Residual Stress

  • What is it

  • Unlike the applied stress, residual stress is induced during processing. There are  two kinds of  residual stresses which are main concerns.

    One  is flow induced residual stress. When in molten state, polymer molecules are unstressed, and they tend to an equilibrium, random coil state. During processing , the polymer is sheared and elongated, and the molecules are oriented  in the flow direction. If solidification occurs before the polymer  molecules are fully relaxed to their state of equilibrium, molecular orientation is locked within the molded part. This kind of stress is often referred to as flow induced residual stress.

    Another is  thermal induced residual stress which arises during the cooling stage. An important fact is polymer shrinks as it cools. During the cooling stage, the polymer cools at different rates from the mold wall to the center. When the polymer starts to cool, the external surface layers start to shrink, while the bulk of polymer at the core is still hot and free to contract. Later when the internal core cools, it's contraction is constrained by the external layers since they are already rigid.  This leads to another kind of residual stress. Usually the stress distribution is tensile in the core and compressive at the surface .

    The following figures , based on an assumed parabolic temperature distribution, can be used to illustrate how the residual stress develops in the injection molded part during cooling process.

    At time t0, cooling starts, the melt temperature is above the glass transition temperature Tg.

    At time t1, the outer layer begins to solidify without any resistance from the liquid core when its temperature decreases to the glass transition temperature Tg.

    At time t2, the second  layer begins to solidify when its temperature decreases to Tg. Since the outer  layer has already solidified,  the shrinkage of the inner layer is resisted by the solidified outer layer, thus leading to a tensile stress in the outer layer and compressive stress in the inner layer.

    At time t3, the third layer begins to solidify when its temperature decreases to Tg.  Again,the shrinkage of the inner layer is resisted by the solidified outer layers. Because its temperature decreases more than the outer layer, so its shrinkage is more than any of the outer layers, thus leading to a tensile stress at the surface and compressive stress in the core.

  • A simple model  [Tim A. Osswald,1998]

  • As is stated above , the thermal induced stress distribution is compressive at the surface and tensile in the core. This conforms to the parabolic temperature distribution in the molded part.  Assuming no residual stress build-up during phase change, a simple function based on the parabolic temperature distribution, can be used to approximate the residual stress distribution in thin sections:

    Here, Tfis the final temperature of the part, E is the modulus, a the thermal expansion coefficient , L the half thickness and Ts denotes the solidification temperature : glass transition temperature for amorphous thermoplastics or the melting temperature for semi crystalline polymers. 

    Enter the thermal expansion coefficient a
    Enter the yougth modulus E
    Enter final temperature  Tf  : 
    Enter solidification temperature Ts  : 
    Enter the half thickness  L
    Enter the position z :
    residual stress : 


    Experiments have been done to compare the function predicted stress and measured stress at the surface, where z=L. The experiment result shows that stress distribution conforms to the function. This model, though  simple, gives reasonable result.

  • Measurement of residual stresses

  • Unlike the  applied stresses, residual  stresses cannot be measured directly. On the contrary, they are calculated indirectly by measuring  the strains that exist within the  material. These strains are generally measured by mechanical or X-ray methods and corresponding stresses calculated from elastic theory formulae.

    General techniques to measure residual stresses are :

  • mechanical

  • Residual stress is in a state of self equilibrium before material  is removed by mechanical means. After a part of the material is removed, the static equilibrium is upset and stress distribution is altered. For example, for a strip with a longitudinal stress distribution, removal of the surface layer generates a moment in the cross section. To balance this moment, the strip bends to some curvature. By measuring the curvature after removing successive layers through the thickness, the initial residual stress distribution can be reconstructed.
  • X-ray

  • Note that residual stress, has effect on the  material structure. Our knowledge of the effects of residual stresses on structure can be obtained by X-ray method. For example, to measure the residual stress in metal, X-ray tells of atomic arrangements; deviations from an ideal arrangement can be interpreted as strain; the state of stress can be reconstructed from this strain.
  • Experimental data compared with the predicted data

  • Some experiments have been done to measure the through thickness residual stresses for SBPs  . To measure the through thickness residual stresses, the layer removal techniques are implemented. After one layer is removed from the bar specimen, the remaining part bends to some curvature. The curvature varies according to the removed thickness .  By removing successive layers through the thickness and measuring the corresponding curvatures at balanced status,  we can get the equation of curvature as a function of thickness z.
    Then using the equation
    we can reconstruct the state of stress.

