Introduction

A thin layer of residual or initial stresses/strains may develop just below the outer surface of a metal part due to the tool used in the machining process (typically referred to as machining induced residual stress/strain) or due to surface treatments (i.e. shot peening). These subsurface residual stresses/strains (SRS) can cause undesired distortion for large parts with thin walls. StressCheck provides a convenient module for predicting the resulting distortion from minimal information about the SRS distribution (i.e., magnitude and depth) obtained with simple experiments or from a complex profile obtained either by simulation or detailed experimental information. Experimental measurements of residual stress/strain may involve X-ray diffraction, for example, or the fitting of machined coupon displacements to shape functions¹. SRS loads are currently only compatible with the linear solver.

Subsurface residual or initial stress/strain profiles can be represented in StressCheck with three pre-defined profile options (step, linear, and polynomial functions) or with formulae for general profiles. Any number of these profiles can be assigned directly to the finite elements or to solid geometry imported from a CAD model. When assigning to solid geometry, StressCheck automatically determines which underlying elements must be associated with an SRS profile assignment. Initial/residual stresses or strains (also referred to as eigenstrains²) can be represented in a body following the theory of elasticity³. If residual/initial stress profiles are entered they are converted to initial strains⁴ in StressCheck during load vector computation:

1. J. B. Castle, S. Nervi, R. B. Hancock and J. Bolin, “Stress function calibration method”. US Patent 8645086, 4 February 2014.
2. S. Timosheno and J. N. Goodier, Theory of Elasticity, 2nd ed., New York: McGraw-Hill Book Company, 1951.
3. T. Mura, Micromechanics of Defects in Solids, 2nd ed., Dordrecht, The Netherlands: Kluwer Academic Publishers, 1987.
4. All possible sources for initial strains are being considered by casting thermal strains and initial stresses as initial strains.

Profile Definitions

Several options are available in StressCheck for applying SRS profiles (Figure 2). Profiles can be defined in the form of initial stress or initial strains. If initial stresses are used they will be converted to initial strains as shown previously. Profiles are defined by selecting “SRS Profiles” from the StressCheck Edit menu. This opens the dialog shown in the below figure. Each profile is defined by a unique ID, a profile type (step, linear, polynomial, or formula), a specification that the profile represents stress or strain, a maximum profile depth (Dmax), and a set of coefficients which are dependent on the profile type. Any number of profiles may be defined. The profile chosen for a particular load assignment is selected at assignment time.

SRS Profile Types

As shown in Figure 3 three pre-defined SRS profile types (step, linear, and polynomial) are available for input in StressCheck and have been optimized for computational efficiency. Additionally, the user may enter arbitrary SRS profiles by using the formula option. For each SRS profile, a maximum subsurface depth (Dmax) must be specified along with several coefficients that define the initial stress or strain magnitudes; the maximum depth is defined as the depth where the profile becomes zero. Depth into the part is represented by the local z-axis so that z = 0 is on the outer part surface(s). For the formula profile type, StressCheck formulae must be selected for each tensor component. Any formula may be defined, but it must be defined as a function of the space variable “z” only: For any profile type, the depth and any coefficient can be specified with a parameter.

For example, the below shows a polynomial profile where several of the coefficients are defined using parameters (SXX1, SXX2, etc.) and several are left to be zero (Figure 4):

Assignment Options

SRS profiles may be assigned to element sets or geometric surfaces. Any combination of assignments can be made. In the Load tab of the Input dialog, the following options are available:

• Object: All Elements > Method: SRS System
• Object: Any Element > Method: SRS System
• Object: Any Surface > Method: SRS System (Figure 5)
• Object: Any Surface > Method: SRS Curve

Application of SRS Profiles to All Elements or Any Element

An SRS profile may be assigned to all elements or a user-defined set of elements in flat regions only. This is useful for finite element meshes without associated underlying geometry or if the user would like to explicitly define which elements to assign a SRS profile. Because CAD geometry is not used for this assignment option, a system must be used as the coordinate reference for specifying the profile orientation. This system must be placed so that the x- and y-axes are flush with the flat wetted face surface and the z-axis points into the component.

Application of SRS Profiles to Any Surface

In the context of SRS applications, any surface refers to the bounding surfaces of a solid body. The outward normal to a surface must point away from the body. For flat or general curved regions, SRS profiles may be assigned to one or more geometric surfaces. Each assignment may contain any number of surfaces. If the surface is flat, a reference system or reference curves may be chosen for specifying the profile orientation. If the surface is curved, one or more curves must be chosen for the reference to ensure proper orientation of the SRS profile. Typically, a bounding curve of the assigned surface will be chosen for the reference, though this is not necessary. In all cases, only the x direction of the system or the tangent direction of the curve is used for orienting the profile; depth is calculated from the surface normal.

