This paper deals with the numerical implementation of the exponential Drucker-Parger plasticity model in the commercial finite element software, ABAQUS, via user subroutine UMAT for adhesive joint simulations. The influence of hydrostatic pressure on adhesive strength was investigated by a modified Arcan fixture designed particularly to induce a different state of hydrostatic pressure within an adhesive layer. The developed user subroutine UMAT, which utilizes an associated plastic flow during a plastic deformation, can provide a good agreement between the simulations and the experimental data. Better numerical stability at highly positive hydrostatic pressure loads for a very high order of exponential function can also be achieved compared to when a non-associated flow is used.

Structural adhesives have been widely used in the aerospace, aviation, shipbuilding, and automotive industries due to a variety of benefits, including multi-material bonding, good load distribution across bonded areas, less environmental degradation, and a simple manufacturing process. It is also relatively inexpensive when compared to other joining methods [

This article presents the numerical implementation of the exponential Drucker-Parger plasticity model in the commercial finite element software ABAQUS through the user subroutine UMAT. The development of subroutines was restricted to elastoplastic behavior, and for yield surface modification, only hydrostatic pressure was considered. The associated flow during plastic deformation was expected to increase the numerical stability for adhesive joint simulations. The experimental findings from specialized testing using the modified Arcan fixture [

From the classical plasticity theory, the total strain increment for an elastoplastic model can be decomposed into elastic and plastic strain increments. For 3D problems, they are represented by 2^{nd} tensor forms as follows:

Based on Hooke’s law, the incrementation of stress-strain relationship in the elastic deformation is written as follows:

A general form of flow rules for plastic behavior is written as:

The associated flow is achieved when the plastic potential function is the same as the yield function (

The flow rules in commercial finite element software are commonly non-associated flows since they can be used for a variety of purposes. In this case, an implicit return mapping algorithm (backward Euler method) is used for high precision calculations into the program [

The hyperbolic function is the only available option for the plastic potential function in ABAQUS when using a general exponent yield criterion. The 2^{nd} order exponential Drucker-Prager yield criteria (parabolic function) is recommended even though the order of exponential Drucker-Prager yield criteria can be arbitrary [

The numerical integration method is required in order to solve the elastoplastic differential equations. The common methods used in computational processes are the forward Euler method, the backward Euler method, and the midpoint method. While the forward Euler method is the simplest computational process, it has a limit on computational stability and high error responses. The backward Euler method is more complex than the forward Euler method but highly accurate for the increase of stress that affects the yield surface expansion. The method is also very stable in terms of computation [

With

The residual vector is then defined as follows:

According to the associated flow rule, the plastic strain increment can be defined as follows:

Then, applying the Newton-Raphson method in the residual vector (

Thus

Substituting

The exponential Drucker-Prager models are expressed as follows:

_{eff} is von Mises stress, _{t} is hardening constant,

The von Mises stress and hydrostatic pressure are defined as

The plastic shear stress equivalent plastic strain relationship in the hardening rule is defined as follows:

The Newton-Raphson method is then applied to yielding function in

Defining

By rearranging

The structural adhesive used in this study is an epoxy-based adhesive called SikaPower-497, manufactured by Sika. The mechanical properties of adhesive were identified from a modified-Arcan test using a universal testing machine (Instron 5567). The specimen deformation was followed by an image correlation system (ARAMIS, manufactured by GOM) as shown in

The modified Arcan fixture was designed to induce a different state of stress within an adhesive layer by varying its angle. A direction of 0° and 90° represents a triaxial and pure shear mode. A direction of 30° and 60° induces a mixed state of stress, which is a tensile-shear mode in an adhesive joint. A direction of 120° represents a compressive-shear mode. The experimental yield and failure surfaces are plotted on the von Mises stress-hydrostatic pressure axis as shown in

Description | Value |
---|---|

Young’s modulus (MPa) [ |
2120 |

Poisson’s ratio [ |
0.36 |

_{t} (MPa) |
|||
---|---|---|---|

Yield surface ( |
7.69 × 10^{−6} |
4 | 17.83 |

The parameters of the nonlinear hardening equation (

_{y} (MPa) |
||||
---|---|---|---|---|

Hardening ( |
29.6 | 9.2 | 19.5 | 62.8 |

The exponential Drucker-Prager model with an associated flow rule was implemented in ABAQUS software via a user subroutine, UMAT. With this developed model, the high-order exponential Drucker-Prager is also possible. It also provides higher simulation stability at high-order exponential and requires fewer model parameters since the yield and plastic potential function are the same (associated flow). In subroutine UMAT, elastic and plastic behavior have to be established from input data (material properties). For elastic behavior, it is the same as in ABAQUS. Concerning plastic behavior, an iterative calculation approach is needed to find the solution of a stress tensor increment for plastic behavior (_{eq}^{pl}) to the program for the next increment calculation. The flow chart of user subroutine UMAT can be summarized in

