In the study of how materials behave under forces, stress is a measure that describes the forces acting during changes in shape. For example, when an object is pulled apart, like a stretched rubber band, it experiences tensile stress and may become longer. When an object is pushed together, like a squeezed sponge, it experiences compressive stress and may become shorter. The larger the force applied and the smaller the area over which the force is spread, the greater the stress. Stress is measured as force divided by area, with standard units of newtons per square meter (N/m²) or pascal (Pa).
Stress refers to the forces that particles within a material push or pull on each other, while strain is the measure of how much the material changes shape. For example, when a vertical rod holds up a heavy weight, each part of the rod pushes against the parts below it. When a liquid is in a sealed container under pressure, each particle is pushed by the particles around it. The container walls and the object applying pressure (like a piston) also push back. These large-scale forces are the result of many tiny forces between molecules. Stress is often written using the lowercase Greek letter sigma (σ).
Strain in a material can happen in different ways, such as when forces from outside (like gravity) or from the surface (like pressure or friction) act on it. When a solid is deformed, it creates internal elastic stress, similar to how a stretched spring tries to return to its original shape. In liquids and gases, only changes in volume cause lasting elastic stress. If a material is deformed slowly over time, even in fluids, there may be resistance from viscous stress. Elastic and viscous stresses are often grouped together as mechanical stress.
Stress can exist even when a material is not visibly deformed, which is a common assumption when studying water flow. Stress can also occur without outside forces, such as in materials like prestressed concrete or tempered glass. Stress can be applied to a material without net forces, for example, due to changes in temperature, chemical changes, or electromagnetic fields (as in piezoelectric or magnetostrictive materials).
The connection between mechanical stress, strain, and how quickly strain changes can be complex, but a simple straight-line relationship may work well for small amounts of stress. If stress becomes too large for a material to handle, it can cause lasting changes (like bending, breaking, or bubble formation) or even alter the material’s structure and composition.
History
Humans have understood stress in materials for a long time, even before written records. Before the 17th century, this knowledge was based on experience and observation, not formal science. This did not stop people from creating advanced tools and structures, such as the composite bow and glassblowing techniques.
Over many centuries, architects and builders learned how to arrange shaped wooden beams and stone blocks to support, transfer, and spread stress effectively. They used creative designs like capitals, arches, cupolas, trusses, and flying buttresses in Gothic cathedrals to achieve this.
Ancient and medieval architects developed basic geometric methods and simple formulas to calculate the correct sizes of pillars and beams. However, a true scientific understanding of stress came only after important tools were created in the 17th and 18th centuries. These included Galileo Galilei’s careful experiments, René Descartes’ coordinate system and analytic geometry, and Isaac Newton’s laws of motion, equilibrium, and calculus. Using these tools, Augustin-Louis Cauchy created the first clear and general mathematical model to describe how elastic materials deform. He showed that the force acting across an imaginary surface depends on the direction of the surface and must be balanced to avoid movement.
Newton also began the study of stress in liquids by developing a formula to calculate friction forces, called shear stress, in smooth, parallel layers of flowing liquid.
Definition
Stress is the force acting across a small boundary divided by the area of that boundary, measured for all possible directions of the boundary. It is based on two concepts: force (a physical quantity) and area (a geometric quantity). Like velocity, torque, or energy, stress is a physical quantity that can be measured and studied without needing to know the specific material or its causes.
In continuum mechanics, stress is a large-scale concept. The particles used in its study must be small enough to be treated as uniform in composition and state, but still large enough to ignore effects from individual atoms and the movement of molecules. The force between two particles is the average of many atomic forces between their molecules. Quantities like mass, velocity, and forces (such as gravity) are assumed to spread smoothly throughout three-dimensional objects. Depending on the situation, particles may also be large enough to average out other small-scale features, like the grains in metal or fibers in wood.
Stress is calculated using the Cauchy traction vector T, which is the traction force F between two parts of a material across a surface S, divided by the area of S. In a fluid at rest, the force is always perpendicular to the surface, which is the familiar concept of pressure. In solids or flowing viscous liquids, the force F may not be perpendicular to S. This means stress must be described as a vector (with both direction and magnitude), not a single number. The direction and strength of stress depend on the orientation of S. To fully describe stress, a tensor called the (Cauchy) stress tensor is used. This tensor is a mathematical tool that connects the normal vector n of a surface S to the traction vector T across S. In any coordinate system, the stress tensor can be written as a 3×3 matrix of real numbers. Even in a uniform material, the stress tensor can change from one location to another and over time, making stress a time-dependent tensor field.
