Specification ASTM D751 includes test methods for covering a wide range of mechanical properties of coated fabrics.  A coated fabric is a material that has at least one layer of a textile and one layer of a polymeric substance.  Examples would be tarps, outdoor clothing, and rain wear.  Some of these tests require specialized equipment (hydrostatic resistance, blocking, wicking, impact, etc.), but most of these tests can be performed using a Universal Testing Machine.  These tests include:

In the ASTM acceptance standard for testing rubber surgical gloves (D3577) and rubber medical gloves (D3578), made of latex, vinyl, and nitrile. The standard describes the standard size, material properties, physical and mechanical property acceptance levels, aqueous protein and antigen protein content, and visual inspection required for a glove to be used in a medical procedure. Other relevant standards include ASTM D6319, ASTM D5250, EN 455-2, ISO 11193-1:2002, and ISO/AWI 11193-2. 

The tear strength of thermoset rubbers, thermoplastic elastomers, and silicones can be measured according to specification ASTM D624.  The test measures the strength required to initiate a tear in a material.  D624 describes multiple different types of sample shapes that can be tested in a standard universal testing machine (tensile testing machine).  Usually materials that are tested to D624 are also tested according to specification ASTM D412 to measure the tensile strength and elongation.  If you are going to perform this test, you should read the entire specification from ASTM. This is a quick summary to decide if this test is right for you, and to point out what equipment you need to perform the test.

Here is a short video of the test:

ASTM D412 (D 412) covers the tensile properties of thermoset rubbers and thermoplastic elastomers. The specification describes two test methods, A and B. Method A is common and can be performed on a universal testing machine (tensile testing machine). If you are going to perform this test, you should read the entire specification from ASTM. This is a quick summary to decide if this test is right for you and to point out what equipment you need to perform the test.