and stresses, being usual practise to consider the stresses as dependent variables [Bertram 08]. Indeed, the theory of constitutive laws is rather complicated, thus the constitutive models are usually classified in two groups: phenomenological models based on experimental observations, often at macroscopic scales [Belytschko 00], and atomistic models motivated by atomic interactions [Mishin 10], [Sauer 07]. Due to their widely practical application and numerical simplicity, only phenomenological models are addressed in this thesis, given particular attention on the elastoplastic constitutive laws for large strains. Many other constitutive laws exist for miscellaneous applications. However, the focus of this thesis is on contact interaction rather than constitutive modelling. Indeed, constitutive laws for the bodies coming into contact can be arbitrary, not affecting the main formulation of contact problems [Wriggers 06].
In many engineering applications involving small strains and rotations, the response of the material may be considered to be linearly elastic (perfectly reversible and path- independent). The corresponding constitutive equation is often referred to as the generalized Hooke's law, which can incorporate fully anisotropic material response. The extension of the small strain linear elasticity to the case of finite strain can be carried out in different ways and different constitutive relations can be established. The constitutive models for large strain elasticity are typically categorized in two classes: hypoelasticity and hyperelasticity. The hypoelastic material laws are formulated in rate form, which relate the rate of stress to the rate of deformation. In order to satisfy the principle of material frame indifference, the stress rate should be objective and should be related to an objective measure of the deformation rate. Due to the properties of the rate of deformation tensor, a hypoelastic material do not strictly reflects the path independence of elasticity. However, if the elastic strains are small, the behaviour is close enough to path independent. Thus, the principal use of hypoelastic constitutive relations is in the representation of the elastic response of elastoplastic constitutive relations, where the elastic deformations are small [Khan 95]. On the other hand, hyperelastic or Green materials are characterized by the existence of a stored energy function. Thus, the work done by a hyperelastic material is independent of the deformation path [Ogden 84]. This description is valid for many
Formulation of Contact Problems and Resolution Methods
17 materials, such as rubbers, foams and biological tissues, which exhibit a nonlinear stress- strain behaviour undergoing finite deformations and an almost incompressible response.
2.1.3.1. Elastoplastic behaviour
Many materials widely used in mechanical applications present a nonlinear behaviour. A large class of nonlinear materials can be described by the assumption of elastoplastic behaviour, referring examples of these materials, metallic ones, such as steels and aluminium alloys. The fundamental idea behind the phenomenological approach to elastoplasticity is that the deformation gradient F is multiplicatively decomposed in two components: p e , F F F (2.14)
where F and e F denotes the elastic and plastic deformation gradients, respectively p [Lee
69], [Simo 88].
Figure 2.3. Schematic representation of the multiplicative decomposition theorem.
The multiplicative decomposition assumes the existence of an intermediate configuration (stress free state), as shown in Figure 2.3. This relaxed configuration is obtained considering that the deformed body has undergone a purely plastic deformation, thus nonconforming configuration with respect to neighbouring material points. Since an arbitrary rigid body rotation can be selected for the intermediate configuration, it is in general not uniquely defined. In order to overcome the uniqueness problem, by convention, all the rigid body rotation is lumped into the plastic deformation gradient F , such that the p
elastic deformation gradient F includes stretch only [e Dunne 05]. Since the elastoplastic
behaviour is a path-dependent deformation process, it is commonly analysed using an
0 Ω Ω ( , )t x X X F
Reference configuration Current configuration
x y z Relaxed configuration e F p F ΩR xR
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incremental procedure. Thus, variables such as displacements, plastic strains, and external loads are written in an incremental form. This requires the use of rate-type measures of deformation, which are derived from the multiplicative decomposition of the deformation gradient (2.14).
The elastoplastic material behaviour is defined by three properties: (i) a yield function; (ii) a flow rule and (iii) a hardening law. The yield function establishes the state of multi- axial stress corresponding to the start of plastic flow, defined in the general form by:
p p
b b Y
( , , ) ( ) ( ),
f σ σ ε Fσ σ σ ε (2.15)
where σ is the Cauchy stress tensor and σb is called back stress tensor, which is related to kinematic hardening. The evolution of the yield stress during plastic flow is defined by the hardening law σY, which depends on the cumulative equivalent plastic strain ε . p Mixed hardening models combine isotropic and kinematic hardening, which results into an evolution of the yield surface by simultaneous translation and expansion. A points is in the elastic domain if the stress state corresponds to f 0. When that point is located on the boundary of the yield surface (f 0) plastic deformation can occur depending on the loading condition, whereas f 0 is inadmissible. The irreversibility of the plastic flow process is expressed by a flow rule. In most metals, an associated flow rule can be considered, where the increment of plastic strain occurs in the normal direction to the yield surface at the load point. This direction is given by the gradient of the yield function (2.15) with respect to the stress tensor:
p f( , b), d dλ σ σ ε σ (2.16)
where dλ is a scalar which determines the size of the plastic strain increment. The model is completed by specifying the loading/unloading conditions for elastoplasticity. These may be given in terms of the classical Karush–Kuhn–Tucker (KKT) conditions for inequality constraints in optimization:
0, 0, 0,
f dλ dλf (2.17)
which ensure that the stress point is not outside the yield surface, that the magnitude of the plastic strain rate is always positive, and that plasticity only occurs when the stress point is on the yield surface [Laursen 02]. The integration of the constitutive law at each material point allows to determine the increments of stress state and the equivalent plastic strain. For further details readers are addressed to see [Alves 03].
Formulation of Contact Problems and Resolution Methods
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