In an assignment with multiple surfaces, each surface is treated separately so that the resulting SRS distribution is the summation of the contributions of profiles from all assigned surfaces. A single assignment with multiple surfaces produces the same SRS distribution as multiple assignments with single surfaces, assuming the reference objects are the same. Additionally, more than one SRS assignment can be made to a surface, and the resulting distribution is the summation of each assignment.

StressCheck automatically determines which elements to load with the SRS profile based on node proximity; if any node of an element lies on an assigned surface then the SRS profile is applied to that element regardless of associativity. It may be the case than an element partially lies under a surface with an SRS assignment and partially extends past the surface.

If an element has a node that lies on an assigned surface (so that an SRS profile is applied to it) but partially extends past the surface boundary, the SRS profile is only applied to the region of the element which lies directly under the surface. The region of the element that extends past the surface may then have no SRS profile assigned, or some other SRS profile from another surface. If an element has only one node which lies on an assigned surface and none of the element lies directly under the surface, then the SRS profile is not applied to the element.

Reference Objects

For all assignment cases, a reference is needed to determine the proper orientation of the SRS profile. This reference can be given by a local system, or for the case of an existing underlying geometry (in particular curved surfaces), a curve object. Computationally, it is more efficient to assign the SRS profile using a system than an adjacent curve as the reference, therefore the former is the recommended option for flat regions.

Reference Cartesian System

When specifying a local Cartesian system as the SRS reference, the given x and y stress/strain components are aligned with the x and y directions of the local system. The shear term xy is then applied with the correct orientation according to the below figure.

For an assignment to All Elements or Any Element, the positive z-direction of the local system provides a reference for SRS profile depth, therefore it is essential that the reference system lie on the wetted face surface which corresponds to the selected elements. For example, if it is desired to apply an SRS profile to the top surface of the flat plate shown below, then a system should be created such that the x- and y-axes of that system are flush with the wetted face surface and the z axis points into the plate (Figure 6):

For an assignment to Any Surface, the location of the system does not matter. Only the x-axis is used for orientation so that the profile x-axis points in the same general direction as the system x-axis. For example, the SRS distribution given by selecting the top surface and reference system in Figure 7 will be exactly the same as that given by selecting an element set and flush system as shown above.

The inward normal direction of an assigned surface gives the local z-axis at each point along the surface, corresponding to profile depth. Suppose the local z-axis is given by u_z and the system x-axis is given by ū_x (which in general may not be tangent to the assigned surface). Two cross-products are required to generate an orthogonal local system where both the x- and y-directions are tangent to the surface:

u_y = u_z x ū_x

u_x = u_y x u_z

The vector u_z remains unchanged as it is the inverse surface normal. The SRS profile is evaluated using this locally defined system, unique to each integration point, in which the z-direction is the inverse normal to the surface and the x-direction points in the same general direction as the selected reference system. The other system axes are not used. Using this technique the system x-axis does not need to lie on the assigned surface or even be parallel to the assigned surface.

Reference Curve

When specifying a curve object as the SRS reference, the local x-direction of the profile across the assigned surface follows the direction of the tangent vector of the curve and the local z-direction is the inverse surface normal. The shear term xy is then applied with the correct orientation according to the above convention. The reference curve is typically a bounding curve of the assigned surface, but does not need to be a bounding curve. If the selected curve in is chosen as a reference for the outer surface, then the local coordinate system will vary as shown in Figure 8. One assignment can include multiple surfaces and multiple reference curves. For every point across each assigned surface, the point is projected to every curve in the assignment and the tangent vector closest to the point out of all reference curves is selected for the tangent vector ū_x. The local profile orientation is then computed using the same technique as explained previously, EQ 60, for a reference system used with surface assignment. The vector u_z remains as the inverse surface normal. The SRS profile is evaluated using this locally defined system, unique to each integration point, in which the z-direction is the inverse normal to the surface and the x-direction points in the same general direction as the closest point out of all selected reference curves.

Note that including multiple curves in one assignment can significantly increase the computational time of the finite element solution, but allows for maximum flexibility in assignment options.