In general, the stress T that one particle P applies on another particle Q across a surface S can act in any direction relative to S. The vector T can be broken into two parts: the normal stress, which acts perpendicular to the surface (either pushing or pulling), and the shear stress, which acts parallel to the surface.
If the normal unit vector n of the surface (pointing from Q to P) is fixed, the normal stress can be represented by a single number, the dot product of T and n. This number is positive if P is pulling Q (tensile stress) and negative if P is pushing Q (compressive stress). The shear stress is then the vector T minus the product of (T · n) and n.
Units
Stress is a type of pressure, and its units of measurement are the same as those used for pressure. In the International System, pressure is measured in pascals (Pa), which means newtons per square meter. In the Imperial system, pressure is measured in pounds per square inch (psi). Because mechanical stress often goes above a million pascals, a unit called megapascal (MPa) is commonly used. One megapascal equals one million pascals.
Causes and effects
Stress in a material can come from many different physical causes, such as external forces and internal processes. Some causes, like gravity, changes in temperature, electromagnetic fields, and phase changes, affect the entire material and change continuously over time and space. Other causes, such as external loads, friction, ambient pressure, and contact forces, create stresses that are focused on specific surfaces, lines, or points. These stresses may also occur over very short time periods, like during collisions. In active matter, tiny particles that move on their own can create stress patterns at a larger scale. Generally, the way stress is spread throughout a material is described as a function that changes in different parts of the material over time.
Stress is often linked to changes in a material's properties, such as birefringence, polarization, and permeability. When stress is applied from outside, it usually causes some deformation, even if it is too small to see. In solid materials, this deformation creates an internal elastic stress, similar to how a stretched spring tries to return to its original shape. Fluids, such as liquids, gases, and plasmas, can only resist changes that affect their volume. If deformation happens over time, even in fluids, there is usually some viscous stress that opposes the change. These stresses can be either shear or normal. The molecular reasons for shear stress in fluids are explained in the article on viscosity. Details about normal viscous stress can be found in Sharma (2019).
The connection between stress and its effects, including deformation and how quickly deformation happens, can be complex. However, a simple approximation might work if the changes are small. If stress exceeds a material's strength limits, it can cause permanent deformation, such as plastic flow, breaking, or the formation of bubbles. It might also change the material's crystal structure or chemical makeup.
Simple types
In some situations, the stress inside a material can be described using a single number or a single vector (a number with a direction). Three common types of stress seen in engineering are uniaxial normal stress, simple shear stress, and isotropic normal stress.
A simple example of stress occurs when a straight rod with uniform material and cross-section is pulled by equal forces (F) acting in opposite directions along its length. If the rod is not moving and its weight is ignored, the force (F) is evenly spread across the cross-sectional area (A). The stress (σ) in this case can be calculated by dividing the force (F) by the area (A): σ = F/A. If the rod is cut along its length, no force exists between the two pieces, so no stress is present there. This type of stress is called simple normal stress or uniaxial stress. If the rod is pushed instead of pulled, the stress is still calculated the same way, but the direction of the force changes. In this case, the stress is called compressive stress.
This calculation assumes the stress is evenly spread across the entire cross-section. In real situations, this might not always be true, especially if the rod is attached or manufactured in a way that causes uneven stress. When this happens, the value σ = F/A is called the average stress, or engineering stress. If the rod is much longer than its diameter and has no major flaws, the stress can be considered evenly spread across any cross-section far from the ends. This idea is known as Saint-Venant's principle.
Normal stress also appears in other situations. For example, when a bar with uniform cross-section is bent, the stress remains perpendicular to the cross-section but varies across it. The outer part of the bar experiences tensile stress, while the inner part experiences compressive stress. Another example is hoop stress, which occurs in the walls of a pressurized cylinder or pipe.
Another type of stress happens when a layer of elastic material, like rubber, is attached to two stiff objects that are pulled in opposite directions by forces (F) parallel to the layer. The stress (τ) in this case is calculated by dividing the force (F) by the cross-sectional area (A): τ = F/A. Unlike normal stress, this shear stress is directed parallel to the cross-section. For any plane perpendicular to the layer, the net internal force and stress are zero.
In practice, shear stress may not be evenly spread across the layer. The ratio F/A then represents an average stress, which is often sufficient for engineering purposes. Shear stress also occurs in a cylindrical bar when opposite torques are applied at its ends. In this case, the shear stress is parallel to the cross-section and increases with distance from the bar’s axis. Shear stress is also found in the middle part of an I-beam under bending loads.
A third type of stress occurs when a material is equally compressed or stretched in all directions. This happens, for example, in a still liquid or gas, or in a cube of elastic material pressed or pulled equally on all sides. In these cases, the stress across any imaginary surface is the same in magnitude and always directed perpendicular to the surface. This type of stress is called isotropic normal stress or isotropic stress. If the stress is compressive, it is called hydrostatic pressure. Gases cannot withstand tensile stress, but some liquids can handle large amounts of tensile stress under certain conditions.
Many engineering parts, such as wheels, axles, pipes, and pillars, have rotational symmetry. The stress patterns in these parts often have rotational or cylindrical symmetry. Engineers use this symmetry to simplify stress analysis by reducing the complexity of the problem.
General types
Mechanical objects often face more than one kind of stress at the same time. This situation is called combined stress. In normal and shear stress, the greatest stress occurs on surfaces that are perpendicular to a specific direction, while no stress exists on surfaces that are parallel to that same direction. If shear stress is only zero on surfaces perpendicular to one direction, the stress is called biaxial. This type of stress can be seen as the combination of two normal or shear stresses. In the most general case, known as triaxial stress, stress is present on all surfaces of the object.
Cauchy tensor
Stress in materials cannot be described using just one vector. Even if a material is stressed the same way throughout its entire volume, the stress across any imaginary surface depends on the direction of that surface in a complex way.
Cauchy discovered that the stress vector across a surface is always a linear function of the surface's normal vector, which is a vector pointing perpendicular to the surface. This means the stress vector $ T $ can be written as $ T = sigma(n) $, where $ sigma $ is a function that follows a rule: $ sigma(alpha u + beta v) = alpha sigma(u) + beta sigma(v) $ for any vectors $ u $, $ v $ and any real numbers $ alpha $, $ beta $. This function $ sigma $, now called the Cauchy stress tensor, fully describes the stress state of a uniformly stressed body. The term "tensor" is now used to describe any linear relationship between two physical vector quantities, as Cauchy first applied it to describe stresses in materials. In tensor calculus, $ sigma $ is classified as a second-order tensor, depending on the convention used.
The stress tensor can be represented as a 3×3 matrix of numbers in any chosen coordinate system. If the coordinates are labeled $ x_1, x_2, x_3 $, the matrix might look like:
$$
begin{bmatrix}
sigma_{11} & sigma_{12} & sigma_{13} \
sigma_{21} & sigma_{22} & sigma_{23} \
sigma_{31} & sigma_{32} & sigma_{33}
end{bmatrix}
$$
If the coordinates are named $ x, y, z $, the matrix might instead be written as:
$$
begin{bmatrix}
sigma_{xx} & sigma_{xy} & sigma_{xz} \
sigma_{yx} & sigma_{yy} & sigma_{yz} \
sigma_{zx} & sigma_{zy} & sigma_{zz}
end{bmatrix}
$$
The stress vector $ T = sigma(n) $ across a surface with normal vector $ n $ (a "row" vector) is calculated using matrix multiplication: $ T = n cdot sigma $, where $ n $ is a row vector and $ sigma $ is the stress tensor matrix. For example:
$$
begin{bmatrix}
T_1 & T_2 & T_3
end{bmatrix}
=
begin{bmatrix}
n_1 & n_2 & n_3
end{bmatrix}
cdot
begin{bmatrix}
sigma_{11} & sigma_{21} & sigma_{31} \
sigma_{12} & sigma_{22} & sigma_{32} \
sigma_{13} & sigma_{23} & sigma_{33}
end{bmatrix}
$$
The linear relationship between $ T $ and $ n $ comes from the laws of physics governing motion and force balance. This relationship is mathematically exact for any material and stress situation. The stress tensor is symmetric, meaning $ sigma_{12} = sigma_{21} $, $ sigma_{13} = sigma_{31} $, and $ sigma_{23} = sigma_{32} $. This symmetry reduces the number of independent parameters needed to describe the stress state from nine to six. These six parameters can be written as:
$$
begin{bmatrix}
sigma_x & tau_{xy} & tau_{xz} \
tau_{xy} & sigma_y & tau_{yz} \
tau_{xz} & tau_{yz} & sigma_z
end{bmatrix}
$$
Here, $ sigma_x, sigma_y, sigma_z $ are normal stresses, and $ tau_{xy}, tau_{xz}, tau_{yz} $ are shear stresses.
The Cauchy stress tensor follows specific rules when the coordinate system changes. A visual tool called Mohr's circle is used to represent this transformation.
As a symmetric 3×3 matrix, the stress tensor has three eigenvectors $ e_1, e_2, e_3 $ and three eigenvalues $ lambda_1, lambda_2, lambda_3 $, such that $ sigma e_i = lambda_i e_i $. In a coordinate system aligned with these eigenvectors, the stress tensor becomes a diagonal matrix with only the three eigenvalues $ lambda_1, lambda_2, lambda_3 $, which are the principal stresses. If all eigenvalues are equal, the stress is uniform in all directions, with no shear stress.
Stress in materials is often not uniform and can change over time. To define stress at a specific point and moment, consider an infinitesimally small particle of the material and average the stresses within that particle.
Human-made objects like plates are often made by cutting, drilling, and welding, which preserve their two-dimensional character. Stress in such objects can be modeled as forces acting on two-dimensional surfaces rather than three-dimensional volumes. In this case, a "particle" is defined as an infinitesimal patch of the plate's surface, and stress is measured as the internal force between adjacent patches divided by the length of their shared boundary. Some tensor components are ignored, but torque (bending stress) must still be considered. These simplifications may not apply at welds or sharp bends.
For thin bars, beams, or wires, stress analysis can be simplified by focusing on cross-sections perpendicular to the object's axis. A "particle" is defined as a tiny segment of the wire between two cross-sections. Stress is reduced to a scalar (tension or compression), but bending and torsional stresses must also be accounted for.
Analysis
Stress analysis is a part of science that studies how forces are spread inside solid objects. It helps engineers design and study structures like tunnels, dams, mechanical parts, and building frames under forces they might experience. It is also used in other fields, such as geology to learn about Earth movements and in biology to understand how living things are built.
Stress analysis usually looks at objects that are not moving and are in balance. According to physics rules, any outside forces acting on an object must be matched by inside forces, which are often forces between tiny parts of the object—called stress. These forces spread through the object, creating a pattern of stress. The main goal of stress analysis is to find these internal stresses when outside forces, such as gravity or the weight of a train wheel, are applied.
In stress analysis, the causes of forces and the details of materials are usually ignored. Instead, it assumes that stress is connected to how materials change shape, using known rules. Stress can be studied by testing real objects or models and measuring the forces they experience. This is often done for safety checks. Most stress analysis uses math, especially during design. The basic problem is solved using physics rules and equations that describe how stress and shape changes are related. These equations involve complex math and are solved using special techniques.
When structures are made of materials that return to their original shape after forces are removed, stress analysis uses simple rules based on how materials stretch or compress. If forces cause permanent changes, more complex rules are needed. Most engineered structures are designed so that the forces they face are within the range where materials behave predictably. In this case, the math becomes easier because the relationship between stress and shape changes is straightforward.
Stress analysis is simpler when the size and forces acting on a structure allow it to be treated as one- or two-dimensional. For example, in truss structures, stress is assumed to be the same in each part. This reduces the math to simple equations. In other cases, three-dimensional problems might be simplified to two dimensions or replaced with easier models, like stretching or twisting.
For two- or three-dimensional problems, complex math is needed. If the shape, rules, and conditions are simple enough, exact solutions can be found. Otherwise, computer methods like the finite element method are used to approximate the answers.
Measures
Other useful stress measures include the first and second Piola–Kirchhoff stress tensors, the Biot stress tensor, and the Kirchhoff stress tensor.