Thermal Stress And Thermal Strain

Edited By Vishal kumar | Updated on Jul 02, 2025 06:37 PM IST

Thermal stress and thermal strain are crucial concepts in material science, referring to the deformation and internal forces that arise within a material when it experiences changes in temperature. When an object is heated or cooled, its dimensions change, but if these dimensional changes are constrained, internal forces develop, leading to thermal stress. This phenomenon is common in everyday life, such as when bridges expand and contract with temperature changes, leading to potential cracks or structural fatigue. Similarly, in electronic devices, thermal stress can cause components to fail if not properly managed. Understanding these effects is essential for designing materials and structures that can withstand temperature fluctuations without compromising their integrity.

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Thermal Stress and Thermal Strain
Solved Examples Based on Thermal Stress and Thermal Strain
Summary

Thermal Stress and Thermal Strain

Thermal stress and thermal strain are essential phenomena that occur when materials undergo temperature changes, leading to expansion or contraction. When a material is heated, it tends to expand, and when cooled, it contracts. However, if the material is restrained or the temperature change is uneven, this can result in internal forces known as thermal stress. Thermal strain, on the other hand, refers to the deformation or change in dimensions of the material due to this stress. These concepts are vital in engineering and construction, as they help explain why bridges, railways, and buildings must be designed to accommodate temperature variations.

Thermal Stress in a Rod Which is Rigidly Fixed

When a rod is rigidly fixed at ends such as to prevent expansion or contraction when its temperature is increased or decreased. Due to preventing its thermal expansion or contraction, compressive or tensile stress is developed in it. As the rod tries to expand or contract, then it applies a reaction force on the rigid support. If the change in temperature of a rod of length L is $\Delta \theta$ then

Thermal strain $=\frac{\Delta L}{L}=\alpha \Delta \theta \quad\left[\right.$ As $\left.\alpha=\frac{\Delta L}{L} \times \frac{1}{\Delta \theta}\right]$

If we know the strain then with the help of Hooke's law, we can find the stress also. If we know the stress, then we can find the force by multiplying the cross-sectional area with stress. Both stress and force can be written as -

So, Thermal stress $=Y \alpha \Delta \theta$
or, Force on the supports $F=Y A \alpha \Delta \theta$

Solved Examples Based on Thermal Stress and Thermal Strain

Example 1: A metal rod of young modulus Y and coefficient of thermal expansion $\alpha$ is held at its two ends such that its length remains invariant. If its temperature is raised by $t^{\circ} \mathrm{C}$ the linear stress developed in it is

1) $\frac{Y}{\alpha t}$
2) $Y \alpha t$
3) $\frac{1}{Y \alpha t}$
4) $\frac{\alpha t}{Y}$

Solution:

Thermal Stress

When a rod whose ends are rigidly fixed such as to prevent expansion or contraction.

wherein

$\operatorname{strain}=\frac{\Delta l}{l}=\alpha \Delta T$

Hence, the answer is the option (2).

Example 3:

A metal rod of young modulus Y and coefficient of thermal expansion $\alpha$ is held at its two ends such that its length remains invariant. If its temperature is raised by $t^0 \mathrm{C}$ the linear stress developed in it is

1) $\frac{Y}{\alpha t}$
2) $Y \alpha \DeltaT$
3) $\frac{1}{Y \alpha t}$
4) $\frac{\alpha t}{Y}$

Solution:
$
\text { Strain }=\frac{\Delta l}{l}=\alpha \Delta T
$
And Thermal stress $=\mathrm{Y} .($ Strain $)=Y \alpha \Delta T$

Hence, the answer is the option (2).

Example 4: A wooden wheel of radius R is made of two semicircular parts (see figure); The two parts are held together by a ring made of a metal strip of cross-sectional area S and length L. L is slightly less than $2 \pi R$ To fit the ring on the wheel, it is heated so that its temperature rises by $\Delta T$ and it just steps over the wheel. As it cools down to the surrounding temperature, it presses the semicircular parts together. If the coefficient of linear expansion of the metal is $\alpha$ and its Youngs' modulus is Y, the force that one part of the wheel applies on the other part is :

1) $2 \pi S Y \alpha \Delta T$
2) $S Y \alpha \Delta T$
3) $\pi S Y \alpha \Delta T$
4) $2 S Y \alpha \Delta T$

Solution:

Increase in length $\Delta L=L \alpha \Delta T$

$
\therefore \quad \frac{\Delta L}{L}=\alpha \Delta T
$

The thermal stress developed is

$\begin{aligned} & \frac{T}{S}=Y \frac{\Delta L}{L}=Y \alpha \Delta T \\ & \text { or } T=S Y \alpha \Delta T\end{aligned}$

From FBD of one part of the wheel

or F = 2T

Where F is the force that one part of the wheel applies on the other part.

