Thermal Stresses and Strains

It is quite common in modem engineering designs, for plastics to be used in conjunction with other materials, particularly metals. In such cases it is wise to consider the possibility of thermal stresses being set up due to the differences in the thermal expansion (or contraction) in each material. 

The change in shape of a material when it is subjected to a change in temperature is determined by the coefficient of thermal expansion, aj- Normally for isotropic materials the value of aj will be the same in all directions. For convenience this is often taken to be the case in plastics but one always needs 

There are standard procedures for determining aj (e.g. ASTM 696) and typical values for plastics are given in Table 1.2. It may be observed that the coefficients of thermal expansion for plastics are higher than those for metals. Thus if 50 mm lengths of polypropylene and stainless steel are each heated up by 60°C the changes in length would be 

If these changes in length take place freely then we will have a thermally induced strain in the material (= 0.3 x 100/50 = 0.6% in the polypropylene) but no stress. However, if the polypropylene was constrained in some way so that the 0.3 mm expansion could not happen when it is heated by 60°C, then there would be a thermally induced stress in the material, i.e. 

If the modulus of the material is 1.2 GN/m at the final temperature, then the stress in the material would be given by 

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TABLE 6.8 Reference Values for the Different Criteria of Thermal Stress and Strain... 

Serviceability limits are considered to determine performance of the product when subjected to service loads and environments. Service conditions represent those maximum or limiting conditions that are expected in service. Examples of serviceability limits that should be considered in the design of RPs include residual deformation, buckling or wrinkling, deflection and deformation, thermal stress and strain, crazing, and weeping. 

Traditionally, the majority of SOFC modeling efforts have taken place at the macroscale and have provided valuable insight into the operation of SOFC cells and stacks [15, 36-38], In macroscale modeling, simplified models of the SOFC multi-physics are used to simulate the operating conditions and performance of the SOFC. Macroscale models are used to investigate the performance of experimental cells and stacks [35, 39], SOFC system-level operation and controls [40, 41], and thermal stresses and strains in the cells and stacks [42-44]. 

Lau, J., Thermal Stress and Strain in Microelectronics Packaging, Lau, J., Ed., Van Nostrand Reinhold, New York, 1993. 

F. H. Murray, Thermal stresses and strains in a finite cylinder with no surface forces, USAEC, AECD-2966, University of Chicago, 1945. 

Thermal expansion induced by insolation may be important in desert areas where rocky outcrops and soil surfaces are barren. In a desert, daily temperature excursions are wide and rocks are heated and cooled rapidly. Each type of mineral in a rock has a different coefficient of thermal expansion. Consequently, when a rock is heated or cooled, its minerals differentially expand and contract, thereby inducing stresses and strains in the rock and causing fractures. Ollier (1969) discussed examples of rock weathering due to insolation. Fire can develop temperatures far in excess of insolation and be quite effective in fracturing rocks (Black-welder, 1927). 

In summary, polarizing microscopy provides a vast amount of information about the composition and three-dimensional structure of a variety of samples. The technique can reveal information about thermal history and the stresses and strains to which a specimen was subjected during formation. Polarizing microscopy is a relatively inexpensive and well accessible investigative and quality control tool. 

The stress submodel results in the stresses and strains in the composite. Stresses and strains are introduced by fiber tension, thermal expansion or contraction, and chemical changes. The effects of fiber tension, temperature and chemical changes may arise simultaneously however, the stresses and strains are analyzed separately. The sum of the stresses and strains caused by each of these factors is the actual stress and strain in the composite. 

A close connection exists between the presence of a flexible polymer skeleton and the flexibility of the bulk material. Macromolecular flexibility is often defined in terms of the glass-transition temperature, Tg. Below this temperature, the polymer is a glass, and the backbone bonds have insufficient thermal energy to undergo significant torsional motions. As the temperature is raised above the Y g, an onset of torsional motion occurs, such that individual molecules can now twist and yield to stress and strain. In this state the polymer is a quasi-liquid (an elastomer) unless the bulk material is stiffened by microcrystalfite formation. Thus, a polymer with a high Tt is believed to have a backbone that offers more resistance to bond torsion than a polymer with a low 7 g. 

Transport phenomena modeling. This type of modeling is applicable when the process is well understood and quantification is possible using physical laws such as the heat, momentum, or diffusion transport equations or others. These cases can be analyzed with principles of transport phenomena and the laws governing the physicochemical changes of matter. Transport phenomena models apply to many cases of heat conduction or mass diffusion or to the flow of fluids under laminar flow conditions. Equivalent principles can be used for other problems, such as the mathematical theory of elasticity for the analysis of mechanical, thermal, or pressure stress and strain in beams, plates, or solids. 

Fig. 7. Comparison of stress-strain curves for carbon-fiber-reinforced epoxy composites of different thermal histories. Error rectangles were drawn to indicate a 95% confidence level for both stress and strain...

S]). The direct piezoelectric effect is the production of electric displacement by the application of a mechanical stress the converse piezoelectric effect results in the production of a strain when an electric field is applied to a piezoelectric crystal. The relation between stress and strain, expressed by Equation 2.7, is indicated by the term Elasticity. Numbers in square brackets show the ranks of the crystal property tensors the piezoelectric coefficients are 3rd-rank tensors, and the elastic stiffnesses are 4th-rank tensors. Numbers in parentheses identify Ist-rank tensors (vectors, such as electric field and electric displacement), and 2nd-rai tensors (stress and strain). Note that one could expand this representation to include thermal variables (see [5]) and magnetic variables. 

The design of pieces that form part of either a structure or a machine often requires an analysis of the distribution of stresses and strains in these pieces. Without taking into account, for the moment, the thermal and calorific effects, the tensions and deformations at each point of the sample should simultaneously satisfy the balance and constitutive equations. 

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