# Stress-resultant constitutive equations for elastoplasticity

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## Concluding remarks and chapter summary

In this chapter the analysis of reinforced concrete plates is considered where we distinguish between the uncracked state I of the plate and the cracked state II of the plate, activated by the tensile failure of the concrete. In state I we consider the standard linear elastic constitutive relations for isotropic material q = CSγ, m = CBκ.
In state II we again consider a linear elastic response for the shear while for the bending part we disregard the influence of Poisson’s ratio and assume independent responses for the two orthogonal directions defined by φ and φ + π 2 of the form m = m(κ).

### Plate elastoplasticity

We consider the following internal variables to describe the irreversible nature of the plastic process during the plate bending: the plastic strain εp, the scalar parameter ξ, which controls the isotropic hardening mechanism, and the strain-like parameters {ij , i, j ∈ {1, 2, 3}, which control the kinematic hardening mechanism. The state variables are functions of position x and pseudo-time t, i.e. εp = εp (x, t), ξ = ξ (x, t) and {ij = {ij (x, t). A usual additive split of reversible (elastic) and irreversible (plastic) strains is assumed ε = εe + εp.

#### Plate elastoviscoplasticity

A stress resultant viscoplastic constitutive equations for plates of Perzyna type are obtained by a modification of the elastoplasticity model presented in the previous section. The basic difference between the viscoplasticity and plasticity is that in the former model the stress states {σ, q,α}, such that φ (σ, q,α) > 0, are permissible, while in the latter are not. The state variables remain the same, except for the viscoplastic strain εvp, which replaces εp. The constrained minimization problem (3.19) for plasticity is here replaced by the penalty form of the principle of maximum plastic dissipation (see e.g. [Simo and Hughes, 1998] section 2.7 and [Ibrahimbegovic et al., 1998] for details), which can be stated as: Find minimum of Lvp (σ, q,α), where Lvp (σ, q,α) = −Dvp (σ, q,α) + 1 η g (φ (σ, q,α)) .

Computational issues for plasticity

As a result of space discretization, addressed in the previous section, the evolution equations (3.20) become ordinary differential equations in time that need to be integrated numerically at each integration point. Backward Euler integration scheme is used for that end. The solution is searched for at discrete pseudo-time points 0 < t1 < . . . tn < tn+1 . . . < T. At a typical pseudo-time increment t = tn+1 − tn and typical integration point located at xh (ξG) ∈ e the problem can be stated as: By knowing the values of the internal variables at the beginning of the pseudo-time increment, i.e. εp n, ξn, {n, find values of the internal variables at the end of the pseudo-time increment, i.e. εp n+1, ξn+1, {n+1, which should satisfy the yield criterion. In the spirit of the operator split method [Ibrahimbegovic, 2006] one assumes that the best iterative guess for strains at the end of the pseudo-time increment, ε(i) n+1, is given data. Here (i) is iteration counter of the (global) Newton-Raphson solution procedure. Prior to the integration of evolution equations, the following test is performed: assume that the pseudo-time step from tn to tn+1 remains elastic and evaluate the trial (test) values of strain-like and stress-like internal variables.

Computational issues for viscoplasticity

The above discussed viscoplastic plate model allows one to define a unified framework for both stress resultant elastoplasticity and stress resultant viscoplasticity for plates. Namely, the integration procedure for the plate viscoplasticity is essentially the same as for the plate plasticity, except that for φtr n+1 > 0 one looks for γn+1 = t η hφn+1i > 0. Its final value is obtained by iterative solution of nonlinear equation − η t γn+1 + φn+1 (γn+1) = 0 → γn+1. (3.52).
The consistent tangent matrix is obtained in the same manner as for plasticity except that one has to replace in its derivation the consistency condition dφn+1 = 0 by dφn+1 − η tdγn+1 = 0. The form of the consistent tangent matrix is the same as (3.50) except that cn+1 is replaced by cn+1 + η t .

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1 Introduction
1.1 Motivation
1.2 Background
1.2.2 Failure analysis
1.3 Goals of the thesis
1.4 Outline of the thesis
2 Limit load analysis of reinforced concrete plates
2.1 Introduction
2.2 Constitutive model for reinforced concrete plates
2.2.1 Basic idea
2.2.2 Moment curvature relationship
2.3 Isotropic and anisotropic reinforcement
2.3.1 Isotropic reinforcement
2.3.2 Anisotropic reinforcement
2.4 Numerical examples
2.4.1 Rectangular simply supported plate with anisotropic reinforcement
2.4.2 Simply supported square plate with anisotropic reinforcement .
2.4.3 Circular clamped plate with isotropic reinforcement
2.4.4 Plate with two free edges
2.4.5 Square plate with point supports in the corners
2.5 Concluding remarks and chapter summary
3 Inelastic analysis of metal plates
3.1 Introduction
3.2 Inelastic plate models
3.2.1 Plate elastoplasticity
3.2.2 Plate elastoviscoplasticity
3.3 Finite element formulation
3.3.1 Space discretization
3.3.2 Computational issues for plasticity
3.3.3 Computational issues for viscoplasticity
3.4 Numerical examples
3.4.1 Limit load analysis of a rectangular plate
3.4.2 Limit load analysis of a circular plate
3.4.3 Elastoplastic analysis of a skew plate
3.4.4 Cyclic analysis of a circular plate
3.4.5 Elastoviscoplastic analysis of a circular plate
3.5 Concluding remarks and chapter summary
4 Inelastic analysis of metal shells
4.1 Introduction
4.2 Inelastic geometrically exact shell formulation
4.2.1 Geometry, kinematics and strains
4.2.2 Variational formulation
4.2.3 Stress-resultant constitutive equations for elastoplasticity
4.3 Finite element formulation
4.3.1 Space-domain discretization
4.3.2 Computational issues for plasticity
4.4 Numerical examples
4.4.1 Pinched cylinder with isotropic hardening
4.4.2 Half of a sphere
4.4.3 Limit load analysis of a rectangular plate
4.5 Concluding remarks and chapter summary
5 Illustration of embedded discontinuity concept for failure analysis
5.1 Introduction
5.2 1D finite element with embedded discontinuity
5.3 Computational procedure for failure analysis
5.4 Numerical examples
5.4.1 Tension test of a bar
5.4.2 Tension test of four parallel bars
5.5 Concluding remarks and chapter summary
6 Failure analysis of metal beams and frames
6.1 Introduction
6.2 Beam element with embedded discontinuity
6.2.1 Kinematics
6.2.2 Equilibrium equations
6.2.3 Constitutive relations
6.3 Computation of beam plasticity material parameters
6.4 Computational procedure
6.5 Examples
6.5.1 Computation of beam plasticity material parameters
6.5.2 Push-over of a symmetric frame
6.5.3 Push-over of an asymmetric frame
6.5.4 Bending of beam under constant axial force
6.5.5 Collapse of a simple frame
6.5.6 Darvall-Mendis frame
6.6 Concluding remarks and chapter summary
7 Failure analysis of 2D solids
7.1 Introduction
7.2 Family of ED elements for planar problems
7.2.1 Kinematics
7.2.2 Equilibrium equations
7.2.3 Constitutive relations
7.3 Computational procedure
7.4 Examples
7.4.1 Tension test
7.4.2 Bending test
7.4.3 Partial tension test
7.4.4 Three point bending test
7.4.5 Four point bending test
7.4.6 Delamination
7.4.7 Elasto-plastic tension test
7.5 Concluding remarks and chapter summary
8 Conclusion
Bibliography

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