$F=2 S Y \alpha \Delta T$

Hence, the answer is the option (4).

Example 5: A force of 20N is applied at one end of a wire of length 2m and has an area of cross-section $10^{-3} \mathrm{~m}^2$. The other end of the wire is rigidly fixed. Its coefficient of linear expansion of the wire is $\alpha=5 \times 10^{-7} /{ }^{\circ} \mathrm{C}$ and Young's modulus $y=2.2 \times 10^9 \mathrm{~m} / \mathrm{m}^2$ and if its temperature is increased by $5^{\circ} \mathrm{C}$ then the increase in the tension of the wire will be.

1) 4.2 N
2) 4.4 N
3) 2.4 N
4) 8.8 N

Solution:

$\begin{aligned} & \text { Increase in the tension of the wire }=Y A \alpha \Delta \theta \\ & \qquad \begin{aligned} & =2.2 \times 10^{11} \times 10^{-2} \times 8 \times 10^{-6} \times 5 \\ & =8.8 \mathrm{~N}\end{aligned}\end{aligned}$

Hence, the answer is the option (4).

Summary

Whenever a material is subjected to a rise or fall of temperature, in which it expands or contracts but is constrained in such a way that the material cannot move freely then it undergoes thermal stress. This constraint has the effect of causing internal forces, which, in turn again, produce the stress within a material. Thermal strain is like a process of deformation or change in shape in the material due to the above-mentioned stress. Engineering and construction have both thermal stress and strain as their concern because they can result in material fatigue, cracking, or failure. For example, it is necessary to design bridges, buildings, and pipes with regard to thermal expansion and contraction to avoid damage. Understanding these phenomena aids engineers in constructing safer and more solid structures that might confront these temperatures variably.

Frequently Asked Questions (FAQs)

1. How does thermal stress affect bridges and other large structures?

Thermal stress significantly affects bridges and large structures as they expand in hot weather and contract in cold weather. Engineers must design expansion joints and use materials with appropriate thermal properties to prevent damage from these temperature-induced stresses.

2. How do thermal stresses affect the design of turbine blades in jet engines?

Thermal stresses significantly impact turbine blade design in jet engines. The blades experience extreme temperature gradients and rapid thermal cycling during operation. Engineers use advanced materials like single-crystal superalloys, implement cooling systems, and design blades with specific geometries to manage thermal stresses. They must balance the need for high-temperature strength with resistance to thermal fatigue and creep.

3. What is thermal stress?

Thermal stress is the internal force that develops within a material when it experiences a change in temperature. This stress occurs because different parts of the material may expand or contract at different rates, leading to internal tension or compression.

4. How does thermal strain differ from thermal stress?

Thermal strain is the actual deformation or change in dimensions of a material due to temperature changes, while thermal stress is the internal force that develops as a result of this deformation. Strain is a measure of the physical change, while stress is the force causing or resisting that change.

5. Why do materials expand when heated?

Materials expand when heated because the increase in temperature causes their atoms to vibrate more vigorously. This increased vibration leads to greater average separation between atoms, resulting in an overall increase in the material's dimensions.

6. Can thermal stress cause a material to break?

Yes, thermal stress can cause a material to break if it exceeds the material's strength. This is especially true for brittle materials or when there are rapid temperature changes, as the internal stresses can build up faster than the material can accommodate, leading to cracks or fractures.

7. What is the coefficient of thermal expansion?

The coefficient of thermal expansion is a measure of how much a material's size changes with temperature. It quantifies the fractional change in size per degree of temperature change. Materials with higher coefficients expand more for a given temperature increase.

8. Why do some materials become more ductile at higher temperatures?

Materials often become more ductile at higher temperatures because the increased thermal energy allows atoms to move more easily. This increased atomic mobility facilitates the movement of dislocations (defects in the crystal structure), making it easier for the material to deform plastically rather than fracture when stressed.