The finite element simulations of the modified Arcan test have been carried out as a validation of the implemented Drucker-Prager model. The four cases were performed: (1) an ABAQUS material model with 2^{nd} order exponential Drucker-Prager yield function and non-associated flow rule (^{nd} order exponential Drucker-Prager yield function and associated flow rule; (3) an ABAQUS material model with 4^{th} order exponential Drucker-Prager yield function and non-associated flow rule (^{th} order exponential Drucker-Prager yield function and associated flow rule. The 4^{th} order in the cases was selected because this high order fits relatively well the experimental data of the initial yield surface (

_{t} (MPa) |
|||
---|---|---|---|

Yield surface order 2 | 1.3 × 10^{−2} |
2 | 19.5 |

Yield surface order 4 | 7.69 × 10^{−6} |
4 | 17.83 |

_{y} (MPa) |
|||
---|---|---|---|

Flow potential ( |
29.6 | 18 | 14.6 |

The modified Arcan fixture and adherends are made of steel (Young’s modulus = 200 GPa, Poisson’s ratio = 0.3 and yield stress = 570 MPa). The detailed geometries and boundary conditions are presented in ^{2}. Because the modified-Arcan fixture is symmetrical in the x-y plane, the finite element model was only constructed in half, and the fixed boundary condition in the z-direction (U3 = 0) across the entire surface is applied. The tie condition is used for the constraint between the fixture and the adhesive layer. The model has 97,860 elements in total, with 350 elements for the adhesive layer. The element volume ranges from 0.3 to 15.5 mm^{3} and is kept constant at 0.3 mm^{3} for the adhesive layer. All fixture components, including an adhesive layer, were assigned a 3D 8-node solid element (C3D8) as shown in

The results of stress field on a modified Arcan fixture loaded in directions of 0°, 60°, 90°, and 120° are shown in

The comparison between experiments and simulations is shown in ^{nd} order exponential Drucker-Prager model.

From the simulations, the ABAQUS material models with 2^{nd} and 4^{th} order exponential Drucker-Prager yield function and non-associated flow rule give almost identical results as the developed UMAT with 2^{nd} and 4^{th} order exponential Drucker-Prager yield function and associated flow rule. For further investigation, higher order exponential model simulations were performed for 120° loading. At the 9th order of exponential function (

The finite element implementation of the exponential Drucker-Prager model for the elastoplastic material has been carried out and validated experimentally with epoxy-based adhesive joints. The numerical implementation of the ABAQUS software was accomplished using a user subroutine, UMAT. This subroutine defines the plastic behavior of the adhesive joint using a yield and plastic potential function. With an associated flow, these functions are identical, providing the advantages of fewer plastic flow parameters and better numerical stability at very high orders of exponential function compared to when a non-associated flow is used. The developed UMAT is also capable of simulating high order exponential function of the Drucker-Prager model which makes it more flexible to apply to other elastoplastic materials.

For the adhesives used in this study (SikaPower-497), its hydrostatic pressure dependent yield surface and plastic behavior have been identified using the modified Arcan tests. The experimental results from different directions of the modified Arcan fixture are used as a validation for the developed UMAT. The simulation result from the 2^{nd} order exponential Drucker-Prager model shows a good agreement with experiments. For the 4^{th} order exponential Drucker-Prager model, it predicts an initial yield surface better than the 2^{nd} order model but poorly interprets the plastic deformation of the adhesive. Concerning the results at direction 120°, neither the ABAQUS material model nor the developed user subroutine UMAT can reproduce the experimental results. This divergence is also expected since the experimental results at direction 120° show the mix of cohesive and adhesive failure modes. In this direction, the compressive stress induced by the fixture strengthens the adhesive joint, so its strength is close to the interface strength leading to the mixed failure mode. The interface element option in ABAQUS [

Total strain increment tensor

^{el}

Elastic strain increment tensor

^{pl}

Plastic strain increment tensor

Stress increment tensor

Constitutive tensor of isotropic liner elastic materials

Plastic multiplier

Plastic potential function

Plastic strain rate direction

Plastic potential function

Eccentricity of hyperbolic function defined an approaching rate of function to its asymptotes

Dilation angle measured in the _{eff} plane

_{eff}

Equivalent von Mises stress

Hydrostatic pressure

_{t}

Hardening constant

_{y}

Yield stress

Material parameters that are independent to plastic deformation

_{11},

_{22},

_{33}

Normal stress components

_{12},

_{13},

_{23}

Shear stress components

_{t}

^{0}

Hardening constant at yield

_{pl}

^{eq}

Equivalent plastic strain

_{pt}

Plastic shear hardening

Hardening coefficient

Material constants of hardening