Coupling model: elasto-plastic-damage tangent modulus

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Materials and Behavior Models

A metal bar is subjected to a uniform uniaxial tension, the load gradually increases. Depending on the materials, there are various behavior models to represent the experimental curve. Material has an elastoplastic behavior if we can represent the experimental stress strain curve by a first segment that is straight line and then by a curve possibly rectilinear in Figure 2.3. If a discharge is performed in a point, the discharge is carried out by the initial Young’s modulus.
There is hardening of the material, that means, variation in its yield with its plastic deformation. As the applied stress is less than the elastic limit σe, the behavior is elastic. When the applied stress reaches the yield stress and the loading is continued to increase, the plastic deformation and the elastic limit increase. When the plastic strain increases, the elastic range is enlarged, the length of the slope segment E increases. But more the yield increases more the material becomes brittle because the increase in the field of elasticity decreases the field of plasticity and increases sensitivity to stress concentrations due to small manufacture defects, the cavities caused by small one-off shocks. In the case of uniaxial tension, we can directly compare the applied stress on the elastic limit as the stress field has a nonzero component in the loading direction. Material has a perfect elastoplastic behavior if the experimental stress-strain curve can be represented by a first straight segment whose slope is the modulus of elasticity, and a horizontal segment. As the applied stress is less than the elastic limit σe, the behavior is elastic. When the applied stress reaches the elastic limit and the loading continues to increase. The total deformation increases but not the stress; there is no hardening of the material in Figure 2.4. When the plastic strain is increased, the elastic field cannot grow; the length of the slope segment E is constant. If a discharge is performed in a point, it is carried out with the initial Young’s modulus. This model of perfectly elastoplastic behavior is adapted in the stretching bearing area for materials such as mild steel.

Control of Plasticity

Assuming that a point in stress space represents the stress state in a material point of the structure. It is within the volume delimited by the surface plasticity. The material has an elastoplastic behavior, the viscosity is not taken into account. Three cases can occur during a load increment in Figure 2.5a.
• The point representing the new state of stress is in the volume defined by the surface plasticity.
• The point representing the new stress state is on the surface of plasticity.
• The point representing the new stress state is outside the volume limited by the surface of plasticity.
In the first case, the behavior was elastic and it still is. For the second, the behavior was elastic and he became plastic. For the third, the behavior was elastic and became inadmissible. The plasticity surface must evolve so that the new point is on a new surface: this is the work hardening of the material. Assume now a point of the stress space representing the stress state in a material point of the structure. It is on the surface in Figure 2.5b.
• The point representing the new stress state came back to the volume delimited by the surface plasticity.
• The point representing the new stress state remains on the plasticity surface.
• The point representing the new stress is outside the volume delimited by the surface plasticity.
For the first case, we have an elastic discharge. For the second, loading is neutral. For the third, the behavior was plastic and it became inadmissible. The plasticity surface must evolve so that the new point is on a new surface: this is the work hardening of the material. Three types of data are needed to treat elastoplasticity:
• The plasticity criterion.
• The flow rule.
• The hardening rule.
The plasticity criterion allows determining the value of the equivalent stress from the components of the stress tensor. If the elastic limit is exceeded by the value of the equivalent stress, the stress state is inadmissible. Plasticity came up or it has developed. A criterion allows determining when the plasticity appears but provides no information about the nature of the plastic behavior. To describe the plasticity surface and its evolution when there is hardening, it is necessary to know the flow law which expresses the relation between increased plastic deformation and increased stress. In the case of plasticity associated, hypothesis of the elastoplasticity theory generally well suited to represent the behavior of metals, the function F, which describes the equation of the plasticity surface, is also called the plastic potential, plastic potential and plasticity surface being identical. The flow law is the relationship between the strain increment and increased stress. The hardening law gives the variation of plasticity with the stress state.
In practice, the current yield is compared to the equivalent stress that is evaluated for the current loading. When viscoplasticity is not taken into account, a point representing a stress state cannot be outside the volume defined (delimited) by the plasticity surface. If the estimated stress during a calculation of iteration is beyond this envelope, the stress state is not physical. The hardening is not involved to exceed the limit but to push it so that the stress reaches the new limit and is therefore admissible. In stress space, work harden a material is to modify its surface plasticity. Surface plasticity φ or intrinsic surface is written in the general form: ¯ p ¯ p ) = 0 (2.4) f (σi j , α i j) − σy(ε ) = φ (σi j , αi j , ε.