9. Why do some materials have a negative coefficient of thermal expansion?

Some materials, like certain ceramics or alloys, have a negative coefficient of thermal expansion, meaning they contract when heated. This unusual behavior is often due to their unique atomic structure or composition, which causes them to become more compact as temperature increases.

10. What is thermal shock?

Thermal shock occurs when a material experiences a sudden, extreme change in temperature, leading to rapid expansion or contraction. This can cause severe thermal stress and potentially lead to cracking or shattering, especially in brittle materials like glass or ceramics.

11. How do composite materials behave under thermal stress?

Composite materials often have complex behavior under thermal stress due to their heterogeneous nature. Different components of the composite may have different thermal expansion coefficients, leading to internal stresses at the interfaces between materials. This can sometimes lead to delamination or other forms of failure if not properly designed.

12. Why do power lines sag more on hot days?

Power lines sag more on hot days due to thermal expansion. As the metal wires heat up, they expand in length. Since they are fixed at the ends, this expansion causes them to sag in the middle, following a catenary curve.

13. How does thermal stress affect the design of spacecraft?

Thermal stress is a critical consideration in spacecraft design due to the extreme temperature variations in space. Engineers must account for thermal expansion and contraction of different components, use materials with appropriate thermal properties, and implement thermal management systems to prevent damage and ensure proper functioning of the spacecraft.

14. What is thermal fatigue?

Thermal fatigue is the weakening of a material caused by repeated heating and cooling cycles. These temperature fluctuations induce repeated stress cycles, which can lead to the formation and growth of cracks over time, even if the individual stress levels are below the material's yield strength.

15. How does the crystal structure of a material affect its thermal expansion?

The crystal structure of a material significantly influences its thermal expansion. Materials with more open crystal structures generally have higher coefficients of thermal expansion. The strength and type of atomic bonds also play a role, with stronger bonds typically resulting in lower thermal expansion.

16. Why do some materials become brittle at low temperatures?

Some materials become brittle at low temperatures because the thermal energy of their atoms decreases, reducing their ability to deform plastically. This can cause the material to fracture rather than bend or stretch when stressed, leading to brittle behavior.

17. How does thermal stress contribute to the weathering of rocks?

Thermal stress contributes to rock weathering through a process called thermal expansion weathering. As rocks heat up during the day and cool at night, they repeatedly expand and contract. This cycling can cause cracks to form and grow over time, eventually leading to the rock breaking apart.

18. What is the difference between isotropic and anisotropic thermal expansion?

Isotropic thermal expansion occurs uniformly in all directions, while anisotropic thermal expansion varies depending on the direction within the material. Many crystalline materials exhibit anisotropic expansion due to their ordered atomic structure, while amorphous materials like glass typically show isotropic expansion.

19. How do thermal stresses affect the performance of electronic devices?

Thermal stresses can significantly impact electronic devices by causing physical deformation, altering electrical properties, and potentially leading to failure. They can cause warping of circuit boards, separation of components from substrates, and changes in resistance or capacitance, all of which can affect device performance or reliability.

20. What is thermal buckling?

Thermal buckling is a phenomenon where a structure suddenly deforms or bends due to thermal stress. It occurs when the compressive thermal stress in a constrained structure exceeds a critical value, causing it to bow or buckle to relieve the stress. This can be particularly problematic in structures like railroad tracks or bridges.

21. How do engineers account for thermal stress in the design of nuclear reactors?

Engineers account for thermal stress in nuclear reactor design through careful material selection, thermal barriers, cooling systems, and structural design. They use materials with appropriate thermal properties, implement complex cooling systems to manage temperature gradients, and design structures to accommodate thermal expansion and contraction without compromising safety or efficiency.

22. What is the relationship between thermal stress and residual stress?

Thermal stress and residual stress are related but distinct concepts. Thermal stress is caused by temperature changes, while residual stress is stress that remains in a material after the original cause of the stress has been removed. However, thermal stress can lead to residual stress if it causes plastic deformation or phase changes in the material.

23. How does thermal stress affect the accuracy of precision instruments?

Thermal stress can significantly affect the accuracy of precision instruments by causing dimensional changes or warping. This can lead to misalignments, changes in calibration, or alterations in the instrument's geometry. To maintain accuracy, precision instruments often require careful temperature control or compensation mechanisms.

24. What is thermal creep?

Thermal creep is the tendency of a material to slowly deform permanently under the influence of stress at high temperatures. It's a time-dependent phenomenon that becomes more pronounced as temperature increases, even when the applied stress is below the material's yield strength at room temperature.

