Stress and Strain Calculation
An EngineersToolbox Calculation Module
The Hooke's Stress and Strain module calculates stresses/strains for a three-dimensional state of strain/stress within the linear elastic range of a homogenous, isotropic material. The module also calculates principal stresses and strains, and strain energy.
An elastic material is one that returns to its original unloaded shape upon the removal of applied forces. A homogeneous material possesses the same material properties at all points. An isotropic material has the same material properties in all directions.
Figure 1 shows the components that define the general state of stress in a three-dimensional solid. Normal stresses sx, sy, and sz act on a plane normal to the axis given by the subscript. Shear stresses txy, txz, tyx, tyz, tzx, and tzy act on a plane normal to the axis given by the first subscript. The second subscript designates the direction in which the shear stress acts. To show static equilibrium, the shear stresses acting on mutually perpendicular planes must be equal:
Thus the general state of stress is completely defined by six components.
Figure 1. Sign conventions and notations for stresses on a solid
In some practical situations, the general state of stress can be reduced to simpler forms. These simplified stess states are described in the following sections.
Aircraft structures are often built from thin metal sheet in which stresses across the thickness are negligible. Assuming that the z axis is in the direction of the thickness, only the x and y faces of the element are subjected to stress. In this case, the general three-dimensional stress model reduces to two dimensions in which sz, txz, and tyz are all zero. This two-dimensional state of stress is called plane stress and is defined by 3 components:
Triaxial Stress, Biaxial Stress, and Uniaxial Stress
Triaxial stress refers to a condition where only normal stresses act on an element and all shear stresses (txy, txz, and tyz) are zero. An example of a triaxial stress state is hydrostatic pressure acting on a small element submerged in a liquid.
A two-dimensional state of stress in which only two normal stresses are present is called biaxial stress. Likewise, a one-dimensional state of stress in which normal stresses act along one direction only is called a uniaxial stress state.
Pure shear refers to a stress state in which an element is subjected to plane shearing stresses only, as shown in Figure 3. Pure shear occurs in elements of a circular shaft under a torsion load.
Figure 3. Element in pure shear
Most structural materials exhibit a linear relationship between stress and strain at low stress levels as shown in Figure 3. This linear elastic region is represented by a straight line on the stress-strain diagram and ends at a point called the proportional limit. For a uniaxial stress state in which normal stress acts only in the x direction, the linear stress-strain relationship is given by Hooke's Law:
The constant E is equal to the slope of the stress-strain line, and is called the Elastic Modulus, or Young's Modulus. Hooke's Law also holds for shear stresses and shear strains in the linearly elastic range:
where the constant G is called the Shear Modulus or the Modulus of Rigidity.
Figure 3. Stress-strain diagram for a material exhibiting elastic-plastic behavior
It is found experimentally that an axial tensile loading induces a lateral strain corresponding to a reduction in a material specimen's cross-sectional area. Similarly, an axial compressive load causes a lateral strain associated with an increase in the cross-sectional area. When the axial stress is removed, the lateral strain disappears along with the axial strain. The ratio of the laterial strain (due to expansion/contraction of the cross-section) to the axial strain is known as Poisson's Ratio and abbreviated using the constant n. For most metals, Poisson's ratio is value between 0.25 and 0.35. In other materials, Poisson's ratio can vary from 0.1 (for some concretes) to 0.5 (for some rubber materials).
The derivation of a generalized Hooke's Law in three dimensions for isotropic materials requires the following assumptions:
Figure 4 shows a two-dimensional element in a homogenous, isotropic material subjected to a biaxial state of stress. The normal stress sx causes the element to elongate sx /E along the x axis. At the same time, the normal stress sy induces a stress of -nsy in the x direction, which causes the element to contract by -nsy / E. Applying superposition, the resulting strain ex is equal to sx /E - nsy / E, as shown in the figure.
Figure 4. Element deformation due to biaxial stress
A generalized Hooke's Law can be established by extending the previous analysis to include normal strains in the y and z directions and including the stress-strain relationships for pure shear (t = Gg). The generalized Hooke's Law, which applies to linearly elastic, homogenous, isotropic materials, is thus given by
where as previously stated, the constant E is the elastic modulus, G is the shear modulus, and n is Poisson's ratio. These 3 material properties can be shown to be related by
Eliminating the constant G from the stress-strain equations, the generalized Hooke's Law can be expressed in matrix form as
or more simply
where [S] is called the compliance matrix. Stresses may be written as a function of the strains by inverting the compliance matrix. The result is
which can be expressed as
where [C] is referred to as the stiffness matrix.