Continuum Mechanics and Thermodynamics

In this section we present the main ingredients for the formulation of a coupled damage-plasticity model in the framework of the thermodynamics of continuum media. In order to describe evolution of elastic properties due to damage and hardening of the material due to plasticity, we consider five different internal variables, namely: ε d is the plastic strain, ξ p a scalar that controls the isotropic hardening, κ p, a deviatoric second order tensor controlling the kinematic hardening, damage compliance tensor D and ξ d , which is a scalar hardening like variable related to damage.
In order to derive governing equations of the model, we consider the following main three groups of ingredients:
• additive decomposition of strain into elastic, damage and plastic strains: ε = ε e + ε p + ε d (3.1).
where ε p corresponds to the unrecoverable strain at zero stress, ε e corresponds to the recoverable part of strains when unloading with the initial elastic modulus. Finally, ε d corresponds to the additional recoverable part of strains due to the evoloution of the elastic modulus during damage. A schematic representation is given in Figure 3.1 for a 1D model.

Plasticity model: yield criterion and consistency condition

The plastic internal variables evolution equations are obtained considering the plastic part of the total dissipation and introducing the plastic Lagrangian and associated plastic Lagrange multiplier γ˙p. The maximization problem is then recast as a minimization problem as: max min L p(σ ∗, qp∗, α ∗, γ˙p) (3.11) γ˙p >0 (σ ∗,qp∗,α ∗) where L p(σ , qp, α , γ˙p) = −D p(σ , qp) + γ˙pφ (σ , qp, α ).

COUPLED DAMAGE-PLASTICITY MODEL : NUMERICAL ASPECTS

The previous equations allow finding the variation of stresses as a function of an increase in infinitely small deformations. In practice, during a finite element analysis, on a computation step, the variation of deformation is not infinitely small. The previous equations must be considered. In this section, we consider an integration point which we know all the features (stress, strain, state variables …) at step (n) and deformations at the step (n + 1) or variations of the total strain between the steps n and (n + 1). At this stage, we are looking for the stresses at the step (n + 1) corresponding to these new strains. The equilibrium is obtained by global iterations on the structure.
First solution for determining the stresses consist in calculating and using the tangential material rule to at the step (n). The stress at the step (n + 1) is calculated simply by: σn+1 = σn +CTe,nΔε (4.1).
If there was a flow at the step, the stresses at the point (n+1) does not obey the plasticity criterion because increase in deformation is finite and not infinitesimal, and the evolution of the tangential material rule has not been taken account at the step. The stresses must be brought on the new surface whose actual equation is φ = 0. It may be possible to cut the variation in deformation at different intervals in order to reduce the error introduced by this scheme. Another commonly used scheme is the elastic predictor / plastic corrector.

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Tangent Matrice, Numerical Integration