25. How do thermal stresses affect the design of heat exchangers?

Thermal stresses significantly impact heat exchanger design. Engineers must account for differential thermal expansion between hot and cold fluids, as well as between different parts of the exchanger. They use flexible designs, appropriate materials, and sometimes stress-relief features to prevent leaks, warping, or fatigue caused by thermal cycling.

26. How does thermal stress contribute to the formation of volcanoes?

Thermal stress plays a role in volcano formation by contributing to the fracturing of rock in the Earth's crust. As magma heats surrounding rock, thermal expansion can cause cracking, creating pathways for magma to move upward. Additionally, the cooling and contraction of lava flows can create distinctive surface features through thermal stress.

27. What is thermal ratcheting?

Thermal ratcheting is a progressive deformation that occurs in materials subjected to cyclic thermal stresses combined with a constant mechanical stress. Each thermal cycle causes a small amount of plastic deformation, which accumulates over time, potentially leading to failure even if individual stress cycles are below the yield strength.

28. How do thermal stresses affect the performance of solar panels?

Thermal stresses significantly impact solar panel performance. Temperature fluctuations can cause expansion and contraction of different components, potentially leading to cell cracking, delamination, or connection failures. Additionally, higher temperatures generally reduce the efficiency of photovoltaic cells, directly affecting power output.

29. What is the difference between linear and volumetric thermal expansion?

Linear thermal expansion refers to the change in length of a material with temperature, while volumetric thermal expansion refers to the change in volume. For isotropic materials, the volumetric expansion coefficient is approximately three times the linear expansion coefficient. However, for anisotropic materials, the relationship can be more complex.

30. How do thermal stresses affect the design of dental fillings?

Thermal stresses significantly influence dental filling design. As teeth and filling materials expand and contract differently with temperature changes (e.g., from hot or cold foods), mismatched thermal expansion can lead to gaps, cracks, or detachment. Dentists and materials scientists work to develop filling materials with thermal properties similar to natural teeth to minimize these issues.

31. What is thermal mismatch?

Thermal mismatch occurs when two materials in contact have different coefficients of thermal expansion. When subjected to temperature changes, they expand or contract at different rates, leading to stress at their interface. This can cause problems in composite materials, coatings, or joints between dissimilar materials.

32. How do thermal stresses contribute to the formation of metamorphic rocks?

Thermal stresses play a crucial role in the formation of metamorphic rocks. As rocks are subjected to high temperatures deep within the Earth, thermal expansion can cause internal stresses. These stresses, combined with pressure and chemical changes, can lead to recrystallization and realignment of minerals, transforming the original rock into a metamorphic rock.

33. What is the zeroth law of thermodynamics and how does it relate to thermal stress?

The zeroth law of thermodynamics states that if two systems are in thermal equilibrium with a third system, they are in thermal equilibrium with each other. This law is fundamental to understanding thermal stress because it governs how heat flows between objects and their environment, which is the underlying cause of thermal expansion and contraction.

34. How do thermal stresses affect the design of aircraft?

Thermal stresses significantly impact aircraft design, especially for high-speed or high-altitude aircraft. Engineers must account for temperature variations that cause expansion and contraction of different parts, particularly in the airframe. They use materials with appropriate thermal properties and design structures to accommodate thermal expansion without compromising aerodynamics or structural integrity.

35. What is thermal shock resistance?

Thermal shock resistance is a material's ability to withstand sudden temperature changes without cracking or failing. Materials with high thermal shock resistance typically have low thermal expansion coefficients, high thermal conductivity, and high strength. This property is crucial for applications involving rapid temperature changes, such as in cookware or aerospace components.

36. How do thermal stresses affect the formation of igneous rocks?

Thermal stresses play a significant role in the formation and characteristics of igneous rocks. As magma cools and solidifies, differential cooling rates can lead to thermal stresses within the rock. These stresses can cause cracking, influence crystal size and orientation, and contribute to the formation of distinctive structures like columnar jointing seen in some basalts.

37. What is the difference between elastic and plastic deformation in the context of thermal stress?

In the context of thermal stress, elastic deformation refers to temporary changes in a material's shape that reverse when the stress is removed. Plastic deformation, on the other hand, involves permanent changes to the material's shape. Thermal stresses can cause either type of deformation depending on the material properties and the magnitude of the stress.

