# Maximum strain theory

It has already been discussed in the previous chapter that strength of machine members is based upon the mechanical properties of the materials used. Since these properties are usually determined from simple tension or compression tests, therefore, predicting failure in members subjected to uniaxial stress is both simple and straight-forward.

But the problem of predicting the failure stresses for members subjected to bi-axial or tri-axial stresses is much more complicated. In fact, the problem is so complicated that a large number of different theories have been formulated. The principal theories of failure for a member subjected to bi-axial stress are as follows:. Since ductile materials usually fail by yielding i.

For ductile materials, the limiting strength is the stress at yield point as determined from simple tension test and it is, assumed to be equal in tension or compression. For brittle materials, the limiting strength is the ultimate stress in tension or compression. According to this theory, the failure or yielding occurs at a point in a member when the maximum principal or normal stress in a bi-axial stress system reaches the limiting strength of the material in a simple tension test.

Since the limiting strength for ductile materials is yield point stress and for brittle materials the limiting strength is ultimate stress, therefore according to the above theory, taking factor of safety F. S for ductile material. S for brittle material. According to this theory, the failure or yielding occurs at a point in a member when the distortion strain energy also called shear strain energy per unit volume in a bi-axial stress system reaches the limiting distortion energy i.

Mathematically, the maximum distortion energy theory for yielding is expressed as. This theory is mostly used for ductile materials in place of maximum strain energy theory. According to this theory, the failure or yielding occurs at a point in a member when the maximum shear stress in a bi-axial stress system reaches a value equal to the shear stress at yield point in a simple tension test.

According to this theory, the failure or yielding occurs at a point in a member when the strain energy per unit volume in a bi-axial stress system reaches the limiting strain energy per unit volume as determined from simple tension test. We know that strain energy per unit volume in a bi-axial stress system. This theory may be used for ductile materials.

According to this theory, the failure or yielding occurs at a point in a member when the maximum principal or normal strain in a bi-axial stress system reaches the limiting value of strain as determined from a simple tensile test.

The maximum principal or normal strain in a bi-axial stress system is given by. According to the above theory. S ………… ……………….

### Theory of Failure considered for Machine Design

From equation iwe may write that. This theory is not used, in general, because it only gives reliable results in particular cases. Site news. Lesson Properties of material, failures and factor of safety. Skip Navigation. Quiz Lesson Failure is generally perceived to be fracture or complete separation of a member. However, failure may also occur due to excessive deformation elastic or inelastic or a variety of other reasons. During the latter part of the 19th century and continuing up to the present, a number of basic failure theories were proposed and tested on a few materials.

Classification System for Mechanical Failure Modes 4. Stress Theories. Applied satisfactorily to many brittle materials, the theory is based on a limiting normal stress. Failure occurs when the normal stress reaches a specified upper limit. Applied satisfactorily to ductile materials, the theory is based on the concept of limiting shearing stress at which failure occurs.

Failure by yielding in a more complicated loading situation is assumed to occur when the maximum shearing stress in the material reaches a value equal to the maximum shearing stress in a tension test at yield. This yield criterion gives good agreement with experimental results for ductile materials; because of its simplicity, it is the most often used yield theory.

Failure by yielding in a more complicated loading situation is assumed to occur when the octahedral shearing stress in the material reaches a value equal to the maximum octahedral shearing stress in a tension test at yield. Strain Theories. The theory is based on the assumption that inelastic behavior or failure is governed by a specified maximum normal strain. Applicable to many types of materials, the theory predicts failure or inelastic action at a point when the strain energy per unit volume exceeds a specified limit.

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The theory is based on a limiting energy of distortion, i. Strain energy can be separated into energy associated with volume change and energy associated with distortion of the body.

The maximum distortion energy failure theory assumes failure by yielding in a more complicated loading situation to occur when the distortion energy in the material reaches the same value as in a tension test at yield. This theory provides the best agreement between experiment and theory and, along the Tresca theory, is very widely used today. Of the failure criteria, the Tresca is the most conservative for all materials, the von Mises the most representative for ductile materials, and the Rankine the best fit for brittle materials.

Laminated-Composite Failure Envelopes M ore on failure theories. Below is a summary of two of most popular theories of failure applied to a simple uniaxial stress state and to a pure shear stress state. Failure Modes 3 Excessive elastic deformation Yielding Fracture stretch, twist, or bending buckling vibration plastic deformation at room temperature creep at elevated temperatures yield stress is the important design factor sudden fracture of brittle materials fatigue progressive fracture stress rupture at elevated temperatures ultimate stress is the important design factor.

Examples Click on image for full size. Failure Modes 3. Failure Criteria.Five different failure theories are provided: four stress-based theories and one strain-based theory.

We denote orthotropic material directions by 1 and 2, with the 1-material direction aligned with the fibers and the 2-material direction transverse to the fibers. For the failure theories to work correctly, the 1- and 2-directions of the user-defined elastic material constants must align with the fiber and the transverse-to-fiber directions, respectively.