In the case of a finite element whose material behavior is elastic and linear, Hooke matrix that  appears in the calculation of the stiffness matrix (or the tangent stiffness matrix for nonlinear geometric analysis) is not only constant for any analysis, but it is uniform in each element. With traditional notations, we can write integral form for the element e: Ke = BT H BdV (4.7).
This matrix is evaluated by numerical integration according to various rules: each term ki j matrix is obtained by sampling the m integration points of the element kiej = BT H N dV = ∑ [Bi]TmHm[B j ]mωm (4.8).
Hm is the Hooke matrix at the point m: it is the same at any point. Wm is the weight function associated with the integration point multiplied by the determinant of the Jacobian matrix of the element. In the case of an analysis in which the plasticity appears, the construction of the stiffness matrix is different. From that point of integration, the equivalent stress is such that there is a plastification, Hooke’s law is replaced by the material tangent law to integration point(s) having plasticity, the law that evolves the level of hardening and stress state as shown The material tangent law in an element is different in each integration point of the element as it depends on the stress level of each integration point. The evaluation of the terms of the tangent stiffness matrix is much more complex than in the case where the material has a linear elastic behavior. The variation of the element terms to be integrated in the tangent stiffness matrix is more important in plasticity than in elasticity. We can therefore ask whether it is advantageous to increase the number of integration points on the element to get a better accuracy. If we increase the number of points, the integral can be better evaluated, but it is a false precision. The plastic solution is less regular than the elastic solution. If we want to increase the precision, it is better to increase the number of elements, because increasing the number of integration points does not change the degree of the deformation field that depends on displacements and not stresses. However, the instant when plasticity begins will be better identified if the number of integration points is increased. Indeed, a linear element with a large strain gradient is considered. If only one point of integration is used, the plasticity begins when the plasticity criterion is reached in the middle of the element. If two points are used, it is when it is achieved about one-quarter of the element. More the number of integration points are big, more they are near the end and the beginning of plasticity is detected. To illustrate this point, there is a quadrangular membrane with an integration rule of 2×2 points. Plasticity must arrive at point P in order to be detected and 14 of the element is then plastic. Plasticity is in fact already appeared in the element, and when the plasticity begins, it is possible that the plasticity is physically won over a quarter of the element , or less than a quarter depending on how it advances into the element as a function of the loading. With a rule 3×3 points for the same item, simply it arrives at point P in order to be detected and ninth element is then plastic . The increase in the number of integration points has significant influence on the cost of calculating the tangent matrix and storage space. And experience shows that this increase does not significantly change the global non-linear behavior. We must ask ourselves what is the purpose of the plastic calculation: determining when the structure begins to plasticize or know how to redistribute the stresses in the spread of plasticity. In the first case, an elastic calculation with a good post-treatment may suffice. In the second case, provided that the mesh is sufficiently fine and is accountable for the development of plasticity, the number of integration points per element has little influence. Anyway, the mesh is not a solution provider, but a developer solution. The calculation code user should know what he wants and where and how the mesh used to represent what he wants.

Numerical Implementation and Operator Split Method for Coupled Damage-Plasticity Model

We use three different levels of computation in order to solve the problem. These levels are separated into global, element and local level. The operator split method (see e.g. [46]) is employed in order to simplify the calculation of the three-level computational task in which the nonlinear equations must be solved. Thus, the Newton iterative procedure is applied until we reach a required convergent tolerance at each level.

Table of contents :

FOREWORD
TABLE OF CONTENTS
LIST OF FIGURES
SUMMARY
ÖZET
RESUME
1. INTRODUCTION
1.1 Problem Statement
1.2 Background and Literature Review
1.3 Scope and Objectives
1.4 Outline
2. FUNDAMENTAL CONCEPTS
2.1 Observations Elastic Limit
2.2 Materials and Behavior Models
2.3 Control of Plasticity
2.4 The Criterion of Plasticity
3. COUPLED DAMAGE-PLASTICITY MODEL : THEORY
3.1 Introduction
3.2 Continuum Mechanics and Thermodynamics
3.2.1 Equations of states
3.2.2 Dissipation potential
3.2.2.1 Plasticity model: yield criterion and consistency condition
3.2.2.2 Damage model: damage criterion and consistency condition
3.2.2.3 Coupling model: elasto-plastic-damage tangent modulus
3.2.3 Hardening models
3.2.4 Flow rule
3.3 Conclusion
4. COUPLED DAMAGE-PLASTICITY MODEL : NUMERICAL ASPECTS 
4.1 Integration Algorithm
4.2 Tangent Matrice, Numerical Integration
4.3 Numerical Implementation and Operator Split Method for Coupled Damage-Plasticity Model
4.3.1 Discretization of the problem
4.3.2 Global computation
4.3.3 Element computation
4.3.4 Local computation and implicit backward Euler scheme
4.3.4.1 Plasticity computation
4.3.4.2 Damage computation
4.4 Numerical Examples
4.4.1 Steel sheet in simple tension test under cycling loading
4.4.2 Steel sheet in bending and shear tests under cyclic loading
4.4.2.1 Bending test
4.4.2.2 Shear test
5. CONCLUSIONS
REFERENCES

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