Linear Elastic with Plastic Hardening |
 
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Linear Elastic with Plastic Hardening |
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The mechanical properties used in an elastic-plastic loading scenario are the elasticity constants Young’s modulus (E) and Poisson’s ratio (ν), and the stress-strain curve (obtained from a tensile test). The latter is used to determine the yield point and the nonlinear isotropic hardening of the material. This information can be also expressed in the form of a stress-plastic strain curve, which starts at the yield point and terminates at the point of fracture. The plastic strain (εpl) can be expressed as:
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(1) |
where σ is stress and ε is total strain.
The constitutive equations governing the 3D beam element are expressed in terms of forces and moments (generalised stresses) and their conjugates strains and curvatures (generalised strains), respectively. This requires that any nonlinear behaviour in bending, axial or torsion be explicitly expressed either independently of each other (Nonlinear Flexible Riser Format) or in some form of dependence (Nonlinear Rigid Riser Format). Elastic deformations can be superposed as they are reversible, but plastic deformations cannot. Plastic deformation is path dependent, which means that the order in which loads are applied to the body is important and will lead to a different yield state.
Flexcom uses a hybrid beam element, where constitutive equations are specified in terms of forces and moments and not stresses. There is no established measure of yielding expressed in terms of forces and moments. The J2 flow theory can be applied selectively to each mode of deformation to control the yielding independently. In this situation, the balancing of the beam capacity has to be performed explicitly. In other words, the bending, axial and torsion loading capacity of the beam must always sum to the same overall value. For example, should the beam becom overstretched, then it cannot support the same bending moment as if it was not loaded at all. Adjusting the capacity is done automatically by the program, which assumes that torsion loads do not produce any plastic deformations. This is reasonable given the nature of the engineering problem, where bending and tensile loads would typically produce plastic deformations. It is considered that the bending moment capacity is dependent on the amount of resultant axial force in the cross-section, and which is calculated by integrating the stress-strain curve over the cross-section. The bending capacity is adjusted at least once at the beginning of an analysis for all beam elements in the model. These adjustments can be performed dynamically at the run-time if the tension changes by more than a specified amount.
Plastic hardening is implemented for the bending and axial responses, while the torsional response is considered to be linear elastic. The shear modulus (G) is calculated from:
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(2) |
The effects of internal and external pressure are included after the axial or bending deformation has taken place. Note that Flexcom does not readjust bending moment capacity as a result of the applied pressure.
If you would like to see an example of the plastic hardening material model in use please refer to Standard Example H05 - Steel Pipe Installation with Plastic Deformation.
•*PLASTIC HARDENING is used to define hardening models for plastic materials.
