Small Deformation Process

Introduction

This process simulates the mechanical behavior of solids under small deformations and is suitable for stress and deformation analyses.

It supports a variety of constitutive models to represent different deformation mechanisms, including elastic, elasto-plastic, viscoelastic, and creep behavior.

Key Features

The major features of this process are:

• Integration with MFront for defining custom constitutive models.
• B-bar method for volumetric locking mitigation.
• Principal stress computation and output.
• Material force calculations and output.

The process handles both 2D and 3D problems with various element types and can account for initial stress conditions, body forces, and complex boundary conditions.

Theoretical background

The small deformation process solves the equilibrium equations in strong form:

where:

• $\boldsymbol{\sigma}$ is the Cauchy stress tensor [M·L⁻¹·T⁻²]
• $\mathbf{b}$ is the body force vector [M·L⁻²·T⁻²].

The constitutive relationship is given by:

where:

• $\mathbf{C}$ is the fourth-order stiffness tensor [M·L⁻¹·T⁻²].
• $\boldsymbol{\varepsilon}$ is the small strain tensor [-]

The strain-displacement relationship for small deformations is:

where:

• $\mathbf{u}$ is the displacement vector [L].

Finite Element Discretization

The equilibrium equation is discretized using the finite element method, resulting in:

where the element matrices are defined as follows.

Element Stiffness Matrix

The element stiffness matrix $\mathbf{K}_e$ [M·L·T⁻²] represents the elastic energy:

where $\mathbf{B}$ is the strain-displacement matrix.

Element Load Vector

The load vector $\mathbf{f}_e$ [M·L·T⁻²] includes body forces and surface tractions:

where:

• $\mathbf{N}$ is the shape function matrix
• $\mathbf{\tau}=\boldsymbol{\sigma}\cdot\mathbf{n}$ is the surface traction vector [M·L⁻¹·T⁻²] with $\mathbf{n}$ the outer normal to the surface.

Definition in the project file

The small deformation process has to be declared in the project file in the processes block. For example in following way:

<process>
    <name>SmallDeformation</name>
    <type>SMALL_DEFORMATION</type>
    <integration_order>2</integration_order>
    <specific_body_force>0 -9.81</specific_body_force>
    <use_b_bar>true</use_b_bar>
    <constitutive_relation>
        <type>LinearElasticIsotropic</type>
        <youngs_modulus>E</youngs_modulus>
        <poissons_ratio>nu</poissons_ratio>
    </constitutive_relation>
    <process_variables>
        <process_variable>displacement</process_variable>
    </process_variables>
    <secondary_variables>
        <secondary_variable name="sigma"/>
        <secondary_variable name="epsilon"/>
    </secondary_variables>
</process>

For more detailed description of tags used in this snippet, please see Processes.

Process variables

The small deformation process requires displacement process variable. For 2D problems, the displacement variable should have 2 components ($u_x$, $u_y$). For 3D problems, the displacement variable should have 3 components ($u_x$, $u_y$, $u_z$). For more details, see Process variables.

Media

The small deformation process requires properties for the solid phase for each medium.

Solid properties

Required solid property

See solid properties for more details on defining them.

Material Models

Material constitutive relations and their parameters are specified in the process section, not in the media section. See the example in Definition in the project file above.

Features

Specific body force

The specific body force vector can be specified to account for gravity effects:

<specific_body_force>0 -9.81</specific_body_force>

B-bar method

The B-bar method can be enabled to mitigate volumetric locking in nearly incompressible materials:

<use_b_bar>true</use_b_bar>

Initial stress conditions

Initial stress conditions can be specified:

<initial_stress>parameter_name</initial_stress>

Reference temperature

Reference temperature can be specified for temperature dependent material models to simulate thermo-mechanical coupling:

<reference_temperature>parameter_name</reference_temperature>

Material Models

The small deformation process supports various constitutive models:

• LinearElastic: Linear elastic material behavior
• MohrCoulomb: Mohr-Coulomb failure criterion
• DruckerPrager: Drucker-Prager plasticity model
• CamClay: Modified Cam Clay model
• Burgers: Viscoelastic Burgers model
• Ehlers: Elastoplastic model with damage
• CreepBGR
• Lubby2
• MFront: Generic material model interface

Each model has specific parameters and capabilities. Refer to the material model documentation for detailed information.

Available benchmarks

To gain more insight into this process, you can investigate small deformation benchmarks.