38. How do thermal stresses affect the design of nuclear waste storage facilities?

Thermal stresses are a critical consideration in designing nuclear waste storage facilities. The decay of radioactive materials generates heat, which can cause thermal expansion of both the waste and containment structures. Engineers must design these facilities to manage and dissipate heat effectively, use materials resistant to thermal degradation, and account for long-term thermal effects to ensure safe containment for thousands of years.

39. What is thermal fatigue cracking?

Thermal fatigue cracking is the formation and growth of cracks due to repeated cycles of heating and cooling. These temperature fluctuations cause repeated expansion and contraction, leading to cyclic stresses. Over time, these stress cycles can initiate and propagate cracks, even if the individual stress levels are below the material's yield strength.

40. How do thermal stresses affect the performance of fuel cells?

Thermal stresses significantly impact fuel cell performance and durability. Temperature gradients within the cell can cause differential expansion, leading to mechanical stresses that may result in cracking or delamination of components. These stresses can affect the integrity of seals, alter the porosity of electrodes, and degrade overall cell efficiency and lifespan.

41. What is the role of thermal stress in the formation of sea ice?

Thermal stress plays a crucial role in the formation and behavior of sea ice. As the surface water freezes, it expands, creating stress within the ice sheet. Temperature fluctuations cause repeated expansion and contraction, leading to cracking and the formation of pressure ridges. These thermal effects contribute to the dynamic nature of sea ice and its interaction with the surrounding environment.

42. How do thermal stresses contribute to the weathering of buildings and monuments?

Thermal stresses significantly contribute to the weathering of buildings and monuments through a process called thermal weathering. Daily and seasonal temperature fluctuations cause repeated expansion and contraction of building materials. This cycling can lead to fatigue, cracking, and eventual disintegration, especially in materials with different thermal expansion coefficients or in structures with poor thermal management.

43. What is the difference between steady-state and transient thermal stress?

Steady-state thermal stress occurs when a material is subjected to a constant temperature gradient, resulting in a stable stress distribution. Transient thermal stress, on the other hand, occurs during temperature changes, where the stress distribution varies with time. Transient stresses are often more severe and can lead to more significant material damage due to the rapid changes in temperature and resulting thermal gradients.

44. What is thermal strain energy?

Thermal strain energy is the energy stored in a material due to its deformation caused by temperature changes. When a material expands or contracts due to temperature variations, it experiences strain, and this deformation stores energy within the material's structure. This stored energy can be released if the material is allowed to return to its original shape and temperature.

45. How do thermal stresses affect the performance of thermoelectric devices?

Thermal stresses significantly impact thermoelectric devices, which generate electricity from temperature differences. The performance and durability of these devices can be compromised by thermal expansion mismatches between different materials, potentially leading to mechanical failure or degradation of electrical contacts. Engineers must carefully design these devices to manage thermal stresses while maintaining efficient heat flow.

46. What is the relationship between thermal conductivity and thermal stress?

Thermal conductivity and thermal stress are closely related. Materials with high thermal conductivity tend to distribute heat more evenly, reducing temperature gradients and thus minimizing thermal stresses. Conversely, materials with low thermal conductivity are more likely to develop significant temperature gradients, leading to higher thermal stresses. This relationship is crucial in designing components that must withstand rapid temperature changes.

47. How do thermal stresses affect the design of brake systems in vehicles?

Thermal stresses significantly impact brake system design in vehicles. During braking, friction generates heat, causing thermal expansion of brake components. Engineers must design brake systems to manage this heat effectively, using materials that can withstand high temperatures and rapid thermal cycling. They also implement cooling mechanisms and design for uniform heat distribution to prevent warping, cracking, or brake fade due to thermal stress.

48. What is thermal annealing and how does it relate to thermal stress?

Thermal annealing is a heat treatment process used to relieve internal stresses in materials. It involves heating a material to a specific temperature, holding it there for a period, then cooling it slowly. This process allows atoms to rearrange themselves, reducing defects and internal stresses. Annealing can be used to mitigate the effects of thermal stress in manufactured parts or to prepare materials for further processing.

49. How do thermal stresses affect the behavior of shape memory alloys?

Thermal stresses play a crucial role in the behavior of shape memory alloys (SMAs). These materials can "remember" and return to their original shape when heated, a property driven by temperature-induced phase transformations. The internal stresses generated during these transformations are key to the shape memory effect. Understanding and managing these thermal stresses is essential for designing and utilizing SMA-based devices effectively.