For applications other than fiber-reinforced composites, the 1- and 2-material directions should represent the strong and weak orthotropic-material directions, respectively.

The input data for the stress-based failure theories are tensile and compressive stress limits, X t and X cin the 1-direction; tensile and compressive stress limits, Y t and Y cin the 2-direction; and shear strength maximum shear stressSin the X — Y plane.

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All four stress-based theories are defined and available with a single definition in Abaqus ; the desired output is chosen by the output variables described at the end of this section. The maximum stress failure criterion requires that. The Tsai-Hill failure criterion requires that. If it is known, then. The Azzi-Tsai-Hill failure theory is the same as the Tsai-Hill theory, except that the absolute value of the cross product term is taken:.

To illustrate the four stress-based failure measures, Figure 1Figure 2and Figure 3 show each failure envelope i. In each case the Tsai-Hill surface is the piecewise continuous elliptical surface with each quadrant of the surface defined by an ellipse centered at the origin. The parallelogram in Figure 1 defines the maximum stress surface. In Figure 2 the Tsai-Wu surface appears as the ellipse. In Figure 3 the Azzi-Tsai-Hill surface differs from the Tsai-Hill surface only in the second and fourth quadrants, where it is the outside bounding surface i.

Since all of the failure theories are calibrated by tensile and compressive failure under uniaxial stress, they all give the same values on the stress axes. The maximum strain failure criterion requires that.

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The plane stress orthotropic failure measures can be used with any plane stress, shell, or membrane element in Abaqus.

Abaqus provides output of the failure index, Rif failure measures are defined with the material description. The definition of the failure index and the different output variables are described below. Failure occurs any time a state of stress is either on or outside this surface. The failure index, Ris used to measure the proximity to the failure surface.

The failure index R is defined similarly for the maximum strain failure theory. Output variables for the stress- and strain-based failure theories are always calculated at the material points of the element. Failure theories Five different failure theories are provided: four stress-based theories and one strain-based theory. In all cases tensile values must be positive and compressive values must be negative.

Stress-based failure theories The input data for the stress-based failure theories are tensile and compressive stress limits, X t and X cin the 1-direction; tensile and compressive stress limits, Y t and Y cin the 2-direction; and shear strength maximum shear stressSin the X — Y plane.Failure criteria for composite materials are usually classified in two categories: non-interactive and interactive theories.

To begin with, a non-interactive failure criterion is that one which only takes into account the effect of one stress or strain component for each failure condition. In other words, it does not consider any interaction between the different components.

For example, the Maximum Stress Theory considers that the material fails when one of the stress components reaches a maximum value. Hence, considering a sample loaded in tension:. Where subindex 1 refers to the fibre direction and 2 corresponds to the transverse direction.

When the stress reaches the limit value measured experimentally under uniaxial stress conditionsthe material fails. It is clear how in that failure criterion only one stress component is considered for each condition. Then, considering again a lamina loaded in tension:. Therefore, when uniaxial loading is considered, these two theories can be correlated:. So far, both of them are, obviously, non-interactive theories. But, what happens when biaxial loading is considered? Let me show you. According to the Maximum Stress Theory, failure occurs when:. On the other hand, the Maximum Strain Theory predicts failure when:. Now it is observed how the Maximum Strain Theory is non-interactive in terms of strain but interactive in terms of stress, since two stress components are present in each expression.

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We should always specify this fact in order to avoid confusions! You are commenting using your WordPress. You are commenting using your Google account. You are commenting using your Twitter account. You are commenting using your Facebook account.In physicsdeformation is the continuum mechanics transformation of a body from a reference configuration to a current configuration.

A deformation may be caused by external loads body forces such as gravity or electromagnetic forcesor changes in temperature, moisture content, or chemical reactions, etc. Strain is a description of deformation in terms of relative displacement of particles in the body that excludes rigid-body motions.

Different equivalent choices may be made for the expression of a strain field depending on whether it is defined with respect to the initial or the final configuration of the body and on whether the metric tensor or its dual is considered. In a continuous body, a deformation field results from a stress field induced by applied forces or is due to changes in the temperature field inside the body. The relation between stresses and induced strains is expressed by constitutive equationse.

Deformations which are recovered after the stress field has been removed are called elastic deformations. In this case, the continuum completely recovers its original configuration. On the other hand, irreversible deformations remain even after stresses have been removed. One type of irreversible deformation is plastic deformationwhich occurs in material bodies after stresses have attained a certain threshold value known as the elastic limit or yield stressand are the result of slipor dislocation mechanisms at the atomic level.

Another type of irreversible deformation is viscous deformationwhich is the irreversible part of viscoelastic deformation. In the case of elastic deformations, the response function linking strain to the deforming stress is the compliance tensor of the material. Strain is a measure of deformation representing the displacement between particles in the body relative to a reference length.

Such a measure does not distinguish between rigid body motions translations and rotations and changes in shape and size of the body. A deformation has units of length. Hence strains are dimensionless and are usually expressed as a decimal fractiona percentage or in parts-per notation. Strains measure how much a given deformation differs locally from a rigid-body deformation. A strain is in general a tensor quantity.

