# Uniaxial stress strain relationship equation

### Distinction between uniaxial stress and uniaxial strain | University of Utah CSM Group

These equation expresses the relationship between stress and strain (Hook's law ) for uniaxial state of stress only when the stress is not greater than the. Two dimensional stress-strain relationships are summarized in the table below. also be applied to material undergoing three dimensional stress (triaxial loading). The development of 3D equations is similar to 2D, sum the total normal strain. The compressive uniaxial stress-strain relationship for the concrete model was obtained using the following equations to compute the multilinear isotropic.

Wave speeds are often miscalculated for the following reasons: The distinction between uniaxial stress and uniaxial strain is even more dramatic when you compare the stress-strain predictions for a simple plasticity model, like von Mises theory shown below: Here, the blue dashed line is probably what you recall from undergraduate strength of materials.

### Mechanics of Materials: Strain » Mechanics of Slender Structures | Boston University

The solid red line is the response in uniaxial strain. Note an important distinction: The fact that there are two distinct slopes in uniaxial strain is the reason why elastic-plastic waves always split into a fast-moving elastic precursor followed later by a slower-moving plastic wave. The stress space path for uniaxial strain loading allows the stress state to slide along the von Mises cylinder parallel to the hydrostat, which is why the post-yield response in uniaxial strain has a slope given by the bulk modulus.

See below to understand that changing the plasticity model also produces significant changes in how the stress state moves through stress space. Uniaxial stress loading also goes by the name unconfined compression assuming that the axial stress is compressive.

Unconfined compression is a special case of so-called triaxial compression in which the stress is still of axial form i. As a rule, material behavior in uniaxial stress is very different from that in uniaxial strain. Shown below, for example, are stress paths predicted by a geomechanics plasticity model showing very different trends in uniaxial stress in comparison to uniaxial strain. Since Poisson's ratio is a ratio of two strains, and strain is dimensionless, Poisson's ratio is also unitless.

Poisson's ratio is a material property. Poisson's ratio can range from a value of -1 to 0.

For most engineering materials, for example steel or aluminum have a Poisson's ratio around 0. Incompressible simply means that any amount you compress it in one direction, it will expand the same amount in it's other directions — hence, its volume will not change. Physically, this means that when you pull on the material in one direction it expands in all directions and vice versa: We can in turn relate this back to stress through Hooke's law.

This is an important note: In reality, structures can be simultaneously loaded in multiple directions, causing stress in those directions. A helpful way to understand this is to imagine a very tiny "cube" of material within an object. That cube can have stresses that are normal to each surface, like this: And, as we now known, stress in one direction causes strain in all three directions.

## Stress–strain analysis

So now we incorporate this idea into Hooke's law, and write down equations for the strain in each direction as: These equations look harder than they really are: Now we equations for how an object will change shape in three orthogonal directions.

Well, if an object changes shape in all three directions, that means it will change its volume.

A simple measure for this volume change can be found by adding up the three normal components of strain: Now that we have an equation for volume change, or dilation, in terms of normal strains, we can rewrite it in terms of normal stresses. A very common type of stress that causes dilation is known as hydrostatic stress. Since it is acting equally, that means: So, in the case of hydrostatic pressure we can reduce our final equation for dilation to the following: This final relationship is important, because it is a constitutive relationship for how a material's volume changes under hydrostatic pressure.

Finally, let's get back to the idea of "incompressible" materials. In stress analysis one normally disregards the physical causes of forces or the precise nature of the materials. Instead, one assumes that the stresses are related to strain of the material by known constitutive equations.

By Newton's laws of motionany external forces that act on a system must be balanced by internal reaction forces, [1] or cause the particles in the affected part to accelerate. In a solid object, all particles must move substantially in concert in order to maintain the object's overall shape. It follows that any force applied to one part of a solid object must give rise to internal reaction forces that propagate from particle to particle throughout an extended part of the system.

With very rare exceptions such as ferromagnetic materials or planet-scale bodiesinternal forces are due to very short range intermolecular interactions, and are therefore manifested as surface contact forces between adjacent particles — that is, as stress.

In principle, that means determining, implicitly or explicitly, the Cauchy stress tensor at every point.

The external forces may be body forces such as gravity or magnetic attractionthat act throughout the volume of a material; [3] or concentrated loads such as friction between an axle and a bearingor the weight of a train wheel on a railthat are imagined to act over a two-dimensional area, or along a line, or at single point.

The same net external force will have a different effect on the local stress depending on whether it is concentrated or spread out. Types of structures[ edit ] In civil engineering applications, one typically considers structures to be in static equilibrium: In mechanical and aerospace engineering, however, stress analysis must often be performed on parts that are far from equilibrium, such as vibrating plates or rapidly spinning wheels and axles.

In those cases, the equations of motion must include terms that account for the acceleration of the particles. In structural design applications, one usually tries to ensure the stresses are everywhere well below the yield strength of the material. In the case of dynamic loads, the material fatigue must also be taken into account. However, these concerns lie outside the scope of stress analysis proper, being covered in materials science under the names strength of materialsfatigue analysis, stress corrosion, creep modeling, and other.

Experimental methods[ edit ] Stress analysis can be performed experimentally by applying forces to a test element or structure and then determining the resulting stress using sensors. In this case the process would more properly be known as testing destructive or non-destructive. Experimental methods may be used in cases where mathematical approaches are cumbersome or inaccurate. Special equipment appropriate to the experimental method is used to apply the static or dynamic loading.

There are a number of experimental methods which may be used: Tensile testing is a fundamental materials science test in which a sample is subjected to uniaxial tension until failure. The results from the test are commonly used to select a material for an application, for quality controlor to predict how a material will react under other types of forces. Properties that are directly measured via a tensile test are the ultimate tensile strengthmaximum elongation and reduction in cross-section area.

From these measurements, properties such as Young's modulusPoisson's ratioyield strengthand the strain-hardening characteristics of the sample can be determined. Strain gauges can be used to experimentally determine the deformation of a physical part. A commonly used type of strain gauge is a thin flat resistor that is affixed to the surface of a part, and which measures the strain in a given direction. From the measurement of strain on a surface in three directions the stress state that developed in the part can be calculated.

Neutron diffraction is a technique that can be used to determine the subsurface strain in a part. Stress in plastic protractor causes birefringence. The photoelastic method relies on the fact that some materials exhibit birefringence on the application of stress, and the magnitude of the refractive indices at each point in the material is directly related to the state of stress at that point.

The stresses in a structure can be determined by making a model of the structure from such a photoelastic material. Dynamic mechanical analysis DMA is a technique used to study and characterize viscoelastic materials, particularly polymers. The viscoelastic property of a polymer is studied by dynamic mechanical analysis where a sinusoidal force stress is applied to a material and the resulting displacement strain is measured.

For a perfectly elastic solid, the resulting strains and the stresses will be perfectly in phase. For a purely viscous fluid, there will be a 90 degree phase lag of strain with respect to stress.