    Residual stresses predicted by equation

    and measured stresses are shown below. As these figures indicate, the experimental data conform to the predicted data very well.

    Some parameters related to  SBP are :
    10% starch SBP:  E = 609.2MPa,          Poisson's ratio = 0.419
    30% starch SBP:  E = 837.0 MPa,         Poisson's ratio = 0.422
    50% starch SBP:  E = 1434.6 MPa,       Poisson's ratio = 0.449
    70% starch SBP:  E = 2163.0 MPa,       Poisson's ratio = 0.479

    Units used for the following figures:
        Residual stress: MPa
        Z: mm

  • An Approximate Model of Thermal Residual Stress in an Injection Molded Part



    Related paper

    Compared with the parabolic model presented above[Tim A. Osswald,1998], this is a more detailed heat transfer thermal-elastic model with phase change. In this model, the thermal problem, a phase change problem, is solved with an explicit enthalpy method. At each time step in this solution the predicted temperature profile is used in a numerical integration step of the stress equation. Results, the stress profile through the part thickness, indicate a low dependence on the ratio of latent heat to specific heat (L/Cp) and a high dependence on the Biot number.

    Efforts have been made to show how a Linear Heat Balance Integral (LHBI) temperature profile approximation and the concept of a Virtual Adjunct Mold (VAM) can be used to arrive at an approximate algebraic model that, on providing material properties and   processing conditions, gives an immediate evaluation of the residual stress through the plate thickness. Results obtained with this model are in very close agreement with a full numerical solution and agree closely with experimental measurements made on a range of starch based biodegradable polymer samples.

    The ability to relate the residual stress in an injected molded part to processing conditions and materials properties is an important step towards process optimization. Experimental measurements on the injection modeling rectangular parts are sufficient to characterize the levels of stress that might be encountered when fabricating a given polymer material under given processing conditions .

    Although many effects contribute to the residual stresses during injection molding of polymers, e.g., packing pressure, viscoelastic relaxation, non-constant material properties, degree crystallinity, etc., its formation may be adequately modeled using a one-dimensional thermo-elastic treatment that assumes constant material properties. This model requires the coupled solution of a thermal solidification problem and evaluation of the integral of stress through the part thickness. In a general setting, a numerical solution is needed, requiring a discretization in both space and time, to fully resolve the model. It has been shown that, on assuming  (1) a linear temperature profile in the solid and (2) a virtual adjunct mold (VAM) to account for convective cooling, the solution of the residual stress model can be carried out analytically to arrive at an explicit relationship for the residual stress (see Eq. (4.4) and (4.5)). This simple model allows for a direct assessment of the level of residual stress that will occur for a given polymer material under given injection molding conditions.  Across a very wide range of conditions the explicit model provides residual stress predictions that are very close to those obtained with a full numerical model. Predictions with the simple model also agree well with a range of experimental measurements made on starch based biodegradable polymers.

    Taken as a whole the results have shown that
    1. The LHBI-VAM treatment does provide an accurate solution of the one-dimensional thermal-elastic residual stress model.
    2. The formation of residual stresses is adequately modeled by a thermal-elastic treatment alone.
    The LHBI-VAM model provides an accurate description of how the residual stress in an injection molding process is effected by process conditions and  material properties.

    Key equations in the Thermal-elastic Model


    To calculate the numerical solution for the thermal-elastic model, the difficult part is to calculate the temperature distribution in the growing solid region. Here are the governing equations:

    In the approximate model, the temperature distribution in solid region is approximated by 
    plug it in (2.3), the residue stress can be approximated by






    Comparison between Model 1 (parabolic model) and Model 2 (thermal -elastic model) when Bi=4.8