Physical insight into strains can be gained by observing that a given strain can be decomposed into normal and shear components. The amount of stretch or compression along material line elements or fibers is the normal strainand the amount of distortion associated with the sliding of plane layers over each other is the shear strainwithin a deforming body. The state of strain at a material point of a continuum body is defined as the totality of all the changes in length of material lines or fibers, the normal strainwhich pass through that point and also the totality of all the changes in the angle between pairs of lines initially perpendicular to each other, the shear strainradiating from this point.

However, it is sufficient to know the normal and shear components of strain on a set of three mutually perpendicular directions. If there is an increase in length of the material line, the normal strain is called tensile strainotherwise, if there is reduction or compression in the length of the material line, it is called compressive strain.

Depending on the amount of strain, or local deformation, the analysis of deformation is subdivided into three deformation theories:.

In each of these theories the strain is then defined differently. The engineering strain is the most common definition applied to materials used in mechanical and structural engineering, which are subjected to very small deformations.

On the other hand, for some materials, e. The Cauchy strain or engineering strain is expressed as the ratio of total deformation to the initial dimension of the material body in which the forces are being applied. The normal strain is positive if the material fibers are stretched and negative if they are compressed. Thus, we have.

Measures of strain are often expressed in parts per million or microstrains. The true shear strain is defined as the change in the angle in radians between two material line elements initially perpendicular to each other in the undeformed or initial configuration. The engineering shear strain is defined as the tangent of that angle, and is equal to the length of deformation at its maximum divided by the perpendicular length in the plane of force application which sometimes makes it easier to calculate.

The stretch ratio or extension ratio is a measure of the extensional or normal strain of a differential line element, which can be defined at either the undeformed configuration or the deformed configuration. It is defined as the ratio between the final length l and the initial length L of the material line. This equation implies that the normal strain is zero, so that there is no deformation when the stretch is equal to unity. The stretch ratio is used in the analysis of materials that exhibit large deformations, such as elastomers, which can sustain stretch ratios of 3 or 4 before they fail.

## WHAT IS MAXIMUM PRINCIPAL STRAIN THEORY

On the other hand, traditional engineering materials, such as concrete or steel, fail at much lower stretch ratios. The logarithmic strain provides the correct measure of the final strain when deformation takes place in a series of increments, taking into account the influence of the strain path.Stress is the ratio of applied force F to a cross section area - defined as " force per unit area ".

Tensile or compressive stress normal to the plane is usually denoted " normal stress " or " direct stress " and can be expressed as. A normal force acts perpendicular to area and is developed whenever external loads tends to push or pull the two segments of a body. A force of 10 kN is acting on a circular rod with diameter 10 mm. The stress in the rod can be calculated as. A compressive load of lb is acting on short square 6 x 6 in post of Douglas fir.

The dressed size of the post is 5. A shear force lies in the plane of an area and is developed when external loads tend to cause the two segments of a body to slide over one another.

Note that strain is a dimensionless unit since it is the ratio of two lengths. The change of length can be calculated by transforming 3 to. Most metals deforms proportional to imposed load over a range of loads. Stress is proportional to load and strain is proportional to deformation as expressed with Hooke's Law.

The Bulk Modulus Elasticity - or Volume Modulus - is a measure of the substance's resistance to uniform compression. Bulk Modulus of Elasticity is the ratio of stress to change in volume of a material subjected to axial loading. Add standard and customized parametric components - like flange beams, lumbers, piping, stairs and more - to your Sketchup model with the Engineering ToolBox - SketchUp Extension - enabled for use with the amazing, fun and free SketchUp Make and SketchUp Pro.

We don't collect information from our users. Only emails and answers are saved in our archive. Cookies are only used in the browser to improve user experience. Some of our calculators and applications let you save application data to your local computer. These applications will - due to browser restrictions - send data between your browser and our server.Rotating the stress state of a stress element can give stresses for any angle. But usually, the maximum normal or shear stresses are the most important.

Thus, this section will find the angle which will give the maximum or minimum normal stress. Start with the basic stress transformation equation for the x or y direction. This gives. Rearranging gives. This angle can be determined by taking a derivative of the shear stress rotation equation with respect to the angle and set equate to zero. When the angle is substituted back into the shear stress transformation equation, the shear stress maximum is.

The minimum shear stress will be the same absolute value as the maximum, but in the opposite direction. The relationships between principal normal stresses and maximum shear stress can be better understood by examining a plot of the stresses as a function of the rotation angle.

However, they will give the same absolute values. In some situations, stresses both normal and shear are known in all three directions. This would give three normal stresses and three shear stresses some may be zero, of course. It is possible to rotate a 3D plane so that there are no shear stresses on that plane.

Stress Analysis. Multimedia Engineering Mechanics. Plane Stress. Principal Stresses. Mohr's Circle for Stress. Pressure Vessels. Principal and Max. Shear Stresses. Case Intro. Case Solution. Beam Stresses. Beam Deflections. Strain Analysis. Basic Math. Basic Equations. Material Properties.