Assignment Examples

Flat Element Set Example

• Create a local reference system so that its origin lays on the surface of interest and the z axis points into the part (negative normal with respect to the reference surface). There are many methods by which a system may be created, but the simplest creation method for this type of problem is “3-Pt. Plane”.
  • In the Geometry tab of the StressCheck Input dialog, choose Create > System > 3-Pt. Plane and select three points on the corners of the surface of interest. The created system will be contained on a plane defined by the three selected points.
  • If there is no underlying geometry (for example in the case of an imported mesh), the location and orientation of the system can be entered manually using Create > System > Locate.
  • Alternatively, points for the “3-Pt. Plane” option can be created using the Locate method. Locations can be determined as follows: On the Mesh tab, Select > Node > Selection and click on a node. The global coordinates of the node will be displayed in the input fields of the Mesh tab.
• On the Loads tab of the StressCheck Input dialog, choose Select > Any Element > SRS System.
• Enter a load ID and ensure that Set is equal to “New set”. Alternatively, if your desired element set is already defined, pick it in the Set dropdown. Note that all loads contributing to a single solution must have the same ID.
• In the Model View, select the desired elements either by marquee-selection or holding Shift and clicking on each element. Note: ensure the Wetted Faces icon is disabled and Select Through icons is enabled for marquee-selections.
  • There is no solution quality penalty for selecting too many elements, so to be safe you may choose the entire region of elements below the SRS surface.
• Use the System dropdown to select the reference system that you created in step 1. It may be helpful to turn on the display of system labels by opening the Display Objects dialog (click on the icon) and checking the box immediately next to Systems.
• Select an SRS profile from the list of available profiles. If there are no profiles defined, press the Edit Profiles button to open the SRS Profile Definition dialog and define any profile.
• Hit the Accept button to finish creating the SRS assignment.

Flat Surface Assignment Example

• Create a local reference system so that its x-axis is roughly tangent to the surface of interest. The system does not need to lie on the surface. There are many methods by which a system may be created, but the simplest creation method for this type of problem is “3-Pt. Plane”. In the Geometry tab of the StressCheck Input dialog, choose Create > System > 3-Pt. Plane and select three points on the corners of the surface of interest. The created system will lay on a plane defined by the three selected points.
• On the Loads tab of the StressCheck Input dialog, choose Select > Any Surface > SRS System.
• Enter a load ID and ensure that Set is equal to “New set”. All loads contributing to a single solution must have the same ID.
• Select the desired surface by clicking on it. It should highlight in red. Multiple surfaces may be selected by holding Shift.
• Select the coordinate system from the pull-down menu in the Load Interface. Alternatively, use the pick arrow next to the System combo to select the system from the Model View.
• Select an SRS profile from the list of available profiles. If there are no profiles defined, press the Edit Profiles button to open the SRS Profile Definition dialog and define any profile.
• Hit the Accept button to finish creating the SRS assignment.

Curved Surface Assignment Example

• Ensure that the surface normal points out of the part. To verify the direction of the normal to a surface, on the Geometry tab of the StressCheck Input dialog, choose Check > Any Surface > Offset and hover the cursor over the surface.
• On the Loads tab of the StressCheck Input dialog, choose Select > Any Surface > SRS Curve.
• Enter a load ID and ensure that Set is equal to “New set”. All loads contributing to a single solution must have the same ID.
• Select the desired surface by clicking on it. It should highlight in red. Multiple surfaces may be selected by holding Shift. For example, the below shows a curved surface highlighted in red (Figure 9):

• Hold Ctrl+Shift and select any number of reference curves that will define the local x-directions for the selected surfaces. As the cursor hovers over the reference curve, a small green arrow will appear indicating the positive direction of the curve.
• The SRS profile x-axis can either be aligned with or opposite to the positive direction of the reference curve. If the profile x-axis points in the positive green arrow direction, select Curve Dir to be Positive. If the profile x-axis points in the opposite direction, select Negative. Note that this is only a concern for profiles with yz or zx shear components.
• Select an SRS profile from the list of available profiles. If there are no profiles defined, press the Edit Profiles button to open the SRS Profile Definition dialog and define any profile.
• Hit the Accept button to finish creating the SRS assignment.

Meshing Guidelines

Standard StressCheck meshing guidelines apply to problems involving SRS assignments. Refer to the Meshing Overview for general tips. In addition, providing a boundary layer of elements below each surface associated with an SRS profile will increase the quality of the approximation. Boundary layer thickness should be equal to SRS profile depth for maximum effectiveness. If using an automesh, boundary layers can be assigned to geometric surfaces by choosing Create > Mesh > Bndry. Layer on the Mesh tab of the StressCheck Input dialog. When creating one layer of elements, enter the desired thickness in the fields “To” and “T-Total”. Be sure to check for element distortion with Check > Mesh > Distortion. If the number of distorted elements is not acceptable (this depends on the particular problem), change global mesh parameters to increase the number of elements. Note that, depending on the number of elements, an automeshed boundary layer may not follow curved surfaces as well as a handmesh.

For an example of SRS assignment for predicting the distortion of a 3D solid, refer to StressCheck Demo: 3D Subsurface Residual Stress (SRS) Distortion Analysis.