Stress-strain relationships such as these are known as constitutive equations. There are a total of 36 elastic constants in the compliance and stiffness matrices. However, the vast majority of engineering materials are conservative and it can be shown that conservative materials have stiffness and compliance matrices that are symmetric. In this case, there is a maximum of 21 elastic constants that are actually independent in the generalized Hooke's law. For an isotropic material, the constants must be identical in all directions and the number of independent elastic constants reduces to 2 (for example, elastic modulus E, and Poisson's Ratio n.
Thermal Stress and Strain
A change in uniform temperature applied to an unconstrained, three-dimensional elastic element produces a expansion or contraction of the element. Free thermal expansion produces normal strains that are related to the change in temperature by
where a is called the coefficient of linear thermal expansion, which is widely tabulated for structural materials. Thermal strain is handled in the same manner as strain due to an applied load. Applying superposition, the thermal strains can be directly added to the stress-strain equations:
Note that free thermal expansion of an isotropic material does not induce angular distortion or shear strains. The corresponding stress equations are
Common engineering solids usually have thermal expansion coefficients that do not vary significantly over the range of temperatures where they are designed to be used, so where extremely high accuracy is not required, calculations can be based on a constant, average, value of the coefficient of expansion.
Stress Transformation in Two Dimensions
The state of stress in Figure 1 is expressed in a predetermined system of axes that that may not be aligned with the direction of maximum stress. Thus it is often necessary to determine the state of stress at other planes in the material. For the two-dimensional stress state shown in Figure 5, the transformation equations for stress at a plane inclined at an angle q to the x-axis are:
where the sign convention for positive shear and normal stresses is shown in Figure 5.
Figure 5. Two-dimensional stresses on an inclined plane
Principal Stresses in Two Dimensions
Angles of q that correspond to maximum or minimum value of sx' can be determined by differentiating sx' with respect to q and setting the derivative equal to zero. This yields an equation for the principal directions qp
The solution to this equation produces two angles that are 90 degrees apart. One angle corresponds to the direction of maximum normal stress smax, and the other corresponding to the direction of minimum normal stress, smin. The planes defined by the angles qp are known as principal planes.
Substituting qp back into the combined stress equation for sx' results in the following equations for the principal stresses:
Here the maximum principal stress is designated s1 and the minimum principal stress is designated s2. Note that s1 is algebraically the greatest stress and s2 is algebraically the least. Therefore when s2 is negative (compressive), it is possible for s2 to be arithmetically greater than s1. To determine the principal directions associated with s1 and s2, one of the two values of qp must be substituted into the combined stress equation for sx'.
Substituting qp into the stress equation for tx'y' gives a result of zero. The shear stress tx'y' is always zero on a principal plane. Planes of maximum shearing stress are determined by differentiating tx'y' with respect to q and setting the derivative equal to zero. This produces the following:
The solution to this equation yields two values of qs which are 90 degrees apart and rotated 45 degrees relative to the planes of principal normal stresses defined by qp. Substituting qs back into the combined stress equation for tx'y' produces equations for the maximum and minimum shear stress:
Here the arithmetically greatest value, regardless of sign, is designated the maximum shear stress. Normal stresses acting on the planes of maximum shear stress can be determined by substituting qs into the stress equations for sx' and sy'. This yields
which clearly states that ss is the average of the normal stresses.
based on a constant, average, value of the coefficient of expansion.
Stress Transformations in Three Dimensions
The three-dimensional state of stress shown in Figure 1 can be expressed in vector notation by the following:
Here the vectors sx, sy, and sz define the stresses on the faces of a cube that has outward normals i, j, and k. For three-dimensional stress transformation, it is necessary to relate these stresses to the normal and shear stresses on an arbitrarily oriented plane defined by the unit normal vector n:
A resultant stress vector sn acting on an oblique plane can be calculated from equilibrium of the tetrahedron shown in Figure 7. In this example, the oblique plane is defined by vertices ABC, and sn is given by:
In cartesian coordinates, sn expands to:
The stress vector sn can similarly be expressed in normal and shear components on ABC. First it is necessary to define a coordinate system x', y', z' in which the x' axis is aligned with the normal vector n and the y' and z' axes lie in the ABC plane. Projecting sn into the x', y', and z' coordinate axes gives:
where p and q are unit vectors corresponding to the y' and z' directions, respectively. The remaining components can be determined in a similar manner by considering a plane perpendicular to ABC in which n coincides with the y' direction.
The preceding 6 expressions form the three-dimensional stress transformation from coordinate system xyz to coordinate system x'y'z', where xyz and z'y'z' are related by the coordinate transformation matrix:
Figure 7. Stress normal to an oblique plane
Principal Stresses in Three Dimensions
Principal stresses for a three-dimensional state of stress are designated s1 ,s2, and s3, where s1 represents the algebraically greatest stress and s3 the least. Principal stresses act parallel to the unit normals; thus the normal stress vector sn may be expressed in terms of principal stress sp by:
Equating this expression to the previous equation for sn yields the following system of equations:
These equations have a non-trivial solution for nx, ny, and nz only if the determinant of the coefficients is zero:
Expanding the determinant leads the following cubic equation:
The quantities I1, I2, and I3 are called stress invariants because they are independent of the coordinate system xyz in which the state of stress is given. Stress invariants are particularly useful in checking the resuls of a stress transformation.
The von Mises failure criterion is based on the stresses acting an octahedral plane, which is a plane whose normal has equal angles relative to the principal axes. There are eight such planes. It can be shown (Ugural 1995) that the normal stress on an octahedral plane is given by
which is referred to as the hydrostatic or volumetric stress. The octahedral shear stress is
According to the von Mises theory of material failure, yielding occurs when the octahdral shear stress given by the above equation is equal to the octahedral shear stress at yield in tension, where s2 = s3 = 0. The yield criterion is thus:
The transformation equations for strain are nearly identical to the stress transformations described previously. All that is necessary is the substitution of normal strains e for normal stresses s and shear strains g/2 for shear stresses t in each derived expression.
Strain Transformation in Two Dimensions
The transformation equations for a two-dimensional state of stress are
For a two-dimensional state of strain, the principal strain directions are
and the corresponding principal strains are
Likewise for a two-dimensional state of strain, the planes of maximum shear are 45 degrees relative to the principal planes and are given by
and the corresponding maximum shearing strains are
Strain Transformation in Three Dimensions
The transformation equations for a three-dimensional state of stress are
The principal strains in three dimensions are the roots of the following cubic equation:
The corresponding strain invariants are:
The work done by external forces in deforming an elastic body is stored within the body in the form of strain energy. Strain energy is a form of potential energy, and thus is a scalar quantity with units such as Joules or foot-pounds. Energy methods in solid mechanics provide an alternate and sometimes more efficient means for formulating governing equation of deformable bodies.
The total work done by combined stresses on an elastic element like that shown in Figure 1 is simply the sum of the work done by each individual component of stress. This approach is valid because, for example, the normal stress sx does no work in the y or z directions. Similarly, the shear stress txy does no work associated with strains gxz, gyz.
Figure 8 shows the equilibrium of an elastic element of dimension dx, dy, and dz, subject to a normal stress sx. The strain energy in the element is calculated by
where du/dx = ex and (sx dy dz) is the force acting in the x direction.
Since dx dy dz represents the volume of the element, the strain energy density, Uo (strain energy per unit volume) due to a normal stress ss sx can be expressed by the following:
Integrating the strain energy equation gives:
Strain energy density represents the area below the stress-strain curve in the same way that work represents the area below a force-displacement curve.
Figure 8. Deformation due to normal stress
The strain energy associated with shear deformation gxy (as shown in Figure 9) can be calculated in a similar manner and is given by
Figure 9. Deformation due to pure shear
As previously stated, the total strain energy associated with a general state of stress can be calculated by simply adding the strain energy due to each of the individual stress components:
Using Hooke's law, strain energy density can be expressed in terms of stress:
Strain energy density can also be expressed in terms of strain:
Components of Strain Energy
A general state of stress can be described in terms of dilatational stresses and distortional stresses as shown in Figure 10. Dilatational stresses, sometimes referred to as volumetric or hydrostatic stresses, result in a volume change without distortion. The dilatational stress tensor is defined by
where sm is the mean stress:
The dilatational strain energy density can be calculated by substituting sm into the expression for strain energy density as a function of stress. The result is
where K is known as the bulk modulus and is given by:
Distortional stresses cause a change in shape without a corresponding change in volume. The distortional stress tensor is defined by
The distortional strain energy density Uod can be calculated by subtracting the dilatational strain energy density Uov from the total strain energy density Uo. The result is:
Figure 10. Superposition of dilatational stresses and distortional stresses
The module's main input form is shown in Figure 11. The Hooke's Law problem is defined by choosing either stress or strain input, and then entering six values of stress/strain on the left side of the input form. Material properties are entered on the right side of the form. The modulus of elasticity E and Poisson's ratio n of the material are required for all problems. If thermal effects are included then the coefficient of thermal expansion and the change in temperature must also be entered. Material properties may be viewed and retrieved from the ETBX Materials Database by clicking the Materials button.
Figure 11. Module input form
Press the Calculate button to generate results. The calculated results will be displayed in the Results Report window shown in Figure 12. The following data is included in the calculation:
Figure 2. Module